## 1. Introduction

Oceanic internal waves are present throughout the stratified ocean, spanning horizontal scales less than about 50 km and vertical scales to about 10 m. The best known forcing mechanisms are surface winds (D’Asaro 1985) and tidal flows interacting with large-scale topography (Sjöberg and Stigebrandt 1992). These processes excite internal waves at relatively large spatial scales and near-inertial and tidal frequencies. Though other mechanisms may also be important, for example direct coupling of internal and surface waves, wave generation in mesoscale rings and fronts (Kunze and Sanford 1984), and interactions with small-scale topography, their magnitude and length and time scales are less well known. Away from direct forcing, the oceanic internal wavefield appears to be remarkably uniform and reasonably well described by the GM model spectrum (Garrett and Munk 1975, 1979), which quantifies the observed distribution of wave energy in (separable) wavenumber and frequency.

Though the amount of energy in the oceanic internal wavefield is small compared to that in the large-scale circulation, internal waves play an important role in the thermodynamics of the ocean. Internal waves propagate spatially and exchange energy with other waves through nonlinear wave interactions. These interactions result in a transfer of energy from large to small vertical scales, and eventually to dissipation as shown schematically in Fig. 1. The qualitative universality of the oceanic internal wavefield suggests that a balance is maintained between forcing at large scales and dissipation at small scales, with nonlinear wave interactions providing an important mechanism for scale transformation. Because dissipation is intimately related to diffusive mixing (Gregg 1987), this implies a balance between wave energy transfers and water mass modification.

Assuming that energy transfer is primarily toward high vertical wavenumbers, energy transfer estimates at nondissipative scales can be used to make predictions of small-scale dissipation rates. Using a weakly nonlinear, resonant interaction theory, McComas and Müller (1981) were able to estimate the rate of energy transfer out of the energy-containing scales in GM wavefields. At smaller vertical scales, where the nonlinearities are thought to be significantly stronger, Henyey et al. (1986) used ray-tracing in a GM background field to estimate the transfer of energy to scales below 5 m. Following Dillon (1982), Kunze et al. (1990) obtained reasonable estimates of the dissipation rate from measurements of 1- to 5-m shear. The approximate location of the wavenumbers across which these energy transfer rates were estimated are shown in Fig. 1.

Here we focus on the dynamics of the larger-scale, energetic internal waves in regions away from direct forcing. We assume that energy is supplied to such regions via the propagation of low-mode internal waves and that the wave spectrum is near equilibrium and well approximated by the GM model. In these regions, the dissipation rate ε should be a function of the GM parameters, in particular the internal wave energy density *Ē.* Observational data (Gregg 1989) suggest that ε ∝ *Ē*^{2}. In this paper, we study nonlinear transfers in GM internal wavefields numerically. Using a three-dimensional nonhydrostatic model that resolves both the energetic low modes as well as a substantial portion of the smaller-scale wave spectrum, we run a series of unforced initial value problems varying the wave energy density *Ē.* The simulations yield quantitative predictions of ε, which are then compared to theory (McComas and Müller 1981; Henyey et al. 1986) and to microstructure measurements taken at six diverse sites in the midlatitude open ocean thermocline (Gregg 1989). Our results demonstrate that, for nearly GM wavefields, quantitatively useful estimates of the dissipation rate can be obtained via direct simulations of internal waves.

In adopting this approach, we make relatively few explicit simplifying approximations. We do not, for example, require that nonlinearities be weak nor necessarily resonant; wave phases are not constrained to be random and vortical modes are permissible solutions to the model equations. The numerical model itself is quite flexible: arbitrary wave spectra and stratification profiles can be specified, the domain can be deep or shallow, and the latitude is input as an arbitrary parameter.

Of course, no approach is entirely satisfactory. In formulating a physically rich model, we have necessarily sacrificed spatial bandwidth. Limiting the bandwith requires that energy be dissipated numerically at scales much larger than the true dissipation scales. It must be justified a posteriori that the available bandwidth is sufficient to adequately represent the dominant wave interactions and make meaningful predictions of ε.

The remainder of the paper is organized as follows. The numerical model, its parameter values, and the initialization procedure are given in section 2. A brief overview of the simulations is given in section 3. Predictions of the dissipation rate and comparison of the predictions with theory and observations are presented in section 4, followed by a discussion of the results in section 5.

## 2. The internal wave model

### a. Model equations

**u**is the velocity vector,

*ρ*′ is the density perturbation from the ambient profile

*ρ̄*(

*z*)

*ρ*

_{0}is the constant reference density,

*g*is the gravitational acceleration, and

**z**is the unit vector in the vertical (positive upward);

The model was used to simulate the flow in an exponentially stratified ocean 2000 m deep over a horizontally periodic domain of size 10 km × 80 km. Stretching the domain permits the inclusion of waves with horizontal scales greater than 10 km, increasing the computational bandwidth considerably. Though domain stretching imposes a preferred direction on the largest-scale modes, choosing a uniform grid spacing in each horizontal direction allows horizontal isotropy to be maintained at scales less than 10 km. This corresponds to an ocean where the large-scale waves are often anisotropic while smaller scales are isotropic.

Stress-free, rigid-lid boundary conditions were applied at both the upper and lower surfaces. Standard pseudospectral methods (e.g., see Canuto et al. 1988) are used to compute spatial derivatives. The discrete forms of the equations are integrated forward in time using a second-order Adams–Bashforth method. All simulations discussed here are unforced; the internal waves are allowed to decay in time. The model parameters are listed in Table 1.

#### 1) Numerical dissipation

Because the numerical model does not resolve the dissipative scales, energy must be removed from the simulations in an ad-hoc manner. In prescribing the dissipation operator

*γ*was chosen by trial and error to minimize the range of damped scales while maintaining numerical stability. The value used (see Table 1) results in the damping of waves with horizontal scales less than about 1 km and/or vertical mode numbers greater than about 40. Larger-scale waves are governed by the inviscid equations of motion. As shown in section 3, the nonlinear dynamics of the undamped waves determine the rate of downscale energy transfer and thus control the rate at which energy is removed from the calculations via

In using this approach, we make some implicit assumptions. First, we assume that upscale energy transfers from the numerically dissipated scales are negligible. Though this assumption cannot be rigorously justified a priori, it seems reasonable if the schematic in Fig. 1 provides a first-order description of the scale transformation processes in the ocean. We further assume that numerically damped waves dissipate locally, that is, without propagating large distances. This approximation seems reasonable because internal wave group velocities are small and interaction times are short at the scales at which numerical dissipation occurs (D’Asaro 1991). To some extent, though clearly not rigorously, both of these assumptions can be justified a posteriori by demonstrating that the simulations yield predictions of the small-scale dissipation rate ε in good agreement with oceanic observations.

The utility of closure schemes based on hyperviscosity has been demonstrated for other flows where similar assumptions are justified. For example, Borue and Orszag (1995a, 1995b) show that for both forced and decaying homogeneous turbulence, the exact manner in which energy is extracted does not appreciably influence the dynamics at inviscid scales and that dissipation rates are determined by the inviscid energy transfers.

### b. Initialization

*ρ̄*/d

*z*

*L*

_{z}, and

*b*are listed in Table 1. The buoyancy frequency

*z*=

*L*

_{z}and decays exponentially with depth.

#### 1) Wave modes

*k*and

*l*and vertical mode number

*j.*Here

*G*

_{+}and

*G*

_{−}are amplitude coefficients to be specified. We obtain the eigenfunctions

*ζ̂*

_{jkl}(

*z*)

*ω*

_{jkl}by solving

*ζ̂*

_{jkl}

*ω*

_{jkl}, determine the velocity and density structure of the discrete wave modes up to an arbitrary amplitude.

#### 2) Wave amplitudes

*α*=

*k*

^{2}+

*l*

^{2}

*j,*is given in Flatté et al. (1979). Following Flatté et al., the GM76 energy density is

*G*

^{2}

*α*

*j*

*EH*

*j*

*B*

*α*

*j*

^{−1}

*E*is specified in joules per square meter and

*G*(

*k, l, j*), we construct complex zero-mean random deviates

*G*

_{+}and

*G*

_{−}such that

*G*

_{+}and

*G*

_{−}separately allows the amplitudes of left and right traveling waves at a given scale to be specified independently.

*M*= 100 vertical modes gives

Here *ζ̂*_{jkl}(*z*)/d*z**k* = 2*π*/*L*_{x}{0, 1, . . . , *nx*/2} and for *l* = 2*π*/*L*_{y}{−*ny*/2 + 1, −*ny*/2 + 2, . . . , *ny*/2 − 1, *ny*/2}. Because all fields are real, modes with negative *k* wavenumbers are not prescribed independently but are related to the positive *k* modes by conjugate symmetry.

The degenerate case *k* = *l* = 0, *j* = 1, 2, . . . , *M* is treated separately. These computational modes are used to represent near inertial waves at horizontal scales larger than *L*_{y} = 80 km. We model these large-scale waves as inertially rotating, horizontal shear currents. The vertical structure of these modes was obtained by numerically solving the linearized equations of motion in the limit *k, l* → 0. The amplitudes of these modes were prescribed by integrating the GM horizontal kinetic energy spectrum over the wavenumber range −*π*/*L*_{x} to *π*/*L*_{x} and from −*π*/*L*_{y} to *π*/*L*_{y}.

Initial conditions for the numerical simulations were calculated by setting *t* = 0 and inverse Fourier transforming (18)–(22). Thus, we represent the midlatitude open ocean internal wavefield as a finite sum of *nx* × *ny* × *M* ≈ 10^{6} horizontally periodic, discrete internal wave modes. Both left and right going waves are specified; each mode satisfies stress-free rigid-lid boundary conditions and is in balance with the stratification profile *N*(*z*). Amplitudes and phases are prescribed randomly, with the expected value of the energy density matching the GM internal wave model.

*z*), the distribution of energy with depth, for the random realization used to initialize simulation I (see Table 2). For comparison, the profile obtained using the GM expected values for the wave amplitudes rather than the random deviates is also shown. The ratio

*E*/

*E*

_{GM}is a measure of the energy density of the simulated fields relative to a GM wavefield. Here

_{xy}is the average over horizontal area

*A*=

*L*

_{x}×

*L*

_{y}, and

*E*

_{GM}is the energy density obtained using the expected values rather than the random deviates for the wave amplitudes. For simulation I,

*E*/

*E*

_{GM}= 1.017 at

*t*= 0.

#### 3) Discretization in modenumber-frequency space

The initialization procedure described in the previous section is based on a uniform discretization of horizontal wavenumber space (*k, l*) with sampling intervals (Δ*k,* Δ*l*) = (2*π*/*L*_{x}, 2*π*/*L*_{y}). Because the internal wave dispersion relation is nonlinear with respect to wavenumber, this discretization leads to a nonuniform sampling of (*j,* *ω*) space. Figure 3 shows the eigenfrequencies determined by numerical solution of (8) for the discrete values *l* = (2*π*/*L*_{y}) {0, 1, 2, · · · , *ny*/2} for *k* = 0. In this figure, the spacing between modes is a measure of how finely the continuous dispersion curves are sampled using the discrete set of numerically resolvable modes. At the smallest resolved scales, both horizontal and vertical, the discretization is quite dense. Relatively few modes, however, are used to represent the largest scales, particularly at high frequencies.

## 3. Simulations

Initial conditions for simulation I were obtained via the procedures described in the section 2. The dynamic evolution of the initial wavefield was modeled by numerically integrating Eqs. (1)–(3) in time. Simulation I was run for eight inertial periods using discrete time steps of 61.1 s. Parallelization of the algorithm was exploited and the simulations were made on a Connection Machine CM-5 with 256 processors and 2 Gbytes of memory. The computational workload for this model is approximately one-half that of a 128^{3} DNS simulation of density-stratified turbulence.

*Ē,*

*V*=

*L*

_{x}×

*L*

_{y}×

*L*

_{z}. The first ¼ inertial period is an initial transient characterized by a rapid decay of the total energy density

*Ē.*The rapid decay at startup results from impulsively applying the dissipation model, which selectively damps waves at small scales, to the initial wave fields, which are prescribed over all computational scales. Waves at horizontal scales less than about 1 km or with vertical modenumbers greater than about 40 are subject to immediate damping via

After about *t* = ½ (all times in inertial periods), the decay rate decreases substantially and by about *t* = 1, we judge the dissipation rate to be insensitive to the details of the damping scheme. After this time, only energy that has undergone scale transformation via nonlinear wave interactions is available for removal by the dissipation operator *t* ≤ 5, well after the decay of initial transients. Within this analysis interval, the dissipation rate is nearly constant, as evidenced by the approximately linear decay in total energy (see Fig. 4).

Figure 5 shows *Ê*(*z*) for simulation I at time *t* = 4, the midpoint of the analysis interval. At this time, the ratio *E*/*E*_{GM} is 0.81, indicating that nearly 20% of the initial energy in the simulation has been dissipated. For reference, the energy density of the undamped GM wavefield (as shown in Fig. 2) is also included. These reference data are also shown after scaling by 0.81. Note that the *N* scaling of the GM spectrum is preserved.

To determine the sensitivity of wave energy transfers to the overall energy level of the wavefield, a series of additional simulations were performed. The velocity and perturbation density fields from simulation I were extracted at *t* = 2.8 and scaled in amplitude by 1/*t* = 2.8. These simulations are designated II, III, and IV, as indicated in Table 2.

Figure 6 shows the energy density as a function of time for simulations I–III. To facilitate comparison, the energy levels resulting from these simulations have been plotted in Fig. 6, rescaled to have the energy level of simulation I. For example, simulation III gives the time history for a wavefield twice as energetic as that in simulation I. The results of simulation III have thus been scaled by a factor of ½ before plotting in Fig. 6. The decay rate for simulation III is greater than that for simulation I. The very low amplitude simulation IV maintains a nearly constant energy level and is not shown in the figure. As expected, the rate of downscale energy transfer increases with increasing energy density of the internal wavefield. A quantitative assessment of the dissipation rates and comparison with theoretical predictions and ocean observations is given in section 4.

## 4. Dissipation rates

### a. Model predictions

*j*greater than about 40. Assuming a constant energy flux to small scales, the calculated values of

For the analysis interval 3 ≤ *t* ≤ 5, the average values of energy density *Ē* and dissipation rate *Ē* = *O*(*Ē*_{GM}), are in good agreement with ε ∝ *Ē*^{2}. In simulation IV, where *Ē* ≪ *Ē*_{GM}, the predicted value is well in excess of the theoretical predictions. In contrast to the *O*(*Ē*_{GM}) simulations, where nonlinearities clearly control the dissipation rates, simulation IV may be unduly influenced by the numerical damping scheme.

### b. Comparison with theory

*Ē,*the internal wave energy density in joules per kilogram, using the parameter values listed in Table 1. Depth-integrated values, normalized by the scale depth

*b,*are shown in Fig. 7.

_{HWF86}are also shown in Fig. 7. For the parameter values in Table 1, ε

_{MM81}is greater than ε

_{HWF86}by a factor of ∼7.5. For

*Ē*=

*O*(

*Ē*

_{GM}), the direct simulation values fall within these two theoretical predictions.

### c. Comparison with observations

*S*

_{10}is the observed shear for waves with vertical wavelengths greater than 10 m and

*S*

_{GM}is the corresponding value for the GM spectrum. We cannot make a direct comparison with the Gregg (1989) scaling because we do not resolve waves to 10-m scale. Gregg (1989), however, found that ε

_{G89}predicts dissipation rates greater then ε

_{HWF86}by a factor of 2 and less than ε

_{MM81}by a factor of 3. The values 2ε

_{HWF86}≤ ε ≤ (⅓)ε

_{MM81}are shaded in Fig. 7 and denoted ε

_{G89}. Figure 7 shows that the magnitudes of the dissipation rates inferred from direct simulation are in reasonable agreement with oceanic observations. It should be noted that the internal wavefields simulated in this study have different GM parameters than those observed by Gregg (1989). In particular,

*j*∗ and

*f*are larger and the total depth smaller in the simulations.

### d. Depth dependence

*N*(

*z*) in Fig. 8. The model values

*N*

^{β}, with

*β*equal to 1.91, 1.90, and 1.90 for simulations I, II, and III respectively. The best-fit slopes are in reasonable agreement with the dynamical predictions of McComas and Müller (1981) and Henyey et al. (1986) and the Gregg (1989) data. The simulation results show poorer agreement with the kinematic scalings of Gargett and Holloway (1984) and Munk (1981), which predict values of 1, −1.5, and 1.5 respectively.

## 5. Discussion

Direct simulation of three-dimensional internal waves in a nonuniformly stratified, rotating ocean can be performed over a limited range of spatial scales. The calculations reported here simulate the inviscid dynamics of internal waves over scales from about 1 to 80 km with vertical modenumbers less than about 40. This computational bandwidth is accessible at moderate computational effort, roughly half that required for 128^{3} simulations of stratified turbulence. Direct calculation of the internal wave energy transfers yields quantitatively useful predictions of the turbulent kinetic energy dissipation rate ε, as determined by comparison with the McComas and Müller (1981) and Henyey et al. (1986) theoretical predictions and the Gregg (1989) observations. These results demonstrate good agreement between the energy flux attributable to nonlinear internal wave interactions at energy containing scales in the GM spectrum and observed rates of turbulent kinetic energy dissipation in the midlatitude open ocean. These results do not bear directly on the dynamics of significantly non-GM wavefields or those near sources or topography.

The apparent link between the nonlinear dynamics of the larger-scale internal waves and ε suggests that it may be possible to construct basin- or global-scale models of the internal wavefield and its associated dissipation and mixing. Such a model could be based on the generation, propagation, and the nonlinear decay of low-mode low-frequency internal waves. Direct simulations at scales and resolutions comparable to those discussed here could lead to a useful parameterization of the nonlinear decay.

## Acknowledgments

We appreciate the many discussions with Mike Gregg, Frank Henyey, and Ren Chieh Lien. This work was supported by the Office of Naval Research (N00014-92-J-1180). Computer resources were provided by the Naval Research Laboratory Center for Computational Science.

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Model parameters.

Simulation descriptions. Time intervals are given in inertial periods. The fields from simulation I were rescaled to initialize simulations II, III, and IV. The scale factors listed give the relative change in energy. The raw velocity and density perturbation fields were scaled by the square root of these factors. 〈*E*〉/*E*_{GM} is the ratio of the average energy density during the analysis interval to the GM value.