Internal wave instability: Wave-wave versus wave

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-

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© 2006 American Institute of Physics

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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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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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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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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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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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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. 1

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