2 Diffusion in wavenumber space

We consider the dynamics of IGWs propagating in a turbulent vortical flow of much larger spatial scale so that the WKB approximation applies. The distribution of wave energy in the $(\boldsymbol{x},\boldsymbol{k})$ phase space is then governed by the conservation

(2.1) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}a+\unicode[STIX]{x1D735}_{\boldsymbol{k}}\unicode[STIX]{x1D6FA}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\boldsymbol{x}}a-\unicode[STIX]{x1D735}_{\boldsymbol{x}}\unicode[STIX]{x1D6FA}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\boldsymbol{k}}a=0\end{eqnarray}$$

of the action density $a(\boldsymbol{x},\boldsymbol{k},t)$ . Here $\unicode[STIX]{x1D6FA}=\unicode[STIX]{x1D714}+\boldsymbol{U}\boldsymbol{\cdot }\boldsymbol{k}$ is the frequency, which sums the intrinsic frequency

(2.2) $$\begin{eqnarray}\unicode[STIX]{x1D714}=\sqrt{f^{2}\cos ^{2}\unicode[STIX]{x1D703}+N^{2}\sin ^{2}\unicode[STIX]{x1D703}},\end{eqnarray}$$

where $f<N$ are the Coriolis and buoyancy frequencies and $\unicode[STIX]{x1D703}$ is the angle between the wavevector $\boldsymbol{k}$ and the vertical, and the Doppler shift $\boldsymbol{U}\boldsymbol{\cdot }\boldsymbol{k}$ , where $\boldsymbol{U}=\boldsymbol{U}(\boldsymbol{x},t)$ is the vortical flow velocity. Assuming that the flow is (i) weak enough that $\unicode[STIX]{x1D714}\gg \boldsymbol{U}\boldsymbol{\cdot }\boldsymbol{k}$ , (ii) evolving on a time scale much longer than $\unicode[STIX]{x1D714}^{-1}$ , and (iii) well modelled by a homogeneous and stationary random field, we can approximate (2.1) by

(2.3) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}a+\boldsymbol{c}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\boldsymbol{x}}a=\unicode[STIX]{x1D735}_{\boldsymbol{k}}\boldsymbol{\cdot }(\unicode[STIX]{x1D63F}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\boldsymbol{k}}a),\end{eqnarray}$$

where $\boldsymbol{c}=\unicode[STIX]{x1D735}_{\boldsymbol{k}}\unicode[STIX]{x1D714}$ is the intrinsic group velocity and $\unicode[STIX]{x1D63F}$ a $\boldsymbol{k}$ -dependent diffusivity tensor (see appendix A for a derivation). The right-hand side of (2.3) captures the scattering of wave action that results from small-but-sustained random Doppler shifting by the flow; in the regime considered, this naturally leads to diffusion in $\boldsymbol{k}$ -space. In Cartesian coordinates, the diffusivity tensor takes the form

(2.4) $$\begin{eqnarray}\unicode[STIX]{x1D60B}_{ij}(\boldsymbol{k})=-\frac{1}{2}k_{m}k_{n}\int _{-\infty }^{\infty }\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D6F1}_{mn}}{\unicode[STIX]{x2202}x_{i}\unicode[STIX]{x2202}x_{j}}(\boldsymbol{c}(\boldsymbol{k})s)\,\text{d}s,\end{eqnarray}$$

where $\unicode[STIX]{x1D6F1}_{mn}(\boldsymbol{x})=\langle U_{m}(\boldsymbol{y}+\boldsymbol{x})U_{n}(\boldsymbol{y})\rangle$ is the velocity correlation tensor, with $\langle \cdot \rangle$ denoting ensemble average, and summation over repeated indices is implied. An analogous expression was obtained by McComas & Bretherton (Reference McComas and Bretherton1977) in the context of wave–wave interactions in the induced-diffusion regime (see Müller et al. Reference Müller, Holloway, Henyey and Pomphrey1986, § 5, for a review). Müller & Olbers (Reference Müller and Olbers1975) and Müller (Reference Müller1976, Reference Müller1977) discussed a flow-induced diffusivity that differs from (2.4) to account heuristically for wave–wave interactions and dissipation.

Figure 1.

(a) IGW energy density $e(\boldsymbol{k})$ in $\boldsymbol{k}$ -space at $t=389f^{-1}$ for the initial-value simulation of § 3 with $\unicode[STIX]{x1D714}=3f$ and $Ro=0.057$ . Red represents the wave modes with $e(\boldsymbol{k})/e_{max}>0.1$ ( $e_{max}$ the maximum energy density), orange represents $0.01<e(\boldsymbol{k})/e_{max}<0.1$ , and blue represents $0.03<e(\boldsymbol{k})/e_{max}<0.01$ . (b) Projection of $e(\boldsymbol{k})$ onto the $(k_{h},k_{v})$ -plane. The constant-frequency cone defined by (2.2) is indicated by grey stripes on the left and dashed lines on the right.

A key property of (2.4) is that $\unicode[STIX]{x1D63F}(\boldsymbol{k})\boldsymbol{\cdot }\boldsymbol{c}(\boldsymbol{k})=0$ since

(2.5) $$\begin{eqnarray}\unicode[STIX]{x1D60B}_{ij}(\boldsymbol{k})\boldsymbol{\cdot }c_{j}(\boldsymbol{k})=-\frac{1}{2}k_{m}k_{n}\int _{-\infty }^{\infty }\frac{\text{d}}{\text{d}s}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F1}_{mn}}{\unicode[STIX]{x2202}x_{i}}(\boldsymbol{c}(\boldsymbol{k})s)\right)\text{d}s=0.\end{eqnarray}$$

Thus there is no diffusion in the direction of the group velocity $\boldsymbol{c}$ . Since $\boldsymbol{c}$ is perpendicular to constant-frequency surfaces, for the IGW dispersion relation (2.2) diffusion is restricted to the cones $\unicode[STIX]{x1D703}=\text{const.}$ (see figure 1). This is because diffusion in $\boldsymbol{k}$ -space stems from resonant-triad interactions between two IGWs and one vortical mode (also termed balanced mode) associated with the flow. The flow is treated as a zero-frequency mode because it evolves slowly compared with $\unicode[STIX]{x1D714}^{-1}$ , so the resonance condition implies that the interacting IGWs have the same frequency. The restriction to a single frequency means that wave action and wave energy only differ by a constant multiple and can be identified with one another.

We particularise (2.4) to IGWs and geostrophic flows using the dispersion relation (2.2) and the geostrophic balance satisfied by the velocity in  $\unicode[STIX]{x1D6F1}_{mn}$ . It is natural to use spherical polar coordinates $(k,\unicode[STIX]{x1D719},\unicode[STIX]{x1D703})$ in $\boldsymbol{k}$ -space and a Fourier counterpart to $\unicode[STIX]{x1D6F1}_{mn}$ in the form of the vortical flow kinetic-energy spectrum $E(K_{h},K_{v})$ , which we assume to be horizontally isotropic so that it only depends on the horizontal and vertical wavenumbers $K_{h}$ and $K_{v}$ (for clarity we systematically use lowercase symbols for coordinates in the IGW wavenumber space and uppercase symbols for coordinates in the flow wavenumber space). Computations detailed in appendix A then reduce (2.3) to

(2.6) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}a=\frac{1}{k^{2}}\unicode[STIX]{x2202}_{k}(k^{2}D_{kk}\unicode[STIX]{x2202}_{k}a)+\frac{D_{\unicode[STIX]{x1D719}\unicode[STIX]{x1D719}}}{k^{2}\sin ^{2}\unicode[STIX]{x1D703}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D719}\unicode[STIX]{x1D719}}a,\end{eqnarray}$$

under the further assumption of spatial homogeneity $\unicode[STIX]{x1D735}_{\boldsymbol{x}}a=0$ . This makes it plain that there is no diffusion in the direction of  $\unicode[STIX]{x1D703}$ . Hence, $\unicode[STIX]{x1D703}$ , or equivalently  $\unicode[STIX]{x1D714}$ , can be treated as a fixed parameter. The only non-zero components of the diffusivity tensor are given by

(2.7a ) $$\begin{eqnarray}\displaystyle & \displaystyle D_{kk}=Bk^{3}\iint _{K_{h}^{2}/K_{v}^{2}>\tan ^{2}\unicode[STIX]{x1D703}}\frac{K_{v}^{2}}{K_{h}}\left(\cot ^{2}\unicode[STIX]{x1D703}-\frac{K_{v}^{2}}{K_{h}^{2}}\right)^{1/2}E(K_{h},K_{v})\,\text{d}K_{h}\,\text{d}K_{v}, & \displaystyle\end{eqnarray}$$

(2.7b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{D_{\unicode[STIX]{x1D719}\unicode[STIX]{x1D719}}}{\sin ^{2}\unicode[STIX]{x1D703}}=Bk^{3}\iint _{K_{h}^{2}/K_{v}^{2}>\tan ^{2}\unicode[STIX]{x1D703}}K_{h}\left(\cot ^{2}\unicode[STIX]{x1D703}-\frac{K_{v}^{2}}{K_{h}^{2}}\right)^{3/2}E(K_{h},K_{v})\,\text{d}K_{h}\,\text{d}K_{v}, & \displaystyle\end{eqnarray}$$

(2.8) $$\begin{eqnarray}B=\frac{\unicode[STIX]{x1D714}\sin ^{2}\unicode[STIX]{x1D703}}{4\unicode[STIX]{x03C0}^{3}(N^{2}-f^{2})|\text{cos}^{5}\,\unicode[STIX]{x1D703}|}\end{eqnarray}$$

depends solely on $\unicode[STIX]{x1D703}$ , $N$ and $f$ .

Equations (2.6)–(2.7) provide a full description of the diffusion of IGW on the constant-frequency cone in $\boldsymbol{k}$ -space for a turbulent flow of given energy spectrum. In the angular $\unicode[STIX]{x1D719}$ -direction, this diffusion leads to an isotropisation of the wave field with rate $D_{\unicode[STIX]{x1D719}\unicode[STIX]{x1D719}}/(k^{2}\sin ^{2}\unicode[STIX]{x1D703})$ . In the radial $k$ -direction, the diffusion leads to a forward cascade of the wave energy to high wavenumbers, where it is efficiently dissipated by viscous processes. Note that wave energy remains confined to one nappe of the cone corresponding to either upward- or downward-propagating IGWs. This is only an approximation; exchanges between upward- and downward-propagating waves do occur, but they are asymptotically small and not captured by the WKB approximation. In what follows, we concentrate on radial diffusion by assuming wave statistics independent of $\unicode[STIX]{x1D719}$ , $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D719}}a=0$ , leaving the study of horizontal isotropisation for future work.

3 Initial-value problem

For $a=a(k,t)$ , we rewrite (2.6) as

(3.1) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}e=\unicode[STIX]{x2202}_{k}(Qk^{5}\unicode[STIX]{x2202}_{k}(k^{-2}e)),\end{eqnarray}$$

where we have introduced $e(k,t)=2\unicode[STIX]{x03C0}k^{2}\sin \unicode[STIX]{x1D703}\unicode[STIX]{x1D714}\,a(k,t)$ and the $k$ -independent parameter $Q=D_{kk}/k^{3}$ . The function $e(k,t)$ is the IGW energy density in $k$ , with $e(k,t)\,\text{d}k$ the energy contained within the interval $[k,k+\text{d}k]$ . We solve (3.1) with initial condition $e(k,0)=\unicode[STIX]{x1D6FF}(k-k_{\ast })$ corresponding to the excitation of IGWs with a single wavenumber  $k_{\ast }$ . (The solution associated with an arbitrary initial condition can be deduced by integration over  $k_{\ast }$ .) We show in § A.3 that

(3.2) $$\begin{eqnarray}e(k,t)=\frac{1}{2}k_{\ast }^{-2}\int _{0}^{\infty }\text{J}_{4}(k^{-1/2}\unicode[STIX]{x1D706})\text{J}_{4}(k_{\ast }^{-1/2}\unicode[STIX]{x1D706})\text{e}^{-Q\unicode[STIX]{x1D706}^{2}t/4}\unicode[STIX]{x1D706}\,\text{d}\unicode[STIX]{x1D706},\end{eqnarray}$$

where J $_{4}$ is a Bessel function of the first kind (DLMF 2018). The large-time behaviour of $e(k,t)$ is readily deduced as $e(k,t)\propto k^{-2}t^{-5}$ away from an asymptotically small neighbourhood of $k=0$ (see § A.3). An inverse diffusion time scale can be read off from (3.2) as  $Qk_{\ast }$ . Using (2.7) this can be written in the dimensionless form

(3.3) $$\begin{eqnarray}\frac{Qk_{\ast }}{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D6E4}\frac{N^{2}}{N^{2}-f^{2}}\frac{k_{\ast }}{K_{h\ast }}Ro^{2},\end{eqnarray}$$

where $\unicode[STIX]{x1D6E4}$ is a dimensionless ‘geometric’ factor that depends only on $\unicode[STIX]{x1D703}$ and the shape (but not the magnitude) of the flow kinetic-energy spectrum and $Ro=K_{h\ast }\langle |\boldsymbol{U}|^{2}\rangle ^{1/2}/f$ is a flow Rossby number. The typical horizontal and vertical inverse flow scales $K_{h\ast }$ and $K_{v\ast }$ are assumed to be related by $K_{v\ast }=NK_{h\ast }/f$ . Equation (3.3) captures the dependence of the diffusion time scale on the Rossby number and on the scale separation between IGWs and flow. The diffusion approximation requires $Qk_{\ast }/\unicode[STIX]{x1D714}\ll 1$ in addition to the WKB conditions $k_{\ast }\sin \unicode[STIX]{x1D703}\gg K_{h}$ and $k_{\ast }\cos \unicode[STIX]{x1D703}\gg K_{v}$ .

We verify the solution of (3.1) against simulations of the three-dimensional non-hydrostatic Boussinesq equations. These are solved using a code adapted from that in Waite & Bartello (Reference Waite and Bartello2006b ) which relies on a de-aliased pseudospectral method and a third-order Adams–Bashforth scheme with time step $0.015/f$ . The triply periodic domain, $(2\unicode[STIX]{x03C0})^{3}$ in the scale coordinates $(x,y,z^{\prime }=Nz/f)$ , is discretised uniformly with $768^{3}$ grid points. A hyperdissipation of the form $-\unicode[STIX]{x1D708}(\unicode[STIX]{x2202}_{x}^{2}+\unicode[STIX]{x2202}_{y}^{2}+\unicode[STIX]{x2202}_{z^{\prime }}^{2})^{4}$ , with $\unicode[STIX]{x1D708}=2\times 10^{-17}$ , is employed in the momentum and density equations. We take $N/f=32$ , a representative value of mid-depth ocean stratification. The initial condition is the superposition of a turbulent flow, obtained by running a quasigeostrophic model to a statistically stationary state, and IGWs. The initial spectrum of the vortical flow peaks at $K_{h\ast }\simeq 4$ and has an inertial subrange scaling approximately as $K_{h}^{-3}$ and $K_{v}^{-3}$ . This spectrum evolves slowly over the IGW-diffusion time scale, and an average is used to calculate $D_{kk}$ in (2.7a ), and hence $Q$ in (3.1). We report experiments with the two Rossby numbers $Ro=K_{h\ast }\langle |\boldsymbol{U}|^{2}\rangle ^{1/2}/f=0.057$ , 0.117 (or $\langle \unicode[STIX]{x1D701}^{2}\rangle ^{1/2}/f=0.1,0.2$ for the alternative Rossby numbers based on the vertical vorticity  $\unicode[STIX]{x1D701}$ ), and the two IGW frequencies $\unicode[STIX]{x1D714}=2f$ , $3f$ . Upward-propagating IGWs are initialised as a ring in $\boldsymbol{k}$ -space with $k_{h\ast }=16$ , $k_{v}=\cot \unicode[STIX]{x1D703}k_{h}$ , random phases, and an initial kinetic energy $\langle |\boldsymbol{u}|^{2}\rangle /2=0.1\langle |\boldsymbol{U}|^{2}\rangle /2$ . The IGW spectrum $e(k,t)$ is computed following the normal-mode decomposition of Bartello (Reference Bartello1995).

Figure 2.

Evolution of the IGW energy $e(k,t)$ in Boussinesq simulations (solid, black) and as predicted by the diffusion approximation (dotted, red) for (a)  $\unicode[STIX]{x1D714}=2f$ , $Ro=0.057$ , (b)  $\unicode[STIX]{x1D714}=2f$ , $Ro=0.117$ , (c)  $\unicode[STIX]{x1D714}=3f$ , $Ro=0.057$ and (d)  $\unicode[STIX]{x1D714}=3f$ , $Ro=0.117$ . Conventionally, $k>0$ ( $k<0$ ) corresponds to upward- (downward-)propagating IGWs.

Figure 3.

Log–log representation of the IGW spectra in figures 2(a) (left, $\unicode[STIX]{x1D714}=2f$ , $Ro=0.057$ ) and 2(b) (right, $\unicode[STIX]{x1D714}=2f$ , $Ro=0.117$ ). The solid lines are the results of the Boussinesq simulation and the dashed lines the predictions of the diffusion approximation. The curves correspond to the times shown in figure 2(a,b) and are successively shifted downwards by half a decade for clarity.

Figure 1, obtained for the lower $Ro$ and $\unicode[STIX]{x1D714}=3f$ , illustrates the confinement of wave energy on the constant-frequency cone, one of the keys to the validity of the diffusion approximation. The confinement is of course not perfect and some energy appears around the cones associated with the harmonic frequencies $2\unicode[STIX]{x1D714}$ and $3\unicode[STIX]{x1D714}$ . Figure 2 shows the evolution of $e(k,t)$ for the four sets of values of $(Ro,\unicode[STIX]{x1D714})$ . The numerical results are compared with the predictions of the diffusion equation obtained by solving (3.1) initialised with the form of $e(k,t_{a})$ extracted from the simulation after an adjustment time $t_{a}>0$ . This procedure accounts for the fact that the diffusion equation (2.3) is only valid after an adjustment period, requiring $t_{a}\gg (K_{\ast }|\boldsymbol{c}|)^{-1}$ , the time to traverse typical eddies at the IGW group speed (cf. Müller et al. Reference Müller, Holloway, Henyey and Pomphrey1986, § 5). The agreement between the numerical simulation and the diffusion approximation is remarkable considering the complexity of the full Boussinesq dynamics and the moderate separation of scales between IGWs and flow. As the diffusion approximation predicts, the simulations with different Rossby numbers behave similarly when $t$ is scaled suitably. The decay is slower for $\unicode[STIX]{x1D714}=3f$ than $\unicode[STIX]{x1D714}=2f$ , consistent with a decrease in $Q$ obtained when evaluating (2.7a ). Scattering from upward-propagating to downward-propagating IGWs, neglected in the diffusion approximation, occurs; it is more substantial for the larger $\unicode[STIX]{x1D714}$ because the two nappes of the constant-frequency cones are closer together, facilitating energy transfers. Figure 3 displays the wave energy spectrum $e(k,t)$ obtained for $\unicode[STIX]{x1D714}=2f$ (figure 2 a,b) in log–log coordinates. It shows that the good agreement between numerical and predicted spectra extends to large wavenumbers for $Ro=0.057$ but not for the larger Rossby, $Ro=0.117$ , at the later times. We note that the wave energy is then very small and may be affected by a contribution associated with spontaneous generation (cf. Kafiabad & Bartello Reference Kafiabad and Bartello2018).

4 Forced response and observed ocean and atmosphere spectra

We now turn to the steady solution of (3.1) in the presence of a forcing of the form $\unicode[STIX]{x1D6FF}(k-k_{\ast })$ . Equation (3.2) admits two steady solutions: the no-flux solution $e(k)\propto k^{2}$ and the constant-flux solution $e(k)\propto k^{-2}$ . Matching these at $k_{\ast }$ yields the steady spectrum

(4.1) $$\begin{eqnarray}e(k)=\frac{1}{4Qk_{\ast }^{2}}\left\{\begin{array}{@{}ll@{}}(k/k_{\ast })^{2}\quad & \text{for}~0<k<k_{\ast }\\ (k_{\ast }/k)^{2}\quad & \text{for}~k>k_{\ast }.\end{array}\right.\end{eqnarray}$$

Note that for IGWs with a single frequency, and correspondingly a single angle  $\unicode[STIX]{x1D703}_{\ast }$ , the horizontal energy spectrum $e_{h}(k_{h})$ satisfies the same power laws as $e(k)$ since

(4.2) $$\begin{eqnarray}e_{h}(k_{h})=\iiint \unicode[STIX]{x1D6FF}(k\sin \unicode[STIX]{x1D703}-k_{h})\frac{e(k)\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{\ast })}{2\unicode[STIX]{x03C0}k^{2}\sin \unicode[STIX]{x1D703}}\,\text{d}\boldsymbol{k}=\csc \unicode[STIX]{x1D703}_{\ast }e(k_{h}\csc \unicode[STIX]{x1D703}_{\ast }),\end{eqnarray}$$

using that the energy density in $\boldsymbol{k}$ -space is $e(k)\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{\ast })/(2\unicode[STIX]{x03C0}k^{2}\sin \unicode[STIX]{x1D703})$ . Thus, (4.1) implies a $k_{h}^{-2}$ horizontal spectrum at large  $k_{h}$ . This remains true for a superposition of IGWs with different frequencies, corresponding to an integration over  $\unicode[STIX]{x1D703}_{\ast }$ .

Figure 4.

Stationary horizontal energy spectrum $e(k_{h})$ for the forced simulations in § 4 with $Ro=0.057$ (a) and 0.117 (b). Straight lines indicate the power laws: $k^{-2}$ (red), $k^{-5/3}$ (blue) and $k^{2}$ (orange).

We confirm the prediction (4.1) by the simulation of the Boussinesq equations in the presence of forcing. In the simulation reported, all the specifications are the same as in § 3 except for the initial condition, which is devoid of IGWs. Instead, an Ornstein–Uhlenbeck forcing with short correlation time (3 timesteps) is applied to the waves with $\unicode[STIX]{x1D714}=2f$ (see Waite Reference Waite2017). The forcing amplitude is adjusted so that the wave energy is approximately 0.01 of the vortical flow energy after reaching a stationary state. Figure 4 shows the stationary spectra for $Ro=0.057$ and 0.117. For the low Rossby number, the prediction (4.1) is well borne out by the simulation results with a clean $k_{h}^{-2}$ spectrum spanning nearly a decade from the forcing scale down to dissipation. For the high Rossby number, the spectrum shallows a little from wavenumber 40 or so to take a shape more consistent with $k_{h}^{-5/3}$ . Two mechanisms can be invoked to explain this shallowing: the Doppler term is not weak compared with the intrinsic IGW frequency, invalidating the diffusion approximation, or nonlinear wave–wave interactions become significant. We can roughly estimate the wavenumbers at which each of these mechanisms is important as

(4.3a,b ) $$\begin{eqnarray}k_{h}\sim \frac{\unicode[STIX]{x1D714}}{\langle |\boldsymbol{U}|^{2}\rangle ^{1/2}}\sim \frac{K_{h\ast }}{Ro}\quad \text{and}\quad k_{h}\sim \frac{K_{h\ast }}{Ro}\frac{\langle |\boldsymbol{U}|^{2}\rangle ^{1/2}}{\langle |\boldsymbol{u}|^{2}\rangle ^{1/2}},\end{eqnarray}$$

corresponding to order-one Rossby numbers based on the wavelength scale $k_{h}^{-1}$ and on the root-mean-square velocity of, respectively, the vortical flow and the IGWs. For the simulation with $Ro=0.117$ , these wavenumbers are approximately 40 and 400, suggesting that the shallowing of the spectrum is associated with the breakdown of the assumption of weak Doppler shift.

The prediction of a $k_{h}^{-2}$ spectrum is significant in view of the ubiquity of this scaling in ocean and atmosphere observations. In the ocean, kinetic-energy spectra show a $k_{h}^{-2}$ dependence in the submesoscale range, say below 20 km, in regions of high mesoscale activity and in a larger range, below 200 km, in less active regions (see Callies & Ferrari (Reference Callies and Ferrari2013) for a comprehensive discussion). Recent analyses by Bühler et al. (Reference Bühler, Callies and Ferrari2014) and Rocha, Chereskin & Gille (Reference Rocha, Chereskin and Gille2016) which separate the contribution of IGWs from that of geostrophic motion indicate that the IGW part of the spectrum follows a $k_{h}^{-2}$ scaling in almost the entirety of its range. Our results above suggest that this may result from IGW energy diffusion by the geostrophic flow. Scales below 10 km or so are the realm of the Garrett & Munk (Reference Garrett and Munk1972) spectrum, also associated with a $k_{h}^{-2}$ dependence. While this spectrum is generally attributed to wave–wave interactions (for example, Müller et al. Reference Müller, Holloway, Henyey and Pomphrey1986; Lvov, Polzin & Yokoyama Reference Lvov, Polzin and Yokoyama2012), interactions with the geostrophic flow may play a significant role, dominating for wavenumbers much smaller than (4.3b ). We emphasise that theories based on linear IGWs, be it the diffusion approximation of this paper or a more general theory accounting for strong Doppler shift, can predict neither the level of the IGW spectrum nor its frequency content since both are determined by the forcing.

In the atmosphere, similarly, there is a broad range of scales, from 500 km to 10 km, where the energy spectrum scales approximately as $k_{h}^{-2}$ . This is the shallow, mesoscale part of the celebrated Nastrom & Gage (Reference Nastrom and Gage1985) spectrum, which is traditionally interpreted as a $k_{h}^{-5/3}$ spectrum but is also consistent with $k_{h}^{-2}$ . There is ongoing debate about the nature of this part of the spectrum: Callies et al. (Reference Callies, Ferrari and Bühler2014, Reference Callies, Bühler and Ferrari2016) attribute it to nearly linear IGWs on the basis of their separation between IGWs and geostrophic motion, but this interpretation is controversial (see Li & Linborg (Reference Li and Linborg2018) for a recent critique). Callies et al. (Reference Callies, Bühler and Ferrari2016) note that ‘the wave interpretation is … not inconsistent with the observed power-law spectra … but an explanation for the spectral shape is so far missing’. Our results provide a possible explanation.

The total spectrum in the high- $Ro$ simulation, shown in figure 5, is reminiscent of atmospheric observations, with a $k_{h}^{-3}$ range at large scales associated with the vortical flow, a $k_{h}^{-2}$ range associated with nearly linear IGWs at intermediate scales, and a further shallowing at small scales (best seen in figure 4 b). While the diffusion approximation explains the $k_{h}^{-2}$ range in our simulations, a degree of caution is required to draw a similar conclusion for the atmospheric spectrum since some of the underlying assumptions – weak flow with homogeneous statistics and relevance of the equilibrium spectrum in particular – are questionable.

Figure 5.

Stationary horizontal energy spectrum for the forced simulations in § 4 with $Ro=0.117$ : total energy (black), vortical energy  $E$ (red), IGW energy $e$ (blue; same as on figure 4 b), and $k_{h}^{-3}$ power law (orange).