2.1 The evolution equations

The balanced flow evolves according to PV advection

(2.7) $$\begin{eqnarray}\displaystyle & \displaystyle q_{t}+J(\unicode[STIX]{x1D713},q)=D_{q}; & \displaystyle\end{eqnarray}$$

the back-rotated velocity satisfies the wave equation

(2.8) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}_{t}+J(\unicode[STIX]{x1D713},\unicode[STIX]{x1D719})+\unicode[STIX]{x1D719}{\displaystyle \frac{\text{i}}{2}}\unicode[STIX]{x1D701}-{\displaystyle \frac{\text{i}}{2}}\unicode[STIX]{x1D702}\unicode[STIX]{x0394}\unicode[STIX]{x1D719}=D_{\unicode[STIX]{x1D719}}. & \displaystyle\end{eqnarray}$$

$D_{q}$ and $D_{\unicode[STIX]{x1D719}}$ in (2.7) and (2.8) are dissipative terms described below.

The wave equation (2.8) is the YBJ model in the case where the near-inertial wave has $\text{e}^{\text{i}mz}$ structure. The back-rotated wave velocity, $\unicode[STIX]{x1D719}$ , evolves through dispersion – the last term on the left of (2.8) – and nonlinear advection and refraction by the second and third terms in (2.8). Without advection, (2.8) is analogous to Schrödinger’s equation (e.g. Landau & Lifshitz Reference Landau and Lifshitz2013, p. 51). The relative vorticity, $\unicode[STIX]{x1D701}=\unicode[STIX]{x0394}\unicode[STIX]{x1D713}$ , is the potential, with negative $\unicode[STIX]{x1D701}$ a well, and the ‘dispersivity’, $f_{0}\unicode[STIX]{x1D706}^{2}$ , is Planck’s constant (Balmforth, Llewellyn Smith & Young Reference Balmforth, Llewellyn Smith and Young1998; Balmforth & Young Reference Balmforth and Young1999; Danioux, Vanneste & Bühler Reference Danioux, Vanneste and Bühler2015). The quantum analogy may be useful for some readers, but it is not necessary for the understanding of the results below.

The terms on the right of (2.7) and (2.8), $D_{q}$ and $D_{\unicode[STIX]{x1D719}}$ , represent small-scale dissipation. Small-scale dissipation is necessary to absorb the forward transfers of potential enstrophy and wave kinetic and potential energies in the numerical solutions reported below. We find that biharmonic diffusion and viscosity,

(2.9a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle D_{q}=-\unicode[STIX]{x1D705}_{e}\unicode[STIX]{x0394}^{2}q\quad \text{and}\quad D_{\unicode[STIX]{x1D719}}=-\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D719}}\unicode[STIX]{x0394}^{2}\unicode[STIX]{x1D719}, & \displaystyle\end{eqnarray}$$

are sufficient to extend the spectral resolution compared to Laplacian dissipation. In practice, we choose $\unicode[STIX]{x1D705}_{e}$ and $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D719}}$ so that the highest 35 % of the modes lie in the dissipation range and aliased wavenumbers are strongly damped.

2.2 The small-amplitude limit and the validity of the NIW-QG approximation

The development of the NIW-QG model is ordered first by assuming that the waves are weak in the sense that

(2.10) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D716}\,\stackrel{def}{=}\,{\displaystyle \frac{U_{w}}{f_{0}L}}\ll 1. & \displaystyle\end{eqnarray}$$

Above, $L$ is a characteristic scale of both waves and balanced flow and $U_{w}$ is a characteristic near-inertial wave velocity. The other small parameter in the expansion is the Rossby number of the balanced flow,

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle Ro\,\stackrel{def}{=}\,{\displaystyle \frac{U_{e}}{f_{0}L}}\ll 1, & \displaystyle\end{eqnarray}$$

where $U_{e}$ is the eddy velocity. The inequalities in (2.10) and (2.11) must be satisfied in order to obtain the NIW-QG system. But XV and Wagner & Young (Reference Wagner and Young2016) make a third assumption: $Ro=\unicode[STIX]{x1D716}^{2}$ , or equivalently that $U_{e}=\unicode[STIX]{x1D716}U_{w}$ . The resulting distinguished limit,

(2.12) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D716}\rightarrow 0,\quad \text{with}~Ro=\unicode[STIX]{x1D716}^{2}, & \displaystyle\end{eqnarray}$$

promotes the importance of wave-averaged effects so that $q^{w}$ appears at an early, and accessible, order in the expansion. Thus (2.12) is for convenience rather than necessity.

The asymptotic ordering in (2.12) does not imply that the NIW-QG system fails for weaker waves, i.e. if $U_{w}$ is comparable to, or even smaller than, $U_{e}$ . Making $U_{w}$ weaker than $U_{e}$ delays wave-averaged effects to longer times – it does not, per se, invalidate the expansion. The main problem with the weak-wave limit is that other physics, not considered in the NIW-QG system, will contend with wave-averaged effects on ultra-long time scales. For example, even without waves, order- $Ro^{2}$ ageostrophic effects modify the evolution of balanced flow and produce departures from QG (e.g. see Muraki, Snyder & Rotunno Reference Muraki, Snyder and Rotunno1999).

To summarize: the main conditions for the validity of the NIW-QG system are (2.10) and (2.11); additionally, validity of the wave equation (2.8) requires that the wave frequency is close of $f_{0}$ . The weak-wave limit $U_{w}/U_{e}\rightarrow 0$ is valid within the NIW-QG framework: in that limit the system reduces to the barotropic potential vorticity equation and the YBJ equation for a passive wave field.

The standard QG approximation is used successfully even when $Ro\sim 1$ (Hoskins Reference Hoskins1975) and we expect the NIW-QG model to enjoy similar success if $\ll$ is replaced by ${<}$ in (2.11). Flows with $Ro>1$ eddies, such as those reported in Barkan et al. (Reference Barkan, Winters and McWilliams2016), evolve on time scales close to $f_{0}^{-1}$ , and an Eulerian decomposition into near-inertial waves and eddies is ill-defined unless there is spatial scale separation between eddies and waves. These large Rossby number flows are outside the purview of the NIW-QG model (XV; Wagner & Young Reference Wagner and Young2016).

2.3 An illustrative solution: the Lamb–Chaplygin dipole

As a preamble to our discussion of stimulated generation in freely evolving two-dimensional turbulence, we consider an example in which the initial QG flow is the Lamb–Chaplygin dipole; see figure 1. This dipole is an exact solution of the Euler equations on an infinite two-dimensional plane where the vorticity is confined to a circle of radius $R$ (Meleshko & Van Heijst Reference Meleshko and Van Heijst1994). The relative vorticity, steady in a frame moving at uniform zonal velocity $U_{e}$ , is

(2.13) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D701}={\displaystyle \frac{2U_{e}\unicode[STIX]{x1D705}}{\text{J}_{0}(\unicode[STIX]{x1D705}R)}}\left\{\begin{array}{@{}ll@{}}\text{J}_{1}(\unicode[STIX]{x1D705}r)\sin \unicode[STIX]{x1D703},\quad & \text{if }r\leqslant R,\\ 0,\quad & \text{if }r\geqslant R.\end{array}\right. & \displaystyle\end{eqnarray}$$

Above $r^{2}=(x-x_{c})^{2}+(y-y_{c})^{2}$ is the radial distance from the centre $(x_{c},y_{c})$ , $\tan \unicode[STIX]{x1D703}=(y-y_{c})/(x-x_{c})$ and $\text{J}_{n}$ is the $n$ th-order Bessel function. The matching condition at $r=R$ is that $\text{J}_{1}(\unicode[STIX]{x1D705}R)=0$ and the smallest solution is $\unicode[STIX]{x1D705}R\approx 3.8317$ . If there is no coupling to the wave $\unicode[STIX]{x1D719}$ , then the dipole (2.13) is a solution of the QG equation (2.7).

Figure 1. Snapshots of the Lamb–Chaplygin dipole solution with parameters presented in table 2. Contours depict potential vorticity, $q/(U_{e}k_{e})=[-1.5,-0.5,0.5,1.5]$ , with dashed lines showing negative values. (a–c) The wave action density $|\unicode[STIX]{x1D719}|^{2}/2f_{0}$ . (d–f) The wave buoyancy; the buoyancy scale is $B=k_{e}mU_{w}f_{0}\unicode[STIX]{x1D706}^{2}$ . These plots only show the central $(1/5)^{2}$ of the simulation domain.

Table 2. Description of parameters of Lamb–Chaplygin simulation. The initial conditions are Rossby number $Ro=U_{e}k_{e}/f_{0}\approx 0.05$ , wave dispersivity $\hslash =f_{0}\unicode[STIX]{x1D706}^{2}k_{e}/U_{e}\approx 1$ and wave amplitude $\unicode[STIX]{x1D6FC}=Ro(U_{w}/Ue)^{2}\approx 3.75$ .

We strongly perturb the dipole in (2.13) by seeding a wave with initial velocity:

(2.14) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}(x,y,t=0)={\displaystyle \frac{1+\text{i}}{\sqrt{2}}}\,U_{w}. & \displaystyle\end{eqnarray}$$

If there was no dipole, this initial condition produces a spatially uniform near-inertial oscillation with speed $U_{w}$ . Further parameters of this solution are summarized in table 2; note $U_{w}=10U_{e}$ .

We integrate the vertical-plane-wave model using a standard collocation Fourier spectral method. We evaluate the quadratic nonlinearities, including in the wave potential vorticity (2.6), in physical space, and transform the product into Fourier space. We time march the spectral equations using an exponential time differencing method with a fourth-order Runge–Kutta scheme – for details, see Cox & Matthews (Reference Cox and Matthews2002) and Kassam & Trefethen (Reference Kassam and Trefethen2005).

With the initial condition in (2.14), $q^{w}$ in (2.6) and $\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}$ are both zero at $t=0$ . The refractive term in the wave equation (2.8), however, immediately imprints itself onto $\unicode[STIX]{x1D719}$ thus creating non-zero $\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}$ and non-zero $q^{w}$ . Once refraction has created gradients in $\unicode[STIX]{x1D719}$ , the advective term in (2.8) can further distort $\unicode[STIX]{x1D719}$ and increase $\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}$ . This scale reduction of $\unicode[STIX]{x1D719}$ is most evident in the wave buoyancy shown in figure 1(d–f). Figure 1 also shows the well-known focusing of waves into the negative vortex. But the wave feedback on the mean flow through $q^{w}$ then results in distortion and shearing of the dipole so that the negative vortex loses its integrity; the lopsided dipole then starts to drift. Once the negative vortex is distorted to small scales it no longer acts as an effective potential well: the trapping of wave energy by the deformed anti-cyclone is weaker than in the initial condition. In fact, figure 1, which shows the materially conserved PV $q$ , understates the development of small scales in the relative vorticity $\unicode[STIX]{x1D701}$ : figure 2 shows that both $\unicode[STIX]{x1D701}$ and $q^{w}$ develop small scales with significant cancellation resulting in the relatively smooth field $q=\unicode[STIX]{x1D701}+q^{w}$ shown in figure 1. Thus at the final time in figure 1 the waves are no longer strongly trapped in the region with $\unicode[STIX]{x1D701}<0$ . This phenomenology, including significant cancellation between $\unicode[STIX]{x1D701}$ and $q^{w}$ , is also characteristic of wave-modified two-dimensional turbulence in § 4.

Figure 2. (a) Snapshot of $q$ (contours) and $\unicode[STIX]{x1D701}$ (colours) at $t\times U_{e}k_{e}=20$ . (b) Snapshot of $q$ (contours) and wave PV $q^{w}$ (colours). Both black lines and colours depict the contour levels $[-1.5,-0.5,0.5,1.5]\times (U_{e}k_{e})$ . Solid lines and reddish colours depict positive values; dashed lines and greenish colours show negative values.

To understand the results in figure 1 and quantify the stimulated generation of wave energy, we need to understand the conservation laws of the vertical-plane-wave model.

3.1 Action conservation equation and action flux

Multiplying the wave equation (2.8) by $\unicode[STIX]{x1D719}^{\star }$ and adding to the complex conjugate, we obtain a conservation equation for action density,

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}{\mathcal{A}}+J(\unicode[STIX]{x1D713},{\mathcal{A}})+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{{\mathcal{F}}}=\underbrace{{\displaystyle \frac{1}{2f_{0}}}(\unicode[STIX]{x1D719}^{\star }D_{\unicode[STIX]{x1D719}}+\unicode[STIX]{x1D719}D_{\unicode[STIX]{x1D719}^{\star }})}_{\stackrel{def}{=}D_{{\mathcal{A}}}}, & \displaystyle\end{eqnarray}$$

where the flux of near-inertial wave action is

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{{\mathcal{F}}}\,\stackrel{def}{=}\,{\displaystyle \frac{\text{i}}{4}}\unicode[STIX]{x1D706}^{2}(\unicode[STIX]{x1D719}\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}^{\star }-\unicode[STIX]{x1D719}^{\star }\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}). & \displaystyle\end{eqnarray}$$

The local conservation law (3.5) shows how the wave action ${\mathcal{A}}$ changes due to geostrophic advection and divergence of the wave flux and dissipation – the second, third and fourth terms in (3.5).

The wave action flux $\boldsymbol{{\mathcal{F}}}$ is analogous to the probability current of quantum mechanics (e.g. Landau & Lifshitz Reference Landau and Lifshitz2013, p. 57). Using the polar representation $\unicode[STIX]{x1D719}=|\unicode[STIX]{x1D719}|\text{e}^{\text{i}\unicode[STIX]{x1D6E9}}$ , the wave action flux $\boldsymbol{{\mathcal{F}}}$ can also be written as

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{{\mathcal{F}}}={\mathcal{A}}\,\unicode[STIX]{x1D702}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6E9}, & \displaystyle\end{eqnarray}$$

where recall that $\unicode[STIX]{x1D702}=f_{0}\unicode[STIX]{x1D706}^{2}$ is the dispersivity. In (3.7), $\unicode[STIX]{x1D702}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6E9}$ is the ‘generalized group velocity’ of hydrostatic near-inertial waves, i.e. $\boldsymbol{{\mathcal{F}}}$ is the generalized group velocity times the action density ${\mathcal{A}}$ . We use the term ‘generalized’ because no WKB-type (Wentzel–Kramers–Brillouin) scale separation is required to obtain the results above. The connection to standard internal-wave group velocity is quickly verified by considering a plane near-inertial wave with $\unicode[STIX]{x1D6E9}=kx+ly$ , yielding $\unicode[STIX]{x1D702}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6E9}=N_{0}^{2}(k,l)/f_{0}m^{2}$ .

Another useful identity involving the action flux $\boldsymbol{{\mathcal{F}}}$ is

(3.8) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }(\hat{\boldsymbol{k}}\times \boldsymbol{{\mathcal{F}}})={\displaystyle \frac{\text{i}}{2}}\unicode[STIX]{x1D706}^{2}J(\unicode[STIX]{x1D719}^{\star },\unicode[STIX]{x1D719}), & \displaystyle\end{eqnarray}$$

where $\hat{\boldsymbol{k}}$ is the unit vector perpendicular to the $(x,y)$ -plane. Using (3.8), and the definition of action in (3.1), the wave PV $q^{w}$ in (2.6) can be written as

(3.9) $$\begin{eqnarray}\displaystyle & \displaystyle q^{w}={\textstyle \frac{1}{2}}\unicode[STIX]{x0394}{\mathcal{A}}+\unicode[STIX]{x1D702}^{-1}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\hat{\boldsymbol{k}}\times \boldsymbol{{\mathcal{F}}}). & \displaystyle\end{eqnarray}$$

Denoting an average over the domain by $\langle \rangle$ , and assuming that the action flux divergence $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{{\mathcal{F}}}$ vanishes after integration, we obtain from (3.5)

(3.10) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\text{d}\langle {\mathcal{A}}\rangle }{\text{d}t}}=\unicode[STIX]{x1D700}_{{\mathcal{A}}}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D700}_{{\mathcal{A}}}\,\stackrel{def}{=}\,\langle \unicode[STIX]{x1D719}^{\star }D_{\unicode[STIX]{x1D719}}+\unicode[STIX]{x1D719}D_{\unicode[STIX]{x1D719}^{\star }}\rangle /(2f_{0})$ is the domain average of the dissipative term on the right of (3.5). In the example shown in figure 1 the total action $\langle {\mathcal{A}}\rangle$ is conserved to within 1 % over the course of the integration.

3.2 Ehrenfest’s theorem

The quantum analogy suggests that we should seek an analogue of Ehrenfest’s theorem (the quantum equivalent of Newton’s law that force equals mass times acceleration). Thus in appendix B we develop a local conservation law for $\boldsymbol{{\mathcal{F}}}$ . The domain average of that result is

(3.11) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\text{d}\langle \boldsymbol{{\mathcal{F}}}\rangle }{\text{d}t}}=\hat{\boldsymbol{k}}\times \langle \unicode[STIX]{x1D701}\,\boldsymbol{{\mathcal{F}}}\rangle -\hat{\boldsymbol{k}}\times \langle (\boldsymbol{{\mathcal{F}}}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\rangle -\unicode[STIX]{x1D702}\left\langle {\mathcal{A}}\,\unicode[STIX]{x1D735}{\displaystyle \frac{1}{2}}\unicode[STIX]{x1D701}\right\rangle +\unicode[STIX]{x1D73A}_{\boldsymbol{{\mathcal{F}}}}. & \displaystyle\end{eqnarray}$$

In the quantum analogy, $\boldsymbol{{\mathcal{F}}}$ is momentum and the left-hand side of (3.11) is mass times acceleration; the forces are on the right of (3.11). Starting from the end, $\unicode[STIX]{x1D73A}_{\boldsymbol{{\mathcal{F}}}}$ is a dissipative term defined in appendix B. The third term on the right of (3.11) is the force due to the gradient of the potential $\unicode[STIX]{x1D701}/2$ . The second term on the right of (3.11) represents the combined effect of stretching and tilting of $\boldsymbol{{\mathcal{F}}}$ by the geostrophic flow. The first term on the right of (3.11) is a ‘vortex force’, again due to $\unicode[STIX]{x1D701}$ , but perpendicular to $\boldsymbol{{\mathcal{F}}}$ . The two $\hat{\boldsymbol{k}}\times$ terms on the right lack quantum analogues.

The results in this section are obtained from the wave equation (2.8) without using the PV equation (2.7). In other words, (3.5)–(3.11) apply to the YBJ equation with $\text{e}^{\text{i}mz}$ structure regardless of the balanced flow dynamics. We turn now to energy conservation and consideration of the PV equation (2.7).

3.3 Energy conservation and conversion

The energy conservation law is considerably more complicated than action conservation. We sequester the details of the local conservation laws to appendix B and present here the simpler results obtained by domain averaging those local conservation laws. The results in (3.12) through (3.16) below are obtained by: (i) multiplying the wave equation (2.8) by $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}^{\star }$ , forming the average $\langle \rangle$ and adding the complex conjugate; and by (ii) multiplying the PV equation (2.7) by $-\unicode[STIX]{x1D713}$ and averaging $\langle \rangle$ .

For the wave potential energy in (3.4) and the balanced kinetic energy in (3.3) we find

(3.12) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\text{d}\langle {\mathcal{P}}\rangle }{\text{d}t}}=\unicode[STIX]{x1D6E4}_{r}+\unicode[STIX]{x1D6E4}_{a}+\unicode[STIX]{x1D700}_{{\mathcal{P}}}, & \displaystyle\end{eqnarray}$$

(3.13) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\text{d}\langle {\mathcal{K}}\rangle }{\text{d}t}}=-\unicode[STIX]{x1D6E4}_{r}-\unicode[STIX]{x1D6E4}_{a}+\unicode[STIX]{x1D6EF}+\unicode[STIX]{x1D700}_{{\mathcal{K}}}, & \displaystyle\end{eqnarray}$$

where the ‘conversion terms’ in (3.12) and (3.13) are

(3.14) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}_{r}\,\stackrel{def}{=}\,\left\langle {\textstyle \frac{1}{2}}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{{\mathcal{F}}}\right\rangle , & \displaystyle\end{eqnarray}$$

and

(3.15) $$\begin{eqnarray}\displaystyle \displaystyle \unicode[STIX]{x1D6E4}_{a}~\stackrel{def}{=}~{\displaystyle \frac{\unicode[STIX]{x1D706}^{2}}{4}}\langle \unicode[STIX]{x0394}\unicode[STIX]{x1D719}^{\star }J(\unicode[STIX]{x1D713},\unicode[STIX]{x1D719})+\unicode[STIX]{x0394}\unicode[STIX]{x1D719}J(\unicode[STIX]{x1D713},\unicode[STIX]{x1D719}^{\star })\rangle & & \displaystyle\end{eqnarray}$$

(3.16)

The dissipative terms, $\unicode[STIX]{x1D700}_{{\mathcal{P}}}$ , $\unicode[STIX]{x1D700}_{{\mathcal{K}}}$ and $\unicode[STIX]{x1D6EF}$ are defined in appendix B. $\unicode[STIX]{x1D6EF}$ in (3.13) is particularly interesting: dissipation of waves $D_{\unicode[STIX]{x1D719}}$ produces balanced kinetic energy. Summing (3.12) and (3.13) the ‘conversion’ terms $\unicode[STIX]{x1D6E4}_{r}$ and $\unicode[STIX]{x1D6E4}_{a}$ cancel, and we obtain the conservation law for total energy $\langle {\mathcal{E}}\rangle =\langle {\mathcal{P}}+{\mathcal{K}}\rangle$ .

Figure 3. Illustration of energy conversion terms in the Lamb–Chaplygin dipole solution with parameters presented in table 2. (a) The action flux $\boldsymbol{{\mathcal{F}}}$ overlaid on contours of relative vorticity $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}/(U_{e}k_{e})=[-1.5,-0.5,0.5,1.5]$ , with dashed lines showing negative values; the scale of the action flux is $F=f_{0}\unicode[STIX]{x1D706}^{2}k_{e}U_{w}^{2}$ . (b) The local advective conversion (i.e. the non-averaged $\unicode[STIX]{x1D6E4}_{a}$ ) overlaid on contours of streamfunction $\unicode[STIX]{x1D713}\times (U_{e}/k_{e})=[-8,-4,-2,0,2,4,8]$ .

The refractive conversion $\unicode[STIX]{x1D6E4}_{r}$ stems from $\text{i}\unicode[STIX]{x1D719}\unicode[STIX]{x1D701}/2$ in the wave equation and is easy to interpret: the convergence of the wave action flux, $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{{\mathcal{F}}}<0$ , into anti-cyclones, $\unicode[STIX]{x1D701}<0$ , is a source of wave potential energy ${\mathcal{P}}$ ; similarly, the divergence of the wave action flux from cyclones is a source of ${\mathcal{P}}$ . Figure 3(a) shows the convergence of $\boldsymbol{{\mathcal{F}}}$ into the anti-cyclone (and divergence from the cyclone) of the dipole solution at $t\times U_{e}k_{e}=1$ , which yields the sharp initial increase of $\langle {\mathcal{P}}\rangle$ discussed below. Ehrenfest’s theorem in (3.11) illuminates the initial structure of $\boldsymbol{{\mathcal{F}}}$ in figure 3. Because $\unicode[STIX]{x1D719}$ is initially uniform, the initial tendency of $\boldsymbol{{\mathcal{F}}}$ is

(3.17) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\boldsymbol{{\mathcal{F}}}(0)=-\unicode[STIX]{x1D702}{\mathcal{A}}\unicode[STIX]{x1D735}{\textstyle \frac{1}{2}}\unicode[STIX]{x1D701}. & \displaystyle\end{eqnarray}$$

Thus the action flux is initially anti-parallel to the gradient of relative vorticity (see figure 3 a), and wave action converges into anti-cyclones (and diverges from cyclones).

Figure 4. Diagnostics of the Lamb–Chaplygin dipole solution with parameters presented in table 2. (a) Energy change about initial condition. (b) Wave potential energy budget (3.12).

The advective conversion $\unicode[STIX]{x1D6E4}_{a}$ in (3.12) and (3.13) stems from the term $J(\unicode[STIX]{x1D713},\unicode[STIX]{x1D719})$ in the wave equation (2.8) and is a source of $\langle {\mathcal{P}}\rangle$ due to straining and deformation of the wave field by the geostrophic flow. The symmetric $2\times 2$ matrix in (3.16) is the strain or deformation tensor of the geostrophic flow. Thus, in analogy with passive-scalar gradient amplification, straining also enhances gradients of the back-rotated near-inertial velocity $\unicode[STIX]{x1D719}$ , thereby generating wave potential energy $\langle {\mathcal{P}}\rangle$ .

3.4 Energetics of the Lamb–Chaplygin dipole solution

Figure 4 shows the energetics of the Lamb–Chaplygin dipole solution in figure 1. In figure 4(a), ${\mathcal{P}}$ increases at the expense of ${\mathcal{K}}$ , while ${\mathcal{A}}$ is conserved. The wave potential energy budget in figure 4(b) shows that this stimulated generation occurs in two stages. First, refraction of the initially uniform wave field causes a dramatic concentration of waves into the anti-cyclone, producing a sharp increase ${\mathcal{P}}$ through $\unicode[STIX]{x1D6E4}_{r}$ . But this rapid initial energy conversion does not last long because the wave feedback deforms the anti-cyclone and dispersion radiates waves away from the dipole (see figure 1). Thus in figure 4(b), $\unicode[STIX]{x1D6E4}_{r}$ decreases sharply, and eventually reverses sign at $t\times U_{e}k_{e}\approx 8$ .

The second stage of stimulated generation starts after refraction has created dipole-scale waves. Advection by the balanced flow can then strain the waves, further reducing their lateral scale (figure 4 b). The ensuing advective conversion, $\unicode[STIX]{x1D6E4}_{a}$ , starts at $t\times U_{e}k_{e}\approx 4$ . Straining by the balanced flow sustains this advective generation of ${\mathcal{P}}$ . The waves eventually escape the straining regions through dispersion and the conversion nearly halts at $t\times U_{e}k_{e}=30$ . The time-integrated $\unicode[STIX]{x1D6E4}_{a}$ accounts for ${\approx}78\,\%$ of the wave potential energy generation; table 3 summarizes the energy budget.

Table 3. The time-integrated budget of wave potential energy and quasi-geostrophic kinetic energy of the Lamb–Chaplygin dipole solutions with parameters provided in table 2. The energy budgets close within $10^{-6}\,\%$ .

3.5 Summary

The expressions for energy conversion in (3.14) and (3.16) clarify the mechanism of stimulated generation triggered by the initially uniform near-inertial wave in (2.14). First, refraction causes a convergence of wave action into anti-cyclones. Then advection strains the waves, reducing their lateral scale. Both processes amplify the lateral gradients of wave amplitude, thereby generating wave potential energy ${\mathcal{P}}$ at the expense of balanced kinetic energy ${\mathcal{K}}$ . Wave action ${\mathcal{A}}$ is conserved throughout this process.

In the remainder of this paper, we describe and quantify stimulated generation in an idealization of an oceanographic post-storm scenario: the uniform initial near-inertial wave in (2.14) interacts with two-dimensional turbulence.

4.1 Relevant parameters

The scaling

(4.4a-d ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{length}\sim k_{e}^{^{-1}},\quad \text{time}\sim (U_{e}k_{e})^{-1},\quad \unicode[STIX]{x1D713}\sim U_{e}k_{e}^{-1},\quad \text{and}\quad \unicode[STIX]{x1D719}\sim U_{w}, & \displaystyle\end{eqnarray}$$

shows that there are two important dimensionless control parameters. The first is

(4.5) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FC}\,\stackrel{def}{=}\,\underbrace{{\displaystyle \frac{U_{e}k_{e}}{f_{0}}}}_{\stackrel{def}{=}Ro}\times \left({\displaystyle \frac{U_{w}}{U_{e}}}\right)^{2}, & \displaystyle\end{eqnarray}$$

which scales the contribution of the wave terms in the potential vorticity (2.6). The second dimensionless parameter is

(4.6) $$\begin{eqnarray}\displaystyle & \displaystyle \hslash \,\stackrel{def}{=}\,\unicode[STIX]{x1D702}\times {\displaystyle \frac{k_{e}}{U_{e}}}=Ro^{-1}\times (\unicode[STIX]{x1D706}k_{e})^{2}, & \displaystyle\end{eqnarray}$$

which scales wave dispersion against the effects of advection and refraction. Assuming that the wave horizontal scale is $k_{e}^{-1}$ , $(\unicode[STIX]{x1D706}k_{e})^{2}$ is the wave Burger number, which is small for near-inertial waves.

Table 4. Description of parameters of the 2-D turbulence simulations. The initial conditions are Rossby number $Ro=U_{e}k_{e}/f_{0}\approx 0.05$ , wave dispersivity $\hslash =f_{0}\unicode[STIX]{x1D706}^{2}k_{e}/U_{e}\approx 0.5{-}2$ and wave amplitude $\unicode[STIX]{x1D6FC}=Ro(U_{w}/U_{e})^{2}\approx 0.1$ .

Figure 5. Snapshots of the turbulence solution with parameters in table 4. (a–c) PV $q$ . (d–f) Wave kinetic energy density $|\unicode[STIX]{x1D719}|^{2}$ . (g–i) Wave buoyancy, with scale $B=k_{e}mU_{w}f_{0}\unicode[STIX]{x1D706}^{2}$ . These plots show $(1/2)^{2}$ of the domain.

Figure 6. Diagnostics of the 2-D turbulence solution with parameters presented in table 4. (a) Energy change about initial condition. (b) Wave potential energy budget (3.12).

4.2 Solution with $\hslash =1$ and $\unicode[STIX]{x1D6FC}=0.1$

Figure 5 shows snapshots of a solution with $\hslash =1$ and $\unicode[STIX]{x1D6FC}=0.1$ with further parameters in table 4. This turbulence solution shares qualitative aspects of the Lamb–Chaplygin solution. Starting from a uniform wave field in (2.14), refraction quickly concentrates the waves into anti-cyclones. Initially the action density ${\mathcal{A}}$ is uniform but by $t\times U_{e}k_{e}\approx 1$ , ${\mathcal{A}}$ varies on eddy scales by a factor of two with significant focusing of waves, indicated by maxima of ${\mathcal{A}}$ , into anti-cyclones (compare (d–f) and (g–i) of figure 5).

Dispersion radiates waves from the vortices; advection enhances the gradients of back-rotated velocity $\unicode[STIX]{x1D719}$ (see figure 5 g–i, which depicts wave buoyancy). By $t\times U_{e}k_{e}\approx 10$ , ${\mathcal{A}}$ varies by a factor of five and the wave buoyancy is amplified by a factor of two. The evolution of potential vorticity $q$ is similar to that in the waveless problem: like-sign vortices merge into bigger vortices. The big vortices keep straining the waves, generating smaller scales in the wave field.

Figure 6(a) shows the inexorable increase in wave potential energy $\langle {\mathcal{P}}\rangle$ and the corresponding decrease in balanced kinetic energy $\langle {\mathcal{K}}\rangle$ . In figure 6(b) quick wave refraction results in an initial sharp generation of $\langle {\mathcal{P}}\rangle$ at the expense of balanced kinetic energy $\langle {\mathcal{K}}\rangle$ . As in the Lamb–Chaplygin solution, the positive refractive conversion, $\unicode[STIX]{x1D6E4}_{r}>0$ , is ephemeral: in figure 6(b), $\unicode[STIX]{x1D6E4}_{r}$ peaks at $t\times U_{e}k_{e}\approx 2$ and then decays rapidly, eventually changing sign at $t\times U_{e}k_{e}\approx 5$ . However, a significant positive advective conversion, $\unicode[STIX]{x1D6E4}_{a}>0$ , sustains stimulated generation so that $\langle {\mathcal{P}}\rangle$ ultimately increases approximately linearly with time.

After 25 eddy turnover time units, the balanced kinetic energy $\langle {\mathcal{K}}\rangle$ has decayed by approximately $14\,\%$ from its initial value. Most of this loss is by stimulated generation of $\langle {\mathcal{P}}\rangle$ . As in the Lamb–Chaplygin solution, advective conversion $\unicode[STIX]{x1D6E4}_{a}$ accounts for most of the energy change. Table 5 presents further details of the energy budget.

The solution illustrates interesting characteristics of stimulated generation. First, the role of refraction is catalytic in that it generates the initial eddy-scale gradients in $\unicode[STIX]{x1D719}$ that are then enhanced by advective straining; the advective conversion, $\unicode[STIX]{x1D6E4}_{a}$ in (3.16), ultimately accounts for most of the energy transfer from turbulence to waves. Second, the approximately linear-in-time growth of wave potential energy $\langle {\mathcal{P}}\rangle$ is very slow in comparison with exponential increase of passive-scalar tracer gradients in turbulent velocity fields. The relatively slow growth of $\langle {\mathcal{P}}\rangle$ suggests that wave dispersion plays an important role in slowing and perhaps opposing advective straining (see § 5 for further discussion of dispersion and ‘wave escape’). To investigate whether these characteristics are general we consider solutions with varying vertical wavelengths and therefore different dispersivities.

Table 5. The time-integrated budget of wave potential energy and QG kinetic energy of the reference 2-D turbulence solution with parameters in 4. The energy budgets close within $0.1\,\%$ .

Figure 7. Snapshots of PV $q$ and its decomposition into relative vorticity $\unicode[STIX]{x1D701}=\unicode[STIX]{x0394}\unicode[STIX]{x1D713}$ , and wave potential vorticity $q^{w}$ . The snapshots were taken at $t\times U_{e}k_{e}=25$ .

Figure 8. The energetics of 2-D turbulence solutions with different dispersivities. (a) Energy change about the initial condition. (b) The energy conversion terms in (3.12). (c) The correlation between relative vorticity and wave kinetic energy. (d) The skewness of relative vorticity.

4.3 Varying dispersivity

Figure 7 shows snapshots of potential vorticity $q$ and its constituents in a set of solutions with varying the vertical wavelength $2\unicode[STIX]{x03C0}\,m^{-1}$ from 280 m to 560 m, yielding dispersivities ranging from 0.5 to 2. (All other parameters are fixed.) The potential vorticity $q$ shows more small-scale filamentation with decreasing dispersivity, but it is otherwise similar across the three solutions. The partition into relative vorticity $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}$ and wave potential vorticity $q^{w}$ , however, depends significantly on dispersivity. In particular, $q^{w}$ develops smaller scales and larger amplitudes with decreasing dispersivity. As anticipated by the dipole example in figure 2, there is cancellation of small-scale features in $q^{w}$ against those in $\unicode[STIX]{x1D701}$ so that $q$ is relatively smooth even in the solution with weak dispersion $\hslash =0.5$ .

Figure 9. Energy-preserving spectra of 2-D turbulence solutions with different dispersivities. The three panels show spectra of balanced kinetic energy ${\mathcal{K}}$ , wave action ${\mathcal{A}}$ and wave potential energy ${\mathcal{P}}$ . All solid lines correspond to spectra at $t\times U_{e}k_{e}=25$ and the dashed line in ${\mathcal{K}}$ is the balanced kinetic energy spectrum at $t\times U_{e}k_{e}=0$ . All spectra are normalized by their total energy, i.e. the area under each curve is one.

The initial evolution of the uniform wave field is similar across dispersivities, with refraction initially generating eddy-scale gradients of the waves; see figure 8. Refraction produces a sharp initial increase of wave potential energy and decrease of balanced kinetic energy, which is almost independent of dispersivity. Figure 8(c) shows that this initial ‘refractive stage’ yields a strongly negative wave–vorticity correlation $r$ ,

(4.7) $$\begin{eqnarray}\displaystyle & \displaystyle r\,\stackrel{def}{=}\,{\displaystyle \frac{\langle \unicode[STIX]{x1D701}{\mathcal{A}}^{\prime }\rangle }{\sqrt{\langle \unicode[STIX]{x1D701}^{2}\rangle \langle {\mathcal{A}}^{\prime 2}\rangle }}}, & \displaystyle\end{eqnarray}$$

where ${\mathcal{A}}^{\prime }\,\stackrel{def}{=}\,(|\unicode[STIX]{x1D719}|^{2}-|\langle \unicode[STIX]{x1D719}\rangle |^{2})/f_{0}$ ; in figure 8(c) the early negative $r$ is nearly independent of dispersivity. Because significant energy exchange takes place in the anti-cyclones due to the initial wave concentration, a positive vorticity skewness ensues (figure 8 d). Once the eddy scales are created, advection strains the waves and generates further wave potential energy at the expense of balanced kinetic energy. It is in this stage that the dependence on dispersivity is pronounced: weakly dispersive waves are strained further than strongly dispersive waves. Thus the advective conversion becomes stronger with decreasing dispersivity (figure 8 b). Advection and dispersion significantly reduce the wave–vorticity correlation; the reduction in correlation increases as the dispersivity decreases (figure 8 c). For the weakest dispersivity considered, the wave–vorticity correlation becomes weakly positive, likely because of the early positive vorticity skewness.

In all solutions reported above, the evolution of the balanced flow is similar to that of waveless 2-D turbulence: there is a transfer of balanced energy towards larger scales driven by merger of like-signed vortices; see figure 9(a). The main difference is that balanced kinetic energy is constantly transformed into wave potential energy via stimulated generation. The stimulated generation process is associated with a forward transfer of wave action ${\mathcal{A}}$ from the infinite horizontal scale in the initial condition (2.14) to the eddy scale; see figure 9(b). The wave potential energy density ${\mathcal{P}}$ in figure 9(c) develops significantly smaller scales than those of the balanced kinetic energy ${\mathcal{K}}$ .

5.1 Strain flow

We are surprised by the successful resistance mounted by the waves to strain-driven exponential amplification of $\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}$ and thus seek to illustrate wave escape with simple flows. We first consider the straining flow $\unicode[STIX]{x1D713}=-\unicode[STIX]{x1D6FC}xy$ . Ignoring dissipation for simplicity, the wave equation (2.8) reduces to

(5.1) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}_{t}+\unicode[STIX]{x1D6FC}x\unicode[STIX]{x1D719}_{x}-\unicode[STIX]{x1D6FC}y\unicode[STIX]{x1D719}_{y}-{\displaystyle \frac{\text{i}}{2}}\unicode[STIX]{x1D702}\unicode[STIX]{x0394}\unicode[STIX]{x1D719}=0, & \displaystyle\end{eqnarray}$$

and the action equation is

(5.2) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{A}}_{t}+\unicode[STIX]{x1D6FC}(x{\mathcal{A}})_{x}-\unicode[STIX]{x1D6FC}(y{\mathcal{A}})_{y}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{{\mathcal{F}}}=0. & \displaystyle\end{eqnarray}$$

Without vorticity and dissipation, Ehrenfest’s theorem (3.11) reduces to

(5.3) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}[\langle {\mathcal{F}}^{x}\rangle ,\,\langle {\mathcal{F}}^{y}\rangle ]=[-\unicode[STIX]{x1D6FC}\langle {\mathcal{F}}^{x}\rangle ,~+\unicode[STIX]{x1D6FC}\langle {\mathcal{F}}^{y}\rangle ], & \displaystyle\end{eqnarray}$$

with solution

(5.4) $$\begin{eqnarray}\displaystyle & \displaystyle [\langle {\mathcal{F}}^{x}\rangle ,\,\langle {\mathcal{F}}^{y}\rangle ]=[\langle {\mathcal{F}}^{x}\rangle _{0}\text{e}^{-\unicode[STIX]{x1D6FC}t},\,\langle {\mathcal{F}}^{y}\rangle _{0}\text{e}^{\unicode[STIX]{x1D6FC}t}], & \displaystyle\end{eqnarray}$$

where the subscript $0$ denotes the initial condition. For a compact wave packet, with a well-defined uniform group velocity $\boldsymbol{c}_{g}$ , the action flux is

(5.5) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \boldsymbol{{\mathcal{F}}}\rangle =\boldsymbol{c}_{g}\langle {\mathcal{A}}\rangle . & \displaystyle\end{eqnarray}$$

Because $\langle {\mathcal{A}}\rangle$ is constant, (5.4) is thus also a solution for $\boldsymbol{c}_{g}(t)$ .

The position of the centre of the packet is

(5.6) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{x}_{c}(t)\,\stackrel{def}{=}\,\langle \boldsymbol{x}{\mathcal{A}}\rangle /\langle {\mathcal{A}}\rangle . & \displaystyle\end{eqnarray}$$

Multiplying (5.2) by $\boldsymbol{x}$ , averaging and using (5.4), yields

(5.7a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}x_{c}-\unicode[STIX]{x1D6FC}x_{c}=c_{g0}^{x}\text{e}^{-\unicode[STIX]{x1D6FC}t},\quad \text{and}\quad \unicode[STIX]{x2202}_{t}y_{c}+\unicode[STIX]{x1D6FC}y_{c}=c_{g0}^{y}\text{e}^{\unicode[STIX]{x1D6FC}t}, & \displaystyle\end{eqnarray}$$

with solution

(5.8) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{x}_{c}(t)=\boldsymbol{x}_{c0}+\boldsymbol{c}_{g0}{\displaystyle \frac{\sinh \unicode[STIX]{x1D6FC}t}{\unicode[STIX]{x1D6FC}}}. & \displaystyle\end{eqnarray}$$

The trajectory of the packet is therefore a straight line following the initial group velocity $\boldsymbol{c}_{g0}$ . The wave packet is not deflected by the hyperbolic streamlines of the straining flow and the packet escapes by accelerating exponentially with time along a straight-line trajectory. This requires that the group velocity $\boldsymbol{c}_{g}(t)$ adjusts in magnitude and direction so as to keep the centre of the packet on the straight and narrow.

The result in (5.8) is so remarkable that it is reassuring to obtain it without invoking Ehrenfest’s theorem. We thus consider the specific example of a wave packet launched in the strain field with the Gaussian initial condition

(5.9) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}(x,y,0)={\displaystyle \frac{2\unicode[STIX]{x03C0}}{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}}\exp \left[-{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{x^{2}}{\unicode[STIX]{x1D707}^{2}}}+{\displaystyle \frac{y^{2}}{\unicode[STIX]{x1D708}^{2}}}\right)+\text{i}(px+qy)\right]. & \displaystyle\end{eqnarray}$$

If $p\unicode[STIX]{x1D707}\gg 1$ and $q\unicode[STIX]{x1D708}\gg 1$ this is a wave packet with initial wavenumber $(p,q)$ and initial group velocity

(5.10) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{c}_{g0}=\unicode[STIX]{x1D702}\,(p,q). & \displaystyle\end{eqnarray}$$

The exact solution to (5.1), subject to the initial condition (5.9), is

(5.11) $$\begin{eqnarray}\displaystyle \displaystyle \unicode[STIX]{x1D719} & = & \displaystyle {\displaystyle \frac{2\unicode[STIX]{x03C0}}{\sqrt{(\unicode[STIX]{x1D707}^{2}+\text{i}\unicode[STIX]{x1D702}f)(\unicode[STIX]{x1D708}^{2}+\text{i}\unicode[STIX]{x1D702}g)}}}\nonumber\\ \displaystyle & & \displaystyle \times \,\exp \left[-{\displaystyle \frac{1}{2}}{\displaystyle \frac{x^{2}\text{e}^{-2\unicode[STIX]{x1D6FC}t}-2\text{i}\unicode[STIX]{x1D707}^{2}px\text{e}^{-\unicode[STIX]{x1D6FC}t}+\text{i}\unicode[STIX]{x1D702}f\unicode[STIX]{x1D707}^{2}p^{2}}{\unicode[STIX]{x1D707}^{2}+\text{i}\unicode[STIX]{x1D702}f}}-{\displaystyle \frac{1}{2}}{\displaystyle \frac{y^{2}\text{e}^{2\unicode[STIX]{x1D6FC}t}-2\text{i}\unicode[STIX]{x1D708}^{2}qy\text{e}^{\unicode[STIX]{x1D6FC}t}+\text{i}\unicode[STIX]{x1D702}g\unicode[STIX]{x1D708}^{2}q^{2}}{\unicode[STIX]{x1D708}^{2}+\text{i}\unicode[STIX]{x1D702}g}}\right],\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

with

(5.12a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle f\,\stackrel{def}{=}\,{\displaystyle \frac{1-\text{e}^{-2\unicode[STIX]{x1D6FC}t}}{2\unicode[STIX]{x1D6FC}}}\quad \text{and}\quad g\,\stackrel{def}{=}\,{\displaystyle \frac{\text{e}^{2\unicode[STIX]{x1D6FC}t}-1}{2\unicode[STIX]{x1D6FC}}}. & \displaystyle\end{eqnarray}$$

Figure 10 illustrates the solution (5.11). The strain flow tilts the packet to align it with the $x$ -axis. The flow then exponentially stretches the packet, which in turn escapes from the saddle point. To calculate the trajectory of the packet we note that

(5.13) $$\begin{eqnarray}\displaystyle & \displaystyle |\unicode[STIX]{x1D719}|^{2}={\displaystyle \frac{(2\unicode[STIX]{x03C0})^{2}}{\sqrt{(\unicode[STIX]{x1D707}^{4}+\unicode[STIX]{x1D702}^{2}f^{2})(\unicode[STIX]{x1D708}^{4}+\unicode[STIX]{x1D702}^{2}g^{2})}}}\,\exp \left[-{\displaystyle \frac{\unicode[STIX]{x1D707}^{2}(x\text{e}^{-\unicode[STIX]{x1D6FC}t}-\unicode[STIX]{x1D702}fp)^{2}}{\unicode[STIX]{x1D707}^{4}+\unicode[STIX]{x1D702}^{2}f^{2}}}-{\displaystyle \frac{\unicode[STIX]{x1D708}^{2}(y\text{e}^{\unicode[STIX]{x1D6FC}t}-\unicode[STIX]{x1D702}gq)^{2}}{\unicode[STIX]{x1D708}^{4}+\unicode[STIX]{x1D702}^{2}g^{2}}}\right], & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

and we recover $\boldsymbol{x}_{c}(t)$ in (5.8) as the centre of the packet in (5.13).

Figure 10. The escape of a Gaussian near-inertial wave packet from the saddle of a strain flow. The Gaussian decay scale is $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D708}$ . The waves are weakly dispersive: $\hslash =\unicode[STIX]{x1D702}/\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D707}^{2}\approx 0.1$ . Black contours show the streamfunction $\unicode[STIX]{x1D713}=-\unicode[STIX]{x1D6FC}xy$ , and colours represent wave zonal velocity; the colour bar limits are fixed. The black dot indicates the centre of the packet and the grey line tracks its trail.

Figure 11. A comparison of passive-scalar and wave solutions ( $\hslash \approx 0.09$ ) with same initial conditions (the wave kinetic energy is equal to the passive-scalar variance) and the same small-scale dissipation. (a) Initial condition of wave back-rotated zonal velocity and of the passive scalar. (b) Wave back-rotated zonal velocity at $t\times U_{e}k_{e}=10$ . (c) Passive-scalar concentration at $t\times U_{e}k_{e}=10$ . (d) Variance of wave velocity or passive-scalar variance. (e) Variance of wave velocity gradient or passive scalar. In (d) and (e), the diagnostics are normalized by their initial values.

5.2 A flow with strain and vorticity

As a second example of wave escape, figure 11 shows a numerical solution for a wave packet launched at the saddle point of a large-scale balanced flow with $\unicode[STIX]{x1D713}=\sin x+\sin y$ . The small-time evolution of the packet is predicted by the strain–flow solution discussed above. Figure 11(a–c) shows that the behaviour of the wave packet is qualitatively different from that of a passive scalar in the same flow: the passive scalar is stretched out along the separatrices while the wave packet escapes into the vortex centres. Thus the passive-scalar packet is strained and quickly diffused into oblivion. On the other hand, the waves are strained just so much, resulting in acceleration and escape from the straining region; the waves finally concentrate in the regions with non-zero vorticity, i.e. in the regions where the Okubo–Weiss criterion indicates no exponential stretching.

On the bottom row, figure 11(d) shows that while wave action $\propto |\unicode[STIX]{x1D719}|^{2}$ is nearly conserved, the analogous passive-scalar variance is strongly dissipated. Figure 11(e) shows that the variance of the passive-scalar gradient at first increases exponentially due to straining and then decays due to diffusion. On the other hand, the potential energy of the waves $\propto |\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}|^{2}$ increases slowly and then oscillates around an equilibrium level. The wave-escape phenomenology in the turbulence solutions of § 4 qualitatively resembles that seen in this simple flow. In particular, the wave potential energy does not reach the dissipative scale (figure 9 c).

6.1 Kinetic energy of the Lagrangian-mean and Eulerian-mean flows

Appendix A.2 shows that in the vertical-plane-wave model, the Stokes drift is horizontally non-divergent, with streamfunction $-{\mathcal{A}}$ . Thus the Stokes velocity is

(6.1) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{u}^{S}=-\hat{\boldsymbol{k}}\times \unicode[STIX]{x1D735}{\mathcal{A}}, & \displaystyle\end{eqnarray}$$

and the Eulerian-mean streamfunction is

(6.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D713}^{\text{E}}\,\stackrel{def}{=}\,\unicode[STIX]{x1D713}+{\mathcal{A}}. & \displaystyle\end{eqnarray}$$

To illustrate the important differences between Lagrangian-mean and Eulerian-mean viewpoints, figure 12 shows a snapshot of the dipole example from § 2.3. This snapshot was taken at $t\times U_{e}k_{e}=10$ , just after the end of the refractive stage of energy conversion, when waves are strongly concentrated in the anti-cyclone. While $\unicode[STIX]{x1D713}$ is fairly symmetric, the Eulerian-mean streamfunction $\unicode[STIX]{x1D713}^{\text{E}}$ displays a stronger anti-cyclone. The Stokes drift is concentrated in the negative $\unicode[STIX]{x1D701}$ region and is anti-parallel to the Eulerian-mean flow. Thus the asymmetry in $\unicode[STIX]{x1D713}^{\text{E}}$ is compensated by a strong ‘Stokes cyclone’, which is set up during the refractive stage and thus the Lagrangian-mean streamfunction $\unicode[STIX]{x1D713}$ is more nearly a symmetric dipole.

Using the decomposition (6.2), the balanced kinetic energy ${\mathcal{K}}=|\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}|^{2}/2$ is

(6.3) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{K}}=\underbrace{{\textstyle \frac{1}{2}}|\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}^{\text{E}}|^{2}}_{\stackrel{def}{=}{\mathcal{K}}^{E}}+\underbrace{{\textstyle \frac{1}{2}}|\unicode[STIX]{x1D735}{\mathcal{A}}|^{2}-\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}^{\text{E}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}{\mathcal{A}}}_{\stackrel{def}{=}{\mathcal{K}}^{S}}. & \displaystyle\end{eqnarray}$$

${\mathcal{K}}$ does not diagonalize, i.e. the cross-term $\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}^{\text{E}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}{\mathcal{A}}$ is non-zero, and the ‘Stokes kinetic energy’ ${\mathcal{K}}^{S}$ is not sign definite.

The first goal here is to obtain an expressions for the time rate of change of ${\mathcal{K}}^{S}$ and ${\mathcal{K}}^{E}$ . We begin with yet another ${\mathcal{A}}$ - $\boldsymbol{{\mathcal{F}}}$ identity: using the definitions of ${\mathcal{A}}$ and $\boldsymbol{{\mathcal{F}}}$ in (3.1) and (3.6) yields

(6.4) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{A}}\,\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\hat{\boldsymbol{k}}\times \boldsymbol{{\mathcal{F}}})=-\boldsymbol{{\mathcal{F}}}\boldsymbol{\cdot }\hat{\boldsymbol{k}}\times \unicode[STIX]{x1D735}{\mathcal{A}}. & \displaystyle\end{eqnarray}$$

The average of (6.4), combined with a standard vector identities, results in

(6.5) $$\begin{eqnarray}\displaystyle & \displaystyle \langle {\mathcal{A}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\hat{\boldsymbol{k}}\times \boldsymbol{{\mathcal{F}}})\rangle =\langle \boldsymbol{{\mathcal{F}}}\boldsymbol{\cdot }\hat{\boldsymbol{k}}\times \unicode[STIX]{x1D735}{\mathcal{A}}\rangle =0. & \displaystyle\end{eqnarray}$$

Equation (6.5) implies that the Stokes velocity $\boldsymbol{u}^{S}$ is, on average, orthogonal to the action flux $\boldsymbol{{\mathcal{F}}}$ .

Forming $\langle {\mathcal{A}}q\rangle$ , and combining the expression for $q^{w}$ in (3.9) with the identity (6.5), yields

(6.6) $$\begin{eqnarray}\displaystyle & \displaystyle \langle {\mathcal{K}}^{S}\rangle =\langle {\mathcal{A}}q\rangle . & \displaystyle\end{eqnarray}$$

An expression for the rate of change of $\langle {\mathcal{K}}^{S}\rangle$ follows by combining the PV advection equation (2.7) with the action conservation (3.5):

(6.7) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\text{d}\langle {\mathcal{K}}^{S}\rangle }{\text{d}t}}=-\unicode[STIX]{x1D6E4}_{S}+\underbrace{\langle {\mathcal{A}}D_{q}\rangle +\langle qD_{{\mathcal{A}}}\rangle }_{\stackrel{def}{=}\unicode[STIX]{x1D700}_{S}}, & \displaystyle\end{eqnarray}$$

where

(6.8) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}_{S}\,\stackrel{def}{=}\,\langle q\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{{\mathcal{F}}}\rangle . & \displaystyle\end{eqnarray}$$

Finally we obtain the rate of change of the Eulerian-mean kinetic energy by combining (6.7) with (3.13):

(6.9) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\text{d}\langle {\mathcal{K}}^{E}\rangle }{\text{d}t}}=\unicode[STIX]{x1D6E4}_{S}-\unicode[STIX]{x1D6E4}_{r}-\unicode[STIX]{x1D6E4}_{a}+\unicode[STIX]{x1D6EF}+\unicode[STIX]{x1D700}_{{\mathcal{K}}}-\unicode[STIX]{x1D700}_{S}. & \displaystyle\end{eqnarray}$$

6.2 Reynolds stresses and buoyancy fluxes

Within the Eulerian framework the horizontal velocity is represented as

(6.10) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{u}=\boldsymbol{u}^{E}+\tilde{\boldsymbol{u}}, & \displaystyle\end{eqnarray}$$

where superscript $E$ denotes the Eulerian-mean and tilde denotes the near-inertial wave velocity. With this decomposition, the Eulerian-mean velocity satisfies

(6.11) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{u}_{t}^{E}+\boldsymbol{u}^{E}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}^{E}+(\tilde{\boldsymbol{u}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\tilde{\boldsymbol{u}})^{E}+\hat{\boldsymbol{k}}\times f_{0}\boldsymbol{u}^{E}+\unicode[STIX]{x1D735}p^{E}=0. & \displaystyle\end{eqnarray}$$

(Dissipation is neglected in the following discussion.) As in the vertical plane-wave model, we confine attention to mean velocities independent of $z$ , so that $\boldsymbol{u}^{E}=(-\unicode[STIX]{x1D713}_{y}^{E},\unicode[STIX]{x1D713}_{x}^{E})$ where $\unicode[STIX]{x1D713}^{E}$ is the Eulerian-mean streamfunction introduced in (6.2). Forming $\langle \!\boldsymbol{u}^{E}\boldsymbol{\cdot }$ (6.11) $\rangle$ yields

(6.12) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\text{d}\langle {\mathcal{K}}^{E}\rangle }{\text{d}t}}=\underbrace{\langle (\tilde{v}^{2}-\tilde{u} ^{2})^{E}\,\unicode[STIX]{x1D713}_{xy}^{E}+(\tilde{u} \tilde{v})^{E}\,(\unicode[STIX]{x1D713}_{xx}^{E}-\unicode[STIX]{x1D713}_{yy}^{E})\rangle }_{\stackrel{def}{=}RSP}. & \displaystyle\end{eqnarray}$$

Above $RSP$ is the Reynolds shear production of Eulerian-mean kinetic energy. Comparing (6.12) with (6.9), and ignoring the dissipative contributions, we see that

(6.13) $$\begin{eqnarray}\displaystyle & \displaystyle RSP=\unicode[STIX]{x1D6E4}_{S}-\unicode[STIX]{x1D6E4}_{r}-\unicode[STIX]{x1D6E4}_{a}. & \displaystyle\end{eqnarray}$$

We can also consider the source of near-inertial potential energy by starting with the wave buoyancy equation

(6.14) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{b}_{t}+\boldsymbol{u}^{E}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\tilde{b}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }[\tilde{\boldsymbol{u}}\tilde{b}-(\tilde{\boldsymbol{u}}\tilde{b})^{E}]+\tilde{w}N^{2}=0, & \displaystyle\end{eqnarray}$$

where we have confined attention to the vertical-plane-wave model so that $b^{E}=0$ . Multiplying (6.14) by $\tilde{b}$ , taking the Eulerian average and then the domain average $\langle \rangle$ , we have

(6.15) $$\begin{eqnarray}\displaystyle \displaystyle {\displaystyle \frac{\text{d}\langle {\mathcal{P}}\rangle }{\text{d}t}} & = & \displaystyle -\underbrace{\langle (\tilde{w}\tilde{b})^{E}\rangle }_{\stackrel{def}{=}BF},\end{eqnarray}$$

(6.16) $$\begin{eqnarray}\displaystyle & = & \displaystyle \unicode[STIX]{x1D6E4}_{r}+\unicode[STIX]{x1D6E4}_{a}.\end{eqnarray}$$

Above ${\mathcal{P}}=(\tilde{b}^{2})^{E}/2N^{2}$ is the wave potential energy and $BF$ is the buoyancy flux; we have used (3.12) to relate $\unicode[STIX]{x1D6E4}_{r}$ and $\unicode[STIX]{x1D6E4}_{a}$ to the Eulerian-mean buoyancy flux.

Finally, from balanced energy equation (3.13) and ${\mathcal{K}}={\mathcal{K}}^{E}+{\mathcal{K}}^{S}$ , we deduce that

(6.17) $$\begin{eqnarray}\displaystyle \displaystyle {\displaystyle \frac{\text{d}\langle {\mathcal{K}}^{S}\rangle }{\text{d}t}} & = & \displaystyle BF-RSP,\end{eqnarray}$$

(6.18) $$\begin{eqnarray}\displaystyle & = & \displaystyle -\unicode[STIX]{x1D6E4}_{S}.\end{eqnarray}$$

Figure 13(a) shows the decomposition of $\langle {\mathcal{K}}\rangle$ into $\langle {\mathcal{K}}^{E}\rangle$ and $\langle {\mathcal{K}}^{S}\rangle$ for the main 2-D turbulence solution discussed in § 4. In the initial condition there is no Stokes flow and ${\mathcal{K}}={\mathcal{K}}^{E}$ . Then in the refractive stage, the kinetic energy of the Eulerian-mean flow $\langle {\mathcal{K}}^{E}\rangle$ increases, while kinetic energy of the Lagrangian-mean flow $\langle {\mathcal{K}}\rangle$ decreases; the ‘Stokes energy’, ${\mathcal{K}}^{S}={\mathcal{K}}-{\mathcal{K}}^{E}$ , is initially zero and becomes negative, due to a positive $\unicode[STIX]{x1D713}^{\text{E}}$ - ${\mathcal{A}}$ correlation (cf. figure 12). As illustrated in figure 12, the refractive stage, which creates the strong spatial modulations in action density ${\mathcal{A}}$ , can also be viewed as setting up the Stokes velocity in (6.1).

Figure 13. Diagnostics of the 2-D turbulence solution with parameters presented in table 4. (a) Kinetic energy change about initial condition. (b) Refractive and advective conversion terms, and shear production.

In the refractive stage, the advective conversion $\unicode[STIX]{x1D6E4}_{a}$ is small and $\unicode[STIX]{x1D6E4}_{S}\approx 2\unicode[STIX]{x1D6E4}_{r}$ . Hence the shear production is due to the refractive conversion, $RSP\approx \unicode[STIX]{x1D6E4}_{r}$ (see figure 13 b). At later times, during the advective stage, $\unicode[STIX]{x1D6E4}_{S}-\unicode[STIX]{x1D6E4}_{r}$ is small, and therefore $RSP\approx BF\approx -\unicode[STIX]{x1D6E4}_{a}$ . At this stage, $\unicode[STIX]{x2202}_{t}\langle {\mathcal{K}}\rangle \approx \unicode[STIX]{x2202}_{t}\langle K^{E}\rangle$ , and the connection between the Eulerian-mean and Lagrangian-mean viewpoints is straightforward: the kinetic energy extracted from the Eulerian-mean flow via Reynolds shear production approximately matches the creation of wave potential energy through buoyancy fluxes. In general, however, the connection is convoluted, involving the rate of change of the ‘Stokes energy’ in (6.17).

7.1 Absence of a direct cascade of wave energy

The solutions reported here introduce the waves at $t=0$ in (2.14) with infinite spatial scale. Wave refraction, $\text{i}\unicode[STIX]{x1D719}\unicode[STIX]{x1D701}/2$ in (2.8), immediately transfers wave energy to the smaller scales of the balanced relative vorticity $\unicode[STIX]{x1D701}$ . This giant leap across wavenumbers is not a direct cascade of wave energy in the sense of Kolmogorov. And because of wave escape, the wave energy that is so efficiently transferred to eddy scales by refraction does not undergo a turbulence-driven direct cascade to the small length scales at which the wave dissipation in (2.9) is effective. This conclusion hinges on the assumption of a barotropic balanced flow and fixed vertical wavenumber. Similar to the vertical-plane-wave model, shallow-water models lack wave capture and an efficient direct cascade of wave energy (e.g. McIntyre Reference McIntyre2009). In less idealized models with balanced baroclinic shear, the vertical wavenumber can increase, and a direct cascade, perhaps resulting in wave capture, can ensue; stimulated generation might then be much stronger than in the vertical-plane-wave model considered here.

7.2 Regimes of wave-modified turbulence

Geostrophic straining accounts for most of the stimulated generation of wave energy in the examples considered in this paper. But refraction plays a fundamental role in these solutions with the uniform initial wave velocity in (2.14), because refraction creates the initial gradients of wave velocity that are then enhanced by geostrophic straining. We experimented by changing the initial condition of $\unicode[STIX]{x1D719}$ to an eddy-scale plane wave and repeated all the 2-D turbulence solutions; the different initial condition significantly suppresses the initial refraction stage, but otherwise yields long-term solutions that are qualitatively similar to the solutions discussed above. Thus, to the extent that the uniform-wave initial condition (2.14) idealizes the generation of large-scale upper-ocean inertial oscillations by storms (e.g. Moehlis & Llewellyn Smith Reference Moehlis and Llewellyn Smith2001; Danioux et al. Reference Danioux, Vanneste and Bühler2015), the initial refraction is a loss of lateral coherence, or a type of inertial pumping (Young & Ben Jelloul Reference Young and Ben Jelloul1997; Klein, Llewellyn Smith & Lapeyre Reference Klein, Llewellyn Smith and Lapeyre2004), which is accompanied by an extraction of energy from the balanced flow by the waves.

Although 10–20 % of the balanced kinetic energy is converted into wave potential energy, and despite the wave breakage of the symmetry between cyclones and anti-cyclones, the wave-modified turbulence in § 4 remarkably resembles waveless two-dimensional turbulence (e.g. McWilliams Reference McWilliams1984): we still observe robust vortices and an increase in vortex length scale due to merger of like-signed vortices. Figure 7 shows small changes in the potential vorticity $q$ and much larger changes in the wave PV $q^{w}$ induced by changing the dispersivity. In this sense, the turbulent evolution is insensitive to wave modification.

To see significant wave modification of the turbulence we increased the amplitude of the initial wave in (2.14) so that $U_{w}=6U_{e}$ (in § 4, $U_{w}=2U_{e}$ ). With this level of wave energy the wave-modified 2-D turbulence differs qualitatively from the waveless variety (not shown). The potential vorticity develops highly filamentary structures with little vortex formation; this inhibition of vortex formation is stronger in the weakly dispersive limit.

7.3 Energy transfers can be bi-directional in non-turbulent balanced flows

In all solutions considered in §§ 2 and 4, the energy transfer is always from the balanced flow into the waves. This positive energy conversion, $\unicode[STIX]{x1D6E4}_{r}+\unicode[STIX]{x1D6E4}_{a}>0$ , remains true for very long turbulence simulations (not shown), because the refractive conversion is small at large time and turbulent stirring always increases lateral wave gradients: $\unicode[STIX]{x1D6E4}_{r}+\unicode[STIX]{x1D6E4}_{a}\approx \unicode[STIX]{x1D6E4}_{a}>0$ .

But the energy transfers can be bi-directional for non-turbulent flows. To illustrate this process, we consider a solution with initially uniform wave and $\unicode[STIX]{x1D713}(\boldsymbol{x},0)=\sin x+\sin y$ . This non-turbulent balanced flow consists of two vortices of opposite signs; there can be no inverse cascade from this initial condition because $\unicode[STIX]{x1D713}(\boldsymbol{x},0)$ is already at the domain scale. Starting from the initial uniform wave, the refractive conversion generates wave potential energy ${\mathcal{P}}$ at the expense of balanced kinetic energy ${\mathcal{K}}$ (figure 14). After this initial refractive stage, however, ${\mathcal{P}}$ and ${\mathcal{K}}$ oscillate about equilibrium levels and there are quasi-periodic energy transfers back and forth between ${\mathcal{P}}$ and ${\mathcal{K}}$ . In this solution, the refractive conversion $\unicode[STIX]{x1D6E4}_{r}$ accounts for most of the quasi-periodic exchanges between ${\mathcal{K}}$ and ${\mathcal{P}}$ .

Figure 14. Diagnostics of an illustrative solution with initially uniform wave and $\unicode[STIX]{x1D713}(t=0)=\sin x+\sin y$ , with $\unicode[STIX]{x1D6FC}=0.1$ and $\hslash =2$ . (a) Energy difference about initial condition. (b) Wave potential energy budget.

7.4 The correlation of wave amplitude with relative vorticity

A secondary result is the strong time dependence of the correlation $r$ , defined in (4.7), between incoherent waves and the relative vorticity; $r$ measures the concentration of waves into cyclones or anti-cyclones (Danioux et al. Reference Danioux, Vanneste and Bühler2015). Refraction concentrates waves into anti-cyclones and expels them from cyclones, thereby generating an initial strong negative $r$ (see figure 8 c). And Ehrenfest’s theorem provides perhaps the simplest explanation for this concentration – see (3.17). As conjectured by Danioux et al. (Reference Danioux, Vanneste and Bühler2015), the subsequent return of $r$ towards zero (and even to positive values in the case $\hslash =0.5$ ) is partially due to the unsteady geostrophic advection. The NIW-QG coupling compounds the unsteady advection: the dramatic initial concentration of waves into anti-cyclones weakens those vortices, with ensuing development of positive skewness of relative vorticity (see figure 8 d); the vorticity skewness increases with decreasing dispersivity because weakly dispersive waves extract more energy from the balanced flow (see figure 8).

7.5 Reconciling action conservation with $RSP$ of wave kinetic energy

A main difficulty in making a connection between the NIW-QG model and earlier studies, such as those in table 1, is that the theory uses the Lagrangian-mean geostrophic flow as a primary variable. Previous studies generally employ an Eulerian wave-mean decomposition. Taylor & Straub (Reference Taylor and Straub2016), for example, use frequency filtering to separate low-frequency motions (Eulerian mean) from high-frequency flow (dominated by near-inertial waves). Taylor & Straub find energy exchange between low-frequency (eddy) kinetic energy and high-frequency (near-inertial) kinetic energy resulting from both vertical and horizontal Reynolds stresses; Taylor & Straub refer to this energy transfer as the ‘advective sink’, meaning a sink of low-frequency, Eulerian-mean kinetic energy. The Reynolds shear production $RSP$ in (6.12) is analogous to the horizontal part of the advective sink.

The Reynolds stresses diagnostics of prior studies in table 1 might seem incompatible with fundamental aspects of the NIW-QG system. For example, the wave action ${\mathcal{A}}$ in (3.1) differs from the wave kinetic energy $|\unicode[STIX]{x1D719}|^{2}/2$ only by the constant factor $f_{0}$ . Thus NIW-QG action conservation seems inconsistent with Reynolds stress transfer of kinetic energy from eddies to waves. But this impression is incorrect: Reynolds stresses are not inconsistent with conservation of ${\mathcal{A}}$ , nor with stimulated generation.

To understand this, note that conservation of ${\mathcal{A}}$ in (3.10) is a statement about the leading-order wave velocities encoded in $\unicode[STIX]{x1D719}$ ; the balanced kinetic energy ${\mathcal{K}}$ also involves higher-order wave kinetic energy in the form of the ‘Stokes kinetic energy’ ${\mathcal{K}}^{S}$ : see (6.3) and the surrounding discussion. A main point from § 6.2 is that the energy transferred from the Eulerian-mean velocity by $RSP$ is accounted for by ${\mathcal{K}}^{S}$ (rather than the leading-order wave kinetic energy $f_{0}{\mathcal{A}}$ ).

At first glance it might seem that because stimulated generation results in an increase in wave potential energy ${\mathcal{P}}$ , it must necessarily involve a transfer from eddy potential energy. The vertical-plane-wave model is a counterexample: the eddies are barotropic and there is therefore no eddy potential energy to transfer. Indeed, if the eddies are barotropic, then $RSP$ is the only pathway between the Eulerian-mean kinetic energy ${\mathcal{K}}^{E}$ and wave energy. This rather obvious fact is deeply hidden by the Lagrangian-mean average: the $RSP$ transfer out of ${\mathcal{K}}^{E}$ passes first through ${\mathcal{K}}^{S}$ on the way to ${\mathcal{P}}$ . The intermediate passage through ${\mathcal{K}}^{S}$ is an inevitable aspect of the Lagrangian-mean formulation of the NIW-QG system. All of these transfers can be diagnosed using $\unicode[STIX]{x1D6E4}_{a}$ , $\unicode[STIX]{x1D6E4}_{r}$ and $\unicode[STIX]{x1D6E4}_{S}$ and (6.13) relates Reynolds shear production to the three $\unicode[STIX]{x1D6E4}$ values.

A full investigation of NIW-QG energetics, including diagnosis of $\unicode[STIX]{x1D6E4}_{r}$ , $\unicode[STIX]{x1D6E4}_{a}$ and $\unicode[STIX]{x1D6E4}_{S}$ in solutions of the Boussinesq equations, is beyond the scope of this article. But we emphasize a crucial simplifying feature of the NIW-QG approximation: there are no important Stokes corrections to pressure and buoyancy, and therefore, to leading order, the Lagrangian-mean velocity is equal to the balanced velocity based on an Eulerian-mean pressure field. In principle, this is a straightforward way of diagnosing Lagrangian-mean velocities from numerical solutions of the Boussinesq equations.

7.6 Final remarks

There are many caveats to the application of our results to the post-storm oceanographic problem. Notably, the lack of geostrophic vertical shear suppresses important mechanisms of vertical refraction and straining, which introduce interesting modifications of the near-inertial wave physics (e.g. Thomas Reference Thomas2012) and can produce strong energy extraction by near-inertial waves (Shakespeare & Hogg Reference Shakespeare and Hogg2017). And our focus on quasi-inviscid initial value problems downplays the role of dissipation, but in forced-dissipative solutions, wave dissipation likely controls the strength of stimulated generation. Finally, better understanding the connection between numerical modelling studies (e.g. Taylor & Straub Reference Taylor and Straub2016) and the NIW-QG theory (XV; Wagner & Young Reference Wagner and Young2016) deserves further investigation. We hope to explore these topics in future work.