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 
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
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.
Examples of resonant triads. The red curves indicate the loci of 
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0278.1
We now further examine the property of (1). Let us classify the interaction terms in (1) depending on whether 
The intervals of integration are determined from the triangle inequality 
The equivalent depth and vertical structure functions are obtained by solving the eigenvalue problem for the linear operator 
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 
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 
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.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0278.1
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.
(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.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0278.1
b. Vertical distribution of interaction intensity
Figure 4 shows vertical cross sections of 
Vertical cross sections of interaction intensity 
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0278.1
(a) Zonal averages of the interaction intensity 
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0278.1
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 
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, 
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
Now we assume that the nonlinearity is sufficiently weak: 
APPENDIX B
Interval of Wavenumber Integration
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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.).
In our 1° resolution calculation, 28.5° is closest to the critical latitude 28.8°.