[1] A vertically two-dimensional internal wave field is forced equally at the near-inertial frequency and the semidiurnal tidal frequency both at the lowest vertical wavenumber. These correspond to wind forcing and internal tide forcing, the main energy sources for the internal wave field. After 5 years of spin-up, a quasi-stationary internal wave field with characteristics of the Garrett-Munk-like spectrum is successfully reproduced. Furthermore, we carry out additional experiments by changing the strength of the semidiurnal tidal forcing relative to the near-inertial forcing. It is demonstrated that the Garrett-Munk-like spectrum is created and maintained only when energy is supplied both from the near-inertial forcing and the semidiurnal tidal forcing. So long as both energy sources are available, nonlinear interactions among internal waves occur such that the resulting internal wave spectrum becomes close to the Garrett-Munk-like spectrum irrespective of the ratio of the near-inertial forcing to the semidiurnal tidal forcing.

[2] The energy available for the deep ocean mixing is originally supplied at large scales (a few tens of kilometers) and then transferred across the internal wave spectrum down to small dissipation scales (a few meters) by nonlinear interactions among internal waves. The intermediate-scale internal wave spectrum, in general, appears to have a universal shape and energy level throughout the global oceans. Garrett and Munk [1972] proposed an analytic form that approximated this universal spectrum (hereafter referred to as the GM spectrum) based on the results of field observations, which was later revised by Munk [1981]. The universality of the internal wave spectrum in the deep ocean is thought to be attributable to nonlinear interactions among internal waves [Müller et al., 1986]. McComas [1977] and McComas and Müller [1981] showed that the GM spectrum is in approximate equilibrium with respect to resonant interactions among internal waves. However they did not examine whether or not the external energy flux can create and maintain the universal GM spectrum by balancing with nonlinear energy transfer to dissipation scales.

[4] We restrict our attentions to vertically two-dimensional internal wave motions by requiring the variability to be independent of one horizontal direction. The numerical model used here solves the Navier-Stokes equations under the Boussinesq approximation given by

where t is time; D/Dt = ∂/∂t + u∂/∂x + w∂/∂z is the total time derivative; u, v, and w are the velocity components in the x, y, and z directions, respectively; ρ′ is the density perturbation from the background density profile (z); p′ is the perturbation pressure; ρ_{0} is a reference density; g is the acceleration due to gravity; F_{u}, F_{v}, F_{w}, and F_{ρ′} are the external forcing terms which will be defined later. The above equations are integrated with a finite difference scheme by applying the leapfrog and Euler-backward scheme. The finite differenced equations are solved on a 128 × 128 grid with resolutions of 865.3 m and 3.9 m in the horizontal and vertical directions, respectively. The subgrid diffusive-dissipative processes are parameterized with a Laplacian operator where eddy viscosity and diffusivity coefficients are assumed to have the same value of 3.7 × 10^{−1} m^{2} s^{−1} in the horizontal and 7.5 × 10^{−6} m^{2} s^{−1} in the vertical. These are the smallest possible values needed to maintain the stability of the calculations. Cyclic boundary conditions are employed at the lateral sides, whereas free-slip top and bottom boundary conditions are employed. We assume the inertial frequency at 30°N, namely, f = 7.27 × 10^{−5} s^{−1} (inertial period T_{i} = 24 hrs) and the constant buoyancy frequency N = 3.49 × 10^{−3} s^{−1} (buoyancy period T_{b} = 30 mins), the typical value observed in the main thermocline.

[5] The numerical experiment starts from the initial state at rest. The model is subsequently forced at the near-inertial frequency as well as the semidiurnal tidal frequency both at the lowest vertical wavenumber in the form,

where u_{1}, v_{1}, w_{1}, and ρ_{1}′ (u_{2}, v_{2}, w_{2}, and ρ_{2}′) are the velocity components and density perturbation at the near-inertial frequency (the semidiurnal tidal frequency) and the lowest vertical wavenumber, α_{i} and β_{i} (i = 1,2) are the forcing coefficients. Then, the kinetic and potential energy input from the near-inertial forcing () and those from the semidiurnal tidal forcing () are given by

with the overbar denoting an average over the model domain.

[6] In order to attain a steady internal wave field, we inject the same amount of energy as dissipated in the model ocean so that

where γ is the ratio of the near-inertial forcing energy to the total forcing energy, and ɛ is the energy dissipation in the model ocean. For the assumed value of N, ɛ is determined using the relationship [Osborn, 1980]

where the eddy diffusivity coefficient κ_{ρ} = 1.0 × 10^{−5} m^{2}s^{−1} is employed following the results of microstructure measurements in the main thermocline away from the topography [Munk and Wunsch, 1998]. The forcing coefficients α_{i}, β_{i} (i = 1,2) are determined using equations (4) and (5) with the assumption,

3. Results and Discussion

3.1. Reproduction of the GM Spectrum

[7]Watanabe and Hibiya [2002] estimated that the wind-induced global energy flux to the surface mixed layer is about 0.7 TW which is comparable to the estimates of global energy flux from internal tides, 0.9 TW, based on astronomical measurements [Munk and Wunsch, 1998]. For this reason, we first carry out a numerical experiment assuming that the lowest vertical wavenumber energy supplied at the near-inertial frequency equals to that supplied at the semidiurnal tidal frequency.

[8] With the start of nonlinear interactions among internal waves, the energy cascades down to high wavenumber portion and hence the structure of the internal wave spectrum begins to be modified (Figure 1). Of special notice is the spectral enhancement at high vertical wavenumber (∼10^{−2} cpm) near–inertial frequency (see the plot at 50 days) which is presumably caused by parametric subharmonic instability [Hibiya et al., 2002; Furuichi et al., 2005]. After about 5 years from the start of calculation, the amount of energy contained in the model ocean becomes saturated and a quasi-stationary state is achieved. Figure 2 shows the comparison of the numerically reproduced quasi-stationary spectrum with the GM spectrum. Except at the low vertical wavenumber portion where forcing is applied and hence the energy level is several times higher than that of the GM spectrum, the reproduced spectrum agrees remarkably well with the GM spectrum both in the energy level and energy distribution. Figure 3 shows the vertical wavenumber quasi-stationary spectrum obtained by integrating the numerically reproduced two-dimensional energy spectrum with respect to horizontal wavenumber. We can confirm that both the energy level and slope of the spectrum agree very well with those of the GM spectrum.

[9] The vertical wavenumber quasi-stationary spectra obtained from the same numerical experiments but for latitudes 20°N, 40°N, and 50°N are also shown in Figure 3. We can find that all the quasi-stationary spectra are very similar with the GM spectrum, nearly irrespective of latitudes.

3.2. Weight of the Energy Sources

[10] The ratio between the near-inertial energy and the semidiurnal tidal energy both supplied at low vertical wavenumber is observed to vary spatially and temporally. Therefore, we next carry out a series of numerical experiments for 30°N increasing the ratio of the near-inertial forcing energy to the total forcing energy from 0% up to 100% with an increment of 20%.

[11]Figure 4 shows the quasi-stationary two-dimensional internal wave spectra for various combinations of the near-inertial forcing and the semidiurnal tidal forcing. Note that all the spectra reach quasi-stationary state, at longest, within 5 years' integration. Of special interest is that no significant difference can be found between the resulting spectra so long as the energy is supplied both from the near-inertial forcing and the semidiurnal tidal forcing. Figure 5 depicts the corresponding quasi-stationary vertical wavenumber spectra where the energy level and slope of all the spectra forced by both the near-inertial energy and the semidiurnal tidal energy are found to agree very well with those of the GM spectrum. This is consistent with the results from field observations that, although the main energy sources differ from region to region, the internal wave spectra in the deep ocean have much the same shape so long as the observations are not made close to a strong source of internal waves.

[12] In contrast, Figures 4 and 5 show that, when the energy is supplied only from the near-inertial forcing, the energy transferred to high wavenumber is reduced such that the resulting quasi-stationary internal wave spectrum is somewhat different from the GM spectrum. The amount of energy transferred to high wavenumber is much more limited when the energy is supplied only from the semidiurnal tidal forcing.

[13] The numerical experiments thus show that the GM-like spectrum can be created and maintained only when energy is supplied from both the near-inertial forcing and the semidiurnal tidal forcing, and not when the energy is supplied from either the near-inertial forcing or the semidiurnal tidal forcing. So long as both energy sources are available, nonlinear interactions among internal waves occur such that the resulting quasi-stationary internal wave spectrum becomes close to the GM-like spectrum irrespective of the ratio of the near-inertial forcing to the semidiurnal tidal forcing.

4. Summary and Conclusions

[14] Using a simple two-dimensional model, we have numerically reproduced the GM-like internal wave spectrum by injecting the near-inertial energy and the semidiurnal tidal energy equally at the lowest vertical wavenumber. In addition, by changing the strength of the semidiurnal tidal forcing relative to the near-inertial forcing, we have shown that the GM-like spectrum is created and maintained only when energy is supplied from both the near-inertial forcing and the semidiurnal tidal forcing, and not when the energy is supplied from either the near-inertial forcing or the semidiurnal tidal forcing. This is consistent with the result from measurements in the Black Sea and the Baltic Seas characterized by almost zero tides that the level of the observed internal wave spectra was about one order of magnitude lower than that of the GM spectrum [Ivanov and Serebryany, 1982; Morozov et al., 2007]. The nonlinear interaction mechanism responsible for the creation and maintenance of the GM-like spectrum irrespective of the ratio of the near-inertial forcing to the semidiurnal tidal forcing remains to be examined in terms of the bispectral analysis [Furuichi et al., 2005].

[15] There are some limitations with the numerical approach in the present study. First, the present numerical experiment is restricted to vertically two-dimensional wave motions in order to reduce the long integral time required to achieve the quasi-steady state. Second, the present model assumes uniform background density stratification over the full ocean depth for simplicity. These compromise the quantitative application of our numerical model 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 motions such as vortical motions, which must become important at small scales [Ramsden and Holloway, 1992]. Although we believe the present study for the vertically two-dimensional problem is a useful first step toward understanding the oceanic internal wave field, considerable follow-on works are absolutely necessary to see if the present results hold in more realistic three-dimensional situations.