Parametric subharmonic instability of internal
gravity wave beams
by
Hussain H. Karimi
B.S., University of California, San Diego (2010)
M.S., Massachusetts Institute of Technology (2012)
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
© Massachusetts Institute of Technology 2015. All rights reserved.
Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Mechanical Engineering
May 8, 2015
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Triantaphyllos R. Akylas
Professor of Mechanical Engineering
Thesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
David E. Hardt
Chairman, Department Committee on Graduate Students
Parametric subharmonic instability of internal gravity wave
beams
by
Hussain H. Karimi
Submitted to the Department of Mechanical Engineering
on May 8, 2015, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Mechanical Engineering
Abstract
Internal gravity wave beams are time-harmonic plane waves with general spatial profile that arise in continuously stratified fluids owing to the anisotropy of this wave
motion. In the last decade, these wave disturbances have been at the forefront of research, both from a fundamental perspective and in connection with various geophysical flow processes. Oceanic internal wave beams, in particular, form the backbone
of the internal tide, generated by the interaction of the barotropic tide with sea-floor
topography. The internal tide breakdown and its role in deep-ocean mixing have
attracted considerable attention. In this context, it is of interest to understand mechanisms by which internal wave beams become unstable and eventually breakdown,
thereby contributing to mixing.
A possible instability mechanism is via resonant triad interactions that amplify
short-scale perturbations with frequency equal to one half of that of the underlying
wave. For spatially and temporally monochromatic internal waves, this so-called
parametric subharmonic instability (PSI) has been studied extensively and indeed can
lead to breakdown. By contrast, the focus here is on understanding how wave beams
with locally confined spatial profile, such as those in the field, may differ, in regard
to PSI, from monochromatic plane waves. To this end, an asymptotic analysis is
made of the interaction of a small-amplitude wave beam with short-scale subharmonic
wavepackets in a nearly inviscid stratified Boussinesq fluid. A novel system of coupled
evolution equations that govern this nonlinear interaction is derived and analyzed.
For beams with general localized profile, unlike monochromatic wavetrains, it is found
that triad interactions are not strong enough to bring about instability in the limited
time that subharmonic perturbations overlap with the beam. On the other hand,
for quasi-monochromatic wave beams whose profile comprises a sinusoidal carrier
modulated by a locally confined envelope, PSI is possible if the beam is wide enough.
In this instance, a stability criterion is proposed which, under given flow conditions,
provides the minimum number of carrier wavelengths a beam of small amplitude must
comprise for instability to arise.
Furthermore, the effect of the Earth’s rotation on PSI of internal wave beams is in2
vestigated. Even though rotation induces transverse motion, plane waves in the form
of beams are still possible. Most importantly, however, in the presence of rotation,
short-scale subharmonic wavepackets may experience prolonged interaction with a
beam of general localized profile, potentially causing instability. This situation arises
when the subharmonic frequency nearly matches the background Coriolis frequency
so the group velocity of subharmonic wavepackets is close to zero. In particular, wave
beams generated by the M2 tidal flow over topography encounter this resonance near
the critical latitude of 28.8◦ (N and S). Coupled evolution equations for subharmonic
wavepackets riding on a beam of general profile under such resonance conditions are
derived. Based on this asymptotic model, it is shown that locally confined beams
above a certain threshold amplitude are unstable to near-inertial subharmonic disturbances. The theoretical predictions are supported by recent field observations
which show that significant energy transfer to subharmonic disturbances does indeed
occur near the critical latitude and not elsewhere.
Thesis Supervisor: Triantaphyllos R. Akylas
Title: Professor of Mechanical Engineering
3
Acknowledgments
My education at MIT would not have been so satisfying had I missed the opportunity
to have Professor Triantaphyllos R. Akylas as my advisor, one of the sharpest minds
I have ever had the pleasure to encounter. A typical discussion of ours would end in
my astonishment at the profound clarity and insight Prof. Akylas is able to provide
with barely a moment’s thought. Indeed, it is an ambition of mine to perform my
future professional duties with a speed, accuracy, and thoroughness that is consistent
with the training I received under his guidance.
The clarity he brings to the scientific community is complemented by his effectiveness as a course instructor. After taking four of his courses and working as his
teaching assistant for four semesters, I have heard countless testimony from classmates conveying their excitement at my fortune to work with such a pedagogical
master of illumination. It is a sentiment I am happy to share.
The numerical work in chapter 4 is an ongoing effort conducted under the supervision of Prof. Chantal Staquet at LEGI in Grenoble. Her generosity and hospitality
during my summer visit was immeasurably granted and left in me a desire to return
to France in any capacity which I can muster. We hope to complete our collaboration
so that the theoretical emphasis of this thesis may be held tangibly in the geophysics
community.
Breadth and scope were added to this thesis by the useful suggestions from thesis
committee members Prof. Tom Peacock and Prof. Kostya Turitsyn. Our discussions
pushed me to further appreciate the implications of our work and its value to a broader
range of scientific knowledge. The scientist attempts to achieve an understanding of
quite complex phenomena, but its societal application relies on the ability to ground
one’s work at an accessible, realizable level.
Assistance has a way of presenting itself in the corners you need it, as evidenced
by the strongly supportive staff in the Mechanical Engineering Department. Leslie
Regan, always the student advocate, contributed swiftly and effectively to a number
of logistical issues that inevitably arose through my time here. The friendly faces of
4
administrators Laura Canfield and Ray Hardin were always available and quick to
handle all office requests.
Perhaps the greatest strength of MIT lies in its cohesive peer network. My professional training has been complemented quite rigorously by my personal development through the many talented colleagues I had the satisfaction of befriending.
From discussing the hyperbolicity of slightly non-linear wave equations to a thorough
breakdown of the best jazz bars in town, the lively discourse between the hallways of
building 3 have been vital to my growth. Although a full list of these friendships is
beyond the scope of this thesis, I would like to thank here Sasan Ghaemsaidi, Usama
Kadri, Tal Cohen, Margaux Martin-Filippi, Maha Haji, Jerry Wang, Ashkan Hosseinloo, Matt Mayser, Kashif Khan and Nils Holzenberger for inspiring me by example
and bringing me daily joy at MIT.
It is a fortunate coincidence that my uncle and his family moved to Cambridge
just one year prior to my arrival. They have been my family away from home while
welcoming me into theirs. Finally, I extend a deep appreciation towards my parents
and brother, who have supported me throughout. From a young age, I was taught
the rewards of perseverance, hard work, and sincerity. The foundations of whatever
success I may experience were prescribed to me as a humble child.
This work was supported in part by the National Science Foundation under grants
DMS-1107335.
5
Contents
Cover page
1
Abstract
2
Acknowledgments
4
1 Introduction and review
14
1.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.2
Brunt-Väisälä frequency . . . . . . . . . . . . . . . . . . . . . . . . .
16
1.3
Boussinesq approximation . . . . . . . . . . . . . . . . . . . . . . . .
18
1.4
Internal gravity waves . . . . . . . . . . . . . . . . . . . . . . . . . .
23
1.5
Wave beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2 PSI of internal gravity wave beams
31
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.2
Long–short wave interaction . . . . . . . . . . . . . . . . . . . . . . .
34
2.3
Nearly monochromatic beam profile . . . . . . . . . . . . . . . . . . .
41
2.4
Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
2.4.1
Sinusoidal wavetrain . . . . . . . . . . . . . . . . . . . . . . .
45
2.4.2
Eigenvalue problem . . . . . . . . . . . . . . . . . . . . . . . .
46
2.5
Top-hat beam envelope . . . . . . . . . . . . . . . . . . . . . . . . . .
48
2.6
Transient disturbance evolution . . . . . . . . . . . . . . . . . . . . .
52
2.7
Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
6
3 Near-inertial PSI of internal wave beams
58
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
3.2
Near-inertial approximation and scalings . . . . . . . . . . . . . . . .
59
3.3
Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
3.3.1
Sinusoidal plane waves . . . . . . . . . . . . . . . . . . . . . .
65
3.3.2
Locally confined beams . . . . . . . . . . . . . . . . . . . . . .
67
3.4
Long-time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
3.5
Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
4 Applications and numerical simulations of PSI in wave beams
78
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
4.2
Application to experiments . . . . . . . . . . . . . . . . . . . . . . . .
79
4.3
Application to beams generated by iTides . . . . . . . . . . . . . . .
83
4.4
Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . .
86
4.4.1
Periodic boundary conditions . . . . . . . . . . . . . . . . . .
87
4.4.2
Qualitative results . . . . . . . . . . . . . . . . . . . . . . . .
88
4.4.3
Remaining challenges . . . . . . . . . . . . . . . . . . . . . . .
94
5 Concluding remarks
98
Appendix
102
A Derivation of wave-interaction equations
102
B Bifurcation of eigensolution branches
105
C Derivation of near-inertial evolution equations
107
D Comparison with Young et al. (2008) of PSI growth rate for sinusoidal plane waves
111
Bibliography
113
7
List of Figures
1-1 Fluid displaced from hydrostatic equilibrium. A fluid parcel initially
at z0 in its equilibrium position having density ρ(z0 ), as shown on
the left, is vertically displaced as shown on the right. The vertical
forces in red are the surface pressure forces and the body weight of the
parcel. A force balance reveals that the displaced fluid parcel oscillates
vertically, much like a mass on a spring, with a frequency dependent
on the strength of the local density stratification. . . . . . . . . . . .
18
1-2 The image on the left is that of an internal gravity wave generated by
the vertical oscillation of cylinder extended into the page [36]. The
observed pattern of wave propagation is sometimes referred to as “St.
Andrew’s Cross”. The vertical rod which supports the oscillating cylinder in the centre is visible as a dark shadow. The resulting beams
emanate at an angle θ to the horizontal. Following the beam that radiates towards the lower right corner, the image on the right indicates
the directions of the group velocity cg and phase velocity c. Lines of
constant phase are shown parallel to the group velocity. . . . . . . . .
27
1-3 The effect of the sign of the magnitude of the wavevector κ when the
direction η is fixed by the dispersion relation. Flipping the sign of κ
also flips the direction of the group velocity cg and phase velocity c. .
8
30
2-1 Geometry of long–short wave interaction. The underlying wave beam
with general locally confined profile of characteristic width L∗ has frequency ω and propagates at an angle θ to the horizontal such that
ω = sin θ. Subharmonic perturbations are short-crested (λ∗ /L∗ 1)
nearly monochromatic wavepackets with frequency close to ω/2 that
propagate at an angle φ to the horizontal, with sin φ = 21 sin θ. . . . .
36
2-2 Schematic of interaction of nearly monochromatic wave beam of frequency ω = sin θ and nondimensional amplitude 1 with subharmonic perturbations of frequency close to 21 ω = sin φ. The beam
profile comprises a sinusoidal carrier modulated by a slowly varying
envelope, Λ∗ /D∗ = O(1/2 ), where Λ∗ denotes the (dimensional) carrier wavelength and D∗ the characteristic width of the envelope. The
perturbations are short-scale wavepackets with (dimensional) carrier
wavelength λ∗ , such that λ∗ /Λ∗ = O(1/2 ). . . . . . . . . . . . . . . .
42
2-3 Plots (—) of the first three eigenvalue branches λ̂(n) (κ̂) of the character(n)
istic equation (2.51), which bifurcate at κ̂c = (2n+1)π for n = 0, 1, 2.
The intersections of the lowest (n = 0) of these modes with the cubic Cκ̂3 ( ), shown here for C = 1.5 × 10−3 , determine the range of
unstable disturbance wavenumbers κ̂l < κ̂ < κ̂u . The dashed lines (–
–) indicate the asymptotic approximations (2.53) and (2.54) of λ̂(0) (κ̂)
(0)
near and far away from the bifurcation point κ̂c , respectively.
. . .
51
2-4 Evolution of wave beam, with initially Gaussian envelope (2.61), and
subharmonic perturbations with the most unstable wavenumber, according to numerical solution of the coupled equations (2.30)–(2.31)
subject to the initial conditions (2.62). The wave envelope magnitudes
of the beam (|q|) and the perturbations (|a|, |b|) are displayed at various
times τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
54
3-1 Geometry of beam–wavepacket interaction. The underlying wave beam
with general locally confined profile of characteristic width L∗ has frequency ω and propagates at an angle θ to the horizontal according to
(3.6). Subharmonic perturbations are short-crested (k∗ L∗ 1) nearly
monochromatic wavepackets with frequency close to ω/2 that propagate at an angle φ to the horizontal given by the dispersion relation
(3.7). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
3-2 Plots of the real part of the first eigenvalue branches λ̂r (κ̂, σ̂) of beam
with Gaussian profile (3.45) for σ̂ = 0 (—), 1 (– –), and 2 (– · –). An
additional eigenvalue branch is shown for σ̂ = 0 which emerges just
before the first branch ends, reaching a slightly larger peak. The intersections of these modes with the quadratic Cκ̂2 , shown here for C =
0.05 ( ) and 0.09 (
), determine the range of unstable disturbance
wavenumbers κ for which (3.43) is satisfied. . . . . . . . . . . . . . .
70
3-3 Evolution of wave beam, with initially Gaussian envelope (3.45), and
subharmonic perturbations with the most unstable wavenumber κ =
1.96, according to numerical solution of the coupled equations (3.16)–
(3.17) subject to the initial conditions (3.50) as shown in figure 3-2.
The real part of wave envelope magnitudes of the beam (Qr ) and the
perturbations (Ar , Br ) are displayed at various times T .
. . . . . . .
74
3-4 Contours of the along-beam velocity component at (a) initialization,
(b) appearance of PSI in the wavefield, and (c) near the end of the
interaction under the assumed asymptotic conditions. . . . . . . . . .
76
4-1 Internal tide generation due to M2 tidal flow over a Gaussian ridge. The
shown horizontal velocity clearly indicates the presence of a discrete
beam. A sample across the beam is taken at the dashed blue line shown
in figure 4-2(a).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
84
4-2 Cross-beam profile and spectra of internal tide. Notice that the crossbeam profile does not appear to contain a carrier wavenumber in (a),
though the spectra reveals a small dominant wavenumber in (b). The
width of the beam, however, is about twice the dominant wavelength.
85
4-3 Contour plot of the vorticity field of a numerical simulation initialized
with small random noise over a sinusoidal wave shown after sufficient
time has passed for instabilities to develop. The dominant mode of
instability appears as fine-scale disturbances with angle of propagation
more shallow than the underlying wave, implying that the frequency
of perturbations are less than that of the underlying wave. . . . . . .
86
4-4 The domain of the numerical simulation is shown in the center, thickoutlined box which contains the underlying beam of interesting (dark
grey). To satisfy periodic boundary conditions in the horizontal and
vertical, two additional beams (light grey) are included in the top-right
and bottom-left corner to include the effects of the dash-outlined boxes
adjacent to the domain. . . . . . . . . . . . . . . . . . . . . . . . . .
87
4-5 Evolution of the total vorticity field is shown as contours. (a) The
underlying beam is initialised with very small random noise. There
are nearly 2 wavelengths contained in the beam of Gaussian envelope.
(b) Development of PSI is clearly visible at t = 1000 as fine-scale
contours are seen to interact with the underlying beam. (c) By t =
1200 PSI wavepackets continuously extract energy from the beam while
transporting energy with more shallow propagation angles. . . . . . .
91
4-6 The total (potential and kinetic) energy of the wave field at the center
of the numerical domain filtered at half the frequency of the underlying
beam. Initially the energy drops off as the randomly seeded disturbance
takes shape as PSI wavepackets. Once formed, the subharmonic perturbations grow at exponential rate, seen here between t = 1000 and
t = 1200. The black line above the energy curve shows the fitting used
to determine the growth rate. . . . . . . . . . . . . . . . . . . . . . .
11
92
4-7 Evolution of the total vorticity field is shown as contours, but for
a beam of general spatial profile lacking the presence of a carrier
wavenumber. (a) The underlying beam is initialised with very small
random noise. (b) Shown at t = 2100, the beam is still completely
intact and random noise is apparently unable to excite any instability mode (including PSI). Compare this to the PSI of figure 4-5 that
appears at t = 1000. . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
4-8 For arbitrary beam angles of propagation, θ, the periodic domain is
rectangular (a) with vertical (Ly ) and horizontal (Lx ) widths. By simple coordinate transformation, y = y 0 tan θ, the domain is square (b)
thereby reducing numerical complexities. . . . . . . . . . . . . . . . .
94
5-1 World map shown with red lines at near-inertial latitudes where energy
transfer to fine-scale subharmonic wave motion is expected to arise. Internal wave generation sites, due to steep topography, near the critical
latitudes are identified by green circles. . . . . . . . . . . . . . . . . . 100
12
List of Tables
1.1
The scaling parameters on the left-hand side are chosen so that they
are appropriate to the wave motion of internal gravity waves. Applied
to the field variables of interest, and to the buoyancy frequency N , we
can easily compare the relative importance of the different terms in the
governing equations simply by inspection. This method allows us to
make clean approximations and clearly judge the limitations of them
in doing so. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
22
Experimental and numerical results of Bourget et al. [3] are summarized. For various beam configurations, the observed PSI wavepacket
characteristics are reported, along with the predictions of our asymptotic analysis in parentheses. . . . . . . . . . . . . . . . . . . . . . . .
13
83
Chapter 1
Introduction and review
1.1
Motivation
Internal gravity waves arise in continuously stratified fluids, such as the ocean and
atmosphere, due to the restoring action of buoyancy forces. Their unique transport
properties cause these waves to contribute to the vertical distribution of energy in
the fluid medium. In the atmosphere, internal gravity waves affect wind speeds by
carrying momentum from the ground to higher altitudes [6]. In the ocean, they are
partially responsible for the gradual vertical temperature gradients [11].
The consequences that follow from the anisotropy of the fluid medium, due to
stratification in the direction of gravity, was first investigated by Görtler [13] and
Mowbray and Rarity [36] in which disturbances to hydrostatic equilibrium were created by an oscillating cylinder in a rectangular tank. In an isotropic medium, such
a source of disturbance would lead to cylindrical wavefronts; however, the vertical
anisotropy here causes wave propagation to take the pattern of four straight arms radiating away from the source. Each arm of this cross pattern, known as St. Andrew’s
Cross (see figure 1-2), carries energy in a distinct direction away from the source
in the form of time-harmonic plane-wave disturbances with a general spatial profile.
These disturbances are known as an internal gravity wave beams.
A common mechanism of internal wave generation in the ocean stems from the
interaction of oscillating tidal flow with underwater topography, such as ridges and
14
trenches [10]. Internal waves generated in this fashion are called internal tides. Deepsea mixing is thought to be influenced by the vertical transport of energy by these
internal tides [48]. The overall effect is believed to be partly responsible for the gradual
temperature variation in the ocean [9]. The processes by which internal waves release
energy into the surrounding system is not fully understood, though various instability
modes of internal waves are suspected to arise in the early stages of possible mixing
and breakdown.
Initial stability analyses on spatially and temporally monochromatic waves found
that internal waves are always unstable in the inviscid limit [7, 31, 35]. The dominant
mode of instability is characterized by short-scale wave disturbances with half the
frequency of the underlying wave and is known as parametric subharmonic instability
(PSI). With the understanding that internal waves may represent the general localized
structure of internal tides, the PSI of monochromatic waves has been often used as a
predictive model for ocean applications.
Internal tides generated in the deep sea, however, propagate as wave beams that
are localized disturbances of finite width and do not necessarily exhibit sinusoidal
behavior in their spatial structure. A more accurate depiction of internal tides may
be drawn by taking advantage of the unique properties that internal waves possess.
Not only are sinusoidal plane waves exact solutions of the nonlinear inviscid equations
of motion, they are also insensitive to nonlinear interactions with plane waves of
different wavelength but equal frequencies. These properties allow the description
of a general wave beam to be given by the superposition of monochromatic waves
with a single identical frequency as presented in Tabaei and Akylas [46]. Since planewave beams are also nonlinear solutions of inviscid stratified fluids, they serve as a
convenient basis for the formulation of analysis that more accurately portray internal
wave motion in the deep ocean.
As pointed out by Sutherland [43], the occurrence of PSI in wave beams is a
much more rare event than suggested by the theoretical predictions of PSI based
on monochromatic waves of infinite extent. Observations from numerical and experimental studies indicate that the width of a realistic internal wave beam plays a
15
decisive role in determining whether or not PSI develops, a parameter that is completely overlooked when considering infinite sinusoidal waves. Our objective here is to
theoretically investigate the conditions under which the PSI of internal wave beams
can instigate breakdown processes that ultimately lead to energy and momentum
deposition.
This thesis is organized as follows. Chapter 1 first presents a brief review of some
well-known, singular properties of internal gravity waves [11, 24, 27, 42]. Although the
treatment in chapter 1 may be found elsewhere, it is included here for the convenience
of the reader. A detailed analysis of PSI as it may occur in internal wave beams
of general spatial profile follows in chapter 2. In chapter 3 we include the effects
of Earth’s rotation, which plays a significant role near critical latitudes. A detailed
application of the theoretical analysis is presented in chapter 4 along with preliminary
work in which we perform numerical simulations over a range of beam configurations.
Finally, the thesis will be closed with a brief concluding discussion in chapter 5.
1.2
Brunt-Väisälä frequency
It is perhaps instructive to first consider the simplest sort of perturbation to a stably stratified fluid under the conditions of hydrostatic equilibrium. A fluid parcel
vertically displaced is acted upon by the restoring buoyancy force which causes the
fluid parcel to accelerate towards its equilibrium position, but the ensuing gain in
momentum results in its overshoot. Still displaced from equilibrium, the restoring
force now acts in the opposite direction in an effort to restore the fluid parcel to its
stable position. This oscillatory motion is much like a mass supported by a spring,
which servers as the restoring force. Let us examine the details of this fluid motion
as a starting point.
In a stratified fluid under hydrostatic equilibrium, the pressure p(z) and density
ρ(z) satisfy
dp
= −gρ,
dz
(1.1)
where z denotes the vertical coordinate, measured upwards. Applying momentum
16
principles to a differential fluid parcel that is initially located at z0 and subsequently
displaced by a short vertical distance δ, as shown in figure 1-1, we find in the vertical
1
1
(ρ(z0 )dV ) δ̈ = p z0 + δ − dz − p z0 + δ + dz dA − gρ(z0 )dV.
2
2
(1.2)
Expanding the pressure terms around the displaced position, z0 + δ, and dividing
through by the parcel volume dV ,


ρ(z0 )δ̈ = 
=−
− 12 dz
dp
(z0 + δ) −
dz
dz
1
dz
2

dp
(z0 + δ)

dz
 − gρ(z0 )
dp
(z0 + δ) − gρ(z0 ).
dz
(1.3)
The hydrostatic pressure gradient is the buoyancy force and can be expressed in terms
of the density stratification by the hydrostatic equilibrium (1.1) as
dp
(z0 + δ) = −gρ(zo + δ).
dz
(1.4)
Now we expand about z0 and substitute ρ(z0 + δ) ≈ ρ(z0 ) + δdρ(z0 )/dz into the
momentum equation to find the equation of motion for simple vertical oscillations to
be
ρ(z0 )δ̈ = g
dρ
(z0 )δ.
dz
(1.5)
Since z0 is arbitrary, that is for any fluid parcel vertically perturbed from its equilibrium position, the preceding equation is valid for any position z and we drop the
point of evaluation z0 . Quite often, the equation of motion is written as
δ̈ + N 2 δ = 0,
(1.6)
g dρ
ρ dz
(1.7)
where
N2 ≡ −
is known as the Brunt-Väisälä frequency, or sometimes simply as the buoyancy fre17
g
Figure 1-1: Fluid displaced from hydrostatic equilibrium. A fluid parcel initially at z0
in its equilibrium position having density ρ(z0 ), as shown on the left, is vertically displaced as shown on the right. The vertical forces in red are the surface pressure forces
and the body weight of the parcel. A force balance reveals that the displaced fluid
parcel oscillates vertically, much like a mass on a spring, with a frequency dependent
on the strength of the local density stratification.
quency. It is a measure of the local density stratification and is useful in characterizing
ocean (and atmospheric) flows. For a stably stratified fluid, density decreases in the
direction opposing gravity so dρ/dz < 0 and N 2 > 0. Typical values of N are
∼ 10−3 s−1 in the ocean and atmosphere [42]. The corresponding period of oscillation
is on the order of 10 hours.
1.3
Boussinesq approximation
There are various ways to present a reasonable approximation to determine the flow
due to perturbations from hydrostatic equilibrium in a stratified fluid [24, 42], each of
which instructively leads to the same result known as the Boussinesq approximation.
However, the reasoning provided between various sources is slightly different, though
the essence is the same. Here, we will provide the details of the approximation as given
by Tabaei and Akylas [46] since we believe that the application of the approximation
18
in this fashion is done so in a manner which picks up all the subtleties and limitations
in a single step. There is also the added advantage of removing the explicit presence of
the hydrostatic density variation from the momentum equations. The mathematical
reward of this simplification will be noted at the end of this section when it becomes
apparent.
First, let us agree on the relevant equations and begin with momentum balance,
%
Du
+ %2Ω × u = −∇P − %gêz + µ∇2 u,
Dt
(1.8)
where D/Dt ≡ ∂/∂t + u · ∇ is the material derivative following a fluid parcel, u =
uêx + vêy + wêz is the vector velocity field, êz is the unit vector oriented in the
positive vertical direction, P is the total pressure, % is the total density, µ is the
dynamic viscosity, and Ω is the local Coriolis parameter. It will prove useful to
invoke the constitutive relation appropriate to oceanography before conserving mass.
That is, the incompressibility condition such that any particular fluid parcel retains
its density, %, throughout the entirety of the flow, is stated as
D%
= 0.
Dt
(1.9)
Now substituting (1.9) into mass conservation
∇·u=−
1 D%
,
% Dt
we find that the right-hand side becomes identically zero, and the continuity equation,
∇ · u = 0,
(1.10)
requires the divergence of the flow field to be zero.
The appropriate scaling parameters are the typical wavelength L of an internal
wave as length scale, 1/N0 as time scale where N0 is a typical value of the BruntVäisälä frequency, and ρ0 a nominal value of density. The relative length scale of
19
density variation L is also important and from (1.7) we find that it scales like
L ∼ g/N02 .
(1.11)
The Boussinesq parameter is the ratio of the two relevant length scales, B ≡ L/L, so
B = LN02 /g.
(1.12)
To show that density perturbations scale like B, consider again the momentum equations (1.8), but with the substitution % = ρ + ρ and P = p + p so that the hydrostatic
variation (1.1) cancels out,
%
Du
+ 2Ω × u
Dt
= −∇p − ρgêz + µ∇2 u.
(1.13)
In gravity waves we expect the inertial terms to balance with buoyancy perturbations,
∼ ρg. Along with the parameters above, then, density perturbations,
ρ Dw
Dt
ρ∼
N02 L
g
ρ0 = Bρ0 .
(1.14)
The result (1.14) anticipates the known result [24, 42].
To write the governing equations in the most convenient form for the ensuing
analysis,
ρ(z)
ρ(x, t),
ρ0
ρ(z)
P (x, t) = p(z) +
p(x, t).
ρ0
%(x, t) = ρ(z) + B
(1.15a)
(1.15b)
Note that the perturbations ρ and p are not simply the variations to hydrostatic
equilibrium since these quantities are scaled with local hydrostatic density; instead,
this locally scaled variable will allow the complete removal of the explicit presence of
the hydrostatic density, ρ(z), within the Boussinesq approximation, from (1.8).
20
To see this, we first introduce (1.15) into (1.8),
Du
ρ
ρ
ρ
+ 2Ω × u = −∇ p + p − gρ 1 + B
ρ 1+B
êz + µ∇2 u.
ρ0
Dt
ρ0
ρ0
(1.16)
The hydrostatic equilibrium terms drop by (1.1), and after dividing through by ρ,
ρ
Du
−∇(ρp)
ρ
1+B
+ 2Ω × u =
− gB êz + ν∇2 u,
ρ0
Dt
ρ0 ρ
ρ0
(1.17)
where ν = µ/ρ is the kinematic viscosity. Distributing the gradient operator in the
first term on the right-hand side and incorporating (1.7),
−∇(ρp)
∇p p(dρ/dz)
∇p N 2
=−
êz = −
pêz ,
−
+
ρ0 ρ
ρ0
ρ0 ρ
ρ0
ρ0 g
(1.18)
the momentum equation yields
Du
∇p
ρ
N2
ρ
+ 2Ω × u = −
p êz + ν∇2 µ.
− gB −
1+B
ρ0
Dt
ρ0
ρ0 ρ0 g
(1.19)
Similarly, (1.15) applied to the incompressibility condition (1.9),
∂
+u·∇
∂t
ρ
ρ(z) 1 + B
= 0.
ρ0
(1.20)
Separating the local and advective derivatives, we distribute the latter per the product
rule,
ρ ∂ρ
ρ
ρ
dρ
+ B
(u · ∇) ρ + 1 +
w
= 0.
B
ρ0 ∂t
ρ0
ρ0
dz
(1.21)
Dividing through by Bρ/ρ0 , the factor in the third term can be simplified with the
use of (1.7) and (1.12) as
ρ0
B
dρ/dz
ρ
=−
N 2 ρ0
N 2 ρ0
=− 2 .
(Bg)
N0 L
21
(1.22)
Scaling parameters
Nondimensional quantities
dimension
based on
quantity
length
typical wavelength
L
time
typical Brunt-Väisälä frequency
N0−1
density
nominal density
ρ0
u/N0 L
ρ/ρ0
P/N02 L2 ρ0
N/N0
Ω/N0
ν/N0 L2
→ u
→ ρ
→ P
→ N
→ Ω
→ ν
Table 1.1: The scaling parameters on the left-hand side are chosen so that they are
appropriate to the wave motion of internal gravity waves. Applied to the field variables of interest, and to the buoyancy frequency N , we can easily compare the relative
importance of the different terms in the governing equations simply by inspection.
This method allows us to make clean approximations and clearly judge the limitations
of them in doing so.
The incompressibility equation can then be written as
N 2 ρ0
ρ
∂
+u·∇ ρ− 2
1+B
w = 0.
∂t
N0 L
ρ0
(1.23)
Thus far, everything is exact. To make an order of magnitude approximation, the
governing equations (1.10), (1.19) and (1.23), must first be appropriately nondimensionalized. The scaling parameters are reiterated in table 1.1 along with a list of the
nondimensional variables which are specifically scaled so that they are all of the same
order. The governing equations then become
(1 + Bρ)
∂
+ u · ∇ + 2Ω× u = −∇p − ρ − BN 2 p êz + ν∇2 u,
∂t
∇ · u = 0,
∂
+ u · ∇ ρ − N 2 (1 + Bρ)w = 0.
∂t
(1.24a)
(1.24b)
(1.24c)
All the field variables are O(1), which leaves the relative magnitude of different terms
solely dependent on B. Furthermore, the Brunt-Väisälä frequency defined by (1.7) is
nondimensionalized to
dρ
= −BρN 2 .
dz
(1.25)
In one fell swoop, we now make the approximation that B → 0, which is to say
22
that the length scale associated with wave motion is much less than the length scale
of relevant hydrostatic density variations. Inherent to this approximation then, is
that ρ → 1 by (1.25). This means that a fluid parcel undergoing wave motion will
experience spatial variations in which the local density is different, however those
changes are considered to be quite small. Note that the entirety of the Boussinesq
approximation is contained in B → 0. The mathematical reward with our particular
form of hydrostatic perturbations (1.15) mentioned earlier is that the momentum
equations have constant coefficients, regardless of the background stratification, ρ.
We will further simplify the fluid system by considering the background stratification
to be uniform (N = 1) so that (1.24) becomes
∂
+ u · ∇ + 2Ω× u = −∇p − ρêz + ν∇2 u,
∂t
∇ · u = 0,
∂
+ u · ∇ ρ = w.
∂t
(1.26a)
(1.26b)
(1.26c)
Recall that we are taking the fluid to be incompressible from its hydrostatic equilibrium. That is, if we follow two different fluid parcels located at z1 and z2 during
equilibrium conditions, they will have a density ρ(z1 ) and ρ(z2 ), respectively, for all
time even when perturbed.
1.4
Internal gravity waves
The purpose of this chapter is to illustrate the physics of internal gravity waves, and
for instructive purposes, we will take the total pressure and density to be P = p + p
and % = ρ + ρ, respectively, in the governing (dimensional) equations (1.8), (1.9), and
(1.10). For the remainder of this review chapter, we will work in the these terms to
avoid confusion by calculating quantities which have very clear, unambiguous interpretations so that the basics of internal gravity are openly understood. Furthermore,
the (weak) effects of viscosity [46] and Earth’s rotation will be neglected for now. We
will return to the nondimensional equations (1.26) in chapters 2 and 3 where it will
23
prove useful as it has in previous works [47].
The Boussinesq approximation, in these dimensional terms, can then be applied
to a uniform background stratification to unveil the equations of wave motion [24, 42]
ρ0
∂
+ u · ∇ u = −∇p − ρgêz ,
∂t
∇ · u = 0,
∂
dρ
+ u · ∇ ρ = −w .
∂t
dz
(1.27a)
(1.27b)
(1.27c)
It is useful to first derive the linear solution which assumes small-amplitude waves
and allows the neglect of the advective terms in (1.27),
ρ0 ut = −∇p − ρgêz ,
(1.28a)
ux + vy + wz = 0,
(1.28b)
ρt = −wρz ,
(1.28c)
where (x, y, z, t)-subscripts denote derivatives. There are five unknowns with the
same number of equations in the set (1.28). First, pressure is eliminated by taking
the curl of the momentum equations [∇×(1.28a)] which yields,
ρ0 (wy − vz )t = −ρy g.
(1.29a)
ρ0 (wx − uz )t = −ρx g.
(1.29b)
uty = vtx .
(1.29c)
Note that (1.29) contains only two linearly independent equations, not three. By
taking cross derivatives of (1.29a) and (1.29b), followed by the difference, we can
re-construct (1.29c). This is expected since the elimination of a variable is associated
with the usage of an equation.
Although it is possible to solve for any of the four remaining variables, it is most
convenient to favour w. We will do so here by focusing on (1.29a) and allow our next
24
steps be guided by the elimination of v followed by u. After taking the z-derivative
of (1.28b), v is expressed as
− vyz = uxy + wzz .
(1.30)
The result (1.30) can be substituted into the y-derivative of (1.29a),
ρ0 (wyy + uxz + wzz )t = −ρyy g
(1.31)
The velocity component u is the next target of elimination and prepared for dispatch
by considering the x-derivative of (1.29b),
ρ0 uzxt = ρ0 wxxt + ρxx g.
(1.32)
Substitution of (1.32) into (1.31) gives
ρ0 (wxx + wyy + wzz )t = −g(ρxx + ρyy )
(1.33)
The last variable to go is density, leaving w isolated. Before invoking (1.28c), it
must have its second x- and y- derivatives taken so to be matched with the t-derivative
of the right-hand side of (1.33),
ρ0 (wxx + wyy + wzz )tt = −g(ρtxx + ρtyy )
= gρz (wxx + wyy ).
(1.34)
Rearranging, the linear solution represented in favour of the vertical velocity component is
where
∇H2
∂2 2
2 2
∇ + N ∇H w = 0.
∂t2
(1.35)
q
≡ ∂ /∂x + ∂ /∂y is the horizontal Laplacian operator and N ≡ − ρg0 dρ
dz
2
2
2
2
is the Brunt-Väisälä frequency as before in §1.2.
25
Assuming a plane wave solution of the form
w = w0 exp {i (k · x − ωt)} = w0 exp {i (kx + ly + mz − ωt)} ,
the dispersion relation is found from (1.35) as
ω2 = N 2
k 2 + l2
= N 2 sin2 θ,
k 2 + l2 + m2
(1.36)
where θ is the angle of the wavevector, k, from the vertical, or equivalently the angle
of propagation from the horizontal as shown in figure 1-2. For simplicity, we will take
l = 0 by rotating our coordinate system by a horizontal angle of tan φ = l/k so that
the waves are now uniform in the new y-coordinate and the y-velocity component is
v = 0. Because the system is horizontally isotropic, there is no loss by making this
rotation.
The ambiguity associated with the signs of θ and ω in (1.36) can be addressed
in different ways, each requiring careful interpretation of the physics involved. Here,
we will always consider ω > 0. Furthermore, θ will be taken from the vertical to the
wavevector such that θ < π/2 (sin θ > 0). In this way, θ and ω are uniquely defined
by (1.36) and can be written, according to our convention, as ω = N sin θ.
To understand this particular geometry more clearly, we first calculate the phase
velocity c and group velocity cg ,
ω
k
êk = N 3 {kêx + mêz } ,
|k|
|k|
m
cg ≡ ∇k ω = N 3 {mêx − kêz } .
|k|
c≡
(1.37a)
(1.37b)
The group velocity and phase velocity are found to be perpendicular to one another,
which equivalently implies the group velocity is also perpendicular to the wavevector,
cg · c = cg · k = 0.
(1.38)
This unusual relationship between the directions of cg and c is attributed to the
26
z
g
x
Figure 1-2: The image on the left is that of an internal gravity wave generated by
the vertical oscillation of cylinder extended into the page [36]. The observed pattern
of wave propagation is sometimes referred to as “St. Andrew’s Cross”. The vertical
rod which supports the oscillating cylinder in the centre is visible as a dark shadow.
The resulting beams emanate at an angle θ to the horizontal. Following the beam
that radiates towards the lower right corner, the image on the right indicates the
directions of the group velocity cg and phase velocity c. Lines of constant phase are
shown parallel to the group velocity.
anisotropy of the system. The density stratification is unique to the direction of gravity resulting in a dispersion relation which requires that the frequency of oscillation is
solely dependent on the orientation of the wavevector, independent of its magnitude.
Also note that the vertical components are exactly opposite,
(c + cg ) · êz = 0.
(1.39)
This is a convenient property to keep in my mind when drawing the wavevector
and group velocity; both vectors are orthogonal with opposing vertical directions.
Physically, the fluid motion is along lines of constant phase unlike the case of the
more familiar surface waves. This is apparent by reconsidering the continuity equation
(1.27b), which can be re-written for the plane wave solution as
k · u = 0.
(1.40)
Revisiting the fully nonlinear governing equations (1.27) in view of the linear plane
27
wave solution, we can expand the advective terms as
u·∇=u
∂
∂
+w
= uk + wm.
∂x
∂z
(1.41)
By recognizing (1.41) is equivalent to (1.40), we find the remarkable result that the
plane wave solution is the fully nonlinear solution since the nonlinear terms identically
vanish [46]. We will return to this crucial point in our construction of wave beams
in section §1.5. For now, we may calculate the other field variables in terms of w0 by
returning to (1.27),

 

m 







−
u



k 











 1 
w 

= w0
exp {i(kx + mz − ωt)} .
ωmρ0 







p
−
2 






 2k 
 




 i N ρ0 
ρ

ωg
1.5
(1.42)
Wave beams
As mentioned at the end of §1.4, the plane wave solution is an exact solution because
the advective terms vanish. This is so because the advective terms, u · ∇, account
for spatial variations in the direction of the flow field; however internal waves are
unique in that there is no spatial variation in this direction. This implies that plane
waves of the same frequency ω and hence, via the dispersion relation (1.36), the same
angle of propagation, do not interact with one another regardless the magnitude of
the wavevector. This is shown explicitly by considering two plane waves with the
same frequency ω,
u = u0,1 ei(k1 ·x−ωt) + u0,2 ei(k2 ·x−ωt) .
(1.43)
The advective acceleration—for the arbitrarily chosen variable w—is
(u · ∇)w = (u1 + u2 )(ik1 w1 + ik2 w2 ) + (v1 + v2 )(il1 w1 + il2 w2 )
+ (w1 + w2 )(im1 w1 + im2 w2 )
28
(1.44)
which can be more insightfully written as
:0
:0
(u · ∇)w = iw1
(k
(k
1 · u1 ) + iw2
2 · u2 ) + i {w1 [k1 · u2 ] + w2 [k2 · u1 ]}
(1.45)
The first two terms are zero by (1.40). Furthermore, by (1.36), the wavevectors k1
and k2 are parallel, as are the velocity vectors u1 and u2 . Then by the same token,
(1.40) requires the remaining two terms in (1.45) to vanish.
We can then superpose a number of plane waves with frequency ω as an integral
sum, which is a so-called wave beam and is also an exact nonlinear solution [46].
To do so, we first introduce the cross-beam coordinate η which is directed along the
wavevector k,
η = x sin θ + y cos θ,
(1.46)
according to the second of figure 1-2. Foreseeing more difficult calculations to follow,
it is worthwhile to introduce the streamfunction which satisfies (1.27b) by using v = 0,
u=
∂ψ
,
∂z
w=−
∂ψ
.
∂x
(1.47)
The wave beam in terms of the streamfunction is
Z
ψ(η, t) =
0
|
∞
iκη
b
Q(κ)e dκ e−iωt + c.c.,
{z
}
(1.48)
≡Q(η)
b
where κ is the magnitude of the wavevector—or simply the wavenumber, Q(κ)
is the
complex amplitude of the plane wave with wavenumber κ, and the integral sum is
b
defined as the η-dependence of the profile as Q(η). We require Q(κ)
to be complex
so that it accounts for the difference in phase associated with the plane waves. The
complex conjugate is included so that ψ is explicitly real and upcoming nonlinear
calculations will be made more feasible.
The limits of integration are such that κ > 0 which is required for uni-directional
beams [47]. To see this more clearly, consider κ → −κ as shown in figure 1-3. The
29
z
z
g
g
x
x
Figure 1-3: The effect of the sign of the magnitude of the wavevector κ when the
direction η is fixed by the dispersion relation. Flipping the sign of κ also flips the
direction of the group velocity cg and phase velocity c.
group velocity and phase velocity of a plane wave with −κ flip in direction by (1.37).
30
Chapter 2
PSI of internal gravity wave beams
Having now a working knowledge of internal gravity wave beams, it is desirable to
understand their interactions within their environment. One such interaction is the
deposition of energy from the wave motion into the surroundings. A freely propagating beam may do so if it is unstable. Here, we build on the stability analysis of
monochromatic internal waves by extending the analysis to internal beams of finite
extent as presented in Karimi and Akylas [20].
2.1
Introduction
There is an extensive literature on the stability of internal gravity waves in continuously stratified fluids with applications to various geophysical processes. Most stability analyses assume uniform stratification in the Boussinesq approximation. Under
these flow conditions, the background buoyancy frequency is constant and, moreover,
sinusoidal plane waves are not only linear solutions, but also exact nonlinear states of
the governing equations in the inviscid limit. A uniformly stratified Boussineq fluid
thus affords a convenient setting for examining the stability of periodic wavetrains of
arbitrary amplitude.
This problem has been addressed by numerous investigations (see Staquet and
Sommeria [41] for a review), and it is now well established that instability of smallamplitude internal waves is instigated by resonant nonlinear wave interactions (see
31
Phillips [40] for a review). Specifically, ignoring dissipation, weakly nonlinear sinusoidal internal waves are unstable to infinitesimal perturbations that form resonant
triads with the underlying wavetrain. In addition, the unstable perturbations singled
out by triad interactions are of short wavelength and have frequency equal to one half
of that of the primary wave. This so-called parametric subharmonic instability (PSI)
has received a great deal of attention as a potential mechanism for transferring energy from large-scale internal waves to small-scale mixing in oceans (see, for example,
Hibiya et al. [15], Koudella and Staquet [23], MacKinnon et al. [29]).
However, as argued by Sutherland [43], the PSI found in stability analyses of spatially and temporally monochromatic internal wavetrains may not be entirely relevant
to ocean internal waves. In fact, an inviscid, uniformly stratified Boussinesq fluid supports time-harmonic plane waves with general spatial profile which propagate along
a direction to the vertical determined by the wave frequency. These disturbances,
often referred to as wave beams, are fundamental to internal wave motion and, like
sinusoidal wavetrains, happen to be exact nonlinear states of the governing equations
[30, 46].
In oceans, wave beams with locally confined profile arise from the interaction
of the barotropic tide with sea-floor topography, as demonstrated by theoretical and
numerical models [1, 22, 25], laboratory experiments [14, 39, 50] and field observations
[5, 18, 26]. In contrast to sinusoidal wavetrains which are generally prone to PSI,
however, no evidence of PSI in wave beams is reported in these studies. Moreover,
the same is true for internal wave beams generated in several laboratory experiments
by oscillating a body in a stratified fluid tank (see, for example, Mowbray and Rarity
[36], Sutherland and Linden [44], Sutherland et al. [45]).
Yet, according to other recent studies, PSI can occur in internal wave beams under
certain circumstances. Similar to earlier experiments, Clark and Sutherland [4] used
a vertically oscillating circular cylinder as wave source in a stratified fluid tank. However, the cylinder oscillations were of relatively large amplitude, resulting in beams
with quasi-monochromatic profile which typically broke down as they propagated
away from the forcing region. Clark and Sutherland [4] indirectly linked this break32
down to PSI, a hypothesis also supported by numerical simulations. Furthermore,
PSI was noted in an experimental–numerical study of a model internal tide [38], as
well as in numerical simulations of the reflection of a localized nearly monochromatic
wave beam from a horizontal surface [51]. Finally, recent experiments [2] have revealed that resonant triad interactions can bring about instability in a localized wave
beam that comprises just three wavelengths of a sinusoidal wavetrain. However, the
observed most unstable perturbations were not short-scale subharmonic disturbances
because the dimensions in the experimental setup, being orders of magnitude smaller
than the typical ocean scales, amplified the effects of viscosity.
The current chapter seeks to understand theoretically the conditions under which
internal wave beams with locally confined profile may suffer PSI in a nearly inviscid,
uniformly stratified Boussinesq fluid. In keeping with the salient features of PSI in
this setting, the analysis focuses on subharmonic disturbances of short wavelength
compared to the beam width, a picture also suggested by the numerical findings of
Clark and Sutherland [4]. Such fine-scale wavepackets are modulated by and also
interact nonlinearly with the underlying large-scale wave beam. To examine the
possibility of PSI as a result of this long–short wave interaction, coupled evolution
equations are derived for the wavepacket envelopes and the beam profile, taking the
beam and the perturbations to have small but finite amplitude.
The analysis brings out the fact that subharmonic wavepackets travel with their
respective group velocities, so their interaction with a locally confined beam has
finite duration; thus PSI hinges upon whether, during this limited time, such perturbations can extract enough energy from the beam to overcome viscous dissipation.
The decisive role, in regard to PSI, of the group velocity of short-scale subharmonic
wavepackets riding on a large-scale internal wave, was first suggested by McEwan and
Plumb [31].
Based on the evolution equations derived here, it is argued that weakly nonlinear beams with general locally confined profile are stable to short-scale subharmonic perturbations, in stark contrast to the well-established PSI of weakly nonlinear
monochromatic plane waves. The reason for this difference is that triad interactions,
33
which are responsible for PSI, are not strong enough to cause instability during the
limited time that the pertubations overlap with a beam of localized profile. An exception arises when the group velocity of subharmonic wavepackets happens to vanish
or nearly so, a condition that can be satisfied when Coriolis effects are taken into
account [12, 49]. PSI of localized beams under this resonance is discussed in the next
chapter.
On the other hand, triad interactions are capable of destabilizing quasi-monochromatic wave beams whose profile consists of a sinusoidal carrier wave modulated
by a locally confined envelope. In this instance, the asymptotic theory reveals that
PSI does occur if a beam is wide enough, and an explicit stability criterion is proposed in terms of the number of carrier wavelengths required for instability to arise.
Although strictly valid for weakly nonlinear slowly modulated beams, the theoretical predictions seem consistent with the experiments and numerical simulations of
Clark and Sutherland [4], which involved finite-amplitude beams with just two carrier
wavelengths.
2.2
Long–short wave interaction
Our analysis assumes two-dimensional disturbances in an incompressible, continuously stratified Boussinesq fluid with constant buoyancy frequency N0 . We shall
work with dimensionless variables, employing 1/N0 as timescale and a characteristic
length L∗ , to be specified later, as lengthscale. With x being the horizontal and y
the vertical coordinate pointing upwards, the steamfunction ψ(x, y, t) for the velocity
field (ψy , −ψx ), and the reduced density ρ(x, y, t) are then governed by
ρt + ψx + J(ρ, ψ) = 0,
(2.1)
∇2 ψt − ρx + J(∇2 ψ, ψ) − ν∇4 ψ = 0,
(2.2)
34
where J(a, b) = ax by − ay bx stands for the Jacobian. The parameter
ν=
ν∗
N0 L2∗
(2.3)
is an inverse Reynolds number, where ν∗ denotes the fluid kinematic viscosity.
In the inviscid limit (ν = 0), equations (2.1) and (2.2) support time-harmonic
plane waves with general spatial profile. These so-called wave beams are manifestations of the anisotropy of internal gravity wave motion: according to the familiar
dispersion relation
ω = sin θ,
(2.4)
the frequency ω of a plane wave with sinusoidal profile depends on the inclination
θ to the vertical, but not the magnitude, of the wavevector. Thus, by superposing
sinusoidal plane waves with wavevectors of different magnitude but pointing in the
same direction, it is possible to construct linear time-harmonic disturbances in the
form of beams. Remarkably, this class of disturbances happen to be also nonlinear
solutions of (2.1) and (2.2) for ν = 0, irrespective of the beam profile [30, 46]. The
dispersion relation (2.4) then links the frequency 0 < ω < 1 of a beam to its direction
θ relative to the horizontal (figure 2-1).
The question of interest here is how wave beams with general locally confined
profile differ from sinusoidal plane waves in regard to PSI. We shall address this
issue via an asymptotic theory for weakly nonlinear beams under nearly inviscid flow
conditions. Specifically, the nondimensional beam amplitude is supposed to be
small:
=
ψ∗
1,
N0 L2∗
(2.5)
where ψ∗ denotes the (dimensional) peak amplitude of the streamfunction and the
lengthscale L∗ is the characteristic width of the beam (figure 2-1). Also, viscous effects
are assumed to be weak relative to nonlinear effects (ν/ 1; see (2.15) below), as
is the case for spatial scales typical of ocean wave beams [2].
Our discussion of PSI focuses on subharmonic perturbations in the form of fine-
35
Figure 2-1: Geometry of long–short wave interaction. The underlying wave beam with
general locally confined profile of characteristic width L∗ has frequency ω and propagates at an angle θ to the horizontal such that ω = sin θ. Subharmonic perturbations
are short-crested (λ∗ /L∗ 1) nearly monochromatic wavepackets with frequency
close to ω/2 that propagate at an angle φ to the horizontal, with sin φ = 12 sin θ.
scale, nearly monochromatic wavepackets with frequency close to one half of the
frequency ω = sin θ of the underlying beam. As discussed in §2.1, this choice is motivated by earlier work on PSI of weakly nonlinear sinusoidal plane waves under nearly
inviscid flow conditions [23, 31], as well as laboratory experiments and numerical
simulations of PSI of quasi-monochromatic wave beams [4]. The dispersion relation
(2.4), then, requires the wavepacket carrier wavevector k to be inclined to the vertical
by φ, such that ω/2 = sin φ, and we write
1
k± = ± êζ .
µ
(2.6)
Here êζ is a unit vector along ζ = x sin φ + y cos φ and µ is a small parameter, to
express the fact that the perturbations are short-crested relative to the beam width:
µ=
λ∗
1,
2πL∗
36
(2.7)
where λ∗ denotes the (dimensional) carrier wavelength of the subharmonic wavepackets (figure 2-1).
Utilizing the presence of these two disparate lengthscales, we shall examine by
asymptotic methods the possibility of the assumed perturbations extracting energy
from the underlying wave beam, leading to instability. This long–short wave interaction is expected to take place on a timescale of O(1/µ), since, according to (2.4), the
group velocities of wavepackets with carrier wavevectors (2.6) are O(µ):
cg ± = ±µ cos2 φ, − sin φ cos φ .
(2.8)
Thus, to study the evolution of the subharmonic perturbations due to their interaction
with the wave beam, we define the ‘slow’ time
T = µt,
(2.9)
and introduce the following expansions for ψ and ρ:
ψ = Q(η, T )e−iωt + c.c. + µδ A(η, T )eiζ/µ + B(η, T )e−iζ/µ e−iωt/2 + c.c. + . . . ,
(2.10a)
ρ = R(η, T )e−iωt + c.c. + δ F (η, T )eiζ/µ + G(η, T )e−iζ/µ e−iωt/2 + c.c. + . . . .
(2.10b)
The first curly bracket in expansions (2.10) represents the underlying wave beam
with amplitude parameter ; the second curly bracket represents the superposed subharmonic wavepackets with amplitude parameter δ 1 and carrier wavevectors given
by (2.6). The beam profile amplitudes Q and R vary in the across-beam direction
η = x sin θ + y cos θ, which is also the spatial modulation variable of the wavepacket
envelopes A, B, F and G. In stability studies based on the so-called ‘pump wave’ approximation, the perturbation amplitude parameter δ is assumed to be infinitesimal
(δ ), and the beam profile is frozen in time. As unstable perturbations grow at the
expense of the underlying beam, however, eventually some feedback is anticipated, so
37
Q and R are allowed to evolve with T in (2.10). The magnitude of δ relative to for
such full coupling to take place is determined below (see (2.16)); and δ, as well as ν
and µ introduced earlier, are treated as independent small parameters at this stage.
Upon substituting expansions (2.10) into the governing equations (2.1) and (2.2),
we collect terms proportional to exp(−iωt) and exp(±iζ/µ) exp(−iωt/2). This results
in six coupled equations for the beam amplitudes Q and R and the subharmonic
wavepacket envelopes A, B, F and G. After consistent elimination of R, F and G,
the following system of equations for Q, A and B is obtained:
µQT −
i 2
δ2
ν
µ QT T + 2 sin χ cos2 21 χAB − Qηη = O(µ3 , µδ 2 /),
2ω
2
(2.11)
i 2 0
1 ν
2 sin2 χ
µ (AT + cAη ) − µ c Aηη +
A−i 2
|Qη |2 A
2
2
2µ
µ ω
3
2
1
∗
∗
∗
+ sin χ
Qηη B + Qη BT + QηT B = O(µ, µ3 , δ 2 , 2 /µ), (2.12a)
2
ω
ω
2
1 ν
sin2 χ
i
B
−
i
|Qη |2 B
µ (BT − cBη ) − µ2 c0 Bηη +
2
2
2
2µ
µ ω
3
2
1
∗
∗
∗
+ sin χ
Qηη A − Qη AT − QηT A = O(µ, µ3 , δ 2 , 2 /µ), (2.12b)
2
ω
ω
where
c=
χ = θ − φ and
∗
ω
2 − cos χ ,
2
c0 =
ω
3 cos2 χ − 4 cos χ − 1 ,
2
(2.13)
denotes complex conjugate. Details of the derivation of this system
are given in Appendix A.
Focusing on equations (2.12) and recalling that T = µt, the leading-order terms
indicate that the envelopes A and B of the subharmonic wavepackets travel across the
beam with speed ±µc = cg ± · êη , the projection of the respective group velocity (2.8)
on the modulation direction η, where êη is a unit vector along η. The higher-order
terms in (2.12) account for the O(µ2 ) effects of dispersion, the O(ν/µ2 ) effects of
viscous dissipation and the nonlinear effects due to the coupling with the underlying
beam. The latter comprise O(2 /µ2 ) cubic terms, which can only affect the phases of
38
the complex envelopes A and B and may be interpreted as nonlinear refraction terms,
as well as O() quadratic interaction terms which may give rise to energy exchange
with the beam.
Based on equations (2.12) and (2.13), we now determine the proper balance between the small parameters , µ, ν and δ so that nonlinear, dispersive and viscous
effects partake equally in the coupled evolution of the subharmonic perturbations
with the underlying beam. From (2.12), the O(µ2 ) dispersive terms are as important as the O() quadratic interaction and the O(2 /µ2 ) nonlinear refraction terms,
if µ ∼ 1/2 . Thus, we put
µ=
1/2
,
κ
(2.14)
where κ = O(1) is a normalized carrier wavenumber of the subharmonic wavepackets.
In view of (2.14), the scaling
ν = 2α2 ,
(2.15)
where α = O(1), then brings the effects of viscous dissipation to the same level
as those of dispersion and nonlinearity. Finally, returning to (2.11), for the beam
amplitude Q to evolve on the same timescale as the wavepacket envelopes A and B,
we set
δ = .
(2.16)
Hence, full nonlinear coupling occurs when the subharmonic perturbations reach an
amplitude comparable to that of the underlying beam. Also, from (2.14)–(2.16), it
is now clear that the O(ν) viscous term in (2.11) is relatively small in comparison to
the quadratic interaction term; naturally, viscous dissipation predominantly affects
the perturbations, as they are of fine scale relative to the beam.
Upon implementing (2.14)–(2.16), equations (2.11) and (2.12), correct to O(1/2 ),
become
QT 0 + 21/2 sin χ cos2
1
χ
2
AB = 0,
c
i
c0
sin2 χ
AT 0 + Aη − 1/2 2 Aηη + 1/2 ακ2 A − i1/2 κ2
|Qη |2 A
κ
2
κ
ω
39
(2.17)
3
c
∗
∗
+ sin χ
Qηη B + 2 Qη Bη = 0,
2
ω
2
0
i
c
c
sin χ
|Qη |2 B
BT 0 − Bη − 1/2 2 Bηη + 1/2 ακ2 B − i1/2 κ2
κ
2
κ
ω
3
c
1/2
∗
∗
+ sin χ
Qηη A + 2 Qη Aη = 0,
2
ω
1/2
(2.18a)
(2.18b)
where
T 0 = 1/2 t = κT.
(2.19)
According to the evolution equations (2.18), subharmonic perturbation wavepackets are expected to travel across a beam of O(1) width virtually intact. As noted
earlier, of the dispersive, nonlinear and viscous effects in (2.18), only the quadratic
interaction terms are potentially destabilizing. These terms, however, being O(1/2 ),
are small relative to the propagation terms associated with the wavepacket group velocities, and cannot bring about instability in the limited time that the perturbations
are in contact with the underlying beam.
More specifically, in the pump-wave approximation where the beam profile Q(η)
does not evolve in time, the nonlinear refraction terms can be removed from (2.18)
by letting
Z
3
1/2 κ
2
(A, B ) → (A, B ) exp i
sin χ
cω
∗
∗
η
2
|Qη | dη .
(2.20)
As they modify only the phases of the wavepacket envelopes A and B, these terms have
no impact on stability. Focusing now on the O(1/2 ) quadratic interaction terms in
(2.18), to stand a chance of causing instability, they must be comparable to the O(c/κ)
propagation terms which control the duration of the interaction of the perturbations
with the beam:
c
= O(1/2 ).
κ
(2.21)
The above requirement could conceivably be satisfied by short-wavelength perturbations with κ = O(−1/2 ); in this limit, however, the phases of the wavepacket
envelopes in (2.20) become O(1/) so they vary on the same scale as the carrier
exp (±iζ/µ) in view of (2.10) and (2.14), violating the premises of the asymptotic
theory. A feasible way to meet (2.21) is by taking c = O(1/2 ), which supposes that
40
the wavepacket group velocities (2.8) nearly vanish; the perturbations then remain
almost stationary and could extract significant energy from the underlying beam to
cause instability. This resonant flow situation, although not possible here as is clear
from (2.13), can arise when Coriolis effects are taken into account and is responsible,
due to the Earth’s rotation, for the instability of internal-tide beams to near-inertial
subharmonic disturbances [12, 49]. Detailed analysis of PSI under such resonant
conditions is presented in the next chapter.
Our conclusion that small-amplitude beams with general locally confined profile
are stable to short-scale subharmonic perturbations may come as a surprise in view
of the well established PSI of weakly nonlinear sinusoidal plane waves. As noted in
§2.1, PSI of a monochromatic wave arises due to subharmonic disturbances that form
resonant triads with the underlying wavetrain. For a localized beam with general
profile of O(1) width, however, this triad mechanism, while still present by virtue of
the quadratic interaction terms in (2.18), cannot cause instability, as perturbations
travel with their respective group velocities and triad interactions have relatively little
time to act. To further clarify this essential difference between sinusoidal waves and
localized beams, we now turn to a discussion of PSI for beams with profile in the form
of a monochromatic carrier with O(1) wavelength, modulated by a locally confined
envelope.
2.3
Nearly monochromatic beam profile
Consider a uniform wave beam of frequency ω = sin θ with nearly monochromatic
profile, involving a carrier modulated by a localized envelope (figure 2-2). Here it is
convenient to choose as the characteristic lengthscale L∗ = Λ∗ /2π, where Λ∗ denotes
the (dimensional) carrier wavelength of the beam profile; thus, the beam carrier
wavevector
k0 = êη ,
(2.22)
where êη is a unit vector in the cross-beam direction, as before. Also, the (dimensional) characteristic width of the beam envelope D∗ satisfies D∗ Λ∗ (see (2.34)
41
Figure 2-2: Schematic of interaction of nearly monochromatic wave beam of frequency
ω = sin θ and nondimensional amplitude 1 with subharmonic perturbations
of frequency close to 21 ω = sin φ. The beam profile comprises a sinusoidal carrier
modulated by a slowly varying envelope, Λ∗ /D∗ = O(1/2 ), where Λ∗ denotes the
(dimensional) carrier wavelength and D∗ the characteristic width of the envelope.
The perturbations are short-scale wavepackets with (dimensional) carrier wavelength
λ∗ , such that λ∗ /Λ∗ = O(1/2 ).
below for the precise scaling of D∗ in terms of Λ∗ ).
The perturbations again are taken to be short-scale (relative to Λ∗ ) wavepackets
with frequency close to 21 ω = sin φ (figure 2-2). Since PSI arises due to subharmonic
disturbances that form resonant triads with the basic wavetrain, recalling (2.6) and
(2.14), the wavepacket carrier wavevectors are chosen as
k± = ±
1
ê
+
k0 ;
ζ
1/2
2
κ
(2.23)
thus, k+ + k− = k0 , as required for the members of a resonant triad. Upon substituting (2.23) in the dispersion relation (2.4) and making use of cg ± · êη = ±1/2 c/κ,
with cg ± and c as given in (2.8) and (2.13), respectively, we find
1
1 1/2
ω± = ω ±
c + O();
2
2 κ
(2.24)
hence, ω+ + ω− = ω + O(). This confirms that k+ , k− and k0 form a resonant triad
42
correct to O(1/2 ) and also suggests that the appropriate ‘slow’ time for the evolution
of the subharmonic perturbation wavepackets is
τ = t.
(2.25)
Returning now to (2.17) and (2.18), we adapt these evolution equations to the
problem at hand: the interaction of a nearly monochromatic beam of frequency ω and
carrier wavevector k0 with two subharmonic wavepackets having carrier wavevectors
(2.23) and frequencies (2.24). Specifically, combining (2.23) and (2.24) with (2.19),
the appropriate expressions for the wavepacket envelopes A(η, T 0 ) and B(η, T 0 ) are
i
c 0
A = exp
η− T
a(ξ, τ ),
2
κ
i
c 0
η+ T
B = exp
b(ξ, τ ).
2
κ
(2.26)
Here, a and b are complex envelopes that evolve on the slow time τ defined in (2.25)
and depend on the ‘stretched’ across-beam coordinate
ξ = 1/2 η,
(2.27)
such that spatial and temporal modulations are equally important. In addition,
the profile amplitude Q(η, T 0 ) of the nearly monochromatic wave beam with carrier
wavevector (2.22) takes the form
Q = q(ξ, τ )eiη ,
(2.28)
where q denotes the beam envelope, which also evolves on τ and depends on ξ; this
ensures strong coupling with the two subharmonic wavepackets, as shown below.
Inserting (2.26) and (2.28) into (2.17) and (2.18), after some simplification making
use of (2.13), we find that a, b and q are governed by
c
aτ + aξ +
κ
c
bτ − bξ +
κ
i c0
κ2
2
a
+
ακ
a
−
i
sin2 χ |q|2 a − sin χ cos2 21 χ qb∗ = 0,
2
8κ
ω
0
∗
i c
κ2
2
2
2
2 1
b
+
ακ
b
−
i
sin
χ
|q|
b
−
sin
χ
cos
χ
qa = 0,
2
8 κ2
ω
43
(2.29a)
(2.29b)
qτ + 2 sin χ cos2
1
χ
2
ab = 0.
(2.30)
Finally, it is possible to remove the terms involving c0 from (2.29),
c
κ2
aτ + aξ + ακ2 a − i sin2 χ |q|2 a − sin χ cos2 21 χ qb∗ = 0,
κ
ω
c
κ2
bτ − bξ + ακ2 b − i sin2 χ |q|2 b − sin χ cos2 21 χ qa∗ = 0,
κ
ω
(2.31a)
(2.31b)
by letting
c0
a → a exp −i
ξ ,
8cκ
c0
b → b exp i
ξ ;
8cκ
(2.32)
this amounts to an O(1/2 ) shift of the carrier wavevectors k± → k± ∓ (c0 /8cκ)1/2 êη
in (2.23).
As a result of the scalings chosen above, no small parameter appears explicitly
in the evolution equations (2.30) and (2.31). In this ‘distinguished limit’, the effects
that control the interaction of the subharmonic wavepacket envelopes a and b with
the beam envelope q, are equally important. Specifically, according to (2.31), the
transport of a and b with their respective group velocities is balanced by viscous and
nonlinear effects, while at the same time q is evolving in response to its nonlinear
coupling with a and b, as described by (2.30).
The system of equations (2.30) and (2.31) forms the basis for the discussion of
PSI of wave beams in the remainder of the paper.
2.4
Stability analysis
A uniform beam corresponds to the steady-state solution
q = q(ξ),
(2.33)
with a = b = 0, of equations (2.30) and (2.31). The linear stability of this state
is examined by assuming that perturbations are small compared to the underlying
beam (|a|, |b| |q|). It then follows from (2.30) that q is frozen in time (pump-wave
44
approximation), so a, b are governed by (2.31) with q = q(ξ).
To bring out the effect of the finite extent of a beam, we let ξ → ξ/D and take
q(ξ) to have fixed O(1) width. Here, D is the scaled width of the beam envelope in
terms of the beam carrier wavelength,
D = 2π
D∗ 1/2
= 2πN 1/2 ,
Λ∗
(2.34)
and N = D∗ /Λ∗ measures the number of carrier wavelengths contained in the beam
(see figure 2-2). It is also convenient to factor out the refraction terms in (2.31),
which have no impact on stability, via a substitution analogous to (2.20)
Z ξ
Dκ3
2
2
0
(a, b ) → (a, b ) exp i
sin χ
|q| dξ .
ωc
∗
∗
(2.35)
Thus, a and b satisfy the reduced system
c
aξ + ακ2 a − γqb∗ = 0,
Dκ
c
bτ −
bξ + ακ2 b − γqa∗ = 0,
Dκ
aτ +
(2.36a)
(2.36b)
with
γ = sin χ cos2
2.4.1
1
χ
2
.
(2.37)
Sinusoidal wavetrain
From (2.36), it is easy to recover the well-known PSI of weakly nonlinear sinusoidal
wavetrains by letting D → ∞ and setting q = 1/2; the peak amplitude of the wave
streamfunction is thus normalized to , according to (2.10a) and (2.28). Normal-mode
solutions, (a, b∗ ) ∝ exp(λτ ), of equations (2.36) then satisfy
λ + ακ2
2
45
1
= γ 2,
4
(2.38)
and the disturbance growth rate is
1
λ = γ − ακ2 .
2
(2.39)
Using (2.37) and recalling that χ = θ − φ with sin θ = 2 sin φ, it can be verified, after
some trigonometry, that (2.39) in the inviscid limit (α = 0) agrees with the growth
rate of inviscid PSI quoted in equation (10) of Koudella and Staquet [23].
According to (2.39), the inviscid growth rate of PSI is independent of the disturbance wavenumber κ so there is no preferred wavelength of instability. Viscous effects
p
stabilize the relatively short waves with κ > γ/2α, and the maximum growth rate
is then obtained for κ = 0. [Strictly, the asymptotic theory, which assumes fine-scale
disturbances obeying (2.14), breaks down for κ 1; under the present weakly nonlinear nearly inviscid flow conditions, the maximum growth rate is realized for finite
but small κ, as illustrated in figure 3 of Koudella and Staquet [23] and in figure 11(a)
of Bourget et al. [2].]
The conclusion that the strongest PSI arises for perturbations with small κ, holds
only for sinusoidal wavetrains, which have infinite extent (D → ∞). As remarked
earlier, in the case of beams with locally confined profile, the duration of the interaction of perturbations with the underlying wave is controlled by the group velocity
c/κ, making instability less likely as κ is decreased; as a result, a low-wavenumber
cut-off is to be expected, in addition to the high-wavenumber cut-off imposed by viscous effects. Thus, instability, if present at all, occurs in an interval of finite κ, and
the maximum growth rate is realized at a certain κ = O(1) within this window. A
detailed discussion of this scenario follows.
2.4.2
Eigenvalue problem
The stability of beams with locally confined envelope, q(ξ) → 0 as ξ → ±∞, hinges
upon finding normal-mode solutions of (2.36)
∗
∗
(a, b ) = â(ξ), b̂ (ξ) eλτ ,
46
(2.40)
with λ = λr + iλi , that decay to zero far from the beam:
â → 0, b̂∗ → 0
(ξ → ±∞) .
(2.41)
Substituting (2.40) into (2.36), â and b̂∗ thus satisfy
âξ + λ̂â − κ̂q b̂∗ = 0,
(2.42a)
b̂∗ξ − λ̂b̂∗ + κ̂q ∗ â = 0,
(2.42b)
with
λ̂ = λ + ακ2
D
κ,
c
κ̂ =
γD
κ.
c
(2.43)
For given envelope profile q(ξ), equations (2.42) along with the boundary conditions (2.41) define an eigenvalue problem, with λ̂ = λ̂r + iλ̂i being the eigenvalue and
κ̂ a parameter that controls the perturbation wavenumber κ. Solving this eigenvalue
problem provides λ̂ = λ̂(κ̂), and the stability of the underlying beam is decided by
the disturbance growth rate λr which follows from (2.43),
λ̂r
αc2 2
λr = γ − 2 2 κ̂ ,
κ̂
D γ
(2.44)
with λr > 0 implying instability.
It is easy to verify that, for given q(ξ) and κ̂, if {λ̂; â, b̂∗ } is an eigensolution of
the problem (2.41)–(2.42), so is {−λ̂∗ ; b̂, −â∗ }. Choosing then the mode with λ̂r > 0,
the first term in (2.44), which derives from the interaction of the disturbance with
the beam, is destabilizing, whereas the second term accounts for viscous effects and is
stabilizing; the stability of an eigensolution thus depends upon which of these terms
prevails. Based on this criterion, a comprehensive stability analysis of a beam with
certain envelope q(ξ) can be carried out by tracing eigensolution branches as κ̂ is
varied.
In the following, for simplicity, the envelope profile q(ξ) will be taken to be real.
In this instance, it can readily be shown (see Appendix B) that a countable infinity
47
of real eigenvalue branches λ̂ = λ̂(n) (κ̂) bifurcate at certain critical values of κ̂:
κ̂(n)
c =
(2n + 1)π
R∞
2 −∞ q(ξ) dξ
(n = 0, 1, 2, . . .).
(2.45)
(0)
In view of (2.43) and (2.44), the lowest of the bifurcation points (2.45), κ̂c , provides
a minimum value of the perturbation wavenumber,
κmin =
c (0)
κ̂ ,
γD c
(2.46)
below which no instability is possible, even in the absence of viscous dissipation.
Clearly, this low-wavenumber cut-off for instability is a characteristic only of beams
with locally confined profile. As expected, decreasing the beam width D increases
κmin , which makes instability less likely. The presence of κmin , combined with the highwavenumber cut-off due to viscous effects in (2.44), confirms that PSI of a localized
beam, if possible at all, is limited to values of κ within a finite window which shrinks
as the beam is made narrower.
2.5
Top-hat beam envelope
As a simple example, we now work out the details of PSI for the top-hat envelope
profile
q(ξ) =


1/2 (|ξ| ≤ 1/2)

 0
(2.47)
(|ξ| > 1/2) ,
which represents a uniform sinusoidal wave of peak amplitude and finite width,
similar to the type of beams studied in the laboratory experiments of Bourget et al.
[2]. For this choice of q(ξ), it is feasible to solve the eigenvalue problem (2.41)–(2.42)
by analytical means.
Briefly, under the normalization â = exp(−λ̂ξ) for ξ > 1/2, the appropriate
48
solution of (2.42) that also satisfies conditions (2.41) takes the form
â = e−λ̂ξ , b̂∗ = 0
b̂∗ = Deλ̂ξ
â = 0,
(ξ > 1/2),
(2.48a)
(ξ < −1/2),
(2.48b)
 
 â 
 
 
 1 
 1 
= D+
eiσξ + D−
e−iσξ
B 
B 
b̂∗ 
(|ξ| < 1/2) ,
(2.49)
−
+
where
σ=
1 2
κ̂
4
− λ̂2
1/2
,
B± =
2
λ̂ ± iσ ,
κ̂
(2.50)
and D+ , D− and D are constants to be determined. Enforcing continuity of â and b̂∗
at ξ = ±1/2 then leads to the characteristic equation
λ̂ sin σ + σ cos σ = 0
(2.51)
for the eigenvalues λ̂ = λ̂(κ̂).
From (2.51), one may verify that a countable infinity of real eigenvalue branches,
λ̂ = λ̂(n) (κ̂), bifurcate at
κ̂(n)
c = (2n + 1)π
(n = 0, 1, 2, . . .),
(2.52)
in agreement with (2.45). On each of these solution branches and κ̂ slightly above
(n)
the bifurcation point κ̂c ,
λ̂(n) =
(2n + 1)π
κ̂ − κ̂c(n) + . . . ,
4
(2.53)
(n)
while, in the other extreme, κ̂ κ̂c ,
1
n2 π 2
λ̂(n) ∼ κ̂ −
+ ....
2
κ̂
(2.54)
In view of (2.43) and (2.44), the leading-order term in (2.54) recovers the growth rate
(2.39) of PSI for a uniform sinusoidal wave; as expected, for very short subharmonic
49
disturbances (κ 1), a nearly monochromatic localized beam behaves like an infinite
sinusoidal wavetrain. Moreover, the same is true when the width of the beam envelope
is increased (D 1) for κ = O(1) fixed, since κ̂ 1 in this limit as well, according
to (2.43).
Returning now to expression (2.44) for the disturbance growth rate λr , instability
(λr > 0) arises if
λ̂(κ̂) > Cκ̂3 ,
(2.55)
αc2
.
γ 3 D2
(2.56)
where
C=
Figure 2-3 shows the first three eigensolution branches λ̂ = λ̂(n) (κ̂) of the characteristic
(n)
equation (2.51), which bifurcate at κ̂c = (2n + 1)π (n = 0, 1, 2) according to (2.52),
along with the cubic in κ̂ on the right-hand side of (2.55), taking C = 1.5 × 10−3
for illustration. For this C, the cubic intersects the first two eigensolution branches
(n = 0, 1), so for a certain range of κ̂ the instability condition (2.55) can be satisfied
by either of these modes; however, the lowest mode (n = 0) always provides the
dominant instability (largest growth rate) since λ̂(0) (κ̂) > λ̂(1) (κ̂). Specifically, there
is a finite range of unstable disturbance wavenumbers, κ̂l < κ̂ < κ̂u , where κ̂l and κ̂u
denote the values of κ̂ at which the cubic Cκ̂3 intersects the lowest-eigenvalue curve
(figure 2-3). Within this finite window of instability, the wavenumber that has the
maximum growth rate (2.44) is expected to emerge from a general initial perturbation
as the preferred scale of PSI.
It is clear from figure 2-3 that, in order for the cubic Cκ̂3 to intersect the eigenvalue
curve λ̂(0) (κ̂), and hence instability to be possible, the parameter C in (2.56) must be
less than a critical value Cc ( = 0.0108),
C < Cc .
50
(2.57)
16
14
12
10
8
6
4
2
0
0
Figure 2-3: Plots (—) of the first three eigenvalue branches λ̂(n) (κ̂) of the characteristic
(n)
equation (2.51), which bifurcate at κ̂c = (2n + 1)π for n = 0, 1, 2. The intersections
of the lowest (n = 0) of these modes with the cubic Cκ̂3 ( ), shown here for C =
1.5 × 10−3 , determine the range of unstable disturbance wavenumbers κ̂l < κ̂ < κ̂u .
The dashed lines (– –) indicate the asymptotic approximations (2.53) and (2.54) of
(0)
λ̂(0) (κ̂) near and far away from the bifurcation point κ̂c , respectively.
Combining (2.56) with (2.15), this condition for instability can be written as
γ 3/2
D>
ν 1/2 c
1
2Cc
1/2
.
(2.58)
Thus, PSI of a locally confined beam is controlled by: /ν 1/2 , the strength of nonlinear
relative to viscous effects; the beam frequency which fixes the beam propagation
direction and hence c and γ according to (2.13) and (2.37); and D, which fixes the
envelope width. More specifically, for given Reynolds number 1/ν, a beam of certain
frequency and amplitude parameter becomes unstable when D exceeds the critical
value
Dc =
1
2Cc
1/2
c ν 1/2
.
γ 3/2 (2.59)
Recalling the scaling (2.34), the critical envelope width Dc translates into a minimum
number of carrier wavelengths,
1
Nc =
2π
1
2Cc
1/2
51
c ν 1/2
,
γ 3/2 3/2
(2.60)
which a weakly nonlinear nearly monochromatic beam must comprise to develop PSI.
Note that, since ν 1/2 / = O(1) according to (2.15), Nc = O(−1/2 ).
Although it was derived for the particular envelope profile (2.47), the instability
condition (2.58) holds in general for real, locally confined envelopes, as suggested
by the bifurcation analysis of the eigenvalue problem (2.41)–(2.42) presented in Appendix B; only the value of Cc depends on the specific envelope shape. Hence, the
stability criterion (2.58) as well as expressions (2.59) and (2.60) for the minimum
envelope width and number of cycles, respectively, required for instability, are also
generally valid.
2.6
Transient disturbance evolution
We now turn attention to the long-time evolution of PSI, when unstable disturbances
are no longer infinitesimal and full coupling with the underlying beam is in effect, as
described by equations (2.30)–(2.31).
As in the simulations of Clark and Sutherland [4], here the unperturbed beam is
taken to have a Gaussian envelope profile, in the normalized form
q(ξ) =
1
exp −ξ 2 ,
2
(2.61)
so that the wave streamfunction has peak amplitude . The assumed initial perturbations consist of small subharmonic disturbances that are locally confined in the beam
vicinity and whose wavenumber κ is within the window of instability predicted by
the linear stability analysis of §2.4.2.
For the envelope profile (2.61), in particular, it follows from the eigenvalue problem
(2.41)–(2.42) that the fundamental eigensolution branch λ̂(0) (κ̂), which bifurcates
√
(0)
at κ̂c = π according to (2.45), intersects with the cubic Cκ̂3 when C < Cc =
2.66 × 10−2 . (The eigenvalues were computed numerically, solving (2.41)–(2.42) by
centred finite differences on a uniform grid with ∆ξ = 0.01 and −10 < ξ < 10.)
Taking C = 5 × 10−3 , instability then arises for 1.81 < κ̂ < 9.15. In addition,
52
we choose D = 1 for the envelope width and θ = π/4 for the beam propagation
√
angle to the horizontal; this, in turn, fixes the beam frequency ω = sin θ = 1/ 2,
the subharmonic propagation angle φ = sin−1 (ω/2) = 0.3614, the group velocity
c = 0.3849 in (2.13), the parameter γ = 0.3932 in (2.37), and α = 2.05 × 10−3 in
view of (2.56). Thus, from (2.43), the disturbance wavenumber κ = 0.9788κ̂, so the
range of unstable wavenumbers is 1.77 < κ < 8.96, with the largest growth rate (2.44)
corresponding to κ = 4.29.
The evolution equations (2.30)–(2.31) were solved numerically using as initial
conditions
q = q(ξ),
a=b=
q(ξ)
100
(τ = 0),
(2.62)
with q(ξ) given by (2.61), and perturbation wavenumber within the unstable range
1.77 < κ < 8.96 determined above. The numerical method used second-order centred
finite differences on a uniform grid, with ∆ξ = 0.02 and −25 < ξ < 25, and fourthorder Runge–Kutta time stepping with ∆τ = 0.005.
As the initial perturbations in (2.62) are small relative to the uniform beam, in
the early stages of our computations the disturbance evolution is governed by the
linearized system (2.36), confirming the predictions of the linear stability analysis.
After an initial adjustment period, a and b adapt to the fundamental instability
eigenmode and grow exponentially in τ with the growth rate (2.44) corresponding to
the chosen value of κ. However, since a and b grow at the expense of q according
to the fully coupled equation system (2.30)–(2.31), this exponential growth cannot
be sustained: as the beam becomes less steep, the subharmonic wavepackets can no
longer stay locked onto it; as a result, the nonlinear wave interaction comes to an end,
and the linearly unstable disturbances eventually decay due to viscous dissipation as
they move away from the beam with their respective group velocities.
This scenario is illustrated in figure 2-4 for subharmonic perturbations with the
most unstable wavenumber, κ = 4.29. Note that the maximum combined amplitude
of these disturbances, reached at τ ≈ 64, is comparable to the beam peak amplitude
at that time, and the final beam peak amplitude, after the perturbations have died
53
0.6
0.4
0.2
0
200
20
100
0
0 −20
0.15
0.15
0.1
0.1
0.05
0.05
0
200
0
200
20
100
20
100
0
0
0
0
−20
−20
Figure 2-4: Evolution of wave beam, with initially Gaussian envelope (2.61), and subharmonic perturbations with the most unstable wavenumber, according to numerical
solution of the coupled equations (2.30)–(2.31) subject to the initial conditions (2.62).
The wave envelope magnitudes of the beam (|q|) and the perturbations (|a|, |b|) are
displayed at various times τ .
out, is roughly 40% of its initial value. Hence, the stability parameter C, which is
inversely proportional to the square of the beam amplitude according to (2.15) and
(2.56), has effectively been increased by roughly a factor of 6, so finally C ≈ 3 × 10−2 ;
as this exceeds the critical value for instability, Cc = 2.66 × 10−2 , the final beam
profile is thus stable according to (2.57).
These findings indicate that the overall effect of PSI in the weakly nonlinear regime
is transfer of energy to short-scale subharmonic perturbations, which ultimately decay
by viscous dissipation, leaving behind a stable beam. However, the rapid growth
of these perturbations and the fact that they become as strong as the underlying
beam, suggest that overturning and/or shear instability leading to breakdown may
be possible due to PSI, in the case of beams with O(1) nondimensional amplitude.
These finite-amplitude phenomena, of course, are beyond the reach of the present
weakly nonlinear theory.
54
2.7
Concluding remarks
As revealed by the preceding analysis, in order for PSI of locally confined internal
wave beams to set in, subharmonic perturbations must overlap with the underlying
beam for long enough time to allow triad interactions to act. Specifically, for a beam
with amplitude parameter 1, the time required for triad interactions to come
into play is O(1/). In the case of localized beams with general profile of O(1) width,
according to the evolution equations (2.18), this nonlinear interaction time scale is
longer than t = O(−1/2 ), the time it takes short-scale subharmonic wavepackets to
travel across the beam. As a result, no PSI is possible save for the resonant situation
where the wavepacket group velocity happens to vanish or nearly so.
On the other hand, when the beam profile is nearly monochromatic, comprising a
sinusoidal carrier with O(1) wavelength modulated by a locally confined envelope of
O(−1/2 ) width, short-scale subharmonic perturbations evolve on the same time scale,
t = O(1/), as triad interactions. Thus, the propagation of subharmonic disturbances
is in balance with nonlinear-interaction effects and weak viscous dissipation, as indicated by the evolution equations (2.29)–(2.30). In this instance, if the beam obeys
condition (2.58), PSI is possible for a finite range of disturbance wavenumbers.
According to the stability criterion (2.58), larger-amplitude and wider beams are
more prone to PSI. Specifically, given the fluid stratification and viscosity, a beam
of certain carrier wavelength and small amplitude can suffer PSI if its width exceeds
the threshold (2.59); thus, the beam profile must comprise a minimum number of
carrier wavelengths for instability to arise. In keeping with our assumption of weakly
nonlinear nearly monochromatic beams, this critical number turns out to be relatively
large, Nc = O(−1/2 ). The theoretical stability criterion (2.58) could be confirmed by
numerical simulations and perhaps laboratory experiments. In fact, in very recent
experimental and numerical work, Bourget et al. [3] have confirmed that the finite
width of a beam does play an important role in resonant-triad instability; however,
as in their earlier study [2], owing to viscous effects, the unstable perturbations were
not of the short-scale subharmonic type discussed here.
55
The conclusion reached here, that nearly monochromatic wave beams can suffer
PSI as opposed to localized beams of O(1) width which were found to be stable, seems
consistent with the experiments and simulations of Clark and Sutherland [4]. As
noted in §2.1, a novel feature of these experiments was that wave beams were induced
via a cylinder performing relatively large-amplitude vertical oscillations (amplitudeto-diameter ratio ≈ 0.33). In response to this forcing, the turbulent oscillatory flow
surrounding the cylinder acted as wave source, and the resulting quasi-monochromatic
wave beams were observed to break down due to PSI. By contrast, no PSI was detected
in relatively thin beams generated by similar means, but with a cylinder performing
smaller-amplitude oscillations (amplitude-to-diameter ratio ≈ 0.10), in which case
the beam width was set by the cylinder radius [44, 45].
Quantitative comparison of the predictions of the asymptotic analysis with Clark
and Sutherland [4], strictly, is not feasible since the experimentally observed beams, as
well as those used as initial condition in companion simulations, had finite amplitude
and involved only roughly two carrier wavelengths. Specifically, using L∗ = 1/kσ ,
where kσ ≈ 0.6 cm−1 is the experimentally observed carrier wavenumber, the wave
amplitude parameter defined in (2.5) of the beam used in these simulations, is
estimated from figure 15(a) of Clark and Sutherland [4] as = 0.79, 0.55 for the
stratified solution of NaCl and NaI, respectively. To convert the Gaussian envelope
profile used in their simulations to the normalized form (2.61), the dimensionless
√
width parameter D = 21/2 kσ σ0 , where σ0 is the (dimensional) standard deviation.
Taking σ0 as a quarter of the beam width, kσ σ0 ≈ 2.6 according to figure 15(a), which
translates into 4σ0 (kσ /2π) ≈ 1.65 carrier wavelengths contained in the beam; also,
D ≈ 3.2, 2.7 for the solution of NaCl and NaI, respectively.
For these , D and θ = π/4 for the beam propagation angle, the values of the
stability parameter in (2.56) for the two stratified solutions turn out to be C ≈
(3.2, 6.4) × 10−4 , well below the critical value Cc = 2.66 × 10−2 for the Gaussian
(2.61). Hence, the beam profile in the simulations is clearly unstable according to
the linear stability criterion (2.57). Moreover, from (2.44), the theoretical maximum
instability growth rate, N0 λr |max ≈ 0.2 s−1 , is about twice the numerical growth
56
rate estimated from figure 15(b,c) of Clark and Sutherland [4]. Also, the theoretical
most unstable wavelength over predicts by a factor of about 2 the preferred instability
wavelength found in the simulations. Given that the beam used as initial condition in
the simulations did not actually have small amplitude and slowly modulated profile,
this rough quantitative agreement with the asymptotic analysis seems reasonable.
57
Chapter 3
Near-inertial PSI of internal wave
beams
In the previous chapter, we presented an asymptotic analysis of PSI absent rotation.
The analysis was possible due to the description of general internal gravity wave
motion in terms of a bundle of monochromatic waves, each with the same frequency
and direction of propagation. This remarkable property holds with the inclusion
of Earth’s rotation, allowing a similar analysis as before. However, the dispersion
relation of internal waves in a rotating system is altered, implying that the group
velocity of internal waves change as well. It turns out this modification can play a large
a role in dictating energy transfer from an internal wave to fine-scale disturbances via
PSI under resonant configurations. We analytically explore this situation now.
3.1
Introduction
The theoretical study of the last chapter (and Karimi and Akylas [20]) and the recent
experimental work Bourget et al. [3] have shown that modulated beams, characterized
by a dominant, carrier wavenumber, are unstable beyond some critical beam width.
Although beams of general spatial profile may be unstable to three-dimensional perturbations [21], they were found to be stable to PSI absent rotation. With the inclusion of rotational effects, however, a special resonant configuration is possible in which
58
the group velocity of subharmonic perturbations vanish, prolonging energy extraction
from the underlying beam. Such a scenario occurs in the ocean at the critical latitude
of 28.8◦ N where the M2 internal tide, produced by the semi-diurnal tidal current, has
twice the local Coriolis frequency. Numerical models and field observations [28, 29]
further suggest that at near-inertial latitudes, significant energy transfer from the M2
internal tide to small-scale subharmonic wave components takes place through PSI.
This process has been investigated analytically in the case of sinusoidal waves and
vertical modes travelling through arbitrary stratification profiles to provide estimates
of disturbance growth rate [49].
In this chapter we will consider the fate of small-amplitude internal wave beams
propagating at near-inertial latitudes in a weakly viscous, uniformly stratified Boussinesq fluid. For these conditions, we derive the evolution equations describing the
interaction of the underlying wave beam with imposed fine-scale subharmonic disturbances following the asymptotic analysis of Karimi & Akylas (2014) [20], though
here we include Coriolis effects due to Earth’s rotation. As expected from previous
global studies [28], it is found that PSI is a means of energy transfer to fine-scale
motion at near-inertial latitudes. Furthermore, the analysis yields the growth rate of
perturbations based on the beam profile and proximity to criticality, allowing comparison to the growth rate of PSI reported in the numerical study of internal-tide
beam generation and propagation near the critical latitude of Gerkema et al. (2006)
[12].
3.2
Near-inertial approximation and scalings
As in the preceding chapter, the analysis here assumes disturbances to an incompressible, continuously stratified Boussinesq fluid with constant buoyancy frequency
N0 . Working with dimensionless variables, we take 1/N0 as time scale and characteristic length L∗ —being the inverse wavenumber for sinusoidal waves and the width
for beams of general spatial profile—as length scale. Assuming no variations in the
transverse (z-) direction, the in-plane flow field may be written as (u, v) = (ψy , −ψx )
59
where ψ(x, y, t) is the streamfunction. Along with the transverse velocity w(x, y, t)
and reduced density ρ(x, y, t), it is governed by
ρt + ψx + J(ρ, ψ) = 0,
(3.1)
wt + J(w, ψ) − f ψy − ν∇2 w = 0,
(3.2)
∇2 ψt − ρx + f wy + J(∇2 ψ, ψ) − ν∇2 ∇2 ψ = 0,
(3.3)
where J(a, b) = ax by − ay bx is the Jacobian. The parameters
ν=
ν∗
N0 L2∗
(3.4)
Ω
sin β
N0
(3.5)
and
f =2
are the inverse Reynolds number and non-dimensional f -plane Coriolis parameter,
respectively, where ν∗ is the kinematic viscosity, Ω the rotation of the Earth, and β
the local latitude.
In the inviscid limit (ν = 0), (3.1)–(3.3) support time-harmonic plane waves with
general spatial profile according to the sinusoidal plane-wave dispersion relation
ω 2 = f 2 + (1 − f 2 ) sin2 θ,
(3.6)
which reflects the anisotropy of internal gravity wave motion. The frequency ω of a
plane wave with sinusoidal profile depends on the inclination θ to the vertical, but
not the magnitude, of the wavevector allowing the superposition of time-harmonic
sinusoidal plane waves to construct a beam of general spatial profile.
To examine the stability of a fully developed wave beam, we consider subharmonic
perturbations with half the frequency of the underlying beam, as observed numerically
and experimentally [4, 12], so that the pair of disturbances propagate with inclination
60
Figure 3-1: Geometry of beam–wavepacket interaction. The underlying wave beam
with general locally confined profile of characteristic width L∗ has frequency ω and
propagates at an angle θ to the horizontal according to (3.6). Subharmonic perturbations are short-crested (k∗ L∗ 1) nearly monochromatic wavepackets with frequency
close to ω/2 that propagate at an angle φ to the horizontal given by the dispersion
relation (3.7).
φ to the horizontal (see figure 3-1), fixed by the dispersion relation
s
sin φ =
ω 2 /4 − f 2
.
1 − f2
(3.7)
The corresponding group velocities of perturbations are
cg± = ±2
1 − f2
sin φ cos φ (cos φêx − sin φêy ) ,
ωk∗ L∗
(3.8)
where k∗ is the (dimensional) magnitude of the modulated-subharmonic carrier wavevector lying along ζ = x sin φ + y cos φ. It is apparent from (3.7) that the group velocities
vanish when ω/2 = f , i.e. at latitude β ≈ 28.8◦ for beams generated by the M2 tidal
current [28].
In the small-amplitude limit,
=
U∗
ψ∗
=
1,
2
N0 L∗
N0 L∗
61
(3.9)
where ψ∗ denotes the (dimensional) peak streamfunction and U∗ the (dimensional)
peak velocity of the underlying beam, it was found in the preceding chapter that
nonlinear effects and dispersive effects are of equal importance when
k∗ =
κ
1/2 L∗
κ = O(1).
,
(3.10)
Furthermore, since the streamfunction of PSI wavepackets scales as 3/2 , the evolution
equations governing its interaction with the underlying beam may be studied with
expansions
i
o
3/2 nh
1/2
1/2
ψ = Q(η, T )e−iωt + c.c. +
A(η, T )eiκζ/ + B(η, T )e−iκζ/ e−iωt/2 + c.c. ,
κ
(3.11a)
nh
i
o
1/2
1/2
ρ = R(η, T )e−iωt + c.c. + F (η, T )eiκζ/ + G(η, T )e−iκζ/ e−iωt/2 + c.c. ,
(3.11b)
nh
i
o
−iωt
iκζ/1/2
−iκζ/1/2 −iωt/2
w = W (η, T )e
+ c.c. + M (η, T )e
+ N (η, T )e
e
+ c.c. .
(3.11c)
The first set of curly brackets in (3.11) represents the underlying wave beam of
amplitude with general-spatial complex profile Q, R, and W varying in the acrossbeam direction η = x sin θ + y cos θ. The second set of curly brackets represents the
pair of perturbation subharmonic wavepackets with carrier wavevector in the ±ζdirection, with complex envelopes A, B, F , G, M , and N spatially modulated in η.
The appropriate ‘slow’ time scales like the group velocity,
T ∼ |cg± |t.
(3.12)
Away from near-inertial conditions, the group velocity scales like 1/2 according to
(3.8) and (3.10) as presented in Karimi & Akylas (2014). However in view of (2.17)–
(2.18) of their paper, envelope propagation comes into equal balance with second-
62
order dispersion and nonlinear effects when
sin φ = 1/2 σ,
(3.13)
where σ = O(1) is a detuning parameter, calculated by putting (3.13) into (3.7),
s
σ=
ω 2 /4 − f 2
.
(1 − f 2 )
(3.14)
The group velocity of PSI, (3.8), scales like under these conditions so the appropriate
slow time variable is
T = t.
(3.15)
Substituting expansions (3.11) into the governing equations (3.1)–(3.3), we collect
terms proportional to exp(−iωt) and exp(±iκζ/1/2 ) exp(−iωt/2). The resulting nine
coupled equations are combined by consistent elimination of density (R, F , G) and
transverse-velocity (W , M , N ) variables in favor of the streamfunctions:
QT + 2γAB = 0,
(3.16)
σc
i c0
Aη −
Aηη + 2ακ2 A − iδκ2 A|Qη |2 + γQηη B ∗ = 0,
κ
2 κ2
i c0
σc
Bηη + 2ακ2 B − iδκ2 B|Qη |2 + γQηη A∗ = 0,
BT − Bη −
κ
2 κ2
AT +
(3.17a)
(3.17b)
where the parameters
p
c = 3(1 − f 2 ),
0
c = 3f,
3f
δ=
,
2(1 − f 2 )
γ=
3f
p
3(1 − 4f 2 )
, (3.18)
4(1 − f 2 )
depend only on the Coriolis parameter f . The parameter α is defined such that viscous
effects primarily affect the fine-scale subarmonic wavepackets and are in balance with
nonlinear and dispersive effects [20],
α=
ν
,
22
63
(3.19)
and
∗
denotes the complex conjugate. Details of the derivation of this system are
given in appendix C.
The first two terms in (3.17) represent the linear propagation of envelopes A and
B with speed ±c/κ = cg± · êη , the projection of the respective group velocity (3.8) on
the modulation direction η, where êη is the unit vector along η. The third and fourth
terms are due to the linear effects of dispersion and viscosity. Coupling between the
evolution equations occurs through the remaining nonlinear terms which allow energy
exchange between the underlying beam and subharmonic perturbations.
3.3
Stability analysis
As written in (3.16)–(3.17), the amplitudes of the three members of the triad interaction are of the same magnitude enabling the study of the evolution of the system
from general initial conditions over long time. However, an underlying beam subject
to small disturbances may be studied by making the so-called ‘pump-wave’ approximation,
|Q| |A|, |B|,
(3.20)
valid during the early stages of interaction. This approximation amounts to linearizing
the system about the underlying beam in steady-state,
Q(η, T ) = Q(η),
(3.21)
remaining static in time, automatically satisfying (3.16). The subsequent behavior
of complex envelopes A and B determine the stability of the beam: if they are able
to extract energy via nonlinear interaction with Q at a rate exceeding the speed of
linear transport, dispersion, and viscous decay, the beam is unstable.
64
3.3.1
Sinusoidal plane waves
The PSI of weakly nonlinear sinusoidal wavetrains corresponds to the profile
Q = 12 eiη ,
(3.22)
in which case the characteristic length is the inverse wavenumber of the beam, L∗ =
1/K∗ , where K∗ is the (dimensional) wavenumber. Putting (3.22) into (3.17), the resulting linear system has harmonic coefficients, susceptible to normal-mode solutions,
A(η, T ) = A0 eλT eiρη ,
B ∗ (η, T ) = B0∗ eλT ei(ρ−1)η ,
(3.23)
where λ is the complex eigenvalue with growth rate given by its real part and ρ is an
O(1) correction to the perturbation wavevectors,



ρ 
κ
ê
k± = ± 1/2 êζ +
1 − ρ η
(3.24)
with êζ the unit vector lying along ζ. If for some κ and ρ the real part of λ is a
maximum, a perturbation of selected spatial scale is expected to emerge. Inserting
(3.22) and (3.23) into (3.17) we obtain the algebraic equations
σc
c0 2 1 2
1
2
(λ + 2ακ ) + i
ρ + 2 ρ − δκ
A0 − γB0∗ = 0,
κ
2κ
4
2
0
σc
c
1
1
(λ + 2ακ2 ) + i − (ρ − 1) − 2 (ρ − 1)2 + δκ2 B0∗ − γA0 = 0.
κ
2κ
4
2
(3.25a)
(3.25b)
For A0 , B0∗ 6= 0 a characteristic equation of the form
(p1 + p2 )(p1 − p2 + p3 ) = p24
(3.26)
arises, which can be factored as
2 2
1
1
p1 + p3 − p2 + p3 = p24 ,
2
2
65
(3.27)
so
)2
0
σc
c
1
(λ + 2ακ2 )+i
−
−ρ
=
2κ 2κ2 2
2
σc
1
c0
1
1 2
1 2
2
.
γ −
ρ−
+ 2 ρ −ρ+
− δκ
4
κ
2
2κ
2
4
(
(3.28)
The real part of λ is maximum when the terms in the curly brackets on the righthand side above vanish—a condition which furnishes ρ in terms of the yet-unknown
leading-order subharmonic wavevector contribution κ,
2
ρ=
κ
c0

v
u

u
c
σc
−1 ± t1 −
− 2
κ
2κ 
0
2c0
κ2
σc
− 2κ
σc
κ
+
−
2
c0
− δκ4
4κ2
c0 2
2
2κ


.
(3.29)

Normal-mode solutions restrict ρ to real values, so the terms under the square root
above must be positive, yielding a lower bound on κ,
δ 4 σ 2 c2 2 1
κ + 02 κ ≥ .
2c0
c
4
(3.30)
Under condition (3.29), the real part of λ is simply λr = 21 γ − 2ακ2 , affirming that
the growth rate is limited by viscous damping. Further maximizing the growth rate,
λr , demands that κ reach the minimum value allowed by (3.30),
κmin
σc
=√
δc0
)1/2
δc03
1+ 4 4 −1
,
2σ c
(r
(3.31)
where the signs of the fourth-order roots from (3.30) are chosen such that κ is real
and positive (without loss, since κ → −κ amounts to A ↔ B). The corresponding
O(1) wavevector correction in (3.29) is
ρmin =
1 σc
− 0 κmin ,
2
c
66
(3.32)
and the corresponding eigenvalue from (3.28),
1
λ = γ − 2ακ2min ,
2
(3.33)
is real, signifying that the assumed subharmonic frequency lacks a correction at O(),
1
ω± = ω + O(3/2 ).
2
(3.34)
The corresponding subharmonic wavevectors
κmin
k± = ± 1/2 êζ +
1 σc
∓ 0 κmin êη + O(1/2 ),
2
c
(3.35)
satisfy the triad resonant condition k+ + k− = K, where K = êη is the wavevector
of the underlying beam, up to O(1/2 ).
Two interesting limits of consideration are the fully detuned (σ → ∞) and perfectly tuned wave (σ = 0). In the former case the wave is far from near-inertial
conditions and κmin = 0, recovering the results of Karimi & Akylas (2014, §4.1) [20]
in which there is no preferred wavelength of instability in the inviscid limit. On the
other hand, the group velocity of perturbations vanishes when the wave is perfectly
tuned and energy transport is due solely to second-order dispersion which evenly
spreads the perturbation energy. This process of energy transport leads to the selection of a preferred wavenumber, κmin = (c0 /2δ)1/4 independent of damping effects,
which may suppress the instability for a sufficiently large damping factor α, or small
by (3.19), permitting the underlying wave to survive PSI. We note that result (3.33)
is identical to the growth rate expression, equation (4.19), of Young et al. (2008) in
the inviscid limit (see appendix D).
3.3.2
Locally confined beams
The width of a locally confined beam serves as the length scale L∗ in the ensuing
stability analysis. The stability of such beams, being finite in space, Q(η) → 0 as
67
η → ±∞, are studied by investigating the normal-modes of (3.17) with Q from (3.21),
B ∗ (η, T ) = B̂ ∗ (η)eλT ,
A(η, T ) = Â(η)eλT ,
(3.36)
where λ = λr + iλi , that decay to zero far from the beam:
 → 0, B̂ ∗ → 0
(η → ±∞).
(3.37)
Substituting (3.36) into (3.17), the mode shapes are found to satisfy
p
3(1 − f 2 )
3
3κ2
Âη − i 2 Âηη − i
|Q |2 Â
f
2κ
2(1 − f 2 ) η
p
3 3(1 − 4f 2 )
+
Qηη B̂ ∗ = 0, (3.38a)
4(1 − f 2 )
p
3(1 − f 2 ) ∗
σ
3 ∗
3κ2
λ + 2ακ2
B̂ ∗ −
B̂η + i 2 B̂ηη
+i
|Q |2 B̂ ∗
f
κ
f
2κ
2(1 − f 2 ) η
p
3 3(1 − 4f 2 ) ∗
+
Qηη Â = 0. (3.38b)
4(1 − f 2 )
λ + 2ακ2
f
σ
 +
κ
Given a beam of some profile, Q(η), detuned by an amount σ from (3.14) with
a local Coriolis frequency f and viscous parameter α, system (3.38) is to be solved
numerically for the complex eigenvalue λ = λ(α, f, σ, κ) = λr +iλi which has maximum
real part for some wavenumber κ. That is, in general, we search for solutions to (3.38)
with a set of fixed parameters σ, f , α, and Q then sweep over κ. However, in ocean
applications f = 2Ω sin β/N0 . 0.1, so
1 − f 2 , 1 − 4f 2 = 1 + O(10−2 ) ≈ 1,
(3.39)
and (3.38) simplifies to
√
√ σ̂
3
3κ2
3
3
λ̂Â + 3 Âη − i 2 Âηη − i
|Qη |2 Â +
Q B̂ ∗ = 0,
κ
2κ
2
4√ ηη
√ σ̂ ∗
3κ2
3 3 ∗
3 ∗
∗
λ̂B̂ − 3 B̂η + i 2 B̂ηη + i
|Qη |2 B̂ ∗ +
Qηη Â = 0,
κ
2κ
2
4
68
(3.40a)
(3.40b)
where
λ̂ =
λ + 2ακ2
,
f
σ̂ =
σ
.
f
(3.41)
Along with boundary conditions (3.37), the reduced eigenvalue problem (3.40)
is numerically solved for a given beam profile Q(η) with normalized detuning σ̂ and
perturbation wavenumber κ being the parameters to the complex eigenvalue λ̂(σ̂, κ) =
λ̂r + iλ̂i . It follows from (3.41) that the growth rate of the perturbation,
λr = f λ̂r (σ̂, κ) − 2ακ2 ,
(3.42)
determines the stability of the beam. For a given Q(η), σ̂, and κ it can be shown
that if (3.40) admits {λ̂; Â, B̂ ∗ } as an eigensolution, so is {−λ̂; −B̂ ∗ , Â} admitted.
Choosing the mode with λ̂r > 0 in (3.42), the instability criteria (λr > 0) is directly
formulated in terms of the eigenvalues,
λ̂r (σ̂, κ) > Cκ2 ,
(3.43)
where
C=
2α
.
f
(3.44)
The left-hand side of the inequality is purely due to the interaction of subharmonic wavepackets with Q(η). To see this, we take Q = 0 in (3.40) and solve the
resulting decoupled equations while keeping in mind that the wavepacket pair  and
B̂ ∗ are bounded in space, finding the result that λ̂ is imaginary. Consequently, (3.43)
indicates the competition between energy extraction from the beam, which varies
with beam profile and proximity to the critical latitude, and viscous effects on the
fine-scale structure of disturbances.
Gaussian profile
Given a beam profile and normalized detuning, σ̂, a wavenumber sweep over κ is
performed in solving the eigenvalue problem (3.37) and (3.40). The beam is unstable
for all κ that satisfy (3.43), and moreover, the growth rate (3.42) is maximized for
69
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
Figure 3-2: Plots of the real part of the first eigenvalue branches λ̂r (κ̂, σ̂) of beam
with Gaussian profile (3.45) for σ̂ = 0 (—), 1 (– –), and 2 (– · –). An additional
eigenvalue branch is shown for σ̂ = 0 which emerges just before the first branch ends,
reaching a slightly larger peak. The intersections of these modes with the quadratic
Cκ̂2 , shown here for C = 0.05 ( ) and 0.09 ( ), determine the range of unstable
disturbance wavenumbers κ for which (3.43) is satisfied.
some selected κ. As a useful example of this procedure, we consider Gaussian profile,
Q = 12 exp −η 2 ,
(3.45)
which fixes the amplitude of the beam streamfunction as and the characteristic
√
width of the beam as L∗ = 2s, where s is the (dimensional) standard deviation.
Beams which are of real and even profile, as in (3.45), can be shown to admit the
additional eigenmodes {λ̂∗ ; −B̂(−η), Â(−η)} and {−λ̂∗ ; Â∗ (−η), B̂(−η)} from (3.40),
revealing that eigenvalues emerge in quartets. Spurious eigenvalues from finite difference solutions to (3.40) with boundary conditions (3.37) can then be filtered out
according to the quartet rule, exposing the true eigenmodes.
Results of this numerical procedure are shown in figure 3-2. For a fixed σ̂, the
real part of the eigenvalue materializes from a finite value of κ, reaching a maximum,
then dropping off to zero whereupon the eigenvalue vanishes. Shown for σ̂ = 0
(though occurring for all σ̂), a second eigenvalue branch appears just before the first
disappears with similar behavior but arriving at slightly larger maximum value than
the first branch. The pattern is repeated with increasing κ so to determine the length
scale of instability, an eigenvalue sweep over a sufficiently large κ range is required.
70
As for the case of beams in the absence of rotation [20], there exists a finite range
of unstable disturbance wavenumbers, κl < κ < κu , where κl and κu denote the
values of κ at which the quadratic Cκ2 intersects the eigenvalue curve (indicated
for C = 0.05 and 0.09 in figure 3-2). Within this finite window of instability, the
wavenumber with maximum growth rate (3.42) is expected to emerge from a general
initial perturbation as the preferred scale of PSI. In order for the quadratic Cκ2 to
intersect the eigenvalue curve λ̂r , and hence instability to be possible, the parameter
C in (3.43) must be less than a critical value Cc (σ̂) set by the first eigenvalue branch
as seen from the κ-gap between peaks of subsequent branches,
C < Cc ,
(3.46)
which depends on the particular beam profile and detuning parameter. Combining
(3.19) with (3.43), this condition of instability can be written in terms of the threshold
amplitude
r
> c =
ν
,
f Cc
(3.47)
or restoring to dimensional form with (3.4) and (3.9)
r
U∗ > U∗c = N0
ν∗
,
f0 Cc (σ̂)
(3.48)
where f0 = f N0 is the dimensional local Coriolis parameter from (3.5).
Though the effects of group velocity are apparent in (3.48) through Cc (σ̂), explicit
effects of beam width do not appear. Not entering the instability criteria, the beam
width simply acts to provides the first-order scaling over which the mode shapes are
spread under near-inertial conditions so long that (3.9) holds. It is important to
note that Cc is not a monotonic function of σ̂. For the eigenvalue branches shown in
figure 3-2, Cc (σ̂ = 0, 1, 2) = 0.088, 0.101, 0.079. Consequently when C is near, but less
than, Cc , the maximum growth rate of perturbation may not be found for a detuned
system. Such cases may occur for beams generated on experimental scales since
viscous effects may be enhanced. An extreme incidence of this possibility is shown
71
for C = 0.09 in figure 3-2 for which the beam is unstable only when detuned, an
event understood by recalling that the angle of subharmonic wavepacket propagation
is determined by σ = f σ̂ in (3.13). The projection of beam width on the subharmonic
direction of propagation, being L∗ / sin(θ − φ) from figure 3-1, increases with the
inclination of perturbation propagation φ. Accordingly, subharmonic disturbances
remain in contact with the beam over a larger distance and may traverse through the
beam for longer duration despite the increased group velocity.
The eigenvalue branches shown in figure 3-2 obtain decreasing peak λr as the
beam is detuned so that the PSI growth rate of beams, (3.42), with small criticality
parameter, C (so Cκ2 is shallow in figure 3-2), is optimal when perfectly tuned (σ̂ =
0). One would expect then, that the energy of M2 internal-tide beams propagating
northward would transfer into subharmonics waves at increasing intensity as it nears
the critical latitude and reach a maximum transfer rate at ≈ 28.8◦ in a manner similar
to that shown in Figure 1c of MacKinnon et al. (2005) [28]. In the limit of complete
detuning (σ̂ → ∞), λr ≤ 0 for all κ implying that beams of general spatial profile
are stable unless they propagate under near-inertial configurations. As the group
velocity of perturbations largely increase (with σ̂), the limited duration of interaction
between disturbance wavepacket and underlying beam is insufficient for sustainable
energy transfer to fine-scale PSI, as remarked in earlier works [20].
The preceding analysis is conveniently applied to the numerical study of the
internal-tide beam propagating under near-inertial conditions by Gerkema et al.
(2006) [12], in which the beam is generated by barotropic tidal flow over a shelf
break. The beam quickly experienced energy loss to PSI, before even reaching the
sea floor. Paragraphs [9] and [10] of their paper provide the dimensional parameters
just below the critical latitude for background frequency N0 = 2 × 10−3 rads/s, beam
frequency ω∗ = 1.405 × 10−4 rads/s, Coriolis parameter f∗ = 6.73 × 10−5 rad/s, and
turbulent viscosity ν∗vert,hor = (10−4 , 10−2 ) m2 /s. Although a strict conversion from
their turbulent viscosity to our kinematic viscosity is not possible, we take the average of their vertical and horizontal scales of turbulent viscosity for the following
comparison, ν∗ = 10−3 m2 /s. From their Figure 1b, the peak velocity is U∗ = 0.2
72
m/s and beam width L∗ = 1 km. Upon non-dimensionalizing according to Sec. 3.2,
it is found that = 0.1 and the parameters of Sec. 3.3.2 are calculated to be σ̂ = 0.95
and C = 1.49 × 10−3 . Assuming the internal-tide beam to have Gaussian profile
(3.45), (3.37) and (3.40) are solved for σ̂ = 0.95 from which it is found that the
growth rate (3.42) is maximum for κ = 12.7 with λr = 1.52 × 10−2 . The corresponding (dimensional) inverse growth rate of subharmonic perturbation streamfunctions
is (N0 λr )−1 = 3.8 days and the growth rate of their kinetic energy is half that, 1.9
days. In comparison, the reported inverse growth of disturbance energy is 2.0 days
in Gerkema et al. (2006), in reasonable compliance with our analysis. The predicted
wavelength of subharmonic wavepackets, 1/2 2πL∗ /κ = 157 m somewhat underestimates the observed instability length scale which appears to be ∼ 250 m in their
Figure 1b.
3.4
Long-time evolution
Unstable disturbances initially grow at an exponential rate as described in the preceding analysis then become finite in amplitude and fully couple with the underlying
beam, adhering to (3.16)–(3.17). Numerically solving the fully nonlinear system with
the unperturbed beam of Sec. 3.3.2,
Q(η, T = 0) = Q(η) = 21 exp −η 2 ,
(3.49)
provides insight into the nature of system evolution over a wide time domain. The
numerical method used second-order centered finite differences on a uniform grid,
with ∆η = 0.04 and −50 < η < 50, and fourth-order Runge-Kutta time stepping with
∆T = 0.002. Taking C = 0.05 and σ̂ = 1, the range of subharmonic wavenumbers for
which the beam is unstable is 1.12 < κ < 2.44, as shown in figure 3-2 and disturbances
with κ = 1.96 develop at the quickest growth rate, λr = 1.86 × 10−2 . With fixed C
and σ̂, we choose f = 0.1 which sets σ = 0.1 and α = 2.5 × 10−3 from (3.41) and
(3.43) in (3.16)–(3.18).
73
0.6
0.4
0.2
0
400
200
0
−20
−10
0.1
0.1
0
0
−0.1
400
−0.1
400
200
0
−20
−10
0
10
20
20
10
0
200
0
−20
−10
0
10
Figure 3-3: Evolution of wave beam, with initially Gaussian envelope (3.45), and
subharmonic perturbations with the most unstable wavenumber κ = 1.96, according
to numerical solution of the coupled equations (3.16)–(3.17) subject to the initial
conditions (3.50) as shown in figure 3-2. The real part of wave envelope magnitudes
of the beam (Qr ) and the perturbations (Ar , Br ) are displayed at various times T .
Numerical solutions to (3.16)–(3.17) are shown in figure 3-3 for the initial subharmonic perturbations
A (η, T = 0) = B (η, T = 0) =
Q(η)
.
100
(3.50)
Conforming to the linearized system (3.40) at the early stages of integration, initial
perturbations of any profile first self-adjust so that they assume the mode shapes
 and B̂. Exponential growth of the disturbances follows according to the selected
κ, chosen such that λr is optimized for comparison to observations, at the expense
of the underlying beam according to the fully nonlinear evolution equations (3.16)–
(3.17). The energy transfer from Q to A and B eventually halts as the underlying
beam drops and perturbations rise in amplitude since the nonlinear wave interaction
weakens. From this time on, the perturbations disperse from the interaction while
decaying due to viscous dissipation and the beam envelope remains static.
74
20
The weakened nonlinear interaction may be understood by considering (3.44) and
using (3.19) to replace α with , which shows that C is inversely proportional to the
square of beam amplitude, . Perturbations acquire their peak amplitude around
T = 220, with a combined amplitude ∼ 36% of the initial beam as shown in figure 33, whereas the beam is ∼ 72% of its initial amplitude. The stability parameter at
this time is C ≈ 0.097 for which κ = 1.96 > κu , outside the window of instability.
We note that C remains less than Cc = 0.101 for σ̂ = 1, allowing a small amount of
additional energy transfer to a slightly lower wavenumber before the beam becomes
stable to all PSI wavenumbers.
Selecting = 0.1, contour plots of the along-beam wave velocity field, computed
from η-derivatives of ψ in (3.11a), are shown in figure 3-4 for times t = 0, 400,
and 2400. In beam periods, 2π/ω = 30.0, where ω is calculated from (3.14), these
snapshots correspond to t/(2π/ω) = 0, 13.3, and 80.0. Within just ten beam periods,
disturbances self-adjust into mode shapes suitable for exponential growth, apparent
from the dispersion of the velocity field on the flanks of the underlying beam in
figure 3-4b. By eighty beam periods, the underlying beam has lost a substantial
amount of energy to the subharmonic perturbations, shown in figure 3-4c, combined
to have about 35% the initial peak velocity. Since viscosity quickly dissipates energy
from the PSI wavepackets, due to a relatively large dissipation factor here, they never
reach a magnitude near that of the underlying wave, which drops to about 65% of its
initial amplitude at the end of interaction.
Although the growth rate of PSI is quite rapid, the fact that they achieve a combined maximum velocity of half the instantaneous underlying beam before dissipating
or dispersing away from the interaction site suggests that the beam may not undergo
breakdown despite significant energy transfer, in a manner consistent with the computations of MacKinnon (2005) et al. [28] and Gerkema et al. (2006) [12] which do
not report breakdown. For a smaller viscous parameter α (or larger amplitude by
virtue of (3.19)), however, it is possible for PSI wavepackets to achieve a combined
peak velocity equal to, or higher, than the instantaneous underlying beam, implying
that breakdown due to overturning and/or shear instability may occur.
75
5
5
5
0.02
0.01
0
0
0
0
−0.01
−5
−20
0
20
−5
−20
0
20
−5
−20
0
20
Figure 3-4: Contours of the along-beam velocity component at (a) initialization, (b)
appearance of PSI in the wavefield, and (c) near the end of the interaction under the
assumed asymptotic conditions.
3.5
Concluding remarks
Consistent with the field observations of MacKinnon et al. (2013) [29], internal-tide
beams are found here to be considerably more susceptible to PSI in the vicinity
of critical latitudes, with a maximum rate of energy loss at criticality. Far from
criticality, only spatially modulated beams may be unstable to PSI since the timescale of nonlinear interaction is insufficiently small to allow for sustainable energy
transfer before perturbations exit the overlapping region with the beam [20]. However,
with the pronounced effect of rotation under near-inertial conditions, it is possible for
resonance configurations to exist in which the group velocity of PSI (nearly) vanishes,
remaining in the overlap region for a duration dictated by weak (group velocity effects
and) dispersion. The proximity to near-inertial conditions is measured by the amount
of detuning.
The effects of detuning play a significant role in the stability characteristics of
wave beams by setting the group velocity of subharmonic disturbances. Beams of
general spatial profile are typically found to be more unstable the closer they are
to perfectly tuned configurations as the interaction is prolonged, and are completely
stable to PSI when they are far from these conditions. The role of beam amplitude is
doubly critical for near-inertial beams, firstly by setting a threshold of instability as
dictated by (3.48). Unstable beams lose energy at a maximum rate to subharmonic
wavepackets of a preferred wavenumber. Secondly, an unstable beam with amplitude
76
−0.02
above the threshold, but not largely so, may not provide PSI with an adequately
large energy extraction rate to onset breakdown processes. In this way, the numerical
solutions of evolution equations (3.16)–(3.17) are consistent with the fact that the
observation of PSI is not necessarily accompanied by breakdown.
Although we may make comparisons to prior numerical results [12, 28], experimental results which include rotation are desired
77
Chapter 4
Applications and numerical
simulations of PSI in wave beams
4.1
Introduction
In the previous two chapters, we provided analytical investigations into the process of energy transfer from an internal wave beam to fine-scale disturbances. The
analysis provided a number of physical insights and quantitative predictions under
weakly nonlinear conditions. Although some applications of the analysis were provided in those chapters, here we will give a detailed application of the theory in
comparison with the recent experimental investigations of Bourget et al. [3], which
accounted for the effects of width on PSI absent rotation. A second application
will be presented in which internal tides generated by the numerical model of iTides
(http://web.mit.edu/endlab) are taken as input parameters to the theoretical calculations of this thesis. Lastly, we report ongoing work in which numerical
simulations of various beam configurations are performed to provide a data set against
which we may quantitatively compare our asymptotic analysis.
78
4.2
Application to experiments
In the experiments of Bourget et al. [3], a nearly-monochromatic wave beam was
generated at the top of a tank filled with salt-stratified water of constant buoyancy
frequency and visualized by synthetic schlieren techniques [45]. The generation mechanism consisted of a horizontal row of thin discs [14, 33] with vertical off-sets to form
a sinusoidal shape. Beam width was controlled by the number of plates included in
the experimental apparatus, beam amplitude by the off-set distance from the center of the discs, and beam propagation angle by cam shaft frequency (via dispersion
relation). Corresponding numerical simulations were run for validation and further
analysis.
Configuration III of their table 1 lists the following beam characteristics:
N0 = 0.91 rad/s,
ω = 0.74,
l0 = 75 m-1 ,
N = 3,
ψ∗
= 33,
ν∗
(4.1)
where N0 is the buoyancy frequency, ω is the non-dimensional beam frequency, l0
is the dimensional horizontal component of the beam wavevector, N is the number
of wavelengths contained in the beam, ψ∗ is the dimensional peak streamfunction,
and ν∗ = 10−6 m2 /s is the kinematic viscosity (of water). The parameters of (4.1)
are written in the notation of chapter 2 since the experiments were conducted in a
non-rotating tank. To put these parameters in the form of inputs to the theoretical
analysis of this thesis, we obtain the beam wavenumber from the dispersion relation,
k∗ =
l0
= 101.3 m-1 ,
ω
(4.2)
and calculate the peak streamfunction explicitly,
ψ∗ = 33ν∗ = 3.3 × 10−5 m-2 /s.
79
(4.3)
The wavenumber sets the lengthscale
L∗ =
1
= 0.0099 m.
k∗
(4.4)
Non-dimensional parameters, calculated according to the scalings of §2.2 and §2.3,
are
ν=
ν∗
= 0.0113,
N0 L2∗
θ = asin(ω) = 0.8331,
=
ψ∗
= 0.3725.
N0 L2∗
(4.5)
Note that the asymptotic condition 1 is not satisfied, so the calculations of this
thesis are not strictly valid. It turns out however, that all of the qualitative features of
their experimental observations are captured in our analysis, and some quantitative
comparisons are within acceptable ranges.
The first-order approximation of the PSI propagation angle, and difference from
that of the underlying beam, are
φ = asin
1
2
sin θ = 0.379,
χ = θ − φ = 0.4541,
(4.6)
and the scaled viscous parameter is
α=
ν
= 0.0407.
22
(4.7)
Coefficients to the evolution equations of PSI wavepacket envelopes in (2.31) are then
calculated as
c=
ω
(2 − cos χ) = 0.4075,
2
c0 =
γ = sin χ cos2
ω
3 cos2 χ − 4 cos χ − 1 = −0.804,
2
1
χ = 0.4164.
(4.8)
2
Given the generation technique and resulting beam shape of their experiments, it is
justified to assume a top-hat velocity profile for the underlying beam, so the scaled
80
width parameter is
D = 2πN 1/2 = 11.50.
(4.9)
The stability parameter is
C=
αc2
= 7.07 × 10−4 ,
3
2
γ D
(4.10)
whereas the criticality parameter for the top-hat profile Cc = 0.0108 > C, implying
that our analysis correctly predicts the experimental beam to be unstable. Note that
we may rewrite the criticality parameter in terms of the non-dimensional amplitude,
c =
αc2
= 0.0244.
4π 2 γ 3 N 2 Cc
(4.11)
The eigenvalue problem of §2.4.2 depends only on the beam shape, here being
the top-hat profile. The growth-rate of the most unstable PSI mode, however, also
requires knowledge of C. Maximizing the growth rate (2.44) from the solution to the
eigenvalue problem, we find
λr
= 0.357,
γ
κ̂ = 9.72,
(4.12)
or by removing the normalization factors introduced in §2.4.2,
κ=
c
κ̂ = 5.77,
γD
λr = 0.357γ = 0.1486.
(4.13)
Note that the value λr /γ is 1/2 for monochromatic waves in inviscid fluids and provides an upper bound for growth rate. The predicted PSI frequencies are
1
1 1/2
ω± = ω ±
c = 0.52, 0.22.
2
2 κ
(4.14)
Combining (2.23), (2.32), and (2.35), the wavevectors of the subharmonic wavepackets
81
are
Dκ3
1
c0 1/2
êη ∓
sin2 χ
k± = ± 1/2 êζ + êη ∓
2
8cκ
ωc
κ
Z
ξ
|q|2 dξ 0 1/2 êη ,
(4.15)
or wavenumbers
q
2
2
+ 2k±,η k±,ζ cos χ,
+ k±,ζ
|k± | = k±,η
(4.16)
where k±,η and k±,ζ are the η- and ζ- components of the subharmonic wavevectors,
respectively. For beams with top-hat envelope, the integral in (4.15) is simply 1/4.
Using κ from in (4.13), the computed wavenumbers are
|k± | = 2.60, 1.64.
(4.17)
Restoring dimensions of the wavenumber, indicated by subscript ∗,
|k±∗ | = L∗ |k± | = 263, 166 m−1
(4.18)
we can compare the predictions of this thesis to the observed PSI characteristics in
the experiments of Bourget et al. [3]. Configuration III of their table 2 report the
observed wavenumbers and frequency of subharmonic wavepackets as
|k±,∗B | = 208, 121 m−1 ,
ω±,B = 0.49, 0.26.
(4.19)
The PSI frequencies predicted in (4.14) agree quite well with those observed experimentally above. Although not as accurate, the predicted wavenumbers (4.18) compare
reasonably to the observed PSI characteristics considering that the small-amplitude
limit of our theory is not satisfied in their experiments. A complete comparison with
the experimental and numerical work of Bourget et al. [3] is given in table 4.1. Note
that it is not possible to compare with configuration I since the width is not precisely
reported. Comparison with configuration V is not useful since the observed instabilities were not of the ω+ + ω− = ω type of triad resonance; rather they relate to the
difference of frequencies (see [3]).
82
Config. Approach of [3]
II
numerical
experimental
III
IV
numerical
N
3
3
3
ω+ /N0
0.64 (0.62)
0.49 (0.52)
0.49 (0.56)
|k+∗ | ( m−1 )
ω− /N0
201 (293)
0.26 (0.23)
208 (263)
0.26 (0.22)
232 (209)
0.25 (0.18)
|k−∗ | ( m−1 )
101 (174)
121 (166)
148 (113)
Table 4.1: Experimental and numerical results of Bourget et al. [3] are summarized.
For various beam configurations, the observed PSI wavepacket characteristics are
reported, along with the predictions of our asymptotic analysis in parentheses.
4.3
Application to beams generated by iTides
Two common mechanisms of ocean internal wave generation are due to forcing by
wind flows over the sea surface and tidal flows over topography [16, 37]. Particularly
relevant to deep-ocean mixing [9] is the latter which has been studied analytically [1],
numerically [8, 17], and experimentally [34]. Internal waves may take quite complicated forms depending on the stratification and structural details of the topography.
Those generated in the deep-ocean, where the stratification is nearly uniform, are
well described by the wave beam formulation.
The use of iTides, an open source numerical model for investigating internal wave
generation, allows a connection to be drawn between the theoretical predictions of
this thesis to wave beams as they occur in the ocean. The iTides model computes
the resulting internal tide due to a barotropic flow over arbitrary topography and
stratification. It will be useful here to compute a typical beam profile from this
generation process. A simple, though illustrative, example is shown in figure 4-1
which displays the internal tide due to the M2 tidal flow over a Gaussian ridge. Here,
the background velocity of the tidal flow is 0.05 m/s with uniform stratification of
N0 = 2 × 10−3 rad/s.
The beam variation is normal to the direction of propagation, denoted by the
cross-beam coordinate η. A sample cross-sectional profile is taken at a distance midway between the generation site and reflection at the upper boundary, shown in
figure 4-2(a). Though there are some small-scale oscillations of velocity amplitude, it
is clear that the internal tide does not arise as a beam profile modulated by a carrier
wavenumber. As discussed at the end of §2.2, PSI is not expected to play a role in
83
Figure 4-1: Internal tide generation due to M2 tidal flow over a Gaussian ridge. The
shown horizontal velocity clearly indicates the presence of a discrete beam. A sample
across the beam is taken at the dashed blue line shown in figure 4-2(a).
energy exchange for such beams.
Nonetheless, we may proceed to make a strict comparison of our theoretical analysis according to the re-scaling procedure of §2.3 by exploring the wavenumber spectra
of the profile in figure 4-2(a), shown in figure 4-2(b). The dominant wavelength here
is Λ∗ = 3700 m, whereas the width of the beam is D∗ = 1750 m < Λ∗ , so only half
a wavelength fits inside the beam width, further supporting the claim that the beam
is not truly modulated by a carrier wavenumber.
Carrying on with the analysis, we find = 0.03 and C = 0.64 which is larger
than Cc = 0.0266 for Gaussian profiles. It is sufficient here to compare to Cc of
the Gaussian profile since the specific shape of beam profile may change Cc no more
than an O(1) factor. The conclusion here, that a typical oceanic wave beam is stable
to energy transfer to fine-scale disturbances via PSI, holds significant consequences
for ocean studies. A more systematic study, taking realistic input topography from
comprehensive field studies, may serve to complement our simple application here; but
considering that C ∼ 25Cc , variations in topography and stratification are expected to
lead to minor changes to our conclusion. Exceptions to this statement may be found
where internal tides of increased amplitude and width are generated. Although broad
topography may generate beams of increased width, the beam amplitude decreases;
84
0.04
6
5
0.03
4
0.02
3
2
0.01
0
1
0
500
1000
1500
2000
0
−0.05
0
0.05
Figure 4-2: Cross-beam profile and spectra of internal tide. Notice that the crossbeam profile does not appear to contain a carrier wavenumber in (a), though the
spectra reveals a small dominant wavenumber in (b). The width of the beam, however,
is about twice the dominant wavelength.
conversely, steep topography induces thinner beams of increased amplitude [10].
These competing effects suggest that it is more important to consider regions of
greater background tidal flows which amplify beam amplitude without considerably
altering the beam profile [8]. Furthermore, by putting (2.15) and (2.34) into (2.56),
the magnified effect of beam amplitude becomes clear from the inversely cubic dependence on of the criticality parameter, C ∼ 1/3 . Indeed, PSI of the internal tide has
been observed on lab scales when the beam amplitude is large as in the experiments
of Pairaud et al. [38] in which the generated beam was of amplitude ≈ 0.3 (taking
the peak velocity in their figure 2(a) as the root mean square and assuming the width
is made up of one wavelength).
An interesting extension of our analysis with iTides can be made by simulating
beams propagating near critical latitudes where the inclusion of rotation significantly
alters their stability to PSI. Internal tide generation is also modified by the presence
of rotation, though accounted for by iTides. Thus, taking realistic topography near
critical latitudes may provide a more faithful depiction of the sort of internal tides
expected to experience PSI. Putting the output beam from iTides as input to the
eigenvalue problem of §3.3, quantitative predictions, such as rate of energy transfer
and wavelength of fine-scale disturbances, may be made.
85
Figure 4-3: Contour plot of the vorticity field of a numerical simulation initialized
with small random noise over a sinusoidal wave shown after sufficient time has passed
for instabilities to develop. The dominant mode of instability appears as fine-scale
disturbances with angle of propagation more shallow than the underlying wave, implying that the frequency of perturbations are less than that of the underlying wave.
4.4
Numerical simulations
A comprehensive study by which the transfer of energy from an underlying monochromatic wave to fine-scale subharmonic wavepackets was presented in Koudella and
Staquet [23]. Having developed an appropriate numerical scheme, it was possible to
investigate the evolution of naturally occurring PSI modes for different wave amplitudes and viscosities. An example output of their scheme is shown in figure 4-3. The
energetics of this process is discussed in detail in their paper.
Here, we adopt their numerical scheme to execute the simulation of a finite-width
beam. The goal of this ongoing project is simulate a number of beam configurations
and compare its behavior with the calculations of chapters 2 and 3. Of particular
interest is to verify the range of parameters over which the asymptotic analysis is
valid, and perhaps to discover if large-amplitude instability mechanisms differ from
the behavior expected of PSI.
86
Figure 4-4: The domain of the numerical simulation is shown in the center, thickoutlined box which contains the underlying beam of interesting (dark grey). To
satisfy periodic boundary conditions in the horizontal and vertical, two additional
beams (light grey) are included in the top-right and bottom-left corner to include the
effects of the dash-outlined boxes adjacent to the domain.
4.4.1
Periodic boundary conditions
Efficiency in computations is gained by making liberal use of Fast Fourier Transform
(FFT) algorithms which require periodic boundary conditions. Though easily implemented for sinusoidal waves, care is required when simulating finite-width beams,
which do not have unbounded periodicity, as shown in the center box of figure 4-4.
Instead, we imagine a field of an infinite number of discrete beams and center in on
one of them with dimensions such that periodicity is satisfied.
The beam of interested is indicated in dark grey in figure 4-4; all other beams,
which are required for periodicity is shaded light grey. Thus adjacent cells, in dashed
outlines, are identical repetitions of the numerical domain. The size of the domain
depends on the width of the beam. Sufficient distance between the beam of interest
and the others must be prescribed to avoid the resulting PSI wavepackets from interfering with each other. For a beam of Gaussian profile, we select three standard
deviations as this buffer length.
In chapter 2, the underlying beam is written in terms of the stream function
ψ(η, t) = Qe−iωt + c.c. ,
87
(4.20)
where Q is the general spatial beam profile evolving at a slower time scale than t.
Our analysis concluded that for 1, the beam is stable to PSI, though nearlymonochromatic beams may suffer energy loss to PSI disturbances. Such a beam is
written as
Q = q (η, t) eiη ,
(4.21)
where q evolves on slow spatial and time scale as deduced from the appropriate
balancing of nonlinear interactions, dispersion, and dissipation effects (see §2.2–2.3).
As initial condition we take the beam profile to be real, which simply amounts to
setting the phase, so the input streamfunction to the Boussinesq solver is
ψ(η, 0) =2q(η, 0) cos(η)
+ {2q(η − d, 0) cos(η − d) + 2q(η + d, 0) cos(η + d)} ,
(4.22)
where the terms in curly brackets, offset by half the diagonal distance of the numerical domain in either direction, are the corner beams that satisfy periodic boundary
conditions (see figure 4-4).
4.4.2
Qualitative results
For the remainder of this study, we will take the Gaussian profile
1
q(η, 0) = exp
2
η2
2σ 2
,
(4.23)
where σ is the standard deviation of the beam non-dimensionalized by the carrier
wavenumber. The corresponding scaled width parameter, from §2.6–2.7,
D=
√
21/2 σ,
(4.24)
allows us to compare numerical observations with the theoretical calculations of chapter 2. A physically useful quantity is the width of the beam in terms of the number
of wavelengths contained within the Gaussian envelope, here defined as the ratio of
88
four standard deviations (dimensional beam width) to carrier wavelength,
Nw =
2 σ∗
2
4σ∗
=
= σ,
Λ∗
π L∗
π
(4.25)
where σ∗ is the dimensional standard deviation.
The number of grid points, n, in the numerical domain required to satisfy the periodic boundary conditions can be estimated based on the requirements for monochromatic waves. From the original code of Koudella and Staquet [23], it is known that
for results to converge nm = 512 grid points are required in the vertical and horizontal
discretization. In the case of a finite-width beam, according to (4.22), the diagonal of
the domain must be a sum of the underlying beam width, the two corner beam widths
(being half that of the full beam), and the spacing between center and corner beams
having a buffer zone of one beam width. Thus, the number of grid points required
in horizontal and vertical directions is the product of number of wavelengths in the
domain and the number of grid points required per wavelength,
Nw
n = Nw + 2
+ 2Nw nm = 4Nw nm .
2
(4.26)
Since each wavelength requires nm = 512 grid points,
n = 2048Nw =
4096
σ,
π
(4.27)
where (4.25) was used. The above is simply an estimate and the FFT algorithms of
the numerical scheme are optimal when the number of grid points are some multiple
of 2. The true resolution of computations required relies upon the smallest scale of
the system, here set by the fine-scale PSI wavepackets (if they develop).
The input to the numerical framework is the non-dimensional beam configuration
which is fully defined by beam amplitude , viscosity ν, angle of propagation θ, beam
profile, and beam width σ for the Gaussian profile. As an example, we take
= 0.1,
ν = 10−4 ,
89
θ = 45◦ ,
σ = 2.
(4.28)
From (4.24), the scaled beam width is
D = 0.895,
(4.29)
and, as in §4.2, the corresponding parameters to the analysis of chapter 2 are
α = 0.005,
c0 = −0.761,
c = 0.385,
γ = 0.393,
C = 0.0152. (4.30)
Containing just Nw = 1.27 wavelengths, calculated from (4.25), this example beam is
fairly thin. Unfortunately, the resources available to us are limited in resolution and
our predictions may only be as accurate as this limitation. Carrying on, the solution
to the eigenvalue problem of §2.4.2 returns the dominant PSI mode, which maximizes
growth rate λr /γ, as
λr
= 0.0930,
γ
κ̂ = 2.80,
(4.31)
from which
ω± = 0.373, 0.333,
k± = 11.6, 10.7.
(4.32)
Note that the integral in (4.15) for the Gaussian profile is
Z
∞
2
0
Z
∞
|q| dξ =
−∞
−∞
1 −ξ02
e
2
2
0
dξ =
r
π
32
(4.33)
Figure 4-5 is the resulting vorticity field taking (4.22), (4.23), and (4.28) as input
to the numerical scheme. Random noise of small amplitude, initialised with the beam,
clearly develop into the coherent wavepacket structure of PSI. The growth rate of PSI
disturbances is found by filtering the wave field around ω/2 and plotting the energy
of the remaining field at the center of the domain as shown in figure 4-6.
The effect of finite-width is clear from figure 4-6 by comparing the analogous
growth rate for unbounded, monochromatic waves, as shown in 4-3. Post-processing
reveals that the growth rate of the beam in figure 4-5 is
λfw = 0.0617,
90
(4.34)
6
6
0.1
5
0.05
4
0.1
5
0.05
4
6
0.2
5
4
0.1
0
3
0
2
−0.05
1
3
3
−0.05
2
−0.1
1
0
2
1
−0.1
0
0
2
4
6
0
0
2
4
6
0
0
−0.1
2
4
6
Figure 4-5: Evolution of the total vorticity field is shown as contours. (a) The
underlying beam is initialised with very small random noise. There are nearly 2
wavelengths contained in the beam of Gaussian envelope. (b) Development of PSI
is clearly visible at t = 1000 as fine-scale contours are seen to interact with the
underlying beam. (c) By t = 1200 PSI wavepackets continuously extract energy from
the beam while transporting energy with more shallow propagation angles.
whereas for the analogous monochromatic it is
λmw = 0.156.
(4.35)
Confinement of the monochromatic wave with an envelope clearly reduces the rate of
energy transfer to fine-scale disturbances. The theoretical growth rate given by (4.31),
λr = 0.0366, underestimates the observed growth rate by roughly 50%. While working through the remaining challenges in §4.4.3, we are also in the process of gaining
access to high-performance computers capable of handling the large resolution required to process wider beams containing more carrier wavelengths. It is expected
that numerical simulations conduced with higher resolution will yield different results
or that the analytical model gains predictive power with Nw . With increased computational capabilities, it will be possible to determine the source of error between the
predicted growth rate and observed growth rate.
A major result of the study presented in chapter 2 was that beams of general
spatial profile were found to be stable. Just as the beam in figure 4-5 was analysed,
we may easily study beams of general spatial profile by modifying (4.22) such that
91
−8
−9
−10
−11
−12
−13
−14
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Figure 4-6: The total (potential and kinetic) energy of the wave field at the center
of the numerical domain filtered at half the frequency of the underlying beam. Initially the energy drops off as the randomly seeded disturbance takes shape as PSI
wavepackets. Once formed, the subharmonic perturbations grow at exponential rate,
seen here between t = 1000 and t = 1200. The black line above the energy curve
shows the fitting used to determine the growth rate.
there is no carrier wavenumber,
ψ(η, 0) =2q(η, 0) + {2q(η − d, 0) + 2q(η + d, 0)} .
(4.36)
That is, the cosine term has been set to unity. Even for a beam less than two full
wavelengths of width, the long-time evolution is drastically different. Plots of the
vorticity contour are show in figure 4-7 with parameters (4.28). The only difference
between the generation of figures 4-5 and 4-7 is the cosine term between (4.22) and
(4.36). Consistent with the discussion at the end of §2.2, the absence of carrier
wavenumber of underlying beam excludes the formation of PSI wavepackets.
In the previous example, θ = 45◦ was chosen for the numerical convenience of having equal resolution in horizontal and vertical coordinates. To avoid the complications
of requiring different resolutions in different directions, which must be multiples of 2,
it is useful to introduce the coordinate transformation
y = y 0 tan θ
92
(4.37)
6
6
0.02
5
4
0.01
3
0.01
5
4
0
3
0
2
−0.01
2
−0.01
1
0
0
2
4
6
−0.02
1
−0.02
0
0
2
4
6
Figure 4-7: Evolution of the total vorticity field is shown as contours, but for a
beam of general spatial profile lacking the presence of a carrier wavenumber. (a) The
underlying beam is initialised with very small random noise. (b) Shown at t = 2100,
the beam is still completely intact and random noise is apparently unable to excite
any instability mode (including PSI). Compare this to the PSI of figure 4-5 that
appears at t = 1000.
so that the transformed vertical length is the same as the original horizontal length,
Ly
= tan θ
Lx
⇒
Ly0
= 1.
Lx
(4.38)
Figure 4-8 illustrates this transformation.
Along with the modification of the numerical domain, density and momentum
equations become
∂ρ ∂ψ
1
∂ρ ∂ψ
∂ρ ∂ψ
+
+
−
= 0,
(4.39)
∂t
∂x tan θ ∂x ∂y 0 ∂y 0 ∂x
2
∂
1
∂2
∂ψ ∂ρ
+
−
2
2
02
∂x
tan θ ∂y
∂t
∂x
( )
2
1
∂2
1
∂ 2 ∂ψ ∂ψ
∂
1
∂2
∂ψ ∂ψ
+
+
−
+
tan θ
∂x2 tan2 θ ∂y 02 ∂x ∂y 0
∂x2 tan2 θ ∂y 02 ∂y 0 ∂x
2
2
∂
1
∂2
∂
1
∂2
−ν
+
+
ψ = 0.
(4.40)
∂x2 tan2 θ ∂y 02
∂x2 tan2 θ ∂y 02
For operations taking place in Fourier space, the transformation above may be re-
93
Figure 4-8: For arbitrary beam angles of propagation, θ, the periodic domain is
rectangular (a) with vertical (Ly ) and horizontal (Lx ) widths. By simple coordinate
transformation, y = y 0 tan θ, the domain is square (b) thereby reducing numerical
complexities.
placed by the operation
ky → ky tan θ,
(4.41)
which is the case for linear terms in the numerical scheme, though non-linear terms
are computed in real space to avoid cumbersome convolution arithmetic. Verification
of the transformations above may be performed for monochromatic waves in an inviscid fluid by comparing numerical results with the analytical growth expression in
Koudella and Staquet [23] and (2.39).
4.4.3
Remaining challenges
A systematic study of distinct beam configurations, including various beam amplitudes, widths, frequencies, and shapes will be carried out awaiting the development
of tools sufficient for quantitative analysis and access to machines capable of performing high-resolution computations in reasonable time frames. Qualitative results show
that the characteristics of PSI are strongly affected by the width of the beam. The
growth rate of a finite-width beam is less than that for a similar monochromatic wave
and PSI is completely absent when the beam is of general spatial profile (i.e. there
does not exist any dominant, carrier wavenumber). To make direct comparisons to
our asymptotic calculations, such as in (4.31) and (4.32), we require precise temporal
and spatial filtering techniques.
94
Frequency analysis
PSI wavepackets propagate at half the frequency of the underlying beam upto leading
order, but the 1/2 correction of (2.24) is not always negligible. When viscous and
finite-amplitude effects are amplified, this correction becomes especially significant,
as in the experiments of Bourget et al. [3] discussed in §4.2. Thus, simply filtering
the flow field resulting from numerical simulations is insufficient to post-process PSI
data for wide parameter ranges.
Time-frequency spectrum and spectrograph methods are quite applicable to the
data sets produced in our numerics, as employed in Joubaud et al. [19] and Bourget
et al. [2]. Taking a moving (in time) Hamming window as smoothing function, they
calculated the contribution to the velocity field of all frequencies in the domain at
every spatial point. Such a Fourier-type analysis introduces a trade-off between the
size of the window and frequency resolution. For unlimited computational capabilities, the ideal case is to obtain data at very high frequencies and select a window size
such that frequency intervals of analysis are sufficiently small. Fortunately, it is easy
to control the data capture intervals in our numerical analysis.
The result of this analysis yields a time-frequency spectrum (see figure 3(a) of
Bourget et al. [2]) which illustrates the evolution of frequency modes and allows the
direct identification of triad members in PSI. Not only does this provide a second
method to measure the growth rate of PSI wavepackets, but it also pre-empts the
wavenumber analysis method of Hilbert transforms.
Wavenumber analysis
Knowing the frequency of the underlying beam, ω, and the two triad disturbances,
ω± , we may then apply the Hilbert transform method of Mercier et al. [32] to each
frequency contribution separately. The Hilbert transformation is a central step in
a method that allows the extraction of wavevector data, which includes directional
information. Firstly, a Fourier transform in time is executed followed by frequency
filtering (here we can easily choose ω, ω+ , or ω− ). After returning to real space,
95
further spatial processing allows the selection of four possible 2-dimensional beams of
a given frequency (according to St. Andrews cross of figure 1-2). As in Bourget et al.
[3], we set the underlying beam to propagate downwards to the right so we know a
priori that PSI disturbances propagate downwards to the right and in the opposite
direction, illustrated vectorially in (4.15). The wavenumber can then be determined
from the filtered flow field for each frequency with beams in set quadrants.
This transformation technique is essentially a sequence of cleverly chosen Fourier
transforms, and though it was originally applied to experimental data, it is particularly well suited for the output of our numerical simulations. The growth rate of each
member of the PSI pair, though expected to be equal (see §2.4.2), can be observed
separately and compared simply by noting the growth rate of any field variable in the
filtered data. The numerical data, however, require truncation techniques to avoid
aliasing errors during the computation. Ensuring that these truncation methods do
not interfere with the Hilbert transform procedure is yet to be completed.
Effects of rotation
Chapter 3 introduced the Earth’s rotation into the asymptotic analysis of chapter 2
and found significant differences around critical latitudes for which the group velocity
of PSI wavepackets nearly vanish. Consistent with global scale DNS simulations [28],
it was found that beams of general spatial profile are vulnerable to instability by
PSI and the nearly-monochromatic requirement discussed at the end of §2.2 may be
relaxed.
For this reason, there is a non-trivial desire to implement the Boussinesq equations
under the thin-layer approximation (3.1)–(3.3) and perform analogous computations
to (4.23), the difference here being that we do not require the cosine factors of (4.22).
The post-processing methods carry over so extending the numerical scheme to account
for Earth’s rotation is simply a matter of properly simulating (3.1)–(3.3). Introduction
of the governing equations with rotation has begun, though verification with the
analytical results of §3.3.1 remains to be done.
Lastly, we wish to perform fully three-dimensional simulations of monochromatic
96
waves and beams with rotational effects. Although it has been well established [41]
that the dominant mode of instability from which internal waves suffer, such is not
necessarily the case once Coriolis effects are considered. It remains to be shown if
the dominant instability mechanism of internal waves under these conditions is twodimensional. Allowing the evolution of waves in the spanwise dimension opens up the
possibility of investigating this question further.
97
Chapter 5
Concluding remarks
In this thesis, we analysed a mechanism by which energy from an internal wave beam
may be transferred to fine-scale disturbances with the purpose of investigating possible deep-ocean mixing processes. Drawing inspiration from the known susceptibility
of monochromatic internal gravity wavetrains to leak energy into finite-scale subharmonic waves, we considered the interaction of a finite-width internal wave beam with
perturbations consisting of fine-scale subharmonic wavepackets. Known as parametric subharmonic instability (PSI), this behavior has been observed in the evolution of
internal wave beams generated on laboratory scale.
Assuming the triad resonance conditions are satisfied, internal wave beams of general spatial profile were taken as input to our investigation in which the exploitation
of the disparity of length scales between the wave beam and perturbation wavepackets
allowed analytical analysis. In the limit of small-amplitude wave beams, it is possible
to derive evolution equations that describe the interaction of the underlying beam
with PSI disturbances. It became evident from these equations that beams of general
spatial profile may undergo PSI under two separate conditions: (1) the beam must be
of a nearly-monochromatic nature possessing a clear carrier wavenumber or (2) the
group velocity of PSI wavepackets must nearly vanish. Although these conditions are
quite distinct, they arise from the same physical phenomena: energy transfer is possible so long that PSI wavepackets overlap with the underlying beam. As a consequence
of having a different frequency from the underlying beam, disturbance wavepackets
98
do not propagate along the beam. Thus, disturbances which cannot extract energy at
a sufficiently rapid rate simply escape the beam, leaving behind an eventless scenario.
Neglecting the rotation of the Earth, possibility (2) above is not possible and
is the subject of chapter 2. The triad resonance conditions are satisfied to higher
order if the underlying beam is characterized by a dominant wavenumber, and if
sufficiently wide, PSI wavepackets interact with the beam over a prolonged duration
permitting sustainable energy extraction to the dominant instability mode. Under
these conditions, it is possible to express an instability criterion based on the beam
amplitude, frequency, shape, and width. It is important to note that the width of
the beam is a critical parameter that directly determines whether PSI is expected
to arise. Though limited to the small number of computational and experimental
work in this subject, application of the theory produces qualitative agreement with
observations. All the defining features of PSI are well captured by the asymptotic
analysis.
Though beams of general profile are found to be stable absent rotation, results of
DNS studies suggest that PSI is a relevant energy transfer mechanism near critical latitudes, which turn out to be where the group velocities of subharmonic wavepackets
vanish. Amplified disturbances of fine-scale structure may further lead to cascading mechanisms which ultimately contribute to deep-ocean mixing. Returning to
the evolution equations, a rescaling is performed in chapter 3 which accounts for
the near-inertial, resonant effects. Without requiring beams to comprise a dominant wavenumber, the asymptotic analysis reveals beams of general spatial profiles
may unstable. The difference in behavior is attributed to the nearly-stationary PSI
wavepackets which remain in the interaction region for extended durations, facilitating energy transfer. Initial comparisons with available numerical work indicate
that the analysis not only yields the qualitative features of PSI, but quantitative
descriptions also appear accurate.
An ongoing, rigorous comparison between our theoretical approach and a direct
numerical approach is the subject of chapter 4. By adapting a numerical scheme
that has proven very insightful to the details of energy transfer when an unbounded
99
Figure 5-1: World map shown with red lines at near-inertial latitudes where energy
transfer to fine-scale subharmonic wave motion is expected to arise. Internal wave
generation sites, due to steep topography, near the critical latitudes are identified by
green circles.
monochromatic wave experiences PSI, we hope to achieve a parameter range over
which the quantitative predictions of the theory are acceptable. It is expected that the
primary factor that controls quantitative agreement is the normalized peak velocity of
the wave beam. Thus far, qualitative comparisons can be made to show that indeed,
beams of general spatial profile are stable.
The implications of our study, as relevant to oceanography, lies in the behavior
of energy transfer and mixing near critical latitudes where PSI is expected to occur.
Considering the generation of internal waves by tidal flow interaction with topography,
a quick search for oceanic cliffs and ridges is performed, as shown in figure 5-1, in
near-critical regions.
An important site to recognize is the Hawaiian ridge, which served as an important
inspiration to some of the initial studies which identified PSI in critical latitudes (see
Hibiya et al. [15] and MacKinnon and Winters [28]). These global-scale numerical simulations observed internal tides generated near the Hawaiian ridge, of latitude ∼ 18◦
N propagating northward towards the critical latitude ∼ 29◦ N where they experience
energy loss to subharmonic wave motion, though do not break down. However, the
100
fine-scale motion, leading to enhanced viscosity and momentum diffusion, may release
energy into the ocean by energy dissipation and contribute to mixing by enhanced
shear motion.
Other possible sites where such energy deposition and enhanced mixing may take
place are shown in the green circles of figure 5-1. These sites include internal wave
generation by the mid-Atlantic Ridge (north and south), topography near the Canary
Islands, the NinetyEast Ridge, the Mariana Trench, then Line Islands Ridge, and the
Kermadec Trench.
101
Appendix A
Derivation of wave-interaction
equations
Here, we provide some intermediate steps in the derivation of the evolution equations (2.11) and (2.12). Interactions between the underlying beam and subharmonic
perturbations appear through nonlinear resonant terms and are best organized by
phase,
J(ρ, ψ) =µδ 2 J(F eiζ/µ , Be−iζ/µ ) + J(Ge−iζ/µ , Aeiζ/µ ) e−iωt
+ δ J(G∗ eiζ/µ , Q) + µJ(R, B ∗ eiζ/µ ) e−iωt/2
+ δ J(F ∗ e−iζ/µ , Q) + µJ(R, A∗ e−iζ/µ ) e−iωt/2 + c.c.,
J(∇2 ψ, ψ) = (µδ)2 J(Aeiζ/µ , Be−iζ/µ ) + J(Be−iζ/µ , Aeiζ/µ ) e−iωt
+ µδ J(Qηη , B ∗ eiζ/µ ) + J(B∗ eiζ/µ , Q) e−iωt/2
+ µδ J(Qηη , A∗ e−iζ/µ ) + J(A∗ e−iζ/µ , Q) e−iωt/2 + c.c.,
(A.1)
(A.2)
where
Ae
−1
i
≡ ∇ Ae
=
A + 2 cos χAη + Aηη eiζ/µ ,
2
µ
µ
−1
i
2
−iζ/µ
≡ ∇ Be
=
B − 2 cos χBη + Bηη e−iζ/µ ,
µ2
µ
iζ/µ
Be−iζ/µ
2
iζ/µ
102
(A.3)
(A.4)
and χ = θ − φ.
The evolution equation (2.11) for Q is derived by substituting expansions (2.10) in
the governing equations (2.1) and (2.2) and collecting terms proportional to exp(−iωt).
Making use of the first set of curly brackets in (A.1), it follows from (2.1) that
i
1
δ 2 sin χ
R = −iQη − µ QηT + µ2 2 QηT T +
(AG − BF )η + O µ3 , µδ 2 / .
ω
ω
ω
(A.5)
Upon substituting (A.5) in (2.2) and using (A.2), one then has
i 2
δ2
ν
µQT −
µ QT T + sin χ (2 cos χAB + BF − AG) − Qηη
2ω
2
2
= O(µ3 , µδ 2 /).
ηη
(A.6)
Next, to derive the evolution equations (2.12), we collect terms proportional to
exp (±iζ/µ − iωt/2). Specifically, making use of (A.1), it follows from (2.1) that
1
4µ2 1
2 sin χ
F =A − 2iµ
AT + Aη −
AT T + AηT −
Qη G∗
ω
ω
ω
µ ω
sin χ
sin χ
Qηη B ∗ + O(µ3 , µ, δ 2 ),
+ 4i 2 (Qη G∗ )T − 2i
ω ω
1
4µ2 1
2 sin χ
G = − B + 2iµ
BT − Bη +
BT T − BηT +
Qη F ∗
ω
ω
ω
µ ω
sin χ
sin χ
− 4i 2 (Qη F ∗ )T + 2i
Qηη A∗ + O(µ3 , µ, δ 2 ).
ω
ω
(A.7)
(A.8)
Also, making use of (A.2), it follows from (2.2) that
1
4 cos χ
2ν
2
A = F − 2iµ
AT − cos χAη + Fη + µ Aηη −
AηT − i 2 A
ω
ω
ωµ
2 sin 2χ
2 sin χ
(A.9)
−
Qη B ∗ − i
Qη Bη∗ + O(µ, µ3 ),
µ ω
ω
1
4 cos χ
2ν
2
B = −G − 2iµ
BT + cos χBη + Gη + µ Bηη +
BηT − i 2 B
ω
ω
ωµ
2 sin χ
2 sin 2χ
+
Qη A∗ + i
Qη A∗η + O(µ, µ3 ).
(A.10)
µ ω
ω
Putting the leading order balance, F = A and G = −B, from above into (A.6)
produces (2.11).
103
Using (A.7)–(A.8) to eliminate F and G from (A.9)–(A.10), we obtain
o
n
3ω
1
ν
ω
2
Aηη + (2 + cos χ) AηT + AT T + 2 A
µ AT + (2 − cos χ) Aη − iµ
2
4
ω
2µ
2 sin2 χ
3
1
+ sin χ Qηη B ∗ + (2 − cos χ) Qη Bη∗ + QηT B ∗ − i 2
|Qη |2 A
2
ω
µ ω
= O µ3 , µ, δ 2 , 2 /µ ,
(A.11)
n
o
3ω
ω
1
ν
µ BT − (2 − cos χ) Bη − iµ2
Bηη − (2 + cos χ) BηT + BT T + 2 B
2
4
ω
2µ
2
2
1
sin χ
3
|Qη |2 B
+ sin χ Qηη A∗ + (2 − cos χ) Qη A∗η − QηT A∗ − i 2
2
ω
µ ω
= O µ3 , µ, δ 2 , 2 /µ .
(A.12)
Finally, to obtain the evolution equations (2.12) for the subharmonic envelopes A
and B, we eliminate AηT , AT T , BηT and BT T in favour of Aηη and Bηη by using the
leading-order balance in (A.11) and (A.12).
104
Appendix B
Bifurcation of eigensolution
branches
Here we show that, for real beam envelope q(ξ), the stability eigenvalue problem
(2.41)–(2.42) admits a countable infinity of real eigenvalue branches, λ̂ = λ̂(n) (κ̂),
(n)
which bifurcate at certain critical values of the wavenumber parameter, κ̂ = κ̂c
(n = 0, 1, 2, . . .).
In the vicinity of each bifurcation point, where 0 < λ̂ 1, we expand
â = â0 + λ̂â1 + . . . ,
b̂∗ = b̂∗0 + λ̂b̂∗1 + . . . ,
(B.1a)
(n)
(B.1b)
with
κ̂ = κ̂c(n) + λ̂κ̂1 + . . . .
Since q(ξ) → 0 (ξ → ±∞), the far-field (outer) solution of (2.41)–(2.42) is taken
in the form
â = e−ξ̃ ,
â = 0,
b̂∗ = 0
(ξ˜ > 0),
(B.2a)
b̂∗ = Keξ̃
(ξ˜ < 0),
(B.2b)
where ξ˜ = λ̂ξ and K is a constant to be specified by matching with the near-field
(inner) solution, valid for ξ = O(1). Specifically, upon substituting (B.1) in (2.42),
105
â0 and b̂0 satisfy
â0ξ − κ̂c(n) q b̂∗0 = 0,
(B.3a)
b̂∗0ξ + κ̂(n)
c qâ0 = 0,
(B.3b)
from which it follows that â20 + b̂∗2
0 is independent of ξ. Thus, to be consistent with
the inner limit (as ξ˜ → 0) of the outer solution (B.2), we set â20 + b̂∗2
0 = 1 so K = ±1
in (B.2b), and the appropriate matching conditions for â0 and b̂∗0 are
b̂∗0 → 0
â0 → 1,
b̂∗0 → ±1
â0 → 0,
(ξ → ∞) ,
(B.4a)
(ξ → −∞) .
(B.4b)
Equations (B.3), subject to (B.4), admit a countable infinity of eigensolutions
Z ξ
(n)
0
0
= (−1) sin κ̂c
q (ξ ) dξ ,
−∞
Z ξ
∗(n)
0
0
n
(n)
q (ξ ) dξ ,
b̂0 = (−1) cos κ̂c
(n)
â0
n
(B.5a)
(B.5b)
−∞
where
κ̂(n)
c =
(2n + 1)π
R∞
2 −∞ q(ξ) dξ
(n = 0, 1, 2, . . .)
are the bifurcation points of the corresponding eigenvalue branches.
106
(B.6)
Appendix C
Derivation of near-inertial
evolution equations
Here, we provide some intermediate steps in the derivation of the evolution equations
(3.16)–(3.17). Interactions between the underlying beam and subharmonic perturbations appear through nonlinear resonant terms and are best organized by phase,
o
5/2 n iκζ/1/2
−iκζ/1/2
−iκζ/1/2
iκζ/1/2
J(ρ, ψ) =
J Fe
, Be
+ J Ge
, Ae
e−iωt
κ
1/2 2
∗ iκζ/1/2
∗ iκζ/1/2
+ J G e
J R, B e
e−iωt/2
,Q +
κ
1/2 ∗ −iκζ/1/2
2
∗ −iκζ/1/2
J R, A e
+ J F e
e−iωt/2 + c.c.,
,Q +
κ
(C.1)
J(w, ψ) =
o
5/2 n iκζ/1/2
1/2
1/2
1/2
J Me
, Be−iκζ/
+ J N e−iκζ/ , Aeiκζ/
e−iωt
κ
1/2 2
∗ iκζ/1/2
∗ iκζ/1/2
J W, B e
e−iωt/2
,Q +
+ J N e
κ
1/2 2
∗ −iκζ/1/2
∗ −iκζ/1/2
+ J M e
,Q +
J W, A e
e−iωt/2 + c.c.
κ
(C.2)
J(∇2 ψ, ψ) =
o
3 n iκζ/1/2
−iκζ/1/2
iκζ/1/2
−iκζ/1/2
J
Ae
,
Be
+
J
Be
,
Ae
e−iωt
κ2
o
5/2 n 1/2
1/2
J Qηη , B ∗ eiκζ/
+ J B∗ eiκζ/ , Q e−iωt/2
+
κ
107
+
o
5/2 n 1/2
1/2
+ J A∗ eiκζ/ , Q e−iωt/2 + c.c.,
J Qηη , A∗ e−iκζ/
κ
(C.3)
where
Ae
iκζ/1/2
1/2
Be−iκζ/
κ2
iκ
1/2
≡ ∇ Ae
= − A + 2 1/2 cos(θ − φ)Aη + Aηη eiκζ/ ,
(C.4)
2
iκ
κ
1/2
2
−iκζ/1/2
≡ ∇ Be
= − B − 2 1/2 cos(θ − φ)Bη + Bηη e−iκζ/ . (C.5)
2
iκζ/1/2
The evolution equation (3.16) for Q is derived by substituting expansions expansions (3.11) in the governing equations (3.1)–(3.3) and collecting terms proportional
to exp(−iωt). Making use of the first set of curly brackets in (C.1) and (C.2), it
follows from (3.1) and (3.2), respectively,
sin θ
sin θ
sin(θ − φ)
R = −i
Qη + − 2 QηT +
(AG − BF )η ,
(C.6)
ω
ω
ω
f cos θ
ν
f cos θ
sin(θ − φ)
W =i
Qη − 2 f cos θQη(3) + (AN − BM )η +
QηT .
ω
ω
ω
ω2
(C.7)
Upon substituting (C.6) and (C.7) in (3.3) and using (C.3), one then has
f cos θ
sin θ
(AG − BF ) +
(AN − BM ) + cos(θ − φ)(AB)
QT + sin(θ − φ) −
2ω
2ω
ν
f 2 cos2 θ
−
1+
Qηη = 0.
ω2
(C.8)
Next, to derive the evolution equations (3.17), we collect terms proportional to
exp(±iκζ/1/2 − iωt/2). Specifically, making use of (C.1), it follows form (3.1) that
2
4κ
2 sin φ
1/2
∗
F =
A+
−i
sin θAη + 2 sin(θ − φ) sin φQη B
ω
ωκ
ω
(
4
8κ2
+ − i 2 sin θ sin(θ − φ)Qη Bη∗ − 3 sin2 (θ − φ) sin φ|Qη |2 A
ω
ω
108
)
4 sin φ
2 sin θ
∗
−i
AT − i 2 sin(θ − φ)Qηη B ,
ω2
ω
2
2 sin φ
4κ
1/2
∗
−i
G=−
B+
sin θBη + 2 sin(θ − φ) sin φQη A
ω
ωκ
ω
(
4
8κ2
+ i 2 sin θ sin(θ − φ)Qη A∗η + 3 sin2 (θ − φ) sin φ|Qη |2 B
ω
ω
)
4 sin φ
2 sin θ
BT + i 2 sin(θ − φ)Qηη A∗ .
+i
ω2
ω
(C.9)
(C.10)
Similarly, using (C.2) in (3.2) yields
2f
2ν 2
4f κ
2f
1/2
∗
cos φA − i
κ M +
cos θAη − 2 sin(θ − φ) cos φQη B
i
M =−
ω
ω
ωκ
ω
(
4f
8κ2
∗
+ i 2 sin(θ − φ) cos θQη Bη + 3 f sin2 (θ − φ) cos φ|Qη |2 A
ω
ω
)
2f
4f
(C.11)
+ i 2 cos φAT + i 2 sin(θ − φ) cos θQηη B ∗ ,
ω
ω
2ν 2
4f κ
2f
2f
1/2
∗
κ N +
cos θBη − 2 sin(θ − φ) cos φQη A
N = cos φB − i
i
ω
ω
ωκ
ω
(
2
i4f
8κ
+ − 2 sin(θ − φ) cos θQη A∗η − 3 f sin2 (θ − φ) cos φ|Qη |2 B
ω
ω
)
2f
4f
(C.12)
− i 2 cos φBT − i 2 sin(θ − φ) cos θQηη A∗ .
ω
ω
Inserting the leading order balance from the above,
F =
2 sin φ
A,
ω
G=−
2 sin φ
B,
ω
M =−
2f
cos φA,
ω
N=
2f
cos φB,
ω
(C.13)
into (C.8) produces
ν
sin(θ − φ) 2
2
QT +
2
sin
θ
sin
φ
+
2f
cos
θ
cos
φ
+
ω
cos(θ
−
φ)
−
ω2
f 2 cos2 θ
1+
Qηη = 0.
ω2
(C.14)
Applying the near-inertial approximation (3.13) to the trigonometric terms by putting
109
(3.6) into (3.14), we find
s
sin θ =
3f 2
1 − f2
s
+ O(),
cos θ =
1 − 4f 2
+ O(),
1 − f2
(C.15)
and (C.14) becomes (3.16) after applying (3.19).
Lastly, making use of (C.3), it follows from (3.3) that
(
2
2i
2i
2i
2
cos(θ − φ)Aη −
sin θFη +
f cos θMη
A = sin φF − f cos φM + 1/2
ω
ω
κ
ωκ
ωκ
)
(
)
2κ
1
4i
ν 2i 2
2i
−
sin(θ − φ)Qη B ∗ + 2 Aηη − AT + sin(θ − φ) cos(θ − φ)Qη Bη∗ −
κ A,
ω
κ
ω
ω
ω
(C.16)
(
2
2
2i
2i
2i
sin φG + f cos φN + 1/2 − cos(θ − φ)Bη −
sin θGη +
f cos θNη
ω
ω
κ
ωκ
ωκ
)
2κ
1
2i
4i
ν 2i 2
∗
∗
+
sin(θ − φ)Qη A + Bηη − BT + sin(θ − φ) cos(θ − φ)Qη Aη −
κ B.
2
ω
κ
ω
ω
ω
B =−
(C.17)
Using (C.9) and (C.11) to eliminate F and M from (C.16), then applying (3.19) and
the near-inertial approximations (3.13) and (C.15), we obtain (3.17a). Likewise, using
(C.10) and (C.12) to eliminate G and N from (C.17) yields (3.17b).
110
Appendix D
Comparison with Young et al.
(2008) of PSI growth rate for
sinusoidal plane waves
The (dimensional) growth rate of perturbation for sinusoidal plane waves (pumpwave) found by Young et al. (2008) is given by their (4.19),
λ∗YTB =
akx2
,
4f0
(D.1)
where kx is the (dimensional) x-component of the wavevector, f0 the (dimensional)
Coriolis parameter, and a the pump-wave amplitude defined in terms of peak xvelocity by
Up,x =
akx ω∗
2akx
cos φ ≈
cos φ,
2
2
ω∗ − f0
3f0
(D.2)
where φ is the phase and ω∗ the dimensional frequency of the pump-wave, and the
near-inertial approximation was employed, ω∗ ≈ 2f0 . Replacing a with Up,x , the
growth rate (D.1) is
3
λ∗YTB = kx Up,x ,
8
111
(D.3)
or in non-dimensional form,
λYTB =
3
3 kx Up,x
= sin θ cos θ.
8 |k| N0 L∗
8
(D.4)
The growth rate derived in this article, (3.33), can be rewritten in a similar form
according to (C.15),
1
3
λ = γ = sin θ cos θ.
2
8
(D.5)
Note that appears explicitly in (D.4), though not in the expression above since it is
absorbed into the slow time scale, T = t, in our analysis.
112
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