## 1. Introduction

Internal gravity waves shape the ocean by mixing density and dissolved substances when they break. This mixing is a key player of Earth’s climate system since it can drive oceanwide flow such as the meridional overturning circulation (Munk and Wunsch 1998; Talley et al. 2003; Wunsch and Ferrari 2004). The wave forcing is mostly due to tides and localized in spectral space, while observations show a spread of energy over a wide range of wavenumbers. It has long been recognized that the spectral wave energy distribution is surprisingly similar at different locations. A series of studies (Garrett and Munk 1972, 1975; Cairns and Williams 1976; Munk 1981) formulated what is known today as the Garrett–Munk (GM) spectrum: a spectral energy distribution characterized by slopes close to minus two in the wavenumber and frequency range of internal gravity waves, superimposed by relatively weak spectral peaks near the inertial and tidal frequencies. Such a continuously populated energy spectrum is similar to isotropic turbulence with its famous −5/3 slope. The apparently fixed slopes therefore point toward energy transfers from the wavenumbers and frequencies where forcing generates the waves, toward regions in spectral space where dissipative processes like wave breaking extract energy from the wave field. Wave–wave interactions by nonlinearities are thought to establish such an energy transfer.

The so-called scattering integral or kinetic equation (Hasselmann 1966; Nazarenko 2011) allows the prediction of the energy transfers by such wave–wave interactions for a given energy spectrum under the assumption of slowly changing wave amplitudes, that is, for weak interactions. Previous studies (Olbers 1976; McComas and Bretherton 1977; Pomphrey et al. 1980; Lvov et al. 2012; Eden et al. 2019b) have evaluated the kinetic equation for internal gravity waves for different versions of the GM spectrum, under different approximations, and using different numerical methods. They share some common features but are also in parts contradictory, which puts doubts on the validity of the method. Furthermore, Holloway in a series of comments (Holloway 1978, 1980, 1982) and Müller et al. (1986) argue that the assumption of weakly changing wave amplitudes might not be justified. It was proposed already by Holloway (1981) that numerical model simulations should be used to clarify the issue. Hibiya et al. (1998) find that forcing applied to a model initialized with the GM spectrum is most effective in the frequency range 2 < *ω*/*f* < 3 to produce energy at high vertical wavenumber, but doubt is also cast on this study since the model is two dimensional but the wave–wave interaction is inherently three dimensional.

The aim of the study presented here is therefore to compare the prediction of the kinetic equation by the use of direct three-dimensional numerical simulations of the interaction of internal gravity waves. We show that the spectral energy transfers predicted by the kinetic equation are indeed similar to the numerical model simulation for certain frequency and wavenumber ranges and speculate how the emerging picture of the coherent spectral shape of the transfers we find may relate to the observed energy and dissipation rates in a global spectral energy budget of the gravity waves in the ocean.

In the following section, we introduce the numerical model and the initialization and diagnostic methods we use to derive the energy transfers by wave–wave interactions in the direct numerical simulations, and compare in section 3 with the predictions of the kinetic equation. Section 4 discusses mechanisms and limits of the kinetic equation. The last section provides a summary and a discussion of the implications of the results for the global wave energy budget.

## 2. Direct numerical simulation

*m*is vertical wavenumber, and

*ω*is frequency, and with the shape functions

*A*(

*x*) =

*n*

_{a}(1 +

*x*

^{r})

^{−1}and

*B*(

*x*) =

*n*

_{b}|

*f*|

*x*

^{−1}(

*x*

^{2}−

*f*

^{2})

^{−1/2}normalized by

*n*

_{a}and

*n*

_{b}such that

*m*

_{*}is given by

*c*

_{*}= 0.5 m s

^{−1}, and the total energy level is set to

*E*

_{0}= 3 × 10

^{−3}m

^{2}s

^{−2}. The parameter

*r*= 2 denotes the (negative) spectral slope in

*m*for large wavenumbers. We show

*ω*,

*m*) from Eq. (1) in Fig. 1a—and later ∂

_{t}

*ωm*

**k**), where

**k**denotes the wavenumber vector

**k**= (

*k*

_{x},

*k*

_{y},

*m*)—we calculate the initial conditions for velocity

*u*,

*υ*,

*w*and buoyancy

*b*in the model in the following way. After applying a three-dimensional Fourier transform, the velocity and buoyancy amplitudes

**q**

^{±}(

**k**) given in appendix B. The sum over the superscript

*s*= ± corresponds to the positive and negative discrete wave frequencies

*ω*

^{±}(

**k**) also given in appendix B. The wave amplitude

**p**

^{±}is the left eigenvector of the linear discrete system, is related to wave energy by

*n*

^{±}(

**k**) also given in appendix B. The wave amplitudes at initial time are then given by

*ζ*

^{s}(

**k**) are realizations of a complex normal number with 0 mean and variance of 1. With different realizations of

*ζ*taken from a random number generator we generate an ensemble of 25 model integrations, which have initially the same spectral energy, but different phases of the waves. An example of the different initial conditions for the model ensemble is shown in Fig. 1b.

The mean wave energy ^{0} = (1/2)|*a*^{0}|^{2}/*n*^{0} are diagnosed and averaged over the ensemble and ∂_{t}_{t}_{t}*λ*_{num} = −∂_{t}*m*, the damping time scale becomes comparable to the inertial period such that we can conclude that the simulated wave–wave interaction is not affected by the numerical damping for simulation periods smaller than a day.

(a) Numerical damping rate *λ*_{num}. The gray area corresponds to *ω* and *m* values not covered by the model grid. (b) Ensemble mean geostrophic energy transfer ∂_{t}^{0} in the numerical model after 0.1 day. (c) Wave energy transfer ∂_{t}*ω*/*f* = 1.25, 2, 3, 5, 10, and 20, where *ω* denotes the wave frequencies on the numerical grid given by Eq. (B2). Dashed lines correspond to the analytical frequencies.

Citation: Journal of Physical Oceanography 50, 4; 10.1175/JPO-D-19-0022.1

(a) Numerical damping rate *λ*_{num}. The gray area corresponds to *ω* and *m* values not covered by the model grid. (b) Ensemble mean geostrophic energy transfer ∂_{t}^{0} in the numerical model after 0.1 day. (c) Wave energy transfer ∂_{t}*ω*/*f* = 1.25, 2, 3, 5, 10, and 20, where *ω* denotes the wave frequencies on the numerical grid given by Eq. (B2). Dashed lines correspond to the analytical frequencies.

Citation: Journal of Physical Oceanography 50, 4; 10.1175/JPO-D-19-0022.1

(a) Numerical damping rate *λ*_{num}. The gray area corresponds to *ω* and *m* values not covered by the model grid. (b) Ensemble mean geostrophic energy transfer ∂_{t}^{0} in the numerical model after 0.1 day. (c) Wave energy transfer ∂_{t}*ω*/*f* = 1.25, 2, 3, 5, 10, and 20, where *ω* denotes the wave frequencies on the numerical grid given by Eq. (B2). Dashed lines correspond to the analytical frequencies.

Citation: Journal of Physical Oceanography 50, 4; 10.1175/JPO-D-19-0022.1

From *t* = 0 to about *t* = 0.1 day there is an increase in the geostrophic energy ^{0} at large vertical wavenumbers and low frequencies (Fig. 2b), and a corresponding loss of wave energy (not shown). Since in the linear model there is no such initial increase in ^{0}, this represents an inverse nonlinear geostrophic adjustment, similar to what is described in Eden et al. (2019a) (cf. their Fig. 6). For *t* > 0.1 day, however, ∂_{t}^{0} becomes small while the wave energy transfer ∂_{t}*ω*/*f* < 3 and energy gain at near-inertial frequencies but larger wavenumbers. Figure 2c shows ∂_{t}*t* = 0.5 day and *t* = 0.1 day; using estimates at other times gives similar results in shape, but the magnitude of ∂_{t}

## 3. Kinetic equation

*a*

^{s}as

*s*

_{0}= 0, ±;

*ω*

_{n}=

*ω*(

**k**

_{n}). The right-hand side of Eq. (4) represents the quadratic nonlinear terms in the equations of motion. They are given by triad interactions between amplitudes at wavenumbers

**k**

_{0},

**k**

_{1}, and

**k**

_{2}satisfying

**k**

_{0}=

**k**

_{1}+

**k**

_{2}. Gravity waves with

*s*

_{0}= ± are interacting only with each other choosing only the triads with

*s*

_{1}= ± and

*s*

_{2}= ±. Such triads represent the gravity wave–wave interaction, but in general triads for

*s*

_{0}= ± can also be formed including the geostrophic mode

*s*

_{1}= 0 or

*s*

_{2}= 0. The same is true for the geostrophic mode with

*s*

_{0}= 0: it can interact only with itself for

*s*

_{1}=

*s*

_{2}= 0, but mixed triads with

*s*

_{1}= ± or

*s*

_{2}= ± are also possible. Such mixed interactions are discussed in Eden et al. (2019a), and apparently also show up here (Fig. 2b). Different models can be represented by Eq. (4), with the interaction coefficients

*c*as the only difference. Here we use the coefficients appropriate to the nonhydrostatic equations of motion given in Eden et al. (2019b). Neglecting the nonlinearities (

*c*= 0), the wave amplitudes stay constant in time, while for

*c*≠ 0 nonlinear interactions will transfer energy between wave triads.

Being just another model representation, Eq. (4) is not yet useful. Using Hasselmann’s weak interaction assumption (Hasselmann 1966; Nazarenko 2011), however, it is possible to cast Eq. (4) into a tendency equation for the energy spectrum ^{s} = (1/2)|*a*^{s}|^{2}/*n*^{s} instead of the amplitudes *a*^{s} as in Eq. (4). The resulting equation is called the scattering integral or the kinetic equation, and its derivation is discussed in more detail in Eden et al. (2019a; see also Hasselmann 1966; Nazarenko 2011). The assumption for the kinetic equation to hold is that wave amplitudes are only slowly (or weakly) changing with respect to the wave periods, a condition which is tested below. The time dependency in Eq. (4) transforms in the kinetic equation into a finite time response function Δ(*s*_{1}*ω*_{1} + *s*_{2}*ω*_{2} − *s*_{0}*ω*_{0}, *t*) [given by Eq. (22) in Eden et al. (2019a)] which converges to *δ*(*s*_{1}*ω*_{1} + *s*_{2}*ω*_{2} − *s*_{0}*ω*_{0}) for time *t* → ∞. Focusing on the gravity wave–wave interaction only, that is, for *s*_{0}, *s*_{1}, *s*_{2} = ±, only resonant triad interactions satisfying *ω*_{0} = *ω*_{1} ± *ω*_{2} are thus important in the long run. For *s*_{0}, *s*_{1}, *s*_{2} = ±, the triad sum in Eq. (4), which stays the same in the kinetic equation, can be resolved into 1) the sum interaction with *s*_{1} = *s*_{2} = +, for which *ω*_{0} = *ω*_{1} + *ω*_{2} and similar for **k** holds, 2) two mixed terms with *s*_{1} = +, *s*_{2} = − or *s*_{1} = −, *s*_{2} = +, for which *ω*_{0} = *ω*_{1} − *ω*_{2} and similar for **k** holds, and 3) the remaining triad interaction, which never becomes resonant. The two mixed terms with *s*_{1} = +, *s*_{2} = − or *s*_{1} = −, *s*_{2} = + are identical and are denoted as difference interaction.

Here we use the same consistent numerical representation of our rederivation of the kinetic equation as detailed in Eden et al. (2019b). Both resonant and nonresonant interactions are calculated on a wavenumber grid corresponding to the model grid for Δ(*ω*, *t*) at different times *t*, using method 2 of Eden et al. (2019b). We choose a grid with half as many grid points but the same domain as the model (the calculations take several days on several thousand processors of the DKRZ supercomputer in Hamburg, Germany), because we except the triads at higher wavenumbers to be effected by the damping in the model. We calculate ∂_{t}^{±} ≡ ∂_{t}*s*_{0} = ±, for the geostrophic mode ∂_{t}^{0} and also the mixed triad interactions between both. Figure 3a shows ∂_{t}^{0} for *t* = 0.04 day, which is close to ∂_{t}^{0} from the numerical model (Fig. 2b). A similar nonlinear adjustment of the GM spectrum has been found in Eden et al. (2019a) for the hydrostatic equations of motion. Their Fig. 6 shows that after about *t* = 0.1 day ∂_{t}^{0} decreases and does not contribute anymore.

(a) Geostrophic energy transfer ∂_{t}^{0} from the kinetic equation for *t* = 0.04 day in the time response function Δ(*ω*, *t*). (b) As in (a), but wave energy transfer ∂_{t}*t* = 0.5 day. (c) The ∂_{t}*ω*/*f* = 1.25, 2, 3, 5, 10, and 20. Regions where |∂_{t}*ω*/(2*π*) are shaded in transparent dark gray, and regions where max(*ν*_{S,D}) > *ω*/(2*π*) are shaded in transparent light gray. In (a)–(c), gray areas without overlaid colors correspond to *ω* and *m* values not covered by the respective grid.

Citation: Journal of Physical Oceanography 50, 4; 10.1175/JPO-D-19-0022.1

(a) Geostrophic energy transfer ∂_{t}^{0} from the kinetic equation for *t* = 0.04 day in the time response function Δ(*ω*, *t*). (b) As in (a), but wave energy transfer ∂_{t}*t* = 0.5 day. (c) The ∂_{t}*ω*/*f* = 1.25, 2, 3, 5, 10, and 20. Regions where |∂_{t}*ω*/(2*π*) are shaded in transparent dark gray, and regions where max(*ν*_{S,D}) > *ω*/(2*π*) are shaded in transparent light gray. In (a)–(c), gray areas without overlaid colors correspond to *ω* and *m* values not covered by the respective grid.

Citation: Journal of Physical Oceanography 50, 4; 10.1175/JPO-D-19-0022.1

(a) Geostrophic energy transfer ∂_{t}^{0} from the kinetic equation for *t* = 0.04 day in the time response function Δ(*ω*, *t*). (b) As in (a), but wave energy transfer ∂_{t}*t* = 0.5 day. (c) The ∂_{t}*ω*/*f* = 1.25, 2, 3, 5, 10, and 20. Regions where |∂_{t}*ω*/(2*π*) are shaded in transparent dark gray, and regions where max(*ν*_{S,D}) > *ω*/(2*π*) are shaded in transparent light gray. In (a)–(c), gray areas without overlaid colors correspond to *ω* and *m* values not covered by the respective grid.

Citation: Journal of Physical Oceanography 50, 4; 10.1175/JPO-D-19-0022.1

Figure 3b shows ∂_{t}*t* = 0.5 day. As in the model, wave energy is transferred predominantly from 2 < *ω*/*f* < 3 to lower frequencies but larger *m*, and also with similar magnitudes as in the numerical model. For *ω* → *N* and *m* > (5–10)*m*_{*}, differences between model and kinetic equation can be noted. Considering only the resonant interactions in the kinetic equation allows for a larger spectral domain since fewer triad interactions need to be calculated. Figure 3c shows the resonant ∂_{t}*π∂T** $\mathcal{E}$ω*) (not shown) agrees with the one in Lvov et al. (2012) (their Fig. 5, for the same spectrum but also using the hydrostatic approximation). The Langevin rate [not shown; Eq. (35) in Müller et al. (1986)] agrees with the one shown in Müller et al. (1986) (their Fig. 6, for the same spectrum as here, but also using the hydrostatic approximation). A difference from the previous calculations is the large transfers showing up for

*ω*→

*N*(hardly seen under the gray shading), but a direct comparison is hampered by the hydrostatic approximation made in the previous studies or because the spectrum was truncated for large

*ω*.

## 4. Mechanism of resonant wave–wave interactions

McComas and Bretherton (1977) suggest three dominant kinds of resonant triad interactions in the scattering integral with different characteristics: elastic scattering (ES) with *ω* ≈ *ω*′ ≫ *ω*″, *m* ≈ −*m*′ ≈ |*m*″/2|; induced diffusion (ID) with *ω* ≈ *ω*′ ≫ *ω*″, *m* ≈ *m*′ ≫ |m″|; and parametric subharmonic instability (PSI) with *ω*′ ≈ *ω*″ ≈ *ω*/2. The triads satisfy the conditions *ω* = *ω*′ ± *ω*″ and *m* = *m*′ ± *m*″, where all three frequencies and wavenumbers can become *ω*_{0,1,2} and *m*_{0,1,2} in the scattering integral. Types ES and ID have their names from well-known processes in particle physics that are also described by a similar scattering integral, whereas PSI is a classic wave–wave interaction.

*T*

^{+}and

*T*

^{−}result from the coefficients

*c*in Eq. (4) and denote the scattering cross sections for sum and difference interaction, respectively, and are given in, for example, Olbers (1976) or Eden et al. (2019b), and where

*D*instead of

*S*. Terms

*ω*and

*m*and represent the frequency and wavenumber of the dominant energy transfers. A part of

*S*(and

*D*) with large magnitude but opposite sign could cancel in the integral in Eq. (5) but could largely affect

*S*| and |

*D*|. To test this effect, we also weighted with max(0,

*S*) and −min(0,

*S*), and similar for

*D*, and found no large differences in

Figure 4 shows *ω*, the PSI mechanism clearly dominates the sum interaction. For small *ω* and/or small |*m*|, *m*|, and if the energy transfer is negative, that is, from *ω* directed toward *m*|. For the difference interaction (Fig. 5), we find for large *ω* and large |*m*| (and thus large *k*) that *ω* ≈ *m*| ≈ *ω* and |*m*| do we find the condition for ID |*m*| ≫ *ω* and large enough |*m*|, we find PSI again, since here *ω*, but with frequencies exchanged relative to the sum interaction. For small *ω* and small |*m*| we find *ω* and

(a) Energy transfer weighted frequency *ω*. (b) As in (a), but wavenumber *m*. (c) Energy transfer ∂_{t}*ω*/*f* = 1.25, 2, 3, 5, 10, and 20. Gray areas correspond to *ω* and *m* values not covered by the grid.

Citation: Journal of Physical Oceanography 50, 4; 10.1175/JPO-D-19-0022.1

(a) Energy transfer weighted frequency *ω*. (b) As in (a), but wavenumber *m*. (c) Energy transfer ∂_{t}*ω*/*f* = 1.25, 2, 3, 5, 10, and 20. Gray areas correspond to *ω* and *m* values not covered by the grid.

Citation: Journal of Physical Oceanography 50, 4; 10.1175/JPO-D-19-0022.1

(a) Energy transfer weighted frequency *ω*. (b) As in (a), but wavenumber *m*. (c) Energy transfer ∂_{t}*ω*/*f* = 1.25, 2, 3, 5, 10, and 20. Gray areas correspond to *ω* and *m* values not covered by the grid.

Citation: Journal of Physical Oceanography 50, 4; 10.1175/JPO-D-19-0022.1

Energy transfer weighted frequency and wavenumber for the difference interaction. Note that *ω* and that *ω*/*f* = 1.25, 2, 3, 5, 10, and 20. Gray areas correspond to *ω* and *m* values not covered by the grid.

Citation: Journal of Physical Oceanography 50, 4; 10.1175/JPO-D-19-0022.1

Energy transfer weighted frequency and wavenumber for the difference interaction. Note that *ω* and that *ω*/*f* = 1.25, 2, 3, 5, 10, and 20. Gray areas correspond to *ω* and *m* values not covered by the grid.

Citation: Journal of Physical Oceanography 50, 4; 10.1175/JPO-D-19-0022.1

Energy transfer weighted frequency and wavenumber for the difference interaction. Note that *ω* and that *ω*/*f* = 1.25, 2, 3, 5, 10, and 20. Gray areas correspond to *ω* and *m* values not covered by the grid.

Citation: Journal of Physical Oceanography 50, 4; 10.1175/JPO-D-19-0022.1

The energy transfer at low frequencies is dominated by the PSI-like interactions. Figure 4c shows ∂_{t}*ω*_{1} − *ω*_{2}| smaller than 30% of (*ω*_{1} + *ω*_{2})/2, and only the positive difference interactions |*ω*_{0} − *ω*_{2}| smaller than 30% of (*ω*_{0} + *ω*_{2})/2. This implies that most of the energy transfers for *m* < (5–10)*m*_{*} seen in the model are also generated by PSI-like mechanisms.

*m*but in particular for ID at large frequencies. Holloway (1978, 1980, 1982) and Müller et al. (1986) argue that the Bolzmann rate ∂

_{t}

*ν*

_{S,D}) <

*ω*

_{0}/(2

*π*) is violated (transparent shading in Fig. 3c) for

*ω*→

*N*and for

*m*> (5–10)

*m*

_{*}, which corresponds for

*m*

_{*}= 0.01 m

^{−1}to a vertical wavelength of ~125–60 m. The differences between model and kinetic equation for

*m*> (5–10)

*m*

_{*}can therefore be explained by the kinetic equation becoming invalid.

## 5. Summary and discussion

We find good agreement between the predictions of the kinetic equation for the spectral energy transfers in the GM spectrum Eq. (1) and direct simulations of internal gravity wave–wave interactions in a three-dimensional, nonhydrostatic numerical model for a certain range of frequencies and wavenumbers. Such an agreement between the kinetic equation for gravity waves and numerical model simulations has not been shown before to our knowledge. Because the signal by wave–wave interactions is relatively small in the model, it is necessary for this validation to remove the effect of the random phase of the initial conditions in physical space by using an ensemble model integration. We also find it necessary to use eigenvalues and eigenvectors appropriate to the discrete numerical model for the initialization with the GM spectrum, for the model diagnostic, and for the calculation of the interaction coefficients for the kinetic equation. Part of the wave energy generates initially the geostrophic mode. After that inverse nonlinear geostrophic adjustment, wave energy is decreased at 2 < *ω*/*f* < 3 and increased at smaller *ω* and larger *m* by the nonlinear terms. The initial increase in the geostrophic mode and the transport pattern of wave energy for small *m* is reproduced by the kinetic equation including nonresonant interactions. The latter is similar to the result including only resonant interactions, but differences show up for both between the kinetic equation and the model simulations for *m* > (5–10)*m*_{*}, where *m*_{*} represents the dominant vertical wavenumber in the GM spectrum.

The resonant energy transfers predicted by the kinetic equation are in agreement with previous studies, although here calculated without using the hydrostatic approximation and with considerably larger numerical effort. Energy transfer weighted frequencies and wavenumbers show that parametric subharmonic instability (PSI) triad interactions are mostly responsible for the energy transfers at low frequencies toward large *m*, but that ID triad interactions become important for larger *ω* and large *m*, but also for small *ω* and very small *m*, as suggested by McComas and Bretherton (1977). The triad interactions for *m* > (5–10)*m*_{*}, however, are found to violate the weak interaction assumption inherent to the kinetic equation, as anticipated by Holloway (1978, 1980, 1982) and Müller et al. (1986), which might explain the difference to the model simulations.

Due to the large computational costs, we were not able to increase the model resolution to fully cover the large scale separation important for ID. However, in the range of *ω* and *m* which we do cover (with considerable computational effort) and which is not affected too much by numerical damping and grid dispersion errors, there are no indications of significant energy transfers beyond the dominant energy transfers from 2 < *ω*/*f* < 3 toward smaller *ω* and larger *m* generated by PSI triad interactions. Our model results thus suggest no large role of ID for energy transfers in the GM spectrum, but this needs to be checked with models with higher resolution in the future. Sugiyama et al. (2009) find in a forced model simulation of internal wave interaction also predominantly energy transfers to near inertial frequencies by PSI triad interactions, but there the model is two dimensional and of coarser resolution as here.

Ignoring the caveat of model resolution and the role of ID, the observed GM spectrum demands energy transfers from 2 < *ω*/*f* < 3 to smaller *ω* and larger *m*. Note that it was shown in Eden et al. (2019b) that varying parameters in the GM spectrum Eq. (1) like the Coriolis parameter *f*, the bandwidth *c*_{*}, or the spectral slope *r*, this coherent energy transfer pattern stays very similar. In steady state, these energy transfers need to be balanced by other terms in a more complete spectral energy balance in physical and spectral space. We believe that this question forms the major challenge to understand the global gravity wave field in the ocean and its effects. Already in the early studies by Müller and Olbers (1975), Olbers (1976), McComas and Bretherton (1977), and Pomphrey et al. (1980) it was envisioned that spectral regions of ∂_{t}_{t}_{t}*m* where waves can break and dissipate this viewpoint is at least partly satisfied.

However, we also know today that the wave field is predominantly generated by the interaction of the barotropic tide with topography, forcing waves with fixed frequency *ω*_{T}. at a rate of 1–2 TW (Wunsch and Ferrari 2004). Recent estimates by Falahat et al. (2014) and Vic et al. (2019) point toward somewhat lower values of 0.5 TW for the first 10 baroclinic modes.

Other forcing processes as inertial pumping at the bottom of the surface mixed layer induced by winds appear to be much smaller (e.g., Rimac et al. 2013). Internal tidal waves can be refracted and scattered at the bottom or the balanced flow (Müller and Xu 1992; Savva and Vanneste 2018), changing their wavenumbers while their frequency *ω*_{T} remains constant. As long as *m* remains small, the tidal waves cannot be directly dissipated by these processes. This route can then only be established by the interaction of the internal tidal waves with the continuum described by the GM spectrum.

We find that the GM spectrum in steady state demands an energy source in the spectral region of 2 < *ω*/*f* < 3. To adjust the Coriolis parameter *f* in order to locate the tidal forcing frequency *ω*_{T} in this frequency interval implies for the most important half-daily tide roughly that 20° < |*ϕ*| < 30° (or 10° < |*ϕ*| < 15° for the daily tide), where *ϕ* denotes geographical latitude. To obtain a balanced energy spectrum in steady state, we could invoke therefore a nonlocal spectral energy balance, in which the tidal wave energy needs to propagate toward the latitudinal window where 2 < *ω*_{T}/*f*(*ϕ*) < 3. Such a nonlocal spectral energy balance is given by the radiative transfer equation for gravity waves discussed in, for example, Olbers et al. (2012), where divergences of energy transports in physical and spectral space balance forcing, dissipation, the energy transfers by the wave–wave interactions, or the rate of change of the spectral energy *ω*_{T}/*f* < 3. We might then also expect an accumulation of wave energy in the corresponding latitudinal window.

*N*

_{fit}is a smooth quadratic fit to the buoyancy frequency profile

*N*(

*z*) and

*m*) are obtained from

*ξ*

_{z}using the polarization relation of internal gravity waves. Integrating

*m*) over a suitable wavenumber range allows to estimate the total wave energy as described in Pollmann et al. (2017). Energy transfer due to wave dissipation is estimated from the integrated strain spectra

*ξ*

_{z}(

*m*) following the so-called finestructure method (e.g., Gregg 1989; Kunze et al. 2006; Polzin et al. 2014). The figure shows that both energy and dissipation feature large meridional variations. In the Pacific Ocean, maxima in the expected latitudinal range can be seen in energy and dissipation, although this is not so clear for the other ocean basins. We suggest that further work using the kinetic equation and observations is needed to explain these large regional variations and to understand how the wave–wave interactions together with propagation, dissipation and forcing shape the global gravity wave energy and its dissipation in the ocean.

Latitudinal variations of (a) internal wave dissipation rates and (b) internal wave energy levels derived from Argo profiles of temperature, salinity, and pressure using the so-called finestructure method (e.g., Gregg 1989; Kunze et al. 2006) following the same procedure as detailed in Pollmann et al. (2017). Dots denote estimates at different longitudinal positions at a depth of 250–500 m, and lines show their average in the different ocean basins. Results deviating by more than a factor of 3 from the average at a given latitude are considered to be outliers and are disregarded in the calculation of the mean.

Citation: Journal of Physical Oceanography 50, 4; 10.1175/JPO-D-19-0022.1

Latitudinal variations of (a) internal wave dissipation rates and (b) internal wave energy levels derived from Argo profiles of temperature, salinity, and pressure using the so-called finestructure method (e.g., Gregg 1989; Kunze et al. 2006) following the same procedure as detailed in Pollmann et al. (2017). Dots denote estimates at different longitudinal positions at a depth of 250–500 m, and lines show their average in the different ocean basins. Results deviating by more than a factor of 3 from the average at a given latitude are considered to be outliers and are disregarded in the calculation of the mean.

Citation: Journal of Physical Oceanography 50, 4; 10.1175/JPO-D-19-0022.1

Latitudinal variations of (a) internal wave dissipation rates and (b) internal wave energy levels derived from Argo profiles of temperature, salinity, and pressure using the so-called finestructure method (e.g., Gregg 1989; Kunze et al. 2006) following the same procedure as detailed in Pollmann et al. (2017). Dots denote estimates at different longitudinal positions at a depth of 250–500 m, and lines show their average in the different ocean basins. Results deviating by more than a factor of 3 from the average at a given latitude are considered to be outliers and are disregarded in the calculation of the mean.

Citation: Journal of Physical Oceanography 50, 4; 10.1175/JPO-D-19-0022.1

## Acknowledgments

This paper is a contribution to the Collaborative Research Centre TRR 181 “Energy Transfer in Atmosphere and Ocean” funded by the Deutsche Forschungsgemeinschaft (DFG, or German Research Foundation) Projektnummer 274762653. The numerical calculations have been performed on the “High Performance Computing system for Earth system research (HLRE-3)” at the Deutsches Klimarechenzentrum (DKRZ) of Hamburg, Germany.

## APPENDIX A

### Numerical Model

We use a model based on the incompressible equations of motions for a rotating and stratified fluid. The Earth’s rotation and stratification stability frequencies, *f* = 10^{−4} s^{−1} and *N* = 50*f*, respectively, are constant and representative for the midlatitude interior ocean. The numerical discretization is given in Eq. (A1). The model domain is triple periodic with 1001 grid points in all directions and 50 km × 50 km × 5 km extent and 50-m lateral and 5-m vertical grid resolution. Dissipation is given by explicit harmonic diffusion and friction with coefficients of 2.5 × 10^{−3} m^{2} s^{−1} (2 × 10^{−5} m^{2} s^{−1}) in lateral (vertical) direction, which can be seen to parameterize the effect of smaller-scale turbulent motions. There is also implicit damping by the time stepping scheme which is chosen as a quasi-second-order Adam–Bashforth interpolation with adjusted weights to allow for stable simulations of gravity waves of highest frequency *N* with a time step of 20 s. The explicit damping affects predominantly the high wavenumbers, while the implicit damping affects the high frequencies, and both effects are proportional to

*i*,

*j*, and

*k*denote discretization in the

*x*,

*y*, and

*z*directions, and finite-differencing operators

_{x}is the grid spacing in the

*x*direction, and averaging operators

## APPENDIX B

### Eigenvalues and Eigenvectors

**k**= (

*k*

_{x},

*k*

_{y},

*m*),

**x**= (

*x*,

*y*,

*z*) = (

*i*Δ

_{x},

*j*Δ

_{y},

*k*Δ

_{z}), and similar for the other variables, the Fourier transform of Eq. (A1) becomes

_{x}→ 0. The left-hand side is linear in the wave amplitudes, and the nonlinear right-hand side involves products of amplitudes integrated over all wavenumbers that are not explicitly shown here.

_{x}, Δ

_{y}, Δ

_{z}→ 0, the finite roots converge to the internal gravity wave dispersion relation. The corresponding right and left eigenvectors are given by

*s*= 0, ± and with the normalization

_{x}, Δ

_{y}, Δ

_{z}→ 0.

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