Abstract

Recent numerical and observational studies have reported that resonant wave–wave interaction may be a crucial process for the energy loss of internal tides and the associated vertical water mixing in the midlatitude deep ocean. Special attention has been directed to the remarkable latitudinal dependence of the resonant interaction intensity; semidiurnal internal tides promptly lose their energy to near-inertial motions through parametric subharmonic instability equatorward of the critical latitudes 29°N/S, where half the tidal frequency coincides with the local inertial frequency. This feature contradicts the classical theoretical prediction that resonant wave–wave interaction does not play a major role in the tidal energy loss in the open ocean. By reformulating the kinetic equation for long internal waves and developing its calculation method, we estimate the energy decay rates of the low-vertical-mode semidiurnal internal tides interacting with the “ubiquitous” oceanic internal wave field. The result shows rapid energy decay of the internal tides, typically within O(10) days for the lowest-mode component, near their critical latitudes. This decay time is severalfold shorter than those in the classical studies and, additionally, varies by a factor of 2 depending on the local depth and density structure. We suggest from this study that the numerical integration of the kinetic equation is a more effective approach than recognized to determine the decay parameter of wave energy, which is indispensable for the global ocean models.

1. Introduction

The internal tide, a type of oceanic internal wave generated in the barotropic tidal flow over bottom topography, is regarded as the most significant energy source for deep ocean mixing. As observed from satellite altimeters, large-scale internal tides generated over steep topography propagate thousands of kilometers in the open ocean (Zhao et al. 2016). Although the generation and propagation of internal tides can be simulated directly by high-resolution numerical ocean models (Simmons et al. 2004; Arbic et al. 2010; Shriver et al. 2012, 2014; Niwa and Hibiya 2011; Müller et al. 2012; Niwa and Hibiya 2014), the dissipation rates of internal tides in the ocean interior have not been well quantified. Using general circulation models, de Lavergne et al. (2016) and Melet et al. (2016) demonstrated that the magnitude and structure of deep ocean circulation are sensitive to the spatial distribution of the mixing intensity associated with far-propagating internal tides. This “far-field tidal mixing” is a frontier topic of ocean mixing today (MacKinnon et al. 2017).

Among the various dynamic processes that lead to the dissipation of internal tides, the resonant wave–wave interaction, called parametric subharmonic instability (PSI), has attracted wide attention in the last couple of decades. In PSI, given a wave frequency , a pair of waves with frequencies and will become unstable if is satisfied. Based on the dispersion relation of inertia–gravity waves, PSI affects internal tides only equatorward of the latitude where half the tidal frequency coincides with the local inertial frequency. For the principal lunar semidiurnal constituent (M2 tide), this critical latitude is 28.8°. Observations suggest that PSI acts most effectively near the critical latitude, with an abrupt switch off at higher latitudes and gradual relaxation toward lower latitudes (Hibiya and Nagasawa 2004; Kunze et al. 2006; Hibiya et al. 2007; Qiu et al. 2012). By using a numerical model, MacKinnon and Winters (2005) reported that a progressive internal wave in a meridional plane loses a large part of its energy when passing through the critical latitude. However, a subsequent observational assessment (Alford et al. 2007; MacKinnon et al. 2013a) did not show such a catastrophic decay of internal tides. Although the energy dissipation rate was largest near 29°N, its value was rather modest compared to that expected. Despite the efforts mentioned above, a consensus on the energy loss of internal tides through wave–wave interactions is yet to be reached.

Previous theoretical studies of PSI have focused mainly on the enhancement of disturbance waves from an infinitesimal state superimposed on parent waves, investigating the “growth rate” of disturbance wave amplitudes (Sonmor and Klaassen 1997; Young et al. 2008; Dauxois et al. 2018). However, when considering the real ocean system, the growth of the disturbance waves ceases long after the onset of PSI, whereas turbulent mixing associated with the decay of internal tides continues. In terms of accurate ocean circulation modeling, what we need to clarify is the global distribution of internal tide energy transferred to dissipation scales. Actually, Oka and Niwa (2013) reproduced the Pacific deep circulation assuming that energy for turbulent mixing is supplied from internal tides propagating with a constant decay time of 30 days (Niwa and Hibiya 2011). Furthermore, recent studies by Ansong et al. (2015) and Buijsman et al. (2016) showed that proper wave drag, which is highly inhomogeneous in space, is indispensable to improve the accuracy of the global internal tide model. Incorporated into the global internal tide model, the “decay rate” of internal tides provides useful information about the subgrid-scale energy dissipation of internal tides in the world’s oceans.

Unlike the growth rate of disturbances, which can be estimated by linear stability analysis, the decay rate of an internal tide interacting with a broadly distributed, continuous spectrum is difficult to assess within the conventional analytical or experimental frameworks. A more suitable theoretical model is the kinetic equation, which describes the energy transfer rate in a continuous wave spectrum in general (Hasselmann 1966; Zakharov et al. 1992).

Using the kinetic equation, Olbers and Pomphrey (1981) and Olbers (1983) estimated the decay rates1 of internal waves and, surprisingly, concluded that the resonant interaction for internal tides “plays no role” in the energy balance of the oceanic internal wave spectrum. Eden and Olbers (2014, hereafter EO14) reassessed the decay rates of internal tides by improving the integration algorithm of the kinetic equation. Nonetheless, their new calculation (Fig. 1c in EO14) still did not show the prominent signals of PSI near 29°N/S. In the end, the discrepancy between the observational and theoretical results has not been resolved.

To conquer this problem, we have reformulated the kinetic equation for long internal waves and, using this, reexamined the energy decay of internal tides. As a result, the modest peaks of the decay rates of internal tides are successfully reproduced at their critical latitudes. Our estimate is completely different from the previous calculation and is compatible with the recent observational evidence. In this article, after a brief description of the formulation and calculation methods, the results and their physical implications are presented.

2. Formulation and calculation method

In most of the previous studies, including EO14, kinetic equations were designed for interactions among vertically propagating waves in a uniformly stratified ocean with infinite depth [reviews can be found in Müller et al. (1986) and Polzin and Lvov (2011)]. However, in the real ocean, the physical properties of low-vertical-mode internal waves depend on the variation of density stratification and vertical boundary conditions. To extend the applicability, we reformulate the kinetic equation for long inertia–gravity waves, as described in  appendix A, as follows:

 
formula

where represents the wave action density of the vertical water column per unit area, a function of horizontal wavevector and vertical mode number j. In the following, we consider only the baroclinic components . Frequency ω is determined by the dispersion relation of inertia–gravity waves:

 
formula

where f is the Coriolis parameter, is the equivalent depth of the jth mode, and is the horizontal wavenumber. The coefficient represents the coupling intensity among each triad as defined by (A7). In an idealized situation where the density stratification is weak and uniform, the coefficient becomes proportional to the delta-like function with respect to ; specifically, it takes a nonzero value only if is satisfied. However, in the real ocean, interaction is always allowed for any combination of , but their coupling intensity varies from place to place depending on the stratification structure and ocean depth.

The kinetic equation describes the statistical evolution of the action spectrum (or energy spectrum) on a time scale much longer than each wave period. Delta functions in the integrand specify the condition of nonlinear resonance and hence restrict the triad interaction on some loci in wavevector space (Fig. 1). In principle, (1) can be integrated to predict the time evolution of the energy spectrum from an arbitrary initial state. Full numerical integration of (1) is, however, highly difficult because of the complexity and singularity of the kernel function.

Fig. 1.

Examples of resonant triads. The red curves indicate the loci of that satisfy (a) and (b) for given and . To create these figures, , and were chosen. Note that different scales are used in the two panels.

Fig. 1.

Examples of resonant triads. The red curves indicate the loci of that satisfy (a) and (b) for given and . To create these figures, , and were chosen. Note that different scales are used in the two panels.

We now further examine the property of (1). Let us classify the interaction terms in (1) depending on whether is involved or not and then rewrite them as , where the first term represents the energy gain of the component supplied through the resonant interactions with and , and the second term corresponds to the energy loss, which is proportional to itself. The coefficient ν, a linear functional of the wave action spectrum , can then be interpreted as the decay rate of wave energy density.

Now, we separate the action density into those of one tidal component and other background waves. By assuming that the background energy spectrum is isotropic in the horizontal directions, representing the wavevector in the polar coordinates , redefining , and performing the integration in (1) with respect to , we may write the decay rate of an internal tide as follows:

 
formula

where

 
formula

is the area of a triangle consisting of . The resonant conditions yield as a function of the integration variable :

 
formula

The first term in (3) originates from the sum interaction , which involves PSI and takes nonzero values only when is satisfied, whereas the second term comes from the difference interaction that occurs at all latitudes. In our calculation, both types of interaction are taken into account so that the decay rates of M2 internal tides take finite values even over 28.8°N/S.

The intervals of integration are determined from the triangle inequality . Compared with the original expression (1), the expression (3) is more simplified, and hence direct numerical computation of it is possible without time integration so long as we have the background action spectrum n, structure function , and equivalent depth for each vertical mode.

The equivalent depth and vertical structure functions are obtained by solving the eigenvalue problem for the linear operator defined as (A3). The geographical data for ocean depth in ETOPO1 (Amante and Eakins 2009) and the climatology of temperature and salinity data in the World Ocean Atlas (Locarnini et al. 2013; Zweng et al. 2013) are used for the calculation. Neutral density is successively calculated from the surface downward and then interpolated in the vertical direction to produce sufficiently smooth and fine density profiles.

In this study, we choose the so-called Garrett–Munk spectrum as the background wave field, which has been believed to be the “ubiquitous” energy spectrum in the open ocean far from major wave generation sites. Munk (1981) defined this spectrum in frequency and vertical mode space as

 
formula

where is the total energy density per unit area, and and are the normalized frequency and vertical mode spectra:

 
formula

where C is the normalization constant. Since hydrostatic approximation eliminates the upper limit of the dispersion relation, we take . Energy density in the ocean has been historically scaled with the local buoyancy frequency as (Polzin and Lvov 2011), so we define as

 
formula

where is a constant, and Z represents the vertical coordinate. Defining representative density and changing the independent variables in (6), the action spectrum in horizontal wavenumber and vertical mode space is derived as

 
formula

In the original expression (Munk 1981), the variable parameters in (7) were chosen as . However, in this study, we adopt as the “standard” values because the recent observational review in Polzin and Lvov (2011) shows that typically varies in the range of 3–15 and Olbers and Eden (2013) also utilized these values for their calculation.

Our target is to estimate the value of decay rate ν in (3) for the lowest five modes of M2 internal tides at latitudes where they can exist as progressive waves. The spherical domain, except for shallow areas with depths less than 500 m, inland seas, and equatorial areas, is partitioned into grid cells. Each water column is then divided into 1024 layers to solve the eigenvalue problem. Interactions between internal tides and background waves of up to 127 vertical modes are examined, with each interval of wavenumber integration discretized into 1024 grids for the calculation.  Appendix B presents a method to determine the intervals of wavenumber integration.

3. Results

a. Spherical distribution of decay rates

In Fig. 2, the decay rates of the lowest five modes of M2 internal tides are depicted on a global map. Note that a logarithmic color scale is utilized. The typical decay time of the lowest mode component is days, the same as that in a classical study (Olbers 1983), except near the critical latitudes 29°N/S, where we find the most rapid decay within roughly 20 days for the lowest mode, which is severalfold shorter than the previous estimates. As the mode number increases, decay rates increase almost monotonically. For the fifth mode, the most rapid decay occurs within a few days, implying that a large part of the energy of high modes is subject to immediate attenuation, resulting in local energy dissipation near the generation sites, as anticipated by St. Laurent and Garrett (2002).

Fig. 2.

Numerically estimated decay rates ν defined in (3) for the lowest five modes of the principal lunar semidiurnal constituent (M2 tide) interacting with the background wave spectrum. Numbers below the color bar indicate the e-folding times of the wave energy.

Fig. 2.

Numerically estimated decay rates ν defined in (3) for the lowest five modes of the principal lunar semidiurnal constituent (M2 tide) interacting with the background wave spectrum. Numbers below the color bar indicate the e-folding times of the wave energy.

The decay rates in the proximity of the critical latitudes2 in the North Pacific vary longitudinally by a factor of 2 (Fig. 3a). This reflects the variation of waveforms that are determined by the depth and density stratification structure. Figure 3b depicts isopycnal surfaces and the bottom topography for the same region as in Fig. 3a. The decay rate of each mode reaches its largest value in shallow areas, especially over the Hawaiian Ridge located in the central Pacific. Another distinct feature is the eastward increase of the decay rates of the lowest mode, which seems to be associated with variation of the density stratification structure. At this latitudinal band, the subtropical gyre causes the pycnocline to become gradually shallower and sharper eastward. In the eastern area, the shallow pycnocline confines the vertical structure of the lowest-mode waves near the surface, reducing the equivalent depth and hence restricting the horizontal wavelength of internal tides. Because shorter waves tend to cause stronger nonlinear interactions, the decay rate of the lowest mode is sensitive to pycnocline depth. Although other higher modes also show longitudinal variations, a clear tendency as found for the lowest mode is no longer recognized. The effect of the variation of stratification and topography is further investigated in the following.

Fig. 3.

(a) Estimated decay rates of the lowest five modes of the M2 internal tide along 28.5°N in the Pacific. The red curve indicates the rate of the lowest mode. (b) The colored contours are isopycnal surfaces of neutral density, successively calculated downward from the surface, taking into account the compressibility of seawater by using data from the World Ocean Atlas. The contour increment is 0.25 kg m−3. The bottom topography in ETOPO1 is also shown. These data are utilized to solve the eigenvalue problem for the vertical modes.

Fig. 3.

(a) Estimated decay rates of the lowest five modes of the M2 internal tide along 28.5°N in the Pacific. The red curve indicates the rate of the lowest mode. (b) The colored contours are isopycnal surfaces of neutral density, successively calculated downward from the surface, taking into account the compressibility of seawater by using data from the World Ocean Atlas. The contour increment is 0.25 kg m−3. The bottom topography in ETOPO1 is also shown. These data are utilized to solve the eigenvalue problem for the vertical modes.

b. Vertical distribution of interaction intensity

Kinetic equation (1) is defined in horizontal wavevector and vertical mode space. In this expression, the coupling coefficient involves the vertical coordinate Z in the form of vertically integrated triple products of vertical structure functions, . To investigate the vertical dependence of the interaction among internal waves, we now heuristically interchange the order of summation and integration so as to rewrite the decay rate of internal tides, ν, as

 
formula

The integrand of this expression, , is interpreted as interaction intensity, specifically written as

 
formula

where we redefined a coupling coefficient (see  appendix A). This expression makes it possible to identify the vertical locations at which internal tides lose energy to the surrounding internal wave field. Note that the Z-dependent factor takes both positive and negative values, whereas its vertical integration is positive definite. Therefore, the interaction intensity occasionally takes negative values, which means energy is flowing into the internal tides. This feature may seem somewhat strange, but there actually exist field observations showing that this backward energy transfer indeed occurs intermittently in the ocean (MacKinnon et al. 2013a).

Figure 4 shows vertical cross sections of in a meridional plane along 179.5°W and a zonal plane along 28.5°N. Black curves represent isopycnal surfaces. Although takes both positive and negative values, Fig. 4 is depicted in a logarithmic scale, the hatched areas of which indicate small negative values. In the left panels, high values on the equatorial side of the critical latitudes appear mainly in the upper ocean, especially for the lowest mode. The horizontal striped patterns of high and low values correspond to the nodes and antinodes of each mode. Effects of bottom topography and stratification are more clearly seen in the right panels. Over the ridges, nodes are lifted upward with their values enhanced. Striped patterns are inclined in the zonal direction synchronized with the isopycnal surfaces, concentrating the interaction intensity near the surface in the eastern area.

Fig. 4.

Vertical cross sections of interaction intensity defined in (11) are depicted along (left) 179.5°W and (right) 28.5°N for the lowest five modes of M2 internal tides. In hatched areas, takes negative values. Black curves show the isopycnal surfaces with an interval of 0.5 kg m−3.

Fig. 4.

Vertical cross sections of interaction intensity defined in (11) are depicted along (left) 179.5°W and (right) 28.5°N for the lowest five modes of M2 internal tides. In hatched areas, takes negative values. Black curves show the isopycnal surfaces with an interval of 0.5 kg m−3.

Figure 5a shows the zonal average along 28.5°N in the North Pacific. Overall, the exponential decay of from the surface down to about 3000-m depth is perceived, except for more rapid decay for the lowest mode above 1000-m depth. Now we further define the cumulative contribution to the zonally averaged decay rate from levels above a specific depth Z as

 
formula

and depict it in Fig. 5b. The gray dotted lines indicate 1000-m depth and . The relative contribution from the upper 1000-m layer overwhelms 80% for and 90% for . In other words, most of the interaction between internal tides and background waves occurs in the upper ocean.

Fig. 5.

(a) Zonal averages of the interaction intensity along 28.5°N in the Pacific. (b) Cumulative contribution of integrated from the surface to each depth, normalized by the value integrated over the total depth, as defined in (12). The gray dotted lines indicate in the horizontal axis and 1000-m depth in the vertical axis. In both panels, those for the lowest five modes of M2 internal tides are plotted.

Fig. 5.

(a) Zonal averages of the interaction intensity along 28.5°N in the Pacific. (b) Cumulative contribution of integrated from the surface to each depth, normalized by the value integrated over the total depth, as defined in (12). The gray dotted lines indicate in the horizontal axis and 1000-m depth in the vertical axis. In both panels, those for the lowest five modes of M2 internal tides are plotted.

4. Discussion and conclusions

Contrary to the previous theoretical arguments, our calculation shows the most rapid energy decay of M2 internal tides near 29°N/S through parametric subharmonic instability, consistent with the results from field observations. This improvement is likely attributed to the difference in background energy spectra. In the vicinity of 29°N/S, the M2 internal tides interact intensively with high-mode near-inertial energy peaks inherent in the oceanic wave spectrum. The near-inertial energy of the background spectrum was largely omitted in the previous estimates by Eden and Olbers (2014, herein EO14); the frequency spectrum was arranged as , which presumably resulted in underestimates of the energy loss of internal tides at the midlatitudes.

The present kinetic equation describes only the resonant triad interaction among internal waves, assuming that each wave component perfectly satisfies linear dispersion relation (2). As nonlinearity becomes strong, this dispersion relation is broadened, and hence near-resonant interaction deviating from the resonant manifold, , comes to contribute to the energy transfer in spectrum space. Lvov et al. (2012) discussed this issue in detail introducing an extended kinetic equation that involves both resonant and near-resonant interactions. However, their methodology is impractical to be used for our purpose owing to the following reasons: (i) to include the near-resonant interaction, we must integrate the kinetic equation within higher-dimensional wavenumber space, which requires too much computation to be applied to global-scale analysis, and (ii) the broadened dispersion relation for each wave component, which is needed to formulate the extended kinetic equation, is inaccessible a priori. As pointed out by Lvov et al. (2012), near-resonant interaction should be considered to explain the stationarity of the internal wave spectrum particularly for the high-vertical-wavenumber and high-frequency content. On the other hand, energy transfer from low-mode internal tides to near-inertial waves, which is the central objective in the present analysis, is expected to be well represented even under the perfect resonance assumption. We anticipate that when the broadening of dispersion relation is taken into account, the sharp peaks of energy decay rates perceived at 28.8°N/S may become indistinct, but the overall structure of the present results will not change.

A remaining issue is how significant resonant wave interaction plays a role in dissipating internal tide energy in comparison with other processes, such as topographic effects or wave–vortex interaction. We here review some recent studies just briefly discussing them. First, from a comprehensive viewpoint, the typical decay time of internal tide energy over the globe has been estimated in some literature. Niwa and Hibiya (2011) found that a linear drag acting on baroclinic current with a decay time of 30 days, which corresponds to the energy decay time of 15 days or longer, is the optimal value to tune their global tide model with the historical current meter records in the world’s oceans. Kelly et al. (2013) calculated the “residence time” of the global internal tide energy in terms of a simple expression of (total internal tide energy)/(internal tide generation rate) as 7–21 days. Zhao et al. (2016) also estimated the residence time of internal tide energy to be 1–1.5 days, much shorter than the previous ones. Caution is needed in that these globally averaged values involve energy dissipation in the coastal regions. Far from lateral boundaries, the typical decay time of internal tides may become substantially longer than the average. In reality, signals of internal tides detected by the satellite altimeters propagate O(1000) km in the open ocean, surviving several tens of days (Zhao et al. 2016).

Based on a semianalytical model, Kelly et al. (2013) estimated the scattering coefficients of mode-1 internal tides over small bottom topography. Their result, defined as the energy loss of an internal tide per unit length, is translated to the decay rate per unit time by multiplying the group velocity of inertia–gravity wave at each location, yielding the typical decay time of internal tides as O(10) days. Mathur et al. (2014) also discussed the scattering of the internal tides due to finite-size topography with more realistic stratification to find that the scattering coefficient is dominantly determined by the largest topography on the path of internal tides. In their calculation, scattering coefficients on the northern side of Hawaii, where no isolated ridges exist, are consistent with those obtained by Kelly et al. (2013). The effect of the slowly varying eddy field is also a key candidate for the energy loss of low-mode internal tides and has been investigated theoretically and numerically (Kerry et al. 2014; Dunphy and Lamb 2014; Dunphy et al. 2017; Wagner et al. 2017). Nevertheless, resources are not enough to see the relative importance between the wave–wave and wave–vortex interactions in various regions. We expect that the recent fundamental efforts mentioned above will be combined with high-resolution eddy-resolving global ocean models (Qiu et al. 2018) and the upcoming Surface Water and Ocean Topography (SWOT) mission (Fu and Ubelmann 2014) to reveal this undetermined process in the near future.

The shortest decay time of the lowest-mode internal tides, 20 days in our estimate, suggests that the role of PSI is still limited even near 28.8°N/S. Nonetheless, the modest peaks of the energy dissipation rates at 29°N on the path of internal tides propagating northward from Hawaii, reported from the Internal Waves Across the Pacific (IWAP) experiment (Alford et al. 2007; MacKinnon et al. 2013a,b), could be reasonably explained in terms of the present result. Moreover, our global estimate suggests that the high energy dissipation rates near the generation sites of internal tides between 20° and 30°N reported by a series of observations by Hibiya and Nagasawa (2004) and Hibiya et al. (2006, 2007) are ascribed to the energy loss of high-mode internal tides due to PSI, which is a more efficient energy pathway than that for far-propagating low modes.

To discuss the “near-field tidal mixing” caused by wave–wave interactions, our theoretical model requires further improvement. In the formulation of the kinetic equation, we treat independently the eigenmodes by assuming that the phase of each wave component is statistically uncorrelated, thus excluding coherent wave structure. In the real ocean, internal tides near the generation site are mostly correlated, sometimes creating a beam structure obliquely emanating from the bottom (Nash et al. 2006). Using a numerical model, Nikurashin and Legg (2011) reproduced local dissipation of internal tides at rough topography, demonstrating bottom-intensified energy dissipation induced by PSI. The present statistical theory cannot cope with this spatially confined interaction process. Occurrence of resonant interaction near an internal-tide generation site was also identified through bispectrum analysis of observation data (Sun and Pinkel 2013). A suitable theoretical model for near-field resonant interaction remains to be constructed.

Finally, the present estimate of decay rates should be interpreted as a lower bound because in this calculation we utilized the most basic spectrum as the background field. At major dissipation sites of internal tides, higher wave energy with a distorted spectrum has been reported (e.g., Hibiya et al. 2012). Enhanced background waves will absorb the internal tide energy more rapidly. To detect the optimal energy balance between the internal tides and the background spectrum, we must construct a time-evolving model as EO14 attempted. Such a model enables identification of the energy transfer route not only in physical space but also in mode-and-wavenumber space, which is linked to the subsequent energy cascade that causes actual wave dissipation and resulting water mixing. This study suggests that the numerical integration of the kinetic equation is a more effective approach than recognized to determine the decay parameter of internal waves, which is indispensable for internal tide models to reproduce deep mixing in the global ocean.

Acknowledgments

The authors express their gratitude to Shinichiro Kida and two anonymous reviewers for their invaluable comments on the original manuscript. Numerical calculations were carried out on the Fujitsu PRIMEHPC FX10 System (Oakleaf-FX) in the Information Technology Center, The University of Tokyo. This study was supported by JSPS KAKENHI Grants JP16H02226 and JP18H04918. This paper forms part of the first author’s doctoral dissertation at The University of Tokyo completed in 2017.

APPENDIX A

Derivation of the Kinetic Equation

Let us consider a rotating stratified ocean with the hydrostatic approximation in the isopycnal coordinate . Here we use the notation of horizontal vector rotation as . The spatial domain is bounded by the flat bottom boundary and the upper free surface. Under the condition that the potential vorticity is uniform on each isopycnal surface, the equation of motion can be described by the canonical equations, in which horizontal velocity potential and vertical thickness are conjugate variables (Lvov and Tabak 2004). Here, for convenience of analysis, the vertical coordinate Z is regarded as the reference level of each isopycnal surface in the state of rest, taking values in . The density ρ is assumed to be a prescribed nonincreasing function of Z. Letting η be the vertical excursion of each isopycnal surface, the thickness variable is defined as . By further employing the Boussinesq approximation, the time evolution of the horizontal velocity potential and the thickness can be written as

 
formula

with the Hamiltonian

 
formula

where . The linear operator is defined such that

 
formula

where is the representative density of seawater. One may find that Montgomery potential appears as a pressure force in the first equation of (A1). The kinetic plus the available potential energy in the whole domain coincides with . It is noted that our formulation is equivalent to that in Lvov and Tabak (2004), except for the vertical boundary conditions. If the vertical domain is extended to and the vertical coordinate is transformed to the density ρ, the Hamiltonian (A2) reduces to the classical one. The linear Hermitian operator defined in (A3) is a key ingredient in the present formulation, which specifies the eigenbasis of the linear internal wave field, as we shall see below.

We now write the eigenvalues and corresponding eigenfunctions of as and , which are generally called equivalent depths and vertical structure functions. Without loss of generality, we can assume that the eigenfunctions are a complete orthogonal set. Let us introduce a complex variable , which is a linear combination of ϕ and γ expanded in vertical eigenmodes and on a horizontal Fourier basis; that is,

 
formula

Here we utilize the dispersion relation of the long inertia–gravity wave , defined in (2). The canonical equations (A1) are consequently rewritten as

 
formula

where the Hamiltonian is expressed in terms of a as

 
formula
 
formula

The coupling coefficients U and V are functions of horizontal wavevectors and vertical mode numbers . For our calculation, only V is needed, and its expression is

 
formula
 
formula
 
formula
 
formula
 
formula

Now we assume that the nonlinearity is sufficiently weak: . An additional assumption that the phase of each component is statistically independent allows us to define the action density n such that , where the angle brackets indicate an ensemble average. The wave energy density in a vertical water column per unit area is approximately . Following the conventional procedure (e.g., Zakharov et al. 1992), kinetic equation (1) is readily derived.

APPENDIX B

Interval of Wavenumber Integration

The interval of wavenumber integration in (3) is specified by the inequality

 
formula

along with dispersion relation (2) and the frequency resonant condition . Replacing the sign of inequality by that of equality, we obtain an equation whose roots correspond to the endpoints of integration. This equation is deformed into a polynomial form:

 
formula
 
formula
 
formula
 
formula
 
formula
 
formula

with the condition of , where we denote x instead of to specify the unknown variable. It is noted that, since the function S in the denominator of the integrand in (3) vanishes at the endpoints, this is a kind of improper integration. That is why (B2) must be solved in high accuracy to perform the numerical integration. In this study, we have solved (B2) for each combination of at each latitude and longitude using the Bairstow method in quadruple precision. In this iterative method, we carefully arrange initial values and the convergence condition to prevent fatal numerical errors. We have verified that the results do not change substantially even when the total number of grids in the integration, 1024 as related in the text, is reduced to half.

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Footnotes

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1

Polzin and Lvov (2011) considered asymptotically the growth rate of near-inertial energy fed by a tidal component through PSI, specifically (53) in their paper, and found that it becomes infinite if the kinetic equation is applied to a situation with a monochromatic internal tide. The decay rate of an internal tide, on the other hand, takes a finite value regardless of the spectrum shape of the tidal component as far as the overall spectrum is sufficiently smooth. A more detailed consideration for the growth rate of PSI in the framework of statistical theory is given by the authors in another paper, Onuki and Hibiya (2018, manuscript submitted to J. Fluid Mech.).

2

In our 1° resolution calculation, 28.5° is closest to the critical latitude 28.8°.