Energy Transfer within the Small-Scale Oceanic ... - AMS Journals

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Energy Transfer within the Small-Scale Oceanic Internal Wave Spectrum RYO FURUE Center for Climate System Research, University of Tokyo, Tokyo, Japan (Manuscript received 8 June 2001, in final form 31 July 2002) ABSTRACT Three-dimensional numerical experiments are conducted to examine energy transfer within the small-scale portion of the Garrett–Munk model spectrum of oceanic internal waves. The rate of energy transfer in the experiments is a significant fraction of observed total transfer rate in the interior main thermocline. This transfer may supplement the previous estimate by the eikonal theory. Because nonlinearity is strong in this spectral region, wavenumber-local interactions dominate the energy transfer rather than scale-separated ones. Transfer to higher horizontal wavenumbers is robust, whereas that to higher vertical wavenumbers seems to depend strongly on the spectral shape. Vortical motions seem to be enhancing energy transfer. All of these suggest that further investigation of this spectral region is important and necessary by means of three-dimensional, fully nonlinear analysis.

1. Introduction Numerical studies using oceanic general circulation models (OGCMs) have shown that the thermohaline circulation is very sensitive to the magnitude and vertical distribution of vertical (or diapycnal or dianeutral) eddy diffusivity of temperature and salinity at the thermocline and deeper depths (Bryan 1987; Cummins 1991; Gargett and Holloway 1992; Hasumi and Suginohara 1999b). Recent studies also show that localized regions of intensified vertical mixing can significantly influence the global-scale circulation (Marotzke 1997; Samelson 1998; Hasumi and Suginohara 1999a). To reproduce such mixing explicitly requires a resolution of centimeters, which is utterly beyond the reach of OGCMs within coming decades. It is therefore crucial to parameterize mixing effects accurately into OGCMs and climate models. Vertical mixing in the deep ocean is generally believed to be supported by small-scale perturbations such as breaking internal gravity waves (IWs). The smallscale perturbations are maintained by energy transfer from large, energy-containing scales, or in some cases by double diffusion (Schmitt 1994, and references therein). The mechanism of the energy transfer is hence an important subject of investigation. Away from boundaries such as the sea surface, bottom, seamounts, and continental boundaries, this energy transfer is generally ascribed to nonlinear interactions among internal gravity Corresponding author address: Dr. Ryo Furue, Center for Climate System Research, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, Japan. E-mail: [email protected]

q 2003 American Meteorological Society

waves (Gregg 1987; Gargett 1989). Again away from boundaries, the IW spectrum in the deep ocean is known to be surprisingly universal and reasonably well described by what is called the Garrett–Munk (GM) spectrum (Garrett and Munk 1972, 1975, 1979; Munk 1981) for vertical scales larger than roughly 10 m (Wunsch 1976; Mu¨ller et al. 1978; Wunsch and Webb 1979). The spectrum is formulated in terms of the frequency and the vertical wavenumber and the flow field is assumed to be a superposition of linear IWs. The GM spectrum has provided the basis for most dynamical calculations. Using a weakly nonlinear theory (Hasselmann 1966, 1967), McComas and Mu¨ller (1981a) estimated the energy transfer rate in the GM spectrum (see also McComas and Bretherton 1977; McComas 1977; McComas and Mu¨ller 1981b; and Mu¨ller et al. 1986). They also identified dominant transfer mechanisms, which turned out to be interactions between large-scale and small-scale waves (scale-separated interactions). At smaller scales, however, nonlinearity becomes stronger and the weakly nonlinear theory may not hold. Henyey et al. (1986) used the ray theory (Henyey and Pomphrey 1983) instead, where the weakness of interaction is not assumed but only the dominance of scale-separated interactions is, and estimated the energy transfer rate (see also Flatte´ et al. 1985). The disadvantage of these semianalytical studies is that they reduce the dynamics included to obtain analytically tractable formalisms. Numerical simulations, on the other hand, include full dynamics, sacrificing the range of scales of motion involved. A typical three-dimensional (3D) simulation can afford only about a hundred grid points in each spatial

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direction, far from adequate to cover the whole spectral range of the IW field. There are a number of numerical studies of sheared and unsheared stratified turbulence not necessarily with implications for the ocean (e.g., Riley et al. 1981; Gerz et al. 1989; Holt et al. 1992; Kaltenbach et al. 1994). Those stratified turbulence studies with implications for the ocean (Shen and Holloway 1986; Ramsden and Holloway 1992; Itsweire et al. 1993; Siegel and Domaradzki 1994) concern mainly small scales of 10 m or less, where the flow field is turbulent or much turbulence-like. Such a flow may occur during and after wave breaking in the real ocean. Carnevale et al. (2001) used a numerical model with a three-dimensional cubic computational domain of a 20m side length. They successfully reproduced transition from the buoyancy range to the intertial range. There are only a few studies, as far as I am aware of, of the large-scale oceanic IW field. Lin et al. (1995), Hibiya et al. (1996), and Winters and D’Asaro (1997) all examined the free evolution of a flow field initialized with the GM spectrum. (See also Lin 1993 and Hibiya et al. 1998). Lin et al.’s simulation, confined to a two-dimensional (2D) vertical plane, resolved wavelengths of 5200 m down to 40.6 m horizontally and 2600 m to 20.3 m vertically (see also Lin 1993). Hibiya et al.’s 2D simulation contained widely disparate scales, namely, from the wavelength of 1.024 3 10 4 m down to 20 m horizontally and from 1280 m to 2.5 m vertically. Winters and D’Asaro’s three-dimensional model covered the wavelengths of 10 4 m through 625 m in one of the horizontal directions, of 8 3 10 4 m through 625 m in the other, and of 2 3 10 3 m through 31.2 m in the vertical. McComas and Mu¨ller’s (1981a) prediction of «, the energy dissipation rate, is larger by a factor of 3 than the observation (Gregg 1989; Winters and D’Asaro 1997). This may be because the weakly nonlinear theory tends to overestimate nonlinear interaction rates at higher wavenumbers (T. Hibiya, Y. Niwa, and K. Fujiwara 1998, personal communication), where the assumption of weak nonlinearity tends to break. In contrast, Henyey et al.’s (1986) prediction of « is smaller by a factor of 2 or more than the observation (Gregg 1989; Winters and D’Asaro 1997). They release a large number of ‘‘test waves’’ (wave packets or rays) at k h 5 10 23 cycles per meter (cpm), 10 22 cpm # | k z | # 10 21 cpm, and 22000 m # z # 0 m, where k h and k z are the horizontal and vertical wavenumbers and z the vertical coordinate, and let those waves evolve in the background flow that is a random superposition of IWs with the amplitude taken from the GM spectrum and the phase randomly chosen. When the vertical wavelength of a test wave becomes less than 5 m, they declare it to have ‘‘reached a critical layer,’’ remove it from the simulation, and count the energy flux through the cutoff wavelength 5 m. This energy flux is the basis of their estimation of «. Although most of the test waves reach a critical layer with a horizontal wavenumber smaller than 10 22 cpm (their Fig.

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5), test waves do fill the wavenumber space beyond 10 22 cpm and has an energy level comparable to the GM level [Flatte´ et al. (1985), their Fig. 5]. We suspect that interactions between test waves (wavenumber-local interactions), ignored in the ray theory, may significantly affect the energy transfer at such small scales, and that this effect may cause additional energy transfer to dissipation scales and supplement the estimation of «. Another concern is the effects of vortical motions on the energy transfer. There is evidence that vortical motions account for a significant fraction of small-scale variabilities (Mu¨ller 1988; Mu¨ller et al. 1988; Lien and Mu¨ller 1992). It has been known that decaying stratified turbulence, initially energetic and three-dimensional, collapses into IWs and vortical motions (Riley et al. 1981; Lilly 1983; Me´tais and Herring 1989). Interactions between vortical and IW motions are studied by using a weakly nonlinear theory (Lelong and Riley 1991). Energy transfer mechanisms involving vortical motions are examined for stratified turbulent flows by using numerical simulations (Riley et al. 1981; Herring and Me´tais 1989; Me´tais and Herring 1989). However, roles of vortical motions in the energy transfer in the realistic oceanic context are yet to be investigated. In this study, we investigate the energy transfer at scales smaller than 100 m both in the vertical and horizontal, using a 3D numerical model. The smallest scales our model can resolve are 2 m in the horizontal and 0.3 m in the vertical; this resolution is restricted by the computer resources. Previous studies for the most part concern either larger or smaller scales as reviewed above. A 3D configuration is necessary to consider interactions between IWs and vortical motions since confining the simulation within a vertical 2D plane would severely restrict those interactions (Winters and D’Asaro 1994). We will suggest that wavenumber-local interactions among small-scale motions efficiently transfer energy to dissipation scales in addition to the critical layer mechanism at larger horizontal scales, thereby supplementing Henyey et al.’s (1986) estimation of «. This paper is organized as follows. The next section describes our model and experimental design. Section 3 presents the results, and section 4 summarizes the work. There is an appendix, which describes how we implement the GM spectrum in the model. 2. Model and experimental design We first outline the numerical model and then describe our experimental design. The reader is referred to Furue (1998) for details of the model. a. Model The governing equations are the usual Boussinesq set with a background density stratification:

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]v 1 r9 1 (v · =)v 5 2 =p9 1 g ]t r0 r0 1 Dy (v), = · v 5 0,

(1) (2)

]r 9 dr˜ 1 (v · =)r9 1 w 5 Dr (r9). ]t dz

(3)

Here v 5 (u, y , w) are the velocity components in the x, y, and z (vertically upward) directions, respectively; r9 is the deviation from the background density stratification r˜ (z), and p9 is the pressure deviation from the hydrostatic pressure determined by r˜ (z). The constant vector g 5 (0, 0, 2g) is the gravitational acceleration; r 0 is a standard, constant value of density. The terms D y (v) and D r (r9) represent subgrid-scale (SGS) dissipation and will be discussed later. We ignore the Coriolis force because the timescale of the system is much shorter than the inertial period, as will be seen in the next section. We consider a finite rectangular domain with side lengths L x , L y , and L z in the x, y, and z directions, respectively. We assume that the background stratification is uniform—that is, dr˜ /dz 5 constant, and that the flow field is statistically uniform in all spatial directions. In this case, the common choice of boundary conditions is periodicity in all the spatial directions. To ensure the incompressibility condition [Eq. (2)] during numerical integration, we transform the three components of the momentum equation (1) into two prognostic equations for nw and z z (Kim et al. 1987), def def where n 5 ] 2 /]x 2 1 ] 2 /]y 2 1 ] 2 /]z 2 , z z 5 ]y /]x 2 ]u/ ]y. Given nw and z z , the three velocity components can be diagnosed by solving the following Poisson equations: nw 5 known, nH y 5 2

nH u 5 2

]2 w ]z 2 z, ]x]z ]y

]2 w ]z 1 z, ]y]z ]x

def

where n H 5 ] 2 /]x 2 1 ] 2 /]y 2 . These equations, however, cannot determine the volume average of w or the horizontal averages of u and y . The information of these averages is in fact lost during the transformation of the momentum equations. We hence retain the horizontal average of the horizontal components of the momentum equation to compute the horizontal averages of u and y . On the other hand, the volume averages of the vertical component of the momentum equation (1) and the density equation (3) become ]w r9 5 2g , ]t r0

]r 9 dr˜ 5 2w , ]t dz

where the overbars denote volume averages. Here we have used the periodicity boundary condition and in-

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voked the fact that the volume averages of the SGS dissipation terms vanish (as is the case for the parameterization shown later). These equations indicate that if w 5 0 and r9 5 0 initially, they remain so forever. We accordingly assume that w 5 r9 [ 0. To solve the governing equations, we use the spectral method (Gottlieb and Orszag 1977), where the equations are spatially Fourier-transformed and the resulting set of ordinary differential equations with respect to time is time-integrated. This method is suitable for a computation, such as a turbulence simulation, requiring accurate evaluation of nonlinear interactions because the method is a very accurate numerical differentiator (Rogallo and Moin 1984). Nonlinear terms are computed in the configuration or physical space to reduce the computation time (‘‘pseudo-spectral method;’’ see Canuto et al. 1988, and references therein). Otherwise the computation would be prohibitively expensive. We use Patterson and Orszag’s (1971) method (‘‘the 3/2 rule;’’ see also Canuto et al. 1988) to achieve an aliasing-free evaluation of the nonlinear terms. This method requires extra spatial resolution for Fourier transformation. This accurate evaluation of nonlinear terms is the most expensive part of the computation, as more than 80% of the CPU time is spent for Fourier transforms. The time advancement is carried out by the implicit scheme for the dissipation terms and by the leapfrog scheme for the other terms (Mesinger and Arakawa 1976). To suppress the computational mode, we apply time filtering at every timestep (Haltiner and Williams 1980, p. 147). The time step is 1.4 s. Since the existing computational resources do not allow a three-dimensional numerical model to resolve both the dissipation scales and the largest scales of the problem, some or other form of SGS parameterization is needed. We use a modified version of the Chollet– Lesieur SGS model (Chollet and Lesieur 1981, hereafter CL81). The CL81 parameterization is defined in the wavenumber space. The original CL81 eddy viscosity/diffusivity has a plateau region; that is, it is nearly constant for wavenumbers sufficiently smaller than the ‘‘cutoff’’ wavenumber or the maximum wavenumber resolved by the numerical model, while it increases rapidly as the wavenumber approaches the cutoff (Domaradzki et al. 1987; Siegel and Domaradzki 1994). We modify the CL81 parameterization in three ways. Since our horizontal and vertical resolutions are different, we use a ‘‘scaled’’ wavenumber. We consider an ellipsoid that passes through the six points (k x , k y , k z ) 5 (6k xc , 0, 0), (0, 6k yc , 0), (0, 0, 6k zc ), where k x , k y , and k z are the wavenumbers in the x, y, and z directions and k xc , k yc , and k zc are the respective cutoff wavenumbers. The def scaled wavenumber is k 5 (k x , k y , k z ) 5 [(k c /k xc )k x , (k c /k yc )k y , (k c /k zc )k z ], where k c is a representative cutoff wavenumber. We use k in the parameterization so that the damping time is constant on the ellipsoid. Another

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modification to the parameterization is that the plateau region is removed from the eddy viscosity/diffusivity. This almost frees large scale motions from dissipation and leaves them almost completely inviscid. The other modification is that we reduce the magnitude of the viscosity/diffusivity coefficient from the original CL81 value (see below). The Fourier transforms of the dissipation operators, D y ( ) and D r ( ), can then be written as

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TABLE 1. List of experiments. GM indicates the relaxed SF (see text). Run designation

Initial SF

Initial WV

Forcing

0 1r 1 F0 F1

0 GM GM9 0 GM9

GM 3 2 GM GM 3 2 GM 3 1.2 GM 3 1.2

No No No Yes Yes

ˆ y (k) 5 D ˆ r (k) 5 2n e (k) | k | 2 . D The eddy viscosity/diffusivity n e is defined as

[

]

E(|k | 5 k c ) n e (k)5 A exp(20.03k c /|k |) kc def

1/2

,

where E( | k | 5 k c ) is the energy spectrum averaged on the ellipsoid. We have assumed that the turbulent Prandtl number is unity. We choose k c 5 k zc , which gives minimum dissipation since, as given later, k zc is the largest among the three cutoff wavenumbers. [Notice that there is a k c in the denominator below E(k) in the formula.] The original value of A is 9.21, but we use A 5 0.921. We have chosen this value by conducting a number of experiments with different values of A: with the original value of A, energy spectra rolled off too fast at higher wavenumbers, and so we reduced A until both horizontal- and vertical-wavenumber energy spectra became straight lines on log–log plots, which is a characteristic of proper energy cascade. The original CL81 parameterization is developed for the isotropic inertial range. If higher wavenumber portion of the resolved spectrum is adequately in the inertial range, as in Carnevale et al. (2001), the use of CL81 is well justified. However, we have found that our model does not seem adequately to resolve the inertial range, as is evidenced from the degree of anisotropy at the highest wavenumber region of the energy spectrum (not shown) in experiments that will be examined in the next section. We nevertheless use the parameterization (in the modified form as described above) primarily as a convenience. Since no reliable parameterization is known for a stratified flow outside the internal range, one must use a parameterization that is more or less questionable. The CL81 parameterization has important benefits over others: 1) the energy removal by dissipation is confined to highest wavenumbers resolved, 2) the strength of dissipation is controlled by the strength of motions at those wavenumbers, 3) it is formulated in the Fourier space, and 4) the parameters involved can be adjusted easily. Besides being benefits, 1 and 2 come from the assumption that energy cascade is (statistically at least) one-way, large scales controlling the cascading, and hence energy’s passing through a wavenumber can be simply replaced with the removal of as much energy as tends to flow into that wavenumber, assumption that is most frequently invoked on eddy viscosity and diffusivity. Our justifications for using CL81 are, thus, this

assumption, the possibility that the inertial range may be at least barely resolved in some of our experiments (shown later), and the reasonableness of the resultant spectral shapes (mentioned above). Other parameter values are as follows. The sizes of the computational domain are L x 5 L y 5 100 m and L z 5 128.1 m. The numbers of Fourier modes are 47, 47, and 427 in the x, y, and z directions, respectively.1 This gives a horizontal resolution of 2.13 m and a vertical resolution of 0.3 m. This resolution is barely adequate to resolve individual breaking waves (Winters and D’Asaro 1994). The cutoff wavenumbers are k xc 5 k yc 5 2p/L x 5 1.45 m 21 and k zc 5 2p/L z 5 10.4 m 21 . The coefficient of time filtering is 2 3 10 23 (nondimensional) the smallest possible value to suppress the computational mode. Here are other parameter values: r0 5 def 1.00 3 10 3 kg m 23 ; g 5 9.80 m s 22 ; N˜ 5 [2(g/ r 0 )dr˜ /dz]1/2 5 5.24 3 10 23 s 21 5 3 cycles per hour, Dt (time step) 5 1.4 s. b. Experimental design Table 1 lists our experiments. We initialize the flow field with the GM spectrum. Here we give only an outline and refer the reader to appendix for details. Let us divide the flow field into two components: those Fourier modes with k h ± 0 and k z ± 0, labeled ‘‘wave-vortical (WV)’’ component; and those with k h def5 0, labeled ‘‘shear flow (SF)’’ component. Here k h 5 (k x2 1 k y2)1/2 is the horizontal wavenumber. We assume that the Fourier modes with k z 5 0 vanish because the GM spectrum does not contain much energy near k z 5 0 and such a motion is unrealistic. We initialize the WV component so that each Fourier mode of the flow field satisfies the polarization relation of IW and has the energy of some multiple of the GM spectrum (Table 1). This component therefore has no linear vortical mode initially (more on this shortly). The GM spectrum might not accurately describe small-scale IW modes because the data on which the GM spectrum is based do not adequately sample the high-wavenumber, high-frequency motions (Mu¨ller et al. 1988). We nevertheless use it in the hope that it gives 1 The resolution for computing nonlinear terms is 72 3 72 3 640 according to the 3/2 rule noted earlier.

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at least a zeroth-order approximation of the oceanic IW field even for such small scales. The GM spectrum sharply increases as k h → 0 and most of the energy is contained around the k h 5 0 axis. Those motions with small k h have much more horizontal kinetic energy than vertical kinetic and potential energy, and have a low frequency. On the other hand, motions with exactly k h 5 0 are simply a steady horizontal flow with vertical shear according to the IW dispersion and polarization relations without the Coriolis force (Gill 1982). We accordingly approximate the unresolved large-scale motions as a horizontal flow with k h 5 0, and initialize the SF component with a k h average of the GM spectrum around k h 5 0 (Lin et al. 1995). Our method of averaging somewhat overestimates the energy of SF (appendix). We compare experiments that have initial SF with ones that do not to assess the effects of the large-scale flow with horizontal scales larger than 100 m on the energy transfer within the small-scale spectrum. This separation of horizontal scales at 100 m is perhaps the most artificial point in our simulations. The separation is, of course, primarily due to the limitation in computer resources, but also can be seen as a crude form of separating wavenumber-local interactions from scale-separated ones. This amounts to assuming that the most significant mode of scale-separated interaction is the deformation (including the critical layer absorption mechanism) of the smaller-scale field by the vertical shear of the larger-scale field. We expect that this assumption is not very wrong because larger-scale field in reality evolves much more slowly than the smallerscale field and is dominated by horizontal motions. The SF taken from the GM spectrum is potentially unstable according to a shear instability criterion (Richardson number ,1/4; see, i.e., LeBlond and Mysak 1978, 411–412). To avoid such initial (potential) instability, we use a relaxed SF, designated GM9 in Table 1, as the initial SF. Here, GM9 is the SF of case 1r after 2.8 3 10 4 s of integration, when an initial energy transfer from SF to WV, a possible indication of instability, has ceased for a long time. Comparing the SF of GM9 with that of GM, we find that they differ only in vertically very small features (&1 m). Here we discuss the vortical mode in the initial condition. The vortical mode is one of the three eigenmodes for each Fourier component of a linearized Boussinesq set with a background stable stratification (Lien and Mu¨ller 1992). The other two are IW modes. In the absence of the Coriolis force, the vortical mode is simply a steady nondivergent horizontal flow without density variance (Furue 1998). The vortical mode has potential vorticity while the IW mode does not (Lelong and Riley 1991). Although our initial conditions do not include linear vortical modes, they do contain potential vorticity. Linear IW modes have horizontal vorticity and undulation of isopycnal surfaces. For a single linear IW mode, =r is perpendicular to its vorticity so that po-

tential vorticity vanishes. When many IW modes are superposed, however, the vorticity vector of one mode may intersect some isopycnal surfaces undulated by other modes, resulting in nonzero potential vorticity. In adition, when we have SF, its strong horizontal vorticity adds to what small potential vorticity the superposition of IW modes has. Potential vorticity is conserved without dissipation and when the vorticity associated with it changes direction it can have a vertical component, which will be projected onto linear vortical modes. Potential vorticity can also be created by the action of viscosity or diffusion, so that the creation is most effective in wave breaking (Winters and D’Asaro 1994). Therefore, vortical modes we will see in the next section consist of vorticity associated with initial potential vorticity and vorticity created by dissipation. There are two types of experiments, labeled ‘‘free’’ and ‘‘forced.’’ In free experiments, the flow field is let freely evolve, or decay, under the influence only of dissipation. The initial energy is twice the GM value. We wait for the flow field to decay and for the energy level to reach the GM level; we then examine the energy transfer (cases 0 and 1). In forced experiments, an external forcing is applied to obtain a statistically steady state. The forcing is to mimic the energy flux from the large-scale spectrum. However, since there is no information about the spatial or temporal scales on which the energy flux is acting, we must use some ad hoc forcing. We use a Gaussian random forcing whose power spectrum is taken from Ramsden and Holloway (1992): A

k3 , (k 2 1 k 02 ) 6

where k is the total wavenumber, (k x2 1 k y2 1 k z2)1/2 , and we set k 0 5 2p/(14 m). This spectrum is isotropic, and it reaches its maximum at the wavelength of 25 m and rapidly rolls off as k 29 for larger k. We do not force components with k $ 2p/(9 m). To avoid exciting the computational mode, we generate the random deviates from a linear Markovian process with a correlation time of 100Dt 5 140 s (Priestley 1981). Then the forcing spectrum is white for frequencies lower than 2p/(140 s) and it decays as v 22 beyond this roll-off frequency. The forcing is incorporated as an additional term on the right-hand sides of the prognostic equations of v and r9 [Eqs. (1) and (3)]. Those forcing terms are chosen to satisfy the polarization relation of IWs, and hence there is no forcing for linear vortical modes. We avoid forcing at k h 5 0 or k z 5 0. By changing the coefficient A, the magnitude of forcing is adjusted to give the GM energy level at the statistically stationary state. As will be seen in the next section, the energy level initially decays very rapidly even with forcing so that the initial energy level is taken to be 1.2 times the GM value to speed up the approach to the stationary state (Table 1).

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FIG. 1. Froude spectra at t 5 0 (dashed line) and at t 5 2.81 3 10 4 s (solid line) for case 1r. The short dashed line indicates the contribution of the shear flow component, and the dashed–dotted line indicates that of the wave-vortical component to the total Froude number at t 5 2.81 3 10 4 s (solid line). All the curves have been smoothed by a moving average.

3. Results a. All GM case (case 1r) First we briefly examine case 1r, where both the SF and WV components are initially set to the corresponding GM values and the flow field is allowed to evolve freely. Figure 1 shows the Froude spectra at t 5 0 and at t 5 2.8 3 10 4 s. The Froude spectrum is defined as def

FFr (k z )5

k z2 [Fu (k z ) 1 Fy (k z )] , ˜2 N

where F u (k z ) and F y (k z ) are the one-sided power spectra of u and y (Gregg et al. 1993). Initially, the Froude spectrum is flat because we use the GM spectrum unmodified even beyond k z 5 2p 3 0.1 m 21 (50.1 cpm), where the GM spectrum is believed to fail to describe the oceanic variance field. At t 5 2.81 3 10 4 s, it is still flat below 0.5 m 21 , but decreases as k 21 beyond z and begins to swell at around 2 m 21 . These features are remarkably similar to those observed (Mu¨ller et al. 1991; Gregg et al. 1993). The roll-off of the Froude spectrum at 0.5 m 21 is due to the change in slope from 22 (the GM value) to 23 in the kinetic energy spectrum of SF, corresponding to the short dashed line in the figure, and the bump or swelling beginning at 2 m 21 is due to the shallowness of the spectral slope of WV kinetic energy there. The roll-off at 0.5 m 21 is unlikely due to instability in SF because the Froude number be 21 low 0.5 m 21 , # 0.5m dk z FFr (k z ), is well below the critical 0 value (Hibiya et al. 1996). The time series of the SF

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and WV energy (not shown here) show that initially, the kinetic energy of SF rapidly decreases and part of that energy is gained by WV. At the same time, the vertical energy spectrum of the SF begins to change its slope at around k z 5 0.5 m 21 (not shown). In case 0, where there is no SF initially, SF gains energy and that gain is mostly confined to vertical wavenumbers below 0.5 m 21 (not shown); the vertical-wavenumber energy spectrum of the SF has a sharp bend at around k z 5 0.5 m 21 . These suggest that for some reason, energy tends to be more efficiently transferred from SF to WV above 0.5 m 21 than below and that the roll-off starts at k z 5 0.5 m 21 because energy is transferred from SF to WV selectively beyond k z 5 0.5 m 21 . The high wavenumber bump in the Froude spectrum in the real ocean is ascribed to the beginning of the turbulent inertial range and is therefore supposed to begin near the Ozmidov wavenumber, (N˜ 3 /«)1/2 (Gargett et al. 1981; Holloway 1983; Mu¨ller et al. 1991). Using the value of « at t 5 1.69 3 10 4 s of ;1 3 10 210 m 2 s 23 (not shown) and N˜ 5 5.2 3 10 23 s 21 , we find an Ozmidov wavenumber k b of 3.7 3 10 m 21 , which is beyond our cutoff wavenumber. However, the kinetic spectrum in the buoyancy range can be written as aN˜ 2 k 23 and that in the inertial range as C K« 2/3 k 25/3 , where a and C K are order-one, nondimensional empirical constants (Carnevale et al. 2001). Using typical values of a ø 0.2 and C K ø 1.5 (Carnevale et al. 2001), the wavenumber where these two spectra meet is k 5 (a/C K ) 3/4 3 (k b ) 5 0.22k b 5 8.2 m 21 , which is not very far from the ;3 m 21 observed in the experiment (Fig. 1). Also, the value of dissipation rate is somewhat ambiguous. As will be shown later, the dissipation rate changes very rapidly from ;10 3 10 210 to ;1 3 10 210 m 2 s 23 within the first 10 4 s. The change is fast particularly in experiments with SF. The spectral bump we are looking at may be a remnant of vigorous overturning motions which created high dissipation at an earlier stage of the development. So, if we take an earlier time, say, t 5 5.6 3 10 3 s, when the dissipation rate is about 10 3 10 210 m 2 s 23 , k b is smaller and the crossover wavenumber is 2.6 m 21 . At this time, the spectral bump is at around 1 m 21 (not shown), which is not far from the estimated 2.6 m 21 . The highest-wavenumber portion of the spectrum may hence be (or in an earlier stage, may have been) in the intertial range although the spectral anisotropy mentioned in the previous section casts some doubts. b. Energy transfer in the unforced cases (cases 0 and 1) We first examine case 0 where the WV component was initialized with twice the GM spectrum but the SF component was initially set to zero. Figure 2 shows the evolution of domain-average energy and dissipation rates for case 0. The total dissipation rate takes a value of about 1 3 10 210 m 2 s 23 at t ø 2.8 3 10 4 s, when the total energy reaches the GM level. The contribution

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FIG. 2. Evolution of (a) domain-average energy and (b) dissipation rates for case 0. For energy, solid 5 total energy, dashed 5 kinetic energy, and dashed–dotted 5 potential energy. For dissipation rates, solid 5 total energy dissipation, dashed 5 kinetic energy dissipation, and dashed– dotted 5 potential energy dissipation. The ordinate range for (b) is narrow so as to focus on the later part of the evolution although the dissipation rates are out of range initially.

of kinetic energy dissipation to the total dissipation is slightly larger than that of potential energy dissipation. This is due to the presence of vortical motions at high wavenumbers. (Remember, there were no linear vortical modes initially.) Figure 3 shows the k h–k z spectra of the total dissipation rate and the nonlinear energy transfer rate at the end of the integration (t 5 2.81 3 10 4 s). The peak of dissipation occurs at the highest horizontal wavenumbers. The lower-wavenumber portion of the nonlinear transfer rate has a complicated structure, but that is statistically insignificant because the period of averaging is shorter than the time scale of those large-scale mo-

tions. However, the transfer rate is negative on average at lower wavenumbers. On the other hand, the higherwavenumber portion is much smoother and consistently positive, and it is statistically significant; the transfer rate does not change much if we take other averaging periods (not shown). Energy is thus transferred from lower to higher wavenumbers via nonlinear interactions. The distribution and magnitude of positive nonlinear transfer rate is very similar to that of the dissipation rate at higher wavenumbers. This indicates that at higher wavenumbers, energy gain from nonlinear interactions is balanced by energy removal by dissipation, that is, the energy spectrum is (statistically) at a quasi equilib-

FIG. 3. The k h–k z spectra of (a) the total dissipation rate and (b) the nonlinear energy transfer rate for case 0. The dissipation rate is the value at t 5 2.81 3 10 4 s, and the energy transfer rate is the value averaged from t 5 2.53 3 10 4 s to t 5 2.81 3 10 4 s. Smoothing has been applied. The contour and color levels are logarithmic; the unit is arbitrary. (b) Cold (green to purple) and warm (green to orange) colors indicate negative and positive values, respectively. The labels on the color bars show the powers of 10. The vertical short dashed lines indicate the horizontal cutoff wavenumber (5 max k x [ max k y ), beyond which the number of resolved Fourier modes rapidly decreases.

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FIG. 4. Evolution of (a) domain-average energy and (b) dissipation rates for case 1. Energy and dissipation rates as in Fig. 2. These energy values do not include those of SF.

rium. Energy transfer toward higher horizontal wavenumber is particularly significant. When the SF is added, energy transfer is enhanced. Figure 4 shows the evolution of domain-average energy and dissipation rates for case 1. The energy level rapidly decreases and reaches the GM value at t ø 1.2 3 10 4 s. (Compare this with t ø 2.8 3 10 4 of case 0.) The dissipation rate then is about 3 3 10 210 m 2 s 23 , three times as large as that of case 0. The k h–k z spectra of the total dissipation rate and the nonlinear energy transfer rate (Fig. 5) again show that energy is transferred to higher wavenumbers by nonlinear interactions and removed by dissipation there, and that the energy spectrum is statistically at a quasi equilibrium. Energy transfer is thus enhanced over that of the case without SF. What is the reason of the enhancement? There seem to be direct effects of energy transfer due to SF, which

can enhance energy transfer toward higher vertical wavenumbers by the critical-layer absorption mechanism. In fact, energy transfer rate at the higher k z and lower k h portion of the wavenumber space is larger in case 1 than that in case 0 (cf. Fig. 5b with Fig. 3b). The k z energy spectrum is flatter in case 1 than that in case 0 (not shown). However, energy transfer toward higher horizontal wavenumber is also larger in case 1 than that in case 0. This cannot be accounted for by direct effects of SF only. As seen in Fig. 4a, kinetic energy rapidly increases during the first 10 3 s of the integration, and kinetic energy remains significantly larger than potential energy in the rest of the integration. This is not seen in case 0 (Fig. 2a). Also, kinetic energy transfer to the higher k h direction is significantly larger (not shown) than potential energy transfer, while the latter is larger than the former in case 0. If the motions consist entirely of linear internal waves modes, kinetic and potential

FIG. 5. The k h–k z spectra of (a) the total dissipation rate and (b) the nonlinear energy transfer rate for case 1. The dissipation rate is the value at t 5 1.12 3 10 4 s, and the energy transfer rate is the value averaged from t 5 1.12 3 10 4 s to t 5 1.41 3 10 4 s. For other information, see the caption to Fig. 3.

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energy should always be the same (remember that we have set the Coriolis parameter to zero). In fact, significant amount of vortical motion is present at higher wavenumbers in case 1, and this seems to be related to the enhanced energy transfer in the case with SF. This point will be taken up later. Last, the sign of vertical density flux has been controversial (Ramsden and Holloway 1992; Carnevale et al. 2001). The vertical density flux here is defined as the real part of rˆ (k)wˆ*(k), where the circumflex denotes Fourier transform. The density flux (not shown) is generally negative at higher k z and k h regions (with an exception dealt with shortly); that is, energy is converted from potential to kinetic energy both in cases 0 and 1. This is consistent with the fact that energy transfer rate is higher for potential energy than for kinetic energy. This result agrees with Ramsden and Holloway (1992) and Carnevale et al. (2001). The only exception is the higher-k h , lower-k z region of case 1 where, as mentioned earlier, the kinetic energy transfer is larger than the potential energy transfer, and energy conversion is accordingly from kinetic to potential (not shown). c. Bispectra To examine how energy is transferred with in the spectrum, we compute bispectra. We outline the definition and meaning of bispectra here, and we refer the interested reader to Furue (1998) for a more detailed discussion. Fourier-transforming the basic equations (1)–(3) and constructing the energy equations, we obtain

O O Y(k9, k0, k) 1 · · · , ] 1 g ˜ |rˆ (k)| 5 O O Q(k9, k0, k) 1 · · · , ]t 2 r N ] 1 |vˆ (k)| 2 5 ]t 2

k9

k0

k9

k0

(4)

2

2

2 0

2

(5)

where def

Y(k9, k0, k) 5 2R{[v(k9) · ik0][vˆ (k0) · vˆ*(k)]d (k9 1 k0 2 k)}, def

Q(k9, k0, k) 5 2

g2 ˜ 2 R{[vˆ (k9) · ik0]rˆ (k0)rˆ ∗(k)d (k9 1 k0 2 k)}. r N 2 0

The circumflex denotes coefficients of Fourier transform, R the real part of a complex number, and d(k) the Kronecker delta. The first terms on the right-hand sides of (4) and (5) come from the nonlinear terms of the basic equations, and they are what we have been calling the nonlinear transfer rates. The quantities Y(k9, k0, k) and Q(k9, k0, k), called bispectra, hence represent how the Fourier mode k gains or loses energy by the interaction with the modes k9 and k0. If we are to focus on a particular k, we examine Y and Q in the sixdimensional k9–k0 space. However, as indicated by the Kronecker delta in Y and Q, only the modes satisfying

FIG. 6. The (a) k9h –k9z , (b) k9h , and (c) k9z plots of the bispectrum (in an arbitrary unit) averaged for the chosen 50 modes and from t 5 2.25 3 10 4 s to t 5 2.81 3 10 4 s for case 0. (a) The contour and color levels are logarithmic; cold and warm colors indicate negative and positive values, as in Fig. 3, respectively. The labels on the color bars show the powers of 10. (b) and (c) The ordinate is logarithmic; positive and negative values are plotted on the upper and lower halves, respectively. (a) and (b) The vertical short dashed line indicates the horizontal cutoff wavenumber.

k 5 k9 1 k0 can directly interact. This allows us to present the bispectrum in the three-dimensional k9-space for the particular k we are focusing on with the understanding that the third mode interacting is the one with wavenumber k 2 k9. We first focus on the region around k h ø 1.0 m 21 , energy transfer to which is most conspicuous in both cases with and without SF. Since bispectra tend to be noisy, we average bispectra for many modes. We choose 50 modes such that k 5 [(17 1 l)Dk x , mDk y , nDk z ] with l 5 0 or def 61, m 5 0 or 61, and n 5 1, 2, . . . , 10, where Dk a 5 2p/L a for each a 5 x, y, z. The central horizontal wavenumber is (k x , k y ) 5 (17Dk x , 0) 5 (1.07 m 21 , 0). We average these 50 bispectra. Since the bispectrum so obtained is still noisy, we also average it in time. Figure 6a shows the k9h–k9z plot of the total bispectrum (i.e., Y 1 Q). There is a negative peak to the left of the modes we are focusing on and a positive peak to the right. This can be seen more clearly in the k9h plot

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shown). But there are not sharp peaks below or above the chosen modes; rather, the interaction is broad, that is, the chosen modes are receiving energy not only from adjacent modes just below but also from lower-k z modes, and they are passing part of that energy onto not only adjacent higher-k z modes but those far above. Another set of bispectra we have computed is for modes with high k h and high k z , namely, those with k 5 [lDk x , mDk y , nDk z ], where l 5 15, . . . , 18, m 5 21, . . . , 2, and n 5 68, . . . , 71. The bispectra for cases 0 and 1 both show energy transfer for higher k h and higher k z ; the tendency toward higher k z is stronger in case 1 than in case 0. The interaction is not very broad. d. Forced experiments (cases F0 and F1)

FIG. 7. Same as Fig. 6 but for case 1 and the averaging period is from t 5 1.12 3 10 4 s to t 5 1.69 3 10 4 s.

(Fig. 6b). This means that the chosen modes are receiving energy from adjacent modes to the left and passing part of that energy to those to the right. There is no such systematic transport in the k9z plot (Fig. 6c). The fact that the chosen modes (k) interact with adjacent modes (k9) implies that the wavenumbers of the third modes (k0) are small. Energy is thus transferred from a large-k h mode to its neighbor to the right with a third small-k0h mode acting as a kind of catalyst, which has much larger energy than the modes we are looking at and hence is little affected by this nonlinear interaction. The enhancement of vertical energy transfer in the case with SF can be also seen in the bispectrum (Fig. 7). In the k9h–k9z plot, the chosen modes are seen to pass energy to the adjacent modes above. This can be clearly seen in the k9z plot. However, similar horizontal transport to that seen in case 0 is also apparent in the k9h plot (Fig. 7b). The tendency for horizontal transport thus seems to be robust. We have computed bispectra for two more regions of the wavenumber space. One is the low-k h and high-k z region with wavenumbers k 5 [lDk x , mDk y , (70 1 n)Dk z ] where l 5 1, 2, . . . , 10, m 5 0 or 61, and n 5 0 or 61. The central vertical wavenumber is about 3.4 m 21 . Here vertical transfer toward higher vertical wavenumbers can be seen in both cases 0 and 1 (not

Although we have shown that the higher wavenumber region is likely to be statistically at equilibrium, the overall wavenumber space may not and the results may be sensitive to that. To explore this possibility, we conduct forced experiments, as described in the previous section. Figure 8 shows the evolution of energy and dissipation rate for the forced case without SF (case F0). The amplitude of forcing was adjusted by trial and error so that the final energy level attains the GM value (about 1.3 3 10 25 m 2 s 22 ). As can be seen from Fig. 8, the remaining trend is small both in the energy and the dissipation rate. The dissipation rate is about 1 3 10 210 m 2 s 23 , which is similar to the value in the corresponding free experiment. The k h–k z spectra of the dissipation rate and the energy transfer rate (Fig. 9) are very similar to those of the corresponding free experiment in pattern and magnitude. This indicates that the energy transfer found in the free experiment is likely to correspond to that of a statistical equilibrium. In the forced case with SF (Fig. 10), there also remains a small trend in the energy (notice the difference in the range of abscissa from that of Fig. 8). The dissipation rate is, however, almost steady. This implies that energy is (slowly) building up at lower wavenumbers. The final total dissipation rate is about 1 3 10 210 m 2 s 23 , which is much smaller than that in the free case, although the total energy level is similar. This seems to be because vertical energy transfer is smaller than that in the free case. This point can be confirmed in the k h– k z plot of energy transfer rate (Fig. 11b): energy transfer to the high-k z and low-k h region is lower here than in the free case (Fig. 5b). The shape of energy spectrum is also different. Figure 12 compares the total energy spectrum of case F1 with those of cases 1 and F0. The spectrum of case 1 is vertically more elongated than the other two, and that of case F1 more resembles that of case F0 than that of case 1. A plausible interpretation is that in the free experiment with SF, there were initially plenty of waves that tend to go to critical layers, but in the forced experiment, there remain not many such

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FIG. 8. Evolution of (a) domain-average energy and (b) dissipation rates for case F0. Energy and dissipation rates as in Fig. 2.

waves at the final stage because of the particular forcing spectrum we have chosen. So at the final state of case F1, the energy transfer is dominated by the horizontal transfer found in case 0. This suggests that we need more detailed information of the energy spectrum to determine how vertical energy transfer works in the presence of vertical shear. Our earlier discussion on the bump of the intertial range in the Froude spectrum is consistent with the above interpretation. We may need stronger forcing or some energy source at higher wavenumbers to obtain vigorous overturning which creates an inertial range and a high disspation rate. This may also account for the difference in dissipation rate between us and Carnevale et al. (2001), who forced the flow field at 20 m of wavelength by imposing an IW motion that has about the same shear as the corresponding GM band. Most of the shear in our simulations, on the other hand, is con-

tained by SF at this scale (Fig. 1) and the forcing is only strong enough to maintain the small energy of the WV field. We note that as in case 1, the kinetic energy is significantly larger than potential energy in case F1 and that the contribution of kinetic energy dissipation is larger than that of potential energy dissipation. All that we stated about the sign of vertical density flux for the unforced cases applies also to these forced cases: the density flux is negative (conversion from potential to kinetic energy) at high wavenumbers except for the high-k h , low-k z region of the forced case with SF (case F1). e. Vortical motions We have seen that in cases 1 and F1, kinetic energy is significantly larger than potential energy, that kinetic

FIG. 9. The k h–k z spectra of (a) the total dissipation rate and (b) the nonlinear energy transfer rate for case F0. The dissipation rate is the value at t 5 1.32 3 10 5 s, and the energy transfer rate is the value averaged from t 5 1.29 3 10 5 s to t 5 1.32 3 10 5 s. For other information, see the caption to Fig. 3.

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FIG. 10. Evolution of (a) domain-average energy and (b) dissipation rates for case F1. Energy and dissipation rates as in Fig. 2.

energy transfer to the higher k h direction is significantly larger than potential energy transfer, and that as a result, the contribution of kinetic energy dissipation is also larger than that of potential energy. This indicates that vortical motions are enhanced in the cases with SF. Figure 13 shows vertical-wavenumber energy spectra of the four experiments; the dashed line indicates the linear vortical mode energy and the dashed–dotted line, the IW energy. There is more vortical energy in the cases with SF than in those without. We have inspected the density fields in the physical space (not shown) of the four cases at the time when their energy levels are of the GM value. We notice that there are more instances of density inversion in the cases with SF than those without. A hypothesis is that SF enhances wave breaking, which creates vortical motions, and those vortical motions somehow enhances energy transfer. Certainly, there should be contributions from initial potentail vorticity. More detailed analyses will be future study.

4. Summary and concluding remarks We have conducted numerical experiments to examine energy transfer within the small-scale oceanic internal wave spectrum. The rate of energy transfer is 1–3 3 10 210 m 2 s 23 when the total energy level is that of the Garrett–Munk spectrum. This value is a significant fraction of the total energy transfer rate observed in the interior main thermocline (Gregg 1989). Energy cascade within this small-scale spectrum can therefore be important. Energy transfer to smaller horizontal scales is robust. This interaction is wavenumber-local because it works with or without the large-scale vertical shear. This transfer may supplement the estimation by Henyey et al. (1986), who considered only the scale-separated critical layer mechanism, which transfers energy to smaller vertical scales owing to vertical shear. With the vertical shear, energy transfer is indeed en-

FIG. 11. The k h–k z spectra of (a) the total dissipation rate and (b) the nonlinear energy transfer rate for case F1. The dissipation rate is the value at t 5 1.97 3 10 5 s, and the energy transfer rate is the value averaged from t 5 1.90 3 10 5 s to t 5 1.97 3 10 5 s. For other information, see the caption to Fig. 3.

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FIG. 12. The k h–k z total energy spectra for (a) case 1 at t 5 1.12 3 10 4 s, (b) case F0 at t 5 1.32 3 10 5 s, and (c) case F1 at t 5 1.97 3 10 5 s. Smoothing has been applied. The contour are logarithmic; the unit is arbitrary. The vertical dotted lines indicate the horizontal cutoff wavenumber.

hanced. One mechanism responsible for it is most likely critical layer absorption as in Henyey et al. (1986). The experiment with an ad hoc forcing suggests that the contribution of this mechanism strongly depends on how energy is supplied to this small-scale spectrum. On the other hand, energy transfer to smaller horizontal scales is also enhanced, which cannot be accounted for by the critical layer mechanism. The key seems to be the presence of vortical motions. With the vertical shear, vortical energy increases and at the same time kinetic energy transfer is enhanced to smaller horizontal scales. But the details are not clear. We only suggest that for these small scales, nonlinearity is strong and so we need to consider interactions between vortical motions and internal waves. From these results, we propose the following scenario. Energy at small wavenumbers is transferred to higher vertical wavenumbers via weakly nonlinear interactions. From there, part of the energy is transferred in the direction of smaller vertical scales by scale-separated interactions such as critical layer mechanism and then ultimately dissipated by wave breaking (Henyey et al. 1986). On the other hand, the other part of the energy is transferred in the direction of smaller horizontal scales first by scale-separated interactions. With increasing

wavenumber, however, wavenumber-local interactions become stronger and finally dominate the energy transfer. These local interactions transfer the energy to smaller horizontal and vertical scales and ultimately up to dissipation scales. In our simulations, vortical motions were generated from the given initial conditions within this small scale spectrum. However, in the real ocean, there may be other sources of vortical motions, such as the cascading from larger scales. If so, the energy transfer within this smallscale spectrum would be more important. Mu¨ller et al. (1988) found in observational data much more vortical energy than wave energy at small horizontal scales such as we have dealt with, although we do not know how universal this result is. We have relied on the Garrett–Munk spectrum, which may not accurately describe small-scale internal wave modes. We also lack information on the vortical mode spectrum. Our assumption that scale-separated interactions can be represented as the shearing of the smallerscale motions by the larger-scale ones (i.e., SF) is yet to be fully justified. To examine the energy transfer within this small-scale portion of the spectrum, we will, after all, need to construct a model that includes larger scales and that computes energy fluxes into the small-

FIG. 13. The k z spectra for cases (a) 0 at t 5 2.81 3 10 4 s, (b) 1 at t 5 1.41 3 10 4 s, (c) F0 at t 5 1.33 3 10 5 s, and (d) F1 at t 5 1.97 3 10 5 s. Solid 5 total energy, long dashed 5 vortical energy, and dashed–dotted 5 wave energy.

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scale region rather than imposing a fixed amount of energy as an initial condition. A three-dimensional numerical model of this type is not feasible with current computer resources. We may need to combine two- and three-dimensional models, the former for examining energy cascade from large to medium scales and the latter for examining that within small scales, where threedimensionality may not be ignored because of vortical motions. Acknowledgments. I would like to thank Dr. Nobuo Suginohara for his advice and patient encouragement and for stimulating and helpful discussions. Discussions with Dr. Toshiyuki Hibiya were enjoyable and invaluable. He also shared generously with me some of the unpublished results and data of the studies he carried out with Dr. Yoshihiro Niwa and Kayo Fujiwara. I am grateful to Dr. George F. Carnevale for helpful comments. Anonymous reviewers’ constructive comments were also helpful and led to significant improvements on the manuscript. Also thanks are due to Drs. Hiroyasu Hasumi, Masaki Kawabe, Yoshihisa Matsuda, and Yoshinobu Tsuji for fruitful discussions. Thanks are extended to Dr. Clive Temperton for providing me with some of his very fast FFT routines, and to Dr. Oaki Iida for helping me with numerical methods for turbulent flow simulations. I used the Hitachi S3800 and SR8000 supercomputers at the Computer Centre of the University of Tokyo for the numerical experiments. The figures were produced with the Dennou Library, developed by the GFD-Dennou Club (information available online at http://www.gfd-dennou.org/). APPENDIX Implementing the GM Spectrum This appendix describes how we initialize the flow field of the numerical model. We make motions with k z 5 0 or k 5 0 vanishing. a. Transforming the spectrum into the (kx , ky , kz ) space We use Munk’s (1981) version of the GM spectrum. The energy spectrum is ˜ 0 )b 2 N 20 EB(v)H( j), F e (v, j) 5 (N/N where v ∈ [ f , N˜ ] is the frequency, j 5 1, 2, . . . , ` the vertical mode number, and N˜ the Brunt–Va¨isa¨la¨ frequency; and b 5 1300 (m), N 0 5 3 cph ø 5.2 3 10 23 s 21 , and E 5 6.3 3 10 25 (dimensionless) are fixed parameters. The functions B(v) and H( j) are defined as def

B(v) 5

2 f , p v (v 2 2 f 2 )1/2

def

H( j) 5

2 21 ( j 2 1 j*) , S

where j* 5 3 (dimensionless) is also a fixed parameter, def ` and S 5 Sj51 ( j 2 1 j*2 ) 21 ø 0.468. Although in our

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model runs we set f 5 0, we use f 5 7.33 3 10 25 s 21, the value of the Coriolis parameter at about 308N, for the GM spectrum to initialize the flow field. We transform this spectrum, written in the (v, j) space, into the (k x , k y , k z ) space as follows. The vertical wavenumber is written as

1

2

˜ 2 2 v 2 1/2 ˜ p N p N j5 j 5 Cj, (A1) 2 b N0 2 v b N0 def where C 5 (p/b)(N˜/N 0 ) (Munk 1981). The second equality is in general an approximation for v K N˜, but is exact for us because we have chosen N˜ 5 N 0 . The dispersion relation is kz 5

v (k h , k z ) 5

1

˜ 2 1 k z2 f 2 k h2 N k h2 1 k z2

2

1/2

.

(A2)

The spectrum written in terms of k h and k z is related to the original spectrum by Fe (k h , k z ) 5

](v, j) F (v, j), ](k h , k z ) e

where the factor before F e (v, j) is the Jacobian of the transformation, which can be computed from (A1) and (A2). The result is ˜ 2Cb 2 N02 E f (N ˜ 2 2 f 2 )1/2 k z2 N 1 Fe (k h , k z ) 5 2 ˜ 2 N0 pS k k h N 1 k z2 f 2 1 . k 1 (Cj*) 2 Since v 5 f and v 5 N˜ respectively correspond to k h 5 0 and k h 5 ` for each k z (A2), the domain of definition is k h ∈ [0, `) and k z ∈ [0.5C, `). (The ‘‘0.5C’’ is because ` of S j5` j51 ø # 0.5C dk z /C.) The three-dimensional spectrum is obtained by using 3

2 z

Fe (k x , k y , k z ) 5

1 Fe (k h , k z ) , 2 2pk h

where k x ∈ (2`, `), k y ∈ (2`, `), and | k z | ∈ [0.5C, `). Notice the factor 1/2, which is introduced here because the domain of independent variables is doubled to include negative vertical wavenumbers. b. Initializing the numerical model We initialize the flow field of our model with internal wave modes, ignoring vortical modes. Since the computational domain is finite, only Fourier modes with wavenumber (k x , k y , k z ) 5 (Dk x n x , Dk y n y , Dk z n z ) are def permissible, where n’s are integers and Dk a 5 2p/L a with a 5 x, y, z. 1) k h ± 0 and k z ± 0 def ˆ5 We initialize the Fourier coefficients X [uˆ (k), yˆ (k), wˆ(k), rˆ (k)] with k h ± 0 and k z ± 0 so that they satisfy the following conditions at t 5 0:

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ˆ 5 X ˆ1 1 X ˆ 2 , where X ˆ 1 satisfies the polarization • X relation of the linear internal wave traveling in the ˆ 2 , that of the internal wave travdirection of k, and X eling in the opposite direction; ˆ \) be given • for each k, the energy per unit mass (\X by the GM spectrum; ˆ 1 and X ˆ 2 be chosen at • the energy partition between X random; ˆ 1 and X ˆ 2 be chosen at random; and • the phases of X • different wave modes have no phase correlation. Although we use the value of the Coriolis parameter at 308N for the GM spectrum, we set f 5 0 in the polarization relation to match the governing equations of the numerical model. 2) SHEAR

FLOW

(k h 5 0 and k z ± 0)

We average the energy of the GM spectrum over the circle centered at k h 5 0 with radius h: 1 ph 2 5

E

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h

dk h Fe (k h , k z )

0

[

]

˜ 2Cb 2 N02 E ˜ 2 2 f 2 )1/2 h 1 N 1 (N arctan . ph 2 N0 pS k z2 1 (Cj*) 2 f (h 2 1 k z2 )1/2

We set h 5 [(Dk x /2) 2 1 (Dk y /2) 2 ]1/2 , and assign this average to the Fourier coefficients of the model variables at t 5 0: 1 ˆ \X(k x 5 0, k y 5 0, k z )\ 2 2 5 (Dk x Dk y Dk z )

1 1 2 ph 2

E

h

dk h Fe (k h , k z ).

(A3)

0

Notice the factor 1/2; this is because F e (k h , k z ) is the spectrum defined for k z . 0. This procedure tends somewhat to overestimate the energy for k h 5 0 because averaging the spectrum over the circle and multiplying the result by the area Dk yDk x gives larger energy than integrating the spectrum over (2Dk x /2, Dk x /2) 3 (2Dk y /2, Dk y /2), which may be more appropriate. (Notice that the smaller the radius h, the larger the average becomes.) To determine each of the Fourier coefficients uˆ, yˆ , wˆ, and rˆ , we use the polarization relations. Although all the modes (two wave modes and one vortical mode) are degenerate for k h 5 0 when f 5 0, we can identify the ˆ 5 (U, V, 0, 0) with the internal wave modes modes X ˆ and X 5 (0, 0, 0, R) with the vortical mode, comparing with those modes for f ± 0. Accordingly, we assign the energy (A3) only to the horizontal velocity components. (Hence, motions with k h 5 0 are a pure horizontal flow.) For each k z , we randomly choose the direction of the flow and also the phase angles of U and V.

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