AN ABSTRACT OF THE DISSERTATION OF
Jeffrey J. Early for the degree of Doctor of Philosophy in Oceanography presented on
October 22, 2009.
Title: Mathematical Approaches to the Physics of Mesoscale Oceanography
Abstract approved:
Roger M. Samelson
The long-term evolution of Gaussian eddies is studied in an equivalent barotropic
model using both linear and nonlinear quasi-geostrophic theory in an attempt to understand westward propagating satellite altimetry tracked mesoscale eddies. By examining
both individual eddies and a large basin seeded with eddies, it is shown that long term eddy
coherence and the zonal wavenumber-frequency power spectral density are best matched
by the nonlinear model. Individual characteristics of the eddies including amplitude decay,
length decay, zonal and meridional propagation speed of a previously unrecognized quasistable state are examined to provide baseline properties for comparison with extended
models.
An analytical technique is then used for evaluating scales of motion of typical
mesoscale eddies in order evaluate the success of existing models and find other more
appropriate theories. Starting from the spherical shallow water equations and assuming
geostrophic dominance, a potential vorticity conservation law is derived in terms of all
four non-dimensional parameters inherent in the equations while retaining the spherical
geometry. By retaining freedom in the parameters, the scales can be determined at which
various theories remain valid. It is argued that the FP equation equation and a new
extension to the FP equation are required to describe the mid-latitude mesoscale eddies.
Analytical solutions to the FP equation are sought using the classical and exterior
differential systems methods of group foliation. Both methods of group foliation are used
to find the cnoidal solution of the Korteweg-de Vries equation, a one-dimensional form
of the FP equation. An exact analytical solution is found for the radial FP equation,
although it does not appear to be of direct geophysical interest, and a reduced quasilinear hyperbolic system is derived for the two-dimensional FP equation.
The forces driving the slow westward propagation of mesoscale eddies also underly
a particle constrained to the surface of the earth, but are quantitatively misunderstood.
Starting with a free particle and successively adding constraints, it is shown that the
particle’s motion is inertial, despite literature to the contrary, and that an accelerometer
trapped in inertial motion would not measure an acceleration.
c
�
Copyright
by Jeffrey J. Early
October 22, 2009
All Rights Reserved
Mathematical Approaches to the Physics of Mesoscale Oceanography
by
Jeffrey J. Early
A DISSERTATION
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Doctor of Philosophy
Presented October 22, 2009
Commencement June 2010
Doctor of Philosophy dissertation of Jeffrey J. Early presented on October 22, 2009
APPROVED:
Major Professor, representing Oceanography
Dean of the College of Oceanic and Atmospheric Sciences
Dean of the Graduate School
I understand that my dissertation will become part of the permanent collection of Oregon
State University libraries. My signature below authorizes release of my dissertation to
any reader upon request.
Jeffrey J. Early, Author
ACKNOWLEDGMENTS
First and foremost I would like to thank my advisor Roger Samelson for allowing me
the freedom to explore my own ideas and subsequently challenging me to further develop
and communicate those ideas. Special thanks also goes to my committee members Andrew
Bennet, Bill Bogley, Dudley Chelton, Juha Pohjanpelto, Bill Smyth and Roland de Szoeke,
each of whom went out of their way to help me at different points during my thesis. I
greatly appreciate the time and energy invested by my advisor and committee members.
Much thanks to my friends at COAS and elsewhere for taking the time to listen, for
engaging in exercise therapy, and for the stimulating conversations over great Oregonian
beers. Graduate school just wouldn’t have been the same without Christopher Wolfe,
Anthony Kirincich, Levi Kilcher, Toshi Kimura, Emily Shroyer, Craig Smith and Sam
Kelly.
The support I have received from my family throughout all my academic endeavors
has been incredible and I thank you all very much. My parents, Sharon and Dennis Early,
deserve special thanks for the incredible time, energy and support they’ve invested in me
that I am only now beginning to appreciate. I am especially grateful to my wife Julie,
who has provided unwavering support even while finishing her own thesis and giving birth
to our son. Finally, thanks to my son, Tevian, for enduring the first seven months of his
life with only half a father.
TABLE OF CONTENTS
Page
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2. LONG-TERM EVOLUTION OF QUASI-GEOSTROPHIC EDDIES . . . . . . . . . 10
2.1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.
Nonlinear Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Waves versus Eddies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Eddy Seeding Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.
12
13
Monopole Regimes: β-gyre, Initialization and the Quasi-Stable State . . . . 20
2.3.1 Formation of the β-gyre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Adjustment Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
22
2.4.
Quasi-Stable State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5.
Zonal Propagation Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6.
Meridional Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.7.
Trapped Fluid Conservation Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.7.1 Eddy Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.2 Eddy Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8.
31
36
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3. NEAR-GEOSTROPHIC APPROXIMATIONS OF THE SPHERICAL
SHALLOW-WATER EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.
Balance Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3.
Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4.
Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5.
Nondimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.5.1 Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
54
TABLE OF CONTENTS (Continued)
Page
3.5.3 Potential Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.
Near-Geostrophic Potential Vorticity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.6.1
3.6.2
3.6.3
3.6.4
3.7.
56
The Complete Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bounding the Cubic Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bounding the Cartesian Approximation . . . . . . . . . . . . . . . . . . . . . . . . .
Magnitude of Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
59
59
60
Equations of Balance Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.7.1
3.7.2
3.7.3
3.7.4
Quasi-Geostrophic Potential Vorticity, f -plane . . . . . . . . . . . . . . . . . . .
Quasi-Geostrophic Potential Vorticity, β-plane . . . . . . . . . . . . . . . . . . .
Flierl-Petviashvili Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
FP + J�� Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
64
65
66
3.8.
Valid Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.9.
Beyond Geostrophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.10. The FP Monopole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.11. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4. GROUP FOLIATION OF EQUATIONS IN GEOPHYSICAL FLUID DYNAMICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2.
Geophysical Fluid Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3.
Korteweg-de Vries Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3.1
4.3.2
4.3.3
4.3.4
4.3.5
4.4.
Nonlinear Wave Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Point Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Classical Method of Group Foliation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
EDS Method of Group Foliation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
85
86
89
91
Radial FP Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.4.1 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
94
TABLE OF CONTENTS (Continued)
Page
4.5.
2-D FP Equation in Polar Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.6.
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5. THE FORCES OF INERTIAL OSCILLATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2.
Free Particle, Inertial Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.3.
Free Particle, Rotating Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.4.
Central Gravitational Field, Rotating Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.5.
Particle on a Rotating Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.6.
Particle on Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.7.
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
APPENDIX A Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
APPENDIX B Exact Solutions to the Inertial Oscillation Problem . . . . . . . . . . . 141
APPENDIX C Disk on the Earth’s Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
APPENDIX D Eddy Propagation Speeds with the FP Equation . . . . . . . . . . . . . 159
LIST OF FIGURES
Figure
2.1
2.2
2.3
2.4
2.5
2.6
2.7
Page
The evolution of an initially Gaussian sea surface height of height 15 cm
and length 80 km. The first column is the linear form of equation (2.1)
with β −1 = 0 while the second column uses values appropriate for the
first baroclinic mode at latitude 24, β −1 = 60. The contours are drawn
for sea-level. The thick black line is the path of the sea surface height
maximum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
Sea surface height 13 years into the eddy seeding experiment. The top
panel shows the evolved state using the linear equation, while the bottom
panel uses the nonlinear equation. Both experiments were seeded with
the exact same eddies at the same time points. This domain shown is a
subset of the entire modeled domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
Variance of the sea surface height versus time for the linear and nonlinear
models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
Zonal frequency-wavenumber spectra for sea surface height of the linear
model (left) and nonlinear model (right) from the eddy seeding experiment. 25 neighboring latitude bands are ensemble averaged and running
median filter is applied. The black line is the maximum (l = 0) Rossby
wave zonal dispersion relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
Zonal frequency-wavenumber spectra for sea surface height from
the merged TOPEX/POSEIDON-ERS satellite altimetry data. The
solid line is computed from the radon transformation. The three
dispersion relations shown are from standard Rossby wave theory,
[Tailleux and McWilliams, 2001] and [Killworth et al., 1997], in order of
increasing frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
Paths of cyclonic (blue) and anticyclonic (red) eddies relative to their
starting position. The linear model (a) shows no preference for equatorward or poleward deflection, while the quasi-geostrophic model (b)
shows cyclones have a poleward preference and anticyclones an equatorward preference. The satellite observations (c) show a pattern similar to
quasi-geostrophic dynamics, but with stronger deflection from anticylonic
eddies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
Time evolution of four properties for a Gaussian initialized 15 cm height,
80 km eddy. Length scale decay rate, height decay rate, zonal speed and
meridional speed. The β-gyre formation (initialization) occurs for times
less than 11 days and is therefore not well described in this figure. All
four properties show the adjustment period of roughly 200 days before
the eddy settles into the quasi-stable state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
LIST OF FIGURES (Continued)
Figure
2.8
2.9
The height, relative vorticity, transported fluid, and fluid velocity for a
Gaussian initialized 15 cm height, 80 km eddy on day 675 of its evolution. The e-fold contour (maximum height divided by e) is shown in
red. The trapped fluid contour is in blue and is determined by finding
the largest closed contour of the stream function in the co-moving frame.
The relative vorticity contour, where ∇2 η = 0, is in black. . . . . . . . . . . . . . . .
Page
25
Zonal speed of eddy versus the eddy amplitude. The black line is the
linear (inverse amplitude) best fit line, cx (A) = 4.2A−1 −4.4 cm/s. Points
are colored with the eddy length scale (in km), suggesting a weak speed
dependence on length scale. The dashed grey line is the maximum zonal
group velocity in linear theory. The results were filtered to only include
eddies in the quasi-stable state that were not found to be interacting with
Rossby waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.10 Meridional speed of eddy versus the eddy amplitude. The black line is
the linear (inverse amplitude) best fit line, cx (A) = −3.7A−1 −0.13 cm/s.
The dashed grey line is the maximum meridional group velocity in linear
theory. Points are colored with the eddy length (in km), suggesting at
most weak speed dependence on length scale. . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.11 Propagation speed of eddy versus eddy amplitude. Same as figures
(2.9) and (2.10), but at latitude 35◦ . The zonal and meridional inverse amplitude best fits lines are cx (A) = 0.93A−1 − 2.0 cm/s and
cy (A) = −0.80A−1 − 0.046 cm/s, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.12 Per particle averages of potential vorticity and energy within the entire
trapped fluid region, the eddy core and the eddy ring of an 80 km, 15
cm Gaussian initialized eddy. The trends are the same for the other
quasi-stable eddies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
2.13 Histogram of the original x-position (a) and the original y-position (b) of
the fluid in the core of an 80 km 15 cm Gaussian initialized eddy on day
675. The eddy extremum is now located at x = −2247 km and y = −330
km. Thus, the core contains only fluid from its starting point over 2000
km away. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
2.14 Location of fluid advected on day 675. (a) Sea surface height with the
trapped fluid contour (blue) and relative voriticty zero contour (black).
Blue circles show the day zero location of the floats in the eddy ring
while red circles show the location of floats in the eddy core. (b) A
passive tracer initial given the value of its location on day 0. . . . . . . . . . . . .
35
LIST OF FIGURES (Continued)
Figure
2.15 Contributions to the total potential vorticity for a float initial at x = 29
km and y = 25 km. The float remains inside the core of the eddy for
all 730 days. On day 730 the float was located at x = −2473 km and
y = −365 km. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Page
36
2.16 Histogram of the original x-position (a) and the original y-position (b) of
the fluid in the ring of an 80 km 15 cm Gaussian initialized eddy on day
675. The eddy extremum is now located at x = −2247 km and y = −330
km. Thus, the core contains a mixture of fluid from throughout its lifetime. 37
2.17 Contributions to the total potential vorticity for a float initial at x = 3
km and y = 69 km. The float begins in the eddy core, cross to the ring,
and is eventually lost by the eddy. On day 730 the float was located at
x = −1900 km and y = −293 km. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
3.2
3.3
38
The magnitude of the terms in equation (3.42) are shown versus the
length scale. The coefficients in the legend can be matched with the
coefficients in equation (3.42). The surface height is fixed at 10 cm, the
latitude at 45 and the fluid velocity varies assuming G = 1. The ratio
of the length scale to the time scale is fixed at 1.46 cm/s, slightly faster
than the long wave Rossby phase speed at that latitude. . . . . . . . . . . . . . . . .
61
Each colored region shows the area in parameter space where a given
equation is valid. An equation is declared valid in a given region if all of
its terms have the greatest magnitude. This is a ‘best case scenario’ for
a equation as one might typically demand that its terms be a factor of
2 or 5 larger than other neglected terms and therefore cause the regions
to shrink. The blue regions require additional terms from either the
next order expansion O(�3 ), or from expansion of trigonometric terms
O(spherical). The surface height is fixed at 10 cm (�H = 0.13), G = 1 and
the ratio of the length scale to the time scale is fixed at the long Rossby
wave phase speed. The equations analyzed are the quasi-geostrophic
potential vorticity equation (3.58) (QG), the FP equation (3.66) (QG +
η 2 ), equation (3.71) (QG + η 2 + J�� ) and a fourth equation extending
QG with the J�� term (QG + J�� ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
Similar to figure (3.2), but now the ratio of the length scale to the time
scale is fixed at 0.8 x the long Rossby wave phase speed in the first panel
and 1.2 x the phase speed in the second panel. This suggests that the
extended QG theories are valid over a range of phase speeds. . . . . . . . . . . . .
70
LIST OF FIGURES (Continued)
Figure
3.4
3.5
3.6
Page
Similar to figure (3.2), but now the surface height is fixed at 5 cm
(�H = 0.63) in the first panel and 15 cm (�H = 0.19) in the second
panel. Quasi-geostrophic theory performs better at the smaller height
perturbations, whereas equation (3.71) (QG + η 2 + J�� dominates at the
larger perturbations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
Similar to figure (3.2), but now the latitude is fixed and the velocity is
allowed to vary (allowing departures from G = 1). The black line is
G = 1. The surface height is fixed at 10 cm (�H = 0.13) and the ratio of
the length scale to the time scale is fixed at 1.5 x the long Rossby wave
phase speed for the top panel at 20 degrees, and 1.0 x the Rossby wave
phase speed for the bottom panel at latitude 45. . . . . . . . . . . . . . . . . . . . . . . . .
73
Similar to figure (3.2), but now the scales are fixed for the FP monopole
solution (3.74). The yellow region indicates the small region in parameter
space where the five terms from the FP equation (3.67) dominate all other
neglected terms. By assumption then, this is the only region where the
FP monopole solution (3.74) is valid. The distance scale shown here is
taken to be the e-fold radius of the FP solution. . . . . . . . . . . . . . . . . . . . . . . . . .
74
5.1
Three example particle paths in a central gravitational field as viewed
from a rotating reference frame. The second figure approximates that of
the International Space Station, while the third is a small deviation from
geosynchronous orbit. All three are inertial and exhibit oscillatory motion. 109
5.2
Force diagram for a particle initially at rest on a rotating sphere. The
net force points towards the equator, but only the normal force, F�N , is a
real, measurable force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.3
Three example particle paths on a sphere as viewed from a rotating reference frame. The first figure depicts the path of a particle released from
rest in the rotating reference frame, while the second and third show
different choices of initially eastward velocities. All three are inertial
tangent to the surface, and oscillatory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.4
Force diagram for a particle initially at rest on the earth. All forces
balance, so the particle will remain at rest in this reference frame. The
outline of a sphere is shown in gray. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.5
Three example particle paths on a earth as viewed from a rotating reference frame. Just as with the sphere, all three are inertial tangent to the
surface and exhibit oscillatory motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
LIST OF FIGURES (Continued)
Figure
Page
B.1 Particle path in case 1, where the particle was launched from 45 N due
east with φ̇ω0 = 0.18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
B.2 Particle path in case 2, where the particle was launched from 45 N due
east with φ̇ω0 = 0.41. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
B.3 Particle path in case 3, where the particle was launched from 45 N due
east with φ̇ω0 = 0.60. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
B.4 Coriolis parameter f = 2ω sin θ and the new coriolis parameter based on
an initial eastward velocity of 25 meters per second. . . . . . . . . . . . . . . . . . . . . . 151
B.5 Eastward drift speed of the particle launched at 25 meters per second to
the East. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
D.6 Propagation speeds of Gaussian initialized eddies at latitude 24 with FP
dynamics. The dashed grey line is the maximum group velocity in linear
theory. Points are colored with the eddy length (in km). The inverse
amplitude best fits lines are drawn in black. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
D.7 Propagation speeds of Gaussian initialized eddies at latitude 35 with FP
dynamics. The dashed grey line is the maximum group velocity in linear
theory. Points are colored with the eddy length (in km).The inverse
amplitude best fits lines are drawn in black. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Mathematical Approaches to the Physics of Mesoscale Oceanography
1.
INTRODUCTION
A multifaceted collection of numerical, theoretical and mathematical studies are
applied towards enhancing our understanding of the mesoscale physics of the ocean. The
primary physical interest in these studies is understanding the underlying physics of slowly
westward propagating eddies. The mathematical interest uses this as a platform to explore
group foliation as a technique for solving problems of physical interest.
Early theoretical models identified Rossby waves as playing a primary role in establishing the large scale circulation properties of the ocean [Anderson and Gill, 1975].
Linear Rossby wave theory assumes that the three dimensional quasi-geostrophic potential vorticity equation can be simplified when the eigenmodes of the vertical structure act
independently. With this assumption, each of these eigenmodes can be interpreted as an
independent layer of fluid with some equivalent depth, D, and free surface perturbation,
η, each satisfying a simplified equation,
� ∂η
�
�
∂ � 2
∇ η−η +
+ β −1 · J η, ∇2 η = 0
∂t
∂x
where J(a, b) = ax by − ay bx . The coefficient β −1 =
√
gD
β0 L2R
(1.1)
where g is the acceleration of
gravity, D is the eigenvalue of the eigenmode, β0 is the variation of the Coriolis parameter
and LR is the Rossby radius of deformation. Mid-latitude mesoscale features typically have
values β −1 = O(100). The linearized quasi-geostrophic potential vorticity equation (β −1 =
0) for each mode admits plane wave solutions obeying a dispersion relation dependent on
D [Pedlosky, 1987].
Satellite altimetry observations of sea surface height showed the existence of global
slowly westward propagating waves [Chelton and Schlax, 1996] and nearly match what
would be expected of first baroclinic eigenmode Rossby waves. These early observations were shown to differ in their predicted phase speed from linearized quasi-geostrophic
2
theory and consequently were followed by attempts to modify the classical theory (e.g.,
[Killworth et al., 1997, Tailleux and McWilliams, 2001, Killworth and Blundell, 2005]).
However, subsequent observations have cast doubt on the original interpretation of the observations as dispersive linear waves [Chelton et al., 2007]. Enhanced altimetry resolution
shows westward propagating coherent structures that are characterized as more eddy-like,
than wave-like. Estimates of nondimensional parameters measuring non-linearity also
suggest that linear theory does not apply.
Chapter 2 directly addresses the issue of whether or not the satellite observations
can be correctly interpreted as linear Rossby waves. A numerical model of the linearized
quasi-geostrophic potential vorticity equation is used in a series of experiments to compare
with the non-linear model that includes the advective terms. It is argued that the long
term coherence and spectral properties of the observed eddies cannot be explained by the
linear theory and must be rejected in favor of the fully non-linear theory.
In order to understand how well non-linear quasi-geostrophic theory compares to
the observations, the model is initialized with isolated eddies over a range of height and
length scales. It is shown that a previously unrecognized quasi-stable, slowly decaying,
eddy state emerges from most initial conditions. Individual characteristics such as amplitude decay, horizontal length scale decay, propagation speed, fluid transport properties,
energy conservation, and potential vorticity conservation are diagnosed. The quantity and
distance of fluid transported by the individual eddies is found to be substantial. This is
in stark contrast to linear Rossby waves, which cannot transport fluid. However, it is also
found that the westward propagation speeds of quasi-geostrophic eddies are consistently
slower than the speeds found in the observations. This suggests that, while non-linear
quasi-geostrophic theory may be better a model for satellite observed eddies than linear
Rossby wave theory, a modified theory is required to explain their properties completely.
A new technique based on scaling analysis is developed in Chapter 3 for evaluating
the scales of motion of the eddy observations in order to find an appropriate model. A
3
typical approach to deriving a new theory is to begin with a general set of equations
and then use our knowledge of an observed phenomenon to apply additional constraints
in hopes of reducing the physical and mathematical complexity of the equations. In the
usual asymptotic derivation of quasi-geostrophic theory, the nondimensional parameters
admitted by the primitive equations are fixed relative to a single nondimensional parameter assumed to be small, �. The dependent variables are then expanded as a perturbation
series with respect to �, and in the limit � �→ 0, only the leading order terms remain.
Quasi-geostrophic theory stems from one particular choice of relationships amongst the
nondimensional parameters. To derive alternative theories, different choices in the relationships between the nondimensional parameters of the primitive equations must be
chosen, and the perturbation series then reapplied.
The approach taken in Chapter 3 shows how a potential vorticity equation can be
derived from the spherical shallow water equations without fixing the nondimensional parameters a priori. The resulting equation can then be reduced to other theories, like quasigeostrophy (1.1), by assuming a precise relationship between the parameters. However,
by retaining the freedom in the parameters it can quickly be deduced which terms in the
equation are necessary for an accurate description of the observed scales. Conversely, we
are also able to determine at which scales the various theories remain valid and at which
scales their assumptions are violated. To specifically address the non-linear effects of
height, fluid speed, and length scale found in the eddy observations [Chelton et al., 2007],
the spherical shallow water equations are chosen as a reasonably generalized starting point.
Chapter 3 concludes by showing that the quasi-geostrophic potential vorticity equation is not a valid equation for the observed scales of the eddies and that the FlierlPetviashvili (FP) equation [Flierl, 1979, Petviashvili, 1980] and a new, extended FP equation are more suitable models. The primary physical difference separating these two
equations from quasi-geostrophy (1.1) is the removal of the restriction that the eddy am-
4
plitudes must be small relative to the equivalent depth, D. The FP equation,
�
�
��
�
�
∂
1 2
∂η
2
∇ η− η− η
+
+ β −1 · J η, ∇2 η = 0.
∂t
2
∂x
has
been
derived
in
this
context
before
(1.2)
[Anderson and Killworth, 1979,
Charney and Flierl, 1981], but also arises in several different contexts including the
Great Red Spot on Jupiter [Petviashvili, 1980], as well as modifications to quasigeostrophy with an exterior mean shear flow [Flierl, 1979] and the thermobaricity of the
equation of state [de Szoeke, 2004]. An axially symmetric solution to (1.2) has been
found numerically [Flierl, 1979] and by analytical approximation [Petviashvili, 1980]
[Boyd, 1991] which propagates zonally slightly faster than the Rossby long wave phase
speed. The faster propagation suggests that the FP and extended FP equations may
directly address the deficiencies of quasi-geostrophic theory shown in chapter 2.
Chapter 4 focuses on finding new solutions to the FP equation (1.2). However,
rather than use the numerical model of Chapter 2, the aim is to find analytical solutions,
with the broader goal of introducing the method of group foliation to the geophysical fluid
dynamics community. One general approach that is applicable to solving many nonlinear
partial differential equations (PDEs), such as the FP equation (1.2), is the method of
symmetry reduction originally proposed by Sophus Lie. This method is used implicitly
by physicists when identifying radial solutions (rotation invariance) or traveling wave
solutions (Galilean invariance), but also explicitly to reveal more complicated symmetry
groups and corresponding invariant solutions (e.g., [Salmon and Hollerbach, 1991]). An
unfortunate consequence of symmetry reduction is that it eliminates any solution not
invariant under the given group, which severally restricts the scope of available solutions
and limits the applications to the physical sciences.
A more general, but less explored, technique to the analytical integration of system of PDEs is the method group foliation. The classical approach developed by Vessiot [Vessiot, 1904] can be applied to the same large class of linear and nonlinear PDEs
5
as symmetry reduction, but it does not suffer from the limitations imposed by group
invariance. Importantly, in contrast with symmetry reduction, Vessiot’s method can
be used to construct any solution of the original system from the reduced equations.
The classical approach to group foliation uses the differential invariants of the symmetry
group to rewrite the equation on the orbit manifold of the symmetry group action (e.g.,
[Martina et al., 2001, Anco and Liu, 2004], while a more modern approach utilizes a reduction process for an exterior differential system (EDS) associated with the equations
(e.g, [Anderson and Fels, 2005, Pohjanpelto, 2008]).
Chapter 4 presents an application of the two different group foliation techniques to
various simplified forms of the FP equation (1.2) including the one-dimensional Kortewegde Vries (KdV) equation. It is shown how the KdV equation reduces from a third order
non-linear ordinary differential equation to a first order ordinary differential equation,
which can then be solved and used to reconstruct the general solution. These techniques
are then applied to find the first known exact analytical solution to a radially symmetric
form of FP equation. Unfortunately, this solution does not appear to be of interest to
the eddy observations. However, exploiting the rotational symmetry of a two-dimensional
form of the FP equation in polar coordinates reduces the elliptic equation to a pair of
quasi-linear hyperbolic equations. In principal, any solution to the two-dimensional FP
equation can be reconstructed from this quasi-linear pair, and therefore may eventually
yield solutions relevant to the eddy observations. Chapter 4 concludes by discussing some
of the challenges that remain for the application of group foliation to more difficult PDEs
such as the full FP equation (1.2).
Chapter 5 explores one of the underlying forces found in mesoscale oceanography.
The slowly westward propagating motions found in the satellite altimetry observations are
caused by latitudinal variations in the Coriolis parameter, known as the β-effect. The
∂η
∂x
term of equations (1.1) and (1.2) can be directly attributed to this effect and is therefore
known to be the cause of the westward signal propagation of Rossby waves and eddies
6
found in these models. Understanding the fundamental forces involved in the β-effect is
unnecessarily complicated with the complete fluid equations of geophysical fluid dynamics.
By reducing the equations from vector field equations to a vector equation, we eliminate
the possibility of finding eddy solutions, but still retain the physics important to the βeffect. Doing this is equivalent to restricting the motion of a particle to the surface of the
earth, where the simplified equations can be solved exactly. The solutions show that the
resulting oscillatory motion of the point particle, known as ‘inertial oscillations’, involves a
slow westward drift directly attributable to the β-effect and the nearly spherical geometry
of the earth [Ripa, 1997].
The name ‘inertial oscillation’ suggests that the oscillatory motion of the particle
is explained entirely by inertial forces, which differ from real forces in that they cannot
be measured. If an observer is not in an inertial frame, he must therefore invent these
fictitious forces to explain the observed motion. This concept was fundamental in the
formulation of Newton’s laws, but was importantly modified twice: once with Einstein’s
theory of special relatively, and then again with Einstein’s theory of general relativity
[Misner et al., 1973]. The name ‘inertial oscillation’ therefore makes the observational
claim that an accelerometer trapped in perfect inertial motion would not measure accelerations. However, [Durran, 1993] argues that there is in fact a force acting on the particle
and suggests removing the word ‘inertial‘ from its name. By this reasoning it is now
repeated [Paldor and Sigalov, 2001] that these oscillations are not inertial.
Starting with free particle and successively adding constraints, Chapter 5 shows that
the particle’s motion is in fact inertial and the accelerometer trapped in inertial motion
would not measure an acceleration in the horizontal plane. However, it is also shown that
the name ‘inertial oscillations’ is ambiguous and perhaps better replaced by the names
‘geopotential oscillation’ or ‘geoinertial oscillation.’
Appendix B computes the exact analytical solutions to the inertial oscillation problem and compares the results to to the typical f -plane approximation. Appendix C extends
7
the calculation of inertial motion from a point mass, to a disc constrained to the earth’s
surface. It shown that even without rotation about its own axis, the disk propagates
slowly westward at the same speed as the particle when given an equivalent amount of
energy. The application of this model to mesoscale eddies has been discussed on several
occasions [McDonald, 1999, Ripa, 2000b, Ripa, 2000a].
8
BIBLIOGRAPHY
Anco and Liu, 2004. Anco, S. and Liu, S. (2004). Exact solutions of semilinear radial
wave equations in n dimensions. Journal reference: J. Math. Anal. Appl, 297:317–342.
Anderson and Gill, 1975. Anderson, D. and Gill, A. (1975). Spin-up of a stratified ocean,
with applications to upwelling. Deep-Sea Res, 22:583–596.
Anderson and Killworth, 1979. Anderson, D. and Killworth, P. (1979). Nonlinear propagation of long rossby waves. Deep-Sea Res, 26:1033–1049.
Anderson and Fels, 2005. Anderson, I. and Fels, M. (2005). Exterior differential systems
with symmetry. Acta Applicandae Mathematicae: An International Survey Journal on
Applying Mathematics and Mathematical Applications, 87(1):3–31.
Boyd, 1991. Boyd, J. (1991). Monopolar and dipolar vortex solitons in two space dimensions. Wave Motion, 13(3):223–241.
Charney and Flierl, 1981. Charney, J. and Flierl, G. (1981). Oceanic analogues of largescale atmospheric motions. Evolution of Physical Oceanography, pages 504–548.
Chelton and Schlax, 1996. Chelton, D. and Schlax, M. (1996). Global observations of
oceanic rossby waves. Science, 272(5259):234.
Chelton et al., 2007. Chelton, D. B., Schlax, M. G., Samelson, R. M., and de Szoeke,
R. A. (2007). Global observations of large oceanic eddies. Geophys. Res. Lett., 34.
de Szoeke, 2004. de Szoeke, R. (2004). An effect of the thermobaric nonlinearity of the
equation of state: A mechanism for sustaining solitary rossby waves. Journal of Physical
Oceanography, 34(9):2042–2056.
Durran, 1993. Durran, D. R. (1993). Is the coriolis force really responsible for the inertial
oscillation? Bulletin of the American Meteorological Society, 74(11):2179–2184.
Flierl, 1979. Flierl, G. (1979). Baroclinic solitary waves with radial symmetry. Dyn.
Atmos. Oceans, 3:15–38.
Killworth and Blundell, 2005. Killworth, P. and Blundell, J. (2005). The dispersion relation for planetary waves in the presence of mean flow and topography. part ii: Twodimensional examples and global results. Journal of Physical Oceanography, 35:2110.
Killworth et al., 1997. Killworth, P., Chelton, D., and de Szoeke, R. (1997). The speed
of observed and theoretical long extratropical planetary waves. Journal of Physical
Oceanography, 27(9):1946–1966.
9
Martina et al., 2001. Martina, L., Sheftel, M., and Winternitz, P. (2001). Group foliation and non-invariant solutions of the heavenly equation. Journal of Physics A:
Mathematical and General, 34(43):9243–9263.
McDonald, 1999. McDonald, N. (1999). The motion of geophysical vortices. Philosophical Transactions: Mathematical, Physical and Engineering Sciences, 357(1763):3427–
3444.
Misner et al., 1973. Misner, C. W., Thorne, K. S., and Wheeler, J. A. (1973). Gravitation. W. H. Freeman and Company.
Paldor and Sigalov, 2001. Paldor, N. and Sigalov, A. (2001). The mechanics of inertial
motion on the earth and on a rotating sphere. Physica D, 160:29–53.
Pedlosky, 1987. Pedlosky, J. (1987). Geophysical Fluid Dynamics. Spring-Verlag, second
edition.
Petviashvili, 1980. Petviashvili, V. (1980). Red spot of jupiter and the drift soliton in a
plasma. Journal of Experimental and Theoretical Physics Letters, 32.
Pohjanpelto, 2008. Pohjanpelto, J. (2008). Reduction of exterior differential systems
with infinite dimensional symmetry groups. BIT Numerical Mathematics, 48(2):337–
355.
Ripa, 1997. Ripa, P. (1997). “inertial” oscillations and the β-plane approximation(s).
Journal of Physical Oceanography, 27:633–647.
Ripa, 2000a. Ripa, P. (2000a). Effects of the earth’s curvature on the dynamics of isolated
objects. part i: The disk. Journal of Physical Oceanography, 30(8):2072–2087.
Ripa, 2000b. Ripa, P. (2000b). Effects of the earth’s curvature on the dynamics of isolated objects. part ii: The uniformly translating vortex. Journal of Physical Oceanography, 30(10):2504–2514.
Salmon and Hollerbach, 1991. Salmon, R. and Hollerbach, R. (1991). Similarity solutions
of the thermocline equations. Journal of marine research, 49(2):249–280.
Tailleux and McWilliams, 2001. Tailleux, R. and McWilliams, J. (2001). The effect of
bottom pressure decoupling on the speed of extratropical, baroclinic rossby waves.
Journal of Physical Oceanography, 31(6):1461–1476.
Vessiot, 1904. Vessiot, E. (1904). Sur l’intégration des systèmes différentiels qui admettent des groupes continus de transformations. Acta Mathematica, 28(1):307–349.
10
2.
LONG-TERM EVOLUTION OF QUASI-GEOSTROPHIC EDDIES
Jeffrey J. Early, Roger M. Samelson and Dudley B. Chelton
Journal of Physical Oceanography
In preparation
11
2.1.
Introduction
Baroclinic Rossby waves have long been believed to play an important role
in the spin-up of the ocean [Anderson and Gill, 1975] and so their apparent direct
observation through satellite altimetry measurements of sea surface height (SSH)
[Chelton and Schlax, 1996] was a well celebrated result. These early observations were
shown to differ in their predicted phase speed from linearized quasi-geostrophic theory and consequently were followed by attempts to modify the classical theory (e.g.,
[Killworth et al., 1997, Tailleux and McWilliams, 2001, Killworth and Blundell, 2005]).
However, subsequent observations from higher resolution SSH fields constructed from multiple satellite altimeters have cast doubt on the original interpretation of the observations
as linear waves [Chelton et al., 2007]. The enhanced observations now show more eddylike characteristics by remaining coherent structures for long durations, and suggesting a
high degree of nonlinearity through several non-dimesional parameters. In light of these
enhanced observations, a strong case needs to be established for rejecting the original
interpretation as linear Rossby waves and showing that they are indeed eddies.
In the first part of the paper we start by examining both linear (β −1 = 0) and
nonlinear (β −1 �= 0) quasi-geostrophic theory in an equivalent barotropic model,
� ∂η
�
�
∂ � 2
∇ η−η +
+ β −1 · J η, ∇2 η = 0
∂t
∂x
where the Jacobian is define as J(a, b) = ax by − ay bx . The coefficient β −1 =
(2.1)
√
gD
β0 L2R
where g
is the acceleration of gravity, D is the fluid depth (or equivalent depth), β0 is the variation
of the Coriolis parameter and LR is the Rossby radius of deformation. Equation (2.1)
is typically written with the nondimensional parameter β associated with the planetary
∂η
vorticity term (β ∂x
); however, the convention used here uses the more natural time scale
of (β0 LR )−1 , rather than f0−1 , and reduces equation (2.1) to a parameter independent
form of the linearized Rossby wave equation when setting β −1 = 0. In one experiment the
12
evolution of individual Gaussian initialized eddies is compared between the two theories
where it is shown that the long term coherence of the observed eddies is best explained
by the nonlinear theory. In an additional experiment, a basin is seeded continuously for
15 years with Gaussian eddies that approximate those eddies observed with altimetry.
By considering the zonal wavenumber-frequency power spectral density, we are able to
compare model results with observations and it is argued that linear theory does not
explain the observed spectra and must be rejected.
In the second part of the paper isolated Gaussian initial conditions are modeled with
the nonlinear equation for time durations longer than previous studies. It is shown that
a previously unrecognized quasi-stable eddy state emerges. Individual characteristics of
these eddies are then diagnosed including amplitude decay, horizontal length scale decay,
propagation speed and fluid transport properties. These provide baseline properties for
comparison with the observations and other theories. It is shown that while transport
properties may reflect those of observed eddies, zonal propagation speeds are too slow by
comparison.
2.2.
Nonlinear Dynamics
2.2.1
Waves versus Eddies
To compare wave-like and eddy-like mesoscale features, consider the single plane-
wave Rossby wave solution to the fully nonlinear quasi-geostrophic equation (2.1),
η(x, y, t) = N0 cos(kx + ly − ωt + φ)
where ω =
−k
k2 +l2 +1
(2.2)
[Pedlosky, 1987]). Although equation (2.2) solves both the linearized
and nonlinear form of equation (2.1), only by setting β −1 = 0 do linear combinations of
Rossby waves succeed in solving the equation because equation (2.1) itself is not linear. For
longer wavelengths, k, l � 1, the linearized form of the equation is only weakly dispersive
13
and so it is not unreasonable to expect coherent features to remain for long durations
as observed in the altimetry data. Approximate analytical solutions to the evolution of
eddies under linear dynamics were found in [Flierl, 1977].
An initial comparison between the linear and nonlinear form of the equation can be
made by considering the evolution of an initially Gaussian sea surface height perturbation.
For all model runs an equivalent depth of D = 79.9 cm (gravity waves phase speed of 2.8
m/s) was used at latitude 24◦ . The resulting deformation radius is 47.2 km with time
scale of 11.7 days and the long-wave Rossby wave speed is therefore cx = 4.7 cm/s. For
the nonlinear case these parameters require setting β −1 = 60, while for the linear case
β −1 = 0.
An initial perturbation of η(x, y, 0) = N0 e−r
2 /L2
with amplitude N0 = 15 cm and
length L = 80 km was modeled for 365 days using the two forms of the equation; the
results are shown in figure (2.1). The linear evolution is dominated by Rossby wave interference patterns, although the sea surface maximum can still be observed to propagated
westward. Conversely, the Gaussian modeled with the nonlinear equation also shows
Rossby wave interference patterns, but is dominated by the coherent westward propagating sea surface maximum. The nonlinear anticyclonic eddy also shows a decrease in the
amplitude decay rate and an equatorward deflection [McWilliams and Flierl, 1979], both
desirable attributes for an appropriate theoretical description of the observations reported
by [Chelton et al., 2007].
2.2.2
Eddy Seeding Experiment
To compare the wavenumber-frequency spectra of observations with those of the
linear and nonlinear models, a basin 11264 km by 5632 km was seeded with Gaussian
eddies. The eddy seeds were of varying height, horizontal length scales and frequency of
occurrence matching the observed statistics. The simulation was run and continuously
seeded for 15 years. To prevent signals from crossing the boundaries, a 400 km thick
14
FIGURE 2.1: The evolution of an initially Gaussian sea surface height of height 15 cm
and length 80 km. The first column is the linear form of equation (2.1) with β −1 = 0
while the second column uses values appropriate for the first baroclinic mode at latitude
24, β −1 = 60. The contours are drawn for sea-level. The thick black line is the path of
the sea surface height maximum.
15
sponge layer was used on all four sides of the basin.
The sea surface height 13 years into the two model runs is shown in figure (2.2).
Because the linear model simply evolves the phases of individual Rossby waves, the energy
at individual wave numbers cannot transfer to other wave numbers and changes only by
virtue of the energy added by the eddy seeds. The sea surface height for the linear model
there reflects an evolving interference pattern of waves with length scales unmodified from
the original eddy seeds. Conversely, the nonlinear model allows interactions and transfers
energy to different scales just as in the study of quasigeostrophic turbulence [Vallis, 2006].
The nonlinear model run shows a clear trend toward longer wavelengths. Further, the
eddies can be observed to interact by changing their propagation paths and merging,
unlike the linear case.
Before we compute the wavenumber-frequency spectra, a suitable spatial and temporal subdomain for the analysis needs to be determined. Because the sea surface is
initially flat, we expect that seeding the model with eddies will require a spin-up period
before a quasi-steady state is reached. Figure (2.3) shows that the variance of the sea
surface height of both the linear and nonlinear models reaches a relative steady value at
about 1500 days. For the spatial domain, analysis was restricted to 1500 km west of the
eastern most-eddy seeds, similarly allowing appropriate spin up time.
The zonal frequency-wavenumber spectra in figure (2.4) show very different behaviors between the two models. For the linear model, the largest signal remains below the
l = 0 zonal Rossby wave dispersion relation. This is consistent with theory, as waves
with given zonal (k) and non-zero meridional wavenumbers (l �= 0) have frequencies that
remain below the l = 0 frequency. The power distributions for the linear and nonlinear
model are substantially different. The signals are essentially non-dispersive for lower wave
numbers for both, while higher wavenumbers remain along the same non-dispersive slope
for the nonlinear model but not the linear model.
The zonal frequency-wavenumber spectrum of the nonlinear model closely matches
16
FIGURE 2.2: Sea surface height 13 years into the eddy seeding experiment. The top
panel shows the evolved state using the linear equation, while the bottom panel uses the
nonlinear equation. Both experiments were seeded with the exact same eddies at the same
time points. This domain shown is a subset of the entire modeled domain.
17
FIGURE 2.3: Variance of the sea surface height versus time for the linear and nonlinear
models.
FIGURE 2.4: Zonal frequency-wavenumber spectra for sea surface height of the linear
model (left) and nonlinear model (right) from the eddy seeding experiment. 25 neighboring
latitude bands are ensemble averaged and running median filter is applied. The black line
is the maximum (l = 0) Rossby wave zonal dispersion relation.
18
FIGURE 2.5: Zonal frequency-wavenumber spectra for sea surface height from the merged
TOPEX/POSEIDON-ERS satellite altimetry data. The solid line is computed from the
radon transformation. The three dispersion relations shown are from standard Rossby
wave theory, [Tailleux and McWilliams, 2001] and [Killworth et al., 1997], in order of increasing frequency.
that of the observations in figure (2.5), while the linear model fails to explain the nondispersive structure observed. As such, linearized quasi-geostrophic theory is not a viable
theory to explain the observed features.
Figure (2.6) shows the paths of the eddies in the eddy seeding experiment relative
to their starting positions. The linear model shows no systematic preference for meridional deflection, matching the results found for the isolated Gaussian in figure (2.1). In
contrast, the eddies tracked in the quasi-geostrophic model show tendencies for poleward
and equatorward deflection for cyclonic and anticyclonic eddies, respectively. This is more
consistent with the observations which shows a similar trend. However, the eddies from the
observations show that anticyclonic eddies have a greater tendency equatorward deflection
than cyclonic eddies do for poleward deflection.
/LQHDU4*0RGHO
/LQHDU4*0RGHO
/LQHDU4*0RGHO
1RQOLQHDU4*0RGHO
1RQOLQHDU4*0RGHO
(a) Linear Dynamics
1RQOLQHDU4*0RGHO
19
2EVHUYDWLRQV
2EVHUYDWLRQV
(b) QG Dynamics
2EVHUYDWLRQV
(c) Observations
FIGURE 2.6: Paths of cyclonic (blue) and anticyclonic (red) eddies relative to their starting position. The linear model (a) shows no preference for equatorward or poleward deflection, while the quasi-geostrophic model (b) shows cyclones have a poleward preference
and anticyclones an equatorward preference. The satellite observations (c) show a pattern similar to quasi-geostrophic dynamics, but with stronger deflection from anticylonic
eddies.
20
2.3.
Monopole Regimes: β-gyre, Initialization and the Quasi-Stable
State
The interest in eddies on a β-plane has a generated a long history of analytical and
numerical models attempting to elucidate some of their basic properties, such as amplitude decay and propagation speeds. The two-dimensional barotropic quasi-geostrophic
potential vorticity equation (2.1) lacks many of the complexities associated with multilayer quasi-geostrophic or primitive equation models, yet remains sufficiently complex
that the evolution properties of Gaussian initialized disturbances are still not completely
understood. In [Korotaev, 1997] the evolution of the initial Gaussian disturbances were
divided into two states, an initialization period followed by what was assumed to be a
quasi-stable state. Here we will argue that there are actually three states: formation of
the β-gyre (initialization), an adjustment period (formally believed to be quasi-stable),
and a third quasi-stable, slowly decaying state that has yet to be well explored.
Typical amplitude (5, 10, 15, 20 cm) and length (40, 60, 80, 100, 120 km) scales
were used to initialize Gaussians at latitude 24◦ . In almost all cases, a quasi-stable westward propagating eddy ultimately emerged as the dominant feature. When the model
was initialized with other non-Gaussian solitary shapes, other solutions were more likely
to emerge (including eastward propagating dipoles), but still included was the quasi-stable
westward propagating eddy, although potentially with very different scales than the initialization shape. These quasi-stable eddies are the focus of this study, but we briefly
consider the other two states as well.
2.3.1
Formation of the β-gyre
The first component of an eddy’s evolution is the formation of the β-gyre, in
which an initially axisymmetric eddy evolves an azimuthal mode-one component due the
β effect [Fiorino and Elsberry, 1989]. The flow associated with the dipole structure of
21
the gyre initially causes a largely meridional deflection of the eddy, and then eventually becomes more zonal. Analytical predictions for the trajectory of an eddy found
good agreement with numerical simulations for time periods less than (β0 LR )−1 , after
which the radiation of Rossby waves strongly alters its evolution [Sutyrin and Flierl, 1994]
[Reznik and Dewar, 1994].
This initialization period must be expected because a Gaussian shape cannot be
a stable solution for the quasi-geostrophic potential vorticity equation (2.1). An radially
symmetric shape like a Gaussian causes the Jacobian to vanish; meaning that the advection
of relative vorticity is trivial (advection is still present, but moves fluid parcels to locations
of fluid parcels with identical attributes). Because the advective nonlinearity is initially
trivial, linear Rossby wave dispersion due to the β effect will necessarily cause the initially
Gaussian shape to become asymmetric (as explored in [Flierl, 1977]). This asymmetry
will in turn induce non trivial advection of relative vorticity through the Jacobian term.
Here we consider an explanation of the formation of the β-gyres valid for this short time
scale initialization period of anti-cyclonic Gaussian eddy.
1. Because the eddy is initially radially symmetric, the first time step is governed
entirely by linear dynamics. The whole eddy moves westward, but due to dispersion,
the signal associated with longer wavelengths will travel westward and decrease the
western slope of the eddy, while the signal associated with shorter wavelengths will
travel eastward and steepen the eastern slope of the eddy.
2. As the grade of the leading edge of the eddy shallows and the trailing edge steepens,
advection plays a larger role (this is the formation of the β-gyre). The stronger flow
to the south and weaker flow to the north cause net southward meridional advection
of fluid, and therefore a meridional deflection of the eddy. The initial meridional
deflection is particularly strong because advection is exactly southward. The eddy
is now translating southwest.
22
3. Eventually the asymmetric shape of the eddy forms to provide a near advectivedispersive balance. Dispersion continues to steepen the trailing edge of the eddy (now
in the northeast) while advection moves fluid to the southwestern region. These two
effects do not completely balance (at an order of magnitude less than their individual
values) and the net effect is to pull the eddy to the southeast while maintaining a
shallow northwestern edge and a steep southeastern edge of the eddy. The linear
dispersion is responsible for slowing the westward propagation of the eddy from
linear long-wave speed [Flierl, 1977], while the advection is responsible for deflecting
the eddy to the south.
2.3.2
Adjustment Period
After time periods of t ∼ (β0 LR )−1 , the eddy’s evolution is largely dictated by its
energy loss due to the excitation of Rossby waves [Flierl, 1984]. Figure (2.7) shows the
rapid changes in horizontal length scale decay rate, height decay rate, zonal speed and
meridional speed over the first 200 days, t ∼ 20(β0 LR )−1 , where the adjustment period
is quite distinct. Numerical models, such as [McWilliams and Flierl, 1979], only consider
times roughly as long this adjustment period, likely owing to the computational power
of the time. Analytical estimates for an eddy’s trajectory and decay rates using the
effects of Rossby wave radiation have been found to be in weak agreement with numerical
simulations during the first parts of this adjustment period [Korotaev, 1997].
For most of the cases considered here, the transition from initialized Gaussian to
quasi-stable, slowly decaying eddy followed the pattern shown in figure (2.7). However,
some of the longer length scale, smaller amplitude Gaussians (which have smaller Umax )
took much longer, or never even reached the quasi-stable state, instead dispersing with
more wave-like characteristics due to the weak nonlinearity of these eddies. During the
adjustment period, the horizontal length scale of all eddies initially increased (therefore
decreasing the ratio of amplitude to length scale) while the maximum fluid velocity in-
23
FIGURE 2.7: Time evolution of four properties for a Gaussian initialized 15 cm height, 80
km eddy. Length scale decay rate, height decay rate, zonal speed and meridional speed.
The β-gyre formation (initialization) occurs for times less than 11 days and is therefore
not well described in this figure. All four properties show the adjustment period of roughly
200 days before the eddy settles into the quasi-stable state.
24
creased. The length of the adjustment period depends on the height and length of the
initial Gaussian, but at approximately 100-200 days, t ∼ 15(β0 LR )−1 , the quasi-stable
state is reached.
2.4.
Quasi-Stable State
The quasi-stable eddy state always forms into a characteristic shape shown in figure
(2.8) necessary for maintaining the near advective-dispersive balance. The height field
is characterized by a steep south-southeastern edge, while the north-northwestern edge is
particularly shallow. The resulting velocity field, determined from geostrophy, follows with
strong flow in the south-southeastern region and weak flow along the north-northeastern
edge. This asymmetry in the height field is easily seen in the red e-fold contour.
In the co-moving frame, the height field (written as a stream-function or pressure
field) becomes far more symmetric. The resulting largest closed contour is shown in blue
and is the instantaneous trapped fluid region. The term“trapped fluid region”, although
commonly used, is a of bit a misnomer as the fluid is only approximately trapped, and
only at that instant. We’ll expand this idea later.
The relative vorticity zero contour, where ∇2 η = 0, shown in black, remains nearly symmetric throughout the eddy lifetime.
Using definitions similar to
[Korotaev and Fedotov, 1994], the inner core of an anti-cyclonic eddy is defined as the
region containing negative relative vorticity, while the outer ring is the surrounding region of positive relative vorticity, separated by this boundary.
25
FIGURE 2.8: The height, relative vorticity, transported fluid, and fluid velocity for a
Gaussian initialized 15 cm height, 80 km eddy on day 675 of its evolution. The e-fold
contour (maximum height divided by e) is shown in red. The trapped fluid contour is in
blue and is determined by finding the largest closed contour of the stream function in the
co-moving frame. The relative vorticity contour, where ∇2 η = 0, is in black.
26
2.5.
Zonal Propagation Speed
The zonal speed of the eddies was found to be dependent on the eddy amplitude
such that larger amplitude eddies propagate faster than smaller amplitude eddiesas shown
in figure (2.9). The figure also shows mild dependence on eddy length scale with smaller
eddies propagating more slowly. In general then, eddies larger in both amplitude and
length scale propagate faster than eddies with small amplitude and length scale. The
inverse amplitude linear best fit line was found to be cx (A) = 4.2A−1 − 4.4 cm/s. This is
suggestive of a lower bound asymptote at the linear long wave speed of 4.7 cm/s.
Previous
studies
have
estimated
the
westward
propagation
speed
of
quasi geostrophic vortices by determining the speed of the center of mass
[McWilliams and Flierl, 1979] [Cushman-Roisin et al., 1990].
However, the center of
mass is determined by integration over the entire domain and doesn’t appear to correlate with the speed of the tracked eddies. Other approaches, such as those found in
[Korotaev, 1997] and [Nycander, 2001], may apply to the initialization period, but do
not appear to apply to the quasi-stable state. One possible deficiency with the approach
of [Nycander, 2001] is that a constant trapped fluid region was assumed.
However,
a shrinking trapped fluid region appears to be one of the defining features of these
quasi-stable eddies.
That the zonal speed of the eddies is less than the linear long wave speed is consistent
with previous experiments. The linear model considered in [Flierl, 1977] suggests that this
should be the case at least for linearized Gaussians. In the case of the nonlinear model considered here, this is also consistent with the notion of ‘wave drag’ caused by the excitation
of Rossby waves forcing a slower propagation speed [Korotaev and Fedotov, 1994].
27
FIGURE 2.9: Zonal speed of eddy versus the eddy amplitude. The black line is the
linear (inverse amplitude) best fit line, cx (A) = 4.2A−1 − 4.4 cm/s. Points are colored
with the eddy length scale (in km), suggesting a weak speed dependence on length scale.
The dashed grey line is the maximum zonal group velocity in linear theory. The results
were filtered to only include eddies in the quasi-stable state that were not found to be
interacting with Rossby waves.
28
FIGURE 2.10: Meridional speed of eddy versus the eddy amplitude. The black line is the
linear (inverse amplitude) best fit line, cx (A) = −3.7A−1 − 0.13 cm/s. The dashed grey
line is the maximum meridional group velocity in linear theory. Points are colored with
the eddy length (in km), suggesting at most weak speed dependence on length scale.
2.6.
Meridional Speed
The meridional speed of the eddy was found to be largely dependent on the amplitude of the eddy as shown in figure (2.10). Just as with the zonal propagation speed, there
appears to be weak dependence on the length scale of the eddy. In order to obtain reliable meridional speeds, some data points were discarded where it was found that zonally
propagating Rossby waves left over from the initialization and adjustment periods were
found to interact with the eddies and dramatically change the meridional deflection.
Both the meridional and zonal speed are dependent on an amplitude scale of about
29
1 cm (the deflection point in the speed dependency plots), but it’s not clear where this
scale arises. The eddy amplitude can be nondimensionalized with another length and the
equivalent depth, 79.9 cm, is a logical choice as a measure of nonlinearity. This can be
then scaled by some nondimensional parameter arising in the system such as the parameter
β −1 , where β −1 = 60 for the scales currently considered. This would mean,
cx (A) =
−β0 L2R
�
D
−1
−1
β ·A
�
(2.3)
or in terms of the height nonlinearity parameter �H ,
�
�
cx (A) = −β0 L2R (β −1 · �H )−1 − 1 .
(2.4)
This hypothesis is consistent with the results in figure (2.11) showing the eddy propagation
speeds at latitude 35◦ .
2.7.
Trapped Fluid Conservation Properties
Because the fluid velocities U in the eddy exceed its translation speed c, we know
that transforming coordinates into the co-moving frame will result in closed circulation
contours within the eddy. The largest of these contours defines the trapped fluid region
where no fluid can escape the region. However, we know this is not the full story. These
quasi-stable eddies have been shown to have slowly decaying amplitude and length scales.
The region of trapped fluid and the amplitude in the trapped fluid are both decreasing,
meaning that the volume of trapped fluid is actually decreasing with time.
Conservation of potential vorticity for a fluid parcel has contributions from three
terms: planetary vorticity, relative vorticity and vortex stretching.


d 


dt 
β0 y
����
planetary vorticity
+
g 2
∇ η
f
� 0 �� �
relative vorticity
−
f0
η
D
����
vortex stretching


=0

(2.5)
30
(a) Zonal propagation speed at latitude 35◦
(b) Meridional propagation speed at latitude 35◦
FIGURE 2.11: Propagation speed of eddy versus eddy amplitude. Same as figures (2.9)
and (2.10), but at latitude 35◦ . The zonal and meridional inverse amplitude best fits lines
are cx (A) = 0.93A−1 − 2.0 cm/s and cy (A) = −0.80A−1 − 0.046 cm/s, respectively.
31
Figure 2.12(a) shows the relative contributions from each of the three terms in
potential vorticity conservation for an 80km, 15 cm Gaussian initialized eddy. The values
are found by summing the terms over the entire trapped fluid region, and then dividing
by the region’s area. The trends for the planetary, vortex stretching and total potential
vorticity are the same for all other eddies reaching the quasi-stable state.
Even though the trapped fluid region is changing in time, it is clear how the planetary and vortex stretching terms should change for the average fluid parcel in the region.
Because the eddy has an southward component of propagation on a β-plane (y decreases),
the contribution from planetary vorticity decreases in time (βy decreases). The decay of
the eddy’s amplitude (η decreases) causes an increase in contribution from vortex stretching (−η increases). That the contribution from relative vorticity remains nearly constant
throughout the eddy life time means that the eddy is maintaining a ratio between the
negative relative vorticity from the eddy core and the positive relative vorticity in the
outer ring.
Energy can be divided into two terms, the kinetic energy,
g2
(η 2 +ηy2 ),
f02 x
and the poten-
tial energy, η 2 . Figure 2.12(b) shows decreasing contribution of both kinetic and potential
energy as the eddy evolves. The initial ratios of kinetic energy to potential depend on the
initial conditions. For example, the 80 km, 15 cm eddy shown in figure (2.12) is initially
dominated by potential energy, while the 40 km, 10 cm eddy is initially dominated by
kinetic energy. Despite the partition differences in the two eddies, both display similar
evolution characteristics, with the average energy per fluid parcel decreasing over time.
This trend is similar to that described by [Korotaev and Fedotov, 1994] [Korotaev, 1997]
and is due to the radiation of energy by Rossby waves.
2.7.1
Eddy Core
Let’s consider the eddy core first (recall that this is region whose outer boundary is
defined as the contour where ∇2 η = 0). Fluid is either entrained, exactly trapped, lost,
32
(a) Trapped Potential Vorticity
(b) Trapped Energy
(c) Core Potential Vorticity
(d) Core Energy
(e) Ring Potential Vorticity
(f) Ring Energy
FIGURE 2.12: Per particle averages of potential vorticity and energy within the entire
trapped fluid region, the eddy core and the eddy ring of an 80 km, 15 cm Gaussian
initialized eddy. The trends are the same for the other quasi-stable eddies.
33
or some combination of entrainment and loss.
Can a new parcel of fluid be entrained in the eddy core? Consider a typical fluid
parcel with no height perturbation and no relative vorticity. In order to join the eddy
core, the fluid parcel must increase its height, and therefore decrease its vortex stretching
contribution to the total potential vorticity. But, in addition, the particle must decrease
its relative vorticity from zero to become negative. Even with no change in relative
vorticity, this means that the particle must come from 35 kilometers north of the eddy for
every one centimeter increase in height. If we consider the eddy at the instant shown in
figure (2.8) where the zero contour is at roughly 4 cm, this would mean that for a parcel
of fluid to even reach the boundary of the core, it must be displaced from its original
rest location 140 km north of the eddy. But, owing the effect of the β-gyre, the eddy is
propagating southwestward and we must therefore conclude that a new fluid parcel will
not be entrained in the eddy core. Exactly this effect can be seen in the x-tracer panel in
figure (2.8) where on day 675 the eddy core still only contains fluid initially trapped when
the eddy was formed.
Do fluid parcels on the eddy core boundary remain on the boundary? For a particles
to remain on the ∇2 η = 0 contour, the fluid flow must be tangential to the contour, there
can be no normal flow. To conserve potential vorticity (2.5), these particles must therefore
obey
β0
dy
f0 dη
=
dt
D dt
(2.6)
throughout their life times. During eddy turnover times the condition is quite reasonable
to meet. As computed before, this only requires a particle to decrease its height by one
centimeter for every 35 kilometers of meridional displacement. In figure (2.8) we can
estimate the north-south extent of the zero contour to be 100 km, and so our condition
would require that the northern edge of the contour be roughly 3 cm higher than the
southern edge contour. Figure (2.8) suggests that this is indeed the approximate difference.
34
FIGURE 2.13: Histogram of the original x-position (a) and the original y-position (b) of
the fluid in the core of an 80 km 15 cm Gaussian initialized eddy on day 675. The eddy
extremum is now located at x = −2247 km and y = −330 km. Thus, the core contains
only fluid from its starting point over 2000 km away.
We can also use equation (2.6) with the parameters from this problem to compute a
condition relating the meridional propagation to the amplitude decay and we find that
9.0
s dy
dη
·
=
.
yr dt
dt
(2.7)
This suggests that that meridional speed shown in figure (2.7) of approximately 0.5 cm/s,
must be offset by a height decay rate of approximately 4.5 cm/yr if a parcel is to remain
on the zero contour. These are both estimates as the values computed for figure (2.7) are
from the eddy maximum, and our condition (2.7) is for the ∇2 η = 0 contour. The eddy
core cannot entrain fluid and if condition (2.7) is not exactly met, then it does not trap
the fluid that defines its boundary, so the eddy core must be shedding fluid (or, similarly,
the boundary of the core is shrinking).
We can validate our entrainment conclusion with the model by considering all of
the floats within the eddy core on day 675 and asking where they were on day zero. This
can be seen in figure (2.7.1) where a histogram of the initial x and y positions of the fluid
shows the fluid in the eddy core contains a subset of the original fluid trapped in the
core during the eddy initialization. Figure 2.14(a) shows theses original float locations as
red dots on top of the sea surface height on day 675. Figure 2.14(b) shows the results of
allowing a passive tracer to advect with the flow. The fluid was given a different color at
35
(a) Original float locations
(b) Passive x tracer
FIGURE 2.14: Location of fluid advected on day 675. (a) Sea surface height with the
trapped fluid contour (blue) and relative voriticty zero contour (black). Blue circles show
the day zero location of the floats in the eddy ring while red circles show the location of
floats in the eddy core. (b) A passive tracer initial given the value of its location on day
0.
each location in x on day zero.
Having established that no new fluid is entrained within the eddy core, we can more
easily interpret figure 2.12(c). Because the total potential vorticity is becoming more
negative on average, this implies that the eddy core is shedding fluid with lower potential
vorticity. Figure 2.15 shows the history of a float initially located in the eddy core which
remains in the eddy core for all 730 days. The parcel of fluid tracked by the float finds
that the total potential vorticity remained conserved, but the surface height adjusted to
compensate for the loss of planetary vorticity while the relative vorticity changed very
36
FIGURE 2.15: Contributions to the total potential vorticity for a float initial at x = 29
km and y = 25 km. The float remains inside the core of the eddy for all 730 days. On
day 730 the float was located at x = −2473 km and y = −365 km.
little. The potential vorticity for the fluid parcel is within 0.1% of its initial value after
730 days. This is excellent confirmation that the model is doing well because individual
contributions vary by well over 50%.
2.7.2
Eddy Ring
The eddy ring consists of fluid with positive relative vorticity, although with mag-
nitude much smaller than the eddy core. The same possibilities for trapped fluid exist
as with the core: fluid is either entrained, exactly trapped, lost, or some combination of
entrainment and loss.
At the very least, the eddy ring will be collecting the fluid shed from the shrinking
37
FIGURE 2.16: Histogram of the original x-position (a) and the original y-position (b) of
the fluid in the ring of an 80 km 15 cm Gaussian initialized eddy on day 675. The eddy
extremum is now located at x = −2247 km and y = −330 km. Thus, the core contains a
mixture of fluid from throughout its lifetime.
boundary of the eddy core. In addition, however, the eddy ring will also entrain new
surrounding fluid. An increase in height and therefore a compensated increase in relative
vorticity is exactly what a fluid parcel requires to join the eddy ring. This can be seen
from the original float locations of floats found in the ring on day 675, shown in figure
(2.16), where it is clear that the eddy ring has collected (and also therefore released) fluid
throughout its lifetime. Figure 2.14(a) shows theses original float locations as blue dots
on top of the sea surface height on day 675
The average PV composition within the eddy ring over time for the 80 km, 15 cm
eddy is shown in figure 2.12(e). Beta decreases, vortex stretching contribution increases;
again, both of these are obvious. The relative vorticity remains flat or mildly increases for
all eddies. The average PV trend always decreases. This makes some sense because the
ring is shedding fluid with higher PV and acquiring new fluid with lower PV, as we can
see from the tracer and figure 2.16.
Figure (2.17) shows the potential vorticity contribution of a float that began in
the eddy core, crossed to the eddy ring and was eventually ejected from the eddy. That
the potential vorticity for this float does not remain perfectly constant is an artifact
of the strong gradients of u and v that are poorly resolved with bilinear interpolation.
While most floats throughout the domain do conserve potential vorticity well, we find
38
FIGURE 2.17: Contributions to the total potential vorticity for a float initial at x = 3
km and y = 69 km. The float begins in the eddy core, cross to the ring, and is eventually
lost by the eddy. On day 730 the float was located at x = −1900 km and y = −293 km.
39
that floats crossing the relative vorticity zero contour often undergo rapid changes in
potential vorticity while crossing the boundary.
2.8.
Conclusions
We found that the individual properties of isolated eddies match the long term
coherence of eddies found in the satellite observations much more closely in the nonlinear
model than in the linear model. Further, the spectral properties of the basin scale eddy
seeding experiment are in excellent agreement with the nonlinear quasi-geostrophic model.
Taken together, we find this a convincing argument that the signals observed in the original
satellite observations [Chelton and Schlax, 1996] are not in fact linear Rossby waves, but
represent eddies obeying nonlinear dynamics, as argued by [Chelton et al., 2007].
In hopes of eventually comparing the individual characteristic of quasi-geostrophic
eddies to the observations, we conducted a study of the long-term evolution of isolated
eddies. Gaussian initialized eddies appear to have three distinct regimes in their evolution,
of which only two have previously been characterized. What was once believed to be a
quasi-stable state, turns out to be well described as an adjustment period and only at life
times of approximately 15(β0 LR )−1 does a true quasi-stable state emerge.
The quasi-stable state is characterized by zonal and meridional propagation rates
strongly dependent on the inverse amplitude of the eddy, with larger amplitudes tending
towards the long wave limit of linear Rossby waves. All propagations rates are slower
than this limit and this is understood to be an effect of the wave drag caused by the
excitation of Rossby waves. These results are in direct conflict with the enhanced eddy
resolving observations [Chelton et al., 2007] which found zonal propagation rates to meet,
and in some cases exceed, the long wave limit. Experiments with different forms of initial
conditions yield the same quasi-stable state and so we do not believe that this is simply
a limitation of not finding the correct initial conditions, but instead a limitation of quasi-
40
geostrophic theory or the neglect of variations in the background mean flow.
The quasi-geostrophic eddies were shown to transport a substantial amount of fluid
over long distances. At any point during an eddy’s lifetime, 100% of the fluid in the core
is from the initialization location, where the core is defined at the relative vorticity zero
contour. This is in contrast to the instantaneously defined trapped fluid region, determined
by transforming into coordinates co-moving with the eddy, which does not well describe the
boundary retaining fluid. In this sense the core of the eddy is a ‘perfect’ transporter of fluid
and carries the same parcels of fluid for thousands of kilometers during its slow decay. The
ring of opposite signed relative vorticity fluid around the eddy transports fluid in a very
different manner. The ring entrains and sheds fluid throughout its lifetime, moving some
parcels of fluid hundreds of kilometers and others thousands of kilometers. In light of our
evidence that the satellite observations are not Rossby waves, these transport properties
have significant implications. Linear Rossby waves cannot transport fluid and therefore
all energy transferred is in the form of kinetic and potential energy. The nonlinear eddies,
in contrast, are capable of transporting fluid and therefore carry energy in the form of
heat, in addition to kinetic and potential energy.
A number of issues of regarding the individual properties of quasi-geostrophic eddies still need to be resolved. Although an empirical relationship between the propagation
speed and the eddy amplitude was found, we have yet to develop a satisfactory analytical
theory for this relationship. Further, we believe that the ideas of radiative Rossby wave energy loss should be applicable outside the adjustment period explored in [Korotaev, 1997]
and [Nycander, 2001]. Analytical formulations for the relationships between eddy amplitude decay rates and propagation speed may be possible.
41
Acknowledgments
This research was supported by the National Science Foundation, Award 0621134,
and by the National Aeronautics and Space Administration, grant NNG05GN98G.
42
BIBLIOGRAPHY
Anderson and Gill, 1975. Anderson, D. and Gill, A. (1975). Spin-up of a stratified ocean,
with applications to upwelling. Deep-Sea Res, 22:583–596.
Chelton and Schlax, 1996. Chelton, D. and Schlax, M. (1996). Global observations of
oceanic rossby waves. Science, 272(5259):234.
Chelton et al., 2007. Chelton, D. B., Schlax, M. G., Samelson, R. M., and de Szoeke,
R. A. (2007). Global observations of large oceanic eddies. Geophys. Res. Lett., 34.
Cushman-Roisin et al., 1990. Cushman-Roisin, B., Tang, B., and Chassignet, E. (1990).
Westward motion of mesoscale eddies. Journal of Physical Oceanography, 20(5):758–
768.
Fiorino and Elsberry, 1989. Fiorino, M. and Elsberry, R. (1989). Some aspects of vortex
structure related to tropical cyclone motion. Journal of the Atmospheric Sciences,
46(7):975–990.
Flierl, 1977. Flierl, G. (1977). The application of linear quasigeostrophic dynamics to
gulf stream rings. Journal of Physical Oceanography, 7(3).
Flierl, 1984. Flierl, G. (1984). Rossby wave radiation from a strongly nonlinear warm
eddy. Journal of Physical Oceanography, 14(1):47–58.
Killworth and Blundell, 2005. Killworth, P. and Blundell, J. (2005). The dispersion relation for planetary waves in the presence of mean flow and topography. part ii: Twodimensional examples and global results. Journal of Physical Oceanography, 35:2110.
Killworth et al., 1997. Killworth, P., Chelton, D., and de Szoeke, R. (1997). The speed
of observed and theoretical long extratropical planetary waves. Journal of Physical
Oceanography, 27(9):1946–1966.
Korotaev, 1997. Korotaev, G. (1997). Radiating vortices in geophysical fluid dynamics.
Surveys in Geophysics, 18(6):567–618.
Korotaev and Fedotov, 1994. Korotaev, G. and Fedotov, A. (1994). Dynamics of an
isolated barotropic vortex on a beta-plane. J. Fluid Mech, 264:277–301.
McWilliams and Flierl, 1979. McWilliams, J. and Flierl, G. (1979). On the evolution of
isolated, nonlinear vortices. Journal of Physical Oceanography, 9(6):1155–1182.
Nycander, 2001. Nycander, J. (2001). Drift velocity of radiating quasigeostrophic vortices. Journal of Physical Oceanography, 31(8):2178–2185.
43
Pedlosky, 1987. Pedlosky, J. (1987). Geophysical Fluid Dynamics. Spring-Verlag, second
edition.
Reznik and Dewar, 1994. Reznik, G. and Dewar, W. (1994). An analytical theory of
distributed axisymmetric barotropic vortices on the -plane. Journal of Fluid Mechanics,
269:301–301.
Sutyrin and Flierl, 1994. Sutyrin, G. and Flierl, G. (1994). Intense vortex motion on the
beta plane: Development of the beta gyres. Journal of the Atmospheric Sciences, 51(5).
Tailleux and McWilliams, 2001. Tailleux, R. and McWilliams, J. (2001). The effect of
bottom pressure decoupling on the speed of extratropical, baroclinic rossby waves.
Journal of Physical Oceanography, 31(6):1461–1476.
Vallis, 2006. Vallis, G. (2006). Atmospheric and oceanic fluid dynamics: fundamentals
and large-scale circulation. Cambridge University Press.
44
3.
NEAR-GEOSTROPHIC APPROXIMATIONS OF THE
SPHERICAL SHALLOW-WATER EQUATIONS
Jeffrey J. Early and Roger M. Samelson
In preparation
45
3.1.
Introduction
Fluid motions in the ocean span a range of scales from millimeters to thousands of
kilometers and operate on time scales from seconds to thousands of years. Even by neglecting all thermodynamic effects and other complicating interactions like the atmosphere
or biological input, we still stand little chance of finding solutions to our mathematical
description without further simplification. A typical approach to simplification is to take
a sufficiently general set of equations like the Navier-Stokes equations and then use our
knowledge of some observed phenomenon to apply additional constraints. For example, by
taking J. Scott Russell’s observations of solitons, we can restrict the Navier Stokes equations to one-dimensional, irrotational, small-amplitude surface waves to find the Korteweg
de Vries equation [Ablowitz and Clarkson, 1991]. If our assumptions about the scales and
physics of the observations are sufficiently accurate, then the simplified equations should
admit solutions similar to the observations. Of course, a number of things can go wrong
during such a derivation, including starting with equations already overly simplified, using bad assumptions about the observations, finding an equation still too difficult to solve
or even finding an equation so simple that it no longer describes the observations. In
all of these cases, we must go back to earlier states of the derivation and reassess the
assumptions made, in hopes of deriving a better theory.
This general process is exactly the pattern followed for satellite altimetry observations of oceanic mesoscales features, which were first compared to linear Rossby
waves and shown to differ from theoretical predictions [Chelton and Schlax, 1996].
This discovery was followed by attempts to modify the classical linear theory (e.g.,
[Killworth et al., 1997]) only to have subsequent observations change the interpretation
of the observations from linear waves to non-linear eddies [Chelton et al., 2007]. Amongst
these observations it was found that these eddies are characterized by a strong degree
of non-linearity as measured by three nondimensional parameters involving eddy height,
46
length scale and typical fluid speed. These nonlinearity parameters are exactly the information we require to modify our assumptions and find a new theory to explain the
observations.
In the original interpretation of the satellite altimetry observations, a linearized form
of quasi-geostrophic (QG) theory that allows the superposition of linear Rossby waves
([Pedlosky, 1987]) is employed to explain the observations. However, as it now believed
that the observations show features that are more eddy-like than wave-like (Chelton, et.
al., in preparation), the fully non-linear quasi-geostrophic equations are a much more likely
candidate theory. On the other hand, given the strong fluid velocities, perhaps a second
order extension to QG theory is required? Alternatively, the strong height non-linearity
suggests QG needs to be extended to include larger amplitudes, as in the Flierl-Petviashvili
equation? Or, perhaps effects from neglecting the earth’s curvature are required, owing
to the long eddy length scales? Any one or all of these cases may be a possibility and an
appropriate theory should demonstrate its validity over the range of observed scales.
In the usual asymptotic derivation of QG, the non-dimensional parameters admitted
by the primitive equations are fixed relative to single parameter, which is typically the
Rossby number defined as � = U/f L where U and L are the characteristic fluid speed and
length scale, respectively, and f is the Coriolis parameter. The dependent variables are
then expanded as a perturbation series with respect to � and in the limit as � �→ 0 only
the leading order remains. Quasi-geostrophic theory stems from one particular choice of
relationships amongst the non-dimensional parameters. However, other choices are possible. In [Williams and Yamagata, 1984] a 5x5 matrix illustrating different choices between
non-dimensional parameters shows the corresponding equations ultimately resulting from
the perturbation expansions. In a similar vein, [Charney and Flierl, 1981] sketch a continuous two dimensional plot of nonlinear parameters showing different regions of behavior
requiring different theoretical descriptions.
In this paper we expand on these ideas by showing how a potential vorticity equa-
47
tion can be derived from the spherical shallow water equations without fixing the nondimensional parameters a priori. The resulting equation can then be reduced to other
theories, such as quasi-geostrophy and the Flierl-Petviashivilli equation, by assuming a
precise relationship between the parameters. However, by retaining the freedom in the
parameters we can quickly deduce which terms in the equation are necessary for an accurate description of the observed scales. Conversely, we are also able to determine at what
scales the various theories remain valid and at what scales their assumptions are violated.
To specifically address the non-linear effects of height, fluid speed, and length scale, the
spherical shallow water equations are a reasonable starting point, although this does neglect other potentially important effects. We show that the quasi-geostrophic potential
vorticity equation is not a valid equation for the observed scales of the eddies and that
the Flierl-Petviashvili equation and an extended FP equation are more suitable models.
This approach removes some of the burden of restating the assumptions when finding
a new theory to describe observations. However, these ideas are not necessarily widely
applicable and owe their success to the particular form of the primitive equations and
the wide range of scales over which geostrophy dominates other effects for mesoscale and
large-scale oceanography. Furthermore, we consider only the single layer shallow water
equations in spherical coordinates, for which a number of (conceivably invalid) assumptions
have already been made to reduce the complexity of the equations.
3.2.
Balance Dynamics
Consider one possible formulation of the primitive equations,
ut + (uux + vuy ) − f v = − ghx
(3.1)
vt + (uvx + vvy ) + f u = − ghy
(3.2)
ht + (uhx + vhy ) = − h (ux + vy ) .
(3.3)
48
Equations (3.1) and (3.2) represent the horizontal momentum equations and (3.3) represents the continuity equation where (u, v) are the x and y fluid speeds and h is the
fluid height. The fluid height h is also thought of as an equivalent depth scale fluid
height for an equivalent barotropic model. Without further simplification, these coupled
partial differential equations are difficult to solve. However, a key principle in oceanography is that mesoscale and large-scale features have pressure fields and velocity fields
largely in geostrophic balance. These features evolve on a ‘slow’ time scale, while the
effects of inertia-gravity waves operate on a ‘fast’ time scale and can often be ignored
[Lighthill, 1952, Ford et al., 2002]. The value in this principle is that it allows for a simplification of primitive equations (3.1)-(3.3). Rather than solving three partial differential
equations for all three variables simultaneously, the equation can be reduced to a set of
balance relations and balance dynamics [Warn et al., 1995] [Vallis, 1996]. For example,
separating the fluid height into a static depth and surface perturbation h = D + η, the
balance relations could be taken as
g
v = ηx
f
g
u = − ηy
f
(3.4)
while the balance dynamics might be the quasi-geostrophic potential vorticity equation
(QGPVE),
∇2 ηt − ηt + J(η, ∇2 η) = 0
(3.5)
where J(a, b) = ax by − ay bx . The evolution of the system is therefore entirely determined
by the evolution of the height field in equation (3.5) and the fluid velocity (u, v) follows
diagnostically from (3.4).
The usual approach to determining the balance relations and balance dynamics is
to expand the variables in a perturbation series as a function of some small parameter.
49
For example, take the nondimensionalized, linear f -plane primitive equations as
v =ηx + �ut
(3.6)
u = − ηy − �vt
(3.7)
�ηt + �(uηx + vηy ) = − (1 + �η) (ux + vy ) .
(3.8)
where � � 1. Given that � � 1, we can write the three field variables as a perturbation
series [Pedlosky, 1987],
� �
v =v 0 + �v 1 + �2 v 2 + O �3
� �
u =u0 + �u1 + �2 u2 + O �3
(3.9)
� �
η =η 0 + �η 1 + �2 η 2 + O �3 .
If equation (3.9) is inserted into the primitive equations (3.6)-(3.8) and terms of the same
order in � are collected, expressions for ui , v i and η i are found. To find a potential vorticity
equation, take the curl of the momentum equations at each order of � individually. The
resulting expression depends on η 0 and the ageostrophic component of the momentum
equations (u1 , v 1 ). The ageostrophic component can be eliminated in favor of the lower
order η 0 to find a potential vorticity equation dependent only on η0 . This is the balance
dynamics. The resulting fluid velocities are determined from the leading order values in
the expansion, u0 = −ηy0 and v0 = ηx0 and provide the balance relations.
Aside from the limitations of this approach due to fixing the non-dimensional parameters a priori that were highlighted in the introduction, [Warn et al., 1995] also showed
that expanding η often leads to secularities in higher order terms. To avoid these secularities, it is suggested to leave the variable η unexpanded. Interestingly, this turns out
to be a similar approach as the iterated models in [Allen, 1993]. The idea is to substitute
equation (3.6) into (3.7) and vice versa so that,
v =ηx − �ηyt + �2 vtt
(3.10)
u = − ηy − �ηxt − �2 utt .
(3.11)
50
Equations (3.10) and (3.11) are still exact, but now depend only on η through order �.
Repeated application of this iterative process allow the fluid speeds to be determined by
η though arbitrary orders of �. By truncating at order n of � one obtains an order n − 1
balance relation for u and v in terms of the height η. Of course, without balance dynamics
of similar order, the balance relations aren’t of much use.
The approach taken here is to find an equation for the balance dynamics from the
potential vorticity conservation law of the spherical shallow water equations. Using the
same iterative approach as for the momentum equation, the potential vorticity law is
similarly used to determine the balance dynamics to arbitrary order. A key feature to
this approach is that a single, general equation can be derived without fixing the nondimensional parameters relative to one another, unlike perturbation expansions.
3.3.
Field Equations
The shallow water equations in spherical coordinates [Gill, 1982] include the momentum equations,
�
∂u
u ∂u
v ∂u � u
g ∂h
+
+
=
tan θ + f v −
∂t
R cos θ ∂φ R ∂θ
R
R cos θ ∂φ
�u
�
∂v
u ∂v
v ∂v
g ∂h
+
+
=−
tan θ + f u −
∂t
R cos θ ∂φ R ∂θ
R
R ∂θ
(3.12)
(3.13)
and with the addition of continuity,
�
�
∂h
u ∂h
v ∂h
h
∂u
∂
+
+
=−
+
(v cos θ) .
∂t
R cos θ ∂φ R ∂θ
R cos θ ∂φ ∂θ
(3.14)
Equations (3.12)-(3.14) have a number of conserved quantities including angular
momentum, energy and potential vorticity [Vallis, 2006]. The conservation of energy can
be written as
∂E
+∇·S=0
∂t
�
�
�
�
where E = 12 gh + u2 + v 2 and S = 12 2gh + u2 + v 2 .
(3.15)
51
If we cross-differentiate (3.12) and (3.13) and use (3.14), then we can show that
potential vorticity is conserved. Namely that,
� �
�
���
d 1
1
∂v
∂
f+
−
(u cos θ)
= 0,
dt h
R cos θ ∂φ ∂θ
where
d
dt
=
∂
∂t
+
u
∂
R cos θ ∂φ
+
v ∂
R ∂θ ,
or equivalently,
� �
��
d 1
u
1 ∂v
1 ∂u
f + tan θ +
−
= 0.
dt h
R
R cos θ ∂φ R ∂θ
3.4.
(3.16)
(3.17)
Coordinates
Before proceeding to derive an equation for the balance dynamics, it is worth considering the coordinates that we’re working in and the associated gradient and laplacian
operators. One of the primary objectives of this study is to assess the validity of approximating the equations of motion to a cartesian plane, so we need to carefully consider our
definition of coordinates.
If we define the (x� , y � , z � ) coordinate system with origin at the center of the earth in
an inertial frame, then in terms of the coordinates (φ, θ, r) (denoted here as geographer’s
coordinates)
x� =(R + r) cos θ cos(φ + ωt),
y � =(R + r) cos θ sin(φ + ωt),
(3.18)
�
z =(R + r) sin θ,
t =t.
Restricting ourselves to motion on the geoid, we can ignore changes in r and neglect the
small metric terms [Gill, 1982]. The gradient operator in the geographer’s coordinates is
∇ζ =
�
1 ∂ζ
1 ∂ζ
,
R ∂θ R cos θ ∂φ
�
(3.19)
52
and the laplacian is
1
∂
∇ ζ= 2
R cos θ ∂θ
2
�
�
∂ζ
1
∂2ζ
cos θ + 2
.
∂θ
R cos2 θ ∂φ2
(3.20)
We have to be careful when trying to write the above operators in typical oceanographer’s coordinates (a local Cartesian plane with (x, y, z)) because in both cases there is
a curvature term that is often ignored. In order to create coordinates resembling a local
Cartesian plane, oceanographers typically define the following horizontal coordinates,
x =Rφ cos θ0 ,
(3.21)
y =R(θ − θ0 ),
where θ0 is the latitude of the origin of the Cartesian plane. We need to use these
coordinates for derivatives, so by the chain rule we find that,
∂
∂
=R cos θ0
∂φ
∂x
∂
∂
=R .
∂θ
∂y
(3.22)
(3.23)
Writing the Laplacian in these coordinates gives us,
∇2 ζ =
cos2 θ0 ∂ 2 ζ
∂2ζ
tan θ ∂ζ
+
−
.
2
2
2
cos θ ∂x
∂y
R ∂y
(3.24)
The coefficient of the x derivative terms stays close to 1 until motions deviate too
far north or south of the cartesian origin. Similarly, the additional tan θ dependent
curvature term can remain fairly small if the distances being considered don’t get too large.
In this study, we will write the Laplacian by keeping the additional curvature term
separate. For the remainder of this paper the following notation will be used,
cos2 θ0 ∂ 2 ζ
∂2ζ
tan θ ∂ζ
+
−
2
2
2
cos θ ∂x
∂y
R ∂y
tan
θ
∂ζ
= ∇2H ζ −
R ∂y
∇2 ζ =
(3.25)
(3.26)
53
and therefore we’ve implicitly defined
∇2H ζ =
3.5.
cos2 θ0 ∂ 2 ζ
∂2ζ
+
.
cos2 θ ∂x2 ∂y 2
(3.27)
Nondimensionalization
The objective of this section is to nondimensionalize the oceanographer’s coordinates
so that we can write the shallow water equations in a form suitable for the iterative scheme.
3.5.1
Coordinates
Simply by inspection we can observe all of the dimensional parameters in the shallow
water equations (3.12)-(3.14). We have t, x, and y for the independent variables, u, v,
and h for the dependent variables and additionally R, g and 2ω appear as constants.
Nondimensionalizing requires that we separate the dimensional part of the variables. We
do this in the following fashion,
t = T t̄,
x = Lx̄,
y = Lȳ,
u = U ū,
v = U v̄,
h = D + N0 η.
(3.28)
The non-dimensional oceanographer’s coordinates are now,
R
x̄ = φ cos θ0 ,
L
R
ȳ = (θ − θ0 ),
L
(3.29)
(3.30)
and so by the chain rule we find that,
∂
R
∂
= cos θ0
∂φ L
∂ x̄
∂
R ∂
=
.
∂θ L ∂ ȳ
(3.31)
(3.32)
Important to note here is that we made the assumption that x and y scale similarly
(they both use L as their dimensional parameter) and that u and v do the same (using
54
U for dimensionality). The result of doing this means that we essentially reduced the
number of dimensional parameters in the problem from 9 down to 7. However, we added
an additional dimension by separating h into a static depth D and a relatively small
variation N0 . We are therefore left with 8 dimensional parameters in the problem. To be
explicit, the dimensional parameters are now T , L, U , D, N0 , R, g and 2ω.
The Buckingham-Pi theorem [Bluman and Anco, 2002] states that because we have
only two fundamental physical units, length and time, and eight dimensional parameters
there are six dimensionless parameters we can construct. However, as soon as we approximate the equations on a cartesian plane, the latitude, θ0 , also counts as a non-dimensional
parameter. There are an infinite number of ways of constructing these parameters, but
we will use the ‘natural’ scaling that arises in our particular problem.
We will use one unusual convention for writing a non-dimensional form of the
trigonometric terms. We write sin θ =
Because θ = θ0 +
L
R ȳ
sin θ
sin θ0
to indicate a normalized form of trig functions.
this means that sin θ ≈ 1 if
L
R
� 1, which can often be the case.
This becomes a convenient way to keep track of which terms are affected by the spherical
geometry of the problem and at what order in our expansion the spherical effects start to
become significant.
3.5.2
Field Equations
To write equations (3.12)-(3.14) in a non-dimensionalized form, use the horizontal
derivatives (3.31)-(3.31), substitute in the non-dimensional scalings (3.28) and then divide
the equations by U · 2 · ω. Traditionally one would also divide by a factor of sin θ0 , but that
ultimately makes the analysis much more difficult. With mild rearranging, the shallow
water equations become,
�
�
G
∂η
1 ∂ ū
1
ū ∂ ū
∂ ū
ū v̄
+ �T ·
+ �R ·
+ v̄
− �β ·
,
∂
x̄
sin
θ
∂
t̄
sin
θ
∂
x̄
∂
ȳ
cos
θ
sin θcos θ
cos θ
�
�
G ∂η
1 ∂v̄
1
ū ∂v̄
∂v̄
ū2
ū = −
− �T ·
− �R ·
+ v̄
− �β ·
.
sin θ ∂ ȳ
sin θ ∂ t̄
sin θ cos θ ∂ x̄
∂ ȳ
cos θ
v̄ =
(3.33)
(3.34)
55
and
�H �T
ū ∂η
∂η
1 + �H η
ηt̄ + �H
+ �H v̄
=−
�R
∂ ȳ
cos θ ∂ x̄
cos θ
where
�
�
�
∂ ū
∂ �
+
v̄cos θ ,
∂ x̄ ∂ ȳ
(3.35)
1
U
U
gN0
N0
, �R =
, �β =
G=
, �H =
(3.36)
2ωT
2ωL
2ωR
2ωU L
D
The first two parameters, �T and �R are the commonly used Rossby numbers without
�T =
the extra factor of sin θ0 . The third parameter, �β is related to the usual parameter
2
β0 by β0 = �β (2ω) Ūcos θ0 . The G parameter is a geostrophic pressure parameter. The
continuity equation (3.35) includes �H which measures the surface perturbation relative
to the equivalent depth. The sixth parameter (required by the Buckingham-Pi theorem)
can be written as the aspect ratio of the ocean, namely the depth scale over the length
scale
D
L.
It does not naturally appear in the equations that we started with, but was
implicitly already used to get the hydrostatic balance. Any of the other standard nondimensional parameters used for these equations can easily be recovered in terms of (3.36).
For example, the Burger parameter F =
�H
G�R
L2
L2R
can be constructed from (3.36) with F =
sin2 θ0 .
Two of the coefficients in equations (3.33)-(3.35) can be varied to match a third
by choosing the units of time and length. In a typical derivation of quasi-geostrophy
the geostrophic parameter G is taken to be 1 and �T is set to equal �R [Pedlosky, 1987].
However, it’s important to note that G could just as well be set equal to �β , and �R set
to 105 or any other value. This is the central idea to nondimensionalization and is done
without loss of generality, provided we only use the two degrees of freedom in dimensions.
Of course, the values of these parameters might as well be chosen to reflect our assumptions
about the resulting solution. So in the case of near geostrophic flows, while setting G = 1
may represent the idea of purely geostrophic motion such that u = − fg0 ηx , the solution is
is not restricted to this assumption owing to the freedom in dimensionality. Using what
we think will be typical values for �T and �R , we may decide that �T = �R , or instead
56
that �1.5
T = �R . Doing this is still without loss of generality, but now provides a concrete
way of ordering the relative magnitude of the terms based on our assumptions of the
resulting solutions. This is how one typically proceeds with a perturbation analysis, but
by continuing the derivation without fixing the units, it becomes easier at a later stage to
estimate the relative sizes of the terms. Specifically, we will also use the freedom in G to
later estimate the range at which departures from pure geostrophy (G = 1) are still valid.
It’s also important to note that the ratio of length scales
This means that we can write
3.5.3
∂
∂ ȳ
=
L
R
can be written as
�β
�R .
�β ∂
�R ∂θ .
Potential Vorticity
�
For notational convenience take ωa = f +
u
R
tan θ +
1
∂v
R cos θ ∂φ
−
can rewrite the potential vorticity conservation law (3.17),
d � ωa �
=0
dt h
1 dωa ωa dh
− 2
=0
h dt
h dt
dωa
d
− ωa ln h = 0.
dt
dt
1 ∂u
R ∂θ
�
so that we
(3.37)
It is this last form of the equation that we will ultimately find most convenient.
Non-dimensionalizing we find
�
ωa = 2ω · sin θ + �R
�
1 ∂v̄
∂ ū
−
cos θ ∂ x̄ ∂ ȳ
�
+ �β tan θū
h = D [1 + �H η]
�
�
��
d
∂
ū ∂
∂
= 2ω · �T
+ �R
+ v̄
.
dt
∂ t̄
∂ ȳ
cos θ ∂ x̄
�
(3.38)
(3.39)
(3.40)
Now taking equations (3.38)-(3.40) and substituting them into equation (3.37) we
57
see that the non-dimensionalized form of potential vorticity conservation is
�
�
�� �
�
�
�
∂
ū ∂
∂
1 ∂v̄
∂ ū
0 = �T
+ �R
+ v̄
sin θ + �R
−
+ �β tan θū
∂ t̄
∂ ȳ
cos θ ∂ x̄
cos θ ∂ x̄ ∂ ȳ
�
�
�
��
�
��
1 ∂v̄
∂ ū
∂
ū ∂
∂
− sin θ + �R
−
+ �β tan θū �T
+ �R
+ v̄
ln [1 + �H η] .
∂ t̄
∂ ȳ
cos θ ∂ x̄ ∂ ȳ
cos θ ∂ x̄
(3.41)
3.6.
Near-Geostrophic Potential Vorticity Equation
The guiding assumption is that the flow is nearly in geostrophic balance, so that
�T , �R , �β � G. We will use the notation O(�n ) to indicate O(�iT �jR �kβ ) where n = i + j + k.
Although �H is generally small for the scales that we are interested in, we do not need to
assume that �H � G in order to proceed and therefore will not include it as part of the
O(�n ) notation.
By substituting equations (3.33) and (3.34) into (3.41) twice, we will derive a single
scalar equation in η through O(�2 ), with combinations of η, ū and v̄ at order O(�3 ). This
equation will still be exact and contain no errors, just as in equations (3.10) and (3.11).
The tedious computation was completed by hand and checked using a symbolic
manipulator.
58
3.6.1
The Complete Equation
0 =�β G·
+ �T �R G·
− �T �β G·
+ �2R G2 ·
+ � R � β G2 ·
− �R �β G·
+ �2β G2 ·
+ �2β G·
− �H �T ·
+ �H �T �R G·
+ �H �T �β G·
− �H �2R G2 ·
− � H � R � β G2 ·
cos θ ηx̄
sin θ cos θ
1
∇2 ηt̄
sin θ
1 + cos2 θ
ηt̄ȳ
sin2 θ cos θ �
�
ηȳ ηx̄x̄x̄
ηx̄ ηx̄x̄ȳ
1
1
−
+ ηx̄ ηȳȳȳ − ηȳ ηx̄ȳȳ +
sin2 θ cos θ
cos2 θ
cos2 θ
�
�
2
ηx̄ ηȳȳ
cos θ
1 + cos θ ηȳ ηx̄ȳ
2 − 3 cos2 θ ηx̄ ηx̄x̄
−3
+
+
cos2 θ cos θ
cos2 θ cos3 θ
sin3 θ
cos θ
ηȳȳ
cos θ
cos4 θ + 3 cos2 θ − 1 ηx̄ ηȳ
sin4 θ cos2 θ
cos θ
1
ηȳ
sin θ
(3.42)
sin θ ηt̄ (1 + �H η)−1
�
� 2
�
�
1
1∂
ηx̄
2
2
+ ηȳ − ηt̄ · ∇ η (1 + �H η)−1
sin θ 2 ∂ t̄ cos2 θ
ηȳ ηt̄
(1 + �H η)−1
2
sin θ cos θ �
�
�
�
��
1
1
ηx̄x̄
ηx̄2
2
η
η
−
η
+
η
η
−
(1 + �H η)−1
x̄ ȳ
ȳ ȳ
x̄ȳ
ȳ
2
2
2
sin θ cos θ
cos θ
cos θ
�
�
cos θ ηx̄ cos 2θ ηx̄2
+ ηȳ2 (1 + �H η)−1 + O(�3 )
sin3 θ cos θ cos2 θ cos2 θ
Equation (3.42) is still exact. We made no approximations from the original potential vorticity equation (3.17) and have simply hidden the remaining finite (but too
numerous to print!) number of terms in the O(�3 ) rather than write them out explicitly.
Unless we perform an additional iteration of inserting equations (3.33) and (3.34) into
(3.42), there are no O(�4 ) terms. In order to find an equation for the balance dynamics
solely in terms of η, we will need to drop at a minimum the O(�3 ) terms because they
contain explicit reference to ū and v̄. For this reason it, it will be necessary to explicitly
59
consider a few of the O(�3 ) terms.
3.6.2
Bounding the Cubic Terms
To determine the magnitude of the O(�3 ) terms, first consider the relative sizes of
�T , �R , and �β . The ratio of parameters �β and �R is the factor
L
R.
Even an eddy with
length scale 600 km will be an order of magnitude smaller than the radius of the earth and
therefore we can consistently regard �β � �R . On the other hand, the condition that �T <
�R requires that L/T < U , a condition not supported by observation [Chelton et al., 2007]
if we take L/T to be the baroclinic wave speed. Closer to the equator �T tends to dominate
while at mid-latitudes �R tends to dominate. This means that the largest O(�3 ) term that
we can construct would be either an �3R or �3T term. However, equation (3.42) contains no
�3T nor �2T �R term and so depending on the latitude, either the �3R or �T �2R term will be the
largest O(�3 ) term.
The O(�3R ) terms look like
�3R G
sin2 θ
�
2ηx̄ ūx̄ v̄x̄t̄
cos3 θ
�
+ ... with the same parameters of
G�3R
sin2 θ
in front of the other nine terms not shown. This term arises in the advection of relative
vorticity. The �T �2R arises from the local change in relative vorticity and is exactly,
�
�T �2R G2 1 ∂ �
2
ηȳȳ ηx̄x̄ + ηx̄ȳ
3
2
sin θ cos θ ∂ t̄
(3.43)
With these we can can place the bound on our error as
3
O(� ) = max
3.6.3
�
�3R G
1
1
2 2
·
4 , �T � R G ·
sin θ
sin3 θ
�
.
(3.44)
Bounding the Cartesian Approximation
Although we have not yet made any assumptions that restrict equation (3.42)
to a Cartesian plane, it is often desirable to make this assumption because the equations become much simpler to solve. A trigonometric factor such as sin θ is defined as
�
�
�
sin θ0 + �Rβ ȳ , using equations (3.30) and (3.36), and a Taylor expansion about θ = θ0
�
approximates this as sin θ ≈ sin θ0 + cos θ0 �Rβ ȳ. Therefore, having already established that
60
�β � �R for the scales that we’re interested in, the leading order terms to the Cartesian
approximation of equation (3.42) are found by replacing sin θ = sin θ0 , cos θ = cos θ0 and
cos θ = 1.
The next to leading order terms in the Cartesian approximation will contain factors
of
�β
�R ȳ
multiplying each term in equation (3.42). If we want to avoid explicit reference to
ȳ when we approximate the equation, we will need to estimate the size of these next to
leading order terms. However, rather than do this for all the terms, it will later be shown
that the �β , �2R and �H �T terms generally dominated the others for the scales that we’re
interested in. The Cartesian approximation for these three terms is bounded by,
�
�
2
�2β
2
2 3 − 5 cos θ0 �H �T �β
O(spherical) = max
G · cot θ0 , �R �β G ·
,
· cos θ0 .
�R
�R
sin3 θ0 cos θ0
3.6.4
(3.45)
Magnitude of Terms
We can now estimate the magnitude of all the terms shown in equation (3.42)
using their coefficients. The scales found in the observations [Chelton et al., 2007] can be
inserted into these coefficients directly. For example, take a near geostrophic (G = 1)
eddy at latitude 45, with height N0 of 10 cm, length scale L of 100 kilometers and time
scale T of 100 days so that L/T is close to the long wave phase speed of a first mode
baroclinic Rossby wave with an equivalent depth D of 80 cm. These choices require that
the fluid velocity U be at 6.7 cm/s in order to satisfy G = 1. The coefficient �β G cot θ0
is approximately 7.2 · 10−5 , whereas �T �R G csc θ0 is an order of magnitude smaller at
5.6 · 10−6 . These are both bigger than the largest O(�3 ) term at 3.9 · 10−7 and the largest
O(spherical) term at 1.1 · 10−6 . Doing this calculation for the all of the remaining terms
in equation (3.42) allows us to estimate which terms are most likely to be important in
describing such an eddy.
For fixed latitude, eddy height, geostrophic parameter and ratio L/T , figure (3.1)
shows the magnitude of the terms in equation (3.42) versus the length scale. Of the 18 O(�)
61
−2.5
εH εT
ε2R
−3
εβ
εT εR
ε2H εT
Magnitude, Log10
−3.5
εH εR εR
εR εβ
−4
−4.5
−5
−5.5
3
O( ε )
−6
O( spherical )
−6.5
50
100
150
200
250
Length Scale (km)
FIGURE 3.1: The magnitude of the terms in equation (3.42) are shown versus the length
scale. The coefficients in the legend can be matched with the coefficients in equation
(3.42). The surface height is fixed at 10 cm, the latitude at 45 and the fluid velocity varies
assuming G = 1. The ratio of the length scale to the time scale is fixed at 1.46 cm/s,
slightly faster than the long wave Rossby phase speed at that latitude.
62
and O(�2 ) terms in equation (3.42), only seven appear above the O(�3 ) and O(spherical)
terms for these scales. The O(�3 ) region shows that at small length scales (and time scales)
the next term in the iteration is needed to keep the equation valid. The O(spherical) terms
indicate that as the length scales increase, the cartesian plane approximation breaks down.
This particular limiting bound arises from the
�H �T �β
�R
· cos θ0 in equation (3.45).
The estimates of the magnitude of terms in equation (3.42) and figures like (3.1)
can be used to help construct new theories.
3.7.
Equations of Balance Dynamics
We can fix the four parameters �H , �T , �R and �β relative to each other using
integral powers of �. The terms in equation (3.42) can then be easily ordered in decreasing
magnitude. Different choices in the relative magnitudes of the four parameters reflect
different choices in the scales being considered and will result in a different ordering of
terms. The leading order terms in the resulting equation will be identical to using the
asymptotic approach and expanding ū, v̄ and η as a perturbation series in �. All higher
order terms would differ, but using the slaving approach by leaving η unexpanded results
in the same equations as our iterative approach [Warn et al., 1995].
It’s important to note that choosing a relationship amongst parameters is not a
mathematical requirement nor physical requirement, but simply an abstract tool used in
various approaches for finding an equation. Once an equation is found we do not require
that all the terms have exactly the same magnitude, just as we do not demand that
the scales actually obey the precise relationship of the parameters. The f -plane quasigeostrophic potential vorticity equation (3.5) when derived using the asymptotic approach
[Pedlosky, 1987] might lead one to believe that all terms will actually be of order �. The
reality is that a solution to a given equation will have different magnitudes for each term at
63
a given point, arguably more like the magnitudes in figure (3.1). The only mathematical
requirement is that these terms sum to zero at all points. The reason for fixing the four
parameters relative to one another is therefore to provide a concrete ordering amongst
terms and in the case of the leading order terms, indicate an equation that can be derived
using the asymptotic approach.
3.7.1
Quasi-Geostrophic Potential Vorticity, f -plane
The f -plane quasi-geostrophic potential vorticity equation [Pedlosky, 1987] can be
obtained by choosing
�H = �,
�T = �,
�R = �,
�β = � 3 .
(3.46)
� �
The leading order terms at O �2 become the equation of balance dynamics which is just
the usual quasi-geostrophic potential vorticity equation on the f -plane,
�
�
� 2 G2
�T �R G
· ∇2 ηt̄ − �H �T sin θ0 · ηt̄ + R2 · Jˆ η, ∇2 η = 0.
sin θ0
sin θ0
(3.47)
Because all three terms in equation (3.47) are order �2 , this equation is identical to what
would be found using the asymptotic approach as is usually done [Pedlosky, 1987]. With
two degrees of freedom in the scales (time and length), we can adjust two of the coefficients
to match the third so that,
�
�
∇2 ηt̄ − ηt̄ + Jˆ η, ∇2 η = 0
(3.48)
where time is now in units of f0−1 and length in units of the Rossby length scale LR =
√
gD
f0 .
Applying the scales (3.46) to the shallow water equations (3.33)-(3.35) we find,
�
�
∂η
∂ ū
∂ ū
∂ ū
sin θ0 v̄ = G
+ �T
+ �R ū
+ v̄
+ O(�2 ),
∂ x̄
∂ t̄
∂ x̄
∂ ȳ
�
�
∂η
∂v̄
∂v̄
∂v̄
sin θ0 ū = −G
− �T
− �R ū
+ v̄
+ O(�2 )
∂ ȳ
∂ t̄
∂ x̄
∂ ȳ
�
�
�
�
�H �T
∂η
∂η
∂ ū ∂v̄
0=
η + �H ū
+ v̄
+ (1 + �H η)
+
+ O(�2 ).
�R t̄
∂ x̄
∂ ȳ
∂ x̄ ∂ ȳ
(3.49)
(3.50)
(3.51)
64
Equations (3.49)-(3.51) are appropriate momentum and continuity equations for these
scales. The balance relations could be constructed from (3.49) and (3.50) by taking the
first iteration so that,
�
�
1 ∂η
�T G ∂ 2 η
�R G ∂η ∂ 2 η
∂η ∂ 2 η
v̄ = G
−
+
−
,
sin θ0 ∂ x̄ sin θ0 ∂ t̄∂ ȳ sin θ0 ∂ ȳ ∂ x̄∂ ȳ ∂ x̄ ∂ ȳ∂ ȳ
�
�
1 ∂η
�T G ∂ 2 η
�R G ∂η ∂ 2 η
∂η ∂ 2 η
ū = −G
−
+
−
.
sin θ0 ∂ ȳ sin θ0 ∂ t̄∂ x̄ sin θ0 ∂ ȳ ∂ x̄∂ x̄ ∂ x̄ ∂ x̄∂ ȳ
(3.52)
(3.53)
However, equation (3.47) is an equation for the height field η only at the leading order
and so there may be variations in η of O(�). Equations (3.52) and (3.53) are really only
valid to leading order and so we must take
1 ∂η
,
sin θ0 ∂ x̄
1 ∂η
ū = −G
,
sin θ0 ∂ ȳ
v̄ = G
(3.54)
(3.55)
as our balance relations.
3.7.2
Quasi-Geostrophic Potential Vorticity, β-plane
The β-plane quasi-geostrophic potential vorticity equation [Pedlosky, 1987] can be
obtained by choosing
�H = �,
�T = �,
�R = �,
�β = � 2 .
� �
All terms are again leading order terms at O �2 so that equation (3.42) becomes
�
�
� 2 G2
�T �R G
· ∇2 ηt̄ − �H �T sin θ0 · ηt̄ + �β G cot θ0 · ηx̄ + R2 · Jˆ η, ∇2 η = 0
sin θ0
sin θ0
(3.56)
(3.57)
With two fundamental units and four coefficients in equation (3.57), we cannot completely
eliminate all the coefficients. Using the same units as with the f -plane, this places a nondimensional coefficient typically denoted as β in front ηx̄ term. However, the time scale
of f0−1 is unnaturally short for mesoscale features. If one takes the time scale as β0−1 L−1
R
then equation (3.57) can be scaled to
�
�
∇2 ηt̄ − ηt̄ + ηx̄ + β −1 · Jˆ η, ∇2 η = 0
(3.58)
65
where β −1 =
√
gD
.
β0 L2R
The non-dimensionalized momentum equations now contain variations
of the Coriolis parameter f¯ = sin θ0 +
�β
�R
cos θ0 ȳ and a metric term γ = 1 −
�β
�R
tanθ0 ȳ,
�
�
G
∂η
∂
ū
∂
ū
∂
ū
f¯v̄ =
+ �T
+ �R ū
+ v̄
+ O(�2 ),
γ ∂ x̄
∂ t̄
∂ x̄
∂ ȳ
�
�
∂η
∂v̄
∂v̄
∂v̄
¯
f ū = −G
− �T
− �R ū
+ v̄
+ O(�2 ),
∂ ȳ
∂ t̄
∂ x̄
∂ ȳ
�
�
�
�
�H �T
ū ∂η
∂η
(1 + �H η) ∂ ū ∂ (v̄γ)
η + �H
+ v̄
=−
+
+ O(�2 ).
�R t̄
γ ∂ x̄
∂ ȳ
γ
∂ x̄
∂ ȳ
(3.59)
(3.60)
(3.61)
The metric term γ arises as the Taylor expansion of cosθ and appears to be occasionally
neglected [Vallis, 2006], perhaps because it does not appear in the potential vorticity
equation (3.57) (see [Pedlosky, 1987] and [Ripa, 1997] for details). The balance relations
remain the same as the f -plane approximation because equation (3.57) again only contains
leading order terms,
1 ∂η
,
sin θ0 ∂ x̄
1 ∂η
ū = −G
.
sin θ0 ∂ ȳ
v̄ = G
3.7.3
(3.62)
(3.63)
Flierl-Petviashvili Equation
The FP equation arises in several different contexts including the Great Red
Spot on Jupiter [Petviashvili, 1980], as well as modifications to quasi-geostrophy
with an exterior mean shear flow [Flierl, 1979], finite amplitude height changes
[Anderson and Killworth, 1979] [Charney and Flierl, 1981] and thermobaricity of the
equation of state [de Szoeke, 2004]. In our case it can be found with the choice of scales
�H = �,
�T = � 2 ,
�R = � 2 ,
�β = � 3 .
(3.64)
Unlike the two quasi-geostrophic potential vorticity equations (3.47) and (3.57), the FP
equation requires both the leading order and next to leading order terms from equation
(3.42). This means that it is not possible to find this equation using the asymptotic
66
� 3�
� 4�
approach and letting � �→ 0. Retaining the O � terms and the O � terms we have,
�
�
� 2 G2
�T �R G 2
· ∇ ηt̄ − �H �T sin θ0 · ηt̄ (1 + �H η)−1 + �β G cot θ0 · ηx̄ + R2 · Jˆ η, ∇2 η = 0. (3.65)
sin θ0
sin θ0
Using the same time and length scale as with equation (3.57), η can be rescaled by a
factor of �−1
H , with the additional requirement that η � 1. In this case the FP equation
becomes,
�
�
ηt̄ − ηt̄ (1 − η) + ηx̄ + β −1 · Jˆ η, ∇2 η = 0,
(3.66)
where we’ve used the fact the η is small to write (1 + η)−1 as (1 − η). The ηt̄ and ηx̄ terms
� �
� �
are O �3 , while the remaining terms are O �4 . Therefore we can multiply the equation
by (1 + η) also consistently write,
�
�
ηt̄ − ηt̄ + ηx̄ (1 + η) + β −1 · Jˆ η, ∇2 η = 0.
(3.67)
The balance relations will now also contain the next to leading order terms of the momentum equations (with factors of
�β
�R
= �),
1 ∂η
v̄ = G ¯
,
f γ ∂ x̄
1 ∂η
ū = −G ¯ .
f ∂ ȳ
3.7.4
(3.68)
(3.69)
FP + J�� Equation
Finally consider the choice of scales,
�H = �,
�T = � 3 ,
�R = � 2 ,
�β = � 4 .
(3.70)
Using the same choice of time and length scale as the FP equation, we have
where J ��
�
�
ηt̄ − ηt̄ (1 − η) + ηx̄ + β −1 · Jˆ η, ∇2 η − β −1 · J �� = 0,
=
1
sin2 θ
�
(3.71)
�
��
ηx̄ ηȳ (ηx̄x̄ − ηȳȳ ) + ηx̄ȳ ηȳ2 − ηx̄2
following the convention of
[Williams and Yamagata, 1984]. This term arises as the advection of η, an ageostrophic
67
effect. To see the origin of this term, take the expression for the advection of η,
�R
�
ū ∂η
∂η
+ v̄
cos θ ∂ x̄
∂ ȳ
�
and substitute in equations (3.33) and (3.34), ignoring all but the �R components. The
resulting equation will still contain ū and v̄ that can then be replaced with their geostrophic
approximations.
The balance relations do not contain the geometric effects found in the FP balance
relations (3.68) and (3.69) because the factor
�β
�R
is O(�2 ). The balance relations are thus,
1 ∂η
,
sin θ0 ∂ x̄
1 ∂η
ū = −G
.
sin θ0 ∂ ȳ
v̄ = G
3.8.
(3.72)
(3.73)
Valid Regimes
Using the information from figure (3.1) we can estimate what terms a theory would
need in describing these eddies and evaluate the a priori appropriateness of an existing
theory. For example, the five terms with the largest magnitude at 150 km in figure (3.1)
are exactly the same terms found in the FP equation (3.66). From about 60 km to 140 km,
the terms from equation (3.71) dominate. Nowhere do the terms from quasi-geostrophy
(3.58) dominate the others. This would suggested that, at least for latitude 45, an extended
theory like equation (3.71) is necessary for capturing the appropriate physics of the eddies.
Figure (3.2) shows the regions where different equations are valid, in the sense that
the terms of those equations dominate all other neglected terms, as a function of length
scale and latitude. Regions of blue indicate areas where none of the theories were valid
without adding additional, neglected terms. From figure (3.2) it is clear that the quasigeostrophic potential vorticity equation (3.58) is only valid for a band of length scales
in the small region around 15 degrees of latitude. Between latitudes 15 degrees and 60
68
70
O(ε3) Terms
Spherical Geometry
Other
QG
QG+η2
60
QG+Jʹ′ ʹ′
QG +η2 + Jʹ′ ʹ′
Latitude
50
40
30
20
10
50
100
150
200
250
Distance Scale (km)
FIGURE 3.2: Each colored region shows the area in parameter space where a given
equation is valid. An equation is declared valid in a given region if all of its terms have
the greatest magnitude. This is a ‘best case scenario’ for a equation as one might typically
demand that its terms be a factor of 2 or 5 larger than other neglected terms and therefore
cause the regions to shrink. The blue regions require additional terms from either the next
order expansion O(�3 ), or from expansion of trigonometric terms O(spherical). The surface
height is fixed at 10 cm (�H = 0.13), G = 1 and the ratio of the length scale to the time
scale is fixed at the long Rossby wave phase speed. The equations analyzed are the quasigeostrophic potential vorticity equation (3.58) (QG), the FP equation (3.66) (QG + η 2 ),
equation (3.71) (QG + η 2 + J�� ) and a fourth equation extending QG with the J�� term
(QG + J�� ).
69
degrees various extensions to QG theory are required with additional terms, all arising
from a relatively large surface height to equivalent depth ratio, �H .
More abstractly, each of the regions shown in figure (3.2) are actually fivedimensional subregions inside the full five-dimensional parameter space defined by the
coordinates (θ0 , N0 , U, L, T ). Figure (3.2) is only showing a particular two-dimensional
slice through the parameter space. The values we assign to the scales are only estimates,
and so a theory occupying a larger subregion around our estimated values is more robust.
Figures (3.3) and (3.4) show nearby regions in the parameter space by varying the L/T
ratio and N0 , respectively. These suggest the extend QG theories occupy a fairly larger
region of parameter space around the scales of interest.
The two equations most likely to describe the observations at mid-latitudes are the
FP equation (3.66) at length scales roughly 125-175 km and a new extension to the FP
equation with the J�� term (3.71) for length scales roughly 60 - 125 km. Although the FP
equation is a subset of terms from (3.71), we find that it is generally not true that the FP
equation alone is valid for the same regions as (3.71) (the J�� term is not the smallest term).
It also worth noting that the length scale hierarchy of the two equations is reflected in
the scaling and balance relations found in the previous section. In particular, the balance
relations (3.68) and (3.69) include a geometric and β-plane effect not found in the balance
relations (3.72) and (3.73), consistent with the notion that the FP equation is valid at
longer length scales ((3.66).
3.9.
Beyond Geostrophy
In the previous section we considered only flows where the scales were chosen to
reflect the geostrophic balance, G = 1. The reality, of course, is that we expect the flows
to deviate from pure geostrophy and hope that our equation remains valid. Because we
left the geostrophic parameter unfixed in our derivation of equation (3.42) we can consider
70
70
3
O(! ) Terms
Spherical Geometry
Other
QG
QG+"2
60
QG+J# #
2
##
QG +" + J
Latitude
50
40
30
20
10
50
100
150
200
250
Distance Scale (km)
70
3
O(! ) Terms
Spherical Geometry
Other
QG
QG+"2
60
QG+J# #
QG +"2 + J# #
Latitude
50
40
30
20
10
50
100
150
200
250
Distance Scale (km)
FIGURE 3.3: Similar to figure (3.2), but now the ratio of the length scale to the time
scale is fixed at 0.8 x the long Rossby wave phase speed in the first panel and 1.2 x the
phase speed in the second panel. This suggests that the extended QG theories are valid
over a range of phase speeds.
71
70
3
O(! ) Terms
Spherical Geometry
Other
QG
QG+"2
60
QG+J# #
2
##
QG +" + J
Latitude
50
40
30
20
10
50
100
150
200
250
Distance Scale (km)
70
3
O(! ) Terms
Spherical Geometry
Other
QG
QG+"2
60
QG+J# #
QG +"2 + J# #
Latitude
50
40
30
20
10
50
100
150
200
250
Distance Scale (km)
FIGURE 3.4: Similar to figure (3.2), but now the surface height is fixed at 5 cm (�H = 0.63)
in the first panel and 15 cm (�H = 0.19) in the second panel. Quasi-geostrophic theory
performs better at the smaller height perturbations, whereas equation (3.71) (QG + η 2 +
J�� dominates at the larger perturbations.
72
flows that depart from G = 1 and determine when our equation remains valid.
Figure (3.5) shows that our theories are valid for a wide range of fluid speeds deviating from a pure geostrophic balance, from G = 10−1 to G = 105 . In fact, it is interesting
to note that when considering flows that depart from geostrophy, suitable equations can
still be found that accurately describe the flows being considered.
3.10.
The FP Monopole
Once an equation has been selected as an appropriate description of some physical
phenomenon, the next task is typically to find solutions admitted by the chosen equation.
However, not all solutions to the given equation are necessarily valid; they may violate
assumptions made during the equation’s derivation. As an extreme example consider
a plane wave of height 100 m and length scale 1 mm as a solution to equation (3.48).
Although it true that such a plane wave indeed solves equation (3.48), it violates a number
of assumptions including the small amplitude assumption and neglecting of viscosity. We
can use the same techniques used above to show the validity of a given equation to assess
the validity of a particular solution to an equation.
The monopole solution to equation (3.67) [Petviashvili, 1980, Flierl, 1979] is anticyclonic, axially symmetry and propagates slightly faster than the long wave Rossby wave
phase speed. This solution is therefore naturally of interest as it may represent some of
the satellite altimetry tracked eddies observed by [Chelton et al., 2007]. The solution can
be written in approximated form [Boyd, 1991] as
�
�
�
x2 + y 2
2gh2
2.3822 + 1.01327r2 − 0.02417r4
ηFP r =
= 2 2 ·
.
L
L f0 α 1 + 0.75r2 + (1/16)r4 + (1/64)r6
(3.74)
The parameter α is determined from the vertical structure and is typically 1.3
[Flierl, 1979]. The FP monopole solution (3.74) has one free parameter, L, which sets
the length scale of the monopole while simultaneously setting the height of the monopole.
73
Time Scale (days)
10
15
20
25
40
O(!3) Terms
Spherical Geometry
Other
35
QG
QG+"2
Velocity Scale (cm/s)
35
30
QG+J# #
QG +"2 + J# #
30
Geostrophic Velocity
25
25
20
20
15
15
10
10
5
5
50
100
150
200
Velocity Scale (cm/s)
5
40
250
Distance Scale (km)
Time Scale (days)
40
60
80
100
35
Velocity Scale (cm/s)
30
120
140
160
40
O(ε3) Terms
Spherical Geometry
Other
35
QG
2
QG+η
30
QG+Jʹ′ ʹ′
2
ʹ′ ʹ′
QG +η + J
Geostrophic Velocity
25
25
20
20
15
15
10
10
5
5
50
100
150
200
Velocity Scale (cm/s)
40
20
250
Distance Scale (km)
FIGURE 3.5: Similar to figure (3.2), but now the latitude is fixed and the velocity is
allowed to vary (allowing departures from G = 1). The black line is G = 1. The surface
height is fixed at 10 cm (�H = 0.13) and the ratio of the length scale to the time scale is
fixed at 1.5 x the long Rossby wave phase speed for the top panel at 20 degrees, and 1.0
x the Rossby wave phase speed for the bottom panel at latitude 45.
74
70
65
60
Latitude
55
50
45
40
35
30
25
20
100
120
140
160
180
200
220
Distance Scale (km)
FIGURE 3.6: Similar to figure (3.2), but now the scales are fixed for the FP monopole
solution (3.74). The yellow region indicates the small region in parameter space where the
five terms from the FP equation (3.67) dominate all other neglected terms. By assumption
then, this is the only region where the FP monopole solution (3.74) is valid. The distance
scale shown here is taken to be the e-fold radius of the FP solution.
If we take the height of the monopole to be 10 centimeters at latitude 24, then we find that
the e-folding diameter of the monopole is 725 km! This unreasonably large and almost
certainly violates a number of assumptions. On the other hand, the same monopole of
height 10 cm at latitude 55 only has an e-folding diameter of 350 km.
To check the FP monopole solution against our assumptions rigorously, we assume
that if the FP monopole (3.74) is a valid solution to equation (3.67), then all five terms in
(3.67) must be of greater magnitude than all the neglected terms in equation (3.42). The
yellow region in figure (3.6) shows that the FP monopole solution (3.74) is therefore valid
for only a vary narrow range of length scales at at high latitudes. The length scales in
75
figure (3.6) were taken to be an approximate e-fold radius of the FP monopole, suggesting
that only monopoles of diameter greater than 320 km are valid. Based on the observations
in [Chelton et al., 2007], this suggests that while the FP equation itself might be valid (as
indicated in figures (3.3) and (3.4)), the FP monopole itself does not likely represent any
sizable number of the observed eddies.
3.11.
Conclusions
By developing a new approach that retains freedom in the scales of motion, we were
able to find that a new extension to the FP equation with the J�� term (3.71) is evidently
required to describe the mid-latitude mesoscale eddies observed by [Chelton et al., 2007].
Eddies with length scales above 125 km are well described by the FP equation (3.66), but
additional corrections to the Cartesian approximation are required for lengths beyond 175
km. Both of these theories extend quasi-geostrophic theory by including effects attributed
to the large surface height displacement relative to the equivalent depth. This directly
pinpoints the small-amplitude assumption as the primary weakness in quasi-geostrophic
theory at these scales.
This technique provides a concrete way of assessing the range of scales over which a
given theory is valid. Although not a rigorous proof a theory’s validity, this does provide
an estimate of when caution needs to be exercised for a theory’s application toward a
particular problem. From these arguments it would appear that equation (3.71) range of
valid scales dominates the regions of interest. Whether this extensions to quasi-geostrophic
theory will show important differences in the evolution of eddies in a model remains to be
shown.
The scales considered here were chosen to reflect the satellite altimetry observations,
but a wide range of other choices could certainly be considered. We expect that quasi-
76
geostrophic theory will dominate in a large region of parameter space around other values,
such as smaller height perturbations and regions where the length scale is not fixed to the
time scale. Similarly, different choices in length scale, time scale, surface height and
latitudes may all yield very different equations suitable for different contexts. Finally, the
lowest order expansions in the cartesian approximation could be used to consider theories
containing weak dependence on latitude.
Acknowledgments
This research was supported by the National Science Foundation, Award 0621134,
and by the National Aeronautics and Space Administration, grant NNG05GN98G.
77
BIBLIOGRAPHY
Ablowitz and Clarkson, 1991. Ablowitz, M. and Clarkson, P. (1991). Solitons, nonlinear
evolution equations and inverse scattering. Cambridge University Press.
Allen, 1993. Allen, J. (1993). Iterated geostrophic intermediate models. Journal of Physical Oceanography, 23(11):2447–2461.
Anderson and Killworth, 1979. Anderson, D. and Killworth, P. (1979). Nonlinear propagation of long rossby waves. Deep-Sea Res, 26:1033–1049.
Bluman and Anco, 2002. Bluman, G. and Anco, S. (2002). Symmetry and integration
methods for differential equations. Springer.
Boyd, 1991. Boyd, J. (1991). Monopolar and dipolar vortex solitons in two space dimensions. Wave Motion, 13(3):223–241.
Charney and Flierl, 1981. Charney, J. and Flierl, G. (1981). Oceanic analogues of largescale atmospheric motions. Evolution of Physical Oceanography, pages 504–548.
Chelton and Schlax, 1996. Chelton, D. and Schlax, M. (1996). Global observations of
oceanic rossby waves. Science, 272(5259):234.
Chelton et al., 2007. Chelton, D. B., Schlax, M. G., Samelson, R. M., and de Szoeke,
R. A. (2007). Global observations of large oceanic eddies. Geophys. Res. Lett., 34.
de Szoeke, 2004. de Szoeke, R. (2004). An effect of the thermobaric nonlinearity of the
equation of state: A mechanism for sustaining solitary rossby waves. Journal of Physical
Oceanography, 34(9):2042–2056.
Flierl, 1979. Flierl, G. (1979). Baroclinic solitary waves with radial symmetry. Dyn.
Atmos. Oceans, 3:15–38.
Ford et al., 2002. Ford, R., McIntyre, M., and Norton, W. (2002). Reply. Journal of the
Atmospheric Sciences, 59(19):2878–2882.
Gill, 1982. Gill, A. (1982). Atmosphere-ocean dynamics. Academic press.
Killworth et al., 1997. Killworth, P., Chelton, D., and de Szoeke, R. (1997). The speed
of observed and theoretical long extratropical planetary waves. Journal of Physical
Oceanography, 27(9):1946–1966.
Lighthill, 1952. Lighthill, M. (1952). On sound generated aerodynamically. i. general theory. Proceedings of the Royal Society of London. Series A, Mathematical and Physical
Sciences, pages 564–587.
78
Pedlosky, 1987. Pedlosky, J. (1987). Geophysical Fluid Dynamics. Spring-Verlag, second
edition.
Petviashvili, 1980. Petviashvili, V. (1980). Red spot of jupiter and the drift soliton in a
plasma. Journal of Experimental and Theoretical Physics Letters, 32.
Ripa, 1997. Ripa, P. (1997). “inertial” oscillations and the β-plane approximation(s).
Journal of Physical Oceanography, 27:633–647.
Vallis, 1996. Vallis, G. (1996). Potential vorticity inversion and balanced equations of
motion for rotating and stratified flows. Quarterly Journal of the Royal Meteorological
Society, 122(529).
Vallis, 2006. Vallis, G. (2006). Atmospheric and oceanic fluid dynamics: fundamentals
and large-scale circulation. Cambridge University Press.
Warn et al., 1995. Warn, T., Bokhove, O., Shepherd, T., and Vallis, G. (1995). Rossby
number expansions, slaving principles, and balance dynamics. Quarterly Journal of the
Royal Meteorological Society, 121(523).
Williams and Yamagata, 1984. Williams, G. and Yamagata, T. (1984). Geostrophic
regimes, intermediate solitary vortices and jovian eddies. Journal of the Atmospheric
Sciences, 41(4):453–478.
79
4.
GROUP FOLIATION OF EQUATIONS IN GEOPHYSICAL
FLUID DYNAMICS
Jeffrey J. Early, Juha Pohjanpelto, and Roger M. Samelson
Discrete and Continuous Dynamical Systems, Series A
Submitted
80
4.1.
Introduction
The main theoretical objects of study in geophysical fluid dynamics are systems
of partial differential equations describing the dynamics and thermodynamics of rotating stratified fluids [Pedlosky, 1987] [Salmon, 1998]. Finding exact analytical solutions to
these nonlinear partial differential equations (PDEs) can be exceptionally difficult, and
increasingly physicists have instead turned to numerical solution methods. The rapid
growth of computing power in the last few decades has made numerical methods more
compelling, but perhaps more importantly, the computational tools now available allow
many partial differential equations to be solved using only a few basic methods. Numerical results, however, can’t provide the same confidence in the understanding of the
underlying physics as analytical solutions. Ideally, standard analytical methods could be
applied towards identifying solutions to nonlinear PDEs with the same level of robustness
as current computational techniques. Unfortunately, in sharp contrast with their linear
counterparts, very few general analytical techniques exist for solving nonlinear PDEs.
One general approach that is applicable to many nonlinear PDEs is the method of
symmetry reduction originally proposed by Sophus Lie. In Lie’s method the point symmetry group of a system of PDEs is first computed algorithmically by solving the linear
determining equations for symmetry generators. Any subgroup of the full symmetry group
can then be used to reduce the system in order to construct solutions to the original system
that are invariant under the action of the subgroup [Olver, 1993]. This method is used
implicitly by physicists when identifying radial solutions (rotation invariance) or traveling
wave solutions (Galilean invariance), but also explicitly to reveal more complicated symmetry groups and corresponding invariant solutions (e.g., [Salmon and Hollerbach, 1991]).
An unfortunate consequence of symmetry reduction is to eliminate any solution not invariant under the given group, which severally restricts the scope of available solutions
and limits the applications to the physical sciences.
81
A more general, but less explored, technique to the analytical integration of PDEs
is the method group foliation. The classical approach developed by Vessiot [Vessiot, 1904]
can be applied to the same large class of linear and nonlinear PDEs as symmetry reduction,
but it does not suffer from the limitations imposed by group invariance. Group foliation
relies on a symmetry group’s action on the space of solutions to the differential equation.
Because the action of symmetry group transforms solutions to solutions, an individual
solution can be transformed into other solutions along the same orbit by applying the
group action. The method of group foliation provides a reduction of the original system
of PDEs by constructing a set of equations whose solutions identify individual orbits.
Solutions found in the space of the reduced coordinates can then be returned to the
full space by solving another system of PDEs. Importantly, in contrast with symmetry
reduction, Vessiot’s method, at least in principle, can be used to construct any solution
of the original system from the reduced equations.
Although the method of group foliation was first put forth more than a century
ago, renewed interest in the technique has only recently begun to develop. What might
be described as the classical method of group foliation can be found in modern form in
[Ovsiannikov, 1982]. here the reduced equations are written in terms of chosen differential
invariants of its symmetry group (the automorphic system) and the integrability conditions are encoded as syzygies between the differential invariants (the resolving system).
Application of this method is still a rarity [Martina et al., 2001], [Anco and Liu, 2004], but
owing to recent advances in the complete algorithmic description of a symmetry group’s
differential invariants and their syzygies [Olver and Pohjanpelto, 2008, Cheh et al., 2008],
the technique has become more readily applicable.
More recent work [Anderson and Fels, 2005, Pohjanpelto, 2008] recasts a system of
PDEs as an exterior differential system (EDS) which is used to transform the system to
reduced equations on a quotient manifold of the group action; a process we call the EDS
method of group foliation. An EDS encodes the differential equation and the integrability
82
conditions as contact forms in an ideal of differential forms, which vanish on pullback by
solutions to the original system. By including only those differential forms annihilated
by the infinitesimal generators of the symmetry group action (the semi-basic forms), the
reduced ideal on the quotient space can be effectively identified, and solutions to the
original system can then be constructed from those of the reduced system via integrating a system of differential equations of generalized Lie type for the group parameters.
Applications of this technique for finite dimensional symmetry groups can be found in
[Anderson and Fels, 2005], while the infinite dimensional case of pseudogroups is treated
in [Pohjanpelto, 2008].
Here we provide an introduction to both the classical and the EDS methods of group
foliation in the context of geophysical fluid dynamics (GFD). Our goal is to introduce these
techniques to the GFD community, by illustrating their application to familiar equations,
and to highlight some of challenges that remain.
4.2.
Geophysical Fluid Equations
The geophysical fluid dynamical equations that we consider here are the onedimensional Korteweg-de Vries (KdV) equation and various forms of the FlierlPetviashivili (FP) equation. Our particular interest is in the Flierl-Petviashivili equation
as an extension to quasi-geostrophic theory. Satellite altimetry observations of oceanic
mesoscale features were initially compared to attributes of those predicted for linear
Rossby waves [Chelton and Schlax, 1996], but higher resolution data now shows that the
observed features are dominated by a high degree of non-linearity and remain coherent
structures for long durations [Chelton et al., 2007]. In particular, the ratio of the observed
eddy heights to the equivalent depth of the first baroclinic mode, strongly violate the small
amplitude assumptions made in quasi-geostrophic theory. The Flierl-Petviashvili equation
extends quasi-geostrophy by allowing for larger amplitude features by introducing an ad-
83
ditional height nonlinearity.
The FP equation arises in several different contexts including the Great Red
Spot on Jupiter [Petviashvili, 1980], as well as modifications to quasi-geostrophy
with an exterior mean shear flow [Flierl, 1979], finite amplitude height changes
[Anderson and Killworth, 1979] [Charney and Flierl, 1981] and thermobaricity of the
equation of state [de Szoeke, 2004]. In a general non-dimensionalized form, the equation
can be written with only one constant β −1 as
�
�
1
∇ φ − φ − φ2
2
2
��
t
+ φx + β −1 · J(∇2 φ, φ) = 0,
(4.1)
where φ(x, y, t) is a two-dimensional time-dependent stream function, J(a, b) = ax by −ay bx
is the 2-dimensional Jacobian, and β −1 = O(100) for oceanic values. This differs from
the quasi-geostrophic potential vorticity equation by the addition of the φ2 term. No
general analytical solutions are known, although the equation has been modeled numerically with β −1 = 1 [Tan and Boyd, 1997]. For a scaled traveling wave, φ(x, y, t) =
�
�
λ2 ψ (λ(x − ct), λy), where λ = 1 + c−1 , and solutions of this form satisfy the reduced
equation
∇2 ψx − ψx + ψψx = 0.
(4.2)
With the additional assumption of radial symmetry the equation becomes
�
�
�
�
∂ 1 ∂
∂ψ
1
r
− ψ + ψ 2 = 0.
∂r r ∂r
∂r
2
(4.3)
Solutions to (4.3) have been found numerically [Flierl, 1979] and by approximation
[Petviashvili, 1980] [Boyd, 1991], but as of yet, no exact analytical solutions seem to have
appeared in the literature. Finally, when equation (4.2) is restricted to only one dimension
[Flierl, 1979] [Qiang et al., 2006] by positing ψ(x, y) = u(x), the equation reduces to the
one-dimensional traveling wave form of the Korteweg-de Vries equation,
∆KdV : uxxx − ux + uux = 0.
(4.4)
84
We begin by solving the KdV equation (4.4) using both the classical and EDS
methods of group foliation. The KdV equation reduces from a third order non-linear
ordinary differential equation to a first order ordinary differential equation on the quotient
manifold. For the radial FP equation (4.3), reduction based on the scaling symmetry turns
out to result in the obvious integration, but the integrated equations, for a particular value
of the constant of integration, admits a symmetry that can used to construct an exact
analytical solution. Finally, we consider the more complicated 2-D FP equation (4.2) in
polar coordinates and reduce it by the EDS method under the rotational symmetry. We
conclude with some remarks on the challenges of applying the EDS method to the full
symmetry group of the FP equation (4.1).
4.3.
4.3.1
Korteweg-de Vries Equation
Nonlinear Wave Solution
The Korteweg-de Vries equation (4.4) is completely integrable and has been thor-
oughly analyzed [Ablowitz and Clarkson, 1991].
The physically interesting (real and
bounded) cnoidal solution can be written in general form as
u(x) = 1 + α
2
�
1−k
2
�
2 − 3 cn
2
�
x − x0 2
α
,k
2
���
.
(4.5)
The constant α scales the length and amplitude, x0 translates the origin, and k varies the
amplitude and the modulus of the elliptic function cn.
In terms of the original stream function φ(x, y, t), these solutions represent a train
�
�−1
of traveling waves with amplitude dependent speed c = α2 − 1 . The well-known sech2
solitary wave (soliton) solution is recovered from (4.5) in the limit k 2 �→ 1, for which the
distance between the periodic waves is infinite.
85
4.3.2
Point Symmetries
Central to both methods of group foliation is the existence of a symmetry group to
the differential equation of study. A basic introduction to analyzing point symmetries of
differential equations can be found in [Olver, 1993].
By definition, the group of point symmetries of a system of differential equations
consist of a local Lie group of transformations acting the space of independent and dependent variables that map any solution of the equations to a new solution. Point symmetries
arising from a local Lie group action are typically identified by solving the determining
equations for the infinitesimal generators of the group action. In the case of the KdV
equation, these are found by prolonging a generic vector field v = ξ(x, u)∂x + η(x, u)∂u on
the base space M = (x, u) to the jet space J 3 = {(x, u, ux , uxx , uxxx )} and then demanding
that the infinitesimal determining equations
Lpr(3) v (uxxx − ux + uux ) = 0 whenever uxxx − ux + uux = 0
are satisfied. One thus finds two independent point symmetries with the infinitesimal
generators
vx =∂x ,
vs = x∂x + 2(1 − u)∂u ,
(4.6)
corresponding to a translation and a scaling. By exponentiating the generators with
(x̃, ũ) = exp[�v](x, u), i.e., by solving a system of ODEs, we can recover the corresponding
finite transformations,
(x, u) �→(x + �, u),
(x, u) �→ (e� x, 1 − e−2� (1 − u)).
(4.7)
As a symmetry of the differential equation, the scaling transformation in (4.7)
implies that if u(x) is a solution to the KdV equation, then so is ũ� (x) = 1 −
e−2� (1 − u(e−� x)). In terms of the solution (4.5) the group action simply rescales α.
86
Indeed, we find that,
2 −2�
ũ� (x) = 1 + α e
�
1−k
2
�
2 − 3 cn
2
�
αe
−� x
+ x0 2
,k
2
���
,
(4.8)
which changes the value of α to αe−� , where � ∈ R. It is in this sense that, given fixed
values of x0 and k, the action of the scaling symmetry traces out an orbit of solutions
belonging to the same equivalence class. Simply by finding a single solution at any point
along the orbit, say when α = 1, we can recover the entire equivalence class of solutions
by applying the scaling transformations. In this sense the entire space of solutions can be
partitioned into orbits, the equivalence classes, and the quotient manifold, denoted Sol/G,
is the space of these equivalence classes.
The central idea of the method of group foliation is to exploit this partitioning afforded to us by the equation’s group structure, and, roughly speaking, rewrite the equation
on a certain quotient manifold determined by its point symmetries. Specifically, for the
KdV equation, we will use the vx scaling of x0 and the vs scaling of α. We should expect
then that we will reduce the KdV equation from a third order ODE to a first order ODE,
with the single constant of integration representing the modulus k of equation (4.5).
4.3.3
Classical Method of Group Foliation
In the first step, the classical method of group foliation requires finding the auto-
morphic system, which describes the orbits of the symmetry group action on the solution
space Sol. The automorphic system is charaterized by the property that all its solutions
can be obtained from a fixed solution by symmetry transformations. As is expounded in
[Ovsiannikov, 1982], the automorphic system is written in terms of differential invariants
of the symmetry group, which are split into new independent and dependent variables.
For the KdV equation (4.4) this is the only equation, but in general, the automorphic
system must be augmented with the resolving system, which encodes the integrability
conditions of the former by way of syzygies among the differential invariants.
87
Differential invariants are functions of the jet space coordinates that are unaffected
by the action of the group. Thus differential invariants ζ(x, u, ux , ...) of the whole group
generated by the two vectors vx and vs must vanish when acted upon by the prolongations
of either vector. The prolongations take the following form
pr(2) vx =∂x ,
(4.9)
pr(2) vs =x∂x + 2(1 − u)∂u − 3ux ∂ux − 4uxx ∂uxx ,
(4.10)
and together the prolonged group action maps coordinates by
�
�
µ(λ, �, x, u, ux , uxx ) = �−1 (x + λ), 1 − �2 (1 − u), �3 ux , �4 uxx .
(4.11)
Invariance under the first vector vx implies that the differential invariants are independent of x. Consequently, the lowest order invariant can be found by looking for the
x independent characteristics of the system pr(2) vs ζ = 0. The lowest order invariant is
therefore found by integrating
du
dux
=
,
2(1 − u) −3ux
whereby
ζ=
ux
3
(1 − u) 2
.
(4.12)
Appealing to our physical sense, using the differential invariant poses a problem because
1 − u may take on negative values in which case ζ will become imaginary. To resolve this,
we simply use the squared value and define the lowest order differential invariant as
y=
u2x
.
(1 − u)3
(4.13)
The next order invariant is similarly found and is given by
w=
uxx
.
(1 − u)2
(4.14)
To verify that y and w are indeed differential invariants, simply apply the group action
(4.11) to see that the group parameters λ and � cancel out, as expected. When writing
equation (4.5) in terms of these invariants, neither x0 nor α will play a role. As the next
88
step in the algorithm we choose y as the new independent and w as the new dependent
variable.
We still need to compute an invariant in order 3 to rewrite the KdV equation in
terms of the new variables. As is well known (see. e.g., [Olver, 1993], proposition 2.53),
the derivative
dw Dx w
2wy 1/2 + uxxx (1 − u)−5/2
=
=
dy
Dx y
2y 1/2 w + 3y 3/2
(4.15)
yields a new differential invariant. For simplicity, we will use w1 = uxxx (1 − u)−5/2 as our
third order differential invariant so that in terms of w and y,
w1 =
�
dw �
2wy 1/2 + 3y 3/2 − 2wy 1/2 .
dy
(4.16)
Now one can easily see that the KdV equation becomes
w1 − y 1/2 = 0.
(4.17)
Writing this using y and w only we see that
(2w + 3y)
dw
− 2w − 1 = 0.
dy
(4.18)
Equation (4.18) can be written in a standard form by letting a = 2w + 3y and z = 5y,
which yields
aaz = a −
6
2
z+ ,
25
5
(4.19)
where a and z denote the differential invariants
a=
2uxx
3u2x
+
,
(1 − u)2 (1 − u)3
z=
5u2x
.
(1 − u)3
(4.20)
Equation (4.19) is an Abel equation of the second kind, a first order ODE, exactly
as anticipated. Any solution to this equation yields a family of solutions to the KdV
equation (4.4) when the original coordinates are reintroduced. Before examining solutions
to (4.19), we will consider the alternative method of group foliation based on exterior
differential systems.
89
4.3.4
EDS Method of Group Foliation
The exterior differential systems (EDS) method of group foliation requires rewriting
the equation in terms of a differential ideal using differential forms on the jet space J n =
X × U (n) . An exterior differential system (also called a differential ideal ) on J n is a subset
I ⊂ Ω∗ (J n ) of the exterior algebra of all differential forms on J n satisfying the following
properties: (i) if ω, ω̄ ∈ I, then ω + f ω̄ ∈ I for any smooth function f on J n , 2) if ω ∈ I
and η is any form in Ω∗ (J n ), then η ∧ ω ∈ I, and 3) if ω ∈ I, then the exterior derivative
dω ∈ I.
A differential equation ∆ : J n → R determines a submanifold S∆ in the jet space
J n = X × U (n) , and so the contact ideal on J n , when restricted to S∆ , encodes the
differential equation and, as a consequence, its solutions. Recall that contact forms are
characterized by the property that their pull-backs under the prolongation (i.e., n-fold
derivative) of any function f : X → U vanishes. For the KdV equation (4.4) it suffices to
consider the contact ideal on the second order jet space J 2 = X × U (2) with coordinates
(x, u, ux , uxx ). Setting the contact forms ω 1 = du − ux dx and ω 2 = dux − uxx dx to zero
essentially require that
ux =
du
dx
and
uxx =
dux
.
dx
The KdV equation (4.4) is therefore described by the exterior differential system I generated by the forms ω 1 , ω 2 and the form ω 3 = duxx − (1 − u)ux dx; in short
I = �ω 1 = du − ux dx, ω 2 = dux − uxx dx, ω 3 = duxx − (1 − u)ux dx�.
(4.21)
The EDS method of group foliation requires us to find a reduced exterior differential
system Ī on the quotient manifold associated with I. In practice Ī is constructed by
identifying the forms in I, the so-called semi-basic forms, that are annihilated by the
prolonged infinitesimal generators of the group action and restricting these to the quotient
manifold. As our prolonged infinitesimal generators are Γ(2) = span{pr(2) vx , pr(2) vs }, the
90
semi-basic one-forms satisfy
�
A1sb = α ∈ I 1 : iX α = 0,
�
X ∈ Γ(2) .
for all
(4.22)
An arbitrary one-form in I takes the form β = β1 ω 1 + β2 ω 2 + β3 ω 3 , where β1 , β2 ,
β3 are smooth functions on J 2 . Now β is semi-basic provided that the two conditions
vx � β = 0,
vs � β = 0,
(4.23)
are satisfied. These equations can be solved to obtain exactly one independent semi-basic
form. For this, we write equations (4.23) in matrix form as
 
 β
 1
(1 − u)ux




β2  = 0.
x(1 − u)ux + 4uxx  
β3
(4.24)
�
�
�
�
��
β = (1 − u) 3u2x + 2uxx (1 − u) − uxx 4uxx + 2(1 − u)2 ω 1
(4.25)



ux
uxx
xux − 2(1 − u) 3ux + xuxx
We can perform basic matrix row reduction to solve for the unknowns, which gives
�
+ux 4uxx + 2(1 − u)
In terms of coordinate differentials, we have
2
�
2
ω −
�
3u2x
�
3
+ 2uxx (1 − u) ω .
�
�
�
�
��
β = (1 − u) 3u2x + 2uxx (1 − u) − uxx 4uxx + 2(1 − u)2 du
�
+ux 4uxx + 2(1 − u)
2
�
dux −
where the dx terms cancel by virtue of (4.23).
�
3u2x
�
(4.26)
+ 2uxx (1 − u) duxx ,
The next step is to choose a cross-subsection and pull back β. In order to write our
equation in the same form as equation (4.19), we choose our cross-section by
�
�
��
�z �1
1
3
2
σ(z, a) = x = 0, u = 0, ux =
, uxx =
a− z
,
5
2
5
(4.27)
where a, z denote the differential invariants (4.20). The du term vanishes on pullback,
giving
1
σ β=
2
∗
��
6
2
a− z+
25
5
�
�
dz − ada .
(4.28)
91
This form generates the ideal Ī on the quotient manifold corresponding to the reduced
equation, which is
aaz = a −
6
2
z+ .
25
5
(4.29)
Equation (4.29) is identical to equation (4.19), which we found by Vessiot’s method
that directly employs differential invariants of the group action. While in the EDS computation it may initially appear as if we had avoided the entire complication of using differential invariants, these do indirectly play a role in our choice of the cross-section (4.27), as
can be seen via a simple application of the moving frames algorithm [Fels and Olver, 1999].
Compare the cross-section of equation (4.27) with the group action (4.11) and use the u
coordinate to solve for the group parameter, which gives � = (1 − u)−1/2 . When this value
of � is substituted into the equations
3
� ux =
�z �1
2
5
,
4
� uxx
1
=
2
�
�
3
a− z ,
5
and the resulting expressions are solved for a and z, the specific choice of differential
invariants is recovered, namely those in (4.20).
4.3.5
Reconstruction
Equation (4.29) possesses the parametric solution [Zaitsev and Polyanin, 2003]
�
�
(5t − 2)2 5
(5t − 2)2
s̄(t) = (z(t), a(t)) = C
+ , Ct
.
(4.30)
(5t − 3)3 3
(5t − 3)3
The constant C is related to the constant k of equation (4.5). All that remains is the
reconstruction problem, that is, recovering solutions of the original equation (4.4) from
(4.30). The classical and EDS methods provide two different approaches.
Starting with the EDS method, we look for solutions of the form
s(t) = µ(γ(t), σ ◦ s̄(t)),
(4.31)
where µ denotes the group action (4.11) and γ(t) is a symmetry transformation to be
chosen so that the resulting s(t) vanishes on the ideal I defined in (4.21). In the standard
92
coordinates of J 2 , equation (4.31) becomes
�
�
�1/2 2 �
��
z(t)
η
(t)
3
s(t) = ξ(t), 1 − η(t), η 3/2 (t)
,
a(t) − z(t)
.
5
2
5
(4.32)
Here the unknown functions ξ(t), η(t) are related to the group parameters λ and � in
(4.11) by
η(t) = �2 ,
ξ(t) = λ/�,
but they are now assumed to explicitly depend on the variable t. Next we simply compute
the pullbacks the generators (4.21) of the EDS I in under s(t) and demand that the
resulting forms vanish. We have
�
�
�
�
�
du
dx
z(t) 1/2 ˙
3/2
σ (du − ux dx) =
− ux
dt = −η̇(t) − η (t)
ξ(t) dt = 0,
dt
dt
5
�
�
dux
dx
∗
σ (dux − uxx dx) =
− uxx
dt
dt
dt
�
�
�
�
�
3 1/2
z(t) 1/2
1 3/2
z(t) −1/2
= η (t)
η̇(t) + η (t)
ż(t)
2
5
10
5
�
�
�
η 2 (t)
3
˙
−
a(t) − z(t) ξ(t) dt = 0.
2
5
�
∗
The two equations can be solved simultaneously to yield
˙ =
ξ(t)
ż(t)
�
,
a(t) 5z(t)η(t)
η̇(t) = −
1 ż(t)
η(t).
5 a(t)
(4.33)
Requiring that the pullback of ω 3 under s(t) vanishes imposes no further constraints on
the functions ξ(t) or η(t).
Reconstruction using Vessiot’s method requires that we solve the expressions (4.20)
for the invariants a, z for u. Given that our solution (4.30) involves a parameter t, we
will also construct x and u as functions of t. The expression for the z invariant in (4.20)
produces the equation
√
5u̇(t)
ẋ(t) = �
.
(1 − u(t))3 z(t)
(4.34)
93
We obtain the second equation by differentiating the z invariant and then using the expression for the a invariant to the resulting equation, which yields
u̇(t) =
1 ż(t)
(1 − u(t)).
5 a(t)
(4.35)
We note that the above equations are equivalent to the equations (4.33) for ξ(t), η(t),
which we will integrate next.
To group parameter α is recovered as the constant of integration from the equation
for η(t) in (4.33). If we use
t=
1 2s + 3
5 s+1
to reparameterize equation (4.30), we find that η(s) = α2 s. It then follows from equation
(4.33) that
√
dx
dξ
3
1
�
=
=−
,
3
ds
ds
α
s − C(s + 1)
(4.36)
which produces the translation group parameter x0 upon integration. To fully recover the
solution (4.5), redefine the constant C =
that
√ �
3
�
x − x0 =
α
δ3
1+δ
and change variables by s = δ − q to see
1
�
−q q 2 − 3δq +
The roots of the quadratic are r± = 32 δ ±
δ
2
�
δ−3
δ+1 .
δ 2 2δ+3
δ+1
� dq.
(4.37)
These are real if δ > 3, in which case
r+ > r− > 0, or if δ < −1, in which case 0 > r− > r+ . For brevity, we only consider the
first case, in which the integral can be written as
√ �
3
1
�
x − x0 =
dq.
α
(r+ − q)(q − r− )q
(4.38)
This can be expressed in terms of a standard elliptic integral [Byrd et al., 1971] as
√
3 2
x − x0 =
sn−1
α r+
��
r+ − q 2 r+ − r−
,k =
r+ − r −
r+
�
.
(4.39)
Solving (4.38) for q and writing the result in terms of u gives exactly the solution (4.5).
94
4.4.
4.4.1
Radial FP Equation
Integration
The approximate and numerical solutions of the radial FP equation appearing in
[Flierl, 1979] [Petviashvili, 1980] [Boyd, 1991] were found by integrating equation (4.3) to
1 ∂
r ∂r
�
∂ψ
r
∂r
�
1
− ψ + ψ 2 = c,
2
(4.40)
and then setting the constant of integration to zero, c = 0. An important difference
between equations (4.3) and (4.40) for the purposes of group foliation is that for c �= −1/2,
equation (4.40) does not possess any point symmetries, while equation (4.3) admits the
same scaling symmetry as was found for the KdV equation, namely,
pr(2) vs = r∂r + 2(1 − ψ)∂ψ − 3ψr ∂ψr − 4ψrr ∂ψrr ,
(4.41)
with the finite action given by
�
�
(r, ψ, ψr , ψrr ) �→ λ−1 r, 1 − λ2 (1 − ψ), λ3 ψr , λ4 ψrr .
(4.42)
However, for the constant value c = − 12 , the point symmetries of equation (4.40) consist
of the transformations (4.42) and in this special case the method of group foliation can
be applied to reduce the equation to a first order ODE.
4.4.2
Solution
The solvable form of the radial FP equation,
1 ∂
r ∂r
�
∂ψ
r
∂r
�
1
+ (ψ − 1)2 = 0,
2
(4.43)
unfortunately, does not appear to be of physical interest as the FP equation is derived
with the assumption ψ � 1, while equation (4.43) would appear to require that as r �→ ∞,
ψrr �→ 0 and ψ �→ 1. Nonetheless, we present the solution, which, in the interest of brevity,
95
we derive directly using standard substitutions, as it is the only known exact analytical
solution of (4.3).
First we let ψ(r) = 1 +
1
x(ln r)
r2
and t = ln r, which transforms (4.43) into
∂2x
∂x
1
−4
+ 4x + x2 = 0.
2
∂t
∂t
2
(4.44)
By designating
y(x) =
1 ∂x
,
4 ∂t
(4.45)
the above equation can be reduced further to an Abel equation of the second kind in
canonical form,
1
1
yyx − y = − x − x2 .
4
32
(4.46)
Solutions to Abel equations of the second kind can be expressed in terms of
the elementary functions only in special cases [Zaitsev and Polyanin, 2003].
How-
ever, a general solution developed in [Panayotounakos et al., 2005] and summarized in
[Panayotounakos et al., 2006] [Panayotounakos et al., 2007] allows the construction of an
exact analytical solution to equation (4.46). Specifically,
�
�
1
1
y(x) = (x + λ0 ) N (x) +
,
2
3
(4.47)
where λ0 is the constant of integration and N (x) is a function to be defined below. Recalling the definition (4.45) for y, the above equation implies that
t(x) =
�
dx
�
� − t0 .
2(x + λ0 ) N (x) + 13
(4.48)
This produces a solution to (4.43) parametric form that can be written in terms of x as
ψ(x) = 1 + xe−2t(x) ,
r(x) = et(x) .
(4.49)
Next we let
�
�
1 (τ sin τ + cos τ ) ci τ + cos2 τ (4τ ci τ + cos τ )
1
1
G(x) =
· (x + λ0 ) − x − x2 , (4.50)
3
16
2
16
(τ ci τ )
96
where τ = ln |x + λ0 |, and write
�
�
7
1 2
p(x) = − + 4G(x) − x − x (x + λ0 )−1 ,
3
8
�
�
20 1
1 2
q(x) = − +
4G(x) − x − x (x + λ0 )−1 .
27 3
8
(4.51)
Then the specific form of the function N (x) in (4.48) depends on the sign of the subsidiary
function
Q(x) =
�
p(x)
3
�3
+
�
q(x)
2
�2
as follows.
1. Q < 0 (p < 0):
2. Q > 0:
�
p
a
N1 (x) = 2 − cos ,
3
3
�
p
a−π
N2 (x) = −2 − cos
,
3
3
�
p
a+π
N3 (x) = −2 − cos
,
3
3
q
where cos a = − �
,
2 −(p/3)3
(4.52)
0 < a < π.
�
�
q �
q �
3
N (x) = − + Q + 3 − Q .
2
2
3. Q ≡ 0:
4.5.
�
q
N1 (x) = 2 3 − ,
2
�
q
N2 (x) = −2 3 − .
2
(4.53)
(4.54)
2-D FP Equation in Polar Coordinates
Applying the method of group foliation to an ordinary differential equation as in
the preceding sections is essentially equivalent to using Lie’s symmetry based method
integrating these equations (see, e.g., [Olver, 1993], section 2.5). While there is no direct
97
extension of Lie’s algorithm to equations with more than one independent variable, the
two group foliation methods can as well be applied to PDEs with the same basic reduction
and reconstruction steps as is described above.
As an example, we consider the once integrated form of the two-dimensional FP
equation (4.2) written in polar coordinates,
�
�
1 ∂
∂ψ
1 ∂ψ 2
1
r
+ 2 2 − ψ + ψ 2 = 0,
r ∂r
∂r
r ∂θ
2
(4.55)
which is encoded in the differential ideal
I =�ω = dψ − ψr dr − ψθ dθ, ωr = dψr − ψrr dr − ψrθ dθ,
�
�
1
1 2
2
ωFP = dψθ − ψrθ dr + r ψrr + ψr − ψ + ψ dθ�.
r
2
(4.56)
on J 2 . Equation (4.55) admits translational symmetries in x and y, as well as a rotational
symmetry, which, in polar coordinates, is simply represented by a translation in the θ
direction and is thus generated by the vector vθ = ∂θ . Here we will apply the EDS
foliation algorithm with the rotational symmetry only to derive the reduced equations for
the radial FP equation.
Taking an arbitrary one-form in the ideal, β = β1 ω + β2 ωr + β3 ωFP , and requiring
that it vanish under the interior product with pr(2) vθ , we find the semi-basic forms
�
�
�
�
1
= pr(2) vθ � ωFP ω − pr(2) vθ � ω ωFP ,
βsb
�
�
�
�
2
βsb
= pr(2) vθ � ωr ω − pr(2) vω � θ ωr ,
(4.57)
(4.58)
which can be written in the coordinate basis as
1
βsb
=(∆ψr − ψrθ ψθ )dr − ∆dψ + ψθ dψθ ,
(4.59)
2
βsb
=(ψrθ ψr − ψrr ψθ )dr − ψrθ dψ + ψθ dψr ,
(4.60)
�
�
where ∆ = −r2 ψrr + 1r ψr − ψ + 12 ψ 2 .
By choosing our cross-section by
σ(q, s, A, B, C) = (r = q, θ = 0, ψ = s, ψr = A, ψθ = B, ψrr = C, ψrθ = F ) ,
(4.61)
98
we find that
1
σ ∗ ωsb
=(∆σ A − F B)dq − ∆σ ds + BdB,
2
σ ∗ ωsb
(4.62)
=(F A − CB)dq − F ds + BdA,
where
∆σ = −q
2
�
�
1
1 2
C + A−s+ s .
q
2
(4.63)
1 and σ ∗ ω 2 in the quotient space with coordinates (q, s), we obtain the
Annihilating σ ∗ ωsb
sb
equations
BBq =F B − ∆σ A,
BBs =∆σ ,
BAq =CB − F A,
BAs =F.
(4.64)
After some algebraic manipulations, these equations together with (4.63) yield the system
Hq =As H − AHs ,
Hs = −q
2
�
�
1
1 2
Aq + AAs + A − s + s ,
q
2
(4.65)
only involving the unknown functions A and H = B 2 /2. Thus, by applying the EDS
method, we are able to reduce the elliptic equation (4.55) to a pair of quasi-linear hyperbolic equations (4.65), and, again, any solution of (4.55), including non-invariant ones,
can, at least in principle, be reconstructed from the solutions of the system (4.65).
4.6.
Conclusions
The classical and EDS methods of group foliation, when applied to the ODEs (4.3)
and (4.4), provide useful techniques for reducing the order of the equations. In the case of
the KdV equation (4.4), the general solution to the third order equation is recovered from
the solutions of the Abel equation of the second kind, a first order ODE. The reduction of
the radial FP equation (4.3) using the scaling symmetry simply results in the integration
apparent by inspection. However, for a particular value of the constant of integration, the
integrated equation (4.43) admits the same symmetry as its progenitor. This additional
99
symmetry allows the equation to be integrated to yield an exact analytical solution to the
radial FP equation.
The application of the two methods of group foliation to PDEs, such as the 2-D FP
equation (4.2), follows the same basic steps as in the case of ODEs. Various forms of the
reduced equations can now be derived by choosing different symmetry subgroups defining
the foliation or different cross-sections to the group orbits, allowing multiple approaches
to solving the equations by the use of the same technique. As an example, the reduction
of the 2-D FP equation written in polar coordinates (4.55) under the rotational symmetry
is considered here. This results in rewriting the elliptic equation (4.55) as a pair of quasilinear hyperbolic equation (4.65) that may be easier to analyze.
In the examples considered in this paper, the EDS method in general proves to be
a simpler and more flexible approach than Vessiot’s classical method. In the application
of the EDS method to the KdV and the radial FP equation, the reduced equations on
the quotient manifold were obtained without the need of having to compute the differential invariants of the symmetry group action, in contrast with the classical method in its
original form that requires explicit expressions for the invariants. Furthermore, choosing
a different cross-section (and therefore a new set of differential invariants) allows one to
rewrite the reduced equations in new coordinates in an efficient manner. The reconstruction step in the EDS method is also straightforward to set up, and amounts to solving a
system of differential equations of Lie type for the symmetry group parameters.
As the equation and its symmetry group increase in complexity, the advantages
of the EDS method over Vessiot’s approach become less pronounced. With the help
of the moving frames method for computing differential invariants and their syzygies
[Olver and Pohjanpelto, 2008, Cheh et al., 2008], the classical method of group foliation
can be similarly used to write the reduced equations on the quotient manifold, identified
as a cross section to the symmetry group orbits, without having to employ the explicit
expressions for the differential invariants. For the symmetry subgroup of the 2-D FP equa-
100
tion (4.2) consisting of scalings and translations, this produces a system of three first order
coupled differential equations. Here the main difficulty with applying the EDS method
is in finding a managable basis for the semi-basic forms. However, if invariant contact
forms are introduced in place of the coordinate differentials, the ensuing computations are
vastly simplied and yield a basis for semi-basic forms that in complexity is comparable to
expressions one encounters in the application of Vessiot’s method. Ultimately a combination of both the classical and EDS methods may be required in order to treat the more
intricate examples.
Acknowledgments
This research was supported by the National Science Foundation, Award 0621134,
and by the National Aeronautics and Space Administration, grant NNG05GN98G.
101
BIBLIOGRAPHY
Ablowitz and Clarkson, 1991. Ablowitz, M. and Clarkson, P. (1991). Solitons, nonlinear
evolution equations and inverse scattering. Cambridge University Press.
Anco and Liu, 2004. Anco, S. and Liu, S. (2004). Exact solutions of semilinear radial
wave equations in n dimensions. Journal reference: J. Math. Anal. Appl, 297:317–342.
Anderson and Killworth, 1979. Anderson, D. and Killworth, P. (1979). Nonlinear propagation of long rossby waves. Deep-Sea Res, 26:1033–1049.
Anderson and Fels, 2005. Anderson, I. and Fels, M. (2005). Exterior differential systems
with symmetry. Acta Applicandae Mathematicae: An International Survey Journal on
Applying Mathematics and Mathematical Applications, 87(1):3–31.
Boyd, 1991. Boyd, J. (1991). Monopolar and dipolar vortex solitons in two space dimensions. Wave Motion, 13(3):223–241.
Byrd et al., 1971. Byrd, P., Friedman, M., and Byrd, P. (1971). Handbook of elliptic
integrals for engineers and scientists. Springer Berlin.
Charney and Flierl, 1981. Charney, J. and Flierl, G. (1981). Oceanic analogues of largescale atmospheric motions. Evolution of Physical Oceanography, pages 504–548.
Cheh et al., 2008. Cheh, J., Olver, P. J., and Pohjanpelto, J. (2008). Algorithms for
differential invariants of symmetry groups of differential equations. Foundations of
Computational Mathematics, 8(4):501–532.
Chelton and Schlax, 1996. Chelton, D. and Schlax, M. (1996). Global observations of
oceanic rossby waves. Science, 272(5259):234.
Chelton et al., 2007. Chelton, D. B., Schlax, M. G., Samelson, R. M., and de Szoeke,
R. A. (2007). Global observations of large oceanic eddies. Geophys. Res. Lett., 34.
de Szoeke, 2004. de Szoeke, R. (2004). An effect of the thermobaric nonlinearity of the
equation of state: A mechanism for sustaining solitary rossby waves. Journal of Physical
Oceanography, 34(9):2042–2056.
Fels and Olver, 1999. Fels, M. and Olver, P. (1999). Moving coframes. i. a practical
algorithm. Acta Appl. Math., 51:161–213.
Flierl, 1979. Flierl, G. (1979). Baroclinic solitary waves with radial symmetry. Dyn.
Atmos. Oceans, 3:15–38.
102
Martina et al., 2001. Martina, L., Sheftel, M., and Winternitz, P. (2001). Group foliation and non-invariant solutions of the heavenly equation. Journal of Physics A:
Mathematical and General, 34(43):9243–9263.
Olver, 1993. Olver, P. (1993).
Springer-Verlag.
Applications of Lie groups to differential equations.
Olver and Pohjanpelto, 2008. Olver, P. and Pohjanpelto, J. (2008). Moving frames for
pseudogroups. Canad. J. Math., 60:1336–1386.
Ovsiannikov, 1982. Ovsiannikov, L. (1982). Group Analysis of Differential Equations.
Academic Press.
Panayotounakos et al., 2005. Panayotounakos, D., Sotiropoulos, N., Sotiropoulou, A.,
and Panayotounakou, N. (2005). Exact analytic solutions of nonlinear boundary value
problems in fluid mechanics (blasius equations). Journal of Mathematical Physics,
46:033101.
Panayotounakos et al., 2007. Panayotounakos, D., Sotiropoulou, A., Sotiropoulos, N.,
and Manios, M. (2007). Exact analytic solutions of the porous media and the gas
pressure diffusion odes in non-linear mechanics. International Journal of Non-Linear
Mechanics, 42(1):157–163.
Panayotounakos et al., 2006. Panayotounakos, D., Theotokoglou, E., and Markakis, M.
(2006). Exact analytic solutions for the damped duffing nonlinear oscillator. Comptes
rendus-Mécanique, 334(5):311–316.
Pedlosky, 1987. Pedlosky, J. (1987). Geophysical Fluid Dynamics. Spring-Verlag, second
edition.
Petviashvili, 1980. Petviashvili, V. (1980). Red spot of jupiter and the drift soliton in a
plasma. Journal of Experimental and Theoretical Physics Letters, 32.
Pohjanpelto, 2008. Pohjanpelto, J. (2008). Reduction of exterior differential systems
with infinite dimensional symmetry groups. BIT Numerical Mathematics, 48(2):337–
355.
Qiang et al., 2006. Qiang, Z., Yuan, Z., and Shi-Kuo, L. (2006). Two-dimensional
rossby waves: Exact solutions to petviashvili equation. Communications in Theoretical
Physics, 45:414–416.
Salmon, 1998. Salmon, R. (1998). Lectures on Geophysical Fluid Dynamics. Oxford
University Press.
Salmon and Hollerbach, 1991. Salmon, R. and Hollerbach, R. (1991). Similarity solutions
of the thermocline equations. Journal of marine research, 49(2):249–280.
103
Tan and Boyd, 1997. Tan, B. and Boyd, J. (1997). Dynamics of the flierl-petviashvili
monopoles in a barotropic model with topographic forcing. Wave Motion, 26(3):239–
251.
Vessiot, 1904. Vessiot, E. (1904). Sur l’intégration des systèmes différentiels qui admettent des groupes continus de transformations. Acta Mathematica, 28(1):307–349.
Zaitsev and Polyanin, 2003. Zaitsev, V. and Polyanin, A. (2003). Handbook of Exact
Solutions for Ordinary Differential Equations. Chapman & Hall/CRC.
104
5.
THE FORCES OF INERTIAL OSCILLATIONS
Jeffrey J. Early and Roger M. Samelson
Bulletin of the American Meteorological Society
In preparation
105
5.1.
Introduction
The primitive equations in oceanography and atmospheric science describe a rich
set of physical phenomena, but their complexity makes analytical solutions exceptionally rare. Only by simplifying the equations do we hope to find such solutions and
fully explain the forcing involved in the resulting solution. The inertial oscillation problem exemplifies this approach by reducing the primitive equations to describe the motion of a particle confined to the surface of the earth. Although nearly inertial motion can be observed in the ocean [Gill, 1982, Stockwell et al., 2004], the interest in this
problem lies also in its simplification of the underlying forcing. Indeed, when Whipple [Whipple, 1917] stated the problem in 1917 he wrote “...the author considers that a
knowledge of the motion of such a particle will prove a useful preliminary to a proper
understanding of the more complicated motion which actually occurs in winds, where the
air particles have other forces besides that of gravity acting upon them.” Nearly a century later the exact solutions for the particle constrained to the earth have been found
[Paldor and Sigalov, 2001, Pennell and Seitter, 1990] but don’t appear to be well known
and the forces responsible for the motion have continued to be debated and discussed
[Durran, 1993, Persson, 1998, Phillips, 2000].
The name “inertial oscillation” suggests that the observed oscillatory motion of
the particle is explained entirely by inertial forces, apparent only because of our rotating
reference frame. This name therefore makes the observational claim that an accelerometer
trapped in perfect inertial motion would not measure accelerations. However, Durran
[Durran, 1993] showed that there must in fact be a force in the inertial frame. The
argument is quite simple and worth repeating. Without forcing, the f -plane approximation
to the equations of motion are simply
dVr
+ 2ω × Vr = 0
dt
(5.1)
106
where Vr is the two-dimensional velocity vector of the according to the observer on the
earth and ω is the angular frequency of rotation of the earth. However, by simple kinematic
computation, the relationship between acceleration in the rotating and fixed frame is
related by
dVf
dVr
=
+ 2ω × V + ω × (ω × R).
dt
dt
(5.2)
If we take equation (5.2) and apply equation (5.1) we see that in the fixed frame,
dVf
= ω × (ω × R).
dt
(5.3)
Because Vf is the velocity in the fixed frame, then by Newton’s second law equation (5.3)
shows there to be a force acting on the particle and hence the forces on the particle are
not inertial. This result suggests a profound difference from the original claim that an
accelerometer would not detect any accelerations. By this reasoning it is now repeated
[Paldor and Sigalov, 2001] that these oscillations are not inertial.
Starting with free particle and successively adding constraints we will show that
the particle’s motion is in fact inertial and the accelerometer trapped in inertial motion
would not measure an acceleration. However, by identifying the underlying force, it will
be clear that that the name inertial oscillation is unsatisfactory. The name we give to
these oscillations reflects our understanding of the phenomenon and to that end Durran
suggested “constant angular momentum oscillation.” This name is problematic in that
it does not reflect the presence of the underlying force involved, but only one of the
conservation laws. Constant angular momentum oscillations could just as easily be the
observed oscillations of satellites in near geosynchronous orbit or even a particle on a
rotating sphere. Both of these phenomena also conserve angular momentum and exhibit
oscillations from the rotating reference frame, yet have distinctly different forcing. We
need a new name.
To understand the forcing we take a different approach than Durran and use the
Lagrangian (Lagrange function) to help write down and understand the forces involved.
107
The advantage to this technique is its simplicity: small changes in our assumptions about
the system effortlessly show changes in the forcing. Starting with the description of a free
particle in spherical coordinates and adding a new assumption at each step, we examine
two other systems nearly identical to the particle on a earth: a particle in orbit and a
particle constrained to a sphere. With this approach we can highlight the subtle differences
that arise when doing physics on the earth and have a clear understanding of source of
each force.
5.2.
Free Particle, Inertial Frame
The Lagrangian for a conservative mechanical system is the kinetic energy minus
the potential energy. By minimizing the action of the functional with the Euler-Lagrange
equations, we obtain the equations of motion exactly as if we’d written out the forces
explicitly. We use the coordinate system (φI , θ, r) in the inertial frame where φˆI points
eastward in the direction of increasing longitude and θ̂ points northward, in the direction
of increasing latitude (locally these coordinates look like (x, y, z)). The Lagrangian for a
free particle is the kinetic energy of the particle,
1
1
1
2
LIfree = ṙ2 + r2 θ̇2 + r2 φ˙I cos2 θ
2
2
2
(5.4)
where dot is used to indicate the time derivative, e.g., ṙ = ∂r
∂t . The Euler-Lagrange
� �
d
∂L
equations, dt
= ∂L
∂ q̇
∂q , provided us with the equations of motion written to resemble
Newton’s second law �a = f� where f� is the specific force,
  
  
aφI  rφ̈I cos θ + 2ṙφ̇I cos θ − 2rφ̇I θ̇ 0
  
  
 a  ≡  rθ̈ + 2ṙθ̇ + rφ̇2 sin θ cos θ  = 0 .
 θ 
  
I
  
  
ar
r̈ − rθ̇2 − rφ̇2I cos2 θ
0
(5.5)
Although these are the equations of motion for a free particle, they appear rather complicated because they’re written in a coordinate system not well suited for the problem,
108
spherical coordinates. It is important to keep in mind that all of these terms are necessary
simply to describe an unforced, and therefore unaccelerated, particle moving in a straight
line. Only additional terms or constraints we add to the Lagrangian will result in terms
that appear as forces.
5.3.
Free Particle, Rotating Frame
In order to identify the so-called “inertial forces”, we need to transform equation
(5.4) into the rotating frame. We apply the transformation φI �→ φ + ωt where ω is the
angular rotation frequency of the earth and φ is the new longitudinal coordinate. The
Lagrangian becomes,
�
1 2 1 2 2 1 2� 2
2
LR
=
ṙ
+
r
θ̇
+
r
φ̇
+
2ω
φ̇
+
ω
cos2 θ,
free
2
2
2
from which we obtain the following equations of motion,
  

−2ṙω cos θ + 2rθ̇ω sin θ
aφ  

  

 a  = −2rφ̇ω sin θ cos θ − rω 2 sin θ cos θ .
 θ 

  

ar
2rφ̇ω cos2 θ + rω 2 cos2 θ
(5.6)
(5.7)
To the rotating observer believing he is in an inertial frame, everything on the left-handside of the equations appear as appropriate acceleration terms for his chosen spherical
coordinate system, while everything on the right-hand-side appear to be forces. These are
the so-called “inertial forces”. The four terms on the right containing a velocity are the
Coriolis force, while the two terms only dependent on position are the centrifugal force.
If we imagine this system to describe a a spaceship far between stars, then in a
practical sense what makes these forces inertial is that when a person on the spaceship
measures his acceleration with an accelerometer, he will find his acceleration to be zero.
By Newton’s second law the person in the spaceship would correctly conclude no forces
are acting on him, despite the rotating observer’s belief to the contrary.
109
FIGURE 5.1: Three example particle paths in a central gravitational field as viewed from
a rotating reference frame. The second figure approximates that of the International
Space Station, while the third is a small deviation from geosynchronous orbit. All three
are inertial and exhibit oscillatory motion.
There are solutions to this system where the particle would appear to undergo
oscillatory motion for the rotating observer. These could certainly be called “inertial
oscillations.”
5.4.
Central Gravitational Field, Rotating Frame
Let’s extend the free particle Lagrangian of equation (5.6) by including a central
gravity field similar to the earth’s,
�
1 2 1 2 2 1 2� 2
GM
LR
φ̇ + 2ω φ̇ + ω 2 cos2 θ +
.
gravity = ṙ + r θ̇ + r
2
2
2
r
(5.8)
The equations of motion for this system differ from equation (5.7) only by the addition of
a gravitational force included in the radial acceleration, ar .
  

−2ṙω cos θ + 2rθ̇ω sin θ
aφ  

  

 a  = −2rφ̇ω sin θ cos θ − rω 2 sin θ cos θ .
 θ 

  

ar
2rφ̇ω cos2 θ + rω 2 cos2 θ − GM
2
r
(5.9)
The solutions to (5.9) are also well known and include motions like the International Space
Station’s (ISS) orbit path and geosynchronous satellites, figure (5.1).
This system too only involves “inertial forces.” Newton would argue that there is
a force of gravity, but by Einstein’s equivalence theorem we know that astronauts aboard
110
the ISS do not detect acceleration with their accelerometers, just as the inertial observer of
the last example. Although we traditionally think of gravity as a force in this equation, we
cannot design a local experiment to distinguish between the cases with and without this
term. We now know that this term is a description of the local geometry of the problem,
and not a force. This is an important point at the heart of general relativity, see chapter
4.3 of [Wald, 1984] or chapter 1 of [Misner et al., 1973].
By this reasoning then, the oscillations of satellites around the earth or oscillations
of slightly perturbed satellites in near geosynchronous orbit could justly be named “inertial
oscillations.”
5.5.
Particle on a Rotating Sphere
We can constrain the particle to the surface of a sphere simply by setting r = R.
However, a more fruitful technique is to apply our constraint with a Lagrange multiplier
so that we analyze the resulting force required to keep the particle on the surface of the
sphere. With λR
sphere now also a function of time, the Lagrangian is
�
1 2 1 2 2 1 2� 2
GM
2
LR
=
ṙ
+
r
θ̇
+
r
φ̇
+
2ω
φ̇
+
ω
cos2 θ +
+ λR
sphere
sphere (r − R).
2
2
2
r
(5.10)
In the inertial frame equation (5.10) would describe geodesic motion on the surface of a
sphere, the path of the great circles. This system was analyzed by [McIntyre, 2000] in
both the rotating and the inertial frame. The equations of motion now include a fourth
equation using λR
sphere as a variable in the Euler-Lagrange equations,

aφ


−2ṙω cos θ + 2rθ̇ω sin θ

  

  

 aθ   −2rφ̇ω sin θ cos θ − rω 2 sin θ cos θ 
  

 =
.
  

GM
2
2
2
R
 ar  2rφ̇ω cos θ + rω cos θ − r2 + λsphere 
  

r
R
(5.11)
111
ω
FN
Fcentrifugal
Fg
FNet
FIGURE 5.2: Force diagram for a particle initially at rest on a rotating sphere. The net
force points towards the equator, but only the normal force, F�N , is a real, measurable
force.
By applying the constraint condition r = R and using the definitions in equation (5.5),
we can now use the ar component of equation (5.11) to find the force of constraint,
λR
sphere =
GM
− 2Rφ̇ω cos2 θ − Rω 2 cos2 θ − Rθ̇2 − Rφ̇2 cos2 θ.
R2
(5.12)
The force of constraint in equation (5.12) is synonymous with the term normal force
(represented F�N ) because it is necessarily perpendicular to the resulting motion. The
normal force is a real force and stands in as a proxy for the collection of electromagnetic
forces that keep our particle from plunging into the depths of this spherical earth. The
accelerations caused by this force are detectable by our accelerometer. The first term is
exactly opposite and equal to the force of gravity, completely negating its effect. The
next two terms in equation (5.12) oppose components of the inertial forces, the second
of which is a centripetal force (opposing the centrifugal force). The last two terms are
perhaps unexpected. They represent the constraint force applied against the particle’s
inertia preventing it from moving in a straight, unaccelerated path. These two terms
would be present even in the inertial frame and without gravity; they distinguish geodesic
motion in Euclidean space from geodesic motion on a sphere and could be called geometric
forces.
Consider the forces acting on the particle initially at rest by setting (φ̇, θ̇, ṙ) =
(0, 0, 0) in equation (5.11). In this case the only forces acting on the particle are gravity,
112
FIGURE 5.3: Three example particle paths on a sphere as viewed from a rotating reference
frame. The first figure depicts the path of a particle released from rest in the rotating
reference frame, while the second and third show different choices of initially eastward
velocities. All three are inertial tangent to the surface, and oscillatory.
the normal force, and the centrifugal force, see figure (5.2). However, we can see from
equation (5.11) that a component of the centrifugal force still points in the −θ̂ direction,
pushing the particle towards the equator. This net force is exactly why we know that
this Lagrangian does not correctly describe a particle on the earth’s surface (a ball on the
table doesn’t roll equator-ward).
The resulting motion shown in figure (5.3) is again periodic, so could this too be
described as inertial oscillations? We have established that there exists a force of constraint, but how would the observer moving with the particle experience this motion?
Assume the observer starts initially at rest with respect to the rotating surface. We can
see from figure (5.3) that the observer will oscillate about the equator, drifting slowly
westward. Because there are only inertial forces in the φ̂ and θ̂ directions, the observer
will not detect any accelerations in those directions. However, the observer’s local vertical
acceleration will show change, directly computable with equation (5.12). There will be
a constant acceleration detected normal to gravity, but in addition there is a position
dependent centripetal force as well as the other velocity dependent geometric forces in
equation (5.12). To the strict three-dimensional observer, any motion described by this
system has detectable forces and therefore is not inertial motion. However, the spirit of
the problem is to restrict the motion, and perhaps therefore also the observations, to only
113
two dimensions. In the remaining two dimensions the motion does only contain inertial
forces and it seems reasonable to describe the oscillatory motion as inertial oscillations.
5.6.
Particle on Earth
In this final case we want to consider a particle under the influence of gravity and
constrained to the surface of the earth. The gravitational potential is parameterized as
the more general G(r, θ) instead of the potential for a point mass (and sphere), − GM
r .
The surface of the earth is described by some function f (r, θ) and therefore our constraint
is described by f (r, θ) = k. The Lagrangian in the rotating frame is
�
1 2 1 2 2 1 2� 2
2
LR
=
ṙ
+
r
θ̇
+
r
φ̇
+
2ω
φ̇
+
ω
cos2 θ − G(r, θ) + λR
earth
earth (f − k) .
2
2
2
(5.13)
This looks identical to equation (5.10) except for the generalizations of the gravitational
potential and the constraint force. The key difference, shown in [Ripa, 1997], is that
1
Vgeo (r, θ) ≡ G(r, θ) − r2 ω 2 cos2 θ
2
must be equal to a constant at the surface of the earth.
(5.14)
According to
[Imagery and Agency, 2000] Vgeo is an effective potential called the total gravity potential and a surface of constant potential is a geopotential surface, or geop.
In
[Imagery and Agency, 2000] it also stated that “the geoid is that particular geop that
is closely associated with the mean ocean surface.” So this means that Vgeo describes the
surface of the earth, which is exactly what we said f does! This should simplify things.
The idea that the earth’s surface is defined by the geopotential can be understood
by considering the previous example of the particle on the rotating sphere. If the earth
were a sphere as described by the system (5.10), then all particles on its surface would
roll towards the equator. So if we imagine the earth were entirely fluid and in static
equilibrium everything would have already rolled towards the equator; its shape would
114
have relaxed to this minimal potential energy state. The reality is that over the oceans
it’s pretty close.
However, now we have a problem with our coordinates. A good set of coordinates
for this problem would be defined with one basis vector normal to the surface and the
other two basis vectors tangent to the surface. Our spherical coordinates do this for a
sphere, but not for a geopotential surface which is clearly a function of both θ and r. The
vector φ̂ points tangent to the surface, but θ̂ and r̂ have components both tangent and
normal to the surface. If we approximate our geopotential as an ellipse, and then use
elliptic coordinates, we can achieve the desired results. However, this turns out not to
change the form of the Lagrangian significantly. The key observation is that if we use a
good set of coordinates where say, ξˆ is normal to the geopotential, then Vgeo is entirely
a function of ξ. The change in direction of the forcing from the difference between the
geometry of a spheroid and an ellipsoid is negligible [Gill, 1982]. At this point we will then
redefine our coordinates so that r̂ is normal to the geopotential and θ̂ is perpendicular
to both r̂ and φ̂. This also allows us to restate our constraint as constant r, rather than
constant Vgeo . The rotating observer standing on the earth experiences −
∂Vgeo
∂r
as the total
force of gravity, so we use equation (5.14) and neglecting those small geometric deviations
to rewrite equation (5.13) as
�
1 2 1 2 2 1 2� 2
LR
=
ṙ
+
r
θ̇
+
r
φ̇
+
2ω
φ̇
cos2 θ − Vgeo (r) + λR
earth
earth (r − R) ,
2
2
2
(5.15)
where R is the approximate radius of the earth.
At this point if we were to ignore the force of constraint and simply proceed to set
r = R we would recover the same Lagrangian as in [Ripa, 1997], and therefore produce the
same force diagrams drawn in figure 1 of both [Ripa, 1997] and [Durran, 1993]. However,
let’s proceed with the force of constraint still under consideration and apply the Euler-
115
Lagrange equations to examine the resulting force. In the rotating frame we find




a
−2ṙω cos θ + 2rθ̇ω sin θ
 φ 

  

 aθ  

−2rφ̇ω sin θ cos θ
  

 =
.
  

∂V
 ar  2rφ̇ω cos2 θ − ∂rgeo + λR
earth 
  

r
R
(5.16)
Equation (5.16) looks almost identical to the equations of motion on sphere, equation
(5.11). The centrifugal and gravitational force in the radial component of equation (5.11)
are wrapped in the definition of Vgeo . However, the centrifugal force in the θ̂ component
of equation (5.11) is missing from equation (5.16). Qualitatively at least this looks a lot
more like the surface of the earth as we know it, objects initially at rest don’t roll towards
the equator. The constraint force is
�
∂Vgeo ��
R
λearth =
− 2Rφ̇ω cos2 θ − Rθ̇2 − Rφ̇2 cos2 θ,
�
∂r �
(5.17)
r=R
which appears nearly identical to the constraint force on the sphere, equation (5.12). For
the particle initially at rest, the constraint force evaluates to
�
�
∂V
�
geo
λR
=
�
earth
∂r �
� r=R
∂G ��
=
− Rω 2 cos2 θ
�
∂r �
(5.18)
r=R
and is depicted in figure (5.4). Because the constraint force is the only real force, we
have already shown that the motion is inertial in the φ̂ and θ̂ directions, just as with the
particle on the sphere. So where’s the discrepancy with Durran’s derivation?
Consider the Lagrangian in the inertial frame, but without using equation (5.14) to
replace the gravitational potential with the total gravity potential,
1
1
1
2
LIearth = ṙ2 + r2 θ̇2 + r2 φ˙I cos2 θ − G(r, θ) + λIearth (r − R) .
2
2
2
(5.19)
116
W
FN
Fcentrifugal
Fg
FIGURE 5.4: Force diagram for a particle initially at rest on the earth. All forces balance,
so the particle will remain at rest in this reference frame. The outline of a sphere is shown
in gray.
Applying the Euler-Lagrange equations we find that
  

aφI
0
  

  

1 ∂G
 aθ  

− r ∂θ
  

 =
.
   ∂G

 ar  − ∂r + λIearth 
  

r
R
(5.20)
Equation (5.20) shows that unlike the case of the sphere, gravity has a component tangent
to the surface and in fact, from equation (5.14), we know the magnitude of this component
is exactly rω 2 sin θ cos θ. The gravitational forces of equation (5.20) are however, not real
forces and it is only the constraint force λIearth found in the r̂ direction that is measured
by the accelerometer. Solving for λIearth in equation (5.20),
λIearth =
∂G
2
− Rθ̇2 − Rφ˙I cos2 θ,
∂r
(5.21)
we see that the constraint force balances only the r̂ component of gravity, and not the
entire force. A particle at rest in the rotating frame, (φ˙I , θ̇, ṙ) = (ω, 0, 0), has a normal
force of
λR
earth
�
∂G ��
=
�
∂r �
r=R
− Rω 2 cos2 θ.
(5.22)
Equations (5.17) and (5.22) are identical, meaning the rotating observer and inertial observer predict the same reading on the accelerometer. The forces as they appear from the
117
FIGURE 5.5: Three example particle paths on a earth as viewed from a rotating reference
frame. Just as with the sphere, all three are inertial tangent to the surface and exhibit
oscillatory motion.
inertial frame are drawn in figure (5.4). For the inertial observer having a non-zero net
force looks correct. The particle is rotating around the polar axis at speed ω and therefore
is accelerating inward.
Consider what would happen if we believed that the θ̂ component of gravity in
equation (5.20) really was a force. This would suggest that a plumb-bob held by an
observer standing on the earth would point not in the local vertical, but would point
slightly poleward! This is certainly not our experience and so we must reject the notion
that gravity is a true force.
The accelerometer trapped in inertial oscillations would therefore not detect any
motion in the φ̂ or θ̂ directions. However, like the particle on the sphere, the accelerometer
would detect changes in acceleration in the local vertical due to the changing magnitude of
the normal force. So again, we could justly call these oscillations are inertial if we restrict
measurements to the two dimensions of the earth’s surface. Representative solutions are
shown in figure (5.5)
118
5.7.
Conclusions
The name “inertial oscillations” turns out not to be so bad after all. The “force”
shown to exist in the θ̂ direction by Durran is gravity and by Einstein’s equivalence
principle it is therefore not measurable by an accelerometer. This additional term should
not strictly be considered a force, but in fact a description of the local geometry. In
our daily lives we often mistakenly think of gravity as the force we feel against our feet,
but the reality is that that it’s the earth pushing us up. The result of this is that an
accelerometer trapped in an inertial oscillation would not detect any accelerations in the
direction tangent to earth’s surface. Only measurement in the local vertical would indicate
that the particle is being forced. So in a very real sense the oscillations described by the
Lagrangian in equation (5.15) are inertial.
Is inertial oscillation really an appropriate name for this phenomenon? In all four
systems we examined, the motion could be described as inertial and all four systems
exhibit oscillations, so the name ”inertial oscillation” isn’t particularly descriptive. Durran
suggested “constant angular momentum oscillation”, but this too suffers from ambiguity.
Because φ does not explicitly enter into any of the four Lagrangians we consider, this means
that angular momentum is conserved in all four cases. The key feature distinguishing
oscillations on the earth from the other three systems is the constraint of the particle to
the geopotential. In order to satisfy equation (5.14) the direction of the total gravity force
and the gravitational force always differ, resulting in a net force in the inertial frame. This
will always be a feature of a geopotential of the earth or another planet and therefore the
name “geopotential oscillation” or “geoinertial oscillation” is perhaps more descriptive.
119
Acknowledgments
Thanks to Emily Shroyer for her helpful discussion regarding an appropriately descriptive name. This research was supported by the National Science Foundation, Award
0621134.
120
BIBLIOGRAPHY
Durran, 1993. Durran, D. R. (1993). Is the coriolis force really responsible for the inertial
oscillation? Bulletin of the American Meteorological Society, 74(11):2179–2184.
Gill, 1982. Gill, A. (1982). Atmosphere-ocean dynamics. Academic press.
Imagery and Agency, 2000. Imagery, N. and Agency, M. (2000). Department of Defense
World Geodetic System 1984, volume 47. NIMA TR8350.2, third edition.
McIntyre, 2000. McIntyre, D. H. (2000). Using great circles to understand motion on a
rotating sphere. American Journal of Physics, 68:1097–1105.
Misner et al., 1973. Misner, C. W., Thorne, K. S., and Wheeler, J. A. (1973). Gravitation. W. H. Freeman and Company.
Paldor and Sigalov, 2001. Paldor, N. and Sigalov, A. (2001). The mechanics of inertial
motion on the earth and on a rotating sphere. Physica D, 160:29–53.
Pennell and Seitter, 1990. Pennell, S. A. and Seitter, K. L. (1990). On inertial motion
on a rotating sphere. Journal of the Atmospheric Sciences, 47(16):2032–2034.
Persson, 1998. Persson, A. (1998). How do we understand the coriolis force? Bulletin of
the American Meteorological Society, 79(7):1373–1384.
Phillips, 2000. Phillips, N. A. (2000). An explication of the coriolis effect. Bulletin of
the American Meteorological Society, 81(2):299–303.
Ripa, 1997. Ripa, P. (1997). “inertial” oscillations and the β-plane approximation(s).
Journal of Physical Oceanography, 27:633–647.
Stockwell et al., 2004. Stockwell, R., Large, W., and Milliff, R. (2004). Resonant inertial
oscillations in moored buoy ocean surface winds. Tellus A, 56(5):536–547.
Wald, 1984. Wald, R. M. (1984). General Relativity. The University of Chicago Press.
Whipple, 1917. Whipple, F. J. W. (1917). Motion of a particle on the surface of a smooth
rotating globe. Monthly Weather Review, 45(9):454.
121
6.
CONCLUSIONS
This dissertation presented a series of studies aimed at understanding the physics of
the slowly westward propagating signal observed with satellite altimetry. Simultaneously,
the concept of group foliation was shown to be a useful mathematical tool for solving
partial differential equations directly relevant to the physical interest.
The conclusions in chapter 2 argued that the satellite observations originally hypothesized to be Rossby waves are, in fact, eddies governed by nonlinear dynamics. However,
it was also shown that the zonal propagation rates of eddies governed by quasi-geostrophic
dynamics are systematically slower than those seen in the observations, suggesting an extended theory would be needed to explain the observations. In order to create a baseline
for comparison between differing theories, the chapter 2 also characterized amplitude decay, length decay, propagation speed, fluid transport properties, energy conservation, and
potential vorticity conservation of isolated eddies. Further work is required to understand
how Rossby wave radiation, or other physical effects, govern the evolution during this
quasi-stable state.
Chapter 3 used theoretical arguments to explore extensions to quasi-geostrophic
theory required to represent the observations. One of these extensions is the well known
Flierl-Petviashvili (FP) equation, and the other equation does not yet appear in the literature. Although not considered in this dissertation, the two candidate theories found in
chapter 3 should be examined with the numerical model developed in chapter 2. These
theories must first be shown to reproduce the spectral results of quasi-geostrophic theory,
but the evolution of isolated eddies can then be compared to the baseline properties of
quasi-geostrophy developed in chapter 2. While a primitive equation model could similarly be used for comparison, using these modifications to quasi-geostrophic theories offer
the advantage that each additional term can be directly attributed to physical differences
122
in the models.
Although this dissertation avoids consideration of the vertical structure of these
eddies, preliminary investigations with a primitive equation model suggest the vertical
structure may be more complicated than assumed. The primitive equation model was
initialized with a stably stratified vertical structure and radially symmetric FP monopole
perturbation to the first baroclinic mode. The monopole did not evolve as a long lasting
eddy structure as is predicated by the single layer equivalent barotropic model, but instead evolved to a more complicated modal structure and consistently lost its eddy-like
structure in times less than 100 days. The vertical structure of these eddies warrants
further investigation.
The technique developed in chapter 3 may have broader implications than the identification of the two new models for the satellite altimetry eddy observations. First, the
technique provides a concrete way of assessing the range of scales over which a given theory
is valid. Although not a rigorous proof a theory’s validity, this does provide an estimate of
when caution needs to be exercised for a theory’s application toward a particular problem.
Second, the application of this technique certainly applies to other choices of scales than
those consider in chapter 3. Different choices in length scale, time scale, surface height
and latitudes may all yield very different equations suitable for different contexts. Finally,
the general technique may be applicable to other equations with similar structure, but
found in other fields of physics.
The exact analytical solution to the FP equation found in chapter 4 appears to
be the first exact solution, but unfortunately is unbounded and therefore not directly
applicable to the eddy observations. However, the reduction of the two-dimensional FP
equation to a pair of quasi-linear equations warrants further investigation. No formal
attempts have yet been made at finding solutions to this form of the equation. In addition
to the equations shown in chapter 4, the foliation of a more complicated form of the FP
equation and also the three-dimensional quasi-geostrophic equation was performed. The
123
latter equation admits a much larger symmetry group than the other equations considered,
but owing to the additional complexity of the equation, further work is required before
this form of the equation becomes useful.
The broader implications of the method of group foliation were discussed in the
introduction of chapter 4. Because it is applicable to a large class partial differential
equations, the method of group foliation may prove to be a useful analytical tool available
to geophysical dynamics community. However, there are still a number of issues that need
to be addressed. In some cases, using the entire symmetry group admitted by the equation
for the foliation proved useful, while in other cases, choosing a suitable subgroup was more
helpful. More general insights into an equation’s resulting structure based on these choices
may be possible with additional experience. In a similar vein, it was noted that different
choices of cross-section can dramatically change the form of the resulting equations and
this freedom can be use to consider alternative forms of the equation. Finally, the issue
of how to choose an appropriate basis for the semi-basic forms in the EDS method could
be further explored and the technique demonstrated with helpful examples.
Chapter 5 contradicts the current literature and concludes that an accelerometer
trapped in perfect inertial motion would not measure accelerations. This conclusion implies that inertial oscillations are the geodesics, the ‘straight lines‘ of motion on the earth.
Slow westward propagation is the natural state of motion for weak energy particles. The
β-effect can therefore be interpreted as an unforced effect which results in slow westward
motion. That there is no real force involved in this effect, implies that no work is done
during the westward propagation and therefore has implications for the energetics of more
realistic westward propagating phenomena like eddies. Of course, the forcing is vastly
more complicated by pressure gradients when considering a fluid and so care must be
taken to separate the β-effect from other effects.
124
BIBLIOGRAPHY
Ablowitz and Clarkson, 1991. Ablowitz, M. and Clarkson, P. (1991). Solitons, nonlinear
evolution equations and inverse scattering. Cambridge University Press.
Allen, 1993. Allen, J. (1993). Iterated geostrophic intermediate models. Journal of Physical Oceanography, 23(11):2447–2461.
Anco and Liu, 2004. Anco, S. and Liu, S. (2004). Exact solutions of semilinear radial
wave equations in n dimensions. Journal reference: J. Math. Anal. Appl, 297:317–
342.
Anderson and Gill, 1975. Anderson, D. and Gill, A. (1975). Spin-up of a stratified ocean,
with applications to upwelling. Deep-Sea Res, 22:583–596.
Anderson and Killworth, 1979. Anderson, D. and Killworth, P. (1979). Nonlinear propagation of long rossby waves. Deep-Sea Res, 26:1033–1049.
Anderson and Fels, 2005. Anderson, I. and Fels, M. (2005). Exterior differential systems
with symmetry. Acta Applicandae Mathematicae: An International Survey Journal
on Applying Mathematics and Mathematical Applications, 87(1):3–31.
Arbic and Flierl, 2003. Arbic, B. and Flierl, G. (2003). Coherent vortices and kinetic
energy ribbons in asymptotic, quasi two-dimensional f-plane turbulence. Physics of
Fluids, 15:2177.
Bluman and Anco, 2002. Bluman, G. and Anco, S. (2002). Symmetry and integration
methods for differential equations. Springer.
Boyd, 1991. Boyd, J. (1991). Monopolar and dipolar vortex solitons in two space dimensions. Wave Motion, 13(3):223–241.
Boyd, 2001. Boyd, J. (2001). Chebyshev and Fourier spectral methods. Dover Pubns.
Byrd et al., 1971. Byrd, P., Friedman, M., and Byrd, P. (1971). Handbook of elliptic
integrals for engineers and scientists. Springer Berlin.
Canuto, 2006. Canuto, C. (2006). Spectral methods: Fundamentals in single domains.
Springer Verlag.
Chan et al., 2005. Chan, C., Psaltis, D., and
”Ozel, F. (2005). Spectral methods for time-dependent studies of accretion
flows. i. two-dimensional, viscous, hydrodynamic disks. The Astrophysical Journal,
628(1):353–367.
125
Charney and Flierl, 1981. Charney, J. and Flierl, G. (1981). Oceanic analogues of largescale atmospheric motions. Evolution of Physical Oceanography, pages 504–548.
Cheh et al., 2008. Cheh, J., Olver, P. J., and Pohjanpelto, J. (2008). Algorithms for
differential invariants of symmetry groups of differential equations. Foundations of
Computational Mathematics, 8(4):501–532.
Chelton and Schlax, 1996. Chelton, D. and Schlax, M. (1996). Global observations of
oceanic rossby waves. Science, 272(5259):234.
Chelton et al., 2007. Chelton, D. B., Schlax, M. G., Samelson, R. M., and de Szoeke,
R. A. (2007). Global observations of large oceanic eddies. Geophys. Res. Lett., 34.
Cushman-Roisin et al., 1990. Cushman-Roisin, B., Tang, B., and Chassignet, E. (1990).
Westward motion of mesoscale eddies. Journal of Physical Oceanography, 20(5):758–
768.
de Szoeke, 2004. de Szoeke, R. (2004). An effect of the thermobaric nonlinearity of the
equation of state: A mechanism for sustaining solitary rossby waves. Journal of
Physical Oceanography, 34(9):2042–2056.
Durran, 1991. Durran, D. (1991). The third-order adams-bashforth method: An attractive alternative to leapfrog time differencing. Monthly Weather Review, 119(3):702–
720.
Durran, 1993. Durran, D. R. (1993). Is the coriolis force really responsible for the inertial
oscillation? Bulletin of the American Meteorological Society, 74(11):2179–2184.
Fels and Olver, 1999. Fels, M. and Olver, P. (1999). Moving coframes. i. a practical
algorithm. Acta Appl. Math., 51:161–213.
Fiorino and Elsberry, 1989. Fiorino, M. and Elsberry, R. (1989). Some aspects of vortex
structure related to tropical cyclone motion. Journal of the Atmospheric Sciences,
46(7):975–990.
Flierl, 1977. Flierl, G. (1977). The application of linear quasigeostrophic dynamics to
gulf stream rings. Journal of Physical Oceanography, 7(3).
Flierl, 1979. Flierl, G. (1979). Baroclinic solitary waves with radial symmetry. Dyn.
Atmos. Oceans, 3:15–38.
Flierl, 1984. Flierl, G. (1984). Rossby wave radiation from a strongly nonlinear warm
eddy. Journal of Physical Oceanography, 14(1):47–58.
Flierl et al., 1980. Flierl, G., Larichev, V., McWilliams, J., and Reznik, G. (1980). The
dynamics of baroclinic and barotropic solitary eddies. Dyn. Atmos. Oceans, 5(1):1–
41.
126
Ford et al., 2002. Ford, R., McIntyre, M., and Norton, W. (2002). Reply. Journal of the
Atmospheric Sciences, 59(19):2878–2882.
Gelb and Gleeson, 2001. Gelb, A. and Gleeson, J. (2001). Spectral viscosity for shallow
water equations in spherical geometry. Monthly Weather Review, 129(9):2346–2360.
Gill, 1982. Gill, A. (1982). Atmosphere-ocean dynamics. Academic press.
Hsu et al., 2005. Hsu, T., Ou, S., Yang, B., and Tseng, I. (2005). On the damping
coefficients of sponge layer in boussinesq equations. Wave Motion, 41(1):45–57.
Imagery and Agency, 2000. Imagery, N. and Agency, M. (2000). Department of Defense
World Geodetic System 1984, volume 47. NIMA TR8350.2, third edition.
Karamanos and Karniadakis, 2000. Karamanos, G. and Karniadakis, G. (2000). A spectral vanishing viscosity method for large-eddy simulations. Journal of Computational
Physics, 163(1):22–50.
Killworth et al., 1997. Killworth, P., Chelton, D., and de Szoeke, R. (1997). The speed
of observed and theoretical long extratropical planetary waves. Journal of Physical
Oceanography, 27(9):1946–1966.
Korotaev, 1997. Korotaev, G. (1997). Radiating vortices in geophysical fluid dynamics.
Surveys in Geophysics, 18(6):567–618.
Korotaev and Fedotov, 1994. Korotaev, G. and Fedotov, A. (1994). Dynamics of an
isolated barotropic vortex on a beta-plane. J. Fluid Mech, 264:277–301.
LaCasce Jr, 1996. LaCasce Jr, J. (1996). BAROCLINIC VORTICES OVER A SLOPING BOTTOM. PhD thesis, Massachusetts Institute of Technology.
Lesieur and Metais, 1996. Lesieur, M. and Metais, O. (1996). New trends in large-eddy
simulations of turbulence. Annual review of fluid mechanics, 28(1):45–82.
Lighthill, 1952. Lighthill, M. (1952). On sound generated aerodynamically. i. general
theory. Proceedings of the Royal Society of London. Series A, Mathematical and
Physical Sciences, pages 564–587.
Maltrud and Vallis, 1993. Maltrud, M. and Vallis, G. (1993). Energy and enstrophy
transfer in numerical simulations of two-dimensional turbulence. Physics of Fluids
A, 5(7):1760–1775.
Martina et al., 2001. Martina, L., Sheftel, M., and Winternitz, P. (2001). Group foliation and non-invariant solutions of the heavenly equation. Journal of Physics A:
Mathematical and General, 34(43):9243–9263.
127
McDonald, 1999. McDonald, N. (1999). The motion of geophysical vortices. Philosophical Transactions: Mathematical, Physical and Engineering Sciences, 357(1763):3427–
3444.
McIntyre, 2000. McIntyre, D. H. (2000). Using great circles to understand motion on a
rotating sphere. American Journal of Physics, 68:1097–1105.
McWilliams and Flierl, 1979. McWilliams, J. and Flierl, G. (1979). On the evolution of
isolated, nonlinear vortices. Journal of Physical Oceanography, 9(6):1155–1182.
Misner et al., 1973. Misner, C. W., Thorne, K. S., and Wheeler, J. A. (1973). Gravitation. W. H. Freeman and Company.
Nycander, 2001. Nycander, J. (2001). Drift velocity of radiating quasigeostrophic vortices. Journal of Physical Oceanography, 31(8):2178–2185.
Olver, 1993. Olver, P. (1993).
Springer-Verlag.
Applications of Lie groups to differential equations.
Olver and Pohjanpelto, 2008. Olver, P. and Pohjanpelto, J. (2008). Moving frames for
pseudogroups. Canad. J. Math., 60:1336–1386.
Ovsiannikov, 1982. Ovsiannikov, L. (1982). Group Analysis of Differential Equations.
Academic Press.
Paldor and Killworth, 1988. Paldor, N. and Killworth, P. D. (1988). Inertial trajectories
on a rotating earth. Journal of the Atmospheric Sciences, 45(24):4013–1019.
Paldor and Sigalov, 2001. Paldor, N. and Sigalov, A. (2001). The mechanics of inertial
motion on the earth and on a rotating sphere. Physica D, 160:29–53.
Panayotounakos et al., 2005. Panayotounakos, D., Sotiropoulos, N., Sotiropoulou, A.,
and Panayotounakou, N. (2005). Exact analytic solutions of nonlinear boundary value
problems in fluid mechanics (blasius equations). Journal of Mathematical Physics,
46:033101.
Panayotounakos et al., 2007. Panayotounakos, D., Sotiropoulou, A., Sotiropoulos, N.,
and Manios, M. (2007). Exact analytic solutions of the porous media and the gas
pressure diffusion odes in non-linear mechanics. International Journal of Non-Linear
Mechanics, 42(1):157–163.
Panayotounakos et al., 2006. Panayotounakos, D., Theotokoglou, E., and Markakis, M.
(2006). Exact analytic solutions for the damped duffing nonlinear oscillator. Comptes
rendus-Mécanique, 334(5):311–316.
Pasquetti, 2005. Pasquetti, R. (2005). High-order les modeling of turbulent incompressible flow. Comptes rendus-Mécanique, 333(1):39–49.
128
Pedlosky, 1987. Pedlosky, J. (1987). Geophysical Fluid Dynamics. Spring-Verlag, second
edition.
Pennell and Seitter, 1990. Pennell, S. A. and Seitter, K. L. (1990). On inertial motion
on a rotating sphere. Journal of the Atmospheric Sciences, 47(16):2032–2034.
Persson, 1998. Persson, A. (1998). How do we understand the coriolis force? Bulletin of
the American Meteorological Society, 79(7):1373–1384.
Petviashvili, 1980. Petviashvili, V. (1980). Red spot of jupiter and the drift soliton in a
plasma. Journal of Experimental and Theoretical Physics Letters, 32.
Phillips, 2000. Phillips, N. A. (2000). An explication of the coriolis effect. Bulletin of
the American Meteorological Society, 81(2):299–303.
Pohjanpelto, 2008. Pohjanpelto, J. (2008). Reduction of exterior differential systems
with infinite dimensional symmetry groups. BIT Numerical Mathematics, 48(2):337–
355.
Qiang et al., 2006. Qiang, Z., Yuan, Z., and Shi-Kuo, L. (2006). Two-dimensional rossby
waves: Exact solutions to petviashvili equation. Communications in Theoretical
Physics, 45:414–416.
Reznik and Dewar, 1994. Reznik, G. and Dewar, W. (1994). An analytical theory of distributed axisymmetric barotropic vortices on the -plane. Journal of Fluid Mechanics,
269:301–301.
Ripa, 1997. Ripa, P. (1997). “inertial” oscillations and the β-plane approximation(s).
Journal of Physical Oceanography, 27:633–647.
Ripa, 2000a. Ripa, P. (2000a). Effects of the earth’s curvature on the dynamics of isolated
objects. part i: The disk. Journal of Physical Oceanography, 30(8):2072–2087.
Ripa, 2000b. Ripa, P. (2000b). Effects of the earth’s curvature on the dynamics of isolated objects. part ii: The uniformly translating vortex. Journal of Physical Oceanography, 30(10):2504–2514.
Salmon, 1998. Salmon, R. (1998). Lectures on Geophysical Fluid Dynamics. Oxford
University Press.
Salmon and Hollerbach, 1991. Salmon, R. and Hollerbach, R. (1991). Similarity solutions
of the thermocline equations. Journal of marine research, 49(2):249–280.
Stockwell et al., 2004. Stockwell, R., Large, W., and Milliff, R. (2004). Resonant inertial
oscillations in moored buoy ocean surface winds. Tellus A, 56(5):536–547.
129
Sutyrin and Flierl, 1994. Sutyrin, G. and Flierl, G. (1994). Intense vortex motion on the
beta plane: Development of the beta gyres. Journal of the Atmospheric Sciences,
51(5).
Tadmor, 1989. Tadmor, E. (1989). Convergence of spectral methods for nonlinear conservation laws. SIAM Journal on Numerical Analysis, 26(1):30–44.
Tan and Boyd, 1997. Tan, B. and Boyd, J. (1997). Dynamics of the flierl-petviashvili
monopoles in a barotropic model with topographic forcing. Wave Motion, 26(3):239–
251.
Vallis, 1996. Vallis, G. (1996). Potential vorticity inversion and balanced equations of
motion for rotating and stratified flows. Quarterly Journal of the Royal Meteorological
Society, 122(529).
Vallis, 2006. Vallis, G. (2006). Atmospheric and oceanic fluid dynamics: fundamentals
and large-scale circulation. Cambridge University Press.
Vessiot, 1904. Vessiot, E. (1904). Sur l’intégration des systèmes différentiels qui admettent des groupes continus de transformations. Acta Mathematica, 28(1):307–349.
Wald, 1984. Wald, R. M. (1984). General Relativity. The University of Chicago Press.
Warn et al., 1995. Warn, T., Bokhove, O., Shepherd, T., and Vallis, G. (1995). Rossby
number expansions, slaving principles, and balance dynamics. Quarterly Journal of
the Royal Meteorological Society, 121(523).
Whipple, 1917. Whipple, F. J. W. (1917). Motion of a particle on the surface of a smooth
rotating globe. Monthly Weather Review, 45(9):454.
Williams and Yamagata, 1984. Williams, G. and Yamagata, T. (1984). Geostrophic
regimes, intermediate solitary vortices and jovian eddies. Journal of the Atmospheric
Sciences, 41(4):453–478.
Zaitsev and Polyanin, 2003. Zaitsev, V. and Polyanin, A. (2003). Handbook of Exact
Solutions for Ordinary Differential Equations. Chapman & Hall/CRC.
130
APPENDICES
131
APPENDIX A Numerical Methods
Introduction
Four distinct potential vorticity conserving equations were modeled using pseudospectral methods and Runge-Kutta time-stepping. Three different damping schemes
were compared for accuracy, of which the spectral vanishing viscosity method was found
to be the best. A sponge layer was used for situation where signal crossing doubly periodic
boundary needed to be suppressed. Passive tracers and float trajectories were also computed during the model runs to diagnose potential vorticity conservation and the quantity
of trapped fluid.
Modeled Equations
Several different equations were modeled,
� ∂η
∂ � 2
∇ η−η +
=0
∂t
∂x
� ∂η
�
�
∂ � 2
∇ η−η +
+ k · J η, ∇2 η =0
∂t
∂x
�
�
�
�
�
∂ � 2
∂
1 2
∇ η−η +
η + η + k · J η, ∇2 η =0
∂t
∂x
2
�
�
�
�
∂
1
∂η
∇2 η − η + η 2 +
+ k · J η, ∇2 η + J �� =0
∂t
2
∂x
(Linear QG)
(QG)
(FP)
(FP + J”)
All of the above equations can be written in the form
∂Q
(x, y, t) = f (x, y, t)
∂t
(A.1)
and can therefore be integrated forward in time using a number of different iterative
techniques. A fourth-order Runge-Kutta algorithm was chosen for this model and remains
one of the more accurate and efficient iterative techniques available [Durran, 1991].
132
Spatial derivatives were computed using pseudospectral methods [Boyd, 2001,
Canuto, 2006]. Given an initial description of the height field η(xm , yn ) on a discretized
grid of M by N points, the Fourier transform is computed as,
η̂(k, l) =
M −1 N −1
1 � �
η(xm , yn ) · e−i2πkxm e−i2πlyn
NM
(A.2)
m=0 n=0
with inverse formula,
M/2−1 N/2−1
η(xm , yn ) =
�
�
k=−M/2 l=−N/2
η̂(k, l) · ei2πkxm ei2πlyn .
(A.3)
Derivatives are computed in the frequency domain with multiplication by the wavenumber,
∂p ∂q
η(xm , yn ) =
∂xp ∂y q
M/2−1 N/2−1
�
�
k=−M/2 l=−N/2
(i2πk)p (i2πl)q η̂(k, l) · ei2πkxm ei2πlyn .
(A.4)
The nonlinear terms are computed by multiplication in the spatial domain in a
pseudospectral model, rather than a convolution in the frequency domain. The advantage
to this technique is the computational efficiency, but without padding the domain, this
has the disadvantage of aliasing high wave numbers into lower numbers. Besides padding
one can also dealias by eliminating wave numbers greater than (2/3)N before multiplying
in the spatial domain; this is known as the “three-halves rule,” (see [Boyd, 2001], p 212).
No appreciable difference was found between model runs with and without dealiasing.
Damping
Neglected from the modeled equations is the viscosity term, νt ∇4 η. The molecular
viscosity νt is approximately 10−6 m2 /s, which non-dimensionalized is 10−10 , much beyond
the resolution of our model. However, because the quadratic non-linearity will increase the
magnitude of the higher wave numbers, we need some mechanism to remove that energy
from our model.
133
One approach to removing energy at the highest wave numbers is simply to increase
the viscosity coefficient to a value high enough to achieve stability in the model. Requiring
the magnitude of the viscosity term to be similar to that of the advection term is equivalent
to requiring the Reynolds number, Re =
UL
νt ,
be order 1. If L is taken to be the grid cell
size, then we want a non-dimensional viscosity of approximately ν̄ = −0.1(∆x̄). In practice
we found that the coefficient could be dropped by an order of magnitude and stability
would still be maintained. The disadvantage to this approach is that is also increases the
viscosity for the larger scales as well, meaning that large scales may be inappropriately
decreased in amplitude.
An alternative is to take the approach [McWilliams and Flierl, 1979] or
[Maltrud and Vallis, 1993] and use higher order operators, also known as hyper-viscosity.
In this case our damping takes the form,
�
�2s−1 2s
Fdamp = (−1)s+1 · C · ∆X
∇ η.
(A.5)
As the order s increase, there is less damping at the lower wave numbers, but the cutoff at
higher wave numbers becomes progressively steeper. Unfortunately the steeper cutoff results in the excessively strong Gibb’s phenomena. Using this technique becomes a balance
of choosing the right coefficient C and s and is discussed in some detail in the appendix
of [Maltrud and Vallis, 1993]. We found that this approach did work, but required fine
tuning of the parameters to prevent excessive decay. This appears to be exactly what happened in [McWilliams and Flierl, 1979] when they concluded that quasi-geostrophic eddy
decay was determined by the damping coefficient, which is now known to be controlled
by Rossby wave radiation [Flierl, 1984].
An alternative approach to hyper-viscosity is to directly define a filter in the
frequency domain applied at each time step [LaCasce Jr, 1996, Arbic and Flierl, 2003,
Chan et al., 2005]. Typically an exponential filter such as,
σ(k) = exp (−αk s )
(A.6)
134
is then applied at each time step, or at certain intervals. This method has the advantage
that one can alter the filter as desired by creating weaker or strong fall offs at the higher
wave number. In some cases, the filter was conditionally applied only to the higher wave
numbers so that the lower wave number are completely undamped [LaCasce Jr, 1996,
Arbic and Flierl, 2003]. A similar variation to this is Kraichnan’s spectral eddy viscosity
[Lesieur and Metais, 1996].
The approach that we found to be the most robust was that of Spectral Vanishing Viscosity (SVV) . The SVV method is very similar to applying a filter at each
time step, but enjoys having a rigorous mathematical treatment showing that, in theory, spectral accuracy is maintained even after applying the filter [Tadmor, 1989]. The
premise is to construct a typical low order hyper-viscosity operator, such as (A.5) where
s = 2, but then to filter this operator such that low wave numbers are completely undamped. This is very much a hybrid of the preceding two methods, but the convergence results originally obtained in [Tadmor, 1989] restrict the choices of coefficients (see
[Karamanos and Karniadakis, 2000] for a nice review). The Fdamp term is filtered with



0
k < σ,
Q(k) =
(A.7)
�
�
�s �


exp −α k−σ
otherwise.
1−σ
where
σ = N θ−1
for θ <
2s − 1
.
2s
Wave numbers below σ are thus left completely undamped. This approach has now been
used in fluid model such as [Gelb and Gleeson, 2001, Pasquetti, 2005] with great success.
To validate the damping schemes, a modon solution was allowed to evolved for 730
days (this idea came from [LaCasce Jr, 1996]). The modon solution is a steady quasigeostrophic dipole that was found analytically [Flierl et al., 1980]. With the spectral vanishing viscosity method, the modon showed no changes in amplitude at greater than three
significant figures after the 730 days (approximately 15,000 time steps).
135
Sponge Layer
One limitation of a periodic domain is that unwanted signal can propagate indefinitely. For example, a monopole initialized in the center of the domain may lose energy
in the form of waves. If these waves travel eastward, they will eventually reappear on the
western boundary of the domain. Having the wave interact with the monopole can be
an undesirable effect when trying to study isolated features. One solution is to insert a
sponge layer along the boundary designed to locally over damp the signal and prevent its
transmission.
Following the approach of [Hsu et al., 2005] we used a sponge layer which modifies
equation (A.8) with the addition of two terms,
∂Q
(x, y, t) = f (x, y, t) + c1 · r(x, y) · Q(x, y, t) + c2 · r(x, y) · ∇2 Q(x, y, t).
∂t
(A.8)
Coefficients c1 and c2 are constant while r(x, y) is a ramp function defined as,
−2π
r(x, y) = exp
�
|
W − x− W
2
2
Lsp
|
�2
· exp
−2π
�
|
H − y− H
2
2
Lsp
|
�2
(A.9)
where W and H are the width and height of the domain and Lsp is the thickness of the
sponge layer.
The first term resembles a simple decay function where
∂Q
∂t
= c1 Q, solved by Q(t) =
ec1 t . If our signal propagates at speed c and over the layer of thickness Lsp , then we expect
the signal to be in the sponge layer for approximately T = Lsp /c. If we want the signal
to be attenuated by two e-folding scales, then this requires that c1 = − L2csp .
The second term resembles viscous damping, but was found to be unnecessary for
these model runs and was generally set to c2 = 0 or c2 = 0.05. These coefficients and
sponge layer thicknesses were chosen based on the domain size and signal to be attenuated.
Setting a thicker sponger layer does more effectively prevent signal from crossing the
136
boundary, but of course consumes a larger portion of the domain. If the sponge layer is
too narrow and steep, then reflections occur at the boundary.
Passive Tracer
In order to diagnose the transport properties of the fluid, two passive tracers were
advected using the velocity field determined from η. A passive tracer, W (x, y, t), does not
affect the flow the fluid motion, has no sources or sinks, and is described by the equation,
∂W
∂W
∂W
+u
+v
=0
∂t
∂x
∂y
(A.10)
where W (x, y, t) is a scalar field. For all of the equations modeled the velocity field (u, v)
is determined directly from the height field, (u, v) = (−kηy , kηx ) at each time step. The
derivatives of W were computed spectrally and equation (A.10) was time-stepped with
the same fourth-order Runge-Kutta algorithm used to solve the dynamical equations.
Only two scalar fields are necessary to create a complete description of the advection
of the fluid, one for each of the spatial dimensions. The following two fields were used as
initial conditions for equation (A.10),

�
W − x− W
2
2
Lpt
�
H − y− H
2
2
Lpt
−2π

X(x, y) =x ∗ 1 − exp


Y (x, y) =y ∗ 1 − exp
−2π
|
|
|
|
�2 


�2 


(A.11)
(A.12)
The initial conditions are designed to give a distinct value at each grid point, but provide
a smooth transition over a length scale of Lpt as the boundary to satisfy the periodic
boundary conditions.
By advecting a passive tracer the origin of the fluid parcel at each grid point is
determined for each time step. However, one disadvantage to using a passive tracer to
137
diagnose trapped fluid is that it suffers strongly from numerical diffusion. A fluid parcel
initially falling exactly on a grid point will, in general, be transported to a new location
that does not fall exactly on a grid point. The scalar value at the nearest grid point to
that parcel will therefore contain an average of other nearby fluid. As the fluid transports
the tracer further from its origin, gradients steepen and this diffusive effect strengthens.
Float Trajectories
An alternative to advecting a passive tracer is to compute float trajectories. The idea
is that an initial position is assigned to a float and the position is then time stepped using
the local fluid velocities. If a float has position (x, y) = (ξ(t), η(t)), then the equations
∂ξ(t)
= u (ξ(t), η(t))
∂t
∂η(t)
= v (ξ(t), η(t))
∂t
(A.13)
(A.14)
are solved using the usual Runge-Kutta fourth order time stepping. The values of u and
v were determined through bilinear interpolation at each time step.
By computing a float trajectory started at each grid point we can paint a relatively
comprehensive picture of the fluid’s transport properties. However, in contrast to advecting a passive tracer, the float positions do not necessarily fall exactly on grid cells and
therefore we cannot necessarily determine the fluid origin of each grid cell at each time
step unless a float happens to fall nearby. The advantage to this technique is that it does
not suffer from the diffusion seen when using passive tracers. The interpolation required
to determine u and v in between grid cells was found to be accurate when gradients in the
velocity field were relatively weak. As noted in chapter 2, for floats being ejected from
the eddy core (where velocity gradients are strong), it was found that potential vorticity
was not well conserved. Higher order interpolation may improve this result.
Computing float trajectories in a translating reference frame brings with it addi-
138
tional complications. In a fixed reference frame, a float can be initialized at each grid
cell and integrated at each time step for the entire model run. However, in a translating
reference frame some floats initially within the spatial domain may quickly be advected
outside the domain. Similarly, some floats of interest may start initially outside the spatial
domain and enter the domain at some later time. To solve this, all floats outside of the
domain of computation were simply given the translational velocities. This is equivalent
to assuming no disturbances exist outside the computation domain.
Tracking
In order to analyze evolving properties of the eddies, relative vorticity extrema were
tracked. This tracking determines both the zonal speed cx and meridional speed cy . The
eddy amplitude, A, was determined by looking for the nearby height extremum. Three
different techniques can be used to determine a length scale: 1) height e-fold contour Le ,
2) relative vorticity zero contour Lrv , or 3) trapped fluid contour Ltrapped . We found the
relative vorticity zero contour to be the most reliable method for determining a length
scale and, unless otherwise specified, this is the length scale we’re referring to. The fluid
velocity was taken to be the maximum fluid velocity within the eddy.
139
BIBLIOGRAPHY
Arbic and Flierl, 2003. Arbic, B. and Flierl, G. (2003). Coherent vortices and kinetic
energy ribbons in asymptotic, quasi two-dimensional f-plane turbulence. Physics of
Fluids, 15:2177.
Boyd, 2001. Boyd, J. (2001). Chebyshev and Fourier spectral methods. Dover Pubns.
Canuto, 2006. Canuto, C. (2006). Spectral methods: Fundamentals in single domains.
Springer Verlag.
Chan et al., 2005. Chan, C., Psaltis, D., and
”Ozel, F. (2005). Spectral methods for time-dependent studies of accretion flows. i.
two-dimensional, viscous, hydrodynamic disks. The Astrophysical Journal, 628(1):353–
367.
Durran, 1991. Durran, D. (1991). The third-order adams-bashforth method: An attractive alternative to leapfrog time differencing. Monthly Weather Review, 119(3):702–720.
Flierl, 1984. Flierl, G. (1984). Rossby wave radiation from a strongly nonlinear warm
eddy. Journal of Physical Oceanography, 14(1):47–58.
Flierl et al., 1980. Flierl, G., Larichev, V., McWilliams, J., and Reznik, G. (1980). The
dynamics of baroclinic and barotropic solitary eddies. Dyn. Atmos. Oceans, 5(1):1–41.
Gelb and Gleeson, 2001. Gelb, A. and Gleeson, J. (2001). Spectral viscosity for shallow
water equations in spherical geometry. Monthly Weather Review, 129(9):2346–2360.
Hsu et al., 2005. Hsu, T., Ou, S., Yang, B., and Tseng, I. (2005). On the damping
coefficients of sponge layer in boussinesq equations. Wave Motion, 41(1):45–57.
Karamanos and Karniadakis, 2000. Karamanos, G. and Karniadakis, G. (2000). A spectral vanishing viscosity method for large-eddy simulations. Journal of Computational
Physics, 163(1):22–50.
LaCasce Jr, 1996. LaCasce Jr, J. (1996). BAROCLINIC VORTICES OVER A SLOPING BOTTOM. PhD thesis, Massachusetts Institute of Technology.
Lesieur and Metais, 1996. Lesieur, M. and Metais, O. (1996). New trends in large-eddy
simulations of turbulence. Annual review of fluid mechanics, 28(1):45–82.
Maltrud and Vallis, 1993. Maltrud, M. and Vallis, G. (1993). Energy and enstrophy
transfer in numerical simulations of two-dimensional turbulence. Physics of Fluids A,
5(7):1760–1775.
140
McWilliams and Flierl, 1979. McWilliams, J. and Flierl, G. (1979). On the evolution of
isolated, nonlinear vortices. Journal of Physical Oceanography, 9(6):1155–1182.
Pasquetti, 2005. Pasquetti, R. (2005). High-order les modeling of turbulent incompressible flow. Comptes rendus-Mécanique, 333(1):39–49.
Tadmor, 1989. Tadmor, E. (1989). Convergence of spectral methods for nonlinear conservation laws. SIAM Journal on Numerical Analysis, 26(1):30–44.
141
APPENDIX B Exact Solutions to the Inertial Oscillation Problem
Numerical solutions to the inertial oscillation problem have been previously considered [Paldor and Killworth, 1988], but the exact solutions were first found by directly
integrating the equations of motion [Pennell and Seitter, 1990] and then later by use of
action-angle variables [Paldor and Sigalov, 2001]. While technically correct, neither of
these computations compare several of the basic properties of the exact solution to that
of the approximate f -plane solution. We demonstrate an alternate solution method and
show the deviations of the exact solution from the commonly understood approximate
solutions.
The f -plane
We already found the exact equations of motion in equation (5.16). After applying
the constraint that r = R, these equations reduce to

 

R
φ̈
cos
θ
2R
φ̇
θ̇
+
2R
θ̇ω
sin
θ

 


=
.
Rθ̈
−Rφ̇2 sin θ cos θ − 2Rφ̇ω sin θ cos θ
(B.1)
We can rewrite this in terms of the velocities (u, v) = (Rφ̇ cos θ, Rθ̇) and we have the more
familiar,
  

�
�
u
u̇  f + R tan θ v 
 = �
� 
u
v̇
− f+R
tan θ u
(B.2)
where f = 2ω sin θ. If you turn the partial time derivatives of equation (B.2) into total
derivatives you have the usual two-dimensional description of a fluid on earth, but without
the forcing. In the typical textbook derivation of inertial oscillations the problem is
formulated on the f -plane, a cartesian plane tangent to the to the earth where the coriolis
force is treated as constant. By assuming the size of the motion is small compared to the
radius of the earth, the two geometric forcing terms,
u
R
tan θ, are neglected. Without the
142
complications of spherical coordinates, this is nothing more than equation (5.1), which
written in component form is,


 
u̇ − f0 v  0

 =  .
v̇ + f0 u
0
(B.3)
A sinusoidal solution of frequency f0 = 2ω sin θ0 is found,

 

u(t)
u
cos
f
t
0 

  0

=

v(t)
u0 sin f0 t
(B.4)
where u0 is the initial particle trajectory. The particle always oscillates anti-cyclonically,
has frequency f0 , and the radius of the oscillation is
u0
f0 .
We will show that these basic
properties will all have slight modifications for the exact solutions.
There is a long history of searching for solutions on the slightly more complicated βplane (see [Paldor and Killworth, 1988] for a good overview). The β-plane approximates
the coriolis parameter as a linear function in latitude, but still neglects the spherical
geometry of the problem. It has now been shown that the β-plane approximation is an
incorrect approximation to this problem because the geometric terms enter at the same
order as the linear variation in the coriolis parameter [Ripa, 1997]. Consistent β-plane
like approximations can be made, we will proceed to the solve the inertial motion exactly.
Particle on a Rotating Sphere
Let’s briefly turn to the particle on the rotating sphere Lagrangian (5.10) because
it’s solution shares many of the same characteristics with the particle bound to the earth.
By applying the constraint directly to the Lagrangian we can begin with a simplified
problem,
�
1 2 2 1 2� 2
2
LR
=
R
θ̇
+
R
φ̇
+
2ω
φ̇
+
ω
cos2 θ.
sphere
2
2
(B.5)
Rather than directly integrate the equations of motion, we can exploit the two conservation law associated with this system, energy and angular momentum. In a conservative
143
mechanical system the energy is determined from Lagrangian by the relation E = q̇ ∂L
∂ q̇ −L,
1
1
1
R
Esphere
= R2 cos2 θφ̇2 + R2 θ̇2 − ω 2 R2 cos2 θ.
2
2
2
Because
dL
dt
∂L
∂φ
= 0 the Euler-Lagrange equations show that L ≡
∂L
∂ φ̇
(B.6)
is a conserved quantity,
= 0. This is found to be
L = R2 cos2 θ(φ̇ + ω).
(B.7)
Substituting this back into the the energy equation to eliminate φ̇ we find that
E=
1
L2
1
+ R2 θ̇2 − ωL
2
2
2 R cos θ 2
(B.8)
Using the two conserved quantities in equations (B.7) and (B.8), the system is now
completely integrable in both variables, φ and θ.
Taking equation (B.8) and solving for θ̇, we can integrate to find
t=
R2
L
�
θ
θ0
�
cos θ
2(E+ωL)R2
L2
dθ.
cos2 θ
(B.9)
−1
By taking our initial conditions to be (u, v) = (Rφ̇0 cos θ0 , 0), the constant quantity
2(E+ωL)R2
L2
is just
1
cos2 θ0
and
R2
L
=
1
.
cos2 θ0 (φ̇0 +ω)
cos θ0 (φ̇0 + ω)t =
�
This means we can rewrite the integral as,
θ
θ0
√
cos θ
dθ.
cos2 θ − cos2 θ0
The radical will remain positive when cos2 θ > cos2 θ0 , or just that −θ0 < θ < θ0 . If we
let s =
sin θ
sin θ0
then integral becomes,
cos θ0 (φ̇0 + ω)t =
�
sin θ
sin θ0
1
√
ds
1 − s2
(B.10)
directly evaluating to our solution,
�
�
sin θ(t) = sin θ0 sin cos θ0 (φ̇0 + ω)t .
(B.11)
144
The expression for φ(t) can be similarly found.
Starting with equation (B.7), we
use the same procedure as before,
L
φ(t) − φ0 = 2
R
�
1
dt − ωt.
cos2 θ(t)
(B.12)
We can find the expression for cos2 θ(t) using equation (B.11). The integral becomes,
φ(t) =
L
R2
�
1
�
� dt − ωt + φ0 .
1 − sin2 θ0 sin2 t(φ̇0 + ω) cos θ0
�
�
Using the substitution u = cos θ0 tan t(φ̇0 + ω) cos θ0 we have that
φ(t) =
�
1
du − ωt + φ0 .
1 + u2
This is now easily evaluated, and the expression for φ(t) is,
�
�
��
φ(t) = tan−1 cos θ0 tan t(φ̇0 + ω) cos θ0 − ωt + φ0 .
(B.13)
Particle on Earth
The exact solution for the particle on the earth proceeds in exactly the same manner
as for that on the sphere. Starting with equation (5.15), but again applying the constraint
that r = R outright, we simplify the Lagrangian to
�
1 2 2 1 2� 2
LR
φ̇ + 2ω φ̇ cos2 θ.
earth = R θ̇ + R
2
2
(B.14)
The angular momentum takes the exact same form as with the sphere,
L = R2 cos2 θ(φ̇ + ω).
but the energy appears different,
1
1
E = r2 θ̇2 + r2 cos2 θφ̇2 .
2
2
(B.15)
145
�
�
However, notice that if we write this in terms of velocities (u, v) = rφ̇ cos θ, rθ̇ , then the
�
�
conserved quantity is E = 12 u2 + v 2 . Again we eliminate φ̇ from the energy equation
by writing it terms of the angular momentum,
E=
1
L2
1
1
+ R2 θ̇2 + R2 ω 2 cos2 θ − ωL.
2
2
2 R cos θ 2
2
(B.16)
If you compare equation (B.16) to that of the spherical case, equation (B.8), you can see
the addition of another term.
Elliptic Integral
The integral for θ is now more complicated, but we simply follow the same procedure
as before and start with the same initial conditions (u, v) = (Rφ̇0 cos θ0 , 0) in order to
evaluate the constant.
ωt =
=
=
�
�
�
θ
θ0
θ
θ0
θ
θ0
�
cos θ
2(E+ωL)
ω 2 R2
cos2 θ −
L2
ω 2 R4
dθ
− cos4 θ
cos θ
�
dθ
(cos2 θ0 (1 + uφ )2 + cos2 θ0 ) cos2 θ − cos4 θ0 (1 + uφ )2 − cos4 θ
cos θ
�
dθ
2
2
(cos θ − cos θ0 ) ((1 + uφ )2 cos2 θ0 − cos2 θ)
where we’ve define uφ =
φ̇0
ω ,
(B.17)
the ratio of the initial velocity of the particle to the tangential
velocity of the earth. This is a useful non-dimesional parameter that will generally be much
less than one for any oceanographic and even atmospheric values. The key to factoring
the radical is to have the initial velocity entirely in the φ̂ direction. We can proceed with
the exact same substitution as before and let s =
sin θ0 ωt =
where k � 2 =
1−(1+uφ )2 cos2 θ0
.
sin2 θ0
�
sin θ
sin θ0
1
sin θ
sin θ0 .
Equation (B.17) becomes,
ds
�
�
�
(1 − s2 ) s2 − k � 2
(B.18)
Equation (B.18) and equation (B.10) illustrate the difference
between a inverse trigonometric integral and an inverse elliptic integral.
146
The parameter k � is called the complementary modulus and is related to the modulus by k 2 + k � 2 = 1 [Byrd et al., 1971]. The modulus is therefore k 2 = cot2 θ0 (2uφ + uφ 2 ),
but it can also be written in terms of more standard oceanographic constants,
�
�
2 = 4 cβ + c2
c = φ̇0 R cos θ0 , f0 = 2ω sin θ0 , and β = 2ω
cos
θ
.
Using
these
values,
k
.
0
2
2
2
R
f
R f
0
0
The modulus separates inverse elliptic integrals from inverse trigonometric integrals.
When the modulus is zero, you tend to recover inverse trigonometric integrals, and when
it is one, you recover inverse hyperbolic functions. The modulus depends on both the
energy via the c2 term and the angular momentum via the c term. It is possible to write
k 2 entirely in terms of E and L, however, only our particular initial conditions appear
to factor and therefore eliminate several square roots. Notice also that the modulus is
dependent on θ0 , the maximum latitude achieved by the particle rather than θmid , the
mid-latitude where u = 0. Using the the two conservation properties one can show that
k 2 takes the exact same form expressed in terms of θmid up to order uφ 3 evaluated at the
mid-latitude. This should avoid any objections to comparing our results to an f -plane
model where one assumes to provide the value of f at the center of the oscillation.
Solutions
There are three cases we need to consider to evaluate equation (B.18): k 2 < 1,
k 2 = 1, and k 2 > 1. As can be seen from the form of k 2 , each of these cases involves
starting the particle with successively greater energy.
Case k 2 < 1
When k 2 < 1, the integral in equation (B.18) is exactly the definition of the inverse
elliptic function dn−1 (u, k).
θ1 (t) = sin−1 (sin θ0 dn(ωt sin θ0 , k))
(B.19)
147
For small values of k, dn(u, k) ≈ 1 − k
2
sin2 u
2
meaning the motion is periodic and stays near
the original latitude. The solution to φ(t) proceeds the same as in the case of the rotating
sphere where we integrate using the angular momentum equation, equation (B.15). In our
case, this equation becomes,
2
φ(t) = cos θ0 (φ̇0 + ω)
�
1
dt − ωt + φ0 .
1 − sin θ0 dn2 (ω � t)
2
We can rewrite dn2 u in terms of sn2 u using the relation dn2 u = 1 − k 2 sn2 u,
2
φ(t) =ω(1 + uφ ) cos θ0
1 + uφ
=
sin θ0
�
sin θ0 ωt
0
�
t
0
dt
cos2 θ
0
+
k 2 sin2 θ
0 sn
2 (ω � t)
− ωt + φ0
ds
− ωt + φ0
1 + k 2 tan2 θ0 sn2 s
(B.20)
The integral is exactly the definition of an elliptic integral of the third kind, see
[Byrd et al., 1971]. Thus,
φ1 (t) =
�
(1 + uφ ) �
Π am (sin θ0 ωt) , −k 2 tan2 θ0 , k − ωt + φ0
sin θ0
(B.21)
In this case, the particle oscillates between two values of θ, namely cos−1 ((1 + uφ ) cos θ0 ) <
θ < θ0 , see figure (B.1).
Case k 2 = 1
Equation (B.18) reduces to the usual integral for the inverse sech function. Perhaps
unsurprisingly, this can can also be seen from equation (B.19) because dn(u, 1) = sech(u).
θ2 (t) = sin−1 (sin θ0 sech(ωt sin θ0 , k))
The integral for φ(t) follows from the angular momentum equation again
φ(t) = ω cos θ0
�
1
dt − ωt + φ0
1 − sin θ0 sech2 (sin θ0 ωt)
2
(B.22)
148
FIGURE B.1: Particle path in case 1, where the particle was launched from 45 N due east
with φ̇ω0 = 0.18.
and can be found by first separating the equation,
φ(t) = ω cos θ0
� �
�
sin2 θ0
dt +
dt − ωt + φ0
cosh2 (sin θ0 ωt) − sin2 θ0
and then using the substitution u = tan θ0 tanh (sin θ0 ωt),
φ(t) = ω cos θ0
�
�
1
−1
t+
tan (tan θ0 tanh (sin θ0 ωt)) − ωt + φ0 .
ω cos θ0
(B.23)
This is a fairly special case that requires a launch of the particle at precisely the
right velocity, uφ = sec θ0 − 1. In this case the particle hits the equator, essentially
remaining in orbit around the equator for its life time, see figure (B.2).
Case k 2 > 1
The modulus k of Jacobi elliptic functions are generally taken to be 0 < k < 1, but
in this case k exceeds 1. We can work around this issue by using the reciprocal of the
149
FIGURE B.2: Particle path in case 2, where the particle was launched from 45 N due east
with φ̇ω0 = 0.41.
1
k,
instead. If we factor a k 2 out of the radical in equation (B.18), then the
�
�
integral is exactly the definition of the inverse of cn u, k1 . This can also be seen directly
�
�
from equation (B.19) using the transform dn(u, k) = cn ku, k1 .
modulus,
θ3 (t) = sin
−1
�
�
1
sin θ0 cn ωt sin θ0 k,
k
��
(B.24)
The φ(t) solution for this case proceeds in the same way as the k 2 < 1 case and yields,
�
1
φ(t) = cos2 θ0 (φ̇0 + ω)
dt − ωt + φ0
1 − sin2 θ0 cn2 (ω � tk)
�
�
(1 + uφ )
1
2
φ2 (t) =
Π am (k sin θ0 ωt) , − tan θ0 ,
− ωt + φ0
k sin θ0
k
(B.25)
(B.26)
In this last case the particle now crosses the equator and oscillates between θ0 and
−θ0 . This solution, figure (B.3) looks very similar to the solution on the rotating sphere,
but with elliptic functions.
150
FIGURE B.3: Particle path in case 3, where the particle was launched from 45 N due east
with φ̇ω0 = 0.60.
Frequency of Oscillation
We can compare the resulting frequency of oscillation that we’ve found to the classical results. Classically, the frequency occurs at f = 2ω sin θ. In our case, if we take
the argument of equation (B.19) we can determine the frequency of oscillation. Elliptic
functions do not oscillate with 2π, but the dn function is in fact 2K(k) periodic where
K(k) is the complete elliptic function and dependent on the parameter k. Also the coriolis
parameter as written is an angular frequency and has a 2π dependence that we need to
eliminate, so we’ll compare using the period. For our function it is true that
�
dn (ω sin θ0 t) = dn ω sin θ0
when
2πω sin θ0
f
�
2π
t+
f
��
= 2K. Therefore,
f (θ0 ) =



 πω sin θ0
K(k)


θ0
 kπω sin
−1
K(k
)
�
θ0 > sec−1 1 +
otherwise.
φ̇0
ω
�
,
(B.27)
151
8Frequency in units of w<
2
1.5
1
0.5
0.25
1
0.5
0.75
1.25
Latitude in radians
1.5
FIGURE B.4: Coriolis parameter f = 2ω sin θ and the new coriolis parameter based on
an initial eastward velocity of 25 meters per second.
where we’ve explicitly indicated that K is a function of the parameter k.
A Taylor expansion for small k of K(k) is K(k) =
π
2
�
1+
k2
4
�
+ O(k 4 ) . For values
of uφ = 0, no particle motion, this reduces so the classical f = 2ω sin θ. However, for non�
� 2 ��
φ̇0
cβ
zero ω , the inertial oscillation frequency differs, f (θ0 ) = f0 1 − f 2 + O Rc2 f 2 .The
0
0
f -plane model therefore predicts too high of a frequency.
In figure (B.4) the new coriolis parameter is plotted along side the classical coriolis parameter. The initial velocity to the east has been scaled to be 25 meters per second
at all latitudes. The cusp is located at θ0 = sec−1 (1 + uφ ), where our second solution
applied.
Particle Drift Speeds
Just looking at the figures depicting the particle motion it is clear that in addition
to oscillation, the particle also translates in φ. This drift speed can be easily characterized
by performing the same integration for φ as before, but instead taking the integral over
152
Drift speed in meters per second, u0=25 m s
20
10
0.25
0.5
0.75
1
1.25
1.5
-10
Latitude in radians
FIGURE B.5: Eastward drift speed of the particle launched at 25 meters per second to
the East.
the whole period and dividing by the length of the period.
�
�
ω + φ̇0
�
�
¯
φ̇1 (t) =
Π −k 2 tan2 θ0 , k − ω
K (k)
¯
φ̇2 (t) =ω(cos θ0 − 1)
�
�
�
�
ω + φ̇0
1
¯
2
� � Π − tan θ0 ,
φ̇3 (t) =
−ω
k
K k1
(B.28)
(B.29)
(B.30)
Figure (B.5) shows eastward drift speed for a particle launched 25 meters per second to the
East. Notice that four basic regimes exist. In the area where the first solution (θ1 (t), φ1 (t))
is valid, the particle only drifts westward. The point where (θ2 (t), φ2 (t)) is valid appears
as a cusp in the figure. The value isn’t so much an average φ̇, but an asymptote. The
third regime is particular interesting. Evidently the third solution (θ3 (t), φ3 (t)) still moves
westward over a very small latitude range. The fourth regime is when the third solution
moves eastward. As we approach the equator the particle eventually moves at exactly the
initial speed we launch it.
Approximating the mid-latitude westward drift speed using a Taylor expansion, we
2
find that ū = − cfβ20 . This approximation has been shown by other means in [Ripa, 1997]
0
and [Paldor and Sigalov, 2001].
153
Oscillation Length Scale
We can estimate and find the true size of the oscillation by using that
�
�
R (θmax − θmin ) = R θmax − cos−1 ((1 + uφ ) cos θmax )
(B.31)
where conservation of angular momentum was used to determine θmin . Performing a
�
�
Taylor expansion, we can see that the radius goes like uf00 1 + uφ cot2 θ0 + ... . To lowest
order this matches the f -plane oscillation length scale, but is again modified at the next
order.
Approximate Motion
Finally, the approximate motion of the particle path can be estimated. We find that
y(t) = R (θ1 (t) − θ0 )
uφ �
u0 �
=
1+
(1 − cos f0 t)
f0
2
(B.32)
(B.33)
and therefore the y(t) looks a lot like the f -plane. Unfortunately though, we are unable
to obtain a an approximation for x(t).
Conclusions
The exact solutions to the inertial oscillation problem are computed. The particle
drift speed, oscillation length scale, frequency of oscillation and approximate motion of
the particle all show deviations from the f -plane approximations.
154
BIBLIOGRAPHY
Byrd et al., 1971. Byrd, P., Friedman, M., and Byrd, P. (1971). Handbook of elliptic
integrals for engineers and scientists. Springer Berlin.
Paldor and Killworth, 1988. Paldor, N. and Killworth, P. D. (1988). Inertial trajectories
on a rotating earth. Journal of the Atmospheric Sciences, 45(24):4013–1019.
Paldor and Sigalov, 2001. Paldor, N. and Sigalov, A. (2001). The mechanics of inertial
motion on the earth and on a rotating sphere. Physica D, 160:29–53.
Pennell and Seitter, 1990. Pennell, S. A. and Seitter, K. L. (1990). On inertial motion
on a rotating sphere. Journal of the Atmospheric Sciences, 47(16):2032–2034.
Ripa, 1997. Ripa, P. (1997). “inertial” oscillations and the β-plane approximation(s).
Journal of Physical Oceanography, 27:633–647.
155
APPENDIX C Disk on the Earth’s Surface
Introduction
One of the three solutions we found for the point mass particle on the geopotential surface was a westward drifting oscillation that did not cross the equator. Let’s
imagine a person standing at the center of that oscillation and swinging the particle
by a tether. If the particle has any spatial extent and were, say, a hockey puck, the
person swinging the puck would notice that the puck is itself spinning about its own
axis. For this example the hockey puck is attached at one end by the tether. When
the person first starts swinging the puck, the person is looking due North at the
hockey puck. The tether is attached to the hockey puck on the South side. After half
an oscillation, the person is now facing due South looking at the hockey puck, and
the tether is attached to the hockey puck on the North side. So after one complete
oscillation, the hockey puck again has its tether leaving from the South side, and
the puck has rotated about its own axis one time. This is a new angular momentum
and if the hockey puck were instead a fluid parcel, we might, by analogy, call this vorticity.
Let’s denote the original angular momentum we found as Lφ and this new angular
momentum about the person holding the tether, Lψ . The quantity Lφ , as we found
before, is a conserved quantity; however, Lψ is not.
In this section, we will explicitly add a new degree of freedom by giving the particle
spatial extent, in the shape of a hockey puck, and allowing it to rotate about its own axis.
The hypothesis being that the hockey puck will now be able to move to the east or west
without having to oscillate in θ because it can oscillate about its own axis, ψ.
156
Lagrangian Formulation
The point particle is now a hockey puck with moment of inertia I2 about the two
non-symmetric axes, and moment I3 about the symmetric axis. In all of the calculations
the moments are actually divided by m, the mass of the particle, but we will absorb this
into the constant itself. The vector ψ̇ points through the middle of the hockey puck while
its magnitude indicates how fast the puck spins. In the inertial frame the lagrangian is
�
1
1
1
1 �
Linertial = ṙ2 + r2 cos2 θφ̇2 + r2 θ̇2 + I2 θ̇2 + φ̇2 cos2 θ
2
2
2
2
�2 GM
1 �
+ I3 ψ̇ + φ̇ sin θ +
.
2
r
(C.1)
Using the same mapping as before to move to the rotating frame, we apply φ �→ φ + ωt,
and find that,
�
� 1
�
�
�
1
1
1 �
Lrot = ṙ2 + r2 cos2 θ φ̇2 + 2ω φ̇ + ω 2 + r2 θ̇2 + I2 θ̇2 + φ̇2 + 2ω φ̇ + ω 2 cos2 θ
2
2
2
2
�
�
�2 GM
1 �
+ I3 ψ̇ + φ̇ + ω sin θ +
.
(C.2)
2
r
Conserved Quantities
This model now has three conserved quantities, the new conservation law being
associated with ψ.
�
�
Lψ = I3 ψ̇ + (φ̇ + ω) sin θ
(C.3)
The quantity Lφ takes a slightly different form, namely,
�
�
�
�
Lφ = r2 + I2 cos2 θ φ̇ + ω + Lψ sin θ.
(C.4)
Finally, the energy is also conserved,
2
� 2 (Lφ − Lψ sin θ)2
1 2 1� 2
1 2 2
1 Lψ
2
E = ṙ +
r + I2 θ̇ +
+ r ω cos θ − ωLφ +
2
2
2 (r2 + I2 ) cos2 θ 2
2 I3
(C.5)
where we have again used the fact that this is on a geopotential surface to replace the
gravitational term. At this point it becomes clear that a reasonable simplifying assumption
157
would be that I2 � r2 . That is to say that the height of the hockey puck is much smaller
than the radius of the earth. Constraining the particle to the surface of a sphere, the
energy is now,
2
(Lφ − Lψ sin θ)2 1 2 2
1
1 Lψ
2
E = R2 θ̇2 +
+
R
ω
cos
θ
−
ωL
+
.
φ
2
2R2 cos2 θ
2
2 I3
(C.6)
We can also rewrite the lagrangian using some of the conserved quantities and the simplifying assumptions.
(Lφ − Lψ sin θ)2 1 2 2
1
Lrot = R2 θ̇2 +
− R ω cos2 θ
2
2R2 cos2 θ
2
(C.7)
Solutions
The primary motivation for adding this new degree of freedom was to find solutions
where the particle can move without oscillating in θ. So let’s look for solutions where
θ̈ = 0 or equivalently,
∂L
∂θ
= 0.
2 (Lφ − Lψ sin θ) (−Lψ cos θ) (Lφ − Lψ sin θ)2 (−2 sin θ)
+
+ ω 2 R2 sin θ cos θ = 0 (C.8)
2R2 cos2 θ
2R2 cos3 θ
(φ̇0 + ω)(−Lψ cos θ) − (φ̇0 + ω)2 (r2 cos2 θ)(tan θ) + ω 2 R2 sin θ cos θ = 0 (C.9)
Where we used the definition of Lφ and Lψ to make appropriate repacements. Factoring
out an ω, this becomes,
�
φ̇0
1+
ω
�2
Lψ
+ 2
R ω sin θ
�
The equation is a simple quadratic in 1 +
φ̇0
ω
�
�
Lψ
φ̇0
= −1 −
±
2
ω
2R ω sin θ
If we make the simplifying assumption that
φ̇0
1+
ω
�
− 1 = 0.
(C.10)
, so can be easily solved. It’s solution is
�
L2ψ
4R4 ω 2 sin2 θ
Lψ
R2 ω sin θ
+1
(C.11)
� 1 (which is reasonable when I3 �
r2 ), then one solution is approximately,
Lψ
φ̇0
=− 2
.
ω
2R ω sin θ
(C.12)
158
In order to compare this solution with the solution for the point mass particle, we need to
find equivalent initial conditions. In the case of the point mass particle we had to set an
initial velocity, c, whereas here we need to set a length scale, I3 . From the energy equation
this is just saying that 12 c2 =
2
1 Lψ
2 I3 .
Lψ can be slightly simplified if we assume that ψ̇ and
˙ 0 is small compared to ω. With these assumptions Lψ = 1 I3 f , where f = 2ω sin θ, the
phi
2
2
2
coriolis parameter. Using this in the energy equation, I3 = 4 fc 2 , so Lψ = 2 cf . Using this
initial condition, the solution we found is
φ̇0 = −
c2
.
R2 sin θf
(C.13)
Angular velocities aren’t very insightful, so let’s convert this into an equivalent u0 by
multiplying through with R cos θ. In this case we have,
u0 = −
where β =
2
R
c2 β
,
f2
(C.14)
cos θ.
Conclusions
Given the same energy, we find that the puck translates westward at the same speed
as a particle.
159
APPENDIX D Eddy Propagation Speeds with the FP Equation
The same set of monopole experiments modeled in Chapter 2 with the quasigeostrophic potential vorticity equation
�
�
∇2 ηt̄ − ηt̄ + ηx̄ + β −1 · Jˆ η, ∇2 η = 0
(D.1)
were repeated with the Flierl-Petviashvili equation,
�
�
ηt̄ − ηt̄ + ηx̄ (1 + η) + β −1 · Jˆ η, ∇2 η = 0.
(D.2)
Because the FP equation (D.1) lacks the cyclone-anticyclone symmetry found in the QG
equation (D.1), the model runs were conducted for eddies of both polarities. The meridional and zonal propagation speeds at latitude 24 are shown in figure (D.6), while those
for latitude 35 are shown in figure (D.7).
A number of preliminary conclusions can be drawn from figures (D.6) and (D.7).
First, it is clear that generally speaking the zonal propagation speed of anticyclonic eddies
is faster than cyclonic eddies. Conversely, the meridional propagation speed is faster for
cyclonic eddies than for anticyclonic eddies, although the effect is less pronounced. This
result is in contrast to quasi-geostrophic dynamics which has the same (absolute) propagation speeds for cyclonic and anticyclonic eddies. Second, unlike the quasi-geostrophic
model, some of these eddies are able to propagate zonally faster than the maximum group
speed. This especially clear for the large (in both length scale and amplitude) anticyclonic
eddies at latitude 35.
The results may, at least in part, address some of the deficiencies with quasigeostrophic theory for explaining the satellite altimetry eddy observations. In particular,
the eddies under FP dynamics are not systematically slower than the maximum group
velocity, unlike like QG. Although whether propagation speedup is strong enough to account for observations should be tested in another eddy seeding model. Additionally, the
160
(a) Linear best fit cx (A) = 4.5A−1 − 4.5 cm/s.
(b) Linear best fit cx (A) = −3.7A−1 − 4.2 cm/s.
(c) Linear best fit cy (A) = −3.4A−1 − 0.13 cm/s. (d) Linear best fit cy (A) = −3.0A−1 + 0.18 cm/s.
FIGURE D.6: Propagation speeds of Gaussian initialized eddies at latitude 24 with FP
dynamics. The dashed grey line is the maximum group velocity in linear theory. Points
are colored with the eddy length (in km). The inverse amplitude best fits lines are drawn
in black.
161
(a) Linear best fit cx (A) = 1.2A−1 − 2.1 cm/s. (b) Linear best fit cx (A) = −0.93A−1 − 1.9 cm/s.
(c) Linear best fit cy (A) = −0.79A−1 − 0.041 (d) Linear best fit cy (A) = −0.75A−1 + 0.058
cm/s.
cm/s.
FIGURE D.7: Propagation speeds of Gaussian initialized eddies at latitude 35 with FP
dynamics. The dashed grey line is the maximum group velocity in linear theory. Points
are colored with the eddy length (in km).The inverse amplitude best fits lines are drawn
in black.
162
asymmetry in propagation speeds between eddies of different polarities may account for
the asymmetry in observed meridional deflection.