### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Numerical Model
- 3. Results and Discussions
- 4. Concluding Remarks
- References

[1] In order to examine how the energy supplied by M_{2} internal tides cascades through the local internal wave spectrum down to dissipation scales, two sets of numerical experiments are carried out where the Garrett-Munk-like quasi-stationary internal wave spectra at 49°N (experiment I) and 28°N (experiment II), respectively, are first reproduced and then perturbed instantaneously in the form of an energy spike at the lowest vertical wave number and M_{2} tidal frequency. These experiments attempt to simulate the nonlinear energy transfer within the quasi-stationary internal wave fields near the Aleutian Ridge and the Hawaiian Ridge, respectively, both of which are generation regions of large-amplitude M_{2} internal tides. In experiment I, the energy spike stays at the lowest wave number, where it is embedded and the spectrum remains quasi-stationary after the energy spike is injected. In experiment II, in contrast, the energy level at high horizontal and vertical wave numbers rapidly increases after the injection of the energy spike, exhibiting strong correlation with the enhancement of high vertical wave number, near-inertial current shear. This implies that as the high vertical wave number, near-inertial current shear is intensified, high horizontal wave number internal waves are efficiently Doppler shifted so that the vertical wave number rapidly increases and enhanced turbulent dissipation takes place. The elevated spectral density in the high vertical wave number, near-inertial frequency band, which plays the key role in cascading energy to dissipation scales, is thought to be caused by parametric subharmonic instability. In experiment I, in contrast, the M_{2} tidal frequency is 1.2 times the inertial frequency at 49°N so that M_{2} internal tide is free from parametric subharmonic instability. Accordingly, even though significant M_{2} internal tidal energy may be generated, it is not available to support local deep water mixing.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Numerical Model
- 3. Results and Discussions
- 4. Concluding Remarks
- References

[2] The pattern and magnitude of the numerically reproduced abyssal oceanic general circulation strongly depend on the value of the vertical eddy diffusivity coefficient [*Bryan*, 1987; *Zhang et al.*, 1999; *Tsujino et al.*, 2000], a measure of turbulent vertical mixing at depth. Turbulent vertical mixing in the stratified ocean interior is considered to be associated with sporadic overturning and breaking of internal waves.

[3] The energy for diapycnal mixing processes in the deep ocean is originally supplied by wind stress fluctuations [*Price*, 1983; *Gill*, 1984; *Greatbatch*, 1984; *D'Asaro*, 1985, 1995; *Kundu*, 1993; *D'Asaro et al.*, 1995; *Nilsson*, 1995; *Niwa and Hibiya*, 1997, 1999; *Hibiya et al.*, 1999; *Nagasawa et al.*, 2000; *Watanabe and Hibiya*, 2002] and tide-topography interactions [*Bell*, 1975; *Baines*, 1982; *Hibiya*, 1986, 1988, 1990; *Matsuura and Hibiya*, 1990; *Sjöberg and Stigebrandt*, 1992; *Morozov*, 1995; *Ray and Mitchum*, 1996; *Cummins and Oey*, 1997; *Kantha and Tierney*, 1997; *Polzin et al.*, 1997; *Xing and Davies*, 1998; *Holloway and Merrifield*, 1999; *Merrifield et al.*, 2001; *Niwa and Hibiya*, 2001a, 2001b]. This energy is then transferred across the local internal wave spectrum down to dissipation scales by nonlinear interactions among internal waves, although such an energy cascade process is not required for mixing directly induced by breaking internal lee waves resulting from tidal interaction with small-scale bottom roughness [*Bell*, 1975; *Polzin et al.*, 1997]. This implies that, in addition to the spatial distributions of tidally generated internal waves [*Sjöberg and Stigebrandt*, 1992; *Kantha and Tierney*, 1997; *Niwa and Hibiya*, 2001a, 2001b] and wind-induced near-inertial internal waves [*Hibiya et al.*, 1999; *Nagasawa et al.*, 2000; *Watanabe and Hibiya*, 2002], nonlinear energy transfer processes within the internal wave spectrum should be examined before the global distribution of diapycnal mixing rates in the deep ocean can be clarified.

[4] Using a vertically two-dimensional, high-resolution numerical model, *Hibiya et al.* [1998] examined the energy transfer processes within the dynamically reproduced Garrett-Munk-like internal wave spectrum obtained by *Hibiya et al.* [1996] (for the universal Garrett-Munk empirical spectrum, see *Garrett and Munk* [1972, 1975] and *Munk* [1981]). However, horizontal size of the employed numerical model was limited to 10.24 km, an order of magnitude smaller than the typical horizontal scale of the realistic forcing in the ocean interior, namely, low vertical mode internal waves propagating from the direct forcing area [*Nagasawa et al.*, 2000; *Niwa and Hibiya*, 2001a, 2001b].

[5] In the present study, the range of horizontal scales resolved by the numerical model is increased up to about 10 times that in the model of *Hibiya et al.* [1998]. This allows us to see the nonlinear energy transfer across a more realistic spectral range, from generation down to dissipation scales. In particular, we present results from two sets of numerical experiments where the Garrett-Munk-like quasi-stationary internal wave spectra at latitudes 28°N and 49°N, respectively, are first reproduced and then perturbed instantaneously in the form of an energy spike at the lowest vertical wave number and M_{2} tidal frequency. These numerical experiments attempt to simulate the nonlinear energy transfer within the quasi-stationary internal wave fields near the Hawaiian Ridge and the Aleutian Ridge, respectively, both of which are known to be generation regions of large-amplitude M_{2} internal tides in the North Pacific [*Niwa and Hibiya*, 2001a, 2001b]. The evolution of each internal wave spectrum after the injection of the energy spike is monitored to see how the energy thus supplied at large scales cascades through the local wave spectrum down to dissipation scales.

### 2. Numerical Model

- Top of page
- Abstract
- 1. Introduction
- 2. Numerical Model
- 3. Results and Discussions
- 4. Concluding Remarks
- References

[6] In order to model at adequate resolution the disparate scale wave interaction, we restrict our attention to vertically two-dimensional wave motions by requiring the variability to be independent of one horizontal coordinate. The two-dimensional Navier-Stokes equations under the Boussinesq approximation are integrated with a finite difference scheme by applying the centered difference and leapfrog scheme. In particular, the Arakawa Jacobian is used for an expression of the advective term [*Arakawa*, 1966] and the subgrid diffusive-dissipative processes are parameterized with a Laplacian operator.

[7] In the present study, two sets of “spike experiments” are carried out. We assume the inertial frequency at 49°N, *f* = 1.14 × 10^{−4} s^{−1} (local inertial period T_{i} = 15.91 hours) in experiment I and the inertial frequency at 28°N, *f* = 6.76 × 10^{−5} s^{−1} (local inertial period T_{i} = 25.82 hours) in experiment II. A constant background buoyancy frequency N = 5.24 × 10^{−3} s^{−1} (buoyancy period T_{b} = 20 min) is assumed in both experiments.

[8] Cyclic boundary conditions are used at the lateral sides of the numerical model, and perfectly reflecting flat bottom and surface are employed. The vertical size of the numerical model is assumed to be 1.3 km for both experiments I and II, and the horizontal size of the numerical model is set to 155 km for experiment I and 110.5 km for experiment II which coincide with one horizontal wavelength of the first vertical mode M_{2} internal tide at 49°N and 28°N, respectively. 8192 and 1024 grid points are used in the horizontal and vertical directions, respectively. Both the background eddy viscosity and diffusivity coefficients are assumed to have a value of 10^{−4} m^{2} s^{−1} in the horizontal direction and 10^{−5} m^{2} s^{−1} in the vertical direction. These are introduced to avoid the pile up of energy at the smallest resolved wave numbers and hence maintain the stability of calculations. Note that these are the smallest possible values determined by trial and error so as to maximize a range of scales in which nonlinearities can be properly represented without being affected by diffusive-dissipative processes.

[9] First, in order to reproduce the quasi-stationary internal wave spectrum at each latitude, we calculate the nonlinear interactions over 10 inertial periods among randomly phased linear internal waves summing over horizontal modes 0–1024 and vertical modes 1–512, the amplitude of each being determined from the Garrett-Munk empirical model. Each quasi-stationary spectrum thus reproduced is then perturbed instantaneously with forcing in the form of an energy spike at the lowest vertical wave number and M_{2} tidal frequency. This experiment simulates the 10-m amplitude, first vertical mode M_{2} internal tide incident on the Garrett-Munk-like quasi-stationary internal wave field which is feasible near the Aleutian Ridge as well as the Hawaiian Ridge [*Niwa and Hibiya*, 2001a, 2001b]. The model is run for 10 inertial periods after the energy spike is injected, using discrete time step of 4 s. In order to avoid numerical instability, the Euler backward scheme is applied every 20 time steps. The outline of the numerical experiments is schematically shown in Figure 1.

### 3. Results and Discussions

- Top of page
- Abstract
- 1. Introduction
- 2. Numerical Model
- 3. Results and Discussions
- 4. Concluding Remarks
- References

[10] With the start of nonlinear interactions among internal waves, the structure of internal wave spectrum begins to be modified. Figure 2 shows time variations of the vertical wave number spectrum of the square of the vertical shear of horizontal current velocity at 49°N normalized by the square of the mean buoyancy frequency (Froude spectrum). It should be noted that each vertical wave number Froude spectrum is an average of the realizations calculated at the 8192 horizontal grid points using data from the top down to the bottom. After about seven inertial periods from the start of calculation, the spectrum becomes quasi-stationary where a change in spectral slope is formed at vertical wave number ∼0.03 cpm, although this “roll-off” wave number is obviously lower than actually observed in the real ocean [*Gregg et al.*, 1993].

[11] In experiment I, the quasi-stationary spectrum at 49°N thus reproduced is perturbed instantaneously at t = 10T_{i} with forcing in the form of an energy spike simulating the 10-m amplitude, first vertical mode M_{2} internal tide. Figure 3 shows time variations of the vertical wave number Froude spectrum after the energy spike is injected. We can see that the energy spike stays at the lowest wave number where it is embedded and the spectrum remains quasi-stationary over 10 inertial periods after the energy spike is injected. This is more directly confirmed in Figure 4 where the ratio of the perturbed spectrum to the unperturbed, freely decaying, reference spectrum is shown in the two-dimensional wave number space for t = 10T_{i}–20T_{i}. No significant increase of spectral intensity can be found in the whole two-dimensional wave number domain. At 49°N, therefore, the energy injected at lowest vertical wave number and M_{2} tidal frequency is not efficiently transferred across the local internal wave spectrum down to dissipation scales.

[12] Figure 5 shows time evolution of the internal wave field for experiment II which is initialized by superposing randomly phased linear internal waves with amplitudes determined from the Garrett-Munk empirical model at 28°N. As in experiment I, after about seven inertial periods from the start of calculation, the vertical wave number Froude spectrum becomes quasi-stationary and the roll-off is seen again at vertical wave number ∼0.03 cpm. In experiment II, the quasi-stationary spectrum at 28°N thus formed is perturbed instantaneously at t = 10T_{i} with the same forcing used in experiment I (Figure 6). In striking contrast to the result of experiment I, the shear spectral amplitudes at vertical wave numbers of 0.01–0.03 cpm and 0.001–0.003 cpm are seen to gradually increase with time exceeding the Garrett-Munk level by up to a factor of about 2. This is more directly confirmed in Figure 7 where the ratio of the perturbed spectrum to the unperturbed, freely decaying, reference spectrum is shown in the two-dimensional wave number space for t = 10T_{i}–20T_{i}. We can see that the enhancement of shear spectral amplitudes at vertical wave numbers of 0.01–0.03 cpm and 0.001–0.003 cpm occurs mainly in the near-inertial frequency (*f* < ω < 2*f*) band. Of special importance is the fact that the energy level at small scales, namely, at horizontal wave numbers >0.006 cpm and vertical wave numbers > 0.02 cpm rapidly increases, exhibiting strong correlation with the enhancement of near-inertial current shear.

[13] These calculated results can be explained in terms of the equation for the evolution of vertical wave number k_{z} of the internal wave propagating through the high vertical wave number, near-inertial background current U_{i}, namely,

where k_{h} is the horizontal wave number. When k_{h} is antiparallel to the background velocity gradient, in particular, ∂k_{z}/∂t becomes positive so that the magnitude of k_{z} becomes higher and higher until dissipative processes eventually set in. This implies that, as ∂U_{i}/∂z increases, the internal waves with large k_{h} are efficiently Doppler shifted so that k_{z} rapidly increases and enhanced turbulent dissipation takes place.

[14] The elevated spectral density in the high vertical wave number, near-inertial frequency band, which plays the key role in transferring energy to dissipation scales, is thought to be caused by the nonlinear interaction termed parametric subharmonic instability [*McComas*, 1977; *McComas and Bretherton*, 1977; *McComas and Müller*, 1981; *Hibiya et al.*, 1996, 1998]. Parametric subharmonic instability is an example of the well-known parametric resonance of a physical system in which the parameter defining the natural frequency is varied with time. In the present case, large-scale internal wave field modulates in time the density stratification that defines the local buoyancy frequency for small-scale internal waves; the amplification of the small-scale internal waves results from the enhancement of the restoring force near the upward and the downward maximum of the isopycnal displacement, namely, at twice the frequency of the small-scale internal waves. Parametric subharmonic instability thus transfers energy from low vertical wave number, double-inertial frequency internal waves directly to high vertical wave number, near-inertial frequency internal waves.

[15] At 28°N, the M_{2} tidal frequency (M_{2} tidal period is 12.42 hours) is 2.1 times the local inertial frequency (inertial period is 25.82 hours) so that M_{2} internal tide is susceptible to parametric subharmonic instability. At 49°N, in contrast, the M_{2} tidal frequency is 1.2 times the local inertial frequency (inertial period is 15.82 hours) so that M_{2} internal tide is free from parametric subharmonic instability. Accordingly, even though significant M_{2} internal tidal energy may be generated over the Aleutian Ridge, it is not available to support local deep water mixing. The difference between the responses at mid and high latitudes is schematically summarized in Figure 8.

### 4. Concluding Remarks

- Top of page
- Abstract
- 1. Introduction
- 2. Numerical Model
- 3. Results and Discussions
- 4. Concluding Remarks
- References

[16] The present numerical experiments have clearly demonstrated that the level of turbulent dissipation in the stratified ocean interior is subject to changes in the intensity of high vertical wave number, near-inertial current shear. This implies that the energy cascade across the internal wave spectrum down to dissipation scales is under strong control by parametric subharmonic instability. It is therefore concluded that an understanding of the sources and variability of low vertical mode internal waves with frequencies over 2*f* is a key in constructing a predictive model of the rates of diapycnal mixing associated with internal wave breaking in the stratified interior of the world oceans.

[17] There are some limitations with the numerical approach in the present study. First, the present numerical experiment is restricted to a vertically two-dimensional wave motions in order to model at adequate resolution the disparate scale wave interaction which is crucial in the internal wave dynamics. Second, the present model assumes an underlying uniform density stratification over the full ocean depth to simplify the analysis of the calculated results. These compromise the quantitative application of our results to the real ocean. Because of the dynamical constraint imposed by the two-dimensionality, in particular, the present model might lack interactions of internal waves with different types of motion such as vortical motions which probably become important at small scales. Nevertheless, we believe that the present study for the vertically two-dimensional problem is the useful first step toward understanding realistic three-dimensional internal waves.

[18] In order to verify whether or not the numerically predicted energy cascade process is actually dominant in the real deep ocean, we have deployed a total of 106 expendable current profilers over a large area in the North Pacific to examine the spatial distribution of high vertical wave number shear. The results of the field observation will be reported in a separate paper [*Nagasawa et al.*, 2002].