PHYSICS OF FLUIDS 18, 074107 共2006兲
Internal wave instability: Wave-wave versus wave-induced mean
flow interactions
B. R. Sutherland
Department of Mathematical and Statistical Sciences, University of Alberta,
Edmonton, Alberta T6G 2G1, Canada
共Received 4 January 2006; accepted 8 June 2006; published online 26 July 2006兲
In continuously stratified fluid, vertically propagating internal gravity waves of moderately large
amplitude can become unstable and possibly break due to a variety of mechanisms including 共with
some overlap兲 modulational instability, parametric subharmonic instability 共PSI兲, self-acceleration,
overturning, and convective instability. In PSI, energy from primary waves is transferred, for
example, to waves with half frequency. Self-acceleration refers to a mechanism whereby a wave
packet induces a mean flow 共analogous to the Stokes drift of surface waves兲 that itself advects the
waves until they become convectively unstable. The simulations presented here show that
self-acceleration dominates over parametric subharmonic instability if the wave packet has a
sufficiently small vertical extent and sufficiently fast frequency. © 2006 American Institute of
Physics. 关DOI: 10.1063/1.2219102兴
I. INTRODUCTION
In the deep ocean interior, fluctuations due to internal
waves span a broad range of spatiotemporal scales that conform to a nearly universal spectrum. This has been represented empirically in models developed primarily by Garrett
and Munk.1–3 Though the models lean heavily on linear
theory to represent the structure and dispersion of the waves,
the very fact that the observed spectrum is universal implies
that nonlinear interactions between waves crucially determine the dynamics by which energy from various sources is
transferred over such a wide and statistically stationary spectral range.
Generally, two approaches have been taken to examine
the nonlinear dynamics underpinning the internal wave spectrum. In one, employed to examine both the atmospheric and
oceanic internal wave spectra, statistical methods examine
the ensemble of waves. These methods make various approximations so that the mathematics are tractable and analytic formulas for the spectra can be reproduced or incorporated in the models.4–9
The second approach has examined interactions between
pairs and triads of waves. For example, ray tracing and direct
numerical methods have revealed that a short quasimonochromatic wave packet interacting with a long, slowly evolving internal wave can be scattered in such a way to produce
a spectrum of waves consistent with observations.10
Triad resonant interactions give an alternate route
through which energy can be transferred to different
scales.11,12 A particular interaction, known as parametric subharmonic instability, results generally when a disturbance of
one frequency imparts energy to disturbances of half that
frequency.13,14 Generally, a plane periodic internal wave is
parametrically unstable, even if its amplitude is infinitesimally small.15–18 In particular, a wave with frequency ␻ and
horizontal wavenumber kx is unstable to the generation of
subharmonic waves with frequency ␻ / 2 and horizontal
1070-6631/2006/18共7兲/074107/8/$23.00
wavenumber 2kx. Consistent with the dispersion relation of
linear theory, the ratio of the vertical wavenumbers, kz⬘ and
kz, of the growing and primary waves, respectively, is
kz⬘
=4
kz
冑
1+
3
cot2 ⌰,
4
共1兲
in which
⌰ ⬅ cos−1共␻/N兲 = cos−1共kx/兩k兩兲
共2兲
is the angle formed between constant-phase lines and the
vertical. In 共2兲, k is the wavenumber vector, ␻ is the wave
frequency, and N is the buoyancy frequency, which is the
natural frequency for vertical oscillations of the fluid. The
corresponding dispersion relationship well describes the internal waves in the atmosphere and ocean, provided their
frequency is much higher than the Coriolis frequency. These
relatively high frequency waves are the ones of interest here.
The vertical spatial scale of subharmonic waves is significantly smaller than that of the primary waves. For example, 共1兲 shows that subharmonic waves with half the frequency 共double the period兲 of the primary waves have a
vertical wavelength at least four times smaller than that of
the primary waves. It is in this way that energy cascades to
small scales. Further evidence for the viability of parametric
subharmonic instability as a mechanism for energy transfer
to small scales has been provided through numerical
simulations18,19 and laboratory experiments.20–22
An alternate mechanism for instability is through interactions between internal waves and the wave-induced mean
flow, 具u典. By contrast with parametric subharmonic instability, in which for the above example a primary wave interacts
with itself to transfer substantial energy to waves at double
the horizontal wavenumber 共kx + kx → 2kx兲, here the primary
wave interacts with itself, thus transferring substantial energy
to a disturbance with a zero horizontal wavenumber 共kx − kx
→ 0kx兲. For waves of sufficiently large amplitude, the result-
18, 074107-1
© 2006 American Institute of Physics
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074107-2
Phys. Fluids 18, 074107 共2006兲
B. R. Sutherland
ing acceleration of the mean flow can be so large that waves
evolve to become convectively unstable through a mechanism called “self-acceleration.”23–26
There are various reasons why self-acceleration has
largely been overlooked as a significant instability mechanism. First, it seems peculiar to nonhydrostatic internal
waves, which are waves with frequency close to the buoyancy frequency, N. Certainly surface waves generate a
Stokes drift, but this is small compared with the horizontal
phase and group speeds of the waves, even for those at
breaking amplitudes. On the other hand, the wave-induced
mean flow for internal waves is larger than the horizontal
group velocity, cgx, even for infinitesimally small amplitude
waves if their frequency is infinitesimally close to but
smaller than the buoyancy frequency. Generally, the condition that 具u典 exceeds cgx is given by the “self-acceleration
condition”26
1
A␰
=
sin 2⌰.
␭ 2冑2␲
共3兲
II. DESCRIPTION OF MODEL
共4兲
Numerical simulations are performed by solving the
fully nonlinear Navier Stokes equations in two dimensions, x
and z, and under the Boussinesq approximation. Details are
provided by Sutherland and Peltier.31
In terms of the density field, ␳ 共the difference of the total
density ␳T and the hydrostatically balanced background density ¯␳兲, and spanwise vorticity field, ␨ ⬅ ⳵u / ⳵z − ⳵w / ⳵x, the
equations are
By comparison, waves are overturning if
1
A␰
=
cot ⌰.
␭ 2␲
Although a single event may be difficult to detect with in situ
instruments and would have a negligible effect upon the general circulation, the local influence of nonhydrostatic wave
generation, propagation, and breaking can non-negligibly affect abyssal ocean mixing and clear-air turbulence.
The point in the work presented here is to examine circumstances under which a wave packet becomes unstable
due to self-acceleration or parametric subharmonic instability as a function of the wave packet frequency, amplitude,
shape, and vertical extent.
The code used for the simulations and the diagnostics
employed for analyses are described in Sec. II. In Sec. III we
present the results of representative simulations that demonstrate the evolution of the wave packet and its associated
wave-induced mean flow profile. Analyses of energy transfers and spectra are provided in Sec. IV and conclusions
given in Sec. V.
So breaking may occur due to self-acceleration for waves
that initially are nonoverturning if their relative frequency
satisfies cos共39.2° 兲 ⯝ 0.78ⱗ ␻ / N ⱕ 1.
Another reason why self-acceleration has been overlooked is that it is not the instability mechanism that dominates for monochromatic internal waves: self-acceleration
does not occur in laboratory experiments in which low
modes are forced in a tank because the waves are confined
laterally by the tank sidewalls and so a mean horizontal flow
cannot develop; self-acceleration does not dominate in the
simulations of plane waves in doubly and triply periodic domains because the predicted wave-induced mean flow is vertically uniform and so, with a simple change in the frame of
reference, can equivalently be taken to be zero.
However, in realistic circumstances internal waves are
not perfectly monochromatic nor are they laterally confined.
Furthermore, in a numerical study examining the evolution
of idealized vertically compact wave packets,26 the energy
transfers between the waves and the wave-induced mean
flow were significantly larger than those due to subharmonic
waves: self-acceleration was the dominant instability mechanism.
A final reason why self-acceleration may not be so well
studied is that observations of the oceanic and atmospheric
internal wave spectra show that energy accumulates at low
frequencies, which is also the regime in which internal waves
are generated by tides in the ocean and by flow over mountain ranges in the atmosphere. This has led some to conclude
that the nonhydrostatic component of the frequency spectrum
for internal waves is unimportant for geophysical flows.
However, recent numerical and experimental studies have
demonstrated that large amplitude, high frequency internal
waves can be generated by mesoscale topography27,28 and in
transient bursts by convective systems29 and by turbulence.30
⳵␨
g ⳵␳
= − u · ⵱␨ +
+ ␯ ⵜ 2␨ ,
⳵t
␳0 ⳵ x
共5兲
⳵␳
d¯␳
= − u · ⵱ ␳ − w + ␬ ⵜ 2␳ .
dz
⳵t
共6兲
Here
uជ ⬅ 共u,w兲 ⬅ 共− ⳵ ␺/⳵ z, ⳵ ␺/⳵ x兲
共7兲
represents the horizontal and vertical components of the velocity field, which are expressed in terms of derivatives of
the streamfunction, ␺. The streamfunction, in turn, is expressed in terms of the vorticity field by inverting the
Poisson equation:
␨ = − ⵜ 2␺ .
The constants in 共5兲 and 共6兲 are the acceleration of gravity, g,
a characteristic density, ␳0, viscosity, ␯, and diffusivity ␬.
The viscosity, ␯ was set to be as small as possible while
maintaining numerical stability. Typically, the corresponding
Reynolds number was over 6000, based on horizontal wavelength and buoyancy period. The diffusivity ␬ was set equal
to ␯. Though not representative of realistic molecular diffusivities of heat or salinity in the ocean, such diffusive processes were irrelevant to the dynamics being examined. Realistic values would require the code to be run at finer
resolution and smaller time steps, but giving no additional
insights into the dynamics of the instability processes of interest.
In all simulations, the background stratification is taken
to be uniform, so that N2 ⬅ −共g / ␳0兲d¯␳ / dz ⬅ N02 is constant.
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074107-3
Phys. Fluids 18, 074107 共2006兲
Internal wave stability
FIG. 1. 共a兲 Gray scale contours showing vertical displacement field associated with wave packet at the start of a typical simulation. Also shown are
vertical profiles of 共b兲 the amplitude envelope and 共c兲 the wave-induced
mean flow associated with the wave packet. Parameters are Lz = 0,
␴ = 10kx−1.
Thus, we may represent density perturbations due to waves
in terms of vertical displacements ␰ through the relation
␳0
d¯␳
␳ = − ␰ = N 02␰ .
dz
g
共8兲
On this background we superimpose a perturbation corresponding to a horizontally periodic wave with fixed horizontal wavenumber, kx. The initial vertical structure is prescribed by the product of an envelope with a vertically
periodic wave. Explicitly, the vertical displacement field is
共the real part of兲
␰共x,z,0兲 = A␰␰ˆ 共z兲eı共kxx+kzz兲 ,
共9兲
in which kz is a constant, representing the vertical wavenumber of waves in the wave packet, A␰ is a real, positive constant representing the wave packet amplitude, and ␰ˆ takes the
following general form:
冉
冦冉
exp −
␰ˆ ⬅
exp −
冊
共z − Lz兲2
,
2␴2
z ⬎ Lz ,
1,
兩z兩 ⱕ Lz ,
共z + Lz兲
2␴2
2
冊
, z ⬍ − Lz .
冣
共10兲
In the case Lz = 0, 共10兲 defines a Gaussian wave packet with
width given by the standard deviation, ␴. In cases with
Lz ⬎ 0, the wave packet structure is said to be “plateaushaped” and the half-width of the wave packet is Lz + ␴. Figure 1 presents an example of a plateau-shaped wave packet
with kz = −0.4kx, A␰ = 0.30/ kx, Lz = 10/ kx, and ␴ = 10/ kx.
Given ␰, the initial density perturbation field is determined using 共8兲, and the initial vorticity field is estimated
from linear theory for plane periodic internal waves:
␨共x,z,0兲 ⯝ N0kx sec ⌰␰共x,z,0兲,
in which ⌰ is given by 共2兲. Using linear theory to establish
the approximate initial condition for moderately large ampli-
tude waves is justifiable because plane internal waves of arbitrarily large amplitude are known to be an exact solution of
the fully nonlinear Euler equations.
If ⌰ = 0, the wave motion is vertically up and down and
the wave frequency matches the background buoyancy frequency. For large ⌰ 共close to ␲ / 2兲, the waves are hydrostatic, having a horizontal extent much larger than their vertical extent. The focus of this study is on nonhydrostatic
waves because it is for these waves that interactions between
the waves and the wave-induced mean flow results in instabilities that grow faster than parametric subharmonic instability.
In the simulations there is no initial background horizontal flow except for that predicted to be induced by waves.
Explicitly the wave-induced mean flow is given by the correlation of the vertical displacement and vorticity fields:32
具u典 = − 具␰␨典.
共11兲
Thus, using 共10兲, the initial background flow is given by
1
N
兩具u典兩t=0 = 共A␰ kx兲2 sec ⌰.
2
k
A plot of this mean flow is given in Fig. 1共c兲. Although 共11兲
is accurate for small amplitude waves, in fully nonlinear numerical simulations of internal waves generated by an unstable shear flows Sutherland33 showed that the approximation is accurate to within numerical errors even for internal
waves with amplitudes close to those for breaking waves.
The simulation proceeds by advancing the vorticity and
density fields in time using 共5兲 and 共6兲. The domain is horizontally periodic with free-slip upper and lower boundaries,
although the vertical extent of the domain is so large that the
waves have negligible amplitude at the boundaries over the
course of the simulations. The vertical resolution is
0.049kx−1 ⯝ 0.0078␭x, which is sufficiently small to capture
the onset and development of the fine-scale subharmonically
excited waves.
While running, the code systematically computes the instantaneous vertical profiles of the mean horizontal velocity
and the predicted value of the wave-induced mean flow using
共11兲. As a measure of the stability of the wave packet to
self-acceleration, these are compared with the horizontal
group velocity predicted by linear theory,
cgx = N0
k z2
N0
cos ⌰ sin2 ⌰.
2
2 3/2 =
共kx + kz 兲
kx
共12兲
III. QUALITATIVE RESULTS
In all the simulations reported upon here, the time scale
is set by N−1 and the length scale is set by kx−1. In practice,
the simulations set N = 1 and kx = 1. But for conceptual convenience, the results are presented in terms of dimensional
quantities relative to N and kx or to the buoyancy period
TB ⬅ 2␲ / N and horizontal wavenumber ␭x ⬅ 2␲ / kx.
We begin by presenting the results in Fig. 2 of a control
simulation of a small-amplitude, nonhydrostatic, Gaussian
wave packet of small vertical extent. Explicitly, the initial
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074107-4
B. R. Sutherland
FIG. 2. Simulation of the Gaussian wave packet prescribed initially with
amplitude A␰ kx = 0.01, vertical wavenumber kz = −0.4kx, and wave packet
width given by the standard deviation ␴ = 10/ kx. Gray scale contours showing normalized vertical displacement associated with the wave packet are
shown at times 共a兲 Nt = 0, 共b兲 Nt = 100 共t ⯝ 16TB兲, and 共c兲 Nt = 200
共t ⯝ 32TB兲. The time evolution of the vertical profile of the normalized
wave-induced mean flow is shown in 共d兲.
vertical displacement field associated with the waves is given
by 共9兲 and 共10兲 with A␰ / ␭x = 0.01/ 2␲ ⯝ 0.0016, kz = −0.4kx
共⌰ ⯝ 21.8° 兲, Lz = 0, and ␴ = 10/ kx ⯝ 1.6␭x.
Over the course of 32 buoyancy periods 共corresponding
to Nt = 200兲 the wave packet moves upward at constant vertical group velocity, consistent with that predicted by linear
theory. The waves undergo weak, linear dispersion evident,
though, the gradual widening of the wave packet and corresponding decrease in its peak amplitude over time. Compared to the examples that follow, the dispersion of the control case is relatively weak, despite the fact that the vertical
extent of the wave packet is comparable to the vertical wavelength 兩2␲ / kz兩, hence resulting in a broad power spectrum.
A. Effect of amplitude
The effects of weakly nonlinear dispersion are demonstrated in Fig. 3, which shows the evolution of a wave packet
having the same wavenumber vector and envelope structure
as that in the control case, but here the amplitude is 30 times
larger so that the half peak-to-peak vertical displacement is
almost 5% of the horizontal wavelength: A␰ / ␭x = 0.30/ 2␲
⯝ 0.048.
A striking difference between this and the control simulation is the initial growth, not a decrease in the maximum
amplitude of the wave. This is consistent with the result
of Sutherland,26 who demonstrated that the amplitude envelope of a vertically compact wave packet increases if its
peak frequency is faster than that of waves with the largest
vertical group velocity. That is, an amplitude increase is expected if ⌰ ⬍ ⌰쐓 ⬅ cos−1关共2 / 3兲1/2兴 ⯝ 35.3°, or, equivalently,
if 兩kz 兩 ⬍ 兩kz쐓 兩 ⬅ 2−1/2kx ⯝ 0.71kx, ␻ ⬍ ␻쐓 ⬅ 共2 / 3兲1/2N ⯝ 0.82N.
The speed of the wave-induced mean flow, 具u典, increases
as the square of the wave packet amplitude. In this simula-
Phys. Fluids 18, 074107 共2006兲
FIG. 3. The same as in Fig. 2 but for a large-amplitude Gaussian wave
packet with amplitude A␰ kx = 0.30 共enhanced with video online for comparison with simulations in Figs. 4 and 5兲.
tion, Fig. 3共d兲 shows that 具u典 increases to almost half the
speed of the horizontal group velocity after approximately 8
buoyancy periods 共Nt ⯝ 50兲. The resulting dispersion of the
wave packet differs substantially from that of the smallamplitude wave packet. The phase lines near the center of
the wave packet tilt upward 关see Fig. 3共b兲兴 and the vertical
advance of the wave packet slows.
These dynamics, namely the self interaction of waves to
produce a mean flow that non-negligibly affects the wave
packet evolution, are the result of self-acceleration. Though
in this simulation the waves are not of such large amplitude
initially that they evolve to overturn and break, the waveinduced mean flow is sufficiently large to advect the waves
horizontally at different speeds with height. This implicitly
imposed shear is what Doppler shifts the waves, locally increasing their frequency near the center of the wave packet.
Because ⌰ ⬍ ⌰쐓, the increase in frequency results in a decrease in the vertical group velocity.
At still later times, the growth of parametrically unstable
disturbances becomes evident. After 32 buoyancy periods
共Nt ⯝ 200兲 small vertical wavelength waves with half the
horizontal wavelength of the initial wave packet have developed beneath the central peak of the wave packet 关Fig. 3共c兲兴.
The corresponding vertical wavenumber is consistent with
that computed using 共1兲.
B. Effect of width and structure
The vertical rate of change of the amplitude envelope
determines the growth rate of the instability due to selfacceleration. This is demonstrated in Fig. 4, which shows the
evolution of a large-amplitude Gaussian wave packet having
the same initial characteristics as that shown in Fig. 3 共that
is, A␰ / ␭x ⯝ 0.048, kz = −0.4kx, and Lz = 0兲, except that the
wave packet width is four times larger: ␴ = 40/ kx.
As in Fig. 3, the peak amplitude of the wave packet
grows in time but at approximately one-quarter the rate.
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074107-5
Internal wave stability
FIG. 4. The same as in Fig. 2 but for a large-amplitude Gaussian wave
packet with amplitude A␰ kx = 0.30 and with initial width ␴ = 40/ kx 共enhanced
with video online兲.
Thus, only after approximately 16 buoyancy periods
关Nt = 100, as shown in Fig. 4共b兲兴 does the vertical tilting of
phase lines become evident, and only after approximately 32
buoyancy periods 关Fig. 4共c兲兴 has the relative frequency been
altered to such an extent that the vertical advance of the
wave packet slows 关Fig. 4共d兲兴.
Parametric subharmonic instability, though expected to
occur eventually, is not manifest to any significant amplitude
by the end of this simulation. Such instability would result in
the development of a fine vertical structure, as given by 共1兲.
Self-acceleration acts to increase the vertical wave number at
the leading flank of the wave packet due to the differential
advection of waves by the wave-induced mean flow.
To re-emphasize, it is not the width but the slope of the
amplitude envelope that determines the growth rate of the
instability. This is demonstrated in Fig. 5, which shows the
evolution of a plateau-shaped wave packet having the same
effective width as the wave packet shown in Fig. 4 but whose
amplitude changes as rapidly on the leading and trailing
flanks as the narrow Gaussian wave packet shown in Fig. 3.
Explicitly, the initial wave packet is prescribed by 共9兲
with A␰ / ␭x ⯝ 0.048 and kz = −0.4kx as before, but here the
amplitude envelope is given by 共10兲 with Lz = 30/ kx and
␴ = 10/ kx.
The initial growth in wave packet amplitude occurs just
as quickly as it does in the narrow large-amplitude Gaussian
wave packet simulation, the increase taking place not at the
wave packet center, where the amplitude is constant, but at
the leading flank.
The wave packet’s trailing flank also grows in amplitude, but at approximately half the rate. Thus, this simulation
demonstrates that the increase in peak amplitude of the
narrow-Gaussian wave packet shown in Fig. 3 is a consequence, not only of the leading edge slowing down as the
intrinsically generated shear Doppler shifts the waves to
higher frequency, but also as the trailing edge speeds up,
Phys. Fluids 18, 074107 共2006兲
FIG. 5. The same as in Fig. 2 but for a large-amplitude plateau-shaped wave
packet with amplitude A␰ kx = 0.30. Initially the constant amplitude region
has half-depth Lz = 30/ kx and decays as a Gaussian on either flank with
standard deviation ␴ = 10/ kx 共enhanced with video online兲.
where the opposite signed shear associated with the waveinduced mean flow Doppler shifts the waves to lower frequencies that are nonetheless larger than ␻쐓.
The long-time development of the wave packet is substantially more complex than in the narrow Gaussian case.
After approximately ten buoyancy periods 共Nt ⯝ 63兲, selfacceleration slows the vertical advance of the leading edge of
the wave packet. Meanwhile, the amplitude of the waves
below the leading edge begins to increase in amplitude, repeating the pattern of growth that occurred initially right at
the leading edge.
After 16 buoyancy periods 共Nt = 100兲, Fig. 5共b兲 illustrates the resulting structure of the waves. The leading edge
has evolved to form a localized packet of waves with nearly
vertical phase lines. Below this level, phase lines of the
waves in the plateau region begin to tilt upward.
The process repeats over time with the wave-induced
mean flow intrinsically changing the phase of the waves
wherever the amplitude envelope varies rapidly with height.
Parametric subharmonic instability is much more pronounced in this simulation, as might be expected because
the span of the wave packet over the plateau region is
plane wave-like—locally it has constant amplitude. Selfacceleration is still the fastest mode of instability developing
at the leading and trailing edges of the wave packet, but once
parametric instability has had time to grow 共after approximately 20 buoyancy periods for which Nt ⯝ 125兲 the wave
packet rapidly disperses with energy being transferred to
waves with ever smaller vertical and horizontal scales. Except near the leading edge, where a fraction of the initial
wave packet, however distorted, continues to progress upward, the majority of the wave packet has disintegrated after
32 buoyancy periods 共Nt ⯝ 200兲, as a consequence of parametric subharmonic instability.
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074107-6
Phys. Fluids 18, 074107 共2006兲
B. R. Sutherland
FIG. 6. The same as in Fig. 5 but for plateau-shaped wave packet with
vertical wavenumber kz = −0.7kx.
C. Effect of wave packet frequency
So far results have been presented exclusively for the
case kz = −0.4kx, corresponding in linear theory to waves with
frequency ␻ ⯝ 0.93N. The results of simulations change dramatically if the initial frequency is smaller than that for
waves with the fastest vertical group velocity 共for which,
␻ = ␻쐓 ⯝ 0.82N兲.
The near-critical case illustrated in Fig. 6, shows the
evolution of a large-amplitude, plateau-shaped wave packet
having the same amplitude and envelope structure as that
shown in Fig. 5, but here kz = −0.7kx.
The wave packet is again unstable with the amplitude
growing at the leading edge, though at a slower rate. Even
though the wave packet amplitude is modulated, it does not
disintegrate to the same extent as the waves in the fastfrequency case. Parametrically unstable disturbances become
apparent after 20 buoyancy periods 共Nt = 125兲, but only near
the wave packet’s trailing edge. These can be seen as the
fine-structure disturbances in Fig. 6共d兲 centered around
z / ␭x ⯝ 5.
For still lower frequency waves, the wave packet is
stable, meaning that its peak amplitude decreases in time, not
just because of linear dispersion, but through enhanced dispersion resulting from weakly nonlinear effects. For example, Fig. 7 shows the evolution of a wave packet with the
same amplitude and envelope structure as that shown in
Fig. 5, but here the vertical wave number is kz = −1.4kx
共␻ ⯝ 0.58N, ⌰ = 54.5°兲.
Just as in the previous two cases, the wave-induced
mean flow is significant. However, here the resulting increase in the intrinsic wave frequency means that the leading
edge of the wave packet moves more quickly upward,
whereas the decreasing frequency at the trailing edge means
that it moves more slowly.
So instability due to self-acceleration does not occur for
waves with ␻ ⬍ ␻쐓. In this simulation, parametric subharmonic instability does develop after about 25 buoyancy pe-
FIG. 7. The same as in Fig. 5 but for a plateau-shaped wave packet with
vertical wavenumber kz = −1.4kx.
riods 关Nt ⯝ 157 in Fig. 7共d兲兴 though it acts to erode the trailing edge of the packet. In simulations with wave packets
initially having a larger amplitude, the instability is initiated
sooner and erodes a greater fraction of the wave packet.
IV. QUANTITATIVE RESULTS
In Sec. III, the unstable evolution of the large-amplitude
wave packets was attributed to a combination of selfacceleration and parametric subharmonic instability. Here,
more rigorous diagnostics are applied to demonstrate the distinction between the two instability mechanisms and to provide more direct numerical comparisons between their relative growth.
Specifically, during the course of each simulation a spectral decomposition of the nonlinear advection terms gives a
measure of the energy transfer from the primary waves, with
horizontal wavenumber kx, to the mean flow and subharmonic waves having horizontal wavenumber 2kx. 共For details, see Sutherland and Peltier.31兲 In theory, parametric subharmonic instability can result in energy transfer among any
triad of waves satisfying k1 + k2 = k3 and ␻共k1兲 + ␻共k2兲
= ␻共k3兲. However, in practice we find that the fastest growing instability transfers energy to waves with double the
horizontal wavenumber and with vertical wavenumber given
by 共1兲.
Figure 8 shows log plots of the normalized energy transfer, TE, as it evolves over time in four simulations. In all
cases the amplitude is A␰ / ␭x ⯝ 0.048 and the vertical wavenumber is kz = −0.4kx, so that self acceleration and parametric
subharmonic instability both occur. Energy transfers are examined as they depend upon the structure and width of the
wave packets.
Figures 8共a兲–8共c兲 examine the effect of extending the
width of a plateau-shaped wave packet. In all three cases, the
transfer of energy between the primary wave and the waveinduced mean flow is the same at early times with small
changes being apparent only near the end of the simulations.
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074107-7
Phys. Fluids 18, 074107 共2006兲
Internal wave stability
FIG. 8. Energy transfer rates from primary waves to mean flow 共solid line兲
and to subharmonic waves 共dashed line兲 computed in simulations of wave
packets prescribed initially with amplitude A␰ kx = 0.30, vertical wavenumber
kz = −0.4kx 共⌰ ⯝ 21.8° 兲, and wave packet width given by 共a兲 Lz = 10/ kx,
␴ = 10/ kx, 共b兲 Lz = 20/ kx, ␴ = 10/ kx, 共c兲 Lz = 30/ kx, ␴ = 10/ kx, 共d兲 Lz = 0,
␴ = 40/ kx.
By contrast, the transfer of energy to subharmonic waves
grows to larger values over time in simulations with successively wider wave packets.
However, the wave packet width alone does not determine the energy transfer rate. As Fig. 8共d兲 demonstrates, a
Gaussian wave packet having the same effective width as the
plateau-shaped wave packet analyzed in Fig. 8共c兲 transfers
energy to the wave-induced mean flow at a lower rate, but is
nonetheless four orders of magnitude larger than the transfer
rate to subharmonic waves over the duration of the simulation.
These results demonstrate that the growth of instabilities
driven by self-acceleration depend upon the rate of change of
the wave packet envelope with height. In contrast, the
growth of parametric subharmonic instability depends upon
the vertical extent over which the wave packet has nearly
constant amplitude.
Once subharmonic waves grow to substantial amplitude,
energy continues to cascade to ever smaller vertical length
scales. The evolution of the resulting spectra as computed
from three simulations are shown in Fig. 9. Each plots the
vertical wavenumber spectra of horizontal velocity computed
at successive times and horizontally offset, as indicated. Because the interest is on the spectral shape and not magnitude,
the units of the power spectrum are arbitrary but are shown
over the same range in all simulations. In all three cases the
power spectrum broadens over time during the growth and
evolution of the instabilities. In the wide-plateau simulation
关Fig. 9共c兲兴, the spectral tail evolves to form a power law with
a −4 slope, smaller than the power observed in the open
ocean, as modeled, for example, by the Garrett-Munk spectrum. It may be that a different spectral slope would develop
in a fully three-dimensional numerical simulation, however,
such an examination is beyond the scope of this study.
FIG. 9. Vertical wavenumber spectra of horizontal velocity variance computed successively over time in simulations of wave packets prescribed
initially with amplitude A␰ kx = 0.30, vertical wavenumber kz = −0.4kx
共⌰ ⯝ 21.8° 兲, and wave packet width given by 共a兲 Lz = 0 / kx, ␴ = 10/ kx, 共b兲
Lz = 0 / kx, ␴ = 40/ kx, 共c兲 Lz = 30/ kx, ␴ = 10/ kx.
V. DISCUSSION AND CONCLUSIONS
This process study has been designed to illustrate the
significance of interactions between moderately large amplitude nonhydrostatic internal waves and their induced mean
flow 共self-acceleration兲 in comparison with interactions between waves and their subharmonics 共parametric subharmonic instability兲. The former are significant, in particular,
for wave packets having frequencies between 共2 / 3兲1/2N and
N. The latter dominate for plane periodic waves and eventually develop within wave packets provided they are sufficiently wide and their amplitude is nearly constant over a
large depth.
Although nonlinear wave-wave interactions undoubtedly
influence the energy cascade from large to dissipative scales,
idealized theoretical, and numerical studies of parametric
subharmonic instability have focused primarily upon plane
periodic waves as the primary mechanism for internal wave
instability in the absence of shear. Self-acceleration is an
alternate instability mechanism that this study shows is
dominant, at least during the initial evolution of nonhydrostatic 共high-frequency兲 internal waves.
ACKNOWLEDGMENT
This research was supported by the Canadian Foundation for Climate and Atmospheric Science 共CFCAS兲.
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