ATMOSPHERIC INTERACTIONS WITH
GULF STREAM RINGS
by
William K. Dewar
B.S. the Ohio State University
(1977)
S.M. the Massachusetts Institute of Technology
(1980)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
and the
WOODS HOLE OCEANOGRAPHIC INSTITUTION
October 1982
Signature of Author_
_
_
_
__
Department of Meteorology and Physical Oceanography,
Institute of Technology and the Joint Program in
Institute of Technology/Woods Hole
Massachusetts
Institution, October, 1982.
__
Massachusetts
Oceanography,
Oceanographic
Certified by
i
Thesis Supervisor
Accepted by
ceanography, Massachusetts
Chairman, Jo t Committee for Physical
Institute of Vechnology/Woods Hole Oceaqographic Institution.
FRO
C y
LR.ITARES
Page -2-
ATMOSPHERIC INTERACTIONS WITH GULF STREAM RINGS
by
William K. Dewar
Submitted to the MIT-WHOI Joint Program
in Physical Oceanography on 8 October, 1982
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
ABSTRACT
are
Rings
Stream
Gulf
concerning
problems
different
Four
The first deals with the particle trajectories of, and
considered.
advection-diffusion by, a dynamic model of a Ring. It is found that the
streaklines computed from the assumptions that the Ring is a steadily
its
describe
accurately
form structure
and permanent
propagating
produces
Ring
the
of
field
The dispersion
Lagrangian trajectories.
east-west asymmetries in the streaklines, not contained in earlier
kinematic studies, which are consistent with observed surface patterns.
In the second problem, we compute the core mixed layer evolution of both
warm and cold Rings, and compare them to the background SST, in an
We demonstrate that
effort to explain observed SST cycles of Rings.
while cold Rings do
identity,
surface
warm Rings retain their anomalous
states of the
atmospheric
local
the
both
in
not, because of differences
structures
layer
mixed
typical
the
and
Slope
the
and
Sargasso
forced
the
concern
problems
fourth
and
third
The
each.
appropriate to
evolution of Gulf Stream Rings as effected by atmospheric interactions.
The
First, we compute the forced spin down of a Gulf Stream Ring.
variations in surface stress across the Ring necessary to spin it down
are caused by the variations in relative air-sea velocity, of which the
From numerical simulations, we find
stress is a quadratric function.
the forced decay rates are comparable to those inferred from Ring
In the final problem, it is suggested that a substantial
observations.
fraction of meridional Ring migration is a forced response, caused by
The warm central
Ring SST and the temperature dependence of stress.
producing
stress,
enhanced
of
regions
waters of anticyclonic Rings are
the
shifts
which
south,
the
to
upwelling to the north, and downwelling
cyclonic
for
computed
is
shift
southward
similar,
A
Ring to the south.
Rings with cold centers, which tends to reconcile their numerically
computed propagation with observations.
Page -3-
TABLE OF CONTENTS
Title Page ................................................
Abstract ......... .......
Chapter I.
2
......
*.....*.......
Table of Contents ......................
.................
Introduction ..................................
Ring Observations and Description ...............
* ... ......... ******
*
Contents ..................
Chapter II.
Preliminaries ................................
3
7
8
10
15
15
a.
Introduction .................................
b.
The Quasi-Geostrophic Horizontal Structure
Equation .................
1
*...*...............
The Equivalent Barotropic Equation .............
A Discussion of Baroclinic Instability .........
c. Advection-Diffusion of a Passive Scalar ...........
15
25
26
27
d.
The Mixed Layer ................................
28
e.
Conservation of Mass and Thermodynamic Energy ...
Momentum Equations .............................
Quasi-Geostrophic Scaling .......................
Ekman Pumping ....................................
Energy Equation .................................
Wind Wave Breaking and Penetrative Convection ...
The Froude Number and Its Value .................
Numerical Techniques .............................
29
31
32
34
35
38
39
40
Particle Trajectories in Numerical Gulf
Stream Rings ..................................
41
a.
Introduction .......................................
..
Ring Model ................*. ...............
41
45
b.
46
49
49
62
64
70
e.
Kinematic Models .................................
Tracer Diffusion in Kinematic Models ............
Tracer Homogenization on Closed Streamlines .....
Advection and Diffusion in a Dynamic Ring Model ..
Dynamic 'Streaklines' ...........................
The Importance of the Dispersion Field ..........
Advection-Diffusion Using Dynamical Advection
Fields ..........................................
Ciritcal Contour ................................
...................
Exterior Streaklines
...............
Considerations
Potential Vorticity
...................................
Exterior
Ring
Implications ....................................
f.
Summary .........................
Chapter III.
c.
d.
........
72
75
75
80
83
84
87
Page -4-
Chapter IV.
An Annual Mixed Layer Model with
Application to Gulf Stream Rings .............
a.
Introduction .......................
Background ........................
b.
c.
An Annual Mixed Layer Model ........
Limit Cycle Calculations ...........
'Typical' Mixed Layers ............
Adjustment Calculations ............
d.
e.
A Bulk Mixed Layer Model .................. 121
121
123
123
124
124
127
The Equations and the Forcing Functions ......
Meteorological and Solar Data ................
Winds ....................
b.
90
90
95
101
103
109
117
Summary ...........................
Appendix A.IV.
a.
~
.........
Air Temperature .............................
Solar Heating ...........................
Initial Experiments with the Thompson Model ..
Appendix B.IV.
90
The Sensitivity of Mixed Layer
Development to Buoyancy Flux .......... ....131
The Reformation of the Thermocline ............. 131
Wintertime Mixed Layers
Appendix C.IV.
........................ 135
Verification of the Annual Mixed Layer
.........................
Equations
...
139
Choice of h ..................................... 139
Validation ...................................... 140
Page -5-
The Wind Forced Spin Down of Gulf Stream
Rings ......................................... 144
Chapter V.
a.
b.
c.
d.
Introduction ..................................... 144
Observations of Ring Decay ...................... 145
Ekman Pumping .................................... 150
Planetary Wave Spin Down ........................ 153
Nonlinear Vortex Spin Down ..................... 157
Barotropic Mode Scaling ......................... 157
Parameters ................................. 158
Unforced Results
e.
f.
g.
.....
....
.
*159
...........
..... 160
Forced Results ...........................
167
..............
Forcing
The Relative Importance of
....169
Parameter Variations ....................
The Spin Down Mechanism .......................... 171
176
....
Summary ...................................
Appendix A.V.
Chapter VI.
Wind Stress in the Presence of Surface
Flows ...................................... 178
Southward Ring Propagation as a Consequence
of Surface Temperature Anomalies ............. 184
a.
Introduction ..................................... 184
b.
Scale Estimates
c.
d.
e.
............................ 187
The Coefficient of Drag ......................... 187
The Ekman Divergence ............................ 189
The Pumping and Its Effects ..................... 189
Governing Equations .............................. 192
Numerical Results ................................ 195
Parameter Studies ............................... 204
.................................... 205
Discussion
Integral Constraints ............................ 206
Zonal Propagation ............................... 209
Meridional Propagation .......................... 210
f. Potential Vorticity Budgets .....................
..
.................
g. Summary .....
211
216
Page -6-
VII.
Summary .................
Suggestions
........................... 220
................
.................
..223
References ................................................ 226
Acknowledgements .......................................... 230
Page -7-
INTRODUCTION
CHAPTER I.
the
shed by
vortices
intense
are
Stream Rings
Gulf
Stream,
Gulf
characterized by velocities up to 150 cm/sec and diameters of about 100
They are commonly found in the Slope Water and the Sargasso, and as
km.
such
Rings transport water between the Slope Water and the Sargasso,
region.
which has led scientists to
in
both
of
2
2
estimated at 1 m /sec , which is the
(the Ring Group,
atmosphere
induced salt and heat flux.
Ring
the
is
of
focus
gradients,
that
likely
due to the
Ring-
to
apply
statements
current
enhancing
Rings
produce
their environment
and
observational
their
of a
diffusive
important
are
Rings
Whether
oceans.
or alter
these estimates,
large as
way,
the
Similar
the
been
has
Rings
by
In addition, the powerful velocities
very
is
of
picture
scale
large
it
Thus,
transport.
caused
example,
same magnitude as that
1981).
tracer
existing
strain
can
For
regions.
Sargasso
the
to
flux
vorticity
potential
they are a dominant component
suggest that
budgets
energy
and
heat
the
in either
dependent phenomena
time
energetic
the most
constitute
to
the
effects as
in a significant
theoretical
effort
(Richardson, 1980).
areas
The
knowledge
transport.
is
essential
Recent
modeling efforts
the
body
of
evolution of
water.
Although
evolving
several
of
questions,
decay,
in
which
our
propagation,
and
1979;
Mied
(McWilliams and Flierl,
Ikeda, 1981; Nof,
freely
these
Ring
include
incomplete,
and Lindemann, 1979;
on
to addressing
1980; Flierl, 1982) have centered
structures
imbedded
these studies
in
a resting
have mentioned
the
Page -8-
potential importance of mean state advection and external forcing, there
have been only a few attempts at including shear (Flierl, 1979) and wind
stress
(Stern, 1965)
in eddy
In the present thesis, we
calculations.
will consider how atmospheric forcing affects the
and
demonstrate
properties
are
several
that
significantly
their
of
influenced
evolution of Rings,
important
oceanographically
air-sea
by
exchange.
In
particular, we shall see that Ring decay and propagation are affected by
wind forcing, and that the evolution of Ring surface waters is sensitive
to diabatic heating.
Also, because many of
the processes involved in
these problems are more naturally discussed in a Lagrangian frame, and
because of
the importance of Ring advection on their surroundings, we
have computed the particle trajectories of a Gulf Stream Ring.
Ring Observations and Description-
Rings are distinguished from the mesoscale variability of the North
Atlantic primarily in two ways.
process:
Gulf
Stream meanders
First, they undergo a unique formation
grow
to
subsequently separate from the current.
sizeable volume of distinctive water.
to occur on both sides
of
finite
amplitude,
close,
and
Second, Rings carry with them a
Ring production has been observed
the stream, producing vortices of positive
(cyclonic) rotation to the south and negative (anticyclonic) rotation to
the north.
Similar structures are found in the vicinity of most major
current systems (Hamon, 1960; Nilsson, Andrews, and Scully-Power,
Kawai,
1979),
although presently,
the North Atlantic.
the literature is most
1977;
complete for
Page -9-
The
first well documented long term observation of a single Ring
the same
(cyclonic) is due to Fuglister (1977), who was able to track
Ring
persist
Rings
catalogued.
many
and
1980)
(Richardson,
Rings
several
date,
To
months.
six
for
their
as
recognizable
properties
physical
common
of
tracked
been
have
structures,
coherent
literally as closed loops of flow, for years at a time (Parker, 1971).
They translate toward the west-southwest at speeds of about 5 cm/sec but
can
Stream.
eastward
rapid
exhibit
As many as
are
and
with
the
Gulf
the
have been
They are formed at a rate of about 7 Rings per
frequently
into
reabsorption
interacting
10 cyclonic and 6 anti-cyclonic Rings
observed to coexist.
year
when
motion
removed
Gulf
Stream
from
the
general
(Richardson, 1980).
circulation
For
by
more
a
complete descriptive review of Rings, see Lai and Richardson (1977).
During
formation,
of
pieces
large
water
are
trapped within
the
closing meanders which results in Rings having a peculiar water mass
For example, a cyclonic Ring in the Sargasso Sea will have
composition.
an interior consisting of Slope Water.
The strong temperature contrasts
between the Slope water and the Sargasso have led to the now standard
labels of
'cold core Ring' for those found in the Sargasso, and
core Ring' for those in the Slope.
some other
terms which will
'warm
The formation process also suggests
be used in the present manuscript.
The
region into which the newly formed Ring moves will be referred to as the
'host region', and the area from which the core waters
be called the
'parent region'.
originated will
As an example, the Sargasso Sea is the
host region of a cold core Ring and the Slope Water the parent region.
Page -10-
Contents-
The following brief summaries of each chapter will serve as a guide to
the new results in this thesis.
Chapter III
(1981)
Flierl
the
computed
particle
trajectories
of
a steadily-
propagating, axisymmetric pressure pattern with closed streamlines.
By
applying this model to Rings, he was able to make many useful statements
with regards to the structure of particle tracks, trapped zone size, and
This study was
averaged Lagrangian velocities.
employed
a
velocity
field
which
turned
purely kinematic, and
to
out
be
dynamically
inconsistent, although it did come from an analysis of Ring data (Olson,
1980). In Chapter III, we conduct a Lagrangian analysis of a dynamically
evolving Ring, the equivalent barotropic Ring model originally proposed
by McWilliams
and
Flierl
Comparisons
(1979).
between the
dynamical
model streaklines and those of the kinematic study are made which point
out where the earlier calculations adequately describe particle motion
and where improvements are needed.
The particle
trajectories
of
the
dynamic Ring are investigated in terms of potential vorticity, and the
importance of
III
with
an
the dispersion field is discussed.
example
of
Ring
interaction
with
We conclude Chapter
tracer
boundaries,
performed with a view towards modeling Ring-thermal front interactions.
Page -11-
Chapter IV
Rings generally
images,
In satellite infra-red
defined pools of anomalously warm or cold water.
to
cycles
be observed by the remote
show up as well-
Thus, one of the first
that of
sensing program was
that cold
It is now documented
annual Ring sea surface temperature.
the
core Ring surface waters do not survive beyond their first summer as an
identifiable cold pool (the Ring Group, 1981); however, warm core Rings,
with
the possible
exception of
throughout
remain visible
summertime,
From XBT data, we find
their lifetime in satellite infra-red images.
evidences of strong air-sea exchange and deep mixed layers in warm core
Rings, and a curious lack of unusual surface water development in cold
core Rings.
In Chapter IV, we consider mixed layer evolution on the
annual time scale, with particular emphasis on explaining the features
of Ring
SST cycles.
dimensional
Using a one
surface
forced response of the core
of
flank, demonstrating what aspects
of
layer
model, we compare
the
of
its
a Ring
the
idea
pervading
that
temperature
responsible for the sea surface
apply
the
infra-red
results
images.
one-dimensionality,
anomalies.
of
this
the
is
it
study
to
The
model,
suggests
how
temperature
the observed surface
This view differs
field can be attributed to local air-sea exchange.
from
that
to
the
dynamics
which
are
(SST) behavior.
We also
of
satellite
interpretation
within
to
Ring
the
objectively
restrictions
interpret
of
SST
Page -12-
Chapter V
Rings
persist
for
years
a
time
(Lai and
Richardson,
a recognizable aging process
although they do experience
1981).
Maillard, and Sanford,
at
estimates
Various
been made using observed subsidence of isotherms
1977),
(Richardson,
decay rates
of
(Parker, 1971)
have
or loss
of potential energy (Cheney and Richardson, 1976) and suggest lifetimes
of
years.
two-three
roughly
One
of
the
evolution concerns the method by which Rings lose their energy.
concluded
by McWilliams
Lindemann
(1979),
strongly
influenced
and
Flierl
as well
(1979),
as
Ring
of
problems
classic
It was
Meid
by
and
that vortex decay in their numerical experiments was
viscosity
by
that
and
dominant
usually
the
dispersive decay mechanism was in large prevented by the strength of the
flow.
The lack of
making
any
definitive
recognition of
closure theory
a well-founded
with
statements
the importance
of
regards
prevented them from
to
beyond
decay
a
the weak non-conservative processes.
In Chapter V, we investigate the possibility that Ring spin down is a
result
of
Ekman
divergence
transfer at the sea surface.
driven
by
local
of stress.
momentum
of
The bulk formula for stress is a quadratic
function of the relative air-sea velocity;
intense surface velocities
variations
therefore,
the presence
of
can induce local, non-negligible, gradients
The dissipative nature
of
the forcing, similar
to bottom
friction, emerges from the calculation of the Ekman pumping; one of the
more useful results is the analytical expression for what corresponds to
the
coefficient
of
viscosity multiplying the
series of numerical experiments,
frictional operator.
A
including the pumping, are performed
and the results compared to oceanic observations of Ring decay.
Page -13-
Chapter VI
In Chapter VI, we consider the effects of the local variations in
stress
of
dependence
by
caused
Ring
a
on
on
drag
of
coefficient
The
field.
temperature
surface
aerodynamic
bulk
the
its
the
temperature difference between the air and water has been documented by
temperature
the
on
contrasts
variations
0(50%)
produces
and
(1968),
Deardorff
of
order
a
few
in
stress
degrees
for
Centigrade.
Ring surface temperature anomalies are such that both Ekman suction and
are
pumping
produced,
forcing
the
Ring
to
the
south.
We
present
numerical experiments, which include surface temperature anomalies, to
demonstrate this effect and discuss the dynamical balances which account
for the meridional propagation.
McWilliams and Flierl (1979) point out
that according to quasi-geostrophic dynamics,
freely evolving cyclonic
vortices (cold core Rings) move northward; a result which is counter to
One of the interesting results of Chapter VI is that
most observations.
both warm and cold core Rings are compelled towards southward motion,
which brings the predicted propagation of cold Rings more into accord
with observations.
Chapters II and VII
The relevant
equations are derived and catalogued in Chapter II.
First, we discuss a two degree of freedom quasigeostrophic model in both
and
layered
review
the
followed
by
continuously
validity
of
derivations
stratified
modal
the equivalent
of
the
forms
barotropic
(Flierl,
1978)
equation.
advection-diffusion
equation
This
and
and
is
the
Page -14-
basic mixed layer equations.
Chapter VII contains a summary together
with a discussion of future research topics suggested by this work.
Page -15-
PRELIMINARIES
CHAPTER II.
Introduction-
II.a
purpose
The
the content
originally
few pages
the next
derivation
the
review
of
formulated
be
will not
quasi-geostrophic
the
of
the
Necessarily,
some
example,
we
equations
as
for
new;
modal
other
the
On
(1978).
by Flierl
catalog
and
derive
to
equations which we will frequently use.
fundamental
of
is
chapter
this
of
rather
hand, a
original derivation of Ekman pumping as the upper boundary condition on
the mesoscale will
In
layer.
contained
all
within
in the
be presented
each
will
we
sections,
equation.
the
out
point
The
dealing with
section
relevant
already
reader
the
mixed
physics
with
familiar
general areas of quasi-geostrophy, advection-diffusion, and mixed layers
can skip directly to Chapter III.
notation employed in this
It
noted however that the
should be
standard,
chapter will become
thus reference
to the tables and sections contained herein should resolve any questions
with respect to symbol definition.
II.b
The Quasi-Geostrophic Horizontal Structure Equation-
The
basic equation describing the dynamics of
the mesoscale
is
quasi-geostrophic psuedo-potential vorticity conservation equation.
dimensional form, this equation is:
2
p + J-
- T--z
)+foY y] = 0
o
(
7
'
-z
" f °
0
]
=
0
Eq.
II.1
the
In
Page -16-
Table II.1
Symbols and Definitions
Symbol
Meaning
Environmental Symbols
t
x
........
........
y ........
z ........
N2 . .. . . . .
fo .......
........
........
. ........
mo
. . . . . . .. . .
g
Time
Zonal coordinate
Meridional coordinate
Depth
Buoyancy frequency
Coriolis parameter
Mixed layer dissipation
Meridional Gradient of f
Coefficient of seawater thermal expansion
Energy equation coefficient
........
o *******
To .......
Gravity
Reference density
Reference temperature
k
Unit vertical vector
.......
D .........
Km ........
Passive scalar Decay rate
Passive scalar coefficient of diffusion
Scales and Nondimensional Parameters
Q ........
U ..........
L ........
H ........
Hi ........
Steepness=Uo/( L2 )
Velocity scale
Horizontal length scale
Depth scale
Average layer thickness
S
Burger number (=(NH) 2 /(foL)2)
Depth ratio (=H /H )
........
........
1
Ro ........
.......
'
fi .......
......
fi
Sijk see**
ijk
.....
2
Rossby number (=U/(foL))
Density step
Continuous separation constant
Layered separation constant
Continuous modal interaction parameter
Layered modal interaction parameter
Variables
u
v
........
........
Zonal velocity
Meridional velocity
w
........
Vertical velocity
ui
......
0*e
vi
P ........
P. .......
F
.... ..
1
Intermediate Layer zonal Velocity
Intermediate Layer meridional velocity
Pressure
Layer pressure
Continuous modal structure
Page -17-
Table II.1
Symbols and Definitions (continued)
Symbol
Meaning
Variables (continued)
Fj(i) .....
Layered modal structure
Continuous barotropic horizontal structure
***........
........ Continuous baroclinic horizontal structure
o........ * Layered barotropic horizontal structure
o
1 ........
........
Layered baroclinic horizontal structure
.- ........
b1 .
Rescaled barotropic horizontal structure
Density
Rescaled baroclinic horizontal structure
Intermediate layer buoyancy
Z
h
Level depth under intermediate layer
Mixed layer depth
S........
...
*...
........
e
d
A
F
........
........
........
.......
F. .......
. ........
T ........
Ta ........
ba ........
Entrainment rate
Isopycnal displacement
Passive scalar concentration function
Internal wave radiation stress
Turbulent density flux
Wind stress
Temperature of seawater
Temperature of air
buoyancy of air
Mathematical Operators and Symbols
Symbol
72
.......
Meaning
2
'/()x)
J(A,B) ....
AxB
curl(A) ...
(Ay)x -
fij
.......
-
+ 3/(4y) 2
Bx y
(Ax)y
Kronecker delta (=0 if i=j,
1 if i=j)
Page -18-
where d/dt, the substantial time derivative, is defined by:
1
a
dt +
d
)
J(P
f
1
9
--
o
(4
+
f
dt
P
).
---
~
Eq. II.2
x
y
y
d" X
o
relative vorticity, vortex
Eqs. II.1 and 2 describe exchanges between
stretching, and planetary vorticity, along the horizontal projection of
such
exchanges
those
that
a particle
trajectory,
vorticity.
For a complete derivation of
this
conserve
potential
equation, see Pedlosky
The proper vertical boundary conditions for Eq. II.1 are on the
(1979).
vertical velocity of the flow:
d/dt(Pz) = -N2 w
We will generally assume a flat bottom (w=0 at z=-H),
at z=0 and -H.
For horizontal boundary conditions,
but allow for a surface divergence.
we shall assume for numerical purposes a doubly periodic domain:
P(x+Lx,y+Ly) = P(x,y).
Non-dimensionalizing x and y by L, t by
(PL) -1 , u and v by
U,
P by foUL, w by U2H/(foL2), and z by H (see Table II.1) returns:
QJ(P,.)][
[-+
St
where
Q
=
U/( 3L2),
(NH)2/(foL)2 .
and
p +
S
S Jz
Jz
is
the
P ] +
x
+ P=0
Burger
0,
number,
Eq. 11.3
defined
The vertical boundary conditions become:
C)
+
[-:Jt
at z=0 and -1.
QJ(P,.) ] -- P
%z
=
-Sw
Eq. 11.4
by
S
=
Page -19-
If
linear
lower boundary
upper and
the
form
of
the
and
the
separable,
mathematically
becomes
II.1
Eq.
conditions are homogeneous,
vertical structure equation takes the Sturm-Liouville form:
j
z
1
( S
2
2/z
F )+
i
1
Eq. 11.5
F. = 0
1
with:
(Fi)z
where
the
0 at z = 0, -1,
-
i
constant,
separation
is
the
non-dimensional
Rossby Deformation Radius corresponding to the ith mode.
We normalize
c
the Fi according to:
FiFjdz = 6 ij.
-I
From Sturm-Liouville
we
theory,
know
the
set
of
functions
[Fil
is
complete, and therefore, we can write the pressure P as:
P(x,y,z,t) =
i(x,y,t)Fi(z)
Eq. 11.6
where
i =
PFidz.
In general we cannot differentiate with respect to z under the summation
sign in Eq. 11.6,
for in the
boundary conditions,
the interval (-1,0).
the
series
case of
non-homogeneous
top
and
bottom
will be non-uniformly convergent
over
To obtain equations for the horizontal structure
functions "ci, we employ a Galerkin approach (Finlayson, 1972), i.e. we
operate on Eq. 11.3 with:
I)
j Fi(Eq. II.3)dz.
Page -20-
The resulting equation for the ith modal amplitude is:
(42 -[
2
7ijkQJ(
)(&i)t +
,(2-;k2)
_
k
+ (ii)x
)
=
Eq. 11.7
= Fi(O)Qwe
0
where:
-
FiFjFkdz
ijk =
is a coefficient representing the non-linear production of mode i from
interactions
modes
of
later discussions,
In
j and k.
the
evolution
equations of Eqs. 11.7 will be referred to as the continuous equations.
It is useful to examine the results of a similar procedure on the
layered
a
to
appropriate
equations
quasi-geostrophic
model.
The
nondimensional equation for the pressure in the ith layer may be written
as:
2
(P7
+ QJ(
it
[
)
]
f
[P
2
L
giHi-1
i
(
-P ) +
i
Eq. 11.8
2
2
fo
0
I
)
(P - P
+--
P. = (forcing).
i+li
where Q = U/(pL2) as before, Hi is the average thickness of layer i, and
'
=
C(i -
i-i)/ o (see Fig. II.1).
As in the continuous equations, we
attempt a separable solution to the linearized form of Eq. 11.8:
i=
Pj(x,y,t)Fj(i),
Page -21-
V
resting
P
depth
interfoce
Figure II.1.
Schematic Diagram of a Two Layer Ocean
The dashed line represents the configuration of the interface for
the
'd(x,y,t)' describes displacements of
the resting state and
Also shown are the average
interface related to geostrophic motion.
layer thicknesses, H 1 and H 2 , and the layer densities -1 and C2- For a
continuously stratified ocean, the density is described by the buoyancy
frequency N2 (z), and 'd' designates the fluctuations of isopycnals away
from the mean state.
Page -22-
where
represents a horizontal structure function, and Fj(i) the jth
<i
eigenmode in the ith layer, which returns a separability condition in
tridiagonal matrix equation
a homogeneous
the form of
for the vector
Fj(i):
f
22
F .(i+1))-
f
f2
2
2
f
f
L
0
] F .(i) +
+
SH.
H
F .(i-1) +
Eq. 11.9
F .(i) = 0.
+
0
i i-1HgiHi
A'1
We normalize the [Fj(i)] by:
Fj(i)Fk(i)Hi
=
Eq. II.10
jk.
i, are:
The horizontal structure equations, governing the
(72 -Fk
2
Y)k +
T ijkJ(ki,(72 - i'j2)cj) + (i)x
Eq. II.11
- Fk(1)Qwe;
we shall refer to Eqs.
II.
=
11 as the layered equations.
In form, Eqs.
II.11 are identical to Eqs. 11.7, however there are important,
differences
and Fk(1)
between
for
the
the two involving
layered
case,
and
continuous equations (see Table 11.2).
the modal parameters,
ijk,
k,
and
subtle
>ijk,
Fk( 0 )
for
k,
the
Consider the baroclinic mode of
a two layer model; all the baroclinic parameters, 5 111,
1, and F1(1),
Page -23-
Table 11.3
Layer and Continuous Modal Parameters
After Flierl (1978)
Continuous
Two Layer
Barotropic
F (z)
....
........................... 1
.1
.....
0
ist Baroclinic
1(0) .....
2.98
(H2/H1)1/2
......................
4.66 x 10- 4 km-
111
.....
1.78
.0
...
0
2
.......
0....
.
..
H1(H2/HI)3/2 x
1/(H)
-H2(H1) 3 /2 x
1/((H2) 3 /2 H)
Rd1
.0000.0.
46.3 km .....
(g'H1H2)1/
2
x
1/(fo 2 H)) 1/2
Page -24-
are specified by a choice of one density step,
g'1,
Hl, assuming a value for the total depth H.
That is, only two of
Fi,
111,
content in
and Fi(0)
Flierl
equations.
the continuous
analogous
reflecting
independent,
are
the
In
independent.
are
three parameters
continuous equations are automatically
and one layer depth,
modal
two
a greater
(1978)
has
the
case,
information
shown that the
'calibrated' because all of
the
information about the mean stratification (in N2) is used to compute the
modal structures, and hence the system parameters.
In the presence of
surface forcing, Eqs. 11.7 are more accurate than the layered equations,
Chapters V and VI which are
and will be used in
concerned with forced
motion.
=
Note
that
if we
II.
7 is
reduced to two ('ijk
in Eq.
unforced
layered
0, the
equations.
number
In
and
this
of
the same number as in
k),
case,
isomorphic to the continuous system and we
calculations in either frame.
continous modal parameters
the
layered
are free
to
system
interpret
the
is
the
Generally, the two layered system is more
intuitive, so it is this system in which we will interpret the advection
diffusion calculations
of Chapter III.
The
conversions between modal
amplitudes and layer pressures for the two layer unforced case are given
by:
c
+S/21
= PI,
Eq. II.12
and:
'o -1/ri
where
= Hi/H
2 .
= P2,
Page -25-
deviation of
d, the
In Chapter V, we will need to calculate
an
The formula we will
isopycnal from its resting depth (see Fig. II.1).
use is:
d = -Pz/N2
= -(T<(Fi))/N2,
Eq.
11.13
which may be obtained by operating on the hydrostatic equation with:
2
[ .f
(Fz)/N2]dz,
and using the quasi-geostrophic equation:
-N 2 d = b.
The Equivalent Barotropic Equation-
We obtain a two degree-of-freedom model if we retain only the two
variety
circumstances,
we
(Flierl,
problems
oceanic
of
Such a model has
11.7.
in Eqs.
lowest modes
simplify
may
the
further
to
a
certain
Under
1978).
equations
study
to
been used
single
a
formula known as the equivalent barotropic equation.
(q2 - 12)Xt + QIIIJ(
,(7
= forcing -
In
this
equations
subsection,
we
2
- p2)c)
+Cx =
Eq. 11.14
dissipation.
will
discuss
the
unforced
isomorphism) to illustrate
(recall the
(we=O)
layered
the
physical
for
a resting
system
that Eq. 11.14 describes.
From Eq.
layer, P2
=
11.12,
we
see
that
the
condition
0, is:
10o =
1C1
.
However, if P2 =0, Eq. 11.8 with i=2 becomes:
(P1)t = 0.
Eq. 11.15
lower
Page -26-
Obviously, it is essential for the existence of time dependent flow that
Eq.
0(
11.15
is
(1979),
0(51/ 2 )
for weak
wave
therefore, "o
the
in
occurred
1/2), as
Flierl
that
where
the
field.
If
= 0(6).
lower
layer
numerical
lower
layer
P1
0(1),
=
interpretation of
A proper
the lower layer not be strictly at rest.
flows,
the
ratio
McWilliams
experiments
of
developed
as
we
see
and
incoherent
an
=
0C1
co/"1 is
/2),
0(
and
Introducing the rescaled modal amplitudes:
and
<1 ~
1/2,
into the layer equations, the lowest order in ~1/2 (<<1) is:
(V2 - -2)t
(q2-)
We
shall
t
+ Qj(-,(-
+ QJ(X, (2
be
2
- -2))
x
+
0, and
=
a.
b.
- r2)X) + ' x = 0.
primarily
concerned
Eq. 11.16
II.16.a,
Eq.
with
the
equivalent
barotropic equation, which we see, if we = 0, is the governing equation
for the first
layer.
baroclinic
fluid with a thin upper
mode of a two layer
Extensions of the present scaling arguments to the case we A 0
will be made later.
A Discussion of Baroclinic Instability-
The
from Eq.
<I
and
lack
11.14.
area
of
mode-mode
excludes
baroclinic
instability
This may be seen by multiplying Eq.
11.7,
with i=1,
flow
at eo
or
averaging,
2
S(((~1)2 + 1- a
assuming
either
no
by
periodic
We obtain:
boundary conditions.
JJ
transfers
2
)/2)dA]t
jc
1
,(
2 -
'i2)'~l)dA,
Eq. II.17
the baroclinic energy equation, the right hand side of which represents
Page -27-
between
conversion
energy
the
barotropic
or
modes,
baroclinic
By previous scaling, the
baroclinic instability processes.
side is O(
and
right hand
) compared to the left hand side, and therefore negligible.
the
While for some applications
baroclinic
lack of
instability might
represent a shortcoming, the problems under consideration in the present
thesis are not likely to be strongly affected.
II.c
Advection-Diffusion of a Passive Scalar-
If a fluid parcel is convecting a passive tracer, A, the evolution
of A is
governed by:
Eq. II.18
V2
At + u.VA = D + Km A
decay and Km molecular diffusivity.
where D symbolizes
If
we average
(< >) Eq. 11.18 in some suitable way, we obtain:
<A>t + <u>. <A> = -V.<u'A'> + <D> + K
with
coherent
fields.
small scale
2
<A>,
providing a source
transport
for
mean
the
We will employ Fickian diffusion as a turbulent closure:
Fi = <u'iA'> = -KijAxj;
Eq. 11.19
therefore, the equation for A becomes:
<A>t +<uj><A>xj = <D> + (Kij<A>xj)xi,
Eq. 11.20
where we have neglected molecular processes.
From
turbulent
measurements
field
in
mixing
stratification,
density surfaces.
and
the
that
and
ocean
tracer
work,
laboratory
is
highly
transport
anisotropic,
occurs
that
it is known
due
principally
to
along
The diffusivity tensor we will use, the only non-zero
elements of which are on the main diagonal, models this anisotropy
by
Page -28-
assigning a value to the vertical mixing coefficient which is orders of
Hence, on
magnitude smaller than those of the horizontal coefficients.
the mesoscale, A is governed by (dropping brackets):
Eq. 11.21
At + u.i7A = K(Axx + Ayy) + D,
where we
have
horizontal
isotropic.
isopycnal)
(along
mixing relative to
(across isopycnal)
ignored vertical
and
assumed
Kij
to
horizontally
be
We shall use only stable tracers, so D will be set to zero.
The Mixed Layer-
II.d
The atmosphere forces the ocean via a layer in which small scale
turbulent processes are important, and computation of their effect has
become
an
computation
area
of
of
the
much
One
effort.
turbulent
method
of
consists
fluctuations.
These
explicit
so-called
deterministic models have proved to be very enlightening, although the
required computational effort is large.
the
observation
that
the
upper
layer
vertical derivatives to be neglected.
A second approach is based on
is
'well mixed',
Bulk models, as
which
allows
the latter are
called, have proved to be reasonably accurate in their prediction of sea
surface temperature, and appear to be simple enough to be included in
large scale ocean models (Adamec, Elsberry, Garwood, and Haney, 1981).
In this thesis, we shall use a bulk mixed layer model and so will
briefly outline the derivation of the bulk equations.
Other discussions
of the mixed layer can be found in Stevenson (1980) and Muller (1981).
Page -29-
Conservation of Mass and Thermodynamic Energy-
We model the ocean as a Boussinesq fluid:
Eq. 11.22
4.u = 0,
Tt + uTx + vTy + wTz = KT ,72T + Qz
where
u
represents
velocity,
T
temperature,
Eq. 11.23
KT
the
coefficient
of
Averaging Eq. 11.23
thermal diffusivity, and Qz internal heat sources.
and using Eq. 11.22 returns:
Tt + uTx + vTy + wTz = KTq2 T+ Qz Here we have made
greater
vertically
(w'T')z.
the standard assumption that
than
(w'q')z where q is an arbitrary variable.
turbulent
transfers are
(u'q')x,
that
or
horizontally,
Eq. 11.24
(v'q')y
<<
A similar scaling will occur
in all mixed layer equations.
We take the equation of state for seawater to be:
f=
where 1o
is
coefficient
salinity.
a
of
Eq.
reference
thermal
11.25
Eq. 11.25
o(1-!(T-To)),
To
density,
expansion
allows
us
for
to
a
reference
seawater,
convert
Eq.
Y the
temperature,
and
we
have
an
to
11.24
ignored
equation
governing buoyancy:
b = -s-fo)/lobt + ubx + vby + wbz = -(w'b')z + K, 2 b + Boz,
where (Bo)z represents
internal buoyancy sources.
We
Eq.
11.26
suppose
turbulent fluxes well-mix the upper layer, so that due to
that the
the lack of z
dependence in the mean state variables, the vertical integration of Eq.
11.26 over the mixed layer depth, h, is trivial:
h(bt + ubx + vby) = -w'b']o + w'b']-h + B(0) -
B(-h),
Eq. 11.27
Page -30-
where
we
have
neglected
variations
surface
free
and
and
diffusion,
dropped the overbars on u,v, and b.
To
turbulent
conditions on the
of
terms
in
equation
this
close
mean
variables,
boundary
The
buoyancy flux need to be specified.
removal from the ocean surface include latent heat
mechanisms of heat
loss, sensible heat loss, and black body radiation, all of which may be
evaluated using bulk empirical formulae and encapsulated in the form:
w'b']o =
where P
V'go(T-Ta)
+ c
Eq. 11.28
the atmospheric
is an empirical coefficient, Ta
temperature,
and c a bias of the heat flux deriving from the fact that evaporation
sea surface.
the
can only
cool
air-sea
temperature
Frankignoul found
difference
of
least
measured
and
p=10-3
a value of
Also, from analysis
From a
cm/sec
bulk meteorological
fluxes (such as in Thompson, 1974),
squares
surface
regression
heat
of
flux,
(personal communication).
formulae for surface heat
P is found to be 1.5 x 10-3 cm/sec.
The calculations performed in this thesis all used 3= 10-3 cm/sec.
At
z=-h, the mixed layer, if it is deepening, entrains cold water:
w'b']-h = (bi-b)e,
Eq. 11.29
where bi is the buoyancy beneath the mixed layer and:
e = ht+uhx+vhy+w]-h
If
the
mixed
layer
is not
deepening,
there
is no
heat
flux at
interface, so:
w'b']-h = 0.
Eq. 11.30.
Using Eqs. 11.28 and 29 in 27 returns:
h(bt + ubx + vby) = -P(b-ba)
+ Bo(0) - Bo(-h) + (bi-b)e, Eq. 11.31
the
Page -31-
where ba is
the buoyancy appropriate to the temperature of the air:
ba = tg (Ta
- T
o)"
Finally, an accurate mixed layer model requires the computation of
field in
the density
i.e.
layer,
intermediate
the so-called
the layer
extending to a depth of deepest wintertime mixed layer penetration, but
which feels direct atmospheric contact for only a fraction of the year.
The intermediate layer buoyancy is governed by:
C)
Sb
-t
i
+ u
+
b. + wb. + v
1
Oy
x i
Eq.
I
z
b
1
= --
J
(w'b')
zz
B
+
o
II.32
Turbulent transport in the intermediate layer is generally weak compared
to those in the mixed layer and to other heat transport processes in the
intermediate layer
interesting
(Stevenson, 1980);
is
comparison
vertical
convection
(Bo)z.
Evaluating
made
be
formula
typical
the
of
radiative
heating,
penetrative
radiation
and
for
A more
them.
strength
the
between
w(bi)z,
heat,
of
a
to
neglect
we shall
(Thompson, 1974) at a depth of 50 m, estimating bz by the N2 value 10
6
sec-2, and w by 10-4 cm/sec (see Chapter V), we see:
wbz/(Bo)z = (10-4cm/sec)(10-6sec-
2
)/(
3 3
.
3
x 10-8cm/sec ) =
= 3 x 10-3 << 1,
indicating that at lowest order, we can neglect vertical heat convection
in the intermediate layer.
Momentum Equations-
The
averaged
momentum
equations
are
(dropping
overbars
convenient and neglecting viscosity):
ut + u.qu + fxu = -;P - (w'u')z + bk.
Eq. 11.33
where
Page -32-
The upper boundary condition on the vertical momentum flux is given by
the wind stress:
-w'u'
0,
~I
while the stress at the base of
intermediate
of
entrainment
Eq. 11.34
the mixed layer consists of both
the
radiation
of
momentum,
layer
the
and
internal waves:
Eq. 11.35
h = (u-ul)e + F.
-w''
In most mixed layer models, the momentum flux by internal wave radiation
is
neglected
important
in
(Niiler
and
determining
(Kantha, 1975).
Bell
Kraus,
the
(1979)
amount
is
available
energy
of
estimates
it
although
1975)
because of
that,
potentially
for
mixing
F, inertial
oscillations are damped out in roughly a week; however, it appears that
for
low
frequencies,
(Pollard, 1970).
o- << fo,
to
loss
momentum
Therefore, we take F = 0.
F
is
unimportant
The vertically integrated
horizontal momentum equations are:
h(ut + uux + vuy + wuz -
fv)
= -hPx + Z-,
(ul-u)e,
Eq. 11.36
and
h(vt + uvx + vvy + wvz + fu) = -hPy +
>+
(vl-v)e,
and the vertical momentum equation is the hydrostatic balance:
Pz
=
b.
Quasi-Geostrophic Scaling-
For large-scale, low-frequency flows,
in Eq. 11.33 are O(Rossby
the inertial momentum terms
number, hereafter Ro)
with respect
to
the
Coriolis acceleration, and can therefore at lowest order be neglected.
Similarly, a scale estimate of the turbulent momentum transport based on
Page -33-
the wind stress, when compared to the Coriolis acceleration, is small,
leaving a lowest order geostrophic balance in the upper layer:
fov = Px,
fou = -Py*
A vertical integration of the hydrostatic balance relates the pressures
at any two depths.
P(z) = P(Z
o
) +jbdz.
Eq. 11.37
We shall choose Zo to correspond to a depth just below the deepest mixed
layer penetration, and therefore a depth governed by quasi-geostrophic
dynamics.
depth
=
Roughly speaking, Zo
the
within
mixed
0(200 m).
'z' will correspond to a
Substituting
layer.
for
in
P
the
zonal
geostrophic balance returns:
u = -PY/fo = -P(Zo)y/fo -
= u(Zo) -
( bdz)y/fo =
Eq.
11.38
(,bdz)y/fo.
The ratio of the two terms on the right hand side of Eq. 11.38 is:
(Ab)Zo/(foUL) = (ab)Zo/(fo 2 L2 Ro),
which will be small if
Ab<< Rofo 2 L2 /Zo.
Eq. 11.39
A typical Rossby number for a
swift, large-scale flow is:
Ro
-
6
4
0(30 cm sec-l/(l0- sec-i 6x10 cm)) = .05;
therefore, for the ratio in Eq. 11.39 to be small:
8
2
10-4) = 1.8.
Db << ((.05) 36x101 )/(10
Note
that
larger.
for a Ring,
u=0(100
cm/sec),
and the
allowable Lb is
even
In any case, restricting our attention to sea surface buoyancy
differences less than 1.8 cm/sec 2 6,T < 9 OC), the lowest order, mixed
layer, geostrophic balance reduces to:
u(z) = u(Z
o
).
Eq. 11.40
Page -34-
The surface can support thermal gradients which, because of the thinness
of seriously perturbing the shallow
of the upper layer, are incapable
pressure field.
Ekman Pumping-
The potential vorticity equation obtained from Eq. 11.33,
which is
valid in the mixed layer, is:
t + ux
=
where
-(Zo) for
vx -
the
+ v'
-
uy, and
upper
meridional velocity.
fowz +
3v = (Ly)xz - (Cx)yz
= w'u'.
(
layer
and
v(Zo)
At Zo, the vorticity
we
11.40,
From Eq.
vorticity
Eq.
for
can
II.41
substitute
the upper
balance is that
of
layer
quasi-
geostrophic dynamics;
It
+ ux + v y + Pv]Zo = fowz(Zo),
which allows us to rewrite Eq. 11.41 as:
Eq. 11.42
-fowz+fowz(Zo) = curl(-)z.
Integrating from Z=O to the level surface z=Z o returns:
-fo(w(O)-w(Zo)) + fowz(Zo)Zo = curl(C(0)) - curl(T(Zo)).
1
11.43
5
4
3
2
Eq.
At depth, turbulent stresses are weak, and we are ignoring internal wave
radiation, hence we can neglect term 5.
Applying the boundary condition
w(O)=O leaves us with terms 2 and 3 on the left hand side of Eq. 11.43.
Term 3 represents a correction to the vertical velocity at depth Zo due
to the quasi-geostrophic
divergences in the
comparing terms 2 and 3 shows:
wz(Zo)Zo/w(Zo)
= O(Zo/H)<<1,
fluid
above
it;
however,
Page -35-
and we obtain the classical Ekman pumping upper boundary condition on
the interior flow:
w(Zo) = w(0) + O(Zo/H)
= k.curl(J(0))/(fo).
Eq. 11.44
Energy Equation-
momentum,
balance,
hydrostatic
conservation) in six unknowns
have
stand, we
equations now
the mixed layer
As
5 equations
The classical
(u,v,p,b,e,h).
mass
and
energy,
thermodynamic
(2
technique
for closing this system of equations uses the overall energy budget of
the mixed layer, careful derivations
of which have been presented in
(1977) and Stevenson
Niiler and Krauss
Here, we shall simply
(1980).
write down the energy equation, and discuss the relative importance of
its several components.
Neglecting
local
of
storage
turbulent
kinetic
the
energy,
bulk
energy equation is:
0
e((b -
b.)h 1
ui
is
the
heat flux, and
3/2+
o
c
b
a
where
= 2m
(u - u )
i
momentum
of
the
the dissipation.
B ho
Eq.
dz,
11.45
-h
d
intermediate
e
layer,
Bo
the
surface
Term 'a' represents a measure of the
energy needed to entrain and mix cold, heavy fluid over the layer's full
vertical
extent.
Term
shear at the naviface.
'b' is the
amount of
energy
available
in the
Term 'c' represents direct turbulence generation
at the surface by the wind, generally thought of as breaking waves, and
Page -36-
term 'd' the flux of potential energy through the surface due to heating
and
turbulent
the
Finally,
cooling.
term
last
the mixed
energy within
the
represents
layer, a
term whose
dissipation
of
importance
in
turbulent erosion models has been pointed out by Stevenson (1979).
The sum of terms a and b represents the energetic stability of the
Consider a simple gravitationally stable two layer system,
mixed layer.
with the upper layer characterized by velocity u and density b, and the
lower
layer
by
ui
and
bi .
The
bulk
potential
energy
of
the
1
system, to a depth of h+#h, is given by:
u
_ -b
PEi =-
z =-h-z=-h-
-
o
zbdz =
-bizdz
+
0
h -
-
ui
bzdz = (b-bi)h2/2 -
bi(h+ h)2/2,
S-bii
and the total kinetic energy by:
KEi = u 2 h/2 + ui2(h)/2.
Now suppose that the system mixes itself (?!)
to a depth h+fh, and that
the new layer is characterized by buoyancy b' and velocity u'.
u' can be computed from the conservation of heat and momentum:
b'
= (bh+biC(h))/(h+ lh),
u'
= (hu+('h)ui)/(h+ h).
and:
The new bulk potential energy is given by:
PEf= -
=
b'zdz = b'(h+Eh)2/2 =
(bh+bi(h)) (h+ h)/
2
,
b' and
Page -37-
and the new kinetic energy by:
KEf= (h=h)u'2/2 =(hu+ hui)2/(2(h+4 h)).
Note that the change in potential energy:
PEf - PE i =
(b - b i )hh/2
+ 0
h)2,
the potential energy has increased because cold fluid has
is positive;
The change in kinetic energy is negative:
mixed up and warm fluid down.
KEf -
KEi = -(u-ui)_h/2 + O( h)2,
in agreement with decreasing the shear in the flow.
The change in the
total energy of the layer is given by:
= ((b-bl)h-(u-ui) 2 )6h/2 .
SEt = (KEf+PEf)-(KEi+PEi)
Et is negative, more kinetic energy has been released than
Clearly, if
Hence, in a system where:
potential energy gained.
(b-b
i
(u-ui)2
)h -
is negative, a perturbation can draw energy from the shear, grow, and
'mix'.
This is the basic dynamic erosion mechanism originally proposed
by Pollard, Rhines, and Thompson (1973), in which the shear at the base
of
layer
the mixed
is
due
to
the
presence
of
wind
driven
inertial
oscillations.
The time rate of change of total energy is:
d/dt(E) =
E/St = ((b-bi)h-(u-ui)2),h/(2St)
((b-bi)h-(u-ui)
2
)e/2,
=
Eq. 11.46
which is a one-dimensional version of the right hand side of Eq. 11.45.
The
only
effect
on
Eq.
11.46
inclusion of a (v-vi)2 term.
mixing
to
stability.
occur,
If
and
drive
of
two
dimensionality
would
be
the
Thus, if Eq. 11.46 is negative, we expect
the
it is positive,
system
i.e.
back
to
a
state
of
dynamic
if there is insufficient kinetic
energy in the shear to generate a mixing event, mixing will occur only
Page -38-
if energy is transported into the region of the mixed layer base.
terms on the right hand side of Eq. 11.45 describe this
The
transport and
identify the sources as wind wave breaking and thermal convection, both
of which we will neglect.
There currently is a difference in opinion
amongst mixed layer modelers as to whether it is appropriate to ignore
these effects, so we now marshal our relevant arguments.
Wind Wave Breaking and Penetrative Convection-
Recently, direct observations of upper layer turbulent dissipation
have been made and numerical experiments which resolve turbulence have
been performed, and some insight into the balance of the dissipation and
energy generating mechanisms has been gained.
For example, Klein and
Coantic (1981) found that the surface wave turbulent field was largely
dissipated
in the upper few meters,
and for mixed layers
deeper than
about 10 m, inclusion of wave breaking made no noticeable difference in
the evolution of the system.
Similarly, Gargett, Sanford, and Osborn
(1980) noted an increased dissipation in the upper
10 m of the ocean,
which they interpreted as a loss of wave driven energy.
Thompson (1981)
demonstrated that the energy in the upper layer caused by a random field
of whitecaps is strongly surface trapped, and conjectured that the most
important property of breaking waves might well lie in their ability to
mix wind momentum downwards.
Hence, we shall equate term
'c' of Eq.
11.45 to a fraction of the total energy dissipation.
As to penetrative convection, Gargett, Sanford, and Osborn observed
that the energy of descending cold water plumes is dissipated prior to
Page -39-
reaching the mixed layer base, and thus does not assist in deepening.
The experiments of Klein and Coantic also exhibit a tendency for buoyant
energy
be balanced
production to
by dissipation, although under weak
winds and strong cooling, an additional few meter deepening in a thirty
meter layer was noted.
Similar small increases in numerical mixed layer
depths, due to penetrative convection, have been noticed by Mellor and
Durbin (1975).
Finally, comparisons of model predicted and observed sea
temperature
surface
penetrative
that
assume
are
convection
the
generally
(Gill and
surface
better
Turner,
potential
Therefore,
1976).
energy
models
using
when
is
flux
without
we shall
by
balanced
dissipation.
The Froude Number Closure and Its Value-
The remaining terms in the energy oequation are:
2
e((b-bi)h-(u-ui) ) =
-k
where
''
removed.
is
Eq. 11.47
d
the dissipation left after the above balances have
been
For C' = 0, Eq. 11.47 reduces to either the Pollard, Rhines,
and Thompson mixing closure:
2
F = ((u-ui) + (v-vi)2)/((b-bi)h) = 1,
a.
Eq. 11.48
We
implement Eq.
or:
b.
e = 0,
and
is the
energetic closure used in this thesis.
11.48 by using 'b' if F < 1, and
'a' otherwise.
and Van Leer (1978) suggest that F = .6.
Note, Price, Mooers,
We have opted to use F = 1 on
the basis of Thompson (1976), who tested a mixed layer model based on
Page -40-
Eqs.
11.48
various
against
other
models
and
it
found
returned
the
highest coherence between predicted and observed SST.
II.e
Numerical Techniques-
employed
We, have
and
(Gottlieb
Orszag,
when
1977)
numerical
perform
to
necessary
spectral methods
and
double Fourier expansions
Time stepping was carried
solutions to the quasi-geostrophic equations.
out using a leap frog scheme with implicit formulation of the viscous
terms; the computational mode was suppressed by substituting a modified
Euler time step at every 50th iteration (Roache, 1977).
numerical
fraction
calculations
of
essentially
the
are
numerical
repeats
of
referenced
calculations
some
earlier
McWilliams and Flierl, the only
finite
difference
within
technique.
the
reported
numerical
The remaining
Finally,
text.
in
this
studies
thesis
conducted
a
are
by
difference being that they employed a
In
Chapters
III,
V,
and
VI,
we
have
referred to these calculations as McWilliams and Flierl's calculations,
although, technically speaking, they have been performed by the author.
Page -41-
PARTICLE TRAJECTORIES IN
CHAPTER III.
NUMERICAL GULF STREAM RINGS
Introduction-
III.a
There is abundant chemical, biological, and physical evidence that
(the Ring
Gulf Stream Rings produce a sizeable net Lagrangian transport
1981),
Group,
in
contrast
the
Given
in which
an
the
Northern
Atlantic
potentially
important
component
across
the
in
transport
Ring
Stream,
Gulf
maintenance
of
water.
quantities
interesting
oceanographically
most
of
carries a volume
Ring
individual
North
is
a
Atlantic
tracer distributions.
Outside
the
of
photographs
satellite
Ring
of
Warm core Rings
particles undergo sizeable excursions.
of
filaments
interactions
of
fraction
fully
the
across
Shelf.
budgets
of
warm
with
these
and
the
Gulf
Stream or
filaments,
the Slope water,
The
the
implications
Slope
are
into
water
cold
or
with
obvious,
fields
that
apparently pull
during
Shelf-Slope Water
'streamers',
and directly
from
Water
Slope
the
the
evidence
temperature
surface
sea
the
have
we
fluid,
'trapped'
observed
are
connect
respect
the
to
although
to
the Gulf
and
heat
date
streamer-flux estimates have appeared in the literature.
no
their
front.
to
A
extend
Stream to
chemical
quantitative
Cold Rings are
observed to interact with the surface temperature expression of the Gulf
Stream in a
Sargasso.
similar manner, pulling
filaments
of
warm water
into
the
Page -42-
Table III.1
Symbols and Definitions
I
Symbol
Meaning
::::
....
111
Kb
o<
S
L
Ls
Uo
Q
......
**1.
00aa
(-1
r
KI
::....
a0
••go
r
oo
ooo
oooo
oooo
,
oooo
•coo
ooo
oooo
oooo
ooo•
Operators
72
....
6 ....
East-west coordinate
North-south coordinate
Time
Averaging time for shear
diffusion
Pattern propagation speed
Non-dimensional biharmonic
viscous coefficient ...............
Non-dimensional Deformation
Radius ............................
Baroclinic self-interaction
coefficient .......................
Linear velocity shear
Non-dimensional diffusion
coefficient for S .................
Dimensional diffusion
5 x 10-4
-1/2
2
2.1
.04
coefficient for S ...............
3x10 6 cm2 /sec
Baroclinic horizontal amplitude
Diffusant concentration function
Length Scale ....................
Shear augmented length scale
Velocity Scale
60 km.
Steepness = Uo/c
0(5-10)
..............
Streamfunction
Streakfunction
Critical Streakfunction
oooo
Xc
Value
Expression
..
....
J2/(cx)2 +
(iy)2 ....................
..............................
(72)3
Meaning
Diffusion
Biharmonic
dissipation
Page -43-
must
we
budgets,
various
the
transport
mass
their
understand
first
on
Rings
of
effects
the
for
account
properly
to
order
In
compute
the particle
trajectories associated with a numerical Gulf Stream Ring.
In addition,
properties, and in the
present chapter, we will
we will discuss a series of advection-diffusion experiments with a view
trapped
the
exchanged between
how fluid may be
towards understanding
zone and the exterior.
The first theoretical pictures of Ring particle trajectories were
in
(1981),
Flierl
by
obtained
axisymmetric pressure pattern.
associated with a steadily-propagating,
Trapped
of
zones
fluid,
propagating
of
a consequence
calculation as
'steepness', U/c where U is a scale for
velocities
and
c
is
speed,
propagation
pattern
the flow.
the strong nonlinearity of
the
this
in
arose
Ring,
the
with
For Rings,
the
streaklines
the
computed
he
which
particle
the
10.
order
of
Outside of the trapped zone, particle trajectories were characterized by
meridional
excursions
the Ring
scale of
on the
(see Fig.
III.1).
weakness of Flierl's calculation derives from the fact that his
was
purely kinematic.
For
by
data
although suggested
example,
velocity
the
(Olson, 1980),
field
not a
is
he
A
study
employed,
to
solution
the
equations of motion; even though it kinematically resembles a Ring, one
must
computed.
well
on
question
dynamical
the
grounds
particle
trajectories
so
Also, the shape of those particle trajectories do not agree
with
those
observations.
The
by
suggested
numerical
satellite
Ring model we
surface
will
employ
temperature
will
evolve
subject to the conservation of potential vorticity, and therefore will
be
dynamically
consistent.
We
shall
also
see
that
its
particle
Page -44-
Figure III.1.
Streaklines
Here we have plotted the streakfunction, -K= 4+cy, appropriate to
Note the critical streakline, stagnation
Olson's model streamfunction.
In this figure, the steepness, Q, equals 10;
point, and trapped zone.
if Q were less than one, all three features would disappear.
Page -45-
Still, most of
trajectories are in better agreement with observations.
the
Ring,
dynamical
that his
reflecting
appear
will
analysis
Flierl's
of
features
interesting
of
assumptions
in
the
steady-propagation
and permanent form are apt.
Ring Model-
Many quasi-geostrophic models of Ring structure have been proposed
McWilliams,
(Flierl, Larichev,
although perhaps
1980)
and Reznik,
the
most successful Ring simulations were performed by McWilliams and Flierl
of
The appealing feature
(1979).
(numerical) is
their model
that
the
pressure field evolves as a 'monopole', which is in agreement with field
of
observations
of
component
barotropic
We
unknown).
presently
a
Ring
will
of
structure
baroclinic
the
also
(whether
character
barotropic
equivalent
their
employ
Ring
monopole
a
has
a
the
is
Ring
model, which was governed by Eq. 11.14:
(0
2
12) , ) +
- p2)lt + Q5111J(,,(2-
x = K76~.
Eq.
11.14
In the next section, after a brief review of Flierl (1981), we will
In section c, we will discuss
extend his results to include diffusion.
a series of numerical experiments involving the advection-diffusion of a
passive
tracer
comparisons
present
some
with
by
McWilliams
the
previous
simulations
front interactions.
of
the
and
Flierl's
kinematic
often
dynamic
results.
observed
Ring
Finally,
and
make
we
will
Ring/Shelf-Slope Water
Page -46-
Kinematic Models-
III.b
Given a steadily propagating streamfunction of the form:
streaklines,
the
that
demonstrated
Flierl
III.1
Eq.
a(x,y,t) = 4(x-ct,y),
<
,
are
motion
particle
of
given by:
S=
Eq. III.2
cy.
F+
The axisymmetric function:
= UoL(1-exp(-3(r-L)/L)) + UoL/2
2
= U o (x2 +y )/(2L)
(1980)
during
maximum swirl
of
the radius
L denotes
where
r>L
r<L,
Eq. III.5
speed, was
found by Olson
Bob, observed
to accurately describe the streamfunction of Ring
this
Using
1977.
experiment,
Ring
cyclonic
the
function
(following Flierl, we will slightly modify Eq. 111.5 by ignoring the
in the exponential) in the definition of ~
X
'3'
returns:
= UoL(1-exp(-(r-L)/L)) + U L/2 + cb
a.
r>L
Eq. III.$
/<=
= Uo(a2+(b+cL/Uo) 2 )/(2L) Eq.
II.4.b
is
(0,-cL/Uo)=(O,Q-1L)
if
this point
Q-1<1),
those
lies
to
relative
within
circles
fluid will result.
for
equation
an
r<L
2
2
2
Uo(a +b )/( L) + cb =
the
close
the
radius
upon
c2 L/(2Uo)
centered
circle
a
center
of
b.
of
the
at
position
Clearly,
Ring.
the maximum velocity, L
themselves,
and
regions
The parameter Q=Uo/c, controlling the
of
(i.e.
trapped
existence of
closed contours, measures flow steepness and the condition that there be
closed contours, Q>1, demonstrates that particle trapping is a kinematic
consequence of strongly nonlinear, coherent flow.
The finite volume of
Page -47-
trapped
fluid
streaklines
the
which
is delimited
by a critical streakline, )kc, outside of
longer
no
At
close.
the
apex
Xc
of
is a
'stagnation point' where, in a frame moving with the Ring at speed c,
In a fixed frame, where we perceive the Ring as moving west at
u=0.
where
occurs
point
stagnation
the
c,
speed
identically matches the pattern velocity, ufixed
=
the
fluid
velocity
(c,0).
In Fig. III.1, we plot the streakline contours associated with Eq.
a steepness
111.4 for
value, Q, of
propagating westward at a speed
speeds
of 50 cm/sec.
10,
corresponding
to a warm Ring
of 5 cm/sec with anti-cyclonic
swirl
The volume of fluid associated with the trapped
zone is roughly three times that of the Ring as defined by the radius of
maximum velocity, and the trapped zone shape is asymmetric to the north
and south.
The equivalent picture for westward propagating cold Rings
may be abstracted from Fig. III.1 by switching north for south.
From Fig. III.1, we see that particles
just north of the critical
streakline are displaced strongly to the north as they move around the
Ring, while those near
to the northern edge of
less dramatic meridional excursions.
shears
near
the
material lines of fluid.
where we
plot
edge
northern
of
Thus, the fluid develops strong
the
An example of
a history of
the Ring undergo much
Ring,
which
acts
to
this is shown in Fig.
distort
111.2,
several material lines as affected by the
streakline field in Fig. III.1.
Page -48-
(0)
-2
e
U
a
(b)
Ring
Cent er
Figure 111.2.
(c)
The Effect of a Ring on Material Lines
Here we demonstrate that, in the vicinity of a Ring, the fluid
We have plotted the relative orientation of
develops strong shears.
In (a), the Ring is
of Ring interaction.
stages
various
at
three lines
Ring, and in (c),
the
of
midst
the
in
are
lines
the
(b),
in
far away,
the Ring has passed. Marked are the Ring center, and the radius where
the velocities are e- 2 of their maximum.
Page -49-
Tracer Diffusion in Kinematic Models-
now
Consider
the
of
advection-diffusion
of
problem
S,
a tracer
with the advection provided by the kinematic Ring model of Eq. 111.3.
We shall model diffusion according to Fick's law:
Flux = -KbVS,
is the exchange
where Kb
pertaining
coefficient
the appropriate equation to solve, in the frame of
circumstances,
Ring, is Eq. 11.22, with the advection field given by
St +J(,S)
=
/.
L/c,
by
time
Scaling
these
Under
to S.
7;
Eq. 11.22
Kbq 2S.
and
UoL,
by
the
x
y
and
the
by
L,
and
K 1 -1
non-dimensional form of Eq. 11.22 becomes:
Eq. III.7
St + QJ(X,S) = K1 72S
where
the
cL/Kb
is
Heath,
steepness
a
Peclet
1975),
number,
Q=Uo/c,
number.
For
K 1 =(.0 4 ),
thus
of
is
Kb=0(10 6
the
order
10,
cm2 /sec)
(Needler
oceanographically
and
interesting
parameter range corresponds to a weak diffusion/strong advection limit.
Tracer Homogenization on Closed Streamlines-
when applied to
evolution
dispersant
of
11.22 and 111.7
some interesting ramifications of Eqs.
There are
an
regions of
arbitrary
imbedded
in
a
closed X(,
initial
linear
with
regards to
condition.
shear
horizontally as:
2
Ls, - (Kbr2t3)1/
flow,
An
short
the
initial
u(x)=rxi,
point
will
time
of
spread
Page -50-
involved
is
shear
across
the
flow,
allowing the advection field to enhance the downstream transport.
If we
1975;
(Csanady,
i.e.
dispersion,
this
that
suppose
diffusion
mechanism
The
1981).
Young,
spreading
shear-augmented
same
material
the
diffusion model
applies
in
a
local sense, we see that diffusion will tend to force arbitrary initial
conditions towards uniformity on closed streaklines.
Scaling the shear
in the closed X{regions by Uo/L, the spread, Ls, of
the initial point
source, will become of order L, and therefore nearly uniform over the
closed contour, at a time:
ta = (LUo/Kb) 1 /3L/Uo.
For scales appropriate to Rings, this time is:
3
ta = [(6x10 6 50)/(3x106)]1/ (6x106/50) = 6.1 days.
An exception to this rule occurs for solid body rotation, in which case
the time scales are controlled by diffusion:
td = L2/K.
In Fig. 111.3, we present a numerical example where both processes are
In this simulation, velocity shear is concentrated near the
occurring.
edges
of
the
trapped
zone,
constant rotation rate.
while
the
center
is
characterized
by
a
Note that by day 14, 0(2 ta), the dispersant has
homogenized near the edge of the trapped zone, but not in the center.
Thus, within the trapped zone, we understand the processes which will
act
on
the
dispersant,
both
of
contours, and in the presence of
along
7.
Therefore, with little
shears, force it
zone.
in which the
it
over
the
closed
towards homogeneity
generality, we
loss of
initial conditions like S(x,y,t=0) = f()
experiments
spread
which
have
chosen
for those advection-diffusion
initial blob was
located inside the
trapped
Page -51-
_..I J i I
O day s
,
1
II
i
i
I
IIii
i
7 days
-
__
i
'i
.
.
.
.
1I I!
-?_
.
1T ! 1]
II [ I I
!!
!
1i i 1 !I
r
V
14 days
I
Figure 111.3.
I It
I
I
II I
r
An Example of Tracer Homogenization
Here are the results of a numerical integration of Eq. 111.7, using
Note that this velocity field is composed of
7as shown in Fig. III.1.
solid body rotation out to the maximum velocity, followed by an
exponential decrease. All of the shear in the velocity field is located
near the critical streakline, and it is there that the tracer has
In the region of solid body rotation, the time scale for
homogenized.
homogenization is the diffusive time scale, which is a much slower
process.
Page -52-
Of course, S at
the critical contour must
dispersant,
of
concentration
external
be consistent with the
111.4
Figs.
to
reference
as
In this experiment, S is assigned the value 1 inside the
demonstrates.
As
trapped zone, and zero elsewhere.
strong non
progresses, a
time
X-dependent component of S develops on the time scale ta, which forces
Note
towards zero in accord with the external conditions.
3X
the S on
the shape of
as well that
the evolving S contours
dominant in-flux of S-free water
A
point.
stagnation
zone-to-exterior fluxes.
similar
For
indicates that
the
of
the
occurs near the western edge
to
applies
statement
example, in Fig. 111.5, we
trapped
the
exhibit
the
results of a numerical integration of Eq. 111.7 subject to the initial
condition:
S(x,y,t=O) = exp(-(r/60 km)2),
where the radial distance r is measured relative to
trapped zone.
the center of the
As time progresses, S appears to exit the Ring near the
tail.
stagnation point, and is left behind in a thin
This experiment
was repeated with different values for Q and K, with the result that the
larger
trapped zone.
cases,
tail with respect
(smaller) the K (Q), the broader the
Smaller
exterior S was
to
the
(larger) K (Q) produced thinner tails.
In all
the
trapped
located dominantly
the
to
south of
zone.
Although both out and in-flux of material becomes apparent near the
stagnation point, it is important to note that exchange is occurring all
along the
values
extent
near
streaklines
the
of
the critical contour.
stagnation
diverge,
allowing
point,
We notice the anomalous
because
the exterior
it
is
dispersant
that
the
advect
away
there
to
S
Page -53-
t17 days
t=O, dys
1-
-
t
Figure 111.4
28 days
Boundary Effects on Tracer Homogenization
The contours in these plots are of dispersant concentration, S.
The initial condition for this series consisted of uniform material
(S=1) inside the trapped zone and S=O outside. As the pattern evolves,
note the region of weak S developing just to the north of the stagnation
point, an indication of an influx of 'S-free' water. Note as well that
although the initial condition of the present simulation was a function
only of streakline, S(x,y,O) = f(-/), the field develops a strongly
The exterior 'S-free'
non-streakline component on a time scale of ta.
water is forcing the dispersant on the critical streakline to zero, and
in turn, interior S is bought by shear dispersion to a functional
dependence on
more in accord with the exterior.
The result about
shear dispersion, quoted in the text, and its applications imply that
the initial condition must meet the exterior condition at the critical
contour.
Page -54-
. .... . . .... ..
Figure 111.5.
....
Advection-Diffusion using the Kinematic Model
Plotted are the results of an advection-diffusion experiment using
the streakfunction of Fig. III.1. This experiment began with a Gaussian
initial condition of scale 60 km, and shown is S at day 113. Here K=.04
'--".
1and Q=10.
Note that
the dispersant fills
out the "trapped volume, and
exits the trapped zone in a thin tail centered on the critical contour.
Page -55-
from the
Ring, and the S-free waters to invade the trapped zone
(see
Fig. III.1).
of Ring-exterior
From these simulations, we see that observations
exchange will largely be confined to the area of the stagnation point,
which for a warm (cold) Ring occurs to the south
(north) of the radius
It is interesting that this behavior, particularly
of maximum velocity.
often observed in satellite pictures
tail formation, is
of warm core
Specifically, three of the four satellite warm
Rings (see Fig. 111.6).
see that
Also, we
core images in Fig. 111.6 have tail-like features.
the cold tongues of water from the Shelf generally end near the southern
edges of the Rings.
By this
simple model, we have
a trapped
of
characteristics
Ring
successfully modeled the
observed
formation.
A less
and tail
zone
successful aspect of this model comes from its east-west symmetry. The
the fluid which ends up
streaklines suggest that
at the southeastern
side of the stagnation point originates near the critical contour, or to
the
southwest
of
'streamers' of
the
Ring.
Satellite
cold water, which
photographs
emanate
from the
of
the
so-called
Shelf-Slope Water
front, suggest that this same fluid originates more to the west of the
Finally, note these simulations imply that the
Ring (see Fig. III.6.d).
fluid which is most likely to exchange with the trapped zone is that to
the southwest
near the
The shape
critical contour.
and size
of
the
probability distribution for exchange is dependent on K, which we do not
know,
therefore
we
have
not
attempted
to
quantify
this
statement.
However, it will be a useful qualitative remark to compare with results
in the coming section.
Page -56-
Figure I11.6.
Satellite Observations of Rings
Here we show several satellite photographs of SST in the vicinity
of warm core Rings. The Rings in a, b, and c have tail-like warm water
streams following behind them, as suggested by our model. There is also
evidence of asymmetric particle trajectories. Note that the temperature
patterns to the left of the Ring are generally broader than those to the
right, indicative of a slower northward motion ahead of the Ring.
Compare the patterns with the dynamic streaklines of Fig. 111.9.
Ring
Shelf-Slope Front
I
(a)
Streamer
I
/Tail
Ring
(b)
Streamer
Tail
Ring
Streamer
l
Slope Water
..P
L
All2
.,
fiAlt:
K
(c)
T il
. . . .<
Page -60-
In d, we see evidence of a trailing low pressure center behind the
Ring, where the streamer of Shelf Water has wound into a cyclonic
swirl. Also, there is a suggestion of a broad northward flow ahead of
the Ring.
Ring
(d)
Streamer
Swirl
Page -62-
Advection and Diffusion in a Dynamic Ring Model-
III.c
The results of the previous section immediately come into question
when we start to investigate the dynamic consistency of
field.
its
the horizontal structure function of a large scale flow,
If %is
be
evolution should
quasi-geostrophy, which
governed by
that X)also be contours of potential vorticity.
of
included a plot
Flierl's
the advection
the
potential
equivalent barotropic Gulf
vorticity,
requires
In Fig. 111.7, we have
q, from McWilliams
Stream Ring simulations.
and
In this
case, q is defined by:
q = (V2-2)
+y/(Q~1 11 )
Eq. 111.8
where o< is the non-dimensional version of
not particularly resemble X,
.
It is obvious that q does
so it seems unlikely from a dynamic point
of view that particles will flow along the streaklines of Fig. III.1,
and therefore, it is not clear that we may employ the intuition gained
from the model of
the previous section.1
The mismatch of
_Cand q is
due to the fact that 7 in Eq. 111.5 is not a solution to the equations
of motion.
We will now infer the particle trajectories from McWilliams
and Flierl's model Ring, which is dynamically consistent,
and compare
them with the kinematic model, as well as analyze particle motion in
1
consistent,
dynamically
of
examples
several
are
There
steadily propagating, permanent-form solutions, for which the dual
conservation by a particle of streakline and potential vorticity is
automatically satisfied. For example, the equation:
q = (i2-,2)l + y/(Q "
11
)
=
f()
is recognized as the governing equation of the modon, or solitary eddy
These
solution, of Flierl, Larichev, McWilliams, and Reznik (1980).
from
patterns
vorticity
potential
and
have very different streakline
7.
and
III.1
those in Figs.
Page -63-
potential
time in
days=10.*
CONTOURED
Figure 111.7.
vorticity or
an
unforced Gulf
Streom Ring,expiv.dot
6
FROM -49.758-1
TO 4.2S5
E-t
RT INTERVALS OF 2.000 E-i
Potential Vorticity
Here we plot q, the potential vorticity, defined by:
2
2
q = ( -r )O( + y/(Q111),
at day 60, from McWilliams and Flierl's equivalent barotropic Ring.
This q configuration was largely retained by the Ring throughout the
experiment (160 days).
Page -64-
terms
of
described
as
character
Lagrangian
result will
principle
The
dynamics.
the
in
be
the
that
section
previous
general
will
be
Also, the importance
retained, but some of the details will be altered.
of friction and wave dispersion will become apparent.
Dynamic 'Streaklines'-
Flierl's
and
from McWilliams
Notice
Ring.
model
of evolution
stages
In Fig. 111.8, we have plotted oC at various
azimuthally
the
asymmetric field, consisting primarily of a single trailing low pressure
and a single leading high pressure, which surrounds the coherent, large
111.2) and
form shape
permanent
its nearly
(see Table
the Ring
The quasi-steady propagation of
amplitude Ring.
suggests
that we compute
streakfunctions for the Ring according to Eq. 111.4, using the average
propagation speed of Table 111.2 for 'c'.
In Fig. 111.9, we plot such a
Xat days 40, 60, and 80 from the streamfunctions in Fig. 111.8.
the
roughly
represents
of
shape
persistent
X
over
turn-over
eddy
8
streakfunction,
corresponding
this
as
period,
even
For
times.
computed
using
Note
80
days
reference,
the
though
Olson's
empirical
streamfunction, but with c as determined by Table 111.2, is also shown
For convenience, we shall refer to the streaklines as
in Fig. 111.9.
111.8
computed
from
Fig.
computed
from
Olson's
streaklines.
in
as
the
dynamic
streamfunction
will
streaklines,
be
called
while
the
those
kinematic
A comparison of the two reveals both many gross features
common, e.g.
the
existence of
a trapped zone, and
contrasts, e.g. the location of the stagnation point.
several marked
Page -65-
(a)
COM
M
FOR4.99t*1
rl 1.9- 1-L
(b)
r0100# AO
o 8ll
I
of i0artav4sf t.46 1-i
t.0
rU
I
-
--
of I M
INTEAVOLS
6
(-1
.9153
t-
t J
F-.999 9
-. 0001
S-.08
~ ~~ .
.8834
'
1i 11
I t I I t.
I I I L- t-
-. 0998
(c)
Figure 111.8.
Dynamic Ring Evolution
Here we plot the horizontal structure of the dynamically evolving
Note the development of the
Ring at days (a) 0, (b) 40, and (c) 80.
azimuthally dependent field, the Ring propagation, and the nearly
The parameters in this experiment were,
permanent form evolution.
K=5x10 - 4 , r2=2,= 111=2.1, and Q=4.76.
Page -66-
Table 111.2
Measured Pattern Propagation Speeds
From McWilliams and Flierl's Equivalent Barotropic Model
y Location
x Location
Day
14.7
....
12.7 ......
.....
40 ....
.................................
45 ....
50 ....
....
55 so**
60 ....
.................
.....
10.2 ......
65 ....
70 ....
11.3
......
.................
.....
9.1 ......
....
...............
7.9 ......
....
.....
..................................
....
.....
6.7 ......
.... .................................
.....
5.4 ......
....
(-2.2,-.2)
(-2.2,-.7)
13.9
75 ................................
80
85
90
95
100
(-2.7,-.7)
14.3
...............
....
14.2
....
Velocity
(-2.4,-.9)
....
13.4
....
13.3
....
13.1
(-2.4,-.2)
(-2.2,-.4)
in grid units (Yx =
iy = 20 km
All locations are expressi
32x32 numerical experiment.
The
dimensionally), and are from
The error on position estimates is
velocities are expressed in km/d
.2 of a grid, which translates to an error for the pattern speed of .4
The average velocity for the whole interval is (-2.4,-.5)
km/day.
km/day, with an error of .04 km/day, which is within the error
associated with all of the above velocities.
Page -67-
(b)
(a)
Chruf ea
.rm
.oF
1
-
f.
.e
to
t
trvA
O es
-2.LtsCo oueaterF1n
t-1
T to
In
o
.-
M ERME-S
IFe see 1,
.......
...
A.
tor...
O.u
.ta
fft
O 1.510I-L
re
----
_
C
-o.
L
(c)
Figure 111.9.
(d)
Dynamic Streakfunctions
Here we have plotted the dynamic streakfunctions appropriate to
days (a) 40, (b) 60, and (c) 80, as computed by an application of the
kinematic formula to the dynamic Ring, and using c = (-2.4 km/day, -.5
km/day). Notice the nearly permanent form of the Ring. For comparison,
in (d) we have included the kinematic streakfunction as computed from
Olson's streamfunction, with c as above.
Page -68-
Perhaps
in Fig.
for the
that
111.9
is
the
Ring
of
ahead
the most apparent difference between the sets of contours
no
are
streakfunctions,
dynamic
mirror
longer
images
of
contours
the
those
it.
behind
Notice, for example, that the stagnation point in Figs. III.9.a, b, and
c has been rotated clockwise with respect to the location of the same in
Also,
Fig. III.9.d.
pulled abruptly
to
the dynamic
streaklines
north of
the south upon crossing to the east
the
Ring, are
of
the Ring,
which is in contrast to their rather gradual rise over the Ring from the
to
Secondly, we notice that the critical streakline originates
west.
the west of the Ring, rather than to the southwest, as in Fig. III.9.d.
Therefore, a particle initially at point 'A' in the dynamic streaklines
would follow a slightly warped version of that path it would take in the
kinematic model, while a particle initially at 'B' would circumnavigate
the Ring to the south, in contrast to that path it would take in the
kinematic Ring.
The
accuracy
will
streaklines
these
of
be
shortly.
addressed
Taken as correct, they imply that the fluid which ultimately reaches the
southeastern side of the stagnation point originates to the west, which
is
a
feature
asymmetry
of
observed
the
in
streaklines,
which
Also,
photographs.
satellite
implies
that
more
of
the
note
fluid
the
passing anticyclonically around the Ring is bought closer to the trapped
zone
than
in the kinematic model.
for
Proximity to the Ring allows
fluid to alter the trapped zone water properties, therefore, relative to
the
kinematic model,
a larger
area
of
exterior fluid
model can exchange with the trapped zone
this
idea, we performed
a pair of
in the
(see Fig. III.10).
dynamic
To
test
advection-diffusion experiments
in
Page -69-
region of exterior fluid
able to invade trapped zone
dynamic
strecklines
tropped
zone
Xc
region ot exterior fluid
able to invade trapped zone
Figure III.10.
An Implication of Asymmetric Streaklines
The area of fluid able to invade the trapped zone in the dynamic
Ring is increased relative to the same for the kinematic Ring due to the
asymmetry of the streaklines.
Page -70-
which an initial spike of marked fluid was situated to the west of the
Ring, near the northern latitude of maximum Ring velocities.
After 70
days, it was found that a greater amount of the marked fluid had entered
the dynamic Ring trapped zone, compared to that for the kinematic Ring?
While this simulation corroborated the basic idea about the larger area
for the origins of entrained fluid, recall that it is
a result which
depends on K and on the Fickian closure.
Finally, the dynamic Ring velocity field is oriented so that
area of the trapped zone is roughly 75%
model.
the
(+ 5%) of that in the kinematic
Although the radii of maximum velocities are matched in these
two ?Kfields, the exterior velocity field of
the
dynamic Ring decays
much more rapidly away from the Ring.
The Importance of the Dispersion Field-
As
pointed
out
McWilliams
by
and
Flierl,
the
pressure
field
neighboring the Ring in Fig. 111.8 consists principally of leading high
and
which are
trailing low centers,
initially
set up
by
dispersion.
Consider the kinematic effect of a trailing low on an otherwise purely
westward moving,
radially
symmetric
pattern
(see
Fig. III.11).
The
trailing low has a cyclonic flow, which augments the azimuthal velocity
between the Ring and the low center, and weakens it otherwise.
In the
steadily-propagating frame of the Ring is a stagnation point, where:
c
=
0 =
If
Ulow
-low
=
0,
the
Uring + Ulow
ring + Vlow
stagnation
point
will
Eq. 111.9
Eq. III.10
occur
on
the
line
x=0,
Page -71-
SC
= Uring
ULow
+
N
Location of
stognation point
Figure III.11.
The Advective Effect of a Trailing Low Pressure
Near to the Ring, the trailing low intensifies the anticyclonic
Ring velocities, which effectively 'pushes' the stagnation point to the
west.
Page -72-
which
intersects
low,
Vlow
Ring
center.
negative,
is
If,
uow
=
0, west
of
the
Eq.
by
point,
stagnation
the
requiring
III.10, to be west of the line x=0.
stagnation point in
however,
The advection by the low pushes the
the
the direction of Ring motion, which explains
position of the stagnation point in Fig. 111.9.
Now
center.
consider the
West of
effects of
analogous
the Ring, the
leading high pressure
the
center produces an anticyclonic flow,
which weakens the northward directed azimuthal flow near the Ring, but
strengthens it farther away.
Fluid parcels west of
the leading high before the trapped zone arrives,
the Ring encounter
and they gain a slow
northward motion, resulting in the gentle rise of the streaklines from
The fact that
the west.
the Ring is subcritical with respect to
fastest available Rossby wave
the front
speeds allows
the
precede the
to
Ring and produce a non-negligible 'upstream influence'.
Hence, the lowest order shape of the streakline field is strongly
influenced
pressure
by
the
centers.
advective
The
existence
relative to the Ring are the
equation,
and
are
the
effects of
of
structures
centers
these
consequences
and
neighboring high
the
of
missing
and
their
low
motion
the governing dynamical
from
the
kinematic
model
necessary to make its particle paths dynamically consistent.
Advection-Diffusion Using Dynamical Advection Fields-
All
results
regarding
the
streakline
field
assume
that
it
accurately describes particle trajectories, although strictly speaking
Page -73-
structure and the
streaklines are only appropriate for a steady state
Ring is
evolving in time.
we will
Now
field
the Lagrangian
verify
through a series of advection-diffusion experiments.
In the kinematic models, there were two very different regions of
Lagrangian flow; a trapped zone associated with the high velocity core,
and an exterior, excluded from the trapped zone by a closed streakline.
Presently, we shall examine the dynamic Ring
for a trapped zone, and
then the exterior for its particle trajectories.
In Fig. 111.12, we plot tracer concentration as determined by Eq.
111.7, where the advection field was provided by the streamfunction in
Fig.
111.8.
In
the
this experiment,
Gaussian with a length scale of
initial tracer distribution was
situated in the Ring.
60 km, and was
The initial tendency for the tracer is to spread radially; however, upon
filling
certain area,
a
spread slows to a halt.
which
propagates
with
Ring,
the
the
radial
Subsequently, the tracer exits via a rather
The fluid velocities
thin tail attached to the Ring southern edge.
in
the vicinity of point 'A' of Fig. 111.12 are roughly the same as the the
pattern
propagation
velocity.
Eulerian velocity at
associated
with
This
'A' and comparing it
motion
the
of
the
by
checked
was
computing
the
to the propagation velocity
maximum
amplitude
streamfunction; both were 0(-2.4 cm/sec, -.5 cm/sec).
in
the
On the basis of
this simulation, we infer that the trapped zone predicted by Fig. 111.9
exists.
Note also that the area of the fluid moving with the Ring in
Fig. 111.12 agrees with the trapped zone area of Fig. 111.9.
interior
streaklines
of
Fig.
111.9
provide
a
reasonable
Hence, the
picture
of
Page -74-
(a
FIaIII.1 II
CIlUTUIlO
TIl1.
Iirlte
Ar
-.
1I1II1I1:
Ilrtliilillliilil
(c)
. 60 1-1Io
Foam
COITOUISID
t3,.24,t-i
or
O
I
l3lTlta
II
."
t
7.I
-1t
ATITAIrtv.a
F
.
71 -I
-L
iU- I _
,
1
"
--
L-
---
F
ElI
OF0.1
[IEIIRVIL
I
(b)
I-I
COlIOUIDF3a 3.6ii
tI
i
Li
,,
333
Figure III.12.
3
it I
oh
iiL
.
l t-i f
,L
Advection-Diffusion by the Dynamic Ring
Here we plot days (a) 0, (b) 40, and (c) 80, from a numerical
Note the
solution to Eq. 111.7 using the dynamic Ring velocity field.
fluid.
of
zone
trapped
the
and
tail formation
Page -75-
particle motion
note
aside,
an
As
region.
that
in
that
the
fluid
velocities of the area where the interior fluid exits the trapped zone
match the propagation velocity of
results.
the Ring, as in the kinematic model
Therefore, the dynamic Ring model also predicts that we should
observe tracer exchange in Rings to occur near the stagnation point, or
for warm (cold) Rings, to
the south
(north) of
the
hydrographic Ring
signature.
Critical Contour-
Recall that the critical contour, (c, divides the exterior fluid
into
two
regons,
one
of
which
circumvents
the
trapped
zone
anti-
In Fig. 111.13, we present
cyclonically, and the other cyclonically.
the results of an experiment designed to test the location of Xc.
An
on
the
initial Gaussian
of
grid
point
scale
was
placed
at a
point
critical streakline predicted by Fig. 111.9 (grid point location (5,9))
It is clear
and its subsequent interaction with the Ring was computed.
in these figures that about half of the marked fluid moves north about
the Ring, and about half south, which supports
the location of 'c
in
therefore,
to
Fig. 111.9.
Exterior Streaklines-
We
test
the
can accurately
exterior
compute solutions
streaklines,
we have
to
Eq.
111.7;
conducted pairs
of
advection-
Page -76-
(a)
COtD
OMOE
FIOA .SA67
E-t
TO7.tL9 1-t
AT STEAVLSOF .7M7E-t
L
'''ii;I
i
L
mpla
i
r
"
i
wi
r
( c)
( b)
CONTOUREO
FAnOI.L
I
E-t
TO
.
3614 [-1
courcUolt FAOR
-* o5Ac- t o a.1i33 t-t
IFAt
ta[§
aLs
AT ?atIatvAS OFl.246E-1
OFI 387? -1
Ii
I
i
5
L
q
Figure 111.13.
-I L! J
'1!
!
Critical Contour
In this experiment, we introduced a blob of fluid on the predicted
critical contour. Notice that the fluid splits, with roughly one half
moving around the trapped zone in either direction indicative of
Shown are days (a) 0, (b) 20, and (c)
behavior near critical contours.
50.
Page -77-
diffusion
Of
the
experiments
two
members
and
inferred
in each
particle
experiment
time-dependent velocity field of
set,
from
trajectories
one member
them. 2
employed
the dynamic Ring, and the
the
other used
the steadily propagating velocity field contained in Fig. III.9.a, i.e.:
O((x-Cx(t-40 days),y-cy(t-40 days), 40 days).
In each experiment set, both members of the pair used the same initial
which
condition,
was
a
grid-scale
Gaussian
spike
of
Different experiment sets were characterized by different
marked
fluid.
locations of
the initial condition, and all the sets together tested the fifth column
of grid points from rows 7 to 15.
In Fig. 111.14, we compare
conditions at (5,15).
the
day 40
quite well.
pairs,
the
pair with initial
This location put the marked fluid in a region of
In Fig. III.14.a and b, we plot the S
strong shear (recall Fig. 111.2).
field at
results of
from which we
note that the
fields
compare visually
(This degree of similarity was typical for all experiment
and for the duration of each.)
In c, we show the result of a
similar experiment using the symmetric field proposed by Olson.
From
this comparison, it is clear that the asymmetric streaklines are better
approximations to the flow field than the symmetric field.
from a and b were also encouraging.
2
Measurements
The maximum value of the tracer and
Haidvogel (personal communication) has demonstrated that the
particle trajectories computed using Eqs. 11.2 are such that the
constraint of q conservation is not satisfied; therefore, we have opted
not to employ that technique to compute particle trajectories.
Page -78-
fa
CeaToulao
.ll 11L
rTa
t.1s98
-iL
ir
liRvA.s of i.iS
ti
TsT
eueollllile,
I
Car
Fai e it*
i- ITI lo
1.81
LI
l
.1 lll
'titftirisi
TT!HIe.
l-
8.11?I-I
,+--
oH
O
--
~
-i
4
--
o-
bF-b --
, .LUL _LL LL
1i
iI
II I
i il iii
LiJ
I.LL !
II1I!
.i
Iil,-
'i ii i I
ti
F_
,-T
L
-
t
pure diffusion
-t
-4
T/
1 .11
-
E
2
59.4?
5.s3
IS o
I.6s
Time (daye)
d
-cC_LI ! I I
r' .LL
Figure 111.14.
1 tl 1I i I!
I I
t
= l
II I
.4
17
The Exterior Streaklines
We compare the results of a pair of experiments initialized with a
marked blob at grid point location (5,15), which is located in a region
The field in (a) used the evolving Ring, and (b) the
of high shear.
dynamic streaklines. Visually, the fields agree well. In (c), we show
the results of the same experiment using
(d) from Fig. 111.9.
Obviously, the dynamic streaklines are a better representation of the
flow of the Ring.
In (d), we compare the history of the amplitude
Also, we plot Smax as determined by
maximum from the experiment pair.
pure diffusion.
It is evident that the solutions are quickly affected
by shear.
Page -79-
its motion are dependent on velocity and shear,3 and were used to test
In d, we compare the
the agreement more of a and b more quantitatively.
values of Smax for the experiment in a and b.
Note we have included a
history of Smax as determined by simple diffusion, which indicates that
the Ring solutions are affected by the shear.
motion
of
Smax,
this
experiment
turned
Even so,
comparison between the pairs.
agreement in this
Finally, with respect to
diagram is representative of all the pairs.
the
The
out
to
the average
have
the
worst
error in velocity
was roughly .3 cm/sec, as compared to the average velocity of 3 cm/sec.
All other comparisons of the motion of Smax were better than this.
On the basis of these comparisons, we
propagating field
accurately
estimates
the steadily
the shear and velocity
evolving Ring outside of the trapped zone.
the
conclude that
of
the
Therefore, we conclude that
streaklines associated with the steadily-propagating field are an
accurate
representation
of
the
particle
trajectories
of
the
dynamic
Ring.
In summary, having tested the critical contour, trapped zone, and
exterior
streaklines
of
Fig.
111.9,
and
shown
them
to
accurately
describe the Ring particle flowfield, we conclude that although the Ring
is
evolving
in
time,
its
variability
does
not
strongly
alter
its
Lagrangian patterns from those of a steadily-propagating structure.
3
to
is sensitive
dispersant
of
concentration
maximum
The
For example, a Gaussian in a linear
strong shear (recall Fig. 111.2).
velocity
profile
eventually
decays
as
t-3/2,
as
compared
to
Using the scales appropriate to a
t- 1/2 for simple diffusion.
Ring, we estimate that the effects of the velocity field will dominate
diffusion after 10 days.
Page -80-
Potential Vorticity Considerations-
II.d
e-in Fig.
of section b,
the assumed pressure field
to
In contrast
111.8 was determined by integrations of an equation of motion, Eq. 11.14
(the present form of the equivalent barotropic equation):
2
= K7 6c,
Eq. 11.14
along a fluid path, potential vorticity,
that
which states
) +.x
-r2)
(72-r2)at + qfllJ(,(J
Eq. 111.8, is altered only by non-conservative processes.
the comparisons of q and the dynamic
defined
by
Consider now
X.
In Fig. 111.7, we show a contour plot of potential vorticity at day
60
the
of
is
which
dynamic
the
negative
large
the
calculations,
Ring
noticeable
most
zone
vorticity
potential
center.
For a Ring, this pool would find its
the Gulf
Stream, having moved northward
feature
origins to the
formation.
during
at
located
Ring
south
The
of
of
second
thing to note in Fig. 111.7 is the strongly perturbed q contours located
just
outside the
the Ring.
look
pool, which
as
shape of
Once set up, this
they have been wrapped around
if
the exterior potential vorticity
field was maintained for the duration of the experiments.
Comparing the potential vorticity (q) of the Ring interior with the
tracer experiments
tracer
in Fig. 111.12, we see
corresponds
closely
Recall, the
potential vorticity.
potential
experiments
vorticity
is
with
of
a
fluid
q.
Given the
region
of
the
trapped
anomalously
mechanism which will
only
parcel
very weak although it
in trapped zone
the
that
is
does
dissipation, which
region
of
negative
alter the
in
these
largely account for the change
slow evolution of the q distribution, we
Page -81-
within
fluid
the
that
conclude
its
retains
largely
core
Ring
the
original composition, there being no sources for potential vorticity of
sufficient strength to allow for much exchange between trapped zone and
exterior.
In the kinematic models of section b, we demonstrated that particle
Strong
Q.
large
of
consequence
a
was
trapping
To see
conjunction with coherent flow, has the same effect dynamically.
this,
consider perturbation expansions of
and the streamfunction,
7Q,
in powers
of
in
nonlinearity,
the potential vorticity, q,
lowest
To
1/Q.
order,
the
potential vorticity becomes:
qo
=
(2
Eq. III.11
+ 0(1/Q)
-,2)-,
which states that within the Ring, the contributions
vorticity by beta are negligible.
to the potential
Similarly, the lowest order dynamical
balance from Eq. 11.14 is:
= 0(1/Q),
J(o,2(,2-r2)ao)
a
demonstrating
potential vorticity and <o.
(o
which
that
states
contours;
Eq.
relationship
functional
=
"o
111.12
is
+ cy/Q
Q
the
i
=
flow.
as
a
dynamical
lowest
order
the
streaklines
condition
consequence
Eq. 111.13
co + O(1/Q),
potential vorticity along streaklines.
trapping
the
between
However, ,7Kto the lowest order in 1/Q is:
large
for
Eq. 111.12
that
the
pressure
parcels
conserve
match
fluid
Hence, we can interpret particle
of
strongly
nonlinear,
coherent
In Fig. 111.15, we plot the potential vorticity from Fig. 111.8
against the baroclinic amplitude
to demonstrate that Eq. III.11, valid
Page -82-
Figure 111.15.
The Functional Relationship of q and Streamfunction
Potential vorticity is plotted against streamfunction as determined
The range .2< <1 characterizes the
by the dynamical Ring at day 40.
Ring. The dashed line is a plot of the initial condition. Note that q
has been randomized, and there is some evidence of q
at small
dissipation inside the Ring.
rv vs a, expOv.dot
-]
.1
*~
+$
EO
4-.•
L
+\
++
10
Ca
!-.00
0.60
-n. d.2n 0 .20
non-dim omp!itude
1.40
Page -83-
asymptotically in the limit of large steepness, applies to the present
calculations where Q:0(5).
'Ring' (cf.
and
In the range .2<:x<1,
characteristic of the
Fig. 111.8), the fluid has a strong correlation between q
'.
Ring Exterior-
Outside of the Ring, the velocities are no longer properly scaled
by the maximum Ring velocities; therefore, the asymptotic expansion in
the inverse of the steepness breaks
the lowest order dynamical statement.
9.b shows
that
outside
of
the
down, and Eq. 111.23 no longer is
An inspection of Figs. III. 7 and
trapped
zone,
the
potential vorticity are not functionally related.
disagreement
between
these
trapped zone, where the
two
fields
is
most
streamfunction
The place where the
obvious
is
potential vorticity contours look as
have been wrapped around the Ring.
and
In the far field
of
near
the
if they
the Ring, q
contours are essentially determined by beta and oriented east-west and
also disagree with the dynamic
Y, which have a slight southward slope.
As the Ring is approached from either the east or west, the q contours
are warped by the dispersion centers, and align well withX.
The discrepancy between q and ,.in
limited
lateral
influence
of
the Ring.
the
far field is due to
In the far field,
the
the
fluid
parcels are not yet affected by the Ring, therefore the ;( do not apply,
and the particles move along q contours.
This is in contrast to very
near the trapped zone, where the particles are under the influence of
the Ring and move on
the dynamic streaklines.
It
is clear
from the
Page -84-
potential vorticity contours at the southwestern corner of the Ring that
fluid parcels
crossing lines
are
of
constant q, for
the Ring
as
is
approached from the west, those contours first turn north, and then back
to
trajectories,
are
velocities,
Ring
The
south.
the
northward
directed
at
and
therefore
that
spot,
fluid
the
from
which
we
conclude that the fluid parcels are altering their potential vorticity.
can
This
vorticity
only
be
a
result
structure near
its
in
weak
is near
in
to
comparison
to
the
potential
grid
point
Still, viscosity is
enhanced viscosity.
effects
the
Indeed
dissipation.
the trapped zone
scale, and thus a region of
relatively
of
advection,
or
to
planetary vorticity, and the fluid reflects this in the particle paths
the
circumnavigating
The
Ring.
gradient
of
q
the
about
Ring
is
minimized azimuthally; therefore the streaklines depicted in Fig. 111.9,
which effectively predict particle flow in the direction of minimum q
gradients,
are
consistent
strong
with
weak
advection
and
some
effect,
potential
vorticity loss.
Even
though
dissipation
is
having
potential vorticity configuration is not sensitive to K.
the
overall
In a run with
smaller K, the only change in the q contours was confined to within a
few grid points of the trapped zone boundary, where the contours were
observed to 'wrap' further around the trapped zone than in Fig. 111.7.
III.e
Implications-
One of the more oceanographically
interesting properties
of Rings
is that they transport water from the Slope Water to the Sargasso Sea.
Page -85-
From this chapter, we have a dynamically
sound understanding of
this
the way fluid moves about the
phenomenon, and a believable picture of
The important question with respect to the effects of Rings on
Ring.
their surroundings
concern the magnitude
of
exchange
the
between the
trapped zone and the exterior.
With regards to this problem, the results of this chapter suggest
that
tracer
rates
flux
are
small,
because
the
constraint
of
q
conservation largely prevents fluid from entering or leaving the Ring.
We have declined to make quantitative estimates of the flux, due to the
parametrization of
crude
viscosity
and diffusion.
This chapter
does
suggest that an important effect of Rings is to act as a moving source
of tracer, distributing quantities like potential vorticity, salt, and
heat to the external fluid along a path which can extend far into the
host regions.
A separate Ring-induced tracer flux which we only begun
to investigate here is that due to the production of streamers.
Finally,
trajectories
we
have
that
demonstrated
are asymmetric
with respect
dynamical
the
to east-west.
compare the critical streaklines in Figs. 111.9.
)(C
particle
For example,
in the dynamic Ring
model is located to the north of the same in the kinematic model.
immediate application of
this result is
to the
interpretation of
The
the
streamers which are noted in thermal images of the sea surface in the
Slope Water.
According to the dynamic streaklines, the water which gets
to the southeastern corner of the Ring, as streamers do, is located west
of the Ring, rather than to the south as in the kinematic streaklines;
hence,
the
dynamic
streaklines
are
in
better
agreement
with
the
Page -86-
(a)
COlriTOUFROM1 LISlS
FRO @.71?S1
COiTOUNII
TO7.61J E-1
(b)
Figure 111.16.
t
NTIaVALS
OFI 74SE-1
TOLI.8
B
ATINTIAVALS
OF . LIi
CMfOa0t FRiO1.657 E1t
TO4.434 E.1
tr
iTr nL.sOf 1.6641-L
(c)
Ring/Shelf-Slope Front Interaction
Plotted are days (a) 0, (b) 20, and (c) 40 from an experiment
designed to test 'streamer' formation. The results are suggestive until
0(40 days), after which the numerical streamers become too broad to
match with observations.
Page -87-
satellite observations.
we
streamers,
of
structure
the
predict
To clarify the degree to which our model can
advection-diffusion
an
show
experiment in which the dynamic Ring was released in the vicinity of an
oriented tracer field.
east-west
results of
The
this
experiment are
For
presented in Fig. 111.16, which shows the growth of a streamer.
short
times,
indicating
that
advection.
thicken
and
are
Rings
After
the
strongly
pictures
the
30
about
pictures
capable
days,
lose
those
resemble
of
their
the
satellites,
production
streamer
the
however,
from
numerical
to
resemblence
by
streamers
oceanic
the
streamers, which are observed to remain thin.
Summary-
III.f
In
this
chapter,
we
have
investigated
the
Lagrangian
associated with dynamical models of Gulf Stream Rings.
equation was
the
equivalent
quasi-conservation of
barotropic
equation, which
flows
The governing
expresses
potential vorticity along fluid paths;
the
the only
non-conservative force in operation was viscosity, which was compelled
to be weak.
It was found that the evolution of the Ring model was slow
enough to allow accurate predictions of particle trajectories to be made
by using Flierl's
(1981)
formula for streaklines.
The
shapes of
the
particle trajectories were somewhat different than those of the radially
symmetric pattern used in earlier kinematic studies.
The present study
predicts that the fluid west of a Ring is most likely to mix into the
trapped zone, and that the Lagrangian fields are east-west asymmetric.
The cause of the asymmetry was shown to be
the high and low pressure
centers created by the evolving Ring, and it was argued from a dynamical
Page -88-
point of view that their presence represented the corrections necessary
to improve earlier kinematic models.
trapping
Particle
investigated from
standpoint
the
of
trajectories
particle
exterior
the
and
and
potential vorticity,
it
were
was
found that dissipation was important to the shape of the q contours near
the trapped zone.
The results suggested trapped zone-exterior exchange
is weak.
Finally, we discussed experiments which demonstrate the ability of
Rings to account for certain satellite observed sea surface temperature
The
patterns.
Rings
tendency for warm core
'tails' was
develop
to
'streamers', or cold
successfully replicated, as was the production of
water advection from the Shelf to the Slope.
Although we were able to duplicate streamer production, there are
still
some
questions
unanswered
about
are observed to
Basically, streamers
long
their
remain thin;
evolution.
term
that is,
after
the
they do not spread
cold water is advected south
in a narrow tongue,
laterally, as in Fig. 111.16.
Rather there appears to be some mechanism
at
work
streamers
which
keeps
the
anomalous
temperatures
the
One possible explanation for
tongue.
confined to a narrow
associated with
this phenomenon takes into account air-sea interaction; if cold surface
water is exposed to a warm atmospheric state, the surface temperature
anomaly is removed
Shelf waters
(see Chapter IV).
would have
production, where
the
them
warm
A hypothetical scenario for the
advecting
air
would
into
erase
the
their
Slope
cold
via
sea
streamer
surface
Page -89-
signature.
The Shelf waters, with all of their unique properties, would
then be injected into the Slope, but thermally modified in the process,
so that they become invisible to satellite infra-red sensors.
of
course, speculation;
whether we
are
discussing
mechanism will be the subject of later study.
a viable
This is,
physical
Page -90-
CHAPTER IV.
AN ANNUAL MIXED LAYER MODEL
WITH APPLICATION TO GULF STREAM RINGS
Introduction-
IV.a.
(SST's) of Gulf
Just after formation, the sea surface temperatures
Stream
Rings
are
different
surrounding water, with
by
cyclonic Rings
anticyclonic Rings by warm.
from
distinguished
those
of
the
by cold SST and
From satellite sea surface imagery, we have
discovered that the subsequent
are different.
degrees
several
evolutions of warm and cold Ring SST's
Quite simply, cold core Rings lose their cold signatures
while warm core Rings do not (the Ring Group, 1981).
Other features of
warm Ring SST behavior include a tendency for their temperature anomaly
to weaken in the summer, but reemerge in the fall (Friedlander, personal
communication).
case, the
Whether cold Rings do the same is less clear.
In any
'survivability' of warm anomalies is apparently much greater
than that of cold, and in this chapter, we will attempt to understand
why this is so.
Background-
Rings are capable of particle trapping and the residence times for
trapped fluid are is estimated to be long
Dewar, 1981).
(Chapter III and Flierl and
Therefore, for most of its lifetime, the upper layer core
Page -91-
waters of a Gulf Stream Ring are primarily those of its parent region.
Repeated hydrographic surveying of individual Rings has verified this.
Still, Rings have been observed to intermittently interact with the Gulf
Stream (Richardson, 1980),
types
into the
different water
injection of
resulting the
to whether
There currently is some question as
core.
in any case
such interactions are normal, but
they leave
general
the
cross-Ring isothermal displacements unaltered.
core
warm
Consider
and
formation
Ring
the
from
propagation
reference frame of a fluid column, initially in the Sargasso, which ends
Prior to formation, the fluid is subjected to
up in the trapped zone.
the Ring is formed, the atmosphere
As
Sargasso Sea air-temperatures.
becomes less Sargasso-like until at separation, and from then on, it is
A comparison of local monthly mean temperatures
Slope Water in nature.
the Sargasso Sea and Slope Water
peculiar to
(Marine Climatic Atlas)
The air above the
shows them to be very different (See Fig. IV.1.a).
Sargasso
is
(Tsarg = 200 C) than that
warmer
over the Slope
(Tslope =
10*C) and is comparatively moderate in annual variation (6Tsarg= 10 OC,
T slope = 20 *C).
IV.1.b and c)
(see Fig.
air
by
is warmed
formation
is
temperature,
reflect
This contrast is characteristic of these two regions
the
the
Gulf
manifested
and
the
change.
as
a
transition
a
cold
to
from
warm
warm
response
of
'typical' Slope
Water
and
water,
Ring
cold
air-
to
temperature
transition
experienced by the surface layer of a cold core Ring.
the
of
column
its sea surface
evolution of
Similarly,
the
To
Stream.
that the
is
and a plausible explanation for it
Sargasso
should
will
be
We will compute
mixed
layers
to
Sargasso and Slope Water forcing, respectively, in an effort to explain
the SST evolution of warm and cold Rings.
Page -92-
30r
28-
40N , 700W - 35 0 N , 65 W
26
24
350N
rJ 650 W
2220
1816
14
12-
40*N
"
J
F
M
A
M
J
I
I
J
A
i,
,
|
I
S
,
I
I
0
,
I
|,
I
N
0
MONTHS
Figure IV.1.
Comparison of Air-Temperature Cycles
In (a), we compare monthly mean air-temperatures from the Slope and
the Sargasso. Note that for the entire year, the Sargasso air is warmer
than the Slope, while the Slope is characterized by much greater
Both the Slope are Sargasso station locations were chosen
variation.
This
because they correspond to points of frequent Ring observation.
the
Slope
of
most
for
characteristic
is
difference in air temperature
Water and Sargasso, as demonstrated in (b) and (c), where we display
In the winter, the
average air temperatures in January and July.
contrasts are great, and in the summer, weak.
Page -93-
Page -94-
Note
budget
surface,
its
through
flux
the
by
determined
the heat
in this model,
that,
of
so
a fluid column
is
employ
a
will
we
A good question to ask is whether we
one-dimensional mixed layer model.
can expect 1-d models to be adequate, especially in view of their poor
to
comparison
in
We consider
fluxes.
surface
the
a mass
is
there
in which
decay,
from Ring
flux
this
of
size
(Schmitt, personal communication).
the trapped zone and with it a heat flux.
transport into
the
comes
error
One possible
problems
other Ring
in
performance
Approximating the Ring by a cylinder, in which case the inflow, ur, is
related to the downwelling, w, by:
ur = wro/2ho,
the
where ro is
depth,
trapped
the
radius of
layer
mixed
the
and ho
zone,
the ratio of the radially directed heat flux to the surface flux
is:
urbr/(Bflux/ho) = wsb/(2 Bflux).
Using an average w of
-4
10-4 cm/sec,
<yb
1 cm/sec 2 ,
of
and a Bflux of
10-3 cm2 /sec 3 , this ratio is:
=
urbr/(Bflux/ho)
It is with less justification that we
and may therefore be neglected.
ignore
heat
the
those
In
Stream.
of
that we
are
due
to
the
applications
intermittent
where
models
1-d
have
explain an
deterministic, cycle in SST.
the cross-Ring
annual,
the Gulf
failed,
the
of
the
therefore
to the fact
most
likely
Also, the results are most dependent on
isopycnal structure, which
interactions with
and
the
In support
1-d models for the problem at hand, we appeal
trying to
with
interactions
such events have upset the budgets.
occurrence of
aptness
flux
.05 << 1i,
Stream or
is
relatively unaffected by
the Shelf Water.
The solutions
Page -95-
will suggest that one dimensional models include the processes necessary
the
to explain the relative 'robustness' of warm Ring SST, as well as
summertime loss and fall reemergence of Ring thermal structure.
An Annual Mixed Layer Model-
IV.b
Mixed layers are regions of very complicated, turbulent activity,
'intermediate layer'
surface layer overlying an
layer
mixed
the
of
example
computed
is
included
and Thompson,
(Pollard, Rhines,
1973)
These
for details).
Appendix A.IV
development
figures
compute
the
conditions.
An
to
the boundary
of
stresses in terms
turbulent Reynolds
in Fig.
each contain
model
bulk
a
by
IV.3
these
graphs
is
the
sudden
Notice
layer.
constant (b and c).
layer
the
that
(a) to
maintains
with
depth
of
at
thermocline
a
development of a thin,
layer
almost
remains
a
depth.
density
discontinuity
the intermediate layer
In
Fig.
the mixed layer
The
being heated rapidly decreases.
IV.3.a
buoyancy-depth profiles from fall/winter.
concerns
this
the
in
occurrence
Proceeding downward from the mixed layer base in b,
intermediate layer, and
gradients
noticeable
of
reorganization
the rate at which the fluid is
mixed
the most
spring, followed by the
shallow depth in early
warm
Perhaps
(c).
layers
(see
buoyancy
10
profiles at 10 day intervals, and range from fall mixed layers
summer mixed
a
IV.2), and
(see Fig.
layer
surface
the
of
'well mixedness'
the
exploit
of
consist
'bulk models',
models, known as
layer
successful mixed
Many
seconds to years.
and exhibit variability on the time scales of
depth and the
are
with
respect
to
the
itself develops strong
typical
model
generated
The interesting behavior here
buoyancy discontinuity
at its
Page -96-
Mixed
Layer structure
Mixed layer
U
,-
-h
Intermediate
layer
Deepest wintertime
penetration
Interior
Figure IV.2.
Structure of Bulk Mixed Layer Models
The upper waters are divided into a well-mixed layer, in direct
The vertical
contact with the atmosphere, and an intermediate layer.
extent of the intermediate layer is defined by the deepest penetration
of the wintertime mixed layer.
Page -97-
23
17
buoyancy
.i.
.. 7
(e/sec-)
-1.3
-l.l
-2.6
2.
a.
buoyancy
-2.2
-2.0
-
(CA/ sao
-1.69
-L.?
-21
I
L
I
'
[
2)
-1.22
l
-4.9s
!
-4.6
I
r
:i
.
20
15
10
,2.
I
Figure IV.3.
-
buoyancy
-2
(cm/
-.
.662
2.4
-6.ia
I
25
0-2)
-4.12
) 2.61
I
i
I
L.Is
I
t.67
I
Temperature Traces from the PRT Bulk Model
Each plot contains 10 traces at intervals of 10 days, and all three
together cover 300 days worth of mixed layer evolution. Of interest is
the summertime mixed layer, which develops nearly as a constant depth
layer, and the fall degradation of the seasonal thermocline. Note also
the change in imtermediate layer structure from (a) to (c), showing that
over one year, this layer has lost heat.
Compared to (a), wintertime
cooling must erode a much stronger seasonal thermocline in (c) prior to
very deep layer formation, and therefore the upcoming winter will
extract less heat from the intermediate layer than the previous winter.
Page -98-
From our point of view,
base, which quickly fades with increasing h.
to
equivalent
is
this
intermediate
layer
determining mixed
buoyancy
of
that
to
the
by matching
depth
layer
the
suggesting
mixed zone,
a
dependence of wintertime SST on deep buoyancy structure.
this
In
direction of
model,
the
the heat flux.
warm and buoyant, and
depth,
layer
mixed
by
governed
is
h,
the
In the summer, the surface waters become
of mixing light water
the requirement
downward
The well-mixed
effectively isolates the surface layer from the deep.
zone subsequently evolves almost as a constant depth layer, underneath
In the wintertime,
of which develops a very strong buoyancy gradient.
convection, driven by
buoyant
the
water, aids in the mixing process.
and the mixed layer penetrates
production
of
cold,
surface
heavy
The seasonal thermocline is eroded,
into the deep buoyancy structure.
The
great depths of the wintertime mixed layer preclude either a sizeable
As air-sea heat
heat flux or buoyancy jump at the mixed layer base.
convection halts
exchange switches sign in the spring, buoyant
thin mixed layer develops anew.
is
layer
heated
year
round
by
We
also note
penetrative
that
and a
the intermediate
radiation
(see
Appendix
A. IV).
A set of equations describing the seasonal character of the mixed
layer/intermediate layer is:
hbt = -5(b-ba) + Bas - Ba(-h),
where h is
Eq. IV.1.a
the mixed layer depth, b the buoyancy,
ba the atmospheric
Page -99-
buoyancy, defined by:
ba(t)
Bas
= gV(Ta-ro)9
surface heat
radiative
the average
flux,
Ba(-h) the
and
average
irradiant heat flux which enters the deep ocean through the mixed layer
The intermediate layer buoyancy, bi, is governed
base (see Table IV.I).
by:
bit = Baz,
Eq. IV.1.b
The mixed layer equation is closed
where Baz is penetrative radiation.
by a specification for h, namely:
h=ho
Eq. IV.1.c
(a constant) during 'spring' and 'summer', or:
b=bi(-h)
Eq. IV.1.d
'Seasons' are delimited by the sign of:
during 'fall' and 'winter'.
F = -P(b-ba)
+ Bas -Ba(-h),
In
i.e. spring/summer if F is positive, and fall/winter if negative.
the
present
penetrative
we have
calculations,
radiation,
seasonal dependence
ignored the
approximating
these
terms
a
by
constant
of
(see
Table IV.1).
Similar
model,
there
equations
was
no
linearly with depth.
were
used
by
'mixed' layer.
although
(1973),
Warren
Incoming
heat
was
in his
distributed
We have chosen the present heat distribution from
an examination of the bulk model, so we believe it is more dynamically
consistent.
Analytical justification for the
contained in Appendix B.IV.
Appendix C.IV,
with
the
time
evolution of h is
The accuracy of Eqs. IV.1 is the subject of
results
that
they
reproduce
averaged SST and mixed layer depth surprisingly well.
the
bulk
model
Page -100-
Table IV.1
Symbol Definitions and Scales
au
Value
Definition
Symbol
Value
Definition
Symbol
11-
Environmental
fo..
S..
z
Into
Coriolis
coefficient..
coefficient
10- 4 sec-1
of thermal
2x10-4
expansion...
vertical
oC-1
g...
gravity.....
103cm/sec 2
o
reference
water
density.....
time........
1 gm/cm 3
coordinate...
....
temperature..
variable
N-S wind
variable
T ...
ro-y
-
--
Atmospheric
hu
ran
-n
.
mixed layer
depth........
velocity
summer time
depth.........
u.o.
ho
• go
..
,,
1
Mixed Laver
h
a
variable
*
variable
10-
gm/cm3
3
-
.
.
variable
E-W wind.....
stress
wind........
velocity
air
density
ua..
stress.......
atmopheric
ba.
ba
o---
buoyancy....
variable
variable
..
temperature.
depth for
wintertime
sample
problem....
variable
ui..
velocity....
10-4
N2
buoyancy
. 0& frequency...
b...
variable
variable
hi
30 m
50 m
Interior
bi ..
buoyancy.....
buoyancy
r ...
gradient ....
sec
__ _
_
±
__
__
__
__
2
variable
Page -100-
Table IV.I
Symbols and Definitions (continued)
.
___________
Symbol
Definition
Symbol
Symbol
Value
I~
4
Definition
Value
4-
_____
Forcing
coefficient
of drag
Cd
E (-)
wind speed
spectrum....
10-3
argument
of random
phase.......
frequency
U=
Ur+iui
X2
Bo(z)
Ho
Ba (-h)
. **
Fourier
coefficients
of wind
attenuation
depth
solar
radiation
buoyancy
flux
average
flux out of
mixed layer
solar zenith
angle
A 1
25 m
10-3cm2
attenuation
depth.......
n,m
coefficients.
in TEM closure.
Bos. .
radiant
surface heating
daily average
radiant flux...
latitude
Bas
sec3
..
daily
frequency
.0004
.0006
40°N
2 /day
variable
coefficient
of solar
radiation
Mo
..
35 cm
fractional
distribution
of radiation
magnitude of
Bas(-h)
Bao
-
-----
.03
---
--
.0004
--
-
Page -101-
In
the
experiments
to
be
discussed,
the
atmospheric
annual
temperature cycle, ba(t), will take the general form:
ba(t)= a 2 + a l cos(rt +t)
Eq. IV.2
with the maximum temperatures occurring at an annual phase of m.
Ba is
taken as:
Ba = Baoexp(z/A2).
Note,
(see Appendix A.IV).
maintenance
production
ho
was
is
explicit
no
reference
wind
to
The effects of the wind, however, enter through both the value
stress.
and
there
of
the
of
chosen
to
the
summertime
seasonal
be
30
m;
mixed
thermocline
the
layer
the
with
rationale
depth,
ho,
of
onset
behind
and
this
the
spring.
choice
is
contained in Appendix C.IV.
IV.c
Limit Cycle Calculations-
The
first
step
average mixed layer
definition of
to
Ring
understanding
characteristics
of
'average' does not exist;
the
SST
is
parent
to
determine
An
region.
however, we have
exact
found that,
subject to a choice of ba, Eqs. IV.1 possess limit cycle solutions.
we assume
that the
the
If
average residence time of fluid in each region is
long, we can interpret the limit cycles as 'average' mixed layers.
This
definition turns out to be impractical, for the limit cycles of Eq. IV.1
represent mixed layers which deepen to infinity at
This
We
is an undesirable feature,
can still
but
it
obtain accurate estimates
turns
for
out
the end of winter.
to
be unimportant.
'average' mixed
terms of 'quasi'-limit cycle solutions as follows.
layers in
Page -102-
The infinite deepening of the late winter part of the limit cycle
is a consequence of the form of the penetrative radiation, because the
heat balance at any level is:
=
bt
between turbulent
i.e.
11.26),
(see Eq.
IV.3
Eq.
-(w'b')z + Baz
and penetrative
fluxes
heat
For any depth to be cycling in a limit state;
radiation.
T-1
At every level,
dt
dt
there is
=
0.
an annual balance between turbulent heat flux
The form we have chosen for the penetrative radiation is
and radiation.
non-zero at
b
all depths;
turbulent heat
therefore, at all depths
flux
(deep winter time mixing) is required.
In the real ocean, radiation is probably not significant beyond the
first
few hundred meters, after which the
is
radiation
correct
this
suspect
fault,
present parameterization of
a series
were
experiments
of
arguments, whatever
to
in which
performed
the radiation
chosen as
depth is
effort
Notice that according
radiation was expunged at some ad-hoc depth, hc.
to the previous
an
In
1981).
(Simpson and Dickey,
cut-off also becomes the depth of deepest winter time mixing.
Several
sea
surface
cut-off
depths
temperature was
were
tested,
unaffected so
giving
as hc
long
e-folding scale of the radiation.
the
was
much
referred to
earlier as
greater
than
the
In the present set of calculations,
we returned to the original formulation, hc =m".
layers,
that
result
The
'typical' mixed
'quasi'-limit cycle mixed layers, were
cycle
was
repeating itself to within a few parts in several decimal places
(see
obtained
Fig IV.4).
by
integrating
Eqs.
IV.1
until
the
numerical
Page -103-
Two properties of limit cycles which are important to the problem
Both are difficult to prove, but
at hand are existence and uniqueness.
apparently
apply
to
the
limit
of
cycles
the
present
For
problem.
example, in all numerical experiments with the same atmospheric forcing,
ba(t), the solutions, regardless of initial condition, converged towards
the same (and therefore apparently unique) limit cycle.
With regards to
existence, we note that the effect of the forcing is to adjust the deep
This may be seen from
buoyancy profile towards limit cycle behavior.
Eq.
Consider
IV.3.
a non-limit
cycle
intermediate
in which
layer,
The
therefore at some depths there are net annual imbalances of heat.
effect
of
solar
radiation is
wintertime mixing to cool it.
always
to warm the
Thus, if at some
water,
and
that
of
depth there is a net
loss of buoyancy, that depth was 'in' the wintertime mixed layer for too
long a time.
Note, however, that the resulting colder profile prevents
the upcoming wintertime mixed layer from deepening as efficiently as the
previous winter
(see Fig. IV.3).
Hence in the winter to
intermediate layer depths will be cooled less.
follow
the
A net heat gain in the
intermediate layer produces a buoyancy profile which allows deep mixing
to occur earlier during winter, and therefore extract greater amounts of
heat from the intermediate layer
in the next
year.
Either
shift
is
closer to an annual heat balance, which is the characteristic of a limit
cycle.
'Typical' Mixed Layers-
In Figs.
IV.4, we
graph mixed
layer
buoyancy,
b, against
mixed
layer depth, h, as a measure of the limit cycle behavior from both the
Sargasso and the Slope.
The air-temperature cycles for each region were
Page -104-
taken to be:
Ta(t)= 10*C + 10 0 Ccos(,-t +)
and:
for the Slope,
Ta(t)= 20
for
the
Sargasso
oC
)
+ 5 °Ccos(.at +
the
contain four years worth of model data;
the width
least
of
line
the plotting
deepest penetration).
IV.4 actually
Notice that Figs.
IV.1).
(see Figs.
repeating
cycle is
(however, observe
to at
the point
of
in Fig. IV.5, we have included several
Finally
model generated lower layer (i.e. depths greater than 30 m) temperature
The Slope Water is characterized by a much greater range of
profiles.
SST than the
Compare
Sargasso,
while
the model profiles
Sargasso
the
actual winter and
with
are much warmer.
profiles
summer
XBT
traces
(Fig. IV.6) taken at locations within the Slope Water and Sargasso where
Rings are frequently observed.
Finally, we note that the SST extremes as
The
cycles match well with observations.
typically 5 °C, as compared to 6
0
C, compared to the model value
predicted by the limit
coldest Slope Water SST is
0
C in the model, and the warmest is 19
of 20
0
C (Colton and Stoddard, 1972).
In the Sargasso south of the Gulf Stream, the range is observed to be
from 20 *C to 26 =C (Fuglister, 1947), and the model predicts 20 °C to
27 OC.
IV.d
Adjustment Calculations-
Given
the
parent
region mixed
layer
structures,
their evolution when subjected to host region forcing.
we
can
compute
Recall that the
sudden change of atmospheric state is meant to model Ring formation.
Page -105-
,2+..
.8
,21.68
2.86
40.01
61.7t
74.46
depth (m)
8 .57
9 .43
depth (m)
a8.8l127.29
118.29
,
153.68
137.14
te6.6
156.80
286.46
14
?10
o
0C
3
N.
N
Figure IV.4.
b Versus h Limit Cycles
Here we plot the limit cycle solutions for the (a) Sargasso and the
(b) Slope Water.
Each graph contains four consecutive years of data,
from which it is apparent that, for the most part, the cycle is
In (a), the cycle
repeating to a few parts in several decimal places.
deepest
wintertime
depth of
at
the
evolving
is
still
weakly
Note the greater range of the Slope water mixed layer
penetration.
temperature in comparison to the warmer Sargasso.
Page -106-
OC
20
-1.68
(cm/sec*-2)
buoyoncy S6
-..a3
-118
-8.
25
8.19
1.8
1
1
II
-
13
5
64
-
Figure IV.5.
-3.37
•
,
buoyancy (cm/sec**2)
-2.S1
-. 94
-1.38
-8.82
I
-8.25
=
8.31
I
Lower Layer Buoyancy Structure
Here we have plotted, in (a) and (b), the buoyancy traces for
depths greater that 30 m from limit cycles (a) and (b), respectively,
of Fig. IV.4. In the present model, the mixed layer is never shallower
The time
than 30 m, so this portion of the column was left out.
months.
two
is
traces
interval between
Page -107-
,0
5, n 0
,2---
2p
--
-
0
-
0
100II-
100 . 12/3/7 6
35" 23.7 N
200 - 65 22.E
300
200.
I
400
500
7M -
sun
7nn.
2/77
SClW/
33 49.3 N.
T.
. = 72 U.i.BN.
---
2
I
!
1
600
ImI
-4116
700
Fb
I
I
6
2
10
26
22
18
14
30
oC
OC
5
to
20
I5
25
30
!55GSF
-
II Mrl t
45C
500
0
Figure IV.6.
Loo. 70400W
-
30
3
15TMPATU
TEMPERATUREVC1
Summer and Winter Slope and Sargasso XBT Measurements
Here are typical upper layer temperature traces from both the
In agreement with the limit cycle buoyancy
Sargasso and the Slope.
traces, the range of the Slope SST is much greater than that of the
Sargasso, but the Sargasso is much warmer.
Page -108-
In Fig. IV.7, we compare the annual cycles of SST for a warm Ring
Included
central mixed layer and a 'typical' Slope Water mixed layer.
which is
are six years worth of Slope Water SST,
1-2
four
and
years
worth
of
warm
core
Note however that Rings are generally only in the host region
response.
for
SST
Sargasso
typical
of
worth
compared to two years
years,
the
so
years
few
first
of
response
the
are
most
In this experiment the Ring was formed at the warmest point
applicable.
Notice the temperature contrast is greatest
in the atmospheric cycle.
in winter, and is roughly 6 OC.
As the year progresses, this contrast
The same information is
first weakens, disappears, and then reemerges.
Note that
plotted in Fig IV. 8 for the analogous case of a cold Ring.
the cold Ring wintertime temperature contrast (0(1*C)) is much weaker
10 C is roughly the error of satellite
than that of the warm core Ring.
measured SST, so these results suggest that in infra-red images, warm
would
Rings
be
evident
much more
than
As
cold.
before,
summertime
erases the cold Ring temperature contrasts, and the onset of the fall
rejuvenates
them.
Recall
that
the
parameters
diabatic forcing
were
chosen to agree with data; therefore, we should attach significance to
the values which the model generates.
The reasons for the difference in the SST contrasts are essentially
(Figs. IV.9 and
contained in the buoyancy profiles from each experiment
10).
Exposing a 'typical' Sargasso Sea mixed layer to the cold Slope
Water
wintertime
mostly
slowly
by
in
results
convective
in
unusually
mixing.
temperature;
equal
Hence,
deep
surface
warm core
extractions
of
layers,
mixed
heat
produced
layers
respond
produce
lesser
decrements in SST owing to the thickening layer from which the heat is
O
o
E
C
00
38.1
39.2
40.3
41.8
43.3
years
Figure IV.7.
Warm Core Ring SST
The first two years
Here we plot a comparison of 6 years of SST.
are a comparison of limit cycle Sargasso and Slope water mixed layers.
The Ring is 'formed' on July 1 of the second year, and the last four
The maximum contrast is roughly 6
years are of warm Ring SST response.
OC.
o
0O
M
q-
Here
we plot
a
comparison
of
6 years
of
SST.
The
first
two
years
are a comparison of limit cycle Sargasso and Slope water mixed layers.
The Ring is
'formed'
on July 1 of the second year, and the last four
years
are
of
cold
Ring
SST response.
The maximum contrast is roughly 1
0C.
0
C.
I,
I~
I
---------- I------------
I
-
Page -111-
removed.
warm
of
Estimates
core
relaxation
layer
mixed
are
times
generally longer than the duration of winter; therefore, being initially
warm, the Sargasso
throughout the
simply
layers
mixed
As winter
cooling season.
their warm identity
maintain
gives
to
way
summer,
thin
layers of depth h o develop and shortly thereafter the mixed layers lose
Note that although surface
memory of their early spring-time buoyancy.
contrasts are covered up in roughly a month, the deep thermal structure
retains its warm identity.
As the layer again progresses into winter,
it exposes its interior buoyancy structure to the atmosphere, and warm
central waters
springtime
In Fig. IV.11, we plot a
reemerge (Figs. IV.7 and 9).
section
XBT
taken
across
a
Ring
core
warm
which
shows
evidences of deep central wintertime mixing and weak surface temperature
gradients.
Conversely, a Slope Water mixed layer moving into the Sargasso Sea
Subsequent
is heated and develops a summertime thermocline.
evolution
of the mixed zone is largely confined to a layer of thickness ho
(see
In Fig. IV.12, we plot an XBT section taken across
cold
Fig. IV.10).
core Ring Bob on which has been drawn a subjective estimate of the mixed
layer
depth.
shallower
at
significant.
there
While
Ring
center,
Still,
is
a
one
in contrast
suggestion
questions
to
that
whether
the warm
layers are no deeper than those of the Sargasso.
the summer, the
the
Ring,
layer
is
difference
is
mixed
the
cold Ring
mixed
After being lost in
contrasts in SST across a cold Ring
do not
reappear
because the surface heat flux mixes into layers of similar depth.
Only
at the end of winter does the unique structure of the core mixed layers
appear.
Note
that
the
observations of
winter mixed
layer
depth are
Page -112-
0
C
L
-2. 12
-t.6
i
I
buoyancy (cm/sec**2)
-.
II
-0.59
I
27
22
17
12
-0.08
0.42
-
0.93
i
L.44
I
%
L7t
0a
10
(l
0O
6
P-
CM
0O
,
NO
Figure IV.9.
Lower Layer buoyancy Traces from a Warm Ring
Here we graph buoyancy against depth for depths greater than 30 m
from an adjusting warm core Ring. The time interval between traces is
The relevant
two months and the first trace corresponds to July 1.
feature of the mixed layer buoyancy evolution concerns the deep,
convectively driven mixing, which forms unusually thick layers thoughout
the year. Also note the deep radiative heating.
Page -113-
1C
10
-3 .72
v
4
cn
-2.94
I
20
15
a
I
I
--
1
I
buoyancy (cm/sec**2)
-2.18
I
-1.42
1
-. 67
~'
I
0.09
'
25
J
0.85
.
1.61
'
0cD
00.
3
v
p,,.
Figure IV.10.
Lower Layer Buoyancy Traces from a Cold Core Ring
Here we graph buoyancy against depth for depths greater than 30 m
from an evolving cold core surface layer.
The time interval between
each corresponds to two months, and the first corresponds to July 1.
Note that these mixed layers tend to remain shallow throughout the year
because the Sargasso atmosphere injects heat into the surface waters.
Page -114-
XBT no.
100
200
300
2
400
500
600
700
43
4.8
4.8
0 km
0 tM
Figure IV.11.
I
4V
50
50
8
48
4.6
4.4
100
I
46
45
4.4
0
150
4
4.4
4.5
200
200
An XBT Transect of a Warm Core Ring
the
of
many
shows
spring,
the
in
taken
transect,
This
characteristics of the solutions we have been computing. Note that the
core waters are well mixed to depths of 0(300 m), and have been capped
This data was
thermocline reformation.
over by early springtime
by his kind
here
reproduced
is
obtained by Dr. Terrence Joyce, and
permission.
SN
0
depth of mixed layer
E
100
+
-c
-
depth of the
15C
isotherm
3001
400
109
III
113
115
117
119
121
123
125
127
129
131
133
xbt#
Figure IV.12.
An XBT Transect of a Cold Core Ring
This transect, taken in December, demonstrates that for a cold
Ring, the mixed layers of the core water resemble the exterior mixed
There is even a hint of the core mixed layers being shallower.
layers.
the Ring presence by the 15 OC isotherm, and have given
denoted
have
We
a rough sketch of the transect with respect to the Ring (Ring Allen, see
in the upper right hand
1979)
Richardson, Maillard, and Sanford,
corner.
Page -116-
those
times
three
roughly
of
intermediate
neglects
model, which
the
Although h is not well represented in late winter, the
layer mixing.
in other seasons and therefore
agreement improves
the SST behavior we
have computed is probably insensitive to this.
With regards to the effects of differing formation dates, a series
of
different
conducted
were
experiments
After
seasons.
contrasts
the
which stress
parent regions.
'formed'
was
Ring
the
developed
adaptation, SST
short
a
in
the
in
is in agreement with the above arguments
This
manner just described.
which
in
in
the
of
temperatures
air
the
and
host
Regardless of formation dates, within a year warm Ring
cores will cool into very deep mixed layers and cold Ring cores will be
heated and develop mixed layer depths of thickness h
note
Finally
that
there
in
difference
is a
the
time
scale
of
approach to the limit cycle states; warm Rings adjust much more quickly
to the atmospheric forcing
(which take
Rings
0(10
than do the
(on times of 0(2 years))
years)).
reason
The
intermediate layer is an integral part
of
the
for
this
is that
structure of
cold
the
limit
the
cycle mixed layer, and the mechanism of approach to its final state is
governed by the loss or gain of heat.
heated,
value.
the
buoyancy
depth
is increasing
towards
its
limit
cycle
The excess heat necessary to do this comes from radiation, which
at depths of,
layer
at
If the adjusting layer is being
requires
say,
150 m is a comparatively weak heat source and the
several
years
to
adjust
(see Fig.
IV.10).
If
the
intermediate layer is being cooled, the loss of heat at depth is due to
wintertime mixed layer cooling.
Thus, the buoyancy structure is altered
Page -117-
by turbulent heat transport, which is stronger than radiation, and the
It is for
time scales of the evolution are the order of a few years.
this reason that we see a more rapid adjustment to the external forcing
by warm pools than cold in Figs. IV.7 and 8, where the initially greater
wintertime anomAly of the anti-cyclonic eddy disappears at a faster rate
than
cold
the
than
long-lived
Thus,
anomaly.
more
are
anomalies
cold
Ring lifetimes
their warm counterparts.
stable
and
are 0(1 yr)
owing to their interactions with the Gulf Stream, and therefore this is
We mention it because
not an important process to Ring-SST evolution.
to longer-lived SST anomalies
of possible applications
in the general
circulation.
IV.e
Summary-
the annual mixed layer was captured in a set of
The evolution of
simple one-dimensional equations which were used to investigate the sea
surface
of
response
the
computed
in
of
terms
budgets
heat
The
Sargasso.
air-sea
local
that
the
transport water between the Slope
important effect of the Ring was to
and
hypothesized
We
cold Rings.
and
warm
of
the
and
exchange,
background SST cycles of the host Region.
mixed layers were
core
compared
to
the
The resulting buoyancy-depth
profiles and temperature contrasts compared well with data.
For warm core
forcing was
which had
that
built
it
the
Rings,
was
the
on
important
the average
structure of
the
evolved, they developed unusually deep
facet
of
the Slope
cooler than the
core.
As
the
diabatic
temperatures
Sargasso layers
layers which had the effect of
Page -118-
For cold core Rings,
impeding the decrease in sea surface temperature.
it was noted that the core mixed layers remained shallow, and therefore
the temperature contrasts across the Ring were decreased in magnitude,
because the Sargasso air temperatures were warm.
From Fig. IV.11 and these results, it is apparent that wintertime
In the
mixed layer development in a warm core Ring is a major event.
present
have
calculation, we
neglected
the
on
Ring,
the
dynamic
and
while
adjusting
buoyancy
field
suggestive
of Ring
thermodynamics, it is clear
will require an active Ring.
of
the
results
are
effects
the
further modeling
that
For example, the buoyancy traces contained
in Figs. IV.4 and 9 indicate a loss of dynamic height.
With respect to
200 decibars, the pressure head of the core over the flank decreases by
roughly 20 dynamic centimeters
30
of
decrease
cm/sec
in
from July to February,
geostrophic
velocity.
resulting in a
Wintertime
the Ring because
carries out a thermodynamic spin-down of
the
mixing
act
of
cooling the intermediate layer removes the depression in the upper layer
isotherms. While the overall annual SST cycle will probably not change
because of
the
inclusion of an evolving Ring, we can expect the Ring
life cycle to be modified by the inclusion of the mixed layer.
The simplicity of
the annual model suggests
that the
results may
Consider
the
implications for the interpretation of satellite infra-red images.
On
extend
the
beyond
basis
of
the
the
scope
present
regardless of whether it is
of
the
chapter,
problem.
present
we
would
interpret
associated with a Ring, as
presence of a deep, warm buoyancy profile.
a warm
pool,
indicating the
With historical information
Page -119-
about geographical variations in density structure, we could even make
some
intelligent
not
are
From large scale infra-red maps,
probably the result of a recent event.
we can search for similar
are
pools
Therefore, any observed cold
are short-lived.
they
view,
anomalies
Cold
origin.
its
structure, although from an observational point of
deep
indicative of
to
as
guesses
and make an objective statement
temperatures
about the origin of the cold pool.
Also, we agree with Simpson and Dickey (1981) about the importance
of penetrative radiation in the upper layers.
most
the intermediate
of
rather
any
by
than
process
solar
operate in the intermediate zones of
manner
as
in
present
the
model
irradiance
occurs
in
observed
to
(as
detrainment
resembling
radiation,
by
restratification occurs
Also,
models).
erosion
turbulent
layer
present model,
In the
is
the real ocean in much
(see
Appendix
the same
the
Given
A.IV).
dependence of SST on intermediate layer buoyancy structure, we recommend
properly
that future models
account
distribution of
for the vertical
radiation.
not
Note, we have
properties,
Rings.
such
as
considered
the
in optical
geographical variations
attenuation
coefficients,
of
the
water
in
Richardson (1980) has reported that upon entering cold core Ring
'Bob', the water color changed and even the odor in the air developed a
Slope Water character.
'dirty'
and
full
of
With respect to Sargasso water, Slope water is
aquatic
microorganisms
irradiation does not penetrate as
water.
deep
and
therefore
in Slope water as
the
solar
in Sargasso
Cold Rings are frequently observed in the summertime to be a few
Page -120-
tenths
of
a
degree
C
warmer
(Vastano, Schmitz, and Hagan,
than
1980);
temporarily become warm core Rings.
the
surrounding
in other words,
Sargasso
Water
cold core Rings
It is interesting to speculate that
this is a manifestation of a more efficient absorption of radiant energy
by
the
murky
Slope
Water
mixed
explanation remains to be seen.
layers;
whether
this
is
a
valid
Page -121-
A Bulk Mixed Layer Model-
Appendix A.IV
The Equations and the Forcing Functions-
A.IV.a
The
is that
employ
shall
model we
layer
mixed
one-dimensional
originally discussed by Pollard, Rhines, and Thompson (1973, hereafter
PRT) and cast into operational form by Thompson (1974, 1976).
in
derived
were
equations
momentum and heat
dimensional
The three
Chapter
II;
their one-dimensional forms are:
P(b-ba) + Bos -
hbt = (bi-b)ht -
(hu)t - fovh = ix,
b. Eq. A.IV.1
(hv)t + fouh = Cy,
c.
(b-bi)h = (u2 +v 2 ),
d.
and
e.
bit = Boz,
where
the
a.
Bo(-h),
of
meaning
symbol
each
Table
in
listed
is
Eqs.
IV.1.
A.IV.1.a-e allow the mixed layer to entrain in two different ways, one
and one
wind driven
caused by fluctuations
mixed
layer,
and
Eq.
in
the
at
the
A.IV.d
is the
stable.
produces turbulence, which mixes
which
cools
depths.
condition
flow
Further acceleration of
the flow
the fluid under the interface up into
the
layer
and
deepens
do not
the
interface
to
cooler,
denser
allow for penetrative convection, so
in density at the mixed layer base can be maintained only
through the dynamic instability of the shear flow.
penetrative
the
for
Second, wintertime cooling at the surface causes convection
Note Eqs. A.IV.1
that a jump
inertial oscillations,
, produce large shears at the base of
interface to be marginally
the layer.
First,
buoyantly driven.
convection
has
been
observed
in
Little evidence of
the
mid
ocean
and
Page -122-
agreement
model-data
if
improves
is
it
neglected
(Gill ahd
Turner,
1976).
Deszoeke and Rhines
(1976) discuss the solution to the mixed layer
equations using an energy closure of the form:
1/2 e(nu*2 + N2 h2 /2 -(u-ui) 2 ) = mu* 3
a so-called
Eq. A.IV.2
They assumed a constant
'turbulent erosion model' closure.
wind stress, chose h=O initially, and neglected diabatic effects,
found
as
that,
the
layer
mixed
Eq.
evolved,
described
A.IV.2
and
four
different balances between the possible energy sources and entrainment.
The first two were associated with the rapid deepening of an initially
unmixed surface layer, the third with deepening due to shear instability
such as
is
governed by Eq.
A.IV.1.d,
the
and
between wind wave breaking and entrainment.
rate
of fluid
first
three
in the last stage was
stages,
they
long-term deepening of
heating
and
instabilities
cooling,
dominate
suggested
the mixed
along
the
energy
Although the entrainment
small compared with that of the
that
layer.
with
fourth with a balance
balance
Here
our view will be
shear
induced
balance, which is
layer which never completely restratifies
described
this
the
that
Kelvin-Helmholtz
consistent
in a
(hf0t and is subjected to a
variable wind stress.
Finally, a test of the present model using data from Ocean Weather
Station
'N' returned a 98% correlation between predicted and
sea surface temperature (Thompson, 1976).
observed
Page -123-
Meteorological and Solar Data-
For the Thompson model to operate properly, detailed information
the
about
forcing
meteorological
In
is required.
this
section, we
discuss the data which we used.
Winds-
Spectra of wind velocity from various ocean weather stations have
been reviewed recently by Muller (1981) who arrived at model spectra for
In a manner similar to Thompson
both zonal and meridional wind speed.
(1973) and Liu and Thompson (1976), we converted those spectra into time
series of wind speed.
number of frequencies.
First, the model spectra were sampled at a finite
Next the
square
roots of
the
spectral values
were multiplied by randomly generated phases:
1/2ei ,
ur(o-) + iui()=u(T)=IE(,)
where
<eil> = 0,
thus composing a complex vector of Fourier amplitudes.
subsequently
fast Fourier
record of wind speed.
transformed
to
physical
The vector was
space, producing
a
By varying the interval and the domain with which
we sampled the spectra, time series of wind speed differing in duration
and
frequency
of
observation were
obtained.
The
data
so
generated,
however, suffers from the defect that the wind speeds are not governed
by a Gaussian parent population, because the magnitudes of the Fourier
coefficients
have
not been
randomized.
Thus,
it
is doubtful
if our
artificial data represents data which would ever be realized in nature
(Wunsch, personal communication).
On the other hand, we did obtain a
gap free data set, which contained the proper amount of energy.
Page -124-
To the stochastic series determined by the spectra, we added mean
to
m/sec
tides
The
east.
the
taken to be 7
Mean wind speed was
winds, tides, and seasonal signals.
cycle
annual
and
were
changed
from
experiment to experiment; their variation seemed not to effect the mixed
layer and in the results here, the only retained deterministic component
was the mean wind.
Finally, stress was computed according to:
CdlIaua,
S
a sample time series of which is contained in Fig. A.IV.I.a.
Air Temperature-
At
coherent
assumed
1981).
low
with
the
fluctuations
the
frequencies,
wind,
meridional
to convect warm air and
The
magnitudes
linearly proportional to
of
the
i.e.
temperature
in
winds
the
from
those from the north,
temperature
were
made
south
were
cold
fluctuations
(Muller,
were
At higher frequencies,
the wind speed.
made
the
coherence with north-south winds was decreased and a purely white noise
component was added.
To the random temperature series we added a yearly
0
mean, an annual cycle, and a weaker ( .5 C) daily signal.
An example of
a model temperature series is included in Fig. A.IV.1.b; note that there
are
energetic
fluctuations
but
that
record
the
is
dominated
by
the
annual variation.
Solar Heating-
In
character,
solar
radiation
is
different
from
surface
heat exchange in that the radiant energy can penetrate into the water
(b)
(a,)
air temperature versus time
E-W wind stress versus time
-18 28
39.81
96.22
193.43
time
218 64
267.85
325.86
382.27
*-t8.93
39 15
96.34
218.71
153.52
time (days)
267.89
325.08
(days)
Figure A.IV.1
Artificial Zonal Wind Stress and Temperature Data
(a). Here we plot one year's worth of wind stress, as computed from
Muller's model zonal wind spectrum by the fast Fourier transform
The mean wind speed of the record was
technique described in the text.
set at 7 m/sec.
temperature computed according
Here is a year's worth of air
(b).
the cycle
this graph,
In
the text.
to the technique outlined in
includes a mean of 20 OC and an annual variation of 10 OC.
382.26
Page -126-
a parametrization of
As
source.
internal heat
as an
column and act
solar irradiance, we employed:
Bo(z)=MoV(t )(Rez//l+(l-R)ez/L2)
The values assigned to the
(Krauss, 1972; Paulson and Simpson, 1977).
Note that the e-folding depth \~ is
constants are listed in Table IV.1.
shallow
very
radiation
the
Operationally,
cm).
(35
exponential was obsorbed within the first meter.
this
from
The e-folding depth
\2 is, however, a function
/\2 is typically much greater; we used 25 m.
of the biological and sediment content of the water and can vary from 10
m
in coastal water
to
33
m in
the
open
ocean
The
(Niiler, 1977).
magnitude of solar radiation is set by both the zenith
angle and the
hour angle of the sun, and in the present model was computed according
to:
V(t) = e-.13/cos();
V(t) = 0
if cos(4)>0
otherwise,
where:
cos (f)=-cos ()cos
with '
the latitude, Q0 = 2z/(1
(+)cos (Lt )+sin (0)sin (),
day), and
6
the zenith angle.
9
was
computed according to:
6? =23*cos(27L*(355-(julian day))/365)
Note that the above formula extinguishes the solar
(Thompson, 1974).
radiation for roughly half of every day.
The
amount
of
energy
which
enters
the
intermediate
layer
by
penetrative radiation is small; however, it is the exclusive (positive)
heat flux in that layer for periods up to 11 months.
The importance of
solar irradiance in the annual mixed layer structure was demonstrated in
the main body of the chapter.
Page -127-
A.IV.b
Initial Experiments with the Thompson Model-
conducted
We
mixed
of
series
a
using
simulations
layer
the
numerical technique documented in Thompson (1976) in order to illustrate
the
processes
experiments
which
govern
annual
the
reported here, we used
a time
all
and a
80 minutes
of
step
In
cycle.
layer
mixed
vertical grid spacing of one meter.
In Fig. A.IV.2.a we display SST and atmospheric temperature against
This graph demonstrates
time from a four year mixed layer simulation.
the cyclical ability of the model.
graph as determined by the Marine
In Fig. A.IV.1.b, we show a similar
several
that
Note
Climatic Atlas.
features of the mixed layer in the data are well reproduced in the model
results.
For example, SST extrema lag atmospheric temperature extrema
by about 30 days.
year,
sea
surface
temperature, which
Secondly, we note that at almost all times of the
temperature
indicates
is
that
greater
than
flux due
to
actually
the
heat
evaporative exchange is directed out of the mixed layer.
the
importance
of
the
solar
radiation,
necessary to warm the mixed layer.
for
it
atmospheric
sensible
and
This indicates
provides
the
heat
Finally we remark on the asymmetric
annual sea surface temperature signal, first discussed by Warren (1973)
and Gill and Turner (1976).
In Fig. IV.3, shown in the main body of
buoyancy profiles as computed by this model.
the
chapter, we
plotted
They correspond to the SST
graph of Fig. A.IV.2.a, and should be compared to Fig. A.IV.3, where we
include some XBT traces from the Slope Water (Dr. Peter Wiebe, personal
(a)
-
b)
SST
30so
28
S0
0
a
35N
26
•
o
65*W
124
-.-
I
3o11
tem1rtu
1
i
SST
22
20
I
I
I
*
18
+
air
3.
-I
12
h
.29
time
(days
Figure A.IV.2.
10
J
24.6.16
1.9
~I
F
M
A
J
J
M
MONTHS
A
x 10)
SST and Atmospheric Temperature Comparisons.
Here we plot a comparison of model generated sea surface
(a).
Note that
temperatures (averaged over one day), and air-temperature.
sensible
that
indicating
SST,
as a general rule, air-temperature exceeds
Also, we see a
and latent heat fluxes are directed out of the layer.
markedly anisotropic annual SST cycle, with the mixed layer cooling
gradually and warming quickly.
(b).
Here we compare monthly mean SST and air-temperature, as
catalogued in the Marine Climatic Atlas, from a location in the Sargasso
Note that, in agreement with (a), the air
Sea (65 oW, 35 ON).
temperatures are generally warmer than SST, and that the SST cycle is
asymmetric.
S
0
N
0
F
Page -129-
S 0
0-
5
I
10
5
I i
I
SFC
-
0
5
0
15
20
5
3
00 -
150-..
250
E
r 250
4w 300-o
200-230
.
-...
300350
350
Lai 39*45 IN
400
45
5000
-
Long
Time
0
b
25
20
5
2O
10
--
TEMPERATURE CC)
TEMPERATURE
Figure A.IV.3.
0'40
6 Jan
400
W
82
3
"
5
10
15
Lot
39e46N
Lo
69959 w
Tlme
12 may 1981
20
25
3(
TEMPERATURE ('C)
C)
TEMPERATURE
(C)
Temperature Traces from the Slope Water
Here are several temperature traces, taken at various times of the
year, from the Slope water. In agreement with the model output in Fig.
IV.3, we see shallow summer mixed layers, and evidences of both deep
winter-time mixing and penetrative radiation. This data was obtained by
Dr. P. Wiebe and is reproduced here by his kind permission.
Page -130-
communication) obtained at various times of the year.
Clearly a strong
vertical gradient of density develops in Fig. A.IV.3 in the upper 50 m
as the layer progresses
penetrative radiation.
through the summer.
We also see evidences of
Comparing the 50 meter depths in Fig. A.IV.4.b
and d shows an increase in temperature of 3 *C.
perfect match of the very deep
(0(500 m))
In view of the almost
thermal structure
traces, it is unlikely that the difference is due to advection.
of
the strong
density
gradient,
it
is
equally
unlikely
of
these
Because
that we
can
account for the additional heat at 50 m in terms of surface exchange,
leaving radiation as the most probable explanation.
Finally, the shape
of the intermediate layer temperature profile in the data is similar to
that of the model.
Page -131-
The Sensitivity of Mixed Layer Development to
Appendix B.IV
Buoyancy Flux-
The behavior of the mixed layer depends critically on the sense of
the heat
models
analytic
simple
the season.
flux, or
In the present appendix, we construct
towards
view
a
with
understanding
the
distinctions between spring and fall mixed layers.
The Reformation of the Thermocline-
The
early
is mixed
depth over which incoming heat
spring
(see Thompson,
1974).
The
first
changes abruptly in
problem
concerns
the
manner by which the new thermocline is established.
Note from the traces in Fig. IV.3 that the late-winter/early-spring
surface
waters
periods of
are
characterized
(' =0)
calm
by
a
deep
uniform
layer.
in the early spring, surface exchange
During
inserts
buoyancy into the upper few meters, which is subsequently mixed by the
resident turbulence (due to wave breaking, and/or Langmuir circulation,
With the onset of a wind event, inertial oscillations
see Fig. B.IV.1).
are generated and the layer commences shear-induced deepening, as in the
problem of Pollard, Rhines, and Thompson (1973, hereafter PRT). We can
solve the momentum equations, Eq. A.IV.I.a and b, to obtain:
u
2
+ v
2
S
-f2h2
2
x
(l-cos(fo t)).
Eq.
B.IV.1
Page -132-
(TrCy
: 0 )
Bflux
Figure B.IV.1.
Schematic of Thermocline Reformation
In early spring, the intermediate layer has been well-mixed by
winter time convection.
During periods of calm, heat is injected and
mixed in the upper few meters.
With the onset of a wind event,
shear-induced deepening mixes the heat downward and creates a new
thermocline.
Page -133-
Neglecting further buoyancy flux (equivalent to the assumption that the
adjustment to the wind will be rapid), the heat equation yields:
(ab)ho = (b-bi)h
where
Eq. B.IV.2
Lb and ho are the initial buoyancy
respectively.
jump and mixed layer depth
From Eq. A.IV.1.d:
Eq. B.IV.3
h2 = 2x2(1-cos(fot))/(fo2(ab)ho).
Eq. B.IV.3 is valid so long as ht>0O, which is true up to time;
fotc =',
the mixed
decelerate and
inertial oscillations
after which the
layer
remains at its maximum depth, given by:
hm = 2Cx/(fo((Ab)ho)
1 2
/ ).
Eq. B.IV.4
(Z3b)ho is proportional to the anomaly of heat in the mixed layer prior
to
onset
the
A
wind.
the
of
flux
heat
0(50 C)
an
to
due
air-sea
temperature difference, operating for 10 hours, produces a (ab)ho of 36
2
wind
Subsequently, a 1 dyne/cm
cm2 /sec 2 .
in
stored
heat
a 30
The interesting point of this result is that it is
meter mixed layer.
the
stress will produce
the
few
upper
determines the final depth of the mixed layer.
exchange
air-sea
by
meters
which
In PRT, the slope of the
Clearly, the energy in
interior buoyancy profile played the same role.
the inertial oscillations is used to mix the buoyant water down into the
wintertime
over
covers
Several
profile.
deep,
the
resisted
energetically
such
well-mixed
by
buoyancy;
A.IV.1.d,
From
30
meters
requires
a
and
entrain,
hence
depth
reach
of
criticality
deeper
a
wind
1
cm/sec 2
stress
the
large
layer which
mixing
stability
gravitational
the
stratification.
a
Eq.
a warm
events produce
of
buoyancy
0(6
is
then
of
the
jump
at
dynes/cm 2 ) to
density
gradient
of
summertime acts as a barrier to shear-induced entrainment, and the fall
Page -134-
I-l
-7r
C
CU)
cn
(D)
-I
Summer
~n/2VV~b
Cu
tsWEC
CW
-t1.2t
43.84
97.89
151.94
time
Figure B.IV.2.
285.99
260.04
314.08
368.t3
(days)
ht Versus Time.
Here we plot numerical measures of ht (seven day averages) as
determined by Thompson's model. Note that with the onset of summer, the
magnitude of ht decreases markedly, indicating that the seasonal
thermocline acts as a barrier to mixing.
Page -135-
cooling season season must
This is further confirmed by Fig. B.IV.2, which
the fluid.
far into
erode it before the mixed layer can extend
shows a numerical measure of ht, where h is a several day average of the
the fluctuations decreases
of
magnitude
the
that
Notice
model.
from a simulation using Thompson's
mixed layer depth,
markedly during the
summer.
Wintertime Mixed Layers-
the
If
efficiently.
sea
surface
We note
from Fig. IV.3
onto
a discontinuity, b=bi,
without
that
the
deepen
mixed
layer
can
a deep
mixed
layer
the
cooled,
is
intermediate
layer
joins
buoyancy,
which indicates that the heat budget of the layer is dominated by the
surface fluxes
(see Eq. IV.1.c).
In the
following problem, we
shall
investigate these features of the wintertime mixed layer.
The lack of a density step may be explained from an energetic point
of view.
From Eq. IV.1.d, the difference in buoyancy at the mixed layer
interface is given by:
(b-bi) = (u2 +v2 )/h = 2Zx2 (1-cos(fot))/foh 3 .
For
Eq. B.IV.5
a two dyne/cm 2 wind stress acting on a 25 m layer, the buoyancy
jump,
b-bi ,
difficult
for
is
.1
thick
cm/sec 2
layers
and
to
falls
maintain
off
as
sizeable
h3 .
It
density
is
very
steps
buoyancy structure, due to the depth over which the momentum is mixed.
in
Page -136-
Now consider the relative importance of entrainment heat flux with
respect to air-sea exchange.
Suppose the layer is subject to a constant
negative buoyancy flux Ho and the density structure of Fig. B.IV.3.
In
the same manner as the thermocline reformation problem, we arrive at:
2h
2=
2
((h2 -
o
2Ht
2Hot
-
16-
) +
LL
(h
2
oo
t ) 2+
- 2Ho2Ht
(1-cos(f t))
2
where ho is the initial mixed layer depth.
Again, this solution remains
valid so long as ht>0, or until time:
fotc = -
-"
Up to that time, we see:
2Hot/(ho 2 r)
= 20 x104/(25 x10 6 ) = .02 << 1,
and:
4
16 Cx2/(rfo2ho ) = 32/625 = .05 <<1,
where we have
and
Ho
=
used -- x = 2 dynes,
10- 3 cm2 /sec 3 , which
ho
allows
= 50 m,
us
to
I
=
10-4sec-2,
expand
the
Hence:
h 2 = ho2 -
2Hot/t
2
+4rx2(1-cos(fot))/(Pfo2ho ).
The entrainment rate is given by:
2
2
ht = -Ho/(ph) + 2 x sin(fot)/(ifoho h),
Eq. B.IV.6
t
=
square
root.
Page -137-
H0
bo
b
t=O
/
b
=-h
o
bi
bi
Figure B.IV.3.
+r
Schematic for Deep Mixed Layer Entrainment
We will consider the
layers, a well-mixed upper
intermediate layer, such
atmosphere is drawing heat
and that the wind is being
upper ocean to be composed initially of two
layer, of depth ho, and a linearly stratified
that -rh o = bo.
We also assume that the
out of the mixed layer at a constant rate,
maintained at a constant value.
Page -138-
therefore, the ratio of entrainment flux to surface flux is:
2 4
(Zb)ht/Ho = -2cx 2 (1-cos(fot)/(fo
h 1')+
Eq. B.IV.7
2 4
+ 4Lx4sin(fot)(1-cos(fot))/(fo3rHoho h ).
The orders of magnitude for the terms in Eq. B.IV.7 are:
-2-x2(1-cos(fot))/(fo2h4r) = 16/625 << 1,
and:
2 4
3
4Zx4 sin(fot)(1-cos(fot))/(fo C1Boho h ) =
= 64/1562.5 << 1i,
so the approximate heat equation governing the layer is:
hbt= -
(b-ba) + Bos -
Bo (
- h )
.
Eq.
B.IV. 8
Page -139-
Verification of the Annual Mixed Layer
Appendix C.IV
Equations-
C.IV.a
Choice of h
The solution to the summertime equations may be compared to data to
The general solution for surface buoyancy during summer is:
specify ho.
b = Aoe-(t
-
tl)/ho+ Ccos(-t + 7t)
Ao = bl-
(C cos(j.tl+Z)
C = p2al/(,.2ho 2 +
g
+ Dsin(Jrt+ Z),
where:
+ Dsin(irtl+ )),
Eq. C.IV.1
2),
and:
2
D = -rhoPa1/(2ho + 2),
where
bl
'spring'.
is
the
initial
Eqs. C.IV.1
sea
surface
buoyancy
at
are valid for tl< t < t2 where t2
transition from heating to cooling, i.e.
the onset of
onset
the
tl,
marks
'fall'.
of
the
Notice
that a mixed layer 'relaxation time':
trl = ho/P
emerges from the solution and defines the time necessary for the summer
mixed layer to lose memory of the late winter surface buoyancy bl.
The onset of fall, t2 , is defined by:
-P(b-ba(t2)) + Bf = 0
Eq. C.IV.2
where we have used the short hand notation:
Bf = Bas(0)-Ba(-h).
Using Eq. IV.2 in Eq. C.IV.2 returns an implicit equation for t2.
Page -140-
Aoe-P(t2-tl)/ho + Ccos(.t2+,)
+ Dsin(1t2 + :) +
Eq. C.IV.3
- alcos(Lrt2 + ')=O.
Assuming that trl is short compared to t2-tl:
exp(-P(t2-tl)/ho) = O(exp(-thtseason/trl) << 1,
and to the neglect of the exponential term:
tan-l((al-C)/D) + n0 = "t2.
Evaluating
argument
the
of
inverse
the
Eq. C.IV.4
tangent,
we
see
that
t2
is
independent of the magnitude of the diabatic forcing and depends only on
system parameters.
tan-l~rho/p) = rt2 + n,-
Eq. C.IV.5
(necessarily
the mixed layer maximum SST
Note from Fig. A.IV.2.b that
occurring at t2) lags the atmospheric maximum temperature by one month.
at
maximum air-temperature occurs
IV.2,
From Eq.
7-,
therefore
the
corresponding value of t2 is 7'/6, which in Eq. C.IV.5, returns a value
of:
r
With P =
2/(10 7
cm/sec
10-3
sec),
we
ho/p
= .6.
(Frankignoul,
obtain
personal
a value for ho
of
communication),
30 m.
It
and
'=
is this
value
mixed
layer
which we have employed throughout the chapter.
C.IV.b
Validation-
Using
the
solution
analytical
to
the
summertime
equations, we can estimate maximum sea surface temperature.
at t2 to the neglect of the exponential terms is:
bmax = (1/2)2al/(g-2ho
2
+
p2 )
+ Bf/p + a2
Eq.
C.IV.6
Using ho=30m and a typical value for Bf, we have:
bmax = a2 + .87al + .6.
Eq. C.IV.7
Eq. C.IV.5
Page -141-
in Eq.
'.6'
The
Eqs. IV.1,
we compared
the maximum SST predicted
depicted in Fig. IV.1.
cm/sec 2 ,
and
This
C.
al=1
1 OC
carried
and
predict
too warm.
with
out,
cycle are a2=0
describing the
The parameters
cm/sec2,
is about
locations were
a
maximum
such checks
Several
result
general
the
SST
of
very
the
simple
the
for
responsible
equations
27.3
at various
predicted
that
temperature is 0(.5*C) too warm, but nowhere grossly incorrect.
of
with
C.IV.7
by Eq.
As an example, consider the Sargasso Sea air-temperature cycle,
data.
0
test of
a
As
radiative heating.
to
due
is
C.IV.7
In view
the
prediction,
agreement was surprising and heartening.
Analogous analytical tests for the wintertime form of the equations
were
not
found;
in
however,
numerical
it
solutions
was
noted
that
relevant sea surface temperature ranges were well predicted, as were the
dates of shift from winter to spring.
A
second
check
independent
of
Eq.
IV.1
was
made
direct
by
In Fig. C.IV.1, we have
comparison with the Thompson numerical model.
plotted two atmospheric temperature cycles, one a smooth version of the
other,
which
unsmoothed
were
version
used
as
was
used
smoothed counterpart Eqs.
the
to
IV.1.
data
force
sets
the
for
the
Thompson
The comparison
comparison.
model,
and
The
its
of predicted SST from
the Thompson model, averaged over one day, and that from Eqs. IV.1
displayed
in
Fig.
C.IV.2;
the
agreement
is
very
good.
is
Similar
comparisons between model mixed layer depths were equally encouraging.
Page -142-
ao -
c
rW)
17
-18
39
9
153
time
Figure C.IV. 1.
210
267
324
381
(days)
Comparison of Raw and Smoothed Air-Temperatures
Here we compare the two air-temperature cycles used to compute the
responses of Thompson's model and Eqs. IV.1. The 'raw' air-temperature
cycle is actually a seven day average of the employed temperature
cycle.
Page -143-
r_
C%
i,,
'0
cJ
a.
a
L t4
(D
.n
hC
Cu"
10
r
-
cn
-1
'-18. 12
38.83
I
95.78
152.73
time
Figure C.IV.2.
2.09.67
266.62
..
323.57
380.52
(days)
Comparison of SST from Thompson's Model and Eqs. IV.1
Here we compare SST as computed by Thompson's model, and by Eqs.
IV.1, subject to the air-temperature cycles displayed in Fig. C.IV.1.
We have neglected penetrative radiation and
In Eqs. IV.1, ho = 30m.
The agreement between the
have run the comparison for one year.
predicted SST's is rather striking.
Page -144-
THE WIND FORCED SPIN DOWN
CHAPTER V.
OF GULF STREAM RINGS
Introduction-
V.a
a
on
oceans
The world's
dispersive
a
constitute
scale
planetary
medium; so an isolated pressure pattern governed by linear dynamics will
rapidly
disperse.
pattern
is
For
decreased.
radiation occurs in 0(6 months)
a
Perhaps
importance.
be
markedly
wave
linear
by
caused
loss
energy
an
most
convincing
most
the
pressure
(Flierl, 1977), while wave radiation by
at
effects
Ring
nonlinear
same
will
spread
a Ring
of
decay
the
example,
the
of
evolution
dispersive
of
rate
the
computed,
non-linear
the
If
of
secondary
this
of
demonstrations
were the numerical Ring simulations of McWilliams and Flierl (1979) and
Mied and Lindemann
In their experiments, it was found that the
(1979).
dynamic tendency for Ring persistence was so strong that the decay which
friction.
a
the subsidence
presence
e.g.
processes,
non-conservative
the
barotropic
sizeable fraction of
the thermocline
the
mode,
model.
Ring
Ring
dispersive
simulations,
of
of
frictional
non-dispersive,
numerical
to
due
mostly
In McWilliams and Flierl's equivalent barotropic simulations,
(79%) of
most
by
was
occurring
was
but
decay
friction
total Ring energy
loss.
could be accounted for
their
In
increased
was
still
layer
two
accounted
by
the
for
Even a reduction
a
of
the viscous coefficient by a factor of 10 was insufficient to remove the
overall dependence of Ring
Lindemann
noted
that
decay
different
on
friction.
viscous
In agreement, Mied and
coefficients
produced markedly
Page -145-
From these results, we would
different time series of Ring amplitude.
decay
in the
conclude that frictional processes play a major role
of
oceanic Rings.
in the
First,
This is both interesting and somewhat unsettling.
numerical experiments, the viscous coefficients were purposely assigned
very small values in order to minimize the influence of non-conservative
seemed
processes, which, therefore,
to be having a disproportionately
Second, the dependence of
large effect with respect to Ring amplitude.
amplitude decay on viscosity is at best bothersome given that viscosity
is
generally
modeled,
crudely
rather
and
currently
is
little
that
In the present chapter, we are going
understood about oceanic friction.
to compute the Ring decay induced by wind forcing, using McWilliams and
Flierl's model, in an effort
model as
the
to
order means
first
oceanic Rings are so affected.
in that
replace the ad-hoc viscosity
of
spin-down, and
Ring
One advantage of
to
decide
doing so comes
if
from
the fact that the relevant coefficients can be computed in terms of well
known quantities.
Observations of Ring Decay-
Real Rings are observed to age/decay, which is characterized by a
relaxation of Ring isotherms towards their resting state depths.
direct
observations,
(1971)
Parker
found
that
average
the
vertical
velocity of the 170C isotherm in cold Rings was roughly 50 cm/day.
recent
direct measurements
(Vastano, Schmitz,
and
of
Hagan,
subsidence
1980).
put
and
More
(60-100)
cm/day
Richardson
(1976)
this at
Cheney
From
Page -146-
concluded from energetic considerations that Rings survive for roughly 2
Notice
Parker's
subsidence
from a
this
that
years.
lifespan
rate applied
resting depth.
agrees
to an
that
with
isotherm initially
Such deflections
of
temperature
from
computed
300 meters
surfaces
are
common in Rings (see Fig. IV.11).
A fuller understanding of Ring decay would allow us to make many
useful statements with regards to property exchange, and therefore the
importance
Rings
of
in
the
general
estimates
Previous
circulation.
indicate that Slope Water/Sargasso Sea exchange due to Rings is large.
Implicit in these has been the assumption that Rings disintegrate after
the anomalous parent
formation, thus leaving in the host region all of
Recently
region material garnered at formation (the Ring Group, 1981).
the general perception of Ring death has shifted from disintegration to
a view that Rings are removed from the circulation through absorption by
the Gulf Stream (Richardson, 1980).
This
being a sort
of
reverse of
formation, that water present in the Ring trapped zone at coalescence,
possibly composed of a substantial fraction of the original water mass,
is
returned
to
the parent
region.
The
total Ring-produced
between the regions is then closely linked to
exchange
the modification of
trapped water mass by the host region (Schmitz and Vastano, 1975).
the
As a
Ring decays, vertical circulations are set up; from an understanding of
the decay, we can estimate where
the flow advects
into or out
of the
trapped zone (see Fig. V.1) and therefore infer what parts of the host
region water
column invade,
or are invaded by,
the Ring.
From such
information, we can begin to make educated guesses about what types of
Slope Water materials enter the Sargasso and vice versa, and perhaps the
quantities involved.
Page -147-
Trapped zone volume of Gulf Stream Ring
C
Surface - -
-1500 m - "
r
K --
' - '
-2500 m -
Figure V.1.
Schematic of a Gulf Stream Ring Trapped Zone
Here we show the vertical distribution of the trapped zone.
The
arrows are indicative of inflow/outflow as set up by warm Ring decay.
The depths at which mass exchange occurs influences the exchange of
biology and chemicals.
Page -148-
Consider how stress is affected by the presence of surface water
velocities.
1 m/sec over the
An eastward wind, blowing at a speed of
surface of water also moving to the east at a speed of 1 m/sec, can not
Conversely, the stress by the same
transfer any momentum to the water.
wind on water flowing to the west at 1 m/sec is
Both statements
water which is motionless.
the
greater than that on
derive from the fact that
relative speed of
the air and
stress
is a nonlinear function of
water.
Consider now the consequences of an eastward wind blowing across
At the northern edge of the Ring, the
a warm core Ring (see Fig. V.2).
water is being accelerated by the stress and hence is capable of flowing
To the south, the fluid is
up the pressure gradient, or into the Ring.
decelerated and in response flows down the gradient, or away from Ring
center.
Since the momentum flux into the ocean from the atmosphere is
greater where the water is flowing against the wind, the southward mass
flux
in
the
surface
boundary
layer
is greater
the
to
Mass
south.
conservation then requires an upwelling in the center of the Ring, which
can either elevate the main thermocline or generate cyclonic relative
vorticity.
Both decelerate the Ring.
The interaction of the wind with a sheared current has also been
considered by Stern (1965, 1966) and Niiler
the divergence
non-constant
driven by variations
local
rotation
rate,
in the
(1969).
Ekman transport
(f+vx-uy),
attempted to explain anomalous isothermal
In these papers,
was
due
computed.
structure under the
to
a
Niiler
surface
layer of the Gulf Stream in terms of the induced upwelling, and Stern
considered the forced response of geostrophic eddies.
is
proportional to
second
order
derivatives
of
This divergence
the velocity and,
as
Page -149-
uo- u small
z
UG
0
0-
.6-
Ua-UN large
N
-
Ekman layer
t
150 C
isotherm
t = 0
Figure V.2.
--
t hermocline
tendency
Spin-down Schematic
In (a), we show the effect of an eastward wind blowing across a
cyclonic Ring, a temperature transect of which is shown in (b). Because
the speed of the wind relative to the water is greater to the south, the
In (b) is a schematic diagram of the effect
stress is greatest there.
of the pumping on the interior. The solid line is the thermocline, and
the dashed line, the tendency of the thermocline due to we*
Page -150-
Stern showed, forces
Therefore, since the
a translation of an eddy.
eddy is moved rather than dissipated, and this divergence operates
smaller
velocity
than
scales
(Stern's
w
that
is
due
to
proportional
to
uxx,
relative
the
of
variations
and
the
on
air-sea
present
w
to
Ux), the dissipation rates computed in the present chapter will not be
affected by Stern's and Niiler's mechanism.
We shall, however, consider
it further in Chapter VI.
V.b
Ekman Pumping-
The formula for wind stress is:
Eq. V.1
= Cd(ua-u) ua-R
where
Cd
is
an empirically
coefficient,
determined drag
ua
the wind
In terms of Fig.
velocity, and u the water velocity (see Table V.1).
V.2, a constant eastward wind blowing across the surface of a Ring:
C
Av)/2
= Cd(ua-u,u)((ua-u)2+v2
2
Eq. V.2
.
Assuming ua >> (u,v), we can expand the square root to obtain:
C = Cdua2(1-2u/ua,-v/va) "
= Io
where IT o is an
Eq. V.3.
- 2Cduaui- Cduavj
'undisturbed' (u=O) surface stress.
The divergence of
the wind driven surface flow is proportional to the curl of Eq. V.3:
Eq. V.4
2
we = (k.(curl(Co))+ Cdua( uy-vx))/fo.
In most oceanographic problems, the purely wind driven divergence,
curl(.Uo), is assumed to overwhelm the second term.
is then determined by an externally
problem, we will assume
that the
Interior evolution
specified stress.
curl of
C,
vanishes,
In the present
i.e. that
the
Page -151-
Table V.1
Symbols and Definitions-
Value
Definition
Symbol
............ Wind Stress
TO
fa
Co
Cd
Cd
Ua
0
00
u
0000*00
f0
0
o
o
L
H
............
...........
Air Density
..........
Water Density ...........
...... Unperturbed Wind Stress ......
........ Coefficient of Drag ........
10- 3gm/cm 3
1 gm/cm 3
constant
10-3
... Normalized Coefficient of Drag ..
........... Wind Velocity
........... Water Velocity
......... Coriolis Parameter ........
. N-S Gradient of Coriolis Parameter
10-6
............ length scale ...........
......... depth of the ocean ........
...........
We
....
Cn(x,y,t)
Fn(z)
E
F(0)
p
S0
0 0
00
modal amplitude function
......
....
........
vertical structure eigenmode
.. eigenvalue of structure equation
surface value of first eigenmode Fl . 3
.. complex planetary wave frequency
.............
Cdua/(PLH) .............. variable
.............
..
r
Q
10-4 sec-1
1.7 x 10-13
cm-1sec-i
60 km.
5 km.
F(0) 2
HI/H
.................
2
. . . . . ............
. . . . . . . ...
.... ratio of Rossby Deformation
Radius to length scale .......
.flow
U/(2L2 ) .......
steepness, .....
variable
small
2
5.5
... baroclinic self interaction
111
FFlFldz/H............. 1.8
coefficient of biharmonic friction
... angle of rotation (appendix A)
K
amplitude tendency due to diffusion
("'Adif)t
(Xf)
t
........
. amplitude tendency due to forcing
Operators
2
S6
.....
.....
2
/)x
2
...........
+ 2d2/ y2
......
5 x 10-5
Page -152-
eastward
is
wind
directed
free
of
Realistically,
shear.
curl(o)
resides at the scales of the basin, and is therefore nearly invisible to
a Ring.
Hence:
Eq.
V.5
Eq. V.5
we = Cdua(2uy-vx)/fo
The situation described in Fig. V.2, of a mean wind and coherent
surface velocity, is far too simple.
The wind fluctuates on time scales
ranging from seconds to years, and excites responses in the upper ocean
of
similar
frequencies.
We
can
for
account
variability
in
the
computation of the mean stress, however the only significant change in
Eq. V.5 is the replacement of ua with an rms measure of the wind,
lual
Without employing this notation, we shall continue
(see Appendix A.V).
to interpret ua as such.
From Eq. V.5, we can obtain an order of magnitude estimate of the
forced vertical velocity.
we* = 0(2Cdua u/(fo'x)) =
= 2x10
4
10-610 3 100/6x10 6 cm/sec =
= 30 cm/day,
where the '*' denotes that we is a dimensional variable.
30 cm/day is a
sizeable surface divergence within a quasi-geostrophic framework.
Wind drift velocities are typically a few centimeters per second,
centimeters
per
velocities
by
the
In terms of the interior streamfunction
H:
while Ring geostrophic velocities are several tens of
second;
therefore, we
geostrophic velocities.
can approximate
We* = -Cdua(2"yy +
the
xx)/fo
surface
=
3dE2
Eq. V.6
Page -153-
where we have denoted (32/3x2+222/y2) by the symbol
2.
Except for the
factor of 2, the form of the pumping looks remarkably like that due to
The
bottom friction.
at
removing
'2' reflects that zonal winds are more effective
in
shear
zonal
velocity
than
in
meridional
velocity.
Non-dimensionally, Eq. V.5 becomes:
=
we =-'dua/(pLH)(27 yy+xx)
with a typical value for
(2yy +xx).
Eq. V.7
D being:
= 10-6103/(6x10 6 5x10 5 10- 13 )
= 0(10-2/3) << 1.
V.c
Planetary Wave Spin Down-
as
From the form of we, it appears that wind forcing will act
viscosity.
a
In order to verify this claim, consider the effect of Eq.
V.5 on a planetary wave.
We assume the interior is governed by quasi-geostrophic dynamics,
Eq. II.1, and that the buoyancy frequency, N(z), is constant.
Consider
a solution to Eq. II.1 of the form:
f
=
F(z)ei( k x + my
)-
pt
Eq. V.8
where p is a possibly complex frequency.
wave will be exponentially damped.
Note that if Real(p) > 0, the
Substituting Eq. V.7 into Eq. II.1
returns an equation for the vertical structure function F(z):
Fzz - E 2 F = 0
Eq. V.9
E2 = S(k2 +m 2 +ik/p)
Eq. V.10
where:
Page -154-
The boundary conditions become:
Fz
and
at z=-1
F z =-FS(k 2 + 2m2 )/ p
at z=0.
The solutions of Eq. V.9 which meet the bottom condition are:
F(z) = Acosh(E(z+l))
where A is an arbitrary constant.
An application of the upper boundary
condition returns the implicit equation:
Etanh(E)= SJ(k2 +
After
expansion
in
powers
of
2 2
m )/p.
T
(<<1),
Eq. V.11
obtain
we
the
lowest
order
equation:
Eotanh(Eo) = 0
Eq.
V.12
which has solutions:
Eo,n = 0, inT-
Eq. V.13
for all integer n > 0, where we have denoted the infinitude of solutions
for Eo with
the second subscript
'n'.
From Eq. V.13, we obtain the
solutions:
Po,n = -ik/(k2+m2 +(ni)
2
/S)
Eq.
V.14
Page -155-
the
lowest order
complex frequency.
surface
pumping, the
po, n
for
are
the
Notice that
Rossby wave
the weak
due to
frequencies,
and
the
lowest order solutions for F from Eq. V.9 are simply the complete set of
Rossby wave modes.
The presence of the non-homogeneous surface boundary condition is
From Eq. V.10 we have:
felt at the next order.
El,n
=
S(k2+2m 2 )/((Po,n)(Eo,n))
Eq. V.15
n > 0,
and, according to Eq. V.11:
Pl,n = 2(Eo,n)(El,n)(po,n)
= (k2+2m2)/(k2+m2 +(ni)
2
2
/S),
/(Sik)
=
Eq. V.16
again for n strictly positive.
In
order
to
recover
P1 ,0,
to the barotropic mode, we return
is
determined
not
the
by
expansion in powers of
9
above
the
complex
to Eq. V.12.
analysis
is breaking down.
is an
frequency
appropriate
The fact
indication
that p1,0
that
our
Such an expansion implicitly
assumes Eoo is 0(1), which contradicts the solution:
Eo,o = 0
from Eq. V.13.
The proper interpretation of the above is that for the
<
barotropic mode, Eo,o < 1.
limit
of
An asymptotic analysis of Eq. V.12, in the
small E, shows E = 0()1/2).
Note however that
the lowest
Page -156-
order
solution
for Po,o
from
Eq.
V.14
does
not
vanish.
Thus,
the
expansion for p appropriate to the barotropic mode will still require a
non-zero
0(1)
term, which
turns
out
to
be
the
barotropic Rossby wave
eigenfrequency:
Po,n = -ik/(k2+m
2
).
The 0(Z1/ 2 ) barotropic solution is:
2
E1,0 = (S(k
+2m 2 )/(Po,n))1
/ 2 ,
Eq. V.17
and the 0(0) correction to p:
Eq. V.18
2
2
2
Pl,n = (k + 2m2)/(k +m ).
Notice that for both the baroclinic and barotropic modes, the correction
to p enters at 0(D); hence, the dimensional spin down time is given by:
tc = 1/(Opl,0
)
= O(PLH/Cdua)
=
= 300/10- 6 secs = 0(3000 days),
where we have used our previous estimate for J.
the surface pumping is to cause the wave to decay.
Clearly, the effect of
Page -157-
V.d
Nonlinear Vortex Spindown-
discuss
In this section, we
the effects of Eq. V.5 on nonlinear,
fully nonlinear
equations,
forced to proceed numerically.
be
we will
of the
Because of the complexity
coherent, quasi-geostrophic systems.
The inclusion of nonlinearity, however, is essential for realistic Ring
modeling,
leading to
long-lived
analytical
solutions
of permanent
as
as well
numerical solutions
exact
(Flierl, Larichev, McWilliams,
form
and Reznick, 1980).
Barotropic Mode Scaling-
model
The
is
use
will
we
equation
interior
the
equivalent
barotropic equation, Eq. 11.14, which governs the baroclinic evolution
With that configuration,
of a two-layer system with a thin upper layer.
the
nonlinear baroclinic/barotropic
however
ignored,
baroclinic
extent
the
to
through
evolution
which
the
investigated on a case by case basis.
friction
as
well as
topographic
and may
interactions are weak
the
barotropic
boundary
affects
mode
conditions
be
must
be
Presently, we are ignoring bottom
effects,
but
have
included
surface
pumping which invokes contributions from both modes:
We = D(Fn'2cn)
=
=
((uoxx + 2iXoyy)Fo+ ( ixx + 2 Syy)F1)
(refer to Table V.1 for definitions).
Eq. V.19
Using the scale analysis which
led to the equivalent barotropic equation, the barotropic amplitude co
Page -158-
is
0(6)
where
= (Hi/H 2 ) in
a two layer system and is a function of N2
for continuous stratification.
small.
6-1/2.
Similarly,
For shallow main thermoclines
is
0/I
0(
1/2)
and
FI(0)
'
will be
scales
as
The latter comes from ti he normalization condition:
SFnFmdz =
combined with the constraint:
0
Fndz = 0.
= 1;
Recall Fo(O)
therefore,
)
to 0(
the barotropic
amplitude
Eq.
in
V.19 is negligible, and the interior equation becomes:
(V2-j2)xt + Q111J(,(,72-;-2)Y)
D = F(0) 2 Cdua/(LH),
where
small
a
to
the
still
dissipation.
necessary
dissipation.
Note
to
include
X =3 $ 2
equivalent
the
decomposition.
modal
+
=
F(0) 2 3,
To
insure
a
higher
Eq. V.20
0
barotropic
the
factor
numerical
order,
plus
equation
of
F(0)2 owing
stability,
it
is
enstrophy-consuming
Following McWilliams and Flierl, we employ:
6
D = Ko o
with K 'small', i.e. its reduction does not affect the present results.
Parameters-
There
solutions.
are
a number
of
parameters
in Eq.
V.20 which
Ring evolution as affected by the parameters
111 has already been the subject of much discussion in
K,
affect
1 2,
its
Q, and
the literature.
We are currently interested in observing the influence of
3 and hence
Page -159-
have adopted a standard set of values for the other parameters:
Q=5.5, r2 = 2, -i111
-5
1.8, and K=5x10
=
which, with exception of K, are the same as
the standard set used by
-4
and we have reduced this
They set K = 5 x 10
McWilliams and Flierl.
and
above values for -111
02 were computed using the mean buoyancy
Note, 1,2 = 2 implies a Ring of scale
profile from MODE (Flierl, 1978).
Due to
60 km and Q=5.5 a characteristic velocity of 33.7 cm/sec.
this
decomposition,
modal
The
10 for reasons which will become apparent.
value by a factor of
scale
corresponds
to
of
a depth
the
0(700m).
Because of the surface intensification of the first baroclinic mode, the
appropriate surface velocity scale computed from 33.7 cm/sec is roughly
100
cm/sec
is
which
an
for
scale
appropriate
velocities.
Ring
0 includes the wind speed ua in its definition, and therefore will be
Finally, r2
subject to variations in the present numerical experiments.
was
altered
from
its
standard
value
of
2, to
values
1 and
of
4,
corresponding to length scales of 45 km and 90 km respectively.
Unforced Results-
McWilliams and Flierl and Mied and Lindemann discussed the unforced
dynamic evolution of their model Gulf Stream Rings.
points
from those
experiments
concerned
the
One of the major
asymmetry
of
the
field.
Although their initial conditions were chosen to be axisymmetric,
Rings quickly developed azimuthal dependence.
the
The largest contributions
to the azimuthal field came from the two closest pressure centers, whose
relative
signs were
determined
by
the
initial
choice
of
Ring
spin.
Anticyclonic Rings developed high pressure centers to their west and low
Page -160-
pressure centers to their east,
both
experiments,
period
the
throughout
Ring
the
with
coherent
remained
extrema
local
In all
and vice versa for cold Rings.
integration and apparently had an important
of
McWilliams and Flierl argued that Ring propagation
effect on its path.
was largely set by the dispersion field.
Forced Results-
repeating McWilliams
After
Flierl's
and
experiments,
to
partly
verify our numerical code and partly to generate data appropriate to an
In Fig.
unforced Ring, we performed a set of forced Ring experiments.
V.3, we plot ac at
them, a
typical forced Ring
remaining
The
section.
Note from Eq. 11.13:
d = -F
represent
plots
Since
resting depth.
the
to as
referred
the
next
is linearly proportional toCK.
Thus
will
be
(x, y, t)/N2 ,
that isopycnal displacement, 'd',
evolution from one of
which will be
experiment
experiments
expl0v.
these
stages of
several different
catalogued
in
Eq. 11.13
deflection
of
a
density
surface
from
its
'd' is of the opposite sign than c<, the Ring in
Fig. V.3 corresponds to a depression in the thermocline.
The initial Ring pattern in explOv was chosen to be a Gaussian:
c<(t=0) = exp(-r2)
in agreement with the initial condition used by McWilliams and Flierl.
Subsequent
development
experiments.
shall
take
of C= was
very
similar to
that noted in their
The paths of the central pressure maximum, xc(t), which we
to
define
Ring
translation,
and
the
neighboring
centers
Page -161-
COMFOULED
FaI
.9ttE*I
L,,
~ ~
I
ro 9.9e
~
i
a7INrEVALSOF
E-1
~,
~
.i18E-1L
~
I
COMfOUNIE
FRon*4.1799-tto8.992 -.
L
f INTERVALS
OF 1.1t1
E-t
-
.899
1.0
-. 12
II
I
I
CONTOUREDO
FlOW*4.48t-L r0 8.641E-L
Figure V.3.
AITMTERVALS
OFL.II E-L
Forced Ring Evolution
Here we have plotted the horizontal
function of time, from forced experiment
Q=5.5, E111=1.8,
correspond to (a)
structure function, ( , as a
expl0v.
In this experiment,
The pictures
K=5x10 - 5 , p 2 = 2, and ua = 7 m/sec.
the initial condition, (b) 50 days, and (c) 75 days.
Page -162-
This was the case in all
matched those of the comparable unforced Ring.
experiments where we were able to compare analogous forced and unforced
We can offer an explanation for this similar to one used by
results.
McWilliams and Flierl to clarify unforced Ring motion.
The
'center of
mass' of the system, X, defined by:
ix
X
xdA
evolves according to:
Xt
=
-1/2
Yt = 0
Note that the evolution of X
which are found by operating on Eq. V.20.
is independent of the forcing.
Deviations of Ring motion from the path
of X are due to the radiation field in the vicinity of the Ring.
Since
the effect of we is not to excite any novel structure in the developing
field,
but
rather
maximum
pressure
to
is
cause
dissipation,
determined
by
the
wave
of
location
dispersion
and
the
Ring
nonlinear
interaction just as in the unforced case.
The
interesting, novel behavior
forcing is
reflected in the history
of
of
the
Ring
caused
by
the wind
the streamfunction amplitudes.
In Fig. V.4, we compare the maximum pressure amplitudes as a function of
time from expl0v and a comparable unforced experiment
and Flierl).
(a la McWilliams
Notice that in the wind forced experiment, there is better
than a factor of two increase in amplitude decay.
After 80 days,
the
pressure maximum of the forced experiment has lost more than 12 % of its
initial value, while that of the free Ring is down by =0(5%).
Page -163-
comparison of amplitude vs. time from
forced experiment expiOv.dat, and free
experiment
exp9v.dat.
k=5e-5, q=5.5
O
C
0
Lf
o .,
E-
C
M
-00
E
O
I
14.43
!
time
Figure V.4.
!
I
49.19
31.81
in
i
66.57
83.95
days
A Comparison of Forced And Unforced Ring Decay
Here we compare the time histories of <c from forced experiment
9
expl0v and unforced experiment exp v. The parameters are standard, with
Note that the 2c in explOv
calculation.
ua = 7 m/sec in the forced
decays roughly 2.5 times faster than that in exp9v.
Page -164-
Fig V.5
CONTOURED
FROM -L.B86E-3
TO 8.458 E-3
AT INTERVALS OF 8. 82 E-3
.0006
o
<
I)
o
0-
-0
--
CONTOURED FROM -7.62tE-3
-0
0
a
o.
-
v""
__
.001
0
TO 2.349 E-3
caif
AT INTERVALS OF
_
1.I8
E-3
-"
b
continued
next page
)t
Page -165-
omplitude tendency due to aoilterms.
ron exptsv.dot
Time In doya=40
CONTOURED FROM -7.688E-1
._,z
i
:
:
:
:
i
:
I
:
TO 8.399 E-l
i
i
i
i
i
i
i
AT INTERVALS OF 1.787 E-t
i
I
i
i
i
i
i
i
i
I
I
i
I
l.,i
.95
.84
t:::
JIJ
Figure V.5.
(a).
K=5x10 -
5
....
...
....
.....
...
....................
i1...1i........i.....±J
Maps of (dif)t
( 'dif)t
=
(,2-
and ("f)t
2)-
Kqlx
from
expl0v,
at
day
40,
with
40,
with
and p 2 =2.
(b).
ua=7m/sec.
(O-f)t
=
( 2
Note that, in the Ring, ( dif)t
(c).
1
(o)t
2)-lwe
<<
from
(f)t.
from expl0v at day 40.
expl0v,
at
day
Page -166-
comparison of amplitude tendencies due
to bihormonic friction and forcing
from explOv.dot
(
C
ff)t
X--a
C
C
o
aE
M
o
-I
dif t
IM
I
2.
14.26
time
Figure V.6.
]
t
31.42
48.58
..
.. -
65.74
1
,
82.9
(days)
("f)t Versus ( dif)t
Here we compare the magnitude of the non-conservative forces at
Plotted are (O<dif)t and (F)t for 80 days from expl0v.
Ring center.
Note that (Cf)t > > (O<dif)t"
Page -167-
The Relative Importance of Forcing-
From Fig. V.4 it is obvious that the maximum amplitude decay has
the
in
significantly increased
experiment,
forced
and
soon we
shall
quantify subsidence rates for comparison with those observed in the open
ocean.
However, in light of previous studies, it is necessary to show
that the decay we are seeing is primarily the sequent of the forcing,
and not of the biharmonic friction.
In
order
friction,
we
numerically
evaluated
the
relative
the
the question of
to address
decay
in
importance
of
due
to
amplitude
friction, defined by:
("dif)t
(see Fig. V.5.a).
=
(,2 -
- 2 )-1K
6
-,
Eq. V.21
Similarly, we computed the amplitude tendency due to
Ekman pumping, given by:
2
("f)t = (,72 - P )-lwe
Eq. V.22
Note that for the value of
(see Fig. V.5.b).
the mean wind speed in
this experiment, 7 m/sec, the magnitude of (cq)t in the vicinity of Ring
center is generally 6 to 8 times greater than ('dif)t (-.009 for (Af)t
center held
as compared to -.001 for (odif)t); a relation which at Ring
throughout explOv, as Fig. V.6 demonstrates.
Away from the maximum in
pressure, ("f)t and ( dif)t are somewhat more comparable, although (~f)t
tends to be larger.
However, as we shall see, away from Ring center,
the influence of both forcing and dissipation is relatively negligible.
Consequently,
we
conclude
coefficient, K = 5 x 10-5,
that
for
this
value
of
the
biharmonic
frictionally induced amplitude decay is
of
secondary importance, and that the dominant non-conservative process is
Page -168-
the forcing.
value
of K as
It was
that we chose
largely for this reason
10-4 as
than 5 x
standard, rather
the above
did McWilliams and
Still, the dominance of forcing as a non-conservative influence
Flierl.
is not overly sensitive to K; in similar experiments with K = 5 x 10-4,
was 0(40%) of (c
('dif)t
Finally,
we
)t.
compare
to
( <f)t
change
the
in
c
caused
by
all
processes, which includes advection and dispersion as well as friction.
The amplitude trend due to all effects is defined by:
("a)t
=
;-2)-1(-J(
,(52-.2) )-
6
x + K. ,
and a map of it is included in Fig. V.5.c.
Eq. V.23
+03 2)
First we should notice that
for the most part, ('a)t is much greater in magnitude than either (df)t
or ('dif)t .
Clearly, away from Ring center, the non-conservative terms
of Eq. V.23 are of negligible importance.
Were we to further dissect
Eq. V.23, we would find that the largest contributions to that equation
are due to advection and dispersion, between which there is even a fair
Near Ring center however,
amount of internal cancellation.
through a zero crossing.
of
the
total amplitude
(cxa)t goes
Therefore, at Ring center, a sizeable fraction
change
can be
effects, or in this case forcing.
attributed
to
non-conservative
By checking the magnitude of
(cCf)t
against (xa)t at xc, it was found that most (0(50-60%)) of the decay in
amplitude at Ring center is
due to the
forcing.
In this
regard, we
agree with McWilliams and Flierl who also attributed the subsidence of
Ring isopycnals to non-conservative processes.
Page -169-
V.e
Subsidence Rates-
It
is
clear
from
Figs.
V.3
that
the
local
pressure
associated with what we identify hydrographically as
'Ring'.
maximum
is
Therefore,
it is the evolution of the amplitude at this point which we will compare
Using Eq. 11.13, we can compute a
to the field estimates of Ring decay.
time series of
'd' at Ring center, and hence the isopycnal subsidence
associated with the numerical Ring.
In order to facilitate comparison
with the relevant Ring observations, the values of the derivative of the
vertical structure function, F(z)z, and of the local buoyancy frequency,
N2, were computed from the MODE data at a depth corresponding
170 C isotherm (Flierl, personal
communication).
A
'd' of
to the
257 m was
computed for the initial Ring thermocline in expl0v, which subsequently
subsided at a rate of .4 m/day.
Parameter Variations-
A
series
determine
variety of
the
)
of
experiments
sensitivity
of
(expOv to
the
and K values were
explOv) were
results
used, and
to
f
2
was
in
D
as
changes
in
ua,
and
to
conditions, used ua = 0, 7, and 14 m/sec.
experiment.
span
the
assigned values
to
A
1
We interpreted
range
of
oceanic
0 was used in the unforced
7 m/sec characterizes relatively quiet
m/sec wintertime conditions.
out
changes.
parameter
(exp7va), 2 (expOv-exp6v, exp8v-expl0v), and 4 (exp7vb).
changes
carried
conditions
and
14
The mean wind was assumed to blow from
west to east, except for exp 4 v, where the vector was oriented north to
south.
The values of K ranged from 0 to 5x10- 4 .
In all experiments the
Page -170-
Subsidence Rates
Table V.2
2
-4
-5
-8
5x10
5x10
5x10
0
K
r
u
a
0
........
-
*g
..*
16
7 ........
.20 . .
..
.30
...
..
.. .
.. . .43
.30
exp0v
exp9v
.50
explv
expl0v
exp8vb
exp2v
.18
.
...
exp8va
2
**
14
..
-
.
.. .
.........
. .50
..
14 ........
..
. .
.70
.
.61
exp5v
exp3v
exp4v
exp6v
.
-
1
exp7va
14 ..........
.. .
.35
exp7vb
4
Across the top of the chart, we have listed the biharmonic
coefficients, and down the side, the values of the rms wind speed ua*
We group the experiments according to their values of C2. All K and r2
are nondimensional, ua are in m/sec, and subsidence rates in m/day.
Note that the subsidence rates agree with observed rates of .6 m/day.
In exp2v, K=0 and after 60 days, the computation began to show
Shortly
signs of significant energy at the grid point scale.
inaccurate
becoming
was
calculation
the
thereafter, it was judged that
and the computation was stopped. K = 5x10-8 improved the performance of
the model, although the potential vorticity fields contained small scale
structure. In these calculations, 6x = y = .333 or 20 km.
Exp3v employed a time step of 14 minutes, as compared to 28
Note that the reduction made no change in the
minutes for the rest.
subsidence rate. In exp4v, the mean wind was blowing from the south to
the north, in contrast to the west to east winds of the others.
Page -171-
shape of the evolving pressure field was very similar to that portrayed
southwest and developing
in Fig. V.3, with the field moving to the
The motions
weak azimuthal asymmetry.
neighboring extrema
the
of
a
in
pressure, and the Ring, matched well between all comparable forced and
free experiments.
The only sizeable dependence of Ring propagation on
(larger)
any parameter was on -2, where it was noticed that smaller
r2
This agrees with the center of mass
resulted in a faster (slower) Ring.
calculations of the previous section.
Subsidence rates from the various
experiments along with the relevant parameters are listed in Table V.2.
Note that the larger decay rates are associated with larger wind speed,
as expected, and that there is a dependence of decay rate on
r2 .
The
smaller i1'2, the more quickly the Ring decays, indicating that the bigger
the Ring with respect to the deformation radius, the more difficult it
Note that in all cases, the
is for surface forcing to decelerate it.
0
computed vertical velocity appropriate to the 17 C isotherm is 10's of
cm/day,
which
cm/day.
The
compares
agreement
with
favorably
Parker's
it
encouraging;
is
vortex spindown is an important
component
of
rate
of
appear
that
forced
would
the
60
observed
evolution of a Gulf
Stream Ring.
V.f
The Spin Down Mechanism-
Because of the agreement between observed and model spindown rates,
it is
Ring
which
the specified technique by which
worthwhile to investigate
is
For
decelerated.
employed
Comparisons of
a K
of
purpose, we
this
5x10- 5
and
(oedif)t and (,<f)t
an
rms
shall
wind
the
investigate exp6v,
speed
of
14
m/sec.
(see previous section) show that,
in
Page -172-
Secondary circulatlons penetrate
deep into the fluid
I
N
®
S
the fhermocline
is deformed by
the pumping
thermocline
N
SEkman
layer
4-
b
Figure V.7.
thermocline
Schematic of Possible Secondary Circulations
(a).
The thermocline gives way to the surface pumping, and the
secondary circulations to close in the deep fluid.
(b).
The thermocline is rigid, which forces the circulations to
close in the upper layers.
Page -173-
exp6v, biharmonic viscosity rarely accounts for as much as 10% of the
spindown analysis are typical
results of this
The
decay in amplitude.
of all the experiments.
A
surface
forced
interior,
resulting in the
Consider a warm
stretching.
tube
and vortex
relative vorticity
production of
in which
Ring,
core
the
of
balance
mass
the
affects
divergence
at Ring
case,
If
center, the surface layers will remove mass from the deeper layers.
fluid
columns
stretching
imposed
the
above
on
(Fig. V.7).
bodily lifted
by
them
the
retains
which
its
rigidity,
the
upper
the thermocline
If it is
will
columns
be
thermocline will
Hence the depression of the
be smoothed out and the Ring will decelerate.
will
thermocline
the
upwelling,
the
resist
to
able
are
thermocline
the
stretched,
be
resulting in the production of cyclonic vorticity which acts to nullify
the
anticyclonic Ring
extant
technique
over
the
other
determines
vertical circulations within
mass
budget
to
close
within
The
domination
the
basic
flow.
structure
upper
layers,
which
entails a shallow, inward-directed circulation.
If
with
the
deformed,
flows.
lower
layer
fluid
is
one
of
A rigid thermocline
the Ring.
the
of
involved
for
the
spin
down
the
weak
forces the
a warm
Ring
thermocline
is
directed
radially
Again for a warm Ring, the lifting of the thermocline creates a
mass deficit in the deep layer which will be filled in from the sides.
From
the
numerical
deceleration is occuring.
solutions,
we
can
decide
which
means
The production of potential vorticity
of
(q) due
Page -174-
8.685 E-2
-L.673E-2
RT
E-2
OF 1.LSL
INTEVRALS
-j
_1
.087
?---
a
-1
4
I
C-ROR
-
-.
-
3
3.7 E-
o
-29.47-3
AT INTERVAAor
4.158L-3
0333
b
.008
Figure V.8.
(a).
Forced Potential Vorticity Production
The generation of
relative vorticity due to
forcing, from
exp6v at day 40, and
(b). the generation of vortex tube stretching by the forcing, also
from exp6v at day 40.
Note that relative vorticity is generated at a rate 3-5 times
greater than stretching, indicating that for this experiment (r2=2), the
thermocline tends to appear rigid.
Page -175-
to the Ekman divergence is given by:
(V2
- 2)
-
t = qt =
e =
2,
Eq.
V.24
where the terms on the left hand side represent productions of relative
The tendency of
vorticity and vortex tube stretching.
the amplitude,
xt, caused by the divergence, is given by Eq. V.22, which in combination
2
with the above formula allows us to solve for , vt:
2
-7 2,t = we + p vt.
2,t and )-2at are provided in Fig. V.8, in which we see
Plots of both
that
for
a
Eq. V.25
core
warm
Ring,
the
forcing
creates
positive
relative
vorticity at Ring center and negative relative vorticity at two points
located at the northern and southern Ring extremities.
Recall that ua
Note that the sign of the forced local rotation
is from west to east.
of the fluid is counter to that of the Ring in all three regions, and
therefore acts to slow the mean flow.
a
to
tendency
maximum).
that
uplift
thermocline
the
relative vorticity production is
of
-2
km),
all
with values
90
(or
the
depreciate
show
pressure
A comparison of the magnitudes in each of the plots indicates
size of vortex stretching.
km to
Similarly, the plots of -2<t
stronger, running 3-5
times the
Similar comparisons were made in experiments
ranging from
of which
1 to
suggested
relative
4 (Ring length
that
the
scales
primary
vorticity production.
from
45
deceleration
Equipartition
mechanism for Rings
is
between these terms
requires a larger structure, 0(200 km),
and since
the present scales are typical of Rings, this spindown mechanism should
Page -176-
Commensurate with this is a secondary circulation
hold for most Rings.
which for a warm Ring tends to be inward in the upper layers and close
through the downwelling regions outside the Ring (see Fig. V.7.b).
Summary-
V.g
In this chapter, we computed the wind-forced spin down of a Gulf
Stream Ring in
order
aspect of its evolution.
if
decide
to
such
a mechanism
is
an
important
The present work was motivated by experiments
of McWilliams and Flierl and Mied and Lindemann, where it was shown that
relaxation
thermocline
strongly
was
by
influenced
non-conservative
The mechanism by which the wind effects a decay derives from
phenomena.
the nonlinear dependence of stress on the relative velocity between the
air and
greater
For a Ring,
the surface water.
occurs
deceleration
where the water flows against the wind than acceleration where the water
flows with it; hence the result is a net loss of Ring energy.
found
gradients
of
the
streamfunction
by
a
the
that
geostrophic
stress
could
differential
resembled that
be
related
operator
of
and
It was
to
the
that
the
a bottom drag
or
a
concomitant divergence
closely
linear frictional law.
First, a sample analytical problem was completed
to verify the tendency for the pumping to effect the decay of interior
motions.
which
Next, a series of
Rings
computed.
were
subjected
Subsidence
model agreed well with
rates
numerical calculations were performed in
to
of
wind
forcing
and
isopycnal surfaces
field estimates,
suggesting
their
from the
decay
was
numerical
that wind forcing
might well play an important role in the life history of a Gulf Stream
Ring.
Page -177-
A further investigation of the model results demonstrated that on
these scales, the thermocline acts as
the mass
thus
deformable, and
balanced in the upper layer.
divergence
than
more rigid
though it were
surface boundary was
in the
Consistent with this vertical circulation,
the primary mechanism of vortex spin down was shown to be the production
of relative vorticity.
Implications of the secondary circulations with
respect to the advection of oceanographic tracers were mentioned.
forced
Wind
with
process,
coefficient,
3,
decay
the
is
added
a
physically
desirable
non-conservative
motivated
feature
that
the
appropriate
does not depend on an unknown eddy diffusivity.
3 does
depend on a parameter, Cd, a constant characterizing
air-sea momentum
that the value of
this quantity is
transfer, but
it is generally felt
well-known (Francis, 1951).
we
demonstrated
have
that
Given reasonable values for rms wind speed,
the
consequences
of
the
induced
Ekman
divergence are significant and in the present calculations overwhelmed
the effects of biharmonic dissipation.
While still necessary to insure
numerical stability in the face of an active enstrophy cascade, we were
able to
effect
conclude that higher order viscosity was having a negligible
on
decay.
Apparently,
we
have
lessened
the
influence
of
biharmonic friction, for which we have no physical justification, and in
its place installed a process for which we do.
We have also apparently
retained the features of biharmonic viscosity which are desirable from a
numerical point of view.
Page -178-
APPENDIX A.V
WIND STRESS IN THE PRESENCE OF
SURFACE FLOWS
high
The
frequency
for
From the formula
the
ocean must
V.1,
requires
as an
remove
turbulent
change
on
several
In
stress.
and
the
of
days
Still,
are
averaged
such
with
temporal
to
sensitive
an
formula, Eq.
as
to
subject
to
such a way
will
winds
on
are
of
scales
averaged
appropriately
the
be
flows
variability
present appendix, we will derive
wind stress, taking into account
in
bulk
Mesoscale
weeks.
and
The
irregular.
speed
input a wind
frequency,
low
days,
highly
fluctuations.
intervals
comparatively
be
therefore
knowledge.
transfer to
momentum
the
that
conclude
we
stress,
common
is
the wind
of
variability
a
of
wind
large scale mean
the high frequency variability of
the
wind and the presence of non-zero surface velocities.
We
denote
the average
of
a variable by an
overbar.
define the averaging process further than to say it is
We shall not
'appropriate' to
the mesoscale, providing structure at the Rossby Deformation scale with
several day variability.
Consider the mean wind stress to be a function
of the averaged surface velocities:
S= Cd(ua- u)Iua-ul
We expect
that the alterations
of
= S(u).
Eq. A.V.1
the mean stress
small compared to the unperturbed (u=0) stress:
caused
by u will
be
Page -179-
S(u) = S(0) +6 S(u)
where
l-s(u)/l s(0)j << 1,
in which case we can expand S(u) about S(0) to obtain an approximate
form for -S(u).
Eq.
S(u) = aSx,Sy ) =
=
u Sx(0)u +Lu x
S ()v
zv
S (0)u +
-
,
x
6U
y
~
CV
A.V.2
S (0)v
y
where all of the partial derivatives of S have been evaluated at u = 0.
In terms of the above formula, the Ekman divergence becomes:
W =
e
SSf
u
y Tx
f
-
fo
(y)
u +AV- Sy
Y (x)
Eq.
v
;x
Ju
A.V.3
--v Sxay v
SxQy
where we have implicitly assumed curl(S(0)) = 0.
The derivatives of S
with respect to u and v can be evaluated using Eq. A.V.1:
)
u
(u v)
=
7-
~uy
v
S
= -C
x
d
aa
uIa
2
u
S
x
=-
(2 ua
d-
2
+
va
b.
Eq.
A.V.4
Page -180-
c
(u 2 + 2v 2 )
a
a
=-C
- S
d
ov y
c.
Iu-a a
Plugging Eqs. A.V.4 into Eq. A.V.3 returns:
iA^
1
e
2u
La
=
fo Cda
L
2 + 22 1
v
a
J
y u
0u
+
A.V.5
Eq.
u
2
a
+ 2v
2
a
ua
-
1
aa
u al
x
Ua
dlua(a -
S-
1
2u v
lual
u - b
Ox
v - c
u)
which with the exception of the last term closely resembles Eq.
form.
For the limiting case appropriate to Fig. V.2,
characterized
by
fluctuations, ((ua')
-1
ua I
a
2
mean
a
(ua,0)
and
i.e. for winds
comparatively
weak
2
, (va')2)<<(ua) , Eq. A.V.4.b is proportional to:
2
a
2
2u a
velocity
V.5 in
2 1/2
= (2 + (0( a )
))IUal
-UaI
Page -181and Eq. A.V.4.c to:
(v2 1/2
2
u
2
+ 2v
a
2
=
(1 + 0(
)
)Iu-a
a
a
while Eq. A.V.4.a is simply:
0(((ua')2)1/2/
ual)lual.
To first order in this special case, the computed Ekman divergence is in
agreement with Eq. V.5.
The coefficient in the last term of Eq. A.V.5, 'c',
is related to
the tilt of the mean stress away from a purely zonal stress and it
always possible to orient our axes such that it will vanish.
example,
'c' is
respect
non-zero with
I
to
one
rotation through an angle A defined by:
2
u
cotan(2a)
=
-
1
of
reference,
for
a
2
v
a
a
frame
If,
is
ava
21u
Eq.
A.V.6
lual
will put us in a frame such that the analogous statistic in that frame
disappears.
frame;
The definition of vector stress is independent of reference
therefore, we conclude
that
the final term, 'c',
of Eq. A.V.5
adds no fundamentally new physics to the production of Ekman pumping.
Without
stress
loss
of
generality,
to be purely
zonal.
we
can
take the
(In fact,
orientation of
the mean
in exp 4 v, the mean stress was
Page -182-
oriented directly to the north, with the result that the subsequent Ring
evolution was unaltered from that reported in this chapter.)
Continuing with Eq. A.V.4.b and c, we assume
that the velocities
are drawn from a parent population governed by a normal
distribution
function:
N(ua,0;ua' 2 ,va' 2 )
with
(va'2)1 /2 .
and
(ua,O) and
vector velocity
mean
of
values
rms
The
standard
north-south
deviations
and
(ua'2) 1/ 2
east-west
wind
Under such conditions, Eq. A.V.4.b
speed can be computed from spectra.
may be written as:
2u
-C
2
+ v2
ad
a
2
Iu
a
2u
-Cd
d
Ua
a
2
uava
2
+
2
2
v
v'2)du dv
,O;u'
N(ua
a
a a
a
a
a
2 1/2
(Ua + v )
a
a
Eq. A.V.7
In Table A.V.1, we have listed a series of values obtained for the
coefficients 'a' and 'b' of Eq. A.V.5 by integrations like those in Eq.
A.V.7.
Notice
that
the
value
of
the
coefficients
do
not
change
dramatically from the standard values a=2 and b=l1 which were used in
Eq. V.5.
Page -183-
Table A.V.1
a
a
7.
7.
0.
7.
7.
0.
0.
7.
0.
0.
u(u21/2
a
a,2 1/2
5.
1.
5.
10.
5.
5.
1.
5.
10.
5.
Values of 'a' and 'b'
coh
a
0
0.
0.
0.
.2
1.81
1.98
1.18
1.63
1.80
b
1.18
1.01
1.81
1.36
1.19
u
a
8.01
7.03
8.01
10.90
8.34
In the above, ua is the mean zoP 4 wind speed, val/tpe mean
the
1
(Va'
and
,
((ua )2)
speed,
wind
meridional
r.m.s. fluctuations about the means, a and b the desired coefficients,
and lual the rms wind speed as computed from the means and
Note that we have included a category labelled 'coh',
fluctuations.
In all integrations except the last, the
which stands for coherence.
implies that meridional winds are
which
zero,
coherence was set to
In the last case, the north
winds.
zonal
the
from
independent
All
south-winds are slightly correlated with the east-west winds.
velocities are in m/sec; coherence, a, and b are non-dimensional.
Page -184-
SOUTHWARD RING PROPAGATION AS A
CHAPTER VI.
CONSEQUENCE OF SURFACE TEMPERATURE ANOMALIES
Introduction
VI.a.
across
blowing
air
Cold
warm
If warm air flows
turbulent convection.
therefore predisposed towards
cold water, vertical motions are gravitationally
over
less
flow is
opposed, and
the
efficient,
and
a more
equal,
being
else
All
turbulent.
and
below
from
heated
is
water
hence greater, transfer of momentum from the wind to the water occurs in
the
empirical
The
flow.
turbulent
formula relating windstress
to air
velocity is:
S= PaCdua I ia o
where L
the
the vector
is
air,
the
(o
proportional
will
In
the
stability
of
in
magnitude
for
of
Table
Although
wind
stress
VI.1,
we
the
have
listed
column;
rather
values
inversely
hence
over
Cd
warm
vague,
bulk
the
basic
corroborate
measured
of
bulk
called
(warm) air
is
1968)
(Deardorff,
air
cold
reasoning
this
so
stress is
the
the above arguments,
static
(decrease)
water.
measurements
idea.
to
increase
(cold)
By
the
Cd
and
density,
density
the
1a
u.a the wind velocity,
water
surface
drag.
coefficient of
stress,
of
Cd
as
a
function of air-sea temperature difference; note the 30% variations for
the relatively small range of temperatures.
The effects of
sea surface temperature
Bunker (1976) in his calculation of
on stress were
included by
basin scale mean wind stress.
As a
Page -185-
Table VI.1
The Drag Coefficient as a Function of Air-Sea
Temperature Difference
(From Bunker (1976))
-3
Drag Coefficient x 10
Air minus Sea Temperature in
Wind
Speed
(m/sec)
.01- 5.
5.-10.
10.-15.
15.-20.
20.-25.
25.-30.
0
C
>5.
4.9
to
1.0
0.9
to
0.2
0.1
to
-0.2
-0.3
to
-1.0
-1.1
to
-4.9
0.06
0.77
1.47
1.95
2.26
2.52
0.60
1.30
1.72
2.04
2.30
2.54
0.98
1.43
1.80
2.10
2.35
2.57
1.20
1.54
1.87
2.16
2.40
2.60
1.32
1.60
1.90
2.22
2.42
2.62
1.56
1.78
2.0
2.25
2.44
2.63
<-5.
1.80
1.86
2.10
2.32
2.48
2.64
Page -186-
result, the Sverdrup transport lines contained a thin, jet-like feature
(Leetma and Bunker, 1978) near the point where the Gulf Stream exits the
coast, which led Reininger, Behringer, and Stommel (1979, hereafter RBS)
to postulate
that
the
maintained in this way.
the
forcing
field,
(1981)
Inherent in
which
drives
transports warm
circulation, which
Veronis
narrowness
of
the
ocean
deep
this idea is
the
Stream
is
a subtle link between
general
water.
Gulf
circulation,
More recently,
and
the
Huynh
and
have used a temperature sensitive coefficient of drag in
model to
a general circulation
demonstrate
influence on poleward
its
heat transport.
Rings, especially warm core Rings, are characterized by several *C
contrasts in sea surface temperature (SST).
Therefore, a constant wind
blowing across the surface of a Ring will develop
varying stability of
become divergent.
gradients
due to the
the air-sea interface, and the surface flow will
This will produce a forced adjustment
of the fluid,
the computation of which will be the focus of the present chapter.
Consider the nature of the forced Ring problem. After Ring genesis,
SST propagation and configuration are governed in large
The
Ring-SST
system's
evolution
is
therefore
'feedback' reminiscent of that studied by RBS.
by advection.
nonlinear,
with
a
The temperature field
produces an Ekman pumping which affects the interior flow; the interior
flow evolves, which in turn alters the SST pattern and the concomitant
vertical
circulation.
There
is,
however,
a
fundamental
between the present problem and that studied by RBS.
RBS computed steady state circulations,
initial value problem.
difference
In their paper,
while the present problem is
an
Page -187-
coherent
a
remains
Ring
the
that
extent
the
to
case,
any
In
structure, a certain component of its evolution will be the result of
temperature
few pages,
In the next
this forcing.
the
of
dependence
suggested that the
it will be
of
coefficient
in
results
drag
Ring
southward migration.
VI.b
Scale Estimates-
The Coefficient of Drag-
In
RBS,
dependence
the
coefficient
the
of
of
air-sea
on
drag
temperature difference was taken to be linear:
Eq. VI.1
Cd =Cdo(l+j(T-Ta))
reference
drag
coefficient, T sea surface temperature, and Ta air temperature.
When
with
j
being
the
coupling
.250/*C)
as
relationship
our model
of
with
Eq. VI.1
necessary, we will employ
for
the
of
used
(RBS
drag.
in Table VI.1
temperature
drag and
j=.1/OC
coefficient
a
Cdo
coefficient,
The
.125
and
functional
is somewhat
more
complicated than Eq. VI.1, which is a crude model, hence, we should be
wary of pressing quantitative interpretations beyond reasonable limits.
Still, Eq.VI.1 captures
dependence,
require only
and
since
that
the spirit of
many
of
the
stress increase
the drag-temperature functional
arguments
presented
in
the
text
(decrease) for a buoyantly unstable
(stable) air-sea interface, qualitative inferences are justified.
Using j=.1, we notice that for IT-TaI=0(10 C), the entries of Table
VI.1
are
reproduced well.
For
5oC
differences,
the errors are more
Page -188-
Table VI.2
Scales and Parameters
Size
Meaning
Notation
-4
fo.........
.. Coriolis Parameter....
......
.. Gradient of f
S........
.........
scale
length
L .............
1/(pL) .... .. time scale ..........
-.........
S-1... .....
k ...........*
..
Kb
..
Q ........
.....
III1
F(0)
1x10
1.7x10
60 km
11.7 days
Deformation Rad. (dim) 42.22 km.
Deformation Rad.(nond) 2.
Viscous coefficient... 5.x10
........ coef, .... .04
Diffusivity
Steepness Parameter... 4.76
Baroclinic self
* .. interaction ..........
2.1
Surface value of
.......... Baroclinic mode ..... 3.0
Lo .....
..
2
Wind Stress Scale .... 1 dyne/cm
bo .......... Buoyancy Scale .........
.. Gravity ..............
g
........
U
j
..
........
..........
..........
1 cm/sec 2
103 cm/sec
2
Thermal Expansion
-4
OC-1
Coefficient .......... 2x10
Velocity Scale ....... 29.1 cm/sec
Coupling Coefficient . .1/*C
Page -189-
estimate and are
to
difficult
strong functions
of air speed, but run
about 10-20%.
The Ekman Divergence-
We can estimate the size of the forced vertical velocity using the
information in Table VI.1.
We
=
We know that:
Eq. VI.2
k.curl()/(fo)
where fo is the Coriolis parameter, k a vertically directed unit vector,
and C the vector wind stress.
we
For
A
Cdo
0(3*C),
=
-6
-6
10
Hence:
= 1/fo(~C/(Vy))
0 m/sec,
1=
, ua
=
a
u aa!(Cd)/(fo
surface
y)*
temperature
contrast
of
4
a length scale of 60 km, and an fo of 10- /sec, we obtain:
we = 8 x 10e-4 cm/sec,
which is clearly
'big enough' to be of dynamical importance.
Recall
that this w is the same order of magnitude as the vertical velocities in
the previous chapter.
The Pumping and it's Effects-
Consider
a
zonal
wind
blowing
across
the
surface
temperature
pattern of a warm core Ring, (Fig VI.1.a) and suppose the unperturbed
stress based on the wind:
Eo = Cdouaj aii
is constant.
The central region of the vortex is warm relative to the
surrounding waters and is therefore a site of enhanced stress.
If we
were to measure stress on a transect commencing south of Ring center and
Page -190-
Magnitude of stress
Cold
U0
600m
Nrth
distance along Iransect
b
-
cold
South
I
Ekman
Layer
Divergence
Isotherm
distance along
transct
South
North
C
Figure Captions Figure VI.1.
Chapter VI.
Schematic of Ring Response to Wind Forcing
An eastward directed wind crossing the top of a warm Ring
(a).
stresses the center of the Ring more than the flanks due to the
destabilizing effect of the warm waters.
(b). In this transect of a warm core Ring, notice the bowl-shaped
isotherms with the deepest penetrations at Ring center.
(c). A schematic of the effect of the divergence produced by the
stress in (a) on the isotherms in (b) is plotted. The direction of we
is upwards to the north, and downwards to the south, producing a
tendency in the thermocline for a shift to the south (the dotted line).
Page -191-
running
due north, we would see irC first
increase and then
decrease.
Therefore, the accompanying vertical divergence has both upwelling and
latter to the south.
appears
former to
the
centers,
downwelling
Ring center and
problem in which
Consider a thought
the
of a warm Ring
The temperature-depth structure
in Fig VI.1.b.
as
north of
the
the
thermocline is flaccid, so that the pumping simply lifts and lowers the
In the upwelling zone
otherwise inextensible upper layer fluid columns.
to the north of Ring center, the thermocline moves upwards and is hence
flattened
relative
to
the
thermocline
further
South of
north.
center, the local divergence from the mixed layer tends to deepen
thermocline.
Ring
the
From Fig. VI.1.c, we see that both trends when applied to
'bowl' of isotherms to move to
the warm Ring of Fig. VI.1.b compel the
the south, suggesting that some fraction of meridional Ring migration is
a forced phenomenon.
Stern (1965) has shown that a uniform wind stress when interacting
with a geostrophic eddy produces an Ekman pumping given by:
w = -7.(kx
where
are
<
)/(f+),
is the geostrophic vorticity.
caused by
variations
in
The gradients in Ekman transport
the local
rotation
produce both up and downwelling centers.
rate,
(f +
, ),
and
The resulting w field is not
unlike that depicted in Fig. VI.1 and, as Stern pointed out, a west to
east wind causes a warm eddy disturbance to translate southward.
Note
that the corrections to the undisturbed stress in the above formula are
O(Ro).
From Table VI.1, one notes the perturbations of the undisturbed
stress caused by the temperature are of the scale of
.5 n).
Therefore,
we
neglect
Stern's
I[t (=(.3 to
mechanism,
as
this
scaling
Page -192-
suggests
is
it
is
It
mechanism.
the present
smaller in effect than
interesting to note, however, that Stern's analysis of the effect of the
divergence agrees with the present ideas.
Governing Equations-
VI.c
The mixed layer buoyancy field is governed by:
bt + ubx + vby = ((bi-b)e + Bo)/h + K 2b,
where b is mixed layer
buoyancy,
layer
bi intermediate
buoyancy,
Bo
surface buoyancy flux, h mixed layer depth, K a coefficient of lateral
It is interesting to
diffusivity, and u and v horizontal velocities.
compare horizontal heat advection to the entrainment flux of heat.
In a
Ring, the horizontal velocities are roughly 100 cm/sec and have a scale
of order 60 km.
rates
of
Warm Ring mixed layers have been observed to deepen at
3 m/day
to
up
Rhines, and Thompson
(Saunders,
(1973),
the
1971),
and
density step
to
according
scales as
Pollard,
(r2/(f2h 2 )).
Hence, the comparison of these two terms yields a ratio:
2
f2 h4 vby/(r 2 ht) = 10-8 1012 10 /(6x10
that
indicating
entrainment
heat
6
3x10-3) = 5x10 5 >> 1,
flux is weak
neglected.
may be
and
Similar comparisons allow us to simplify this equation further.
wind-driven
velocities
are
roughly
1
cm/sec
and,
compared
geostrophic Ring velocities of 100 cm/sec, are negligible.
Typical
to
the
Surface heat
fluxes, represented by Bo/h, are greater than entrainment heat flux:
2
Bohf 2 /(hC ht) = 104/3 >> 1i,
and
while
represent
still
the
apparently
strongest
small
compared
non-conservative
heat
to
advective
flux
to
the
effects,
surface
Page -193-
layer.
Thus, at lowest order, the surface buoyancy equation becomes:
Eq. VI.3
bt + J(,b) = Bo/h + K-2b,
where J is the Jacobian operator, and It the geostrophic streamfunction.
As for the final term, there are few good estimates of K appropriate to
the
layer,
mixed
minimize
to
attempt
K's
values.
using
by
unfluence
its
presence
calculations
the
in
therefore
however
necessary
is
to
the
to
we
follow,
shall
possible
smallest
numerical
insure
stability.
In the present experiments, Bo has been neglected as an influence
Ring SST cycles have been studied in
on the surface buoyancy pattern.
Chapter IV, where it was shown that the strongest temperature contrasts
occur in the winter, and that Ring mixed layers produce the variations
in heat release necessary to affect the stability of the overlying air,
and
construct
weaken;
non-zero
variations
therefore,
stress
surface
buoyancy
In
stress.
in
should
more
become
contrasts
the
and
Specifying
uniform.
allowing
contrasts
SST
summer,
to
them
evolve
0
according to Eq. VI.3 with BoO more resembles and is meant as a model
of wintertime Ring
SST.
field will be taken from the
III.
Ekman
divergences
initial shape assigned to
The
tracer distributions computed in Chapter
based
these
on
evolution suggestive of wintertime Rings.
in terms
of
flux, it is
the buoyancy
reduced computing needs.
evident from Eq. VI.3
SST
fields
should
produce
This approximation gains much
By neglecting surface
that we no longer have
to
buoyancy
compute
mixed layer depth, ostensibly reducing the number of dependent variables
from three to two.
In fact,
the reduction is better as the equation
Page -194-
governing mixed layer depth requires the knowledge of at least one more
variable (usually interior buoyancy).
dynamic
upper
the
of
divergence
The
and the
the surface
link of
provides
transport
mass
layer
the
The Ekman divergence,
interior.
Eq. VI.2, becomes:
We = j(C(y)bx -
C(x)by)/(fog ),
Eq. VI.4
where we have used a linear equation of state of the form:
to
temperature
convert
constant.
buoyancy
to
Eq. VI.5
;(T-To))
and
assumed
the air
temperature
In Eq. VI.4:
(x)
S(y)
and g is gravity.
The
'
-
= ,o(1
interior
=
Cdol alua,
=
Cdojalva,
In the present experiments,
the
by
governed
be
will
Z(y) = 0.
equivalent
barotropic
equation:
(-2-,2),t + (fo)-1
llJ(/, (,72- 2)z) +cx
=
f
2
F(0)we/H
Eq.II.14
where p2 is the eigenvalue corresponding to the first eigenmode, F(0)
the surface value of the baroclinic eigenmode, H the depth of the fluid,
and 3111
the baroclinic
self
interaction coefficient.
Note that the
expressions for the surface velocities of Eq. VI.3 are:
u
=
-(0)
(cy)/fo,
and:
v = F(0)(
x)/fo.
The nonlinearity of the problem, i.e. the
transport
of
the
surface waters
and
connection between the
the forcing of
the
interior, is
Page -195-
from
apparent
4, and
VI.3,
Eqs.
such,
As
7.
we
way
no
have
of
linearizing the equations and retaining the correct physical processes,
and for this reason, no analytical solutions to this set have yet been
It
found.
evolution
however, somewhat enlightening
is,
of
initially
an
motionless
ocean,
to consider
the
possessing
a
linear
surface
temperature field, which is subjected to the onset of an eastward wind.
Under these assumptions, the mixed layer will translate southward at the
Ekman velocity, advecting with it the surface temperature structure, and
therefore the horizontal divergence:
We = we(x,y-C(x)t/(foho)).
To
interior,
the
pattern.
pumping
the
related Rossby
The
translating
a
resembles
wave excitation
wind
stress
is well known
problem
(Flierl, 1978), from which we expect the forcing to resonate with those
planetary
waves
phase
whose
speed
matches
the
drift.
Ekman
The
interesting point of this problem is that an otherwise constant wind can
interact with a
surface
temperature
distribution
interior
to produce
motions.
VI.d.
Numerical Results-
The
non-dimensional
set
of
equations
which
describe
the
Ring
evolution are:
(,2-'2)cet
+ Q111J(<,(Q 2 -F 2 ))
+
=x
= -. by + K;o,.,
Eq. VI.6
and:
bt + QF(O)J(,,b)
where
Q=U/(pL 2 ),
a
measure
= Kbo 2 b,
of
the
nonlinearity
of
the
flow,
X=-F(O)jobo/(gHT.LU), and we have employed a biharmonic friction with
Page -196-
Typical values
for a west
Notice that
coefficient K.
the parameters
of all
(
to east wind,
and scales
is negative.
listed
are
in Table
The values of F(0), f111, and 112 computed by Flierl (1978) from
VI.2.
the MODE buoyancy profile were used and the remainder of the scales were
chosen
to agree with
there is currently no
'small'.
The
Here K=5x10-4;
those of McWilliams and Flierl.
justification for this choice, other
results
the
of
computations
to
than it is
discussed
be
in
this
chapter were largely unaffected by a reduction of K to 5 x 10-5.
In Fig. VI.2, we compare the results of two experiments which will
be referred to as
exp0b and explb.
the Ring to evolve freely.
to .1.
figure caption.
The
a control experiment
the Ekman pumping, i.e. set
which we have turned off
j was set
ExpOb is
in
j=0, and allowed
In explb, the value of the coupling constant
other
numerical parameters
listed
are
in the
The initial conditions for both experiments, contained
in the panels a and d, were chosen as Gaussians of e-folding scales 60
km, for the baroclinic amplitude, and 90 km, for the surface buoyancy
field.
field and the
Clearly, the shapes of the sea surface temperature
interior pressure field compare favorably between the two experiments.
By day 60, however, we see evidences that the pressure field in explb is
travelling at a different rate than that in expOb.
Both of the above
observations are reinforced by the graphs of Fig. VI.3 and 4 in which
the histories of
compared.
the maxima
The numerically
in pressure and the path of
generated positions
of
the
the Ring are
local
pressure
maxima, which are graphed in Fig. VI.4, are listed in Table VI.3.
From
Page -197-
ol
arou.,IFa
Ii l l i*a l.ai - i
lat
*I
I"
go
a
I ate
.
tI
rhk5aslI Iu1.@
l at
l tll
retFae
i n.
*l
111,ts
w a to
7
i
L
i_
-I
-4
Ej
I
,tg III I
I
L
ieI-I
i l.l*
I
I-I
1.
-._
7.
Ii
COUTOUffe
Fl i
.t91i-I
tIftllllllll
itI -1
I
TI1.111
C-
I
'
i-
I
i
F I.S
I-t
ii
i
.92
.08
fiJ~i
ru
CO-teilatFO 1 81IL.
ii
-
-
I
To S.LM -1
!
I
I
. 1
lIiNtIVKS
a 1.814E-t
.9 2
1-r~T~mTfrFFrn
.72
I
-7
\/-
-
/2
I
..1
-i
j1
Il.
~iI
i~
F-I
Fig. VI.2
continued next page
-. 095
-1
Page -198-
Co01R
l
LN 1
R
Ot 9le tE
at INY
lAO
I|
ots
(l
-
3
(,
L
L
";
or ,TEva.SOFeae1#t
r
C
1.
I II
Cram e
I
teoifE
cOTIMIS oesCtle te
4Sol('jl~
i
1/ L(
I I I
w 2,
,.7,
I .
1-t
I
ra ?
s
I ! I
i
C-1
4 taltlo.s
I; l
IJ1 I I
coatemlne
ra II.ot
we4.8"1-1
ll
I
I I I t
[-I
iiiIIIil
re * lot [-a
Ir .
Av s 1lAt
1-t
t
iI l ll
.94
-. 084
7
C'NWWI
P
toi
.61
9
I'I
I . w1
2
I
COXt51140
FR1n
4 22t
? *t t
CrW
I
I
Fig VI.2.
I I
II
IIl(t
I
I
9.6T (-1
t*9 t
' 1
MV9. f t.114
8 1-t
Inl-
Forced Ring Evolution
Plotted are the baroclinic hori zontal amplitude functions and the
sea surface temperatures from expOb, after (a) 0 days, (b) 40 days, and
(c)
60 days.
In this set, Q=4.76,
the coupling coefficient, j, was set to
are from exp1b, with all parameters the
o t.1.
Again, the plots are for (d)
days.
E 1 1 1 = 2 . 1 , K=5x10 -4 ,I
2=2, and
0. In the second set, the plots
same except for j, which is set
0 days, (e) 40 days, and (f) 60
Page -199-
comparison of free versus forced amplitude
evoIution.
Eps=.1,
and eps=0.
ke5.e-4, q= 4 .7 6 , psi=
2
.1
4-
o"1
CO
C -
expOb
0
explb
c
E
O
C
0%11
i
I
42.5
8.39
i
19.24
i1
30.08
Time
Figure VI.3.
4,.92
St.76
62.61
73.45
in days
A Comparison of Ring Amplitudes
Here we compare time histories of
from exp0b and exp1b.
Note that the
unaffected by the forcing; after 70
thermocline depression is 0(5 m).
The
the uncertainty involved in the location
the maximum pressure amplitudes
amplitude evolution is largely
days, the difference in the
wiggles in the plot are due to
of the maximum.
Page -200-
motion of Ring center from free
and forced experiments exp1b and
expOb, for
110 days
expO b
3.
+-
cm
0
0 -C,-
explb
C
64
I
I
.63
6.22
x
Figure VI.4.
8.82
position
I
I
It.41
(grid
14. 00
I
16.59
units)
Forced and Free Ring Trajectories
Trajectories of Ring center are plotted from exp0b and exp1b. Ring
Notice the
center was defined as the location of the pressure maximum.
decreased westward drift and increased southward drift in the forced
experiment. All locations are to an accuracy of 4 km.
Page -201-
Table VI.3
Locations of the Local Maximum of Pressure versus Time
Explb
ExpOb
Time
Xrel
Yrel
0. ..... 0.000
0.000
- 4.444
-13.333
-17.777
-26.667
-35.555
-51.111
-55.555
-66.666
-82.222
-95.555
-105.44
-115.55
-122.22
-137.77
-4.444
-6.666
-11.111
-13.332
-15.555
-22.222
-24.444
-26.666
-28.888
-33.555
-35.777
-35.555
-42.222
-42.222
5....
10. ...
15. ...
20. ...
25. ...
30. ...
35. ...
40. ...
45. ...
50. ...
55. ...
60. ...
65. ...
70. ...
(days)
(km)
(km)
Time
0. ....
5 .. *
10. .. .
15. ..
20. ..
25. ..
30. ..
35. ..
40. ..
45...
50. ..
55. ..
60. ..
65. ..
70. ..
(days)
Xrel
Yrel
0.000
0.000
-4.444
-6.666
-17.777
-22.222
-28.888
-46.666
-53.222
-64.444
-68.888
-86.666
-95.555
-102.22
-115.55
-126.66
-4.444
-6.666
-8.888
-17.777
-22.222
-26.666
-33.333
-35.555
-42.222
-46.666
-53.333
-57.777
-64.444
-66.666
(km)
(km)
All distances measured relative to initial center.
Page -202-
the first graph, we see that the amplitude decay is nearly identical in
the experiments; therefore, we conclude that, unlike the results of the
induced
the
chapter,
previous
pumping is not seriously affecting
From Table VI.3, we calculate
rate at which the Ring is losing energy.
an average
of
pressure
(-2
(-2.4 km/day, -.5 km/day) in exp0b.
km/day, -1 km/day) in exp1b, and
The forced Ring of
in
maximum
the
for
propagation velocity
the
explb is moving to
the south at
nearly twice the
speed of the Ring in exp0b, but to the west at a lesser rate.
Finally, note from the plots in Fig. VI.2 that the pressure fields
develop azimuthally dependent structures, which consist mostly of a high
east.
of
center west
pressure
The
McWilliams
generation
and
isolated high.
Flierl
of
and
those
is
structures
due
to
has
to
center
each Ring and a low pressure
on by
been commented
dispersion
from
the
the
initial,
Both centers remain coherent with the Ring throughout
the experiments.
Comparing the locations of the high and low pressure centers from
each experiment, we
notice
they are somewhat different.
in a counter-
those in exp0b, the local centers in exp1b are rotated
clockwise
sense
about
the
pronounced for later times.
Ring,
a
difference
However, the
centers differs by only a few percent.
which
amplitudes
Relative to
of
becomes
each
of
more
the
For example, the low pressure of
exp0b at day 40 is -.08, compared to that of exp1b which is -.084.
Page -203-
Table VI.4
Experiments and Parameter Settings
In all experiments, the time step was .02 of a day, and the spatial
increment was 20 km. All began with an initial condition in pressure of
a Gaussian with length scale 60 km. The temperature initial condition
was a Gaussian of scale lb.
The coupling coefficient was j, and F
K is the lateral
measured the surface intensification of the Ring.
'mpr' stands for
diffusivity.
of
coefficient
temperature
surface
rate.
propagation
meridional
from
Note that we can obtain the motion of a freely evolving cold Ring
The governing equation is invariant to the
this experiment.
transformation (po,y) (-c,-y), so warm Ring simulations become cold Ring
simulations with a reverse of north and south and the sign of the
pressure. Therefore, freely evolving cold Rings move north at a rate of
.5 km/day.
Page -204-
Parameter Studies-
Three other experiments were conducted in order to test a range of
Ring types (see Table VI.4).
ExplOb was designed as a cold core Ring experiment, with a central
SST contrasts across cold Rings
low pressure covered by a cold pool.
are generally weak (Chapter IV, and Vastano, Schmitz, and Hagan, 1980),
and in expl0b, we have reduced the size of the temperature contrast to
2.5 OC
(in the others, ST was
freely
evolving
northward,
cold
0
5
nonlinear interactions
dispersion
their
with
Rings
The
C).
them
move
field
of
(McWilliams and Flierl, 1979), and by the inclusion of the
forcing, this
been
tendency has
The forced Ring
reduced.
of
expl0b
moves northward at a rate of .3 km/day, as compared to the meridional
motion of the free Ring of .5 km/day.
In expllb, we have reduced the
pool
by
choosing
addition, the
an
size of
initial
the
Gaussian
surface
the propagation
on these
60
of
km
length
scale.
In
trapped zone has been reduced from
that in exp1b by setting F(O) to 1.
of
area covered by the warm surface
The results indicate a dependence
parameters,
as
southward Ring
motion in
expllb as been reduced from -1. km/day, to -.7 km/day.
Finally, in expl2b, we set the value of the lateral coefficient of
heat diffusivity to .4, and noticed no significant change in propagation
or Ring evolution from that in exp1b.
Page -205-
In all experiments,
the pressure and SST fields evolved in such a
way that they visually resembled those in Fig. VI.2, with the only major
change being a reduction in the size of the SST pattern when the size of
Therefore, Figs. VI.2 are representative
the trapped zone was reduced.
It
of the pressure and SST configurations and extrema for all tests.
would be interesting to expand the
combinations
exploring the possible
experiments
by
more fully
set
of
of
trapped zone size and initial
buoyancy structure.
VI.e.
Discussion-
Both warm and cold Rings are observed to propagate southwestward,
understanding
although
evaluating
their
they
Why
1978).
and Worthington,
the
influence
move
trajectory
on
the
(Richardson, Cheney,
km/day
with a meridional component of up to -1.5
is
south
of
currently
Ring
a
surrounding
is
unknown,
important
waters.
Mean
to
state
advection has been suggested as a mechanism for southward Ring motion
cold
because
Ring
motion
agrees
circulation (the Ring Group, 1981).
is less clear.
Flierl,
with
the
sense
of
the
general
Whether this is true for warm Rings
From the work of Meid and Lindemann and McWilliams and
we know model Rings
north and warm south.
can self-propel,
with
cold Rings
moving
Topography is also a possibly important steering
mechanism, probably playing a stronger role for warm Rings than cold.
According to the results here, we apparently have another mechanism for
producing
southward
motion
produced, and its magnitude.
and
we
will
now
investigate
how
it
is
Page -206-
Intregral Constraints-
that the
demonstrate
strictly
propagation.
to
its
of
Similar arguments prove enlightening in the present case.
The center of mass (COM) vector of o<,
X=
Predictive
much
for
responsible
were
configuration
radial
able
their freely evolving Ring from a
of
departures
were
Flierl
and
McWilliams
constraints,
integral
From
X = (X,Y) is defined as:
Eq. VI.7
rxxdA/( f<dA).
for
equations
both
components
of
obtained
be
X may
from
Eq. VI.6 with the assumptions that both & and b vanish at infinity:
Xt = -1/p 2 ,
Eq. VI.8
and:
d((bdA)/((/fodA)C2),
Yt =
where we have used we from Eq. VI.4.
mass,
X, moves
at
long Rossby wave
the
applies in the absence of pumping.
net
meridional
drift
Note that
which
is,
phase
the
speed,
zonal center
a result
of
which
The y center of mass experiences a
quite
evidently,
response.
a forced
=0 .
Contrast this with unforced evolution, where Yt
Note the sign of Yt depends on the relative sign of
buoyancy with respect to the average baroclinic amplitude.
the average
Since )
is
negative, a warm pool overlying a solitary high pressure center forces Y
to the south.
low
pressure.
The same southward tendency holds for a cold pool over
Both
of
these
SST-baroclinic
pressure
configurations
describe Rings, the former warm core and the latter cold core.
Page -207-
Note
the pressure
the propagation speed of
from Table VI.3 that
maximum in explb is not that prescribed by Eq. VI.8, and the discrepancy
can be shown to be related to the amount of baroclinic 'mass' which has
The difference between X and Ring center
been radiated by the Ring.
(Xc,y c ) is, by definition:
X - Xc =
(x-xc) dA/(
.dA),
Eq. VI.9
and:
Y - yc =
Clearly,
if c' were
presence
above
the
symmetric,
radially
would
expressions
The two radially asymmetric features in C( of greatest
reduce to zero.
magnitude
ji(y-yc)dA/(c dA).
the
are
leading
and
high
the
is having a profound influence
trailing
their
hence,
low;
Again
on Ring propagation.
following McWilliams and Flierl, we define a 'departure field', o<', of
the
baroclinic
amplitude
streamfunction
from
a
purely
radial
configuration:
-
=-c-(r)
'
2
e-[(x-xc)2 + (y-yc) ],
c
where, in the second equality, we have chosen a Gaussian based on local
maximum pressure,'Oc, for the radial function.
In terms of c<',
Eqs.
VI.9 become:
X -
xc =-
(x-xch:dA/(
dA),
Eq. VI.10
and:
Y - yc = -f(y-yc)b
dA(/
dA)
which demonstrates the importance of the departure field.
Leading highs
and/or trailing lows cause the Ring center to lag behind the zonal COM.
Similarly, northerly highs in conjunction with southerly lows produce a
net southward Ring displacement relative to Y.
independent
of
the dynamics,
and hence
the
Note that Eq. VI.12 is
dispersion
field has
the
CONTOURED FROM -8.835E-2
T~ll
Il1T
I
TO 7.594
T
F
CONTOURED FRON -7.9tSE-2
AT INTERVALS OF 1.954 E-2
E-2
T
ilT
TO 6.8t9 E-2
AT
INTERVALS OF 1.741
TI
exp0b
''i"
i i'
i
7.768 E-2
.654E-2/
. i
i i
Figure VI.5.
I I
.
iI.
Departure Fields
Here we have plotted -', the departure of the horizontal structure
from a Gaussian, from exp0b and explb at day 60.
Note that the local
pressure centers of explb are rotated cyclonically about the Ring
relative to the same in exp0b.
E-2
Page -209-
The
of the existence of forcing.
above effect on the Ring regardless
integral constraints in Eq. VI.10 convey only the kinematic tendencies
of
the
The
departure field.
clockwise
rotation of
a high pressure
to
center tends to push eastern fluid to the south and southern fluid
the
The above integrals measure
the west, and vice-versa for a low.
effect that the departure field is imparting to the Ring.
In Fig. VI.5, we plot
departure fields for both expOb and explb.
The important distinction therein concerns the relative positions of the
local extrema.
As noted in section c, the low and the high in Explb are
rotated cyclonically about Ring center relative to the same in exp0b.
Zonal Propagation-
For the duration of the experiments, the high pressure of
to lag behind zonal COM, although the free Ring of
exp0b eventually moves west faster
Table VI.3).
is to
Therefore, we
the west and the low pressure to the east of Ring center.
expect both Rings
x'
The acceleration of
than its counterpart in exp1b
(see
the free Ring was first noticed by
McWilliams and Flierl who pointed out that the zonal propagation speed
of the free Ring appeared to be asymptoting to the phase speed of a long
Rossby wave.
This was
interpreted to
be a result of
the slow anti-
cyclonic motion of the dispersion centers about the Ring, which stopped
just short of lining them up meridionally with Ring center.
Clearly, as
the centers move about the Ring, the zonal weight assigned to either the
leading
high
accelerate.
or
the
trailing
low
diminishes,
causing
the
Ring
to
In explb, the tendency for the rotation of the centers is
Page -210-
that
so
decreased,
the
center
each
of
maxima
local
remains
nearly
stationary, and are unable, through their circulations, to augment Ring
Thus the westward pattern speed of the forced Ring
zonal propagation.
is less than that of the free.
Meridional Propagation-
By an area average of the amplitude and buoyancy equations, we see:
(fbdA)t
= 0,
(jadA)t
= 0,
and:
so
that
forced
the
propagation in
COM
meridional
can
VI.10
Eq.
be
For exp1b, this becomes:
computed from the initial conditions.
Yt = 9/I8,
or dimensionally:
Yt = -.6 km/day,
which should be compared with the computed Ring center propagation of
Thus, we conclude that roughly one-half of the meridional
-1.0 km/day.
The difference
migration of the numerical Ring is due to the forcing.
between the motion of the Ring center and that of the COM is -.4 km/day,
and is due
drift
roughly
is
observation
experiments
modeled
to the dispersion field.
that
is
the
the
same
as
comparable.
in
that
strength
of
exp0b,
the
the
Given
the temperature dependence
Note that
the magnitude of this
in
agreement
dispersion
crudeness
with
field
with which
of the coefficient of
the
in
both
we
have
drag, we can
not say much more than the forced southward migration of the Ring is on
the
order
of
.5
km/day;
however,
this
is
the
same
order
as
the
Page -211-
meridional
interaction.
in
enough
Ring-dispersion
to
due
propagation
the self-induced
circulation and
general
the
of
meridional component
nonlinear
field
The important point is that the present mechanism is large
its
effects
significant
a
for
account
to
fraction
of
of
the
meridional Ring motion.
Potential Vorticity Budgets-
VI.f
The
forcing, which must
explb is
propagation in
novel Ring
therefore affect
result
a direct
the Ring's
vorticity.
potential
(q) budget of a particle, outside of
Consider the potential vorticity
the trapped zone, about to interact with a Ring.
In the following examination of q, we will use
expllb.
In
this
experiment, j = .1 and, with the exception of F(O), all other parameters
matched those in explb.
This has the effect
In expllb, F(O) was set to 1, rather than 3.
of decreasing
the trapped zone size, owing to the
reduced surface velocites, which are computed according to:
u = -F(O)xx,
and
v = F(0)-y.
The
initial conditions
for
*- in expllb were the same as
The SST condition was also chosen as a
Gaussian of length scale 60 km.
Gaussian of
scale 60 km, which is a reduction from the 90 km Gaussian
used in exp1b.
As a result,
the forced component of
migration was reduced from -.6 km/day to
total
Ring
constitutes
in explb, a
migration
from
-1.
a sizeable increase
km/day
to
meridional Ring
roughly -.3 km/day, and the
-.7
km/day.
This
still
in southward Ring migration over the
Page -212-
volocity du~ to torcing,
vertico!
Timo
in
rrom axpltb.dot.
F(O)=t.
days=39.t
CONTOURED FROM -3.431E-2
TO 4.h76 E-2
AT INTERVRLS OF 0.879 E-2
..
.... ................
.
---.
;....1~
:........
--;----
.~..~.,
,........
,-.,.,.....
--..
I-;
---
i-
-- ..1...;
- --:.......
.-.....
----.~1...;
. ... ...........
Figure VI.6.
the
. .. . i
....
...... ................ ... ......... .... .....................
...............
. ....
.....
........
...
.
.............
i.............
..
Forced Divergence
Here we have plotted a graph of the Ekman divergence produced by
In this
surface temperature field at day 40 from expllb.
experiment,
r2=2,
K=5x1 0 - 4,
111i= 2 .1,
Q=4.
7
6,
F(0)=I,
and Kb='
04
.
the regions of upwelling to the north and downwelling to the south.
Note
Page -213-
-.5 km/day,
unforced speed of
and indicates
that the results of
this
chapter are not overly sensitive to parameter variations.
In
Fig.
according
VI.6, we plot
to Eq.
a
map
Although
VI.4.
of
the
divergence
Ekman
the
pumping
intensifies
computed
the
behind
Ring, in the 'tail' emanating from the trapped zone, the largest centers
lie over the Ring
of vertical velocity
the
that
the
of
half
northern
Ring
Note
thermocline expression.
experiences
upwelling
and
the
southern half downwelling.
Ekman
divergence
can
produce
potential
vorticity
through
vortex
tube stretching or relative vorticity generation:
qt = [;2 - P2],t,
From we, we can compute the forced trend in amplitude production, Gt, by
solving the elliptic equation:
[Q2-r2]dt = -by*
The generation of vortex tube stretching is given by:
which may then be combined with we to compute the forced production of
relative vorticity:
et = -Ay + r 2t.
Plots
of
these
fields
are
contained
in
Fig.
VI.7.
Note
that
production of q is equally divided between these two components.
principle
reason
for
this
comes
from
the near
match
of
the
the
The
Rossby
Deformation Radius, C-1, and the scale of the pumping (cf Fig. VI.6).
CONTOURED FROM -2. t28E-2
I
i-t I
TO 2.084
IilI~ii~ i<
E-2
AT INTERVALS OF
0.527
CONTOURED FROM -t.529E-2
E-2
Yi1 Tf........tIT
TO 1.768 E-2
AT
i
--ti
INTERVALS OF 8.366 E-2
i
i I I
-7
0f
o
E-
relative
I
vorticity generation1..1.. .
relative
vorticity
Figure VI.7.
generation
vortex
tube
stretching
Forced Production of Potential Vorticity
Here we have plotted maps of the forced production of relative
The
vorticity and vortex tube stretching, as computed in expllb.
interesting points from these plots are that the fields are dipole in
nature, as was we in Fig. VI.6, and that the magnitudes from each are
The forcing produces equal amounts of relative vorticity
comparable.
and vortex tube stretching.
Page -215-
If
the
Ring were
not
forced,
i.e.
We=0,
parcels
moving
north
around the Ring, in order to conserve potential vorticity, would develop
negative relative vorticity to cancel the increase of q due to beta.
In
terms of the pressure field, this is equivalent to the production of a
high, which is
from
mass
forcing.
the
layer.
boundary
the
a
As
column
fluid
moves
the north, it initially
the forced Ring towards
from the southwest of
receives
by
caused
center
freely
the high
consider the alteration of
evolving Ring is maintained. Now
pressure
the
of
how the local maximum to the northwest
The
by
responds
column
The Coriolis force in
thickening vertically and expanding horizontally.
conjunction with the diverging radial velocities of the column produces
negative relative
thermocline.
high.
vertical
presses
thickening
the
Both of these forced responses augment the production of a
the
To
the
vorticity, and
north
of Ring
center,
the
columns
lose
mass
to
the
boundary layer, part of which is supplied by a horizontal convergence in
the interior and the
rest
responds to the shrinkage by uplifting and
by positive
convergence
The thermocline
by vortex tube shrinkage.
relative vorticity
the
fluid responds
production;
both of
to the
which
In total, the location of the high to
counter the production of a high.
the west of the Ring is shifted to the south of its unforced position.
Similar
arguments
apply
to
the
trailing
low.
The
effect
of
upwelling on a parcel is to increase its potential vorticity, while that
of
downwelling
is to
decrease
it.
Therefore,
for
particles
moving
south, the forcing tends initially to compensate for the loss of q due
to beta.
Once south of Ring center however, the forcing changes sign
and supplements the loss of q.
For a southbound column, the development
Page -216-
To the observer in a
of the low is first enhanced and then suppressed.
stationary frame, the location of the minimum is shifted to the north.
Note that the above forced trends in pressure, highs shifting south
and
lows
north,
are
evolution
the
precisely
of
(cf. Eq. VI.10).
results in the southward shift of the center of mass
describe
arguments
Similar
and
budgets
vorticity
potential
the
which
interior
the
the
associated drifts of the Ring trapped zone.
Again consider a particle in a clockwise transit, but within the
trapped region.
In the northern half of the Ring, as the particle is
moving west, the mass flux is directed up into the boundary layer (Fig.
As
VI.6).
negative
a column of
a result,
relative
vorticity,
In the
thermocline.
south,
and
the
fluid
is both
which
lifted,
mass
creating
stretched,
upwells
the
the interior
flux into
locally
depresses the thermocline and creates negative relative vorticity.
each succeeding pass of
a column, the thermocline
north and deepened to the south;
deep
With
is flattened to the
therefore the pressure pattern shifts
to the south.
Summary-
VI.g.
In
this
chapter,
we
have
dependence of the coefficient of
results
in
a
forced
propagation
demonstrated
drag in the
of
Gulf
that
the
temperature
formula for wind stress
Stream Rings.
The
large
temperature contrasts associated with Rings produce regions of enhanced
and
suppressed wind
stress
and creates both convergent
and divergent
Page -217-
For warm and cold core Rings, the response of the Ring is to move
flow.
to the south.
From Chapter
the strength
that
anticipate
IV, we
forced
the
of
of cold core Rings will be less than that of warm core, owing
motion
largely
to
the
SST
reduced
Nonetheless,
contrasts.
forced
the
propagation of a cold core Ring will be in opposition to that caused by
the departure field.
(Recall that
the freely evolving warm core Ring
simulations presented here may be immediately extrapolated to cold core
Rings by substituting low pressure
Eq. VI.6).
(cf.
for high and north for south
This is not so if we include the forcing term, /by.)
In one
experiment using a cold core Ring and a weak SST contrast, the resulting
northward motion of
reduced by :50%,
the Ring was
There are several mechanisms which potentially
such
as
mean
advection
and
but
interesting to
all three.
as
southward
moved
by
and
self-propagation,
all
three.
reversed.
affect Ring migration,
in
we suggest that forced migration is as large as either.
possibly
not
For
cold
this
chapter
Warm Rings are
it
rings,
speculate that their southward drift is a resultant
is
of
Cold ring northward self-propagation is about the same size
southward mean advection and
the
remaining
forced
tendency
could
to
their
account for observed motion.
The
ability
of
Rings
to
transport
fluid
is essential
systematic, forced, meridional drift.
The forcing moves the Ring south,
which in turn shifts the trapped zone.
The movement of the trapped zone
transports the temperature field which is responsible for the creation
Page -218-
of the pumping.
The system, composed of the interior and the surface
trapped zone, interacts with the wind to self-propel.
We have not included all possible large scale mixed layer processes
in the present
problem.
Most
importantly, we have neglected
surface
heat exchange, both in order to pose a problem that we had the resources
to solve, and because it seemed as if we had little to learn from its
inclusion.
The resulting computations correspond to fall or winter Gulf
Stream Rings.
It seems from Chapter IV that the inclusion of air-sea
exchange would
simply
the
modulate
of
intensity
the
vertical
flow.
During the summer, we would expect the pumping to be weakened by the
degradation of the SST anomaly; with the reemergence of the anomaly in
From this standpoint, we
the fall, southward motion should increase.
really have nothing new to learn by including the surface buoyancy flux
in the
problem, and
for
this
reason it
possible that the seasonality of
was neglected.
Still
it is
the forcing could produce some novel
results and it would be interesting to include a more active mixed layer
in the calculation as a test of its importance.
The equations which were
chapter assume
solved in this
that the
mixed layer remains relatively shallow, in which case the alteration of
the
surface pressure
field from that
of
the
interior is
negligible.
Recall from Chapter II that the pressure field in the mixed layer begins
to assume an independent character from that of the interior if either
the
temperature difference
layer becomes deep.
across
the Ring
During various periods
becomes
of
large,
the year,
or if
the
the surface
Page -219-
of
should
endeavor
layer
thin
the
relax
to
therefore we
criteria,
these
both of
meet
warm core Rings
layers
the
However,
assumption.
problem we obtain almost necessitates a move away from quasi-geostrophic
theory.
Perhaps the most blatant ageostrophic effect comes from those
isotherms which are deep in Ring center, but which rise to the base of
the
influence
produces
buoyancy.
Under
governed
structure,
by
large
such
circumstances,
of
a
strong
the
upper
mean,
layer
intermediate
and
buoyancy
diabatic
annual
is
no
longer
solely
density
time-independent
i.e.:
-wN2 .
bt
Nor
in
changes
advection
vertical
Second,
flank.
Ring
mixed layer at
does this problem easily lend
itself
to
numerical
To
solution.
compute the response of a mixed layer requires fine vertical resolution;
when investigating two horizontal dimensions as well, the computational
requirements will be immense.
Still, this is an important problem; we
should see vertical circulations
density structure.
If
this
develop as
is so, we need
the fluid adjusts
to
the
to understand how such a
circulation affects Ring evolution and to assess its importance.
It
is
clear
that
there
are
other
in
processes
need
of
investigation, and it is the hope of the author to continue working on
all of the mentioned areas.
It is equally clear, however, that the SST
field of a Ring will affect the stress pattern in a manner similiar to
that
modeled here.
The
interesting
result
of
this
chapter
is
the
self-propellant nature of the Ring-SST system caused by the temperature
dependent
coefficient
of
drag.
Further
experimentation
will
enlightening with respect to relative importance of this mechanism.
prove
Page -220-
CHAPTER VII.
dealing
importance
evolution
the
with
Rings
of
distributions
property
oceanic
to
The
suggest
present work
for the
primary motivation
large
the
and
scale circulation is currently unknown, although many estimates
their effects are large.
The
Rings.
core
cold
and
warm
both
of
problems
several
investigated
have
we
thesis,
present
the
In
SUMMARY
comes from a desire to better quantify their influence and, this being a
difficult problem, we have made some small progress in this direction.
to
the processes
that
demonstrates
component
field
which
Ring
of
affecting
their
sea
This
do.
they
surface
For example, we have
regards
thesis
important
an
are
interactions
Ring-atmosphere
dynamic evolution.
and their
behave as
to
cause Rings
behavior,
with
definitive statements
We have attempted to make some
temperature
demonstrated
that spindown induced by frictional loss to the wind is strong enough to
account for something between 30-100% of
mechanisms
decay
of
which
(Huang, personal
the
have been demonstrated to
communication) none
have as large an effect as wind forcing.
lateral
exception of
drag, and with the
dissipation, bottom drag, and wave
latter,
include
proposed
been
have
Other
observed Ring spindown.
In addition, wind forcing can
be computed in terms of known quantities.
We have also shown that Ring
SST cycles may be accounted for in terms of air-sea heat exchange, which
differs
from
the
for those cycles.
the
observed
pervading feeling
that Ring
dynamics
are
responsible
Finally, we have shown that a significant fraction of
southward
motion
of
Rings
can
be
ascribed
to
the
Page -221-
The size of the induced propagation
interaction of wind with Ring SST.
mechanisms
suggested
other
to
due
that
with
compares
observed
and
pattern propagation rates.
We have tested each of the processes individually, yet they affect
of
aspects
different
(propagation,
evolution
Ring
decays,
shallow
temperature field) which are probably independent at first order.
can however combine the principal
to get an more global Ring
results
The study of particle trajectories in Rings strongly supports
picture.
the
We
one-dimensional
the
of
aptness
models
layer
mixed
for
the Ring
The SST results from Chapter IV also indicate that the
trapped zone.
surface buoyancy equation used in Chapter VI, in which air-sea heat flux
was neglected, is a reasonable model of wintertime Ring SST, and that
warm Rings
should be more affected by the propagation mechanism than
calculations
The
cold.
ought
configuration
Lagrangian
field
of
a
set
be
to
Ring
up,
proved
indicated
III
Chapter
of
information
the
and
useful
how
in
SST
about
the
dynamical
final
the
the
understanding of the effects of the atmospheric forcing.
On the
basis
of
the
calculations, we
with regards to oceanic observations.
calculation
which
significantly through heat
layer response
suggested
predictions
In Chapter IV, we performed a
that
warm
loss to the atmosphere.
in cold Rings
some
We have suggested that both warm
and cold Rings are spun down by the wind.
simple
can make
Rings
are
weakened
Conversely, mixed
is confined to the near surface, so this
energy loss mechanism is not in operation for them.
We then can predict
Page -222-
that
should
warm Rings
to
owing
Rings,
cold
than
faster
decay
the
With respect to the
combination of mechanical and thermal energy loss.
questions about how significant Ring effects are on their surroundings,
we can make a few qualitative statements.
of
analysis
corresponds
Chapter
to a
q, and
vorticity,
potential
anomalous
pool of
Ring
Stream
Gulf
a
of
zone
trapped
the
III,
According to the Lagrangian
for
The methods by which this can
particles to exit, their q must change.
occur are limited to non-conservative processes, which unfortunately we
are presently unable
rates,
q
which
parametrizations,
biharmonic
alteration
Using crude
to model in a satisfactory fashion.
only
is
yield
roughly
particle exchange at the trapped zone boundary.
limited
a
for
allow
to
enough
strong
decay
observed
the
If we investigate Rings
which decay by wind forcing, the results are similar to those in the
According to advection-diffusion experiments,
biharmonic experiments.
fluxes in and out of the Ring are largely controlled by the size of the
In summary, tracer transport, from Ring core to
diffusion coefficients.
exterior, is apparently controlled by the non-conservative and diffusive
process about which we know little.
It is generally felt, however that
these processes are weak, especially in comparison to the estimates of
the potential effect of Ring advection fields (Flierl and Dewar, 1981),
so that Rings have probably been overestimated in their importance to
Slope Water-Sargasso
exchange
depend
on
confidence
what
their
experiments
of
region
of
Chapter
a warm Ring
values
are;
Group,
it is
coefficients,
the
quantities
(the Ring
however,
vertical
the
are
consistent with
a
to
difficult
in
velocities
V,
Because
1981).
loss
the
in
of
say
the
with
wind-forced
the
upwelling
10-20%
of
the
Page -223-
Thus, published estimates
trapped zone volume over a period of a year.
which assume total Ring loss of material might be off by a factor of
non-negligible
Also
note that
lesser
a
of
While
5-10.
material
source
faster
the
decay
warm Rings
of
Rings
as
a
region.
Water-Sargasso
Slope
the
for
leaves
still
this
magnitude,
is consistent
with
a
greater trapped zone-to-Slope Water flux, so perhaps Ring flux is more
important to the Slope.
With
respect
Ring
to
it
fluid,
exterior
on
effects
has
been
For
postulated that Rings warp existing gradients, thus enhancing flux.
Rings travelling west, the advection-diffusion experiments indicate that
the length of time during which individual particles interact with any
Ring
is
short
The
enhanced.
significant
enough
effects
accumulated
(Flierl
however
and
several
of
are
fluxes
north-south
that
Ring
passages
East-west
1981).
Dewar,
only
slightly
can
be
fluxes,
affected by the larger zonal displacements of particles by Rings have
not
really been tested here.
that north-south excursions of
passage, and thus
Also, recall that we have demonstrated
particles
can be
the presence of fronts,
sizeable
such as
during Ring
occur in the Slope
Water, can enlarge meridional property flux.
Suggestions-
The
formation
of
deep
wintertime mixed
layers most
likely
is a
strong influence for warm core Rings,
and in this thesis, we have not
considered its dynamical implications.
What we have learned is that the
layers are formed by local air-sea exchange and anomalous deep buoyancy
Page -224-
careful
a
so
centimeters),
dynamic
20
of
(loss
wintertime
during
that a warm Ring will be strongly spun down
We calculated
structure.
modeling of the effects of buoyancy forcing on Rings is important to the
of
estimates
grid
large,
Therefore,
will
and
with
to
as
layer-quasi-
mixed
dimensions,
horizontal
two
well
as
author's plan
the
it is
costly
be
combined
a
perform
calculation,
interior
geostrophic
very
to
necessary
computation
beta,
but
commence on the above problem
probably prove informative
will
is
analyze.
to
difficult
Such a model
using a quasi-two dimensional (horizontal-vertical) model.
neglects
the
response,
layer
mixed
computed
accurately
to
required
dense
the
of
Because
Slope.
the
to
Ring-induced flux
of
terms
in
Ring
response.
From
temperature
the
XBT
water
Slope
data,
note
we
from which we
inversions in the upper 50 meters,
sizeable vertical salt gradient.
in and out of the deep
gets well-mixed
cold
Therefore, whatever is causing
Rings
shows
frequently
temperature-salt structure at the base
currently known how
infer a
(0(100m)) levels and that the temperature never
to these depths.
core
of
Second, we note that heat is getting
mixing is different in different parts of the column.
from
development
the
to model
Hydrographic data
compensating,
similar,
It is not
of the mixed layer.
these mixed
layer
salt
the
and
temperature
traces, or parameterize their effect on mixed layer density, but with
respect
to
predicting
sea
surface
temperature,
it
is
important
understand how the incoming heat is distributed with depth.
to
The data
suggests that it is not a linear distribution, and that the presence of
Page -225-
salt is having some influence.
are responsible
clear.
or
can maintain
Whether double diffusive instabilities
the
variations
in
turbulence
is not
Page -226-
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Page -230-
ACKNOWLEDGEMENTS
It has been my good fortune, during the past few years, to have
labored on my thesis under the thoughtful tutelage of Dr. Glenn R.
privilege, to
and unique
my pleasure,
is
it
now
And
Flierl.
as an
students
doctoral
up-and-coming
to
him
unequivocally recommend
the
possessing
and
aptitude,
and
probity,
academic
impeccable
of
advisor
him
count
to
honored
I
am
thesis.
a
of
vision essential to the steerage
In addition, I have
and his wife, Norma Kroll, amongst my friends.
thorough committee,
and
stimulating
most
a
of
enjoyed the attentions
Carl Wunsch, all
and
Richardson,
Philip
composed of Drs. Dale Haidvogel,
of whom gave an early version of this manuscript a careful reading and
Special thanks
offered an abundance of muchly appreciated criticisms.
go to Dr. Haidvogel, from whom I have learned computational methods, and
to Dr. Terry Joyce for chairing my thesis defense.
I would certainly be remiss were I not to mention the many valuable
comments I have recieved from and discussions I have had with Drs.
William Schmitz, Paola Rizzoli, and Mark Cane, and I gratefully
acknowledge Drs. Harry Bryden, Ed Harrison, and Mel Briscoe for their
sage advice at critical points during my graduate student tenure.
This thesis has benefited from two summers' worth of my unofficial
participation in the Geophysical Fluid Dynamics summer school, at which
times I met and worked with Drs. Rory Thompson and Don Olson.
The Joint Program in Oceanography is endowed with as fine a
Two fellow
collection of people as I suspect I shall ever meet.
students, Stephen Meacham and Rui X. Huang, have had more to do with the
content of this thesis than I think either of them realize. I have also
profited handsomely from discussions with and the friendship of Karen
Beggs, M. Benno Blumenthal, Al Campbell, Affonso Mascarenhas, Randy
I have been inspired by my companions and
Patton, and Sophie Wacogne.
contemporaries Drs. William R. Young, Lawrence Pratt, and Lynne Talley,
and have been so lucky as to share the final throes of completion with
Dr. Teresa Chereskin.
Several of the figures in this thesis have been expertly drafted by
of
set
cryptic
a
many
deciphered
patiently
who
Kole,
Isabelle
the
of
number
A
scrawl.
illegible
nearly
my
in
instructions written
sections in the final draft have been cast into shape by the efforts of
Virginia I. Mills.
The Joint Program is most ably administrated by A. Lawrence Pierson
and Mary Athanis, and the Department of Meteorology and Physical
Oceanography by Jane McNabb and the aforementioned Miss Mills, all of
whose efforts, I must say, have made my transit through MIT both
pleasant and remarkably free of non-academic challenge.
Dr. and Mrs.
Finally I thank my wife, Melinda, and my parents,
and endured
supported,
William D. Dewar, who have patiently listened to,
month, and
one
years,
five
for
William K. Dewar, the graduate student,
its trying
without
not
say,
we
shall
been,
24 days, a job which has
moments.
The present research has been conducted under NOAA contract #
The computational costs
NA80AA-D-0057 and NSF contract # OCE-8240455.
and ONR grant #
Mollo-Christenson
Erik
have been generously borne by Dr.
operation of an
coherent
and
ONR-c N00014-79-C-0838, and the maintenance
inveterately disagreeable computer must be credited to the able and
dogged persistence of Ken Morey.