3.1. Wave-kinetic equation

The derivation of the kinetic equation is a long, but standard exercise in WWT (Hasselmann Reference Hasselmann1966; Zakharov et al. Reference Zakharov, L'vov and Falkovich1992; Nazarenko Reference Nazarenko2011; Galtier Reference Galtier2022). Since the present system does not have a canonical Hamiltonian structure, the final results cannot be anticipated. Assuming a time scale separation between the linear and the nonlinear times, we introduce the interaction representation variable as

(3.1)\begin{equation} c_{\boldsymbol{k}} = \frac{a_{\boldsymbol{k}}}{\epsilon} {\rm e}^{{\rm i} \omega_{\boldsymbol{k}} t}, \end{equation}

where the parameter $\epsilon \ll 1$ quantifies the strength of the nonlinearity. For stratified flows, $\epsilon$ corresponds to the Froude number $Fr$. Using (2.10), we obtain

(3.2)\begin{align} \dot{c}_{\boldsymbol{k}} ={-} {\rm i} \epsilon \sum_{1,2} V_{12}^{\boldsymbol{k}} \delta_{12}^{\boldsymbol{k}} c_1 c_2 {\rm e}^{{\rm i} \omega_{12}^{\boldsymbol{k}}t} - 2 {\rm i} \epsilon \sum_{1,2} V_{12}^{\boldsymbol{k}} \delta_{\boldsymbol{k}2}^1 c_1 c_2^* {\rm e}^{- {\rm i} \omega_{\boldsymbol{k}2}^{1}t} - {\rm i} \epsilon \sum_{1,2} V_{12}^{\boldsymbol{k}} \delta_{\boldsymbol{k}12} c_1^* c_2^* {\rm e}^{- {\rm i} \omega_{\boldsymbol{k} 1 2}t}, \end{align}

where $\omega _{12}^{\boldsymbol {k}} \equiv \omega _{\boldsymbol {k}} - \omega _{1} - \omega _{2}$ and $\omega _{\boldsymbol {k} 1 2} \equiv - \omega _{\boldsymbol {k}} - \omega _{1} - \omega _{2}$. The derivation of the kinetic equation describing the evolution of the wave-action spectrum

(3.3)\begin{equation} n_{\boldsymbol{k}} \equiv \left( \frac{2 {\rm \pi}}{L} \right)^3 \left\langle |c_{\boldsymbol{k}}|^2 \right\rangle , \end{equation}

in the infinite size limit ($L \rightarrow \infty$) and in the small nonlinearity limit ($\epsilon \rightarrow 0$) is given in Appendix A. These computations follow the steps described in Nazarenko (Reference Nazarenko2011). The final result is

(3.4)$$\begin{gather} \dot{n}_{\boldsymbol{k}} = St_{\boldsymbol{k}} = \int \left[ \mathcal{R}_{12}^{\boldsymbol{k}} - \mathcal{Q}_{\boldsymbol{k}2}^{1} - \mathcal{Q}_{\boldsymbol{k}1}^{2} \right] \,\mathrm{d}^3\boldsymbol{k}_1 \,\mathrm{d}^3\boldsymbol{k}_2, \end{gather}$$

(3.5)$$\begin{gather}\mathcal{R}_{12}^{\boldsymbol{k}} = 4 {\rm \pi}\epsilon^2 \delta(\boldsymbol{k} - \boldsymbol{k}_1 - \boldsymbol{k}_2) \delta(\omega_{12}^{\boldsymbol{k}}) V_{12}^{\boldsymbol{k}} \left( V_{12}^{\boldsymbol{k}} n_1 n_2 - V_{\boldsymbol{k}1}^{2} n_{\boldsymbol{k}} n_1 - V_{\boldsymbol{k}2}^{1} n_{\boldsymbol{k}} n_2 \right) + O(\epsilon^3), \end{gather}$$

(3.6)$$\begin{gather}\mathcal{Q}_{\boldsymbol{k}2}^{1} = 4 {\rm \pi}\epsilon^2 \delta(\boldsymbol{k}_1 - \boldsymbol{k} - \boldsymbol{k}_2) \delta(\omega_{\boldsymbol{k}2}^{1}) V_{12}^{\boldsymbol{k}} \left( V_{\boldsymbol{k}2}^{1} n_{\boldsymbol{k}} n_2 - V_{\boldsymbol{k}1}^{2} n_1 n_{\boldsymbol{k}} - V_{12}^{\boldsymbol{k}} n_1 n_2 \right) + O(\epsilon^3). \end{gather}$$

It can be shown that this kinetic equation is equivalent to the one derived by Caillol & Zeitlin (Reference Caillol and Zeitlin2000). Equations ((3.4)–(3.6)) are relatively compact and more suitable for theoretical treatments. The total wave energy balance equation is

(3.7)\begin{align} \dot{E} &= \int \omega_{\boldsymbol{k}} \dot{n}_{\boldsymbol{k}} \,\mathrm{d}^3\boldsymbol{k} = \int \omega_{\boldsymbol{k}} \left[ \mathcal{R}_{12}^{\boldsymbol{k}} - \mathcal{Q}_{\boldsymbol{k}2}^{1} - \mathcal{Q}_{\boldsymbol{k}1}^{2} \right] \,\mathrm{d}^3\boldsymbol{k} \,\mathrm{d}^3\boldsymbol{k}_1 \,\mathrm{d}^3\boldsymbol{k}_2 \end{align}

(3.8)\begin{align} &= \int \left[ \omega_{\boldsymbol{k}} \mathcal{R}_{12}^{\boldsymbol{k}} - \omega_{1} \mathcal{Q}_{12}^{\boldsymbol{k}} - \omega_{2} \mathcal{Q}_{21}^{\boldsymbol{k}} \right] \,\mathrm{d}^3\boldsymbol{k} \,\mathrm{d}^3\boldsymbol{k}_1 \,\mathrm{d}^3\boldsymbol{k}_2 \end{align}

(3.9)\begin{align} &= 4 {\rm \pi}\epsilon^2 \int \delta(\boldsymbol{k} - \boldsymbol{k}_1 - \boldsymbol{k}_2) \delta(\omega_{12}^{\boldsymbol{k}}) \nonumber\\&\quad \times \left( \omega_{\boldsymbol{k}} V_{12}^{\boldsymbol{k}} - \omega_{1} V_{\boldsymbol{k}2}^{1} - \omega_{2} V_{\boldsymbol{k}1}^{2} \right) \left( V_{12}^{\boldsymbol{k}} n_1 n_2 - V_{\boldsymbol{k}1}^{2} n_{\boldsymbol{k}} n_1 - V_{\boldsymbol{k}2}^{1} n_{\boldsymbol{k}} n_2 \right) \,\mathrm{d}^3\boldsymbol{k} \,\mathrm{d}^3\boldsymbol{k}_1 \,\mathrm{d}^3\boldsymbol{k}_2, \end{align}

which is zero because of the symmetry of the interaction coefficients (2.12a,b). Note that it is not necessary to use the resonance condition in frequencies to prove that the kinetic equation conserves energy; $E$ is also an invariant of the Boussinesq equations ((2.1)–(2.3)) and of the wave-action equation (2.10).

For now, the kinetic equation ((3.4)–(3.6)) does not take a standard form typical for canonical Hamiltonian systems, except if the relation $\mathcal {R}_{12}^{\boldsymbol {k}} = \mathcal {Q}_{12}^{\boldsymbol {k}}$ holds. For this to happen, $V_{12}^{\boldsymbol {k}}$ should be fully symmetric with respect to the $3$ indices when evaluated on the resonant manifold $\boldsymbol {k} = \boldsymbol {k}_1 + \boldsymbol {k}_2$, $\omega _{\boldsymbol {k}} = \omega _{1} + \omega _{2}$ (or permutations) such that we could write $V_{12}^{\boldsymbol {k}} = V_{\boldsymbol{k} 2}^{1} = V_{\boldsymbol{k} 1}^{2}$ in $\mathcal {R}_{12}^{\boldsymbol {k}}$, $\mathcal {Q}_{\boldsymbol{k} 2}^{1}$ and $\mathcal {Q}_{\boldsymbol{k} 1}^{2}$. Such a symmetry could be expected since the kinetic equation obtained in the Lagrangian formalism is almost canonical (Hasselmann Reference Hasselmann1967; Müller & Olbers Reference Müller and Olbers1975; Olbers Reference Olbers1976). As already shown in figure 2(b), a resonance condition in wave vectors is not sufficient to have $V_{12}^{\boldsymbol {k}} = V_{\boldsymbol{k} 2}^{1} = V_{\boldsymbol{k} 1}^{2}$. However, adding the constraint of resonance condition in frequencies eventually allows to satisfy this symmetry, as will be explained in the § 3.3.

3.2. Steady, scale-invariant spectra

Despite its compact form, analytical solution to the kinetic equation (3.4) are not easy to find. The only exception is for $n_{\boldsymbol {k}} \propto 1/\omega _{\boldsymbol {k}}$, corresponding to the equilibrium (RJ) spectrum with equipartition of energy. It is worth mentioning that this RJ spectrum can be obtained without using the resonance condition in frequencies, but only the symmetry of the interaction coefficients (2.12a,b) when the wave vector resonance condition is satisfied. We can try to find other steady-state solutions to the kinetic equation, in the non-hydrostatic case, by using the ansatz

(3.10)\begin{equation} n_{\boldsymbol{k}} = k^{\nu} f(\theta_{\boldsymbol{k}},\varphi_{\boldsymbol{k}}) , \end{equation}

corresponding to a separable, scale-invariant spectrum. We adapt the computations of Shavit, Bühler & Shatah (Reference Shavit, Bühler and Shatah2024) in order to find a possible value for the exponent $\nu$. We first write the evolution equation for the energy density averaged over angles $e(k,t)$. Namely,

(3.11) \begin{align} \dot{e}(k,t) &= \int \omega_{\boldsymbol{k}} \dot{n}_{\boldsymbol{k}} \sin \theta_{\boldsymbol{k}} \,\mathrm{d} \theta_{\boldsymbol{k}} \,\mathrm{d} \varphi_{\boldsymbol{k}}\nonumber\\ &= \int \left[ \omega_{\boldsymbol{k}} \mathcal{R}_{12}^{\boldsymbol{k}} - \omega_{\boldsymbol{k}} \mathcal{Q}_{\boldsymbol{k}2}^{1} - \omega_{\boldsymbol{k}} \mathcal{Q}_{\boldsymbol{k}1}^{2} \right] \,\mathrm{d}^3\boldsymbol{k}_1 \,\mathrm{d}^3\boldsymbol{k}_2 \sin \theta_{\boldsymbol{k}} \,\mathrm{d} \theta_{\boldsymbol{k}} \,\mathrm{d} \varphi_{\boldsymbol{k}}. \end{align}

In steady state, the integral on the right-hand side of (3.11) must be zero. If we assume (3.10), this integral is also only a function of $k$ and $\nu$ and depends on the angular variable via some function $f$. To find possible values for $\nu$, we adapt the Zakharov transformation, so we replace

(3.12)\begin{equation} k_1 \rightarrow \frac{k}{k_1} k, \quad k_2 \rightarrow \frac{k}{k_1} k_2, \quad (\theta_{\boldsymbol{k}},\varphi_{\boldsymbol{k}}) \leftrightarrow (\theta_{1},\varphi_{1}) , \end{equation}

in the integral with $\mathcal {Q}_{\boldsymbol{k} 2}^{1}$, and a similar transformation for the integral with $\mathcal {Q}_{\boldsymbol{k} 1}^{2}$ in (3.11). We then obtain the stationarity condition

(3.13)\begin{align} 0 &= 4 {\rm \pi}\epsilon^2 \int \delta(\boldsymbol{k} - \boldsymbol{k}_1 - \boldsymbol{k}_2) \delta(\omega_{12}^{\boldsymbol{k}}) \nonumber\\ & \quad \times \left[ \omega_{\boldsymbol{k}} V_{12}^{\boldsymbol{k}} - \left( \frac{k_1}{k} \right)^{\chi} \omega_{1} V_{\boldsymbol{k}2}^{1} - \left( \frac{k_2}{k} \right)^{\chi} \omega_{2} V_{\boldsymbol{k}1}^{2} \right]\nonumber\\ & \quad \times \left( V_{12}^{\boldsymbol{k}} n_1 n_2 - V_{\boldsymbol{k}1}^{2} n_{\boldsymbol{k}} n_1 - V_{\boldsymbol{k}2}^{1} n_{\boldsymbol{k}} n_2 \right) \,\mathrm{d}^3\boldsymbol{k}_1 \,\mathrm{d}^3\boldsymbol{k}_2 \sin \theta_{\boldsymbol{k}} \,\mathrm{d} \theta_{\boldsymbol{k}} \,\mathrm{d} \varphi_{\boldsymbol{k}} , \end{align}

where $\chi \equiv \alpha - 2 \beta - 2 \nu - 2 d$, with $d=3$ being the number of spatial dimensions. Condition (3.13) is satisfied when $\chi = 0$ due to symmetry of the interaction coefficients (2.12a,b). It leads to the exponent $\nu = -4$, which is consistent with the high-frequency limit of the Garrett–Munk spectrum ($\propto k_h^{-4} |k_z|^{0}$), the PR spectrum ($\propto k_h^{-7/2} |k_z|^{-1/2}$), the spectrum obtained in the hydrostatic limit by considering ID triads only (Lanchon & Cortet Reference Lanchon and Cortet2023) ($\propto k_h^{-3} |k_z|^{-1}$) and oceanic measurements (Lvov et al. Reference Lvov, Polzin and Tabak2004).

Testing the validity of $\nu = -4$ is beyond the scope of this study. For this, we need to show that the collision integral converges in the vicinity of $\nu = -4$, for some yet unknown $f(\theta _{\boldsymbol {k}},\varphi _{\boldsymbol {k}})$. This analysis has been done in the hydrostatic limit for bi-homogenous spectra $\propto k_h^{\nu _h} |k_z|^{\nu _z}$, and it has been found that the line $\nu = \nu _h + \nu _z = -4$ corresponds to non-physical spectra because of collision integral divergences (Lvov et al. Reference Lvov, Polzin, Tabak and Yokoyama2010; Dematteis & Lvov Reference Dematteis and Lvov2021). Yet, the divergence in the hydrostatic limit does not imply that $\nu =-4$ is unrealizable outside this limit because the convergence conditions could be less restrictive in the non-hydrostatic case. Moreover, the ansatz (3.10) is more general than $n_{\boldsymbol {k}} = C k_h^{\nu _h} |k_z|^{\nu _z}$, which may allow other local spectra. Finally, it is important to note that the angular dependence $f(\theta _{\boldsymbol {k}},\varphi _{\boldsymbol {k}})$ is embedded everywhere in the integrand in a non-trivial way and could lead to a cancellation of the collisional integral for $\nu \neq -4$. Indeed, the solution obtained by Lvov et al. (Reference Lvov, Polzin, Tabak and Yokoyama2010) and Dematteis & Lvov (Reference Dematteis and Lvov2021) in the hydrostatic limit is of such a type.

3.3. Canonical form and resonant manifold

We proceed now to show that $V_{12}^{\boldsymbol {k}} = V_{\boldsymbol{k} 2}^{1} = V_{\boldsymbol{k} 1}^{2}$ on the resonant manifold, and thus that the kinetic equation (3.4) can be written in a canonical form. We first use the symmetry (2.12a,b), together with the resonance condition in frequencies $\omega _{\boldsymbol {k}} = \omega _{1} + \omega _{2}$, to readily show that

(3.14)\begin{equation} \delta(\boldsymbol{k}-\boldsymbol{k}_1-\boldsymbol{k}_2) \delta(\omega_{12}^{\boldsymbol{k}}) \left[ \omega_{1} (V_{12}^{\boldsymbol{k}} - V_{\boldsymbol{k}2}^{1}) + \omega_{2} (V_{12}^{\boldsymbol{k}} - V_{\boldsymbol{k}1}^{2}) \right] = 0. \end{equation}

Equation (3.14) has a simple geometrical meaning: the vectors $(V_{12}^{\boldsymbol {k}} - V_{\boldsymbol{k} 2}^{1}, V_{12}^{\boldsymbol {k}} - V_{\boldsymbol{k} 1}^{2})$ and $(\omega _{1}, \omega _{2}=\omega _{\boldsymbol {k}}-\omega _{1})$ are orthogonal for all points of the resonant manifold. Because the $\omega _{\boldsymbol {k}}$ values only depend on the angles $\theta _{\boldsymbol {k}}$, to satisfy (3.14) while varying $k$ and $k_1$ (at fixed angles), the vectors $(V_{12}^{\boldsymbol {k}} - V_{\boldsymbol{k} 2}^{1}, V_{12}^{\boldsymbol {k}} - V_{\boldsymbol{k} 1}^{2})$ must remain co-linear. The orthogonality condition (3.14) thus leads to

(3.15) \begin{align} \left. \begin{pmatrix} V_{12}^{\boldsymbol{k}} - V_{\boldsymbol{k}2}^{1} \\ V_{12}^{\boldsymbol{k}} - V_{\boldsymbol{k}1}^{2} \end{pmatrix} \right|_{(k,k_1,\theta_{\boldsymbol{k}},\theta_{1})} = g(k,k_1) \left. \begin{pmatrix} V_{12}^{\boldsymbol{k}} - V_{\boldsymbol{k}2}^{1} \\ V_{12}^{\boldsymbol{k}} - V_{\boldsymbol{k}1}^{2} \end{pmatrix}\right|_{(1,1,\theta_{\boldsymbol{k}},\theta_{1})} \perp \begin{pmatrix} \omega_{1} \\ \omega_{2} \end{pmatrix} \quad \forall (\omega_{1}, \omega_{2}), \end{align}

where $g$ is an unknown function that depends on $k$ and $k_1$ only. We have used here the fact that the resonant manifold can be parametrized using the variables $(k,k_1,\theta _{\boldsymbol {k}},\theta _{1})$. Using (3.15) in (3.14), we see that whether $g(k,k_1)=0$, or $\omega _{1} (V_{12}^{\boldsymbol {k}} - V_{\boldsymbol{k} 2}^{1})|_{(1,1,\theta _{\boldsymbol {k}},\theta _{1})} + \omega _{2} (V_{12}^{\boldsymbol {k}} - V_{\boldsymbol{k} 1}^{2})|_{(1,1,\theta _{\boldsymbol {k}},\theta _{1})}= 0$. Equation (3.15) therefore allows us to reduce the analysis to $(k,k_1)=(1,1)$. Using the symbolic computational capabilities of the Mathematica software (Wolfram Research, Inc. 2024), we show in the supplementary material that

(3.16) \begin{align} &\left.\left( V_{12}^{\boldsymbol{k}} - V_{\boldsymbol{k}2}^{1} \right)\right|_{(1,1,\theta_{\boldsymbol{k}},\theta_{1})} ={-}\sqrt{\frac{N}{128}} \frac{1}{\cos(2\theta_{\boldsymbol{k}}) + \cos(2\theta_{1}) + 4\sin\theta_{\boldsymbol{k}} \sin\theta_{1}}\nonumber\\ &\quad \times \left\{ \left[ \csc\left(\frac{\theta_{\boldsymbol{k}}-\theta_{1}}{2}\right) \sin\left( \frac{\theta_{\boldsymbol{k}} + \theta_{1}}{2}\right) \right. \right.\nonumber\\ &\quad \times (9 \cos(2\theta_{\boldsymbol{k}}) - 3\cos(\theta_{\boldsymbol{k}}-3\theta_{1}) - 8\cos(2\theta_{\boldsymbol{k}}-2\theta_{1}) \nonumber\\ &\quad + 16\cos(\theta_{\boldsymbol{k}}-\theta_{1}) \nonumber\\ &\quad- 3 (4 + \cos\left(3\theta_{\boldsymbol{k}}-\theta_{1}\right) - 3\cos(2\theta_{1}) + 4\cos(\theta_{\boldsymbol{k}}+\theta_{1}))) \nonumber\\ &\quad \times\left. \left. \left( \sqrt{-\csc\theta_{\boldsymbol{k}} + \csc\theta_{1}} - \csc\theta_{1} \sqrt{\sin\theta_{1} - \csc\theta_{\boldsymbol{k}} \sin^2\theta_{1}} \right) \right] \right\} = 0. \end{align}

It then follows from ((3.14)–(3.15)) that

(3.17)\begin{align} \delta(\boldsymbol{k}-\boldsymbol{k}_1-\boldsymbol{k}_2) \delta(\omega_{12}^{\boldsymbol{k}}) V_{12}^{\boldsymbol{k}} = \delta(\boldsymbol{k}-\boldsymbol{k}_1-\boldsymbol{k}_2) \delta(\omega_{12}^{\boldsymbol{k}}) V_{\boldsymbol{k}2}^{1} = \delta(\boldsymbol{k}-\boldsymbol{k}_1-\boldsymbol{k}_2) \delta(\omega_{12}^{\boldsymbol{k}}) V_{\boldsymbol{k}1}^{2}, \end{align}

which is the desired symmetry to put the kinetic equation in a canonical form.

The kinetic equation (3.4) can therefore be rewritten

(3.18)$$\begin{gather} \dot{n}_{\boldsymbol{k}} = St_{\boldsymbol{k}} = \int \left[ \mathcal{R}_{12}^{\boldsymbol{k}} - \mathcal{R}_{\boldsymbol{k}2}^{1} - \mathcal{R}_{\boldsymbol{k}1}^{2} \right] \,\mathrm{d}^3\boldsymbol{k}_1 \,\mathrm{d}^3\boldsymbol{k}_2, \end{gather}$$

(3.19)$$\begin{gather}\mathcal{R}_{12}^{\boldsymbol{k}} = 4 {\rm \pi}\epsilon^2 \delta(\boldsymbol{k} - \boldsymbol{k}_1 - \boldsymbol{k}_2) \delta(\omega_{12}^{\boldsymbol{k}}) |V_{12}^{\boldsymbol{k}}|^2 \left( n_1 n_2 - n_{\boldsymbol{k}} n_1 - n_{\boldsymbol{k}} n_2 \right). \end{gather}$$

Similarly to Rossby waves (Nazarenko Reference Nazarenko2011), the wave-action dynamics (2.10) does not have a canonical Hamiltonian structure, but the kinetic equation (3.18) is the same as if the system were canonical because of the resonance condition in frequencies. Namely, we could have obtained ((3.18)–(3.19)) by using the Hamilton equation ${\rm i}\dot {a}_{\boldsymbol {k}} = {\delta H_{eff}}/{\delta a_{\boldsymbol {k}}^*}$ together with the effective Hamiltonian $H_{eff} = \tfrac {1}{2} \sum _3 \omega _3 |a_3|^3 + \sum _{1,2,3} ( \delta _{12}^3 V_{12}^3 a_1 a_2 a_3^* + \delta _{23}^1 V_{23}^1 a_1^* a_2 a_3^* )$. Yet, it is not equivalent to the original wave mode equation (2.10) outside the resonant manifold.

In the case of a wave-action spectrum invariant under rotation around the stratification axis, i.e. $n_{\boldsymbol {k}}=n(k_h,k_z,t)$, the kinetic equation ((3.18)–(3.19)) takes a simpler form after integrating over azimuthal angles $(\varphi _{1},\varphi _{2})$

(3.20)$$\begin{gather} \dot{n}_{\boldsymbol{k}} = St_{\boldsymbol{k}}^{(h)} = \int \left[ \mathcal{R}_{12}^{\boldsymbol{k}(h)} - \mathcal{R}_{\boldsymbol{k}2}^{1(h)} - \mathcal{R}_{\boldsymbol{k}1}^{2(h)} \right] k_{1h} k_{2h} \,\mathrm{d} k_{1h} \,\mathrm{d} k_{1z} \,\mathrm{d} k_{2h} \,\mathrm{d} k_{2z}, \end{gather}$$

(3.21)$$\begin{gather}\mathcal{R}_{12}^{\boldsymbol{k}(h)} = 8 {\rm \pi}\epsilon^2 \delta(k_z - k_{1z} - k_{2z}) \delta(\omega_{12}^{\boldsymbol{k}}) \frac{|V_{12}^{\boldsymbol{k}}|^2}{\varDelta} \left( n_1 n_2 - n_{\boldsymbol{k}} n_1 - n_{\boldsymbol{k}} n_2 \right), \end{gather}$$

(3.22)$$\begin{gather}\varDelta = \frac{1}{2} \sqrt{(- k_h + k_{1h} + k_{2h}) (k_h - k_{1h} + k_{2h}) (k_h + k_{1h} - k_{2h}) (k_h + k_{1h} + k_{2h})}. \end{gather}$$

In the latter equations, the interaction coefficients are evaluated using $\boldsymbol {k}_h = \boldsymbol {k}_{1h} + \boldsymbol {k}_{2h}$, and similar relations obtained by permutations of wave vectors. It follows that they are functions of $(k_h, k_z, k_{1h}, k_{1z}, k_{2h}, k_{2z})$ only. The factor 2 arising in $\mathcal {R}_{12}^{\boldsymbol {k}(h)}$ (3.21) when compared with $\mathcal {R}_{12}^{\boldsymbol {k}}$ (3.19) comes from the fact that there are two solutions $(\varphi _{1},\varphi _{2})$ to $\boldsymbol {k}_h = \boldsymbol {k}_{1h} + \boldsymbol {k}_{2h}$ (or permutations of wave vectors) for each $(k_h,k_z,k_{1h},k_{1z},k_{2h},k_{2z})$. Here, $\varDelta$ is the area of the triangle formed by the horizontal wave vectors. It arises from the average of $\delta (\boldsymbol {k}_h - \boldsymbol {k}_{1h} - \boldsymbol {k}_{2h})$. We can further simplify the collision integral (3.20) by working with polar coordinates $(k_h,k_z)=k(\sin \theta _{\boldsymbol {k}}, \cos \theta _{\boldsymbol {k}})$, $(k_{1h},k_{1z})=k_1(\sin \theta _{1}, \cos \theta _{1})$ and $(k_{2h},k_{2z})=k_2(\sin \theta _{2}, \cos \theta _{2})$. Because the last two terms of the collision integral are symmetric with respect to $\boldsymbol {k}_1 \leftrightarrow \boldsymbol {k}_2$, we obtain

(3.23) \begin{gather} St_{\boldsymbol{k}}^{(h)} = \mathcal{I}_{\boldsymbol{k}} - 2 \mathcal{J}_{\boldsymbol{k}}, \end{gather}

(3.24) \begin{align}\mathcal{I}_{\boldsymbol{k}} &= 8 {\rm \pi}\epsilon^2 \int \delta(k \cos \theta_{\boldsymbol{k}} - k_1 \cos \theta_{1} - k_2 \cos \theta_{2}) \delta(N(\sin \theta_{\boldsymbol{k}} - \sin \theta_{1} - \sin \theta_{2}))\nonumber\\ &\quad \times \frac{|V_{12}^{\boldsymbol{k}}|^2}{\varDelta} \left( n_1 n_2 - n_{\boldsymbol{k}} n_1 - n_{\boldsymbol{k}} n_2 \right) k_1^2 \sin \theta_{1} k_2^2 \sin \theta_{2} \,\mathrm{d} \theta_{1} \,\mathrm{d} k_1 \,\mathrm{d} \theta_{2} \,\mathrm{d} k_2, \end{align}

(3.25) \begin{align}\mathcal{J}_{\boldsymbol{k}} &= 8 {\rm \pi}\epsilon^2 \int \delta(k_1 \cos \theta_{1} - k \cos \theta_{\boldsymbol{k}} - k_2 \cos \theta_{2}) \delta(N(\sin \theta_{1} - \sin \theta_{\boldsymbol{k}} - \sin \theta_{2}))\nonumber\\ &\quad\times \frac{|V_{12}^{\boldsymbol{k}}|^2}{\varDelta} \left( n_{\boldsymbol{k}} n_2 - n_1 n_{\boldsymbol{k}} - n_1 n_2 \right) k_1^2 \sin \theta_{1} k_2^2 \sin \theta_{2} \,\mathrm{d} \theta_{1} \,\mathrm{d} k_1 \,\mathrm{d} \theta_{2} \,\mathrm{d} k_2. \end{align}

To go further, we need to parametrize the resonant manifold, corresponding to the set of wave vectors $(\boldsymbol {k}, \boldsymbol {k}_1, \boldsymbol {k}_2)$ satisfying the resonant conditions

(3.26)\begin{equation} \begin{cases} \boldsymbol{k} = \boldsymbol{k}_1 + \boldsymbol{k}_2 \\ \omega_{\boldsymbol{k}} = \omega_{1} + \omega_{2} \end{cases} \quad \text{or} \quad \begin{cases} \boldsymbol{k}_1 = \boldsymbol{k} + \boldsymbol{k}_2 \\ \omega_{1} = \omega_{\boldsymbol{k}} + \omega_{2} \end{cases}. \end{equation}

It appears that it is relatively easy to find $(k_{2h}, k_{2z})$ as a function of $(k_h, k_z, k_{1h}, k_{1z})$. This leads to

(3.27)\begin{equation} \begin{cases} k_{2z} = k_z - k_{1z} \\ k_{2h} = |k_z - k_{1z}||\tan \theta_{2}| \end{cases} \quad \text{or} \quad \begin{cases} k_{2z} = k_{1z} - k_z \\ k_{2h} = |k_{1z} - k_z||\tan \theta_{2}| , \end{cases} \end{equation}

with $|\tan \theta _{2}| = {|\sin \theta _{\boldsymbol {k}} - \sin \theta _{1}|}/{\sqrt {1-(\sin \theta _{\boldsymbol {k}} - \sin \theta _{1})^2}}$ and $\sin \theta _{1} = k_{1h}/k_1$. These solutions are valid if and only if

(3.28)\begin{equation} \begin{cases} \omega_{\boldsymbol{k}} \geq \omega_{1} \\ k_h \leq k_{1h} + k_{2h} \\ k_{1h} \leq k_{2h} + k_h \\ k_{2h} \leq k_h + k_{1h} \end{cases} \quad \text{or} \quad \begin{cases} \omega_{1} \geq \omega_{\boldsymbol{k}} \\ k_{1h} \leq k_h + k_{2h} \\ k_h \leq k_{2h} + k_{1h} \\ k_{2h} \leq k_{1h} + k_h , \end{cases} \end{equation}

otherwise there is no solution. We can use the $\delta$-Dirac function to perform integration over $k_{2h}$ and $k_{2z}$ such that we finally obtain

(3.29)$$\begin{gather} \mathcal{I}_{\boldsymbol{k}} = \frac{8 {\rm \pi}\epsilon^2}{N} \int_{\mathcal{D}_{\mathcal{I}}} \frac{k_1^2 \sin \theta_{1} k_2^2 \sin \theta_{2}}{\cos^2 \theta_{2} \varDelta} |V_{12}^{\boldsymbol{k}}|^2 \left( n_1 n_2 - n_{\boldsymbol{k}} n_1 - n_{\boldsymbol{k}} n_2 \right) \,\mathrm{d} \theta_{1} \,\mathrm{d} k_1, \end{gather}$$

(3.30)$$\begin{gather}\mathcal{J}_{\boldsymbol{k}} = \frac{8 {\rm \pi}\epsilon^2}{N} \int_{\mathcal{D}_{\mathcal{J}}} \frac{k_1^2 \sin \theta_{1} k_2^2 \sin \theta_{2}}{\cos^2 \theta_{2} \varDelta} |V_{12}^{\boldsymbol{k}}|^2 \left( n_{\boldsymbol{k}} n_2 - n_1 n_{\boldsymbol{k}} - n_1 n_2 \right) \,\mathrm{d} \theta_{1} \,\mathrm{d} k_1, \end{gather}$$

where $\mathcal {D}_{\mathcal {I}}$ and $\mathcal {D}_{\mathcal {J}}$ are the integration domains of the resonant manifold given by conditions (3.28) detailed later and shown in figure 3. Note that it is possible to parameterize the resonant manifold with the variables $(k_1,k_2)$ instead of $(k_1,\theta _{1})$. However, it requires solution of a polynomial equation of order 4 to find the angles $(\theta _{1},\theta _{2})$. Despite the fact that it is mathematically possible, it leads to equations that are much more difficult to use than the one obtained when we parameterize the resonant manifold using $(k_1,\theta _{1})$.

Figure 3. Transfer coefficient on the integration domain of the collisional integral for several $\theta _{\boldsymbol {k}} \in [0;{\rm \pi} /2]$: (a) $\theta _{\boldsymbol {k}}={\rm \pi} /12$, (b) $\theta _{\boldsymbol {k}}={\rm \pi} /6$, (c) $\theta _{\boldsymbol {k}}={\rm \pi} /4$ and (d) $\theta _{\boldsymbol {k}}={\rm \pi} /2$. The borders of the domains are given by the critical lines $r_{\alpha }$, $r_{\beta }$, $r_{\gamma }$ and $r_{\delta }$ ((3.31)–(3.34)). Here, $\mathcal {D}_{\mathcal {I}}$ lies outside $\omega _{1} = \omega _{\boldsymbol {k}}$ lines (white background), while $\mathcal {D}_{\mathcal {J}}$ is contained inside $\omega _{1} = \omega _{\boldsymbol {k}}$ lines (green background).

For simplicity, in order to define analytically $\mathcal {D}_{\mathcal {I}}$ and $\mathcal {D}_{\mathcal {J}}$, we will assume that the wave-action spectrum is invariant under the transformation $k_z \rightarrow - k_z$, i.e. $n(k_h, k_z) = n(k_h, -k_z)$. In this way, we can restrict the study of the collision integral to $k_z \geq 0$ (i.e. $0 \leq \theta _{\boldsymbol {k}} \leq {\rm \pi}/2$). The study of the resonant surface when the wave-action spectrum does not have this symmetry requires consideration of the case with $k_z \leq 0$ (i.e. ${\rm \pi} /2 \leq \theta _{\boldsymbol {k}} \leq {\rm \pi}$), which is longer but technically not more difficult. The borders of the integration domains $\mathcal {D}_{\mathcal {I}}$ and $\mathcal {D}_{\mathcal {J}}$ correspond to the zeros of $\varDelta$ given by (3.22). Note that the borders of the resonant manifold are attained for wave triads with co-linear horizontal projection, i.e. waves on the same vertical plane. They are easily obtained in the variable $k_1/k$ as a function of $(\theta _{\boldsymbol {k}},\theta _{1})$. For each $(\theta _{\boldsymbol {k}},\theta _{1})$, the zeros of $\varDelta$ are attained on two of the following lines:

(3.31)$$\begin{gather} \frac{k_1}{k} = r_{\alpha}(\theta_{\boldsymbol{k}}, \theta_{1}) \equiv \frac{\sqrt{1-(\sin \theta_{\boldsymbol{k}} - \sin \theta_{1})^2}\sin(\theta_{\boldsymbol{k}})-\sin(\theta_{1})\cos(\theta_{\boldsymbol{k}})+\sin(2\theta_{\boldsymbol{k}})/2}{\sqrt{1-(\sin \theta_{\boldsymbol{k}} - \sin \theta_{1})^2}\sin(\theta_{1})+\sin(\theta_{\boldsymbol{k}})\cos(\theta_{1})-\sin(2\theta_{1})/2}, \end{gather}$$

(3.32)$$\begin{gather}\frac{k_1}{k} = r_{\beta}(\theta_{\boldsymbol{k}}, \theta_{1}) \equiv \frac{\sqrt{1-(\sin \theta_{\boldsymbol{k}} - \sin \theta_{1})^2}\sin(\theta_{\boldsymbol{k}})+\sin(\theta_{1})\cos(\theta_{\boldsymbol{k}})-\sin(2\theta_{\boldsymbol{k}})/2}{\sqrt{1-(\sin \theta_{\boldsymbol{k}} - \sin \theta_{1})^2}\sin(\theta_{1})-\sin(\theta_{\boldsymbol{k}})\cos(\theta_{1})+\sin(2\theta_{1})/2}, \end{gather}$$

(3.33)$$\begin{gather}\frac{k_1}{k} = r_{\gamma}(\theta_{\boldsymbol{k}}, \theta_{1}) \equiv \frac{-\sqrt{1-(\sin \theta_{\boldsymbol{k}} - \sin \theta_{1})^2}\sin(\theta_{\boldsymbol{k}})-\sin(\theta_{1})\cos(\theta_{\boldsymbol{k}})+\sin(2\theta_{\boldsymbol{k}})/2}{\sqrt{1-(\sin \theta_{\boldsymbol{k}} - \sin \theta_{1})^2}\sin(\theta_{1})+\sin(\theta_{\boldsymbol{k}})\cos(\theta_{1})-\sin(2\theta_{1})/2}, \end{gather}$$

(3.34)$$\begin{gather}\frac{k_1}{k} = r_{\delta}(\theta_{\boldsymbol{k}}, \theta_{1}) \equiv \frac{-\sqrt{1-(\sin \theta_{\boldsymbol{k}} - \sin \theta_{1})^2}\sin(\theta_{\boldsymbol{k}})+\sin(\theta_{1})\cos(\theta_{\boldsymbol{k}})-\sin(2\theta_{\boldsymbol{k}})/2}{\sqrt{1-(\sin \theta_{\boldsymbol{k}} - \sin \theta_{1})^2}\sin(\theta_{1})-\sin(\theta_{\boldsymbol{k}})\cos(\theta_{1})+\sin(2\theta_{1})/2}, \end{gather}$$

leading to the subdomains of $\mathcal {D}_{\mathcal {I}}$ and $\mathcal {D}_{\mathcal {J}}$ that are listed in table 1. It is worth mentioning that the description of the resonant manifold of internal waves (accounting for rotation) in the $(k_1/k, \theta _{\boldsymbol {k}}, \theta _{1})$ variables is available in Olbers (Reference Olbers1974) (see § 4 of this reference).

Table 1. Subdomains of the resonant manifolds $\mathcal {D}_{\mathcal {I}}$ and $\mathcal {D}_{\mathcal {J}}$ for $\theta _{\boldsymbol {k}} \in [0;{\rm \pi} /2]$.

In figure 3, we represent the transfer coefficient $\mathcal {T}_{12}^{\boldsymbol {k}} \equiv k_1^2 \sin \theta _{1} k_2^2 \sin \theta _{2} |V_{12}^{\boldsymbol {k}}|^2 / (\cos ^2 \theta _{2} \varDelta )$ on the integration domains $\mathcal {D}_{\mathcal {I}}$ and $\mathcal {D}_{\mathcal {J}}$. Here, $\mathcal {T}_{12}^{\boldsymbol {k}}$ measures the strength of the interaction of the triad ((3.29)–(3.30)), which is important information for studying the evolution of IGWs. As can be expected from earlier studies (McComas & Bretherton Reference McComas and Bretherton1977; Müller et al. Reference Müller, Holloway, Henyey and Pomphrey1986; Lvov et al. Reference Lvov, Polzin, Tabak and Yokoyama2010; Eden, Pollmann & Olbers Reference Eden, Pollmann and Olbers2019; Olbers, Pollmann & Eden Reference Olbers, Pollmann and Eden2020; Lanchon & Cortet Reference Lanchon and Cortet2023), the transfer coefficient is important for triads with large scale separation, i.e. when $k_1$ or $k_2 \gg k$. The transfer coefficient is also large for triads with $k \sim k_1 \sim k_2$ at the border of the domain, corresponding to triads contained in a vertical plane. It is in line with Dematteis & Lvov (Reference Dematteis and Lvov2021), who showed that the dominant contribution to the collision integral for the steady-state spectra, in the hydrostatic limit, is due to the horizontally co-linear wave triads. It has also been shown that local interactions correspond to an important part of the energy transfers for the Garrett–Munk spectrum (Wu & Pan Reference Wu and Pan2023).

3.4. Hydrostatic limit

In strongly stratified flows, the energy tends to concentrate in modes with wave vectors such that the vertical component is much larger than the horizontal one. It is therefore worth considering the almost vertical propagation hypothesis (or hydrostatic limit) with $|k_z| \gg k_h$, and so $\omega _{\boldsymbol {k}} = N k_h/k \simeq N k_h/|k_z|$. In that limit, our kinetic equation is equivalent to other kinetic equations of IGW turbulence (Müller & Olbers Reference Müller and Olbers1975; Caillol & Zeitlin Reference Caillol and Zeitlin2000; Lvov & Tabak Reference Lvov and Tabak2001), as shown in the review of Lvov et al. (Reference Lvov, Polzin and Yokoyama2012). We have checked that our interaction coefficients are equal to the one of Lvov & Tabak (Reference Lvov and Tabak2001) up to machine precision in the hydrostatic limit, when evaluated on the resonant manifold. In the hydrostatic limit, we can find stationary, axisymmetric, bi-homogeneous solutions $n_{\boldsymbol {k}} = n(k_h,k_z) \propto k_h^{\nu _h} |k_z|^{\nu _z}$ to the wave-kinetic equation. This theoretical and numerical work has already been achieved in earlier studies (Pelinovsky & Raevsky Reference Pelinovsky and Raevsky1977; Caillol & Zeitlin Reference Caillol and Zeitlin2000; Lvov & Tabak Reference Lvov and Tabak2001; Lvov et al. Reference Lvov, Polzin and Tabak2004, Reference Lvov, Polzin, Tabak and Yokoyama2010; Dematteis & Lvov Reference Dematteis and Lvov2021), so we will not repeat all the computations here. Instead, we show that the KZ spectrum can be obtained without involving resonance condition in frequencies.

In the hydrostatic limit, we have the following simplifications:

(3.35)$$\begin{gather} \boldsymbol{e}_{p1} \boldsymbol{\cdot} \boldsymbol{k}_2 = \frac{k_{1z}}{k_1 k_{1h}} \boldsymbol{k}_{1h} \boldsymbol{\cdot} \boldsymbol{k}_{2h} - \frac{k_{1h} k_{2z}}{k_1} \simeq \frac{k_{1z}}{|k_{1z}| k_{1h}} \boldsymbol{k}_{1h} \boldsymbol{\cdot} \boldsymbol{k}_{2h} - \frac{k_{1h} k_{2z}}{|k_{1z}|}, \end{gather}$$

(3.36)$$\begin{gather}\boldsymbol{e}_{p1} \boldsymbol{\cdot} \boldsymbol{e}_{p2} = \frac{1}{k_1 k_2} \left( k_{1z} k_{2z} \frac{\boldsymbol{k}_{1h} \boldsymbol{\cdot} \boldsymbol{k}_{2h}}{k_{1h} k_{2h}} + k_{1h} k_{2h} \right) \simeq \frac{k_{1z} k_{2z}}{|k_{1z}| |k_{2z}|} \frac{\boldsymbol{k}_{1h} \boldsymbol{\cdot} \boldsymbol{k}_{2h}}{k_{1h} k_{2h}}, \end{gather}$$

and similar relations after permutations of wave vectors. Note that the scalar products of horizontal wave vectors (e.g. $\boldsymbol {k}_{1h} \boldsymbol {\cdot } \boldsymbol {k}_{2h}$) are determined by the resonance condition for horizontal wave vectors, and are only functions of $(k_h,k_{1h},k_{2h})$. Outside the hydrostatic limit, the wave frequency and interaction coefficients are homogeneous in wave-vector amplitudes $(k,k_1,k_2)$, but not in $(k_h,k_{1h},k_{2h})$ and $(|k_z|,|k_{1z}|,|k_{2z}|)$ separately. Contrarily, in the hydrostatic limit, the wave frequency and interaction coefficients are bi-homogeneous, i.e. are homogeneous in $(k_h,k_{1h},k_{2h})$ and $(|k_z|,|k_{1z}|,|k_{2z}|)$ separately. More precisely, the transformation $(k_h,k_{1h},k_{2h}) \rightarrow \mu _h (k_h,k_{1h},k_{2h})$, $(k_z,k_{1z},k_{2z}) \rightarrow \mu _z (k_z,k_{1z},k_{2z})$ changes the frequency and interaction coefficients in the following way:

(3.37a,b)\begin{equation} \omega_{\boldsymbol{k}} \rightarrow \boldsymbol{\mu}^{\boldsymbol{\alpha}} \omega_{\boldsymbol{k}} \quad \text{and} \quad V_{12}^{\boldsymbol{k}} \rightarrow \boldsymbol{\mu}^{\boldsymbol{\beta}} V_{12}^{\boldsymbol{k}}, \end{equation}

where we have used the notation of Balk & Nazarenko (Reference Balk and Nazarenko1990) $\boldsymbol {\mu }^{\boldsymbol {\alpha }} \equiv \mu _h^{\alpha _h} \mu _z^{\alpha _z}$, with $\boldsymbol {\alpha } = (1,-1)$ and $\boldsymbol {\beta } = ( \tfrac {3}{2}, -\tfrac {1}{2} )$. The homogeneity degrees inside and outside the hydrostatic limit are linked by $\alpha _h+\alpha _z=\alpha =0$ and $\beta _h+\beta _z=\beta =1$.

We now perform the Zakharov-Kuznetsov transformation (Kuznetsov Reference Kuznetsov1972) for bi-homogenous spectra

(3.38a–d)\begin{equation} k_{1h} \rightarrow \frac{k_h}{k_{1h}} k_h, \quad k_{2h} \rightarrow \frac{k_h}{k_{1h}} k_{2h}, \quad k_{1z} \rightarrow \frac{k_z}{k_{1z}} k_z, \quad k_{2z} \rightarrow \frac{k_z}{k_{1z}} k_{2z} \end{equation}

to the integral with $\mathcal {Q}_{\boldsymbol{k} 2}^{1}$, and a similar transformation for the integral with $\mathcal {Q}_{\boldsymbol{k} 1}^{2}$ in the wave-kinetic equation (3.4). After the ZK transformation (3.37a,b), the collision integral becomes

(3.39)\begin{align} &St_{\boldsymbol{k}} \rightarrow 4 {\rm \pi}\epsilon^2 \int \delta(\boldsymbol{k} - \boldsymbol{k}_1 - \boldsymbol{k}_2) \delta(\omega_{12}^{\boldsymbol{k}}) \nonumber\\ &\qquad \times \left[ V_{12}^{\boldsymbol{k}} - V_{\boldsymbol{k}2}^{1} \left( \frac{\boldsymbol{k}_1}{\boldsymbol{k}} \right)^{\boldsymbol{\chi}} - V_{\boldsymbol{k}1}^{2} \left( \frac{\boldsymbol{k}_2}{\boldsymbol{k}} \right)^{\boldsymbol{\chi}} \right] \left( V_{12}^{\boldsymbol{k}} n_1 n_2 - V_{\boldsymbol{k}2}^{1} n_{\boldsymbol{k}} n_2 - V_{\boldsymbol{k}1}^{2} n_{\boldsymbol{k}} n_1 \right) \,\mathrm{d} \boldsymbol{k}_1 \,\mathrm{d} \boldsymbol{k}_2 , \end{align}

with $\boldsymbol {d} = (2,1)$ and $\boldsymbol {\chi } = \boldsymbol {\alpha } - 2 \boldsymbol {\beta } - 2 \boldsymbol {\nu } - 2 \boldsymbol {d}$. The integrand in (3.39) is zero when $-\boldsymbol {\nu } = \boldsymbol {\alpha }$ ($n_{\boldsymbol {k}} \propto 1/\omega _{\boldsymbol {k}}$), which corresponds to the RJ spectrum, and when $\boldsymbol {\chi } = \boldsymbol {\alpha }$, which leads to the KZ spectrum $n_{\boldsymbol {k}} \propto k_h^{-7/2} |k_z|^{-1/2}$. Interestingly, we obtain the RJ and the KZ spectra by using the symmetry of the interaction coefficients (2.12a,b), and not the resonance on frequencies, which is unusual in WWT theory. As said in the introduction, the same theoretical spectrum was first obtained by Pelinovsky & Raevsky (Reference Pelinovsky and Raevsky1977), and later by Caillol & Zeitlin (Reference Caillol and Zeitlin2000) and Lvov & Tabak (Reference Lvov and Tabak2001) using different formalisms. Yet, the ZK transformation (3.38a,b) is a non-identity transformation that takes the limit of zero wavenumbers to infinity and vice versa, which may lead to the cancellation of oppositely signed divergences. If the original collision integral converges and is equal to zero, then the found spectrum is indeed a valid solution, in which case the spectrum is called local. If not, then the spectrum in question is a spurious solution; it is called a non-local spectrum (Nazarenko Reference Nazarenko2011) and is not physically realizable. It turned out that the KZ (or PR) spectrum is non-local (Caillol & Zeitlin Reference Caillol and Zeitlin2000).

The collision integral is expected to have other zeros. This situation is typical for anisotropic media (Kuznetsov Reference Kuznetsov1972; Balk & Nazarenko Reference Balk and Nazarenko1990; Lvov et al. Reference Lvov, Polzin and Tabak2004). To find these spectra, we need to compute the collision integral to find its zeros in the $(\nu _h,\nu _z)$ plane. It is important to note that the integrand in the collision integral can diverge if $k_{1h}$ or $k_{2h} \rightarrow 0$ which corresponds to infrared (IR) divergence, and when $k_{1h}, k_{2h} \rightarrow \infty$ which correspond to ultraviolet (UV) divergence. Depending on $(\nu _h,\nu _z)$, the divergences can be integrable (in which case the spectrum is local), or non-integrable (in which case the spectrum is non-local). To identify these divergences, we need to analyse the terms of the collision integral and see for which values of $(\nu _h,\nu _z)$ they lead to non-integrable divergences. The locality conditions have been obtained by Lvov et al. (Reference Lvov, Polzin, Tabak and Yokoyama2010), and it was shown that another steady-state solution exists due to the opposite signs of IR and UV divergences. More precisely, the collision integral converges only on the segment $\nu _z = 0$, $-4 < \nu _h < -3$ and the steady-state spectrum $n_{\boldsymbol {k}} \propto k_h^{-3.69}$ has been obtained by finding the zero of the collision integral on this segment numerically (Lvov et al. Reference Lvov, Polzin, Tabak and Yokoyama2010; Dematteis & Lvov Reference Dematteis and Lvov2021). Yet, any spectrum with $\nu _z \neq 0$ around that solution would lead to a divergent collision integral. It means that the collision integral is not differentiable with respect to $\nu _z$ and $\nu _h$ for spectra with $\nu _z=0$ and, therefore, the energy flux integrals are divergent (see Zakharov et al. Reference Zakharov, L'vov and Falkovich1992, § 3.32). This rules out realizability of these spectra. Since our formalism is equivalent to Lvov & Tabak (Reference Lvov and Tabak2001) in the hydrostatic limit, the same results are expected.