1. Introduction
a. Motivation
Breaking internal gravity waves (IWs) are one of the most important processes underlying diapycnal mixing, which is thought to be a primary mechanism for closing and regulating the meridional overturning circulation (Wunsch and Ferrari 2004); in turn, the overturning circulation is a key regulator of the Earth system’s climate (IPCC 2013). IW energy spectra are readily observable along one dimension in the ocean via dropped, floated, or dragged CTD probes and moored profilers (Alford et al. 2017). Phenomenological models of energy spectra associated with these observations were developed in the 1970s by Garrett and Munk (GM) (Garrett and Munk 1972, 1975, 1979), and theoretical models of the detailed dynamics that give rise to such spectra have been the subject of research in the ensuing decades (McComas 1977; Müller et al. 1986; Dematteis et al. 2022; Olbers 1976). The theoretical basis of IW dynamical processes inform stateoftheart diapycnal mixing models [such as Internal Wave Dissipation, Energy and Mixing (IDEMIX); Olbers and Eden 2013; Eden and Olbers 2014] in global ocean simulations. However, verification of such theory via observations is difficult due to the intractability of rapidly measuring detailed two or threedimensional flow fields.
More recently, global (e.g., Arbic et al. 2010, 2018; Müller et al. 2015; Rocha et al. 2016b; Arbic 2022) and regional (e.g., Nelson et al. 2020; Pan et al. 2020; Thakur et al. 2022) ocean models have begun to permit detailed representations of the largest scales of the IW continuum due to a combination of tidal forcing, coupling to atmospheric wind and buoyancy forcing fields, and higher horizontal and vertical resolution (Arbic 2022). The highestresolution regional models are able to permit detailed fourdimensional sampling of realistic supertidal internal wave fields (Thakur et al. 2022; Pan et al. 2020; Eden et al. 2020) that is not possible from observations, potentially a watershed breakthrough for IW modeling and theoretical innovation.
This paper will work to bring theoretical understanding of IW interactions into alignment with the internal wave dynamics as revealed in a highresolution regional model using a novel asymmetric spectral flux decomposition [described in section 2b(2)]. It will also use both the spectral flux methods and numerical model output to interpret observations of spectral transfers that are made in fewer than three spatial dimensions, specifically in Sun and Pinkel (2012). A distinguishing feature of this work is that it will focus on the supertidal band of internal gravity waves and on interactions of internal waves with the mesoscale eddy field.
b. Resonant nonlinear interactions in theoretical IW models
The GM spectrum (Garrett and Munk 1975) does not attempt to describe the means of forcing, dissipation, spectral inhomogeneities due to tidal harmonics, or spatial inhomogeneities. It also is completely uninvolved with details of the nonlinear interactions that give rise to such a spectrum. McComas and Bretherton (1977) identified different types of nonlinearinteraction mechanisms likely to be important to the development of the IW continuum spectrum, including elastic scattering (ES), parametric subharmonic instability (PSI), and induced diffusion (ID). Dematteis and Lvov (2021) and Dematteis et al. (2022) provide a more restrictive definition of ID and introduce spectrally local interactions (LI) as an additional mechanism of consequence. This paper will focus on ID and, to a lesser extent, LI in part because they are most easily diagnosed from numerical model output with asymmetric spectral fluxes.
ID involves highwavenumber, highfrequency waves scattering off of nearinertial, lowwavenumber waves and inducing diffusion of wave action,
This paper differentiates between such conflicting accounts of ID by decomposing the vertical^{2} spectral kinetic energy flux into components based on the frequency band of the energy source. Vertical spectral fluxes are used because the various theoretical models of IW cascades are consistent in their predictions that within the IW “inertial range,” energy flow is to smaller vertical wavenumbers (Müller et al. 1986; Dematteis et al. 2022). Our use of frequencydecomposed spectral fluxes also allows for the traditional definition of ID (which conserves and diffuses wave action in the supertidal band) to be decomposed into two components: 1) downscale kinetic energy diffusion within the supertidal band (ID_{diff}) and 2) the compensating energy exchanged with nearinertial and tidal frequencies (ID_{comp}). A separate but conceptually related mechanism, downscale kinetic energy diffusion that is induced by eddy fields instead of wave fields, is referred to as ID_{eddy}. Such labels are included here for reference, but they are not rigorously defined until section 2b(2).
c. Observations of resonant nonlinear interactions
Sun and Pinkel (2012) used sonar and CTD observations near Hawaii to directly quantify nonlinear energy transfers among supertidal internal waves. They found coherent energy transfers that were inconsistent with ID. We will label these energy transfers as a separate mechanism, “SP,” throughout the paper but will hold off on discussing the physical meaning of this signal. They were also unable to find a signal where they expected to see ID on verticalwavenumber bispectra. This paper produces a similar bispectrum in section 3b in an attempt to reproduce the most prominent feature from their observations while also explaining why Sun and Pinkel (2012) do not observe ID. In computing bispectra from model output, the assumption of Sun and Pinkel (2012) that the verticalgradient component of the spectral KE transfer represents the entire transfer is tested.
d. Resonant nonlinear interactions in IWpermitting simulations
Regional models that permit a partially resolved IW continuum must account for significant spatial energy flux of IWs across the regional boundaries (Mazloff et al. 2020). Numerous regional models have a vigorous mesoscale eddy field but lack robust internal wave and internal tide forcing at the boundary conditions, (e.g., Nugroho et al. 2018; Renault et al. 2021; Wang et al. 2021). A straightforward means of accounting for this is to use IWpermitting globalmodel output to set the regional boundary conditions. Global models require simultaneous atmospheric and tidal forcing as well as sufficient horizontal and vertical resolution in order to energize a robust IW field, conditions that have only begun to be implemented recently (Arbic et al. 2010; Müller et al. 2015; Rocha et al. 2016b,a; Arbic et al. 2018; Arbic 2022). Nelson et al. (2020), Pan et al. (2020), and Thakur et al. (2022) used a regional model with tidal and wind forcing, and, importantly, imposed IW forcing at the boundaries from an IWpermitting global model. These are the first and, along with two recent papers (Siyanbola et al. 2023; Delpech et al. 2023), are the only published regional models that attempt to resolve the IW boundary flux issue noted by Mazloff et al. (2020) at the time of submission. The present paper will continue using this approach.
Idealized simulations of flow realizations of the GM IW spectrum present an alternative approach for studying IW theory through numerical simulations. Eden et al. (2020) use this technique with a very fine 50m horizontal grid spacing and are able to sidestep the problem of boundary conditions by diagnosing IW interactions on a periodic domain. They find PSI to be the dominant mechanism for moving energy from tidal frequencies in the range of 2f_{0}–2.5f_{0} (based on their Fig. 4c) and ID to play a relatively small role. This represents a promising approach of future study to which the asymmetric spectral fluxes of this paper could be applied.
As IWpermitting models adopt higher resolutions, they begin to resolve more of the IW continuum, both in global models (Müller et al. 2015; Savage et al. 2017) and regional models (Nelson et al. 2020). Pan et al. (2020) finds that IW dispersion curves in a highresolution regional model are clearly defined over two orders of magnitude of frequency and horizontal wavenumbers, much greater than in available global models. They further find evidence for a dynamically significant ID in the IW continuum, which motivates the work of the present paper to study such dynamical processes.
Highresolution IWpermitting regional models now have the potential to validate recent advances in theoretical understanding and to study challenges faced in observations of internal wave scattering. Some open questions that we will address are:

At what model resolutions do ID and LI become significant?

Is ID associated with forward or inverse frequency cascade?

How does ID_{diff} compare to ID_{comp}?

Why do Sun and Pinkel (2012) not see ID in their observations? What do they see?

How important are ID_{eddy} and other mechanisms that do not fit into existing categories of nonlinear wave–wave interactions?
2. Methods
a. The model
The numerical model simulates a region north of the Hawaiian archipelago (Fig. 1). It includes wind and astronomical forcing as well as rigidly imposed nested Dirichlet boundary conditions for all prognostic fields, which are provided from a similarly forced global ocean simulation of the Massachusetts Institute of Technology general circulation model (MITgcm), often denoted LLC4320 (Rocha et al. 2016b; Arbic et al. 2018).^{3} As a result, it partially resolves the spectrum of nearinertial oscillations, internal tides and supertidal internal waves. The study area features a northwardpropagating internaltidal beam generated in the French Frigate Shoals as well as interactions with the Musician Seamounts. The region also features areas north and south of the critical latitude for PSI at 28.8°N (Alford et al. 2007). We use a hydrostatic formulation of MITgcm (Adcroft et al. 2015).
Previous work based on wavenumber and frequency spectra has demonstrated that this regional Hawaii simulation permits a partial representation of the internal wave cascade (Nelson et al. 2020; Pan et al. 2020; Thakur et al. 2022). Additionally, there is evidence that the model accurately represents the dispersion relation of the IWs, as indicated by the energy spectra in Fig. 8 of Pan et al. (2020). The mechanisms of wave damping in this model are the subject of ongoing research; most recently Thakur et al. (2022) found that the highverticalwavenumber end of the IW continuum is made more accurate by permitting greater energization at those scales. Despite having an incomplete picture of the exact details of IW damping, the general argument to be made here is that downscale spectral flux of IW KE will be in balance with damping at the highest vertical wavenumbers and will permit an approximately physically accurate picture of the nonlinear interactions at intermediate vertical wavenumbers. We do not explore spectral balance in horizontal wavenumber in this paper.
Another notable limitation of the model at representing IW dynamics is that it is hydrostatic. This will introduce error in the limit that ω → N, where N is the buoyancy frequency, and with
The simulation employs a finitevolume solver. The horizontal eddyviscosity scheme is a modified form of the Leith scheme (FoxKemper and Menemenlis 2008), while interior and mixed layer dissipation and diffusivity is handled by the κprofile parameterization (KPP; Large et al. 1994). The Leith scheme and KPP are the dominant mechanisms acting in the ocean interior to dissipate KE in the modeled IW continuum. Wind forcing is updated every 6 h, while boundary conditions in the regional model are updated hourly. A staggered, secondorder (explicit) Adams–Bashforth timestepping method is used for all terms except the vertical viscosity (from the KPP scheme), which is treated implicitly using a backward timestepping scheme. The model grid is stretched substantially in the vertical direction (vertical grid spacing is 0.333 m at the surface and 133 m at the domain bottom in the highestresolution run). The regional model was run for 106 days, from 1 March 2012 until 15 June 2012, and output was collected every 20 min for the last 7 days of this period and used for the analysis presented in this work. The regional model was run with four different resolutions [the same ones as in Nelson et al. (2020)]:

88 vertical levels and 2km horizontal grid spacing,

88 vertical levels and ∼250m horizontal grid spacing,

264 vertical levels and 2km horizontal grid spacing, and

264 vertical levels and ∼250m horizontal grid spacing.
b. Spectral KE budgets
The present work is focused on attaining a picture of nonlinear IW triadic scattering mechanisms as far from spectrally inhomogeneous forcing as possible, that is, deep within the “inertial range” of the IW continuum. In particular, nonlinear interactions that are most active and easily discernible in this spectral range, ID and LI, will be diagnosed, while PSI and ES will not be. The strategy for attaining such a picture is to construct a partial spectral KE budget of terms that are expected to be dominant at midtohigh vertical wavenumbers in the IW continuum: nonlinear fluxes, decomposed into contributions from different mechanisms based on frequency, and dissipation. This study will focus exclusively on KE. In future work, the asymmetric spectral flux methods used in this paper [see section 2b(2)] could be extended to apply to nonlinear scattering of buoyancy perturbations as well, thereby handling potential energy budgets.
The flow field is decomposed into spectral bands: the lowpass band (LP) with ω < 0.8f_{0}, a bandpass of nearinertial and tidal frequencies (BP) with 0.8f_{0} < ω < 2.5f_{0}, and a supertidal highpass band (HP) with 2.5f_{0} < ω. The HP band at high vertical wavenumber is of primary interest for the IWcontinuum energetics. To the extent that an inertial range exists in the IW continuum, forcing and dissipation will exhibit scale separation and energy will move through the intermediate scales via nonlinear interactions among waves. The dissipation mechanisms and the vertical spectral flux should be in approximate balance at small scales. To compute these vertical spectral kinetic energy budgets, the velocity evolution due to each term in the governing equations must first be interpolated to a uniform vertical grid that has the same number of grid points. This sampling rate is based on numerical experiments with synthetic spectra.
The LP band includes both mesoscale eddies and the background timemean flow. Because a mean flow is included in the flow solution and is not removed for analysis, IW frequencies are subject to Doppler shifting. The Doppler shift, if large, could affect the frequency analysis. However, in our regional model, this appears not to be the case–consider the following: In Pan et al. (2020) and Nelson et al. (2020), the dispersion relation and consistency relation of the IW field were studied, which agree with the relations caused only by waves and do not show obvious Doppler shift effects. When the Doppler shift is small, it can be considered as a small and random correction of the dispersion relation, under which a diffusion equation of wave action can be derived as in Savva and Vanneste (2018) and Dong et al. (2020). This diffusion effect is exactly what we studied in the paper (i.e., we have considered weak Doppler shift in our analysis.)
1) Terms and boundary conditions
2) Spectral flux and transfers
Spectralenergyflux methods can be used to study the direction of energy cascades across scales in ocean models and observations (e.g., Scott and Wang 2005; Schlösser and Eden 2007; Capet et al. 2008; Sun and Pinkel 2012; Arbic et al. 2013). This paper will utilize a decomposed version of the spectral energy fluxes and transfers that has some distinct benefits over previously used versions: the isolation of different spectral bands, a decomposition into interaction diagrams, and spatial resolution of the spectral exchanges in the direction orthogonal to the spectra computed [see section 2b(4)]. Also, while many other spectral methods, including implementations of waveturbulence theory (WTT), spectral fluxes, and coarsegraining, often^{4} consider energy exchange among sets of three modes (see appendix A), the spectralflux decomposition presented in this paper will use an asymmetric version of scattering diagrams that isolates the exchange of energy between two modes mediated by a third mode, allowing for the direction and magnitude of energy exchange between any two scales to be exactly discerned.
Induced diffusion, as it is described in McComas and Bretherton (1977), can be broken up into two subprocesses, ID_{diff} and ID_{comp}. Each has a corresponding spectral flux:
Separately, one can consider KE diffusion that is induced by catalyst modes in the eddy field (LP) rather than the wave field (BP), which we label
3) Bispectra
Now that these bispectra have been defined, they can be interpreted along the same lines as the spectral flux decomposition in section 2b(2). Induced diffusion, as defined in McComas and Bretherton (1977), would be contained in
A likely reason that the bispectra of Sun and Pinkel (2012) was not identified by the authors as measuring ID_{comp} is that in some versions of WTT the scattering coefficients are symmetric between the source and catalyst wavenumbers, as discussed in appendix A. The observations of Sun and Pinkel (2012) do not symmetrize between the catalyst and source modes, meaning these two bispectra may have actually been substantially different from one another and would not necessarily correspond to WTT predictions. Rather, they attempted to measure bispectra corresponding to
4) Spatial distributions of integrated spectral budgets
5) Spectral transfers versus wavenumber and frequency
3. Results and discussion
All results in this paper are computed from an average of five subregions of interest, shown in Fig. 3. Subregions, rather than the entire model domain, are used to compute results because of the prohibitive computational cost and data management requirements associated with the highresolution case, which has 2.3 billion points of 3D data over the entire model domain. These subregions were chosen to capture a sample of rough and flat topography (visible in the contours) and a range of supertidalfrequency (HP) kinetic energies (Fig. 3a), supertidal (HP) vertical spectral kinetic energy fluxes (Fig. 3c), and ratios of nearinertial and tidal to eddy (BP/LP) kinetic energy (Fig. 3b). The five subregions in question were the only ones results were computed for. All results in this paper are computed with units of inverse volume as opposed to mass. We use a background density of ρ_{0} = 1027.5 kg m^{−3}. When applicable, the domainaveraged Coriolis frequency, f_{0} = 6.85 × 10^{−5} rad s^{−1} is also used. Also, on the basis of testing different values of ξ to define scale separation in ID, LI, and SP (appendix F), we choose to use ξ = 2.
The frequency spectrum of the highresolution case (∼250 m × 264 vertical levels) is shown in Fig. 4a with shading and separate curves indicating the three frequency bands discussed in section 2b. The band lines are shown at all frequencies to convey the overlap left after a 12thorder bandpass filter is applied. Nelson et al. (2020) have previously compared frequency spectra of these runs with observations and the GM spectrum (Garrett and Munk 1975). The supertidal frequency spectrum in the highresolution simulation is slightly steeper than the asymptotic ω^{−2} prediction of GM while the spectrum from the lowresolution model (not shown) falls off much more steeply. In vertical wavenumber space (Fig. 4b), the eddy (LP) frequencies are energetically dominant at low and high wavenumbers while the BP frequencies are energetically dominant at intermediate wavenumbers. HP contains the least kinetic energy—an order of magnitude less than BP. There is some energy buildup at the vertical gridscale in both simulations in Fig. 4b. This buildup is likely a reflection of the sharp gradients that exist at the top and bottom of the domain and of the (intentional) choice to analyze the entire water column rather than tapering it, as is done in Thakur et al. (2022), Nelson et al. (2020), Pan et al. (2020). Finally, note that KE of the HP band is much greater at higher resolution, while the energy levels of BP and LP do not change very much between the different resolution cases.
a. Integrated vertical spectral KE budgets
As described in section 2b, the downscale vertical spectral KE flux and integrated vertical spectral dissipation transfers should be in balance at high vertical wavenumbers, at least to the extent that scale separation exists in the internal wave continuum. Assessing details of the budget at these wavenumbers is of primary importance. (In contrast, forcing terms are active at the lowest vertical wavenumbers such that the spectral flux and dissipation should not be in balance at those wavenumbers). To emphasize details at high wavenumbers, budgets are presented on linear axes as opposed to logarithmic axes, the latter of which would emphasize the lowest wavenumbers.
The partial integrated vertical spectral kinetic energy budget of the highresolution case is shown in Fig. 5, with the left and right panels corresponding to all frequencies and HP frequencies. The overall amount of dissipation can be assessed by the value of the dissipation curve at m = 0, while the amount of energy that is dissipated at a given vertical wavenumber is proportional to the slope of the dissipation curve. The dissipation, spectral flux, and advection into the domain are in balance at the highest wavenumbers for both all frequencies (Fig. 5a) and the supertidal (Fig. 5b) frequencies. The latter indicates that the internal waves are energized at small and intermediate but not high vertical wavenumbers; energy in the IW field at high vertical wavenumbers gets there through nonlinear interactions among waves and eddies in the flow. This does not indicate that scale separation exists between forcing and dissipation; there is no classic inertial range at the resolutions used in this paper. However, the absence of forcing at the highest wavenumbers indicates that the IW continuum has some cascadelike properties and that an LEStype IW closure may be appropriate.
The frequencydecomposed vertical spectral KEflux reflects specific types of nonlinear interactions that underlie the ocean’s IW continuum. Within the HP band, ID_{diff} and LI are expected to play a dominant role (Dematteis and Lvov 2021; Dematteis et al. 2022), at least among wave interactions. As described in section 2b(2), spectral flux into the HP band is decomposed into seven components [see Eq. (15) and Fig. 2]. These seven components are presented in Fig. 5b along with dissipation and boundary advection. The HPtoHPflux components are exchanges between large and small vertical scales on either side of vertical wavenumber m. On the other hand, the BPtoHP and LPtoHP flux components can come from any vertical wavenumber; these directional exchanges are depicted in wavenumber–frequency space in Fig. 6. Taken together, the components of the spectral flux shown in Figs. 5b and 6 constitute a complete decomposition of all advective energy exchange with supertidal (HP) modes above a given vertical wavenumber, m.
The individual components of spectral KE flux can be more easily compared across simulations and regions by looking at just a single vertical wavenumber. Integrated spectral budgets through a specific wavenumbers of m = 3.8 and m = 12.5 cycles per kilometer are compared across resolutions and subregions, respectively, in Fig. 7. The wavenumber of m = 3.8 cycles per kilometer was chosen to capture a strong signal from the nonlinear KE scattering mechanisms in both the highresolution run (in which the peak Π_{LI} is at a slightly higher vertical wavenumber) and the lowresolution run (in which the peak Π_{LI} is at a slightly lower vertical wavenumber). In all cases, BPtoHP flux, which is decomposed into
In all resolutions in Fig. 7,
For vertical spectral KE flux through 12.5 cycles per meter in the highresolution case,
The decomposition in Fig. 7a indicates that the spectral flux due to eddyinduced diffusion (
Combined, the HPtoHP spectral flux (
An important implication of the resolution dependence of HPtoHP spectral flux is that at the lower resolutions (2 km × 88 levels, a resolution that is computationally feasible in an IWpermitting global model such as MITgcm LLC4320), the IW continuum will be approximately (generalized) quasilinear^{7} (Marston et al. 2016) around the lowerfrequency tides, nearinertial waves, and eddy fields in which interactions of the type
The vertical spectral KE flux decomposition in the highresolution case is compared across the different subregions in Fig. 7b. Boxes B and D, which are over rugged topography, clearly contribute the most to vertical spectral KE flux and dissipation of the five regions. In particular, both
The boxes (B and D) overlying rough topography also have negative contributions of HP KE to their budgets from the boundary advection (BC) term. Supertidal (HP) IW energy is generated in these regions and moves outward into the rest of the domain (such as boxes A and C).
b. Bispectra of nonlinear scattering mechanisms
Bispectra, introduced in Eq. (25) in section 2b(2), are displayed in Fig. 8. The
The
The bispectrum of
The bispectra containing compensating energy,
c. m–ω spectra
Twodimensional wavenumber–frequency local spectral budgets, introduced in section 2b(5) are shown in Fig. 9. The primary motivation for computing these is to discern the direction of the frequency cascade associated with different nonlinear wave interactions. The advective spectra are decomposed in Figs. 9a–g. The energy transfer decomposition of the supertidal (HP) frequencies exists in the top portion of each figure, with ω > 2.5f_{0}. Note that these spectra constitute a local (or transfer), as opposed to an integrated (or flux), budget. Also note that the plots for
For
The
Energy transfer from BP (nearinertial and tidal frequencies) into HP (the supertidal band), depicted in Fig. 9c is positive throughout the HP band. Energy transfer from LP (eddy frequencies) into HP, depicted in Fig. 9d, shows some KE transfer from HP to LP at the lower vertical wavenumbers and only forwardfrequency (LPtoHP) transfer at the highest vertical wavenumbers. Note that in these two plots, the transfer need not sum to zero in the supertidal (HP) band.
KE advected into the regions of study is also decomposed into twodimensional wavenumber–frequency spectra in Fig. 9f. This implies that, at least for the five subregions of interest, there is advection of supertidal KE into these regions at lowtomid vertical wavenumber and advection out of these regions at high vertical wavenumbers.
Some theories of the IW continuum (e.g., McComas 1977) suggest that wind forcing injects energy at high ω, low m, at which point KE is moved through nonlinear interactions among waves to smaller ω. Signatures of this are not visible as an inverse frequency cascade at low vertical wavenumber in the advective spectra of transfer within the HP band (Figs. 9a,b,e), where KE might be leaving the highest frequencies. They are also not visible in the top left of the PEtoKE conversion spectral, Fig. 9g, where some energy might be injected from SSH perturbations. These characteristics of the m–ω spectral transfers may be because wind forcing is limited to being updated only every 6 h. Higherfrequency forcing, or perhaps coupling to an atmospheric model, are likely necessary to force the flow in a manner consistent with the picture developed by McComas (1977) and will be explored in a future study. Such highfrequency forcing could also impact the overall direction of the ID or LI cascades that are observed. On the other hand, there is a PSI signal at tidal frequencies. The plot of KE transfer from BP (Fig. 9c) conserves energy within the BP band and the strong negative signal at low vertical wavenumber and the semidiurnal frequency is consistent with PSI. This likely PSI signal is about an order of magnitude lower in box E, which is north of the critical latitude for PSI, than in boxes B, C, and D, which are south of the critical latitude (not shown). The signal in box A, which straddles the critical latitude, is also much larger than in box E.
d. Interpretation of and comparison with observations of Sun and Pinkel (2012)
Sun and Pinkel (2012) use observational data to compute bispectra in an attempt to identify ID. Specifically, they compute bispectra using the catalyst mode and using only the vertical component of the gradient in the advective scattering [see Eq. (23)]. Related bispectra [Eq. (24)] are reproduced from the present model data (averaged over the 5 regions of interest in the domain, depicted here in Fig. 3) in Fig. 11. To reiterate important points made in section 2b(3), an advantage of the method used in this paper is that unlike in Sun and Pinkel (2012), the bispectra can include the vertical and horizontal gradients (as in Fig. 11c). At the same time, the method used in this paper requires averaging over positive and negative values of the source and destination modes, implying that it cannot identify vertical anisotropies in scattering mechanisms such as through ES. Additionally, Sun and Pinkel (2012) separate their supertidal (HP) band from their lowpass background field (LP) by removing intermediate frequencies whereas the bispectra in Fig. 11 do not.
It is worth discussing whether it is appropriate to use only the verticalgradient component of a bispectrum such as in Eq. (23). Sun and Pinkel (2012) point out that such an approximation is only valid in horizontally homogeneous flows and noted that would not be applicable in their region of study. We further point out that the horizontalgradient part of these bispectra and advective tendencies in general should only be zero in horizontally homogeneous flows if the source and destination fields are the same [as it is in Gargett and Holloway (1984), which Sun and Pinkel (2012) cite to support their assumption]. To put this another way, if f, g, and h are distinct general incompressible threedimensional velocity fields, then under horizontally homogeneous statistics,
Sun and Pinkel’s (2012) assumption that the horizontalgradient contribution to the bispectra vanishes is also verified explicitly by comparing the bispectra with and without the horizontalgradient contribution in Figs. 11a and 11b, respectively. We also compute the vertical and horizontalgradient spectral fluxes of
Sun and Pinkel (2012) found a positive signal at m_{cat} ≪ m_{dest}; this behavior is not expected from ID. Rather, a signal at m_{cat} ≈ m_{dest}, which would be consistent with ID, was absent or very weak and not bicoherent. These features of Sun and Pinkel’s (2012) results are generally consistent with the bispectra computed from the present model output in Fig. 11a, which also shows a strong positive signal at m_{cat} ≪ m_{dest}, similar to Sun and Pinkel (2012). This signal indicates a transfer of energy from highverticalwavenumber nearinertial and tidal waves to highverticalwavenumber supertidal IWs via scattering off a lowverticalwavenumber supertidal IWs. As this signal has been reproduced in the present numerical simulation, it is referred to as a distinct mechanism, SP, following the name of Sun and Pinkel. SP, occurring at m_{cat} ≪ m_{dest}, reflects a transfer of energy from large vertical wavenumbers at nearinertial and tidal frequencies (BP) into supertidal modes at similar vertical wavenumbers (HP) via scattering off of catalyst modes of low vertical wavenumber in the supertidal band (HP). It is noteworthy that in Fig. 7b, in which
The plots in Fig. 11a show a larger range of catalyst modes and a smaller range of destination modes than those in Sun and Pinkel (2012), thereby making the aspect ratio of m_{cat} ≈ m_{dest} easily identifiable as the diagonal. The bispectra of Sun and Pinkel (2012) are presented on a linear scale which makes it only easy to discern about one order of magnitude in their signal, possibly reflecting a observational noise floor. The present numerical results appear to have a lower noise floor (although we did not compute a bicoherence analysis.) The bispectra are presented on a logscale color bar to reveal the weaker signal. The far left of the left panel of Fig. 16 in Sun and Pinkel (2012) shows alternatingsigned bands near the origin. In Fig. 11a of this paper, a similar signal can be seen to extend weakly along the diagonal, m_{cat} ≈ m_{dest}. This pattern should include ID_{comp} and indicates that compensating energy is primarily coming from modes that are catalysts in the associated ID_{diff} diagrams that are inducing KE diffusion downscale from this mode (positive above the diagonal) as opposed to from upscale into this mode (negative below the diagonal), which actually act weakly in the opposite direction.
4. Conclusions
With the increasing availability of highresolution IWpermitting numerical models, waveturbulence phenomenologists and theorists have acquired a validation tool capable of filling in gaps that are not easily accessible from observational data. At the same time, IWpermitting numerical models can enable more meaningful interpretation of observations, which are constrained by budgetary and other limitations from fully sampling complex fourdimensional IW fields. Another useful tool for these purposes is the asymmetric scattering diagram, which readily serves as a point of comparison with observations of triads [such as in Sun and Pinkel (2012)] and also allows for the study of resonant nonlinear mechanisms, such as induced diffusion and local interactions, that govern the IW cascade.
As a starting point, integrated vertical KE spectral budgets were computed for decomposition of both nonlinear advective scattering and dissipation. Advection and dissipation are found to be nearly in balance at high vertical wavenumbers, although an inertial range in which forcing and dissipation are absent was not present. Further results indicate that the extent of nonlinearity in such an IW continuum is highly dependent on the resolution of the model. In coarse 2 km resolutions (common in the highestresolution global models), the IW continuum is approximately (generalized) quasilinear around lowerfrequency (ω < 2.5f_{0}) background waves and eddies, meaning very little energy is transferred within the supertidal IW continuum. At such resolutions, most energy in the IW continuum has been transferred to a highfrequency mode from a lowfrequency mode through a single nonlinear interaction rather than a series of cascadelike processes (such as through ID or LI). Thus, fully nonlinear aspects of the IW continuum are largely small in global models and at best can be parameterized based on available quasilinear flow information.
A central motivation of this work is to better understand nonlinear IW scattering mechanisms in a realistic ocean model. On one hand, theoretical questions in WTT pertaining to ID and LI in McComas and Bretherton (1977) and Dematteis et al. (2022) can be tested as well as decomposed using asymmetric spectral transfer diagrams. On the other hand, mechanisms that involve the eddy field or that do not fit neatly into existing frameworks can be measured, with the efficient computation of bispectra in position space being particularly instrumental for the latter activity. We find that ID is associated with a small forward frequency cascade and is strongest over rough topography. This stands in contrast to the theoretical framework of McComas and Müller (1981) who predicted an inverse frequency cascade associated with ID. Nonlinear spectral KE fluxes within the supertidal band involving only the wave part of the flow are also approximately decomposed into spectrally scaleseparated (ID) and local (LI) components. The vast majority of these fluxes are local (LI) at low resolution. In addition, spectral fluxes are highly spatially inhomogeneous, and are largest within the subregion studied that had rough topography at depth.
The method of decomposing asymmetric spectral fluxes into catalyst, source, and destination modes enabled the separation of two distinct exchanges within ID: 1) ID_{diff}, the supertidal energy diffusion, and 2) ID_{comp}, energy compensation from the nearinertial and tidal frequencies. Partial spectral flux budgets reveal that
In addition, the vertical spectral KE flux of a mechanism analogous to ID_{diff} in which eddy fields catalyze supertidal energy diffusion (termed
In summary, the present findings identify eddyinduced IW KE diffusion (
A natural next step will be to direct attention to dissipation mechanisms [building on Thakur et al. (2022)] and parameterizations of the IW continuum along with the spectral transfer decompositions presented in this paper. In particular, we will examine vertical and horizontal spatial distributions. This strategy will hopefully reveal different modeling strategies for quasilinear and fully nonlinear mechanisms that, in the real ocean, would move energy beyond the grid scale of a numerical model. Looking forward, we anticipate highresolution regional models, such as this one, to become increasingly important in probing details of the IW cascade and potentially developing model parameterizations for such details.
Dematteis et al. (2022) use a specific set of parameters to define the IW spectrum that is consistent with the stationary solution of the waveturbulence theory collision integral. For a frequency spectrum of E(ω) ∼ ω^{−2}, their parameter choices imply a vertical wavenumber m energy spectrum that is E(m) ∼ m^{−2}, which is not necessarily consistent with all parts of the ocean or the simulations in this paper (cf. with Fig. 4b). However, Dematteis et al. (2022) argue that it is necessary for this set of parameters to reflect dominant processes in the ocean so as to resolve the “oceanic ultraviolet catastrophe,” i.e., for the spectral frequency fluxes associated with induced diffusion to be forward and not inverse given the lack of highfrequency energy sources to feed the IW continuum (Polzin and Lvov 2017). For this reason, we compare the present numerical simulations with predictions made for E(m) ∼ m^{−2} in Dematteis et al. (2022).
A positive vertical spectral energy flux will denote energy transfer from low to high vertical wavenumber. The “vertical” strictly refers to the direction of the wavenumber of spectral modes; vertical spectral fluxes referenced in this paper therefore have no relation to hypothetical spectra computed from vertical spatial energy fluxes. This paper will sometimes drop the “energy” designation; e.g., “vertical spectral flux” implies a spectral flux of kinetic energy.
Details of the boundary forcing are described in Nelson et al. (2020).
One exception to this is a recent paper, Dematteis and Lvov (2023), that determines a realization of energy and waveaction exchanges between pairs of modes mediated by a third in the context of catalystsourcesymmetric WTT; see appendix A. The problem addressed in Dematteis and Lvov (2023) is similar to the one addressed with asymmetric spectral diagrams in this paper. Here we use symmetries evident in the governing equations to identify a realization of such energy exchanges. A comparison with and implications of using a catalystsource symmetric system of equations to arrive at a realization of energy exchanges will not be explored in the present paper but offers a rich avenue of future enquiry.
This follows from the requirement that
If
“Quasilinear” is a technical term that does not mean approximately linear.
The direction of the frequency cascade will be determined in section 3c.
To reprise some relevant points previously made in sections 1b and 2b(5), wave action is expected to be conserved in the supertidal band under ID. The definition of wave action, Α = E/ω, implies the direction of compensating energy that must be supplied from the catalyst band to conserve wave action is the same as that of the frequency cascade associated with ID, a point that is made clear in the discussion of Fig. 5 in McComas and Müller (1981). Dematteis et al. (2022) study spectra that they expect share properties with the real ocean that have both a weak forward frequency cascade (top row in Fig. 5 of that paper) and neutral frequency cascades (middle row in Fig. 5 of that paper) under ID, indicating compensating energy from the catalyst band to the supertidal band that is absent or weak.
The error introduced by defining scale separation in
Acknowledgments.
Authors Skitka and Arbic acknowledge support from Office of Naval Research Grant N000141912712. Authors Thakur and Arbic acknowledge support from National Science Foundation Grant OCE1851164 and NASA Grant 80NSSC20K1135. We thank Cynthia Wu for insightful discussion of energy transfers among triads of modes. We thank both anonymous reviewers for improving the paper with helpful comments. For instance, one anonymous reviewer identified the point in the derivation of the kinetic equation of waveturbulence theory in which symmetrization between the catalyst and source modes occurs, and the other anonymous reviewer encouraged us to apply scale separation in the vertical wavenumber to our definition of induced diffusion and related mechanisms.
Data availability statement.
Data and dataprocessing scripts are publicly available online through Harvard Dataverse (https://doi.org/10.7910/DVN/FSMM1O; Skitka et al. 2022).
APPENDIX A
Asymmetric KE Transfer within Advective Triads
Various spectral methods, including spectral fluxes, coarse graining (Aluie et al. 2018), and WTT (e.g., Lvov et al. 2010; Dematteis et al. 2022; Dematteis and Lvov 2021), are sometimes implemented such that scattering is symmetric between the source and catalyst modes. This symmetry between catalyst and source modes is most clearly indicated in the diagrams in Fig. A3. The ideas discussed in this appendix are extended and interpreted in appendix B.
APPENDIX B
Extension to Spectral Frequency Transfers
Therefore, the extension of spectral fluxes and transfers to frequency space as used by Arbic et al. (2014) and Müller et al. (2015), as well as in Eqs. (5), (12), (24), and so on, can be interpreted thanks to Morten (2015) as the exchange of energy from a wave of one frequency to another as the sample window over which the frequencies are computed is moved forward in time.
APPENDIX C
Local Spectral Budget
APPENDIX D
Divergence Correction
APPENDIX E
Budget Term Diagnosis Implementation
Below are details of the numerical methods used to diagnose various terms in the spectral budgets used in this paper. Also, see Adcroft et al. (2015) and Nelson et al. (2020) for further information on the solver details and simulation settings.

Advection (
$\mathsf{A}$ ) is a flux form centered secondorder operator. In our analysis, we compute this term exactly as in the model, except that it is computed on a vertically uniform grid. Experiments suggest that this procedure yields better energy conservation than interpolation after the velocity tendencies are computed. All other terms are computed on the nonuniform grid and then interpolated. Horizontal boundary conditions are handled by assuming periodicity and explicitly removing the energy sources/sinks associated with the resulting unphysical (three dimensional) flow divergence, as described in appendix D. 
Advection of kinetic energy into the simulation domain from the boundaries (BCs) is computed explicitly at the surfaces of the region being analyzed. This is done with the same fluxform discretization as for the previously mentioned volume advection operator.

KPP background and shear (
$\mathsf{D}$ _{KPP_BG} and$\mathsf{D}$ _{KPP_Shear}) are computed using the same discretizations as in the model below 20 m depth, using a 30m onesided Tukey taper from 20 to 50 m. The taper excludes the vast majority of the diagnosed mixed layer, which always lies within the upper 50 m during the weeklong model output period across the domain. The mixed layer dissipation ($\mathsf{D}$ _{KPP_ML}) is entirely omitted in the hope that it will prove to be unimportant in the general balance of the IW field and any other conclusions drawn from this work. The findings of this work generally support this assumption. All other terms are computed throughout the entirety of the water column. 
The Leith scheme (
$\mathsf{D}$ _{Leith}) is computed using the same discretizations as in the model. 
Quadratic bottom boundary layer drag (
$\mathsf{D}$ _{QBD}) is computed exactly as it is in the model. This term implicitly handles the noslip bottom boundary condition. 
A noslip side boundary condition is computed exactly as it is computed in the model (BCs). This contribution was found to be negligible and is therefore omitted from the results.

Bottom scattering and the pressure boundary condition (i.e., the nonormal flow condition, BCs) are handled implicitly in the other terms, such as advection.

The pressure term (
$\mathsf{P}$ ) mediates the linear transfer of potential energy into kinetic energy via the hydrostatic pressure but will not transfer kinetic energy among scales away from the boundary conditions. This potentialenergytokineticenergy conversion was computed through the pressure in a manner that is a good approximation of how it is represented in the model. The conversion turns out to not have a significant impact for higher vertical wavenumbers and is not included in any budget plots. 
The Coriolis force (
$\mathsf{C}$ ) rotates the modes containing KE but, as a linear term, does not transfer it among scales. Thus the Coriolis term is not computed for spectral budgets in this paper. The contribution to the drag from the scattering of the Coriolis tendency off of the bottom topography is not computed but has been determined to be very small in a different test case. 
In some figures, the total energy dissipation is given. This total dissipation is the sum of dissipations associated with the KPP background and shear terms, the Leith scheme, and quadratic bottom drag, but omits the KPP mixed layer term.

Last, a residual is computed as the sum of the aforementioned terms. The residual includes the combined interpolation and discretization errors of each term, any omitted forcing, the KPP mixed layer dissipation, the small contributions of intentionally neglected terms that were previously mentioned, as well as energy buildup over time.
The KE budget terms are computed across the entirety of the water column (with the exception of KPP) in part because the flow is vertically inhomogeneous. In terms of spectral analysis, the flow resembles a transient signal in position space, and it is appropriate to use a rectangular taper function (i.e., no taper function) to capture parts of the signal at the top and bottom of the domain. Put another way, no taper is needed for this analysis because the analysis does not compute the spectrum from a sample of an infinite dimension. Rather, the spectrum is computed on the entire domain on which the signal is defined. The exception to this, as previously mentioned, is KPP’s interior dissipation, which tapered from 20 to 50 m while KPP’s mixed layer dissipation is simply omitted. Also note that a Tukey taper is applied in time.
APPENDIX F
Definitions of ScaleSeparated Interactions
The scaleseparation factor ξ is introduced in Eq. (14) to identify triads with two modes that are much larger in vertical wavenumber that the third. The breakdown of energy in
Figure F1 also shows an estimate^{F1} of the error introduced by our approximate definition of scale separation for
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