2.1 The generalized Hasegawa–Mima model

We consider a magnetized plasma in a uniform magnetic field in the $z$ direction and with an equilibrium local gradient of the plasma density. Upon assuming a quasi-adiabatic response for the electrons and a fluid description for the ion dynamics, we model electrostatic two-dimensional (2-D) turbulent flows in the $xy$ plane using the gHME (Smolyakov & Diamond Reference Smolyakov and Diamond1999; Krommes & Kim Reference Krommes and Kim2000), which in normalized units is given by

(2.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}w+\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}w+\unicode[STIX]{x1D6FD}\,\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D713}=Q. & & \displaystyle\end{eqnarray}$$

Here $\boldsymbol{x}=(x,y)$ is a 2-D coordinate, the $x$ axis is the ZF direction and the $y$ axis is the direction of the local gradient of the background plasma density, which is measured by the constant $\unicode[STIX]{x1D6FD}$ . In (2.1), time is measured in units of the inverse ion cyclotron frequency while length is measured in units of the ion sound radius. (See, e.g. Zhu et al. (Reference Zhu, Zhou, Ruiz and Dodin2018c ) for more details.) The function $\unicode[STIX]{x1D713}(t,\boldsymbol{x})$ is the electric potential, $\boldsymbol{v}\doteq \boldsymbol{e}_{z}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D713}$ is the ion fluid velocity on the $xy$ plane and $\boldsymbol{e}_{z}$ is a unit vector normal to this plane. (The symbol $\doteq$ denotes definitions.) The generalized vorticity $w(t,\boldsymbol{x})$ is given by

(2.2) $$\begin{eqnarray}\displaystyle w(t,\boldsymbol{x})\doteq (\unicode[STIX]{x1D6FB}^{2}-L_{\text{D}}^{-2}\widehat{a})\unicode[STIX]{x1D713}, & & \displaystyle\end{eqnarray}$$

where $\widehat{a}$ is an operator such that $\widehat{a}=1$ in parts of the spectrum corresponding to DWs and $\widehat{a}=0$ in those corresponding to ZFs. The constant $L_{\text{D}}$ is the ion sound radius. ( $L_{\text{D}}=1$ in normalized units.) The term $Q(t,\boldsymbol{x})$ in (2.1) represents external forces and dissipation. Notably, equation (2.1) with $L_{\text{D}}\rightarrow \infty$ also describes Rossby turbulence in planetary atmospheres (Farrell & Ioannou Reference Farrell and Ioannou2003, Reference Farrell and Ioannou2007; Marston, Conover & Schneider Reference Marston, Conover and Schneider2008; Srinivasan & Young Reference Srinivasan and Young2012; Ait-Chaalal et al. Reference Ait-Chaalal, Schneider, Meyer and Marston2016).

For isolated systems, where $Q=0$ , equation (2.1) conserves two nonlinear invariants: the enstrophy ${\mathcal{Z}}$ and the energy ${\mathcal{E}}$ (strictly speaking, free energy). These are defined as

(2.3a,b ) $$\begin{eqnarray}\displaystyle {\mathcal{Z}}(t)\doteq \frac{1}{2}\int \text{d}^{2}\boldsymbol{x}\,w^{2},\quad {\mathcal{E}}(t)\doteq -\frac{1}{2}\int \text{d}^{2}\boldsymbol{x}\,w\unicode[STIX]{x1D713}. & & \displaystyle\end{eqnarray}$$

The statistical model that we shall derive conserves both of these nonlinear invariants.

2.2 Separating the mean and fluctuating components of the fields

Let us decompose the fields $\unicode[STIX]{x1D713}$ and $w$ into their mean and fluctuating components, which will be denoted by bars and tildes, respectively. For any arbitrary field $g(t,\boldsymbol{x})$ , the mean part is defined as $\overline{g}(t,y)\doteq \int \text{d}x\,\boldsymbol{\langle }\!\langle g\rangle \!\boldsymbol{\rangle }/L_{x}$ , where $L_{x}$ is the system’s length along $x$ and $\boldsymbol{\langle }\!\langle \cdot \rangle \!\boldsymbol{\rangle }$ denotes the statistical average over realizations of initial conditions or of the external random forcing. The mean part will describe the coherent ZF dynamics. In contrast, the fluctuating quantities will describe the incoherent DW dynamics.

We consider the fluctuating quantities to be small in amplitude. The small-amplitude ordering is denoted by using the small dimensionless parameter $\unicode[STIX]{x1D716}_{\text{nl}}\ll 1$ . In particular,

(2.4a,b ) $$\begin{eqnarray}\displaystyle w=\overline{w}+\unicode[STIX]{x1D716}_{\text{nl}}\,\widetilde{w},\quad \boldsymbol{v}=U\boldsymbol{e}_{x}+\unicode[STIX]{x1D716}_{\text{nl}}\widetilde{\boldsymbol{v}}, & & \displaystyle\end{eqnarray}$$

where $U(t,y)\doteq -\unicode[STIX]{x2202}_{y}\overline{\unicode[STIX]{x1D713}}$ is the ZF velocity field and the two components of the generalized vorticity are related to $\unicode[STIX]{x1D713}$ as (Parker & Krommes Reference Parker and Krommes2013)

(2.5a,b ) $$\begin{eqnarray}\displaystyle \overline{w}=\unicode[STIX]{x1D6FB}^{2}\overline{\unicode[STIX]{x1D713}},\quad \widetilde{w}=\unicode[STIX]{x1D6FB}_{\text{D}}^{2}\widetilde{\unicode[STIX]{x1D713}}. & & \displaystyle\end{eqnarray}$$

Here $\unicode[STIX]{x1D6FB}_{\text{D}}^{2}\doteq \unicode[STIX]{x1D6FB}^{2}-1$ . Also, $\overline{\widetilde{w}}=0$ and $\overline{\widetilde{\boldsymbol{v}}}=0$ by definition.

From (2.1), we then derive the governing equations for the fluctuating and the mean fields (Srinivasan & Young Reference Srinivasan and Young2012):

(2.6a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\widetilde{w}+U\unicode[STIX]{x2202}_{x}\widetilde{w}+[\unicode[STIX]{x1D6FD}-(\unicode[STIX]{x2202}_{y}^{2}U)]\unicode[STIX]{x2202}_{x}\widetilde{\unicode[STIX]{x1D713}}+\unicode[STIX]{x1D716}_{\text{nl}}\,f_{\text{eddy}}=\widetilde{Q}, & \displaystyle\end{eqnarray}$$

(2.6b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}U+\unicode[STIX]{x1D707}_{\text{zf}}U+\unicode[STIX]{x1D716}_{\text{nl}}^{2}\,\unicode[STIX]{x2202}_{y}\overline{\widetilde{v}_{x}\widetilde{v}_{y}}=\overline{Q}, & \displaystyle\end{eqnarray}$$

$f_{\text{eddy}}(t,\boldsymbol{x})\doteq \widetilde{\boldsymbol{v}}\cdot \unicode[STIX]{x1D735}\widetilde{w}-\overline{\widetilde{\boldsymbol{v}}\cdot \unicode[STIX]{x1D735}\widetilde{w}}$

$\unicode[STIX]{x1D716}_{\text{nl}}$

$Q=\overline{Q}+\unicode[STIX]{x1D716}_{\text{nl}}\,\widetilde{Q}$

2.3 Temporal and spatial scale separation of fluctuating and mean fields

In the model derived below, we shall assume that there is a temporal and spatial scale separation between the DW and ZF fields. Specifically, let $\unicode[STIX]{x1D70F}_{\text{dw}}$ and $\unicode[STIX]{x1D706}_{\text{dw}}$ respectively denote the characteristic period and wavelength of the DWs. In a similar manner, the characteristic time and length scales of the ZFs are given by $T_{\text{zf}}$ and $L_{\text{zf}}$ , respectively. Upon following the discussion in Ruiz et al. (Reference Ruiz, Parker, Shi and Dodin2016), we shall characterize the scale separation between DWs and ZFs by introducing the geometrical-optics (GO) parameter

(2.7) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D716}\doteq \text{max}\left(\frac{\unicode[STIX]{x1D70F}_{\text{dw}}}{T_{\text{zf}}},\frac{\unicode[STIX]{x1D706}_{\text{dw}}}{L_{\text{zf}}},\frac{L_{\text{D}}}{L_{\text{zf}}}\right)\ll 1. & & \displaystyle\end{eqnarray}$$

In addition, we assume that the DWs are weakly damped and weakly dissipated so that $\widetilde{Q}=\unicode[STIX]{x1D716}\,\widetilde{\unicode[STIX]{x1D709}}-\unicode[STIX]{x1D716}\,\unicode[STIX]{x1D707}_{\text{dw}}\widetilde{w}$ , where $\widetilde{\unicode[STIX]{x1D709}}$ is some white-noise external forcing with zero mean and $\unicode[STIX]{x1D707}_{\text{dw}}$ is intended to emulate the dissipation of DWs caused by the external environment. The external dissipation and random forcing are scaled using the GO parameter (McDonald & Kaufman Reference McDonald and Kaufman1985). For the mean quantities, we consider $\overline{Q}=-\unicode[STIX]{x1D707}_{\text{zf}}\,\overline{w}$ , where $\unicode[STIX]{x1D707}_{\text{zf}}$ represents dissipation on the mean flows.

Upon using the orderings above, we write (2.6) as

(2.8a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D716}\unicode[STIX]{x2202}_{t}\widetilde{w}+\unicode[STIX]{x1D716}U\unicode[STIX]{x2202}_{x}\widetilde{w}+\unicode[STIX]{x1D716}[\unicode[STIX]{x1D6FD}-(\unicode[STIX]{x2202}_{y}^{2}U)]\unicode[STIX]{x2202}_{x}\widetilde{\unicode[STIX]{x1D713}}+\unicode[STIX]{x1D716}\unicode[STIX]{x1D707}_{\text{dw}}\widetilde{w}+\unicode[STIX]{x1D716}_{\text{nl}}f_{\text{eddy}}=\unicode[STIX]{x1D716}\widetilde{\unicode[STIX]{x1D709}}, & \displaystyle\end{eqnarray}$$

(2.8b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}U+\unicode[STIX]{x1D707}_{\text{zf}}U+\unicode[STIX]{x1D716}_{\text{nl}}^{2}\,\unicode[STIX]{x2202}_{y}\overline{\widetilde{v}_{x}\widetilde{v}_{y}}=0, & \displaystyle\end{eqnarray}$$

$\unicode[STIX]{x1D716}$

$\unicode[STIX]{x1D716}$

$\hbar$

$\widetilde{\boldsymbol{v}}=\boldsymbol{e}_{z}\times \unicode[STIX]{x1D716}\unicode[STIX]{x1D735}\widetilde{\unicode[STIX]{x1D713}}$

$\widetilde{w}=(\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D6FB}^{2}-1)\widetilde{\unicode[STIX]{x1D713}}$

$f_{\text{eddy}}=\widetilde{\boldsymbol{v}}\cdot \unicode[STIX]{x1D716}\unicode[STIX]{x1D735}\widetilde{w}-\overline{\widetilde{\boldsymbol{v}}\cdot \unicode[STIX]{x1D716}\unicode[STIX]{x1D735}\widetilde{w}}$

$\unicode[STIX]{x2202}_{y}\overline{\widetilde{v}_{x}\widetilde{v}_{y}}\sim O(\unicode[STIX]{x1D716}^{0})$

$\overline{\widetilde{v}_{x}\widetilde{v}_{y}}$

$\widetilde{w}(t,\boldsymbol{x})$

$U(t,y)$

2.4 Abstract vector representation

Let us write the fluctuating fields as elements of an abstract Hilbert space $L^{2}(\mathbb{R}^{3})$ of wave states with inner product (Dodin Reference Dodin2014; Littlejohn & Winston Reference Littlejohn and Winston1993)

(2.9) $$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D719}\mid \unicode[STIX]{x1D713}\rangle =\int \text{d}t\,\text{d}^{2}\boldsymbol{x}\,\unicode[STIX]{x1D719}^{\ast }(t,\boldsymbol{x})\,\unicode[STIX]{x1D713}(t,\boldsymbol{x}), & & \displaystyle\end{eqnarray}$$

where the integrals are taken over $\mathbb{R}^{3}$ . Let $|t,\boldsymbol{x}\rangle$ be the eigenstates of the time and position operators $\widehat{t}$ and $\widehat{\boldsymbol{x}}$ . (Considering $\widehat{t}$ as an operator will allow us to write the statistical closure in § 3 in a simple manner.) Hence, $\widetilde{w}(t,\boldsymbol{x})$ is written as $\widetilde{w}(t,\boldsymbol{x})=\langle t,\boldsymbol{x}\mid \widetilde{w}\rangle$ . Since $\widetilde{w}(t,\boldsymbol{x})$ is real, then $\langle \widetilde{w}\mid t,\boldsymbol{x}\rangle =\langle t,\boldsymbol{x}\mid \widetilde{w}\rangle$ . In the following, we shall use the notation $\unicode[STIX]{x1D639}\doteq (t,\boldsymbol{x})$ to denote space–time coordinates. Likewise, $|\unicode[STIX]{x1D639}\rangle \doteq |t,\boldsymbol{x}\rangle$ is the corresponding space–time eigenstate, and $\text{d}^{3}\unicode[STIX]{x1D639}\doteq \text{d}t\,\text{d}^{2}\boldsymbol{x}$ is the space–time differential volume.

In addition to the time and position operators, we introduce the frequency operator $\widehat{\unicode[STIX]{x1D714}}$ , such that $\widehat{\unicode[STIX]{x1D714}}\doteq \text{i}\unicode[STIX]{x1D716}\unicode[STIX]{x2202}_{t}$ in the coordinate representation. Likewise, the wavevector operator $\widehat{\boldsymbol{k}}$ is defined as $\widehat{\boldsymbol{k}}\doteq -\text{i}\unicode[STIX]{x1D716}\unicode[STIX]{x1D735}$ . In particular, these operators are Hermitian, and one has $\langle \unicode[STIX]{x1D639}\mid \widehat{\unicode[STIX]{x1D714}}\mid \widetilde{w}\rangle =\text{i}\unicode[STIX]{x1D716}\unicode[STIX]{x2202}_{t}\widetilde{w}$ and $\langle \unicode[STIX]{x1D639}\mid \widehat{\boldsymbol{k}}\mid \widetilde{w}\rangle =-\text{i}\unicode[STIX]{x1D716}\unicode[STIX]{x1D735}\widetilde{w}$ . Using the relation between the fluctuating generalized vorticity and electric potential, one has $|\widetilde{w}\rangle =-\widehat{k}_{\text{D}}^{2}|\widetilde{\unicode[STIX]{x1D713}}\rangle$ , where

(2.10a,b ) $$\begin{eqnarray}\displaystyle \widehat{k}_{\text{D}}^{2}\doteq \widehat{k}^{2}+1,\quad \widehat{k}^{2}\doteq \widehat{\boldsymbol{k}}\cdot \widehat{\boldsymbol{k}}. & & \displaystyle\end{eqnarray}$$

Let us write the fluctuating quantities appearing in (2.8) in terms of the abstract states and operators. One finds that (2.8a ) can be written as

(2.11) $$\begin{eqnarray}\displaystyle \widehat{\mathscr{D}}|\widetilde{w}\rangle =\frac{\text{i}\unicode[STIX]{x1D716}_{\text{nl}}}{2}|f_{\text{nl}}[\widetilde{w},\widetilde{w}]\rangle -\frac{\text{i}\unicode[STIX]{x1D716}_{\text{nl}}}{2}|\overline{f_{\text{nl}}[\widetilde{w},\widetilde{w}]}\rangle +\text{i}\unicode[STIX]{x1D716}|\widetilde{\unicode[STIX]{x1D709}}\rangle . & & \displaystyle\end{eqnarray}$$

Here $\widehat{\mathscr{D}}$ is the DW dispersion operator

(2.12) $$\begin{eqnarray}\displaystyle \widehat{\mathscr{D}}\doteq \widehat{\unicode[STIX]{x1D714}}-\widehat{U}\widehat{k}_{x}+(\unicode[STIX]{x1D6FD}-\widehat{U}^{\prime \prime })\widehat{k}_{x}\widehat{k}_{\text{D}}^{-2}+\text{i}\unicode[STIX]{x1D716}\unicode[STIX]{x1D707}_{\text{dw}}, & & \displaystyle\end{eqnarray}$$

where $\widehat{U}\doteq U(\widehat{t},\widehat{y})$ , and the prime above $U$ henceforth denotes $\unicode[STIX]{x2202}_{y}$ ; in particular, $\widehat{U}^{\prime \prime }\doteq \unicode[STIX]{x2202}_{y}^{2}\,U(\widehat{t},\widehat{y})$ . Also, $|\widetilde{\unicode[STIX]{x1D709}}\rangle$ is the ket corresponding to the random forcing $\widetilde{\unicode[STIX]{x1D709}}$ . It is to be noted that the $\widehat{\mathscr{D}}$ includes the nonlinear coupling between the ZFs and the DWs.

Additionally in (2.11), the kets $|\,f_{\text{nl}}[\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}]\rangle$ and $|\,\overline{f_{\text{nl}}[\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}]}\rangle$ are given by

(2.13) $$\begin{eqnarray}\displaystyle & \displaystyle |f_{\text{nl}}[\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}]\rangle \doteq \int \text{d}^{3}\unicode[STIX]{x1D639}\,|\unicode[STIX]{x1D639}\rangle \langle \unicode[STIX]{x1D719}\mid \widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\mid \unicode[STIX]{x1D713}\rangle , & \displaystyle\end{eqnarray}$$

(2.14) $$\begin{eqnarray}\displaystyle & \displaystyle |\overline{f_{\text{nl}}[\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}]}\rangle \doteq \int \text{d}^{3}\unicode[STIX]{x1D639}\,|\unicode[STIX]{x1D639}\rangle \overline{\langle \unicode[STIX]{x1D719}\mid \widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\mid \unicode[STIX]{x1D713}\rangle }, & \displaystyle\end{eqnarray}$$

where $\widehat{\mathscr{K}}(\unicode[STIX]{x1D639})$ is an operator describing the nonlinear eddy–eddy interactions

(2.15) $$\begin{eqnarray}\displaystyle \widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\doteq \widehat{\mathscr{L}}_{j}|\unicode[STIX]{x1D639}\rangle \langle \unicode[STIX]{x1D639}|\widehat{\mathscr{R}}_{j}+\widehat{\mathscr{R}}_{j}|\unicode[STIX]{x1D639}\rangle \langle \unicode[STIX]{x1D639}|\widehat{\mathscr{L}}_{j}. & & \displaystyle\end{eqnarray}$$

(The summation over repeating indices is assumed.) The operators $\widehat{\mathscr{L}}_{j}$ and $\widehat{\mathscr{R}}_{j}$ are given by

(2.16a,b ) $$\begin{eqnarray}\displaystyle \widehat{\mathscr{L}}_{j}\doteq (\boldsymbol{e}_{z}\times \widehat{\boldsymbol{k}})_{j}k_{\text{D}}^{-2},\quad \widehat{\mathscr{R}}_{j}\doteq \widehat{\boldsymbol{k}}_{j}. & & \displaystyle\end{eqnarray}$$

Indeed, we can verify that $|f_{\text{nl}}[\widetilde{w},\widetilde{w}]\rangle$ represents nonlinear eddy–eddy interactions. When projecting onto the coordinate eigenstates, one obtains

(2.17) $$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D639}\mid f_{\text{nl}}[\widetilde{w},\widetilde{w}]\rangle & = & \displaystyle \langle \widetilde{w}\mid \widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\mid \widetilde{w}\rangle \nonumber\\ \displaystyle & = & \displaystyle \langle \widetilde{w}\mid \widehat{k}_{\text{D}}^{-2}(\boldsymbol{e}_{z}\times \widehat{\boldsymbol{k}})_{j}\mid \unicode[STIX]{x1D639}\rangle \langle \unicode[STIX]{x1D639}\mid \widehat{\boldsymbol{k}}_{j}\mid \widetilde{w}\rangle +\langle \widetilde{w}\mid \widehat{\boldsymbol{k}}_{j}\mid \unicode[STIX]{x1D639}\rangle \langle \unicode[STIX]{x1D639}\mid (\boldsymbol{e}_{z}\times \widehat{\boldsymbol{k}})_{j}\widetilde{k}_{\text{D}}^{-2}\mid \widetilde{w}\rangle \nonumber\\ \displaystyle & = & \displaystyle -(\langle \unicode[STIX]{x1D639}\mid (\boldsymbol{e}_{z}\times \widehat{\boldsymbol{k}})_{j}\mid \widetilde{\unicode[STIX]{x1D713}}\rangle )^{\ast }\langle \unicode[STIX]{x1D639}\mid \widehat{\boldsymbol{k}}_{j}\mid \widetilde{w}\rangle -(\langle \unicode[STIX]{x1D639}\mid \widehat{\boldsymbol{k}}_{j}\mid \widetilde{w}\rangle )^{\ast }\langle \unicode[STIX]{x1D639}\mid (\boldsymbol{e}_{z}\times \widehat{\boldsymbol{k}})_{j}\mid \widetilde{\unicode[STIX]{x1D713}}\rangle \nonumber\\ \displaystyle & = & \displaystyle -2(\boldsymbol{e}_{z}\times \unicode[STIX]{x1D716}\unicode[STIX]{x1D735}\widetilde{\unicode[STIX]{x1D713}})\cdot \unicode[STIX]{x1D716}\unicode[STIX]{x1D735}\widetilde{w}\nonumber\\ \displaystyle & = & \displaystyle -2\widetilde{\boldsymbol{v}}\times \unicode[STIX]{x1D716}\unicode[STIX]{x1D735}\widetilde{w},\end{eqnarray}$$

which is one of the nonlinear terms appearing in (2.8a ). Here we substituted the coordinate representation for the wavevector operator and used $|\widetilde{w}\rangle =-\widehat{k}_{\text{D}}^{2}|\widetilde{\unicode[STIX]{x1D713}}\rangle$ . Note that the factor 2 appears because we have written the ket $|f_{\text{nl}}[\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}]\rangle$ in a symmetric form so that

(2.18) $$\begin{eqnarray}\displaystyle |f_{\text{nl}}[\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}]\rangle =|f_{\text{nl}}[\unicode[STIX]{x1D713},\unicode[STIX]{x1D719}]\rangle & & \displaystyle\end{eqnarray}$$

for any two real fields $\unicode[STIX]{x1D719}$ and $\unicode[STIX]{x1D713}$ .

Following Ruiz et al. (Reference Ruiz, Parker, Shi and Dodin2016), the fluctuating terms appearing in (2.8b ) can also be rewritten in the abstract representation. Upon noting that $\widetilde{v}_{x}\widetilde{v}_{y}=(-\unicode[STIX]{x1D716}\unicode[STIX]{x2202}_{y}\widetilde{\unicode[STIX]{x1D713}})(\unicode[STIX]{x1D716}\unicode[STIX]{x2202}_{x}\widetilde{\unicode[STIX]{x1D713}})=-\langle \unicode[STIX]{x1D639}\mid \widehat{k}_{x}\mid \widetilde{\unicode[STIX]{x1D713}}\rangle \langle \widetilde{\unicode[STIX]{x1D713}}\mid \widehat{k}_{y}\mid \unicode[STIX]{x1D639}\rangle$ , one obtains

(2.19) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}U+\unicode[STIX]{x1D707}_{\text{zf}}U=\unicode[STIX]{x1D716}_{\text{nl}}^{2}\,\unicode[STIX]{x2202}_{y}\overline{\langle \unicode[STIX]{x1D639}\mid \widehat{k}_{x}\widehat{k}_{\text{D}}^{-2}\mid \widetilde{w}\rangle \langle \widetilde{w}\mid \widehat{k}_{\text{D}}^{-2}\widehat{k}_{y}\mid \unicode[STIX]{x1D639}\rangle }. & & \displaystyle\end{eqnarray}$$

3.1 Statistical-closure problem

Let us now introduce the correlation operator for the fluctuating vorticity field:

(3.1) $$\begin{eqnarray}\displaystyle \widehat{\mathscr{W}}\doteq \overline{|\widetilde{w}\rangle \langle \widetilde{w}|}, & & \displaystyle\end{eqnarray}$$

where the Hermitian operator $|\widetilde{w}\rangle \langle \widetilde{w}|$ can be interpreted as the fluctuating-vorticity density operator by analogy with quantum mechanics (Ruiz et al. Reference Ruiz, Parker, Shi and Dodin2016). It is to be noted that $\widehat{\mathscr{W}}$ is defined as the zonal average of the fluctuating-vorticity density operator. The zonal average of an abstract operator is discussed in § A.2. By using the identity (A 20), one can write (2.19) in terms of the correlation operator:

(3.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}U+\unicode[STIX]{x1D707}_{\text{zf}}U=\unicode[STIX]{x1D716}_{\text{nl}}^{2}\,\unicode[STIX]{x2202}_{y}\langle \unicode[STIX]{x1D639}\mid \widehat{k}_{x}\widehat{k}_{\text{D}}^{-2}\widehat{\mathscr{W}}\widehat{k}_{\text{D}}^{-2}\widehat{k}_{y}\mid \unicode[STIX]{x1D639}\rangle . & & \displaystyle\end{eqnarray}$$

Now, let us obtain the governing equation for $\widehat{\mathscr{W}}$ . Multiplying (2.11) by $\langle \widetilde{w}|$ from the right and zonal averaging leads to an equation for the correlation operator:

(3.3) $$\begin{eqnarray}\displaystyle \widehat{\mathscr{D}}\widehat{\mathscr{W}}=\frac{\text{i}\unicode[STIX]{x1D716}_{\text{nl}}}{2}\,\overline{|f_{\text{nl}}[\widetilde{w},\widetilde{w}]\rangle \langle \widetilde{w}|}+\text{i}\unicode[STIX]{x1D716}\,\overline{|\widetilde{\unicode[STIX]{x1D709}}\rangle \langle \widetilde{w}|}. & & \displaystyle\end{eqnarray}$$

Subtracting from (3.3) its Hermitian conjugate gives

(3.4) $$\begin{eqnarray}\displaystyle [\widehat{\mathscr{D}}_{\text{H}},\widehat{\mathscr{W}}]_{-}+\text{i}[\widehat{\mathscr{D}}_{\text{A}},\widehat{\mathscr{W}}]_{+}=\text{i}\unicode[STIX]{x1D716}_{\text{nl}}\,[\overline{|f_{\text{nl}}[\widetilde{w},\widetilde{w}]\rangle \langle \widetilde{w}|}]_{\text{H}}+2\text{i}\unicode[STIX]{x1D716}\,[\overline{|\widetilde{\unicode[STIX]{x1D709}}\rangle \langle \widetilde{w}|}]_{\text{H}}, & & \displaystyle\end{eqnarray}$$

where the subscripts ‘H’ and ‘A’ denote the Hermitian and anti-Hermitian parts of an arbitrary operator; i.e. $\widehat{\mathscr{A}}_{\text{H}}\doteq (\widehat{\mathscr{A}}+\widehat{\mathscr{A}}^{\dagger })/2$ and $\widehat{\mathscr{A}}_{\text{A}}\doteq (\widehat{\mathscr{A}}-\widehat{\mathscr{A}}^{\dagger })/(2\text{i})$ . Specifically, the operator $\widehat{\mathscr{D}}$ is decomposed as $\widehat{\mathscr{D}}=\widehat{\mathscr{D}}_{\text{H}}+\text{i}\widehat{\mathscr{D}}_{\text{A}}$ , where

(3.5a ) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{\mathscr{D}}_{\text{H}}=\widehat{\unicode[STIX]{x1D714}}-\widehat{U}\widehat{k}_{x}+[\unicode[STIX]{x1D6FD}-\widehat{U}^{\prime \prime },\widehat{k}_{x}\widehat{k}_{\text{D}}^{-2}]_{+}/2, & \displaystyle\end{eqnarray}$$

(3.5b ) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{\mathscr{D}}_{\text{A}}=\text{i}[\widehat{U}^{\prime \prime },\widehat{k}_{x}\widehat{k}_{\text{D}}^{-2}]_{-}/2+\unicode[STIX]{x1D716}\unicode[STIX]{x1D707}_{\text{dw}}. & \displaystyle\end{eqnarray}$$

$\widehat{\mathscr{D}}_{\text{H}}$

$\widehat{\mathscr{D}}_{\text{A}}$

$[\cdot ,\cdot ]_{\mp }$

$[\widehat{\mathscr{A}},\widehat{\mathscr{B}}]_{-}=\widehat{\mathscr{A}}\widehat{\mathscr{B}}-\widehat{\mathscr{B}}\widehat{\mathscr{A}}$

$[\widehat{\mathscr{A}},\widehat{\mathscr{B}}]_{+}=\widehat{\mathscr{A}}\widehat{\mathscr{B}}+\widehat{\mathscr{B}}\widehat{\mathscr{A}}$

Equations (3.2) and (3.4) are not closed. The left-hand side of (3.4) is written in terms of $\widehat{\mathscr{W}}$ , which is bilinear in the fluctuating vorticity. However, the right-hand side of (3.4) contains terms that are linear and cubic with respect to $\widetilde{w}$ . This is the fundamental statistical closure problem (Frisch & Kolmogorov Reference Frisch and Kolmogorov1995; Kraichnan Reference Kraichnan2013; Krommes Reference Krommes2002). The next step is to introduce a statistical closure in order to express (3.4) in terms of the correlation operator $\widehat{\mathscr{W}}$ only.

3.2 A comment on the quasilinear approximation

One possibility is to neglect the first term on the right-hand side of (3.4). Then, one linearizes (2.11) so that

(3.6) $$\begin{eqnarray}\displaystyle \widehat{\mathscr{D}}|\widetilde{w}\rangle \simeq \text{i}\unicode[STIX]{x1D716}|\widetilde{\unicode[STIX]{x1D709}}\rangle . & & \displaystyle\end{eqnarray}$$

After formally inverting $\widehat{\mathscr{D}}$ , we have $|\widetilde{w}\rangle \simeq \text{i}\unicode[STIX]{x1D716}\widehat{\mathscr{D}}^{-1}|\widetilde{\unicode[STIX]{x1D709}}\rangle$ . Substituting into (3.4) leads to the closed equation

(3.7) $$\begin{eqnarray}\displaystyle [\widehat{\mathscr{D}}_{\text{H}},\widehat{\mathscr{W}}]_{-}+\text{i}[\widehat{\mathscr{D}}_{\text{A}},\widehat{\mathscr{W}}]_{+}=2\text{i}\unicode[STIX]{x1D716}^{2}\,[\widehat{\mathscr{S}}(\widehat{\mathscr{D}}^{-1})^{\dagger }]_{\text{A}}, & & \displaystyle\end{eqnarray}$$

where $\widehat{\mathscr{S}}\doteq \overline{|\widetilde{\unicode[STIX]{x1D709}}\rangle \langle \widetilde{\unicode[STIX]{x1D709}}|}$ is the zonal-averaged density operator associated with the random external forcing. We also used the fact that, for any operator $\widehat{\mathscr{A}}$ , one has $(-\text{i}\,\widehat{\mathscr{A}})_{\text{H}}=\widehat{\mathscr{A}}_{\text{A}}$ .

Equations (3.2) and (3.7) now form a closed system. In this approximation, the equation for the DW fluctuations was linearized in (3.6), but the DW nonlinearity is only kept in the equation for the mean field (3.2). This constitutes the QL approximation. If one projects (3.7) on the double-physical coordinate space $(\unicode[STIX]{x1D639},\unicode[STIX]{x1D639}^{\prime })$ using multiplication by $\langle t,\boldsymbol{x}|$ and $|t,\boldsymbol{x}^{\prime }\rangle$ , then one obtains the so-called CE2 equations (Farrell & Ioannou Reference Farrell and Ioannou2003, Reference Farrell and Ioannou2007; Marston et al. Reference Marston, Conover and Schneider2008; Srinivasan & Young Reference Srinivasan and Young2012; Ait-Chaalal et al. Reference Ait-Chaalal, Schneider, Meyer and Marston2016). Likewise, if one projects (3.7) on the ray space $(t,\boldsymbol{x},\unicode[STIX]{x1D714},\boldsymbol{k})$ using the Weyl transform (Weyl Reference Weyl1931), then one obtains the Wigner–Moyal model discussed in Ruiz et al. (Reference Ruiz, Parker, Shi and Dodin2016), Parker (Reference Parker2018) and Zhu et al. (Reference Zhu, Zhou, Ruiz and Dodin2018c ).

3.3 A statistical closure beyond the quasilinear approximation

We extend our theory beyond the QL approximation in order to retain wave–wave scattering, namely, the term $f_{\text{nl}}$ in (3.4). Let us separate $\widetilde{w}$ into three components:

(3.8) $$\begin{eqnarray}\displaystyle |\widetilde{w}\rangle =|\widetilde{w}_{0}\rangle +\unicode[STIX]{x1D716}_{\text{nl}}|\widetilde{\unicode[STIX]{x1D719}}\rangle +\unicode[STIX]{x1D716}|\widetilde{\unicode[STIX]{x1D711}}\rangle . & & \displaystyle\end{eqnarray}$$

Here $|\widetilde{w}_{0}\rangle =O(1)$ is chosen to satisfy the linear part of (2.11); i.e. $\widehat{\mathscr{D}}|\widetilde{w}_{0}\rangle =0$ . The fluctuations in $\widetilde{w}_{0}$ are due to random initial conditions, whose statistics are considered to be uncorrelated to those of the random forcing $\widetilde{\unicode[STIX]{x1D709}}$ . When formally inverting $\widehat{\mathscr{D}}$ in (2.11), one finds that, to lowest order in $\unicode[STIX]{x1D716}_{\text{nl}}$ and $\unicode[STIX]{x1D716}$ ,

(3.9a ) $$\begin{eqnarray}\displaystyle & \displaystyle |\widetilde{\unicode[STIX]{x1D719}}\rangle \simeq \frac{\text{i}}{2}\,\widehat{\mathscr{D}}^{-1}\{|f_{\text{nl}}[\widetilde{w}_{0},\widetilde{w}_{0}]\rangle -\overline{|f_{\text{nl}}[\widetilde{w}_{0},\widetilde{w}_{0}]\rangle }\}, & \displaystyle\end{eqnarray}$$

(3.9b ) $$\begin{eqnarray}\displaystyle & \displaystyle |\widetilde{\unicode[STIX]{x1D711}}\rangle \simeq \text{i}\widehat{\mathscr{D}}^{-1}|\widetilde{\unicode[STIX]{x1D709}}\rangle . & \displaystyle\end{eqnarray}$$

(3.10) $$\begin{eqnarray}\displaystyle [\widehat{\mathscr{D}}_{\text{H}},\widehat{\mathscr{W}}]_{-}+\text{i}[\widehat{\mathscr{D}}_{\text{A}},\widehat{\mathscr{W}}]_{+} & = & \displaystyle \text{i}\unicode[STIX]{x1D716}_{\text{nl}}\{\overline{|f_{\text{nl}}[\widetilde{w}_{0},\widetilde{w}_{0}]\rangle \langle \widetilde{w}_{0}|}\}_{\text{H}}+2\text{i}\unicode[STIX]{x1D716}_{\text{nl}}^{2}\{\overline{|f_{\text{nl}}[\widetilde{\unicode[STIX]{x1D719}},\widetilde{w}_{0}]\rangle \langle \widetilde{w}_{0}|}\}_{\text{H}}\nonumber\\ \displaystyle & & \displaystyle +\,\text{i}\unicode[STIX]{x1D716}_{\text{nl}}^{2}\{\overline{|f_{\text{nl}}[\widetilde{w}_{0},\widetilde{w}_{0}]\rangle \langle \widetilde{\unicode[STIX]{x1D719}}|}\}_{\text{H}}+2\text{i}\unicode[STIX]{x1D716}^{2}[\widehat{\mathscr{S}}(\widehat{\mathscr{D}}^{-1})^{\dagger }]_{\text{A}}\nonumber\\ \displaystyle & & \displaystyle +\,O(\unicode[STIX]{x1D716}_{\text{nl}}^{3},\unicode[STIX]{x1D716}_{\text{nl}}^{2}\unicode[STIX]{x1D716},\unicode[STIX]{x1D716}_{\text{nl}}\unicode[STIX]{x1D716}^{2}).\end{eqnarray}$$

Here we neglected $O(\unicode[STIX]{x1D716}_{\text{nl}}^{3},\unicode[STIX]{x1D716}_{\text{nl}}^{2}\unicode[STIX]{x1D716},\unicode[STIX]{x1D716}_{\text{nl}}\unicode[STIX]{x1D716}^{2})$ terms, e.g. $\unicode[STIX]{x1D716}_{\text{nl}}^{3}|f_{\text{nl}}[\widetilde{w}_{0},\widetilde{\unicode[STIX]{x1D719}}]\rangle \langle \widetilde{\unicode[STIX]{x1D719}}|$ and $\unicode[STIX]{x1D716}_{\text{nl}}^{2}\unicode[STIX]{x1D716}|f_{\text{nl}}[\widetilde{w}_{0},\widetilde{\unicode[STIX]{x1D719}}]\rangle \langle \widetilde{\unicode[STIX]{x1D711}}|$ . Note that the zonal averages of quantities involving both $\widetilde{w}_{0}$ and $\widetilde{\unicode[STIX]{x1D709}}$ (e.g. $\unicode[STIX]{x1D716}_{\text{nl}}|\widetilde{\unicode[STIX]{x1D709}}\rangle \langle \widetilde{w}_{0}|$ and $\unicode[STIX]{x1D716}_{\text{nl}}\unicode[STIX]{x1D716}|f_{\text{nl}}[\widetilde{w}_{0},\widetilde{w}_{0}]\rangle \langle \widetilde{\unicode[STIX]{x1D709}}|$ ) are zero since the statistics of $\widetilde{w}_{0}$ and $\widetilde{\unicode[STIX]{x1D709}}$ are independent. The factor of two in the second term on the right-hand side of (3.10) is due to the symmetry property (2.18).

Now, let us explicitly calculate the statistical average of the nonlinear terms in (3.10). To do this, we shall use the quasinormal approximation which expresses higher-order $(n\geqslant 3)$ statistical moments of $\widetilde{w}_{0}$ in terms of the lower-order moments. A further discussion on the validity of this approximation and its relation to the widely used random-phase approximation in homogeneous wave turbulence theory is given in § 5. When adopting the quasinormal approximation, one specifically obtains

(3.11) $$\begin{eqnarray}\displaystyle \overline{\widetilde{w}_{0}(\unicode[STIX]{x1D639}_{1})\widetilde{w}_{0}(\unicode[STIX]{x1D639}_{2})\widetilde{w}_{0}(\unicode[STIX]{x1D639}_{3})}=0, & & \displaystyle\end{eqnarray}$$

(3.12) $$\begin{eqnarray}\displaystyle \overline{\widetilde{w}_{0}(\unicode[STIX]{x1D639}_{1})\widetilde{w}_{0}(\unicode[STIX]{x1D639}_{2})\widetilde{w}_{0}(\unicode[STIX]{x1D639}_{3})\widetilde{w}_{0}(\unicode[STIX]{x1D639}_{4})} & = & \displaystyle \langle \unicode[STIX]{x1D639}_{1}\mid \widehat{\mathscr{W}}_{0}\mid \unicode[STIX]{x1D639}_{2}\rangle \,\langle \unicode[STIX]{x1D639}_{3}\mid \widehat{\mathscr{W}}_{0}\mid \unicode[STIX]{x1D639}_{4}\rangle \nonumber\\ \displaystyle & & \displaystyle +\,\langle \unicode[STIX]{x1D639}_{1}\mid \widehat{\mathscr{W}}_{0}\mid \unicode[STIX]{x1D639}_{3}\rangle \,\langle \unicode[STIX]{x1D639}_{2}\mid \widehat{\mathscr{W}}_{0}\mid \unicode[STIX]{x1D639}_{4}\rangle \nonumber\\ \displaystyle & & \displaystyle +\,\langle \unicode[STIX]{x1D639}_{1}\mid \widehat{\mathscr{W}}_{0}\mid \unicode[STIX]{x1D639}_{4}\rangle \,\langle \unicode[STIX]{x1D639}_{2}\mid \widehat{\mathscr{W}}_{0}\mid \unicode[STIX]{x1D639}_{3}\rangle ,\end{eqnarray}$$

where $\widehat{\mathscr{W}}_{0}\doteq \overline{|\widetilde{w}_{0}\rangle \langle \widetilde{w}_{0}|}$ and $\unicode[STIX]{x1D639}_{i}=(t_{i},\boldsymbol{x}_{i})$ denotes a space–time coordinate.

Upon substituting (3.11), one finds that the first term appearing in the right-hand side of (3.10) vanishes; i.e. $\overline{|f_{\text{nl}}[\widetilde{w}_{0},\widetilde{w}_{0}]\rangle \langle \widetilde{w}_{0}|}=0$ . After substituting (2.13) and (3.9a ), one obtains the following expression for the second term on the right-hand side of (3.10):

(3.13) $$\begin{eqnarray}\displaystyle & & \displaystyle \overline{|f_{\text{nl}}[\widetilde{\unicode[STIX]{x1D719}},\widetilde{w}_{0}]\rangle \langle \widetilde{w}_{0}|}=\int \text{d}^{3}\unicode[STIX]{x1D639}\,|\unicode[STIX]{x1D639}\rangle \overline{\langle \widetilde{\unicode[STIX]{x1D719}}\mid \widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\mid \widetilde{w}_{0}\rangle \langle \widetilde{w}_{0}|}=-\frac{\text{i}}{2}\int \text{d}^{3}\unicode[STIX]{x1D639}\,\text{d}^{3}\unicode[STIX]{x1D63A}\,|\unicode[STIX]{x1D639}\rangle \nonumber\\ \displaystyle & & \displaystyle \quad \times \,\overline{[\langle \widetilde{w}_{0}\mid \widehat{\mathscr{K}}^{\dagger }(\unicode[STIX]{x1D63A})\mid \widetilde{w}_{0}\rangle -\overline{\langle \widetilde{w}_{0}\mid \widehat{\mathscr{K}}^{\dagger }(\unicode[STIX]{x1D63A})\mid \widetilde{w}_{0}\rangle }]\langle \unicode[STIX]{x1D63A}\mid (\widehat{\mathscr{D}}^{-1})^{\dagger }\,\widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\mid \widetilde{w}_{0}\rangle \langle \widetilde{w}_{0}|},\end{eqnarray}$$

where $|\unicode[STIX]{x1D639}\rangle$ and $|\unicode[STIX]{x1D63A}\rangle$ are eigenstates of the coordinate operator. Inserting (3.12) in the abstract representation gives

(3.14) $$\begin{eqnarray}\displaystyle & & \displaystyle \overline{[\langle \widetilde{w}_{0}\mid \widehat{\mathscr{K}}^{\dagger }(\unicode[STIX]{x1D63A})\mid \widetilde{w}_{0}\rangle -\overline{\langle \widetilde{w}_{0}\mid \widehat{\mathscr{K}}^{\dagger }(\unicode[STIX]{x1D63A})\mid \widetilde{w}_{0}\rangle }]\langle \unicode[STIX]{x1D63A}\mid (\widehat{\mathscr{D}}^{-1})^{\dagger }\,\widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\mid \widetilde{w}_{0}\rangle \langle \widetilde{w}_{0}|}\nonumber\\ \displaystyle & & \displaystyle \quad =\langle \widetilde{w}_{0}\mid \widehat{\mathscr{K}}^{\dagger }(\unicode[STIX]{x1D63A})\mid \widetilde{w}_{0}\rangle \langle \unicode[STIX]{x1D63A}\mid (\widehat{\mathscr{D}}^{-1})^{\dagger }\,\widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\mid \widetilde{w}_{0}\rangle \langle \widetilde{w}_{0}|\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\langle \widetilde{w}_{0}\mid \widehat{\mathscr{K}}^{\dagger }(\unicode[STIX]{x1D63A})\mid \widetilde{w}_{0}\rangle \langle \unicode[STIX]{x1D63A}\mid (\widehat{\mathscr{D}}^{-1})^{\dagger }\,\widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\mid \widetilde{w}_{0}\rangle \langle \widetilde{w}_{0}|\nonumber\\ \displaystyle & & \displaystyle \quad =2\langle \unicode[STIX]{x1D63A}\mid (\widehat{\mathscr{D}}^{-1})^{\dagger }\,\widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\mid \widetilde{w}_{0}\rangle \langle \widetilde{w}_{0}\mid \widehat{\mathscr{K}}^{\dagger }(\unicode[STIX]{x1D63A})\mid \widetilde{w}_{0}\rangle \langle \widetilde{w}_{0}|\nonumber\\ \displaystyle & & \displaystyle \quad =2\langle \unicode[STIX]{x1D63A}|\,(\widehat{\mathscr{D}}^{-1})^{\dagger }\,\widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\,\overline{|\widetilde{w}_{0}\rangle \langle \widetilde{w}_{0}|}\,\widehat{\mathscr{K}}^{\dagger }(\unicode[STIX]{x1D63A})\,\overline{|\widetilde{w}_{0}\rangle \langle \widetilde{w}_{0}|}\nonumber\\ \displaystyle & & \displaystyle \quad =2\langle \unicode[STIX]{x1D63A}|\,(\widehat{\mathscr{D}}^{-1})^{\dagger }\,\widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\,\widehat{\mathscr{W}}_{0}\,\widehat{\mathscr{K}}^{\dagger }(\unicode[STIX]{x1D63A})\,\widehat{\mathscr{W}}_{0},\end{eqnarray}$$

where the connecting overlines indicate the correlations of the terms that contribute to the final result. (This notation is also commonly used in quantum field theory (Peskin & Schroeder Reference Peskin and Schroeder1995).) It is to be noted that this result can also be obtained by inserting the completeness relation $\widehat{1}=\int \text{d}^{3}\unicode[STIX]{x1D639}^{\prime }|\unicode[STIX]{x1D639}^{\prime }\rangle \langle \unicode[STIX]{x1D639}^{\prime }|$ and then using (3.12) explicitly. In the third line, we used the symmetry property given in (2.18). Substituting this result into (3.13) gives

(3.15) $$\begin{eqnarray}\displaystyle \overline{|f_{\text{nl}}[\widetilde{\unicode[STIX]{x1D719}},\widetilde{w}_{0}]\rangle \langle \widetilde{w}_{0}|}=-\text{i}\int \text{d}^{3}\unicode[STIX]{x1D639}\,\text{d}^{3}\unicode[STIX]{x1D63A}\,|\unicode[STIX]{x1D639}\rangle \langle \unicode[STIX]{x1D63A}|(\widehat{\mathscr{D}}^{-1})^{\dagger }\,\widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\,\widehat{\mathscr{W}}_{0}\,\widehat{\mathscr{K}}^{\dagger }(\unicode[STIX]{x1D63A})\,\widehat{\mathscr{W}}_{0}. & & \displaystyle\end{eqnarray}$$

In a similar manner, substituting (3.9a ) into the third term in (3.10) leads to

(3.16) $$\begin{eqnarray}\displaystyle \overline{|f_{\text{nl}}[\widetilde{w}_{0},\widetilde{w}_{0}]\rangle \langle \widetilde{\unicode[STIX]{x1D719}}|} & = & \displaystyle \int \text{d}^{3}\unicode[STIX]{x1D639}\,|\unicode[STIX]{x1D639}\rangle \overline{\langle \widetilde{w}_{0}\mid \widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\mid \widetilde{w}_{0}\rangle \langle \widetilde{\unicode[STIX]{x1D719}}|}=-\frac{\text{i}}{2}\int \text{d}^{3}\unicode[STIX]{x1D639}\,\text{d}^{3}\unicode[STIX]{x1D63A}\,|\unicode[STIX]{x1D639}\rangle \langle \unicode[STIX]{x1D63A}|(\widehat{\mathscr{D}}^{-1})^{\dagger }\nonumber\\ \displaystyle & & \displaystyle \times \,\overline{\langle \widetilde{w}_{0}\mid \widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\mid \widetilde{w}_{0}\rangle [\langle \widetilde{w}_{0}\mid \widehat{\mathscr{K}}^{\dagger }(\unicode[STIX]{x1D63A})\mid \widetilde{w}_{0}\rangle -\overline{\langle \widetilde{w}_{0}\mid \widehat{\mathscr{K}}^{\dagger }(\unicode[STIX]{x1D63A})\mid \widetilde{w}_{0}\rangle }]}.\end{eqnarray}$$

As before, when using the quasinormal approximation, we obtain

(3.17) $$\begin{eqnarray}\displaystyle & & \displaystyle \overline{\langle \widetilde{w}_{0}\mid \widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\mid \widetilde{w}_{0}\rangle \,[\langle \widetilde{w}_{0}\mid \widehat{\mathscr{K}}^{\dagger }(\unicode[STIX]{x1D63A})\mid \widetilde{w}_{0}\rangle -\overline{\langle \widetilde{w}_{0}\mid \widehat{\mathscr{K}}^{\dagger }(\unicode[STIX]{x1D63A})\mid \widetilde{w}_{0}\rangle }]}\nonumber\\ \displaystyle & & \displaystyle \quad =\langle \widetilde{w}_{0}\mid \widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\mid \widetilde{w}_{0}\rangle \langle \widetilde{w}_{0}\mid \widehat{\mathscr{K}}^{\dagger }(\unicode[STIX]{x1D63A})\mid \widetilde{w}_{0}\rangle +\langle \widetilde{w}_{0}\mid \widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\mid \widetilde{w}_{0}\rangle \langle \widetilde{w}_{0}\mid \widehat{\mathscr{K}}^{\dagger }(\unicode[STIX]{x1D63A})\mid \widetilde{w}_{0}\rangle \nonumber\\ \displaystyle & & \displaystyle \quad =2\langle \widetilde{w}_{0}\mid \widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\mid \widetilde{w}_{0}\rangle \langle \widetilde{w}_{0}\mid \widehat{\mathscr{K}}^{\dagger }(\unicode[STIX]{x1D63A})\mid \widetilde{w}_{0}\rangle \nonumber\\ \displaystyle & & \displaystyle \quad =2\,\text{Tr}[\widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\,\overline{|\widetilde{w}_{0}\rangle \langle \widetilde{w}_{0}|}\,\widehat{\mathscr{K}}^{\dagger }(\unicode[STIX]{x1D63A})\,\overline{|\widetilde{w}_{0}\rangle \langle \widetilde{w}_{0}|}]\nonumber\\ \displaystyle & & \displaystyle \quad =2\,\text{Tr}[\widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\,\widehat{\mathscr{W}}_{0}\,\widehat{\mathscr{K}}^{\dagger }(\unicode[STIX]{x1D63A})\,\widehat{\mathscr{W}}_{0}],\end{eqnarray}$$

where we introduced the trace operation $\text{Tr}\,\widehat{\mathscr{A}}=\int \text{d}^{3}\unicode[STIX]{x1D639}\langle \unicode[STIX]{x1D639}\mid \widehat{\mathscr{A}}\mid \unicode[STIX]{x1D639}\rangle$ . By using the identity operator $\widehat{1}=\int \text{d}^{3}\unicode[STIX]{x1D639}^{\prime }|\unicode[STIX]{x1D639}^{\prime }\rangle \langle \unicode[STIX]{x1D639}^{\prime }|$ , we are able to write $\langle w_{0}\mid \widehat{\mathscr{A}}\mid w_{0}\rangle =\text{Tr}(\widehat{\mathscr{A}}|w_{0}\rangle \langle w_{0}|)$ . Also, note that the trace $\text{Tr}[\widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\,\widehat{\mathscr{W}}_{0}\,\widehat{\mathscr{K}}^{\dagger }(\unicode[STIX]{x1D63A})\,\widehat{\mathscr{W}}_{0}]$ is not an operator but rather a function of the space–time coordinates $\unicode[STIX]{x1D639}$ and $\unicode[STIX]{x1D63A}$ . Upon inserting (3.17) into (3.16), we obtain

(3.18) $$\begin{eqnarray}\displaystyle \overline{|f_{\text{nl}}[\widetilde{w}_{0},\widetilde{w}_{0}]\rangle \langle \widetilde{\unicode[STIX]{x1D719}}|}=-\text{i}\int \text{d}^{3}\unicode[STIX]{x1D639}\,\text{d}^{3}\unicode[STIX]{x1D63A}\,|\unicode[STIX]{x1D639}\rangle \langle \unicode[STIX]{x1D63A}|(\widehat{\mathscr{D}}^{-1})^{\dagger }\,\text{Tr}[\widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\,\widehat{\mathscr{W}}_{0}\,\widehat{\mathscr{K}}^{\dagger }(\unicode[STIX]{x1D63A})\,\widehat{\mathscr{W}}_{0}]. & & \displaystyle\end{eqnarray}$$

We then substitute (3.15) and (3.18) into (3.10). Approximating $\widehat{\mathscr{W}}_{0}\simeq \widehat{\mathscr{W}}$ , which is valid to the leading order of accuracy of the theory, gives a closed equation for the correlation operator:

(3.19) $$\begin{eqnarray}\displaystyle [\widehat{\mathscr{D}}_{\text{H}},\widehat{\mathscr{W}}]_{-}+\text{i}[\widehat{\mathscr{D}}_{\text{A}},\widehat{\mathscr{W}}]_{+}=2\text{i}\unicode[STIX]{x1D716}_{\text{nl}}^{2}[\widehat{\mathscr{F}}\,(\widehat{\mathscr{D}}^{-1})^{\dagger }]_{\text{A}}-2\text{i}\unicode[STIX]{x1D716}_{\text{nl}}^{2}[\widehat{\unicode[STIX]{x1D702}}\,\widehat{\mathscr{W}}]_{\text{A}}+2\text{i}\unicode[STIX]{x1D716}^{2}[\widehat{\mathscr{S}}(\widehat{\mathscr{D}}^{-1})^{\dagger }]_{\text{A}}, & & \displaystyle\end{eqnarray}$$

where

(3.20a ) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{\unicode[STIX]{x1D702}}\doteq -\int \text{d}^{3}\unicode[STIX]{x1D639}\,\text{d}^{3}\unicode[STIX]{x1D63A}\,|\unicode[STIX]{x1D639}\rangle \langle \unicode[STIX]{x1D63A}|(\widehat{\mathscr{D}}^{-1})^{\dagger }\widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\,\widehat{\mathscr{W}}\,\widehat{\mathscr{K}}^{\dagger }(\unicode[STIX]{x1D63A}), & \displaystyle\end{eqnarray}$$

(3.20b ) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{\mathscr{F}}\doteq \frac{1}{2}\int \text{d}^{3}\unicode[STIX]{x1D639}\,\text{d}^{3}\unicode[STIX]{x1D63A}\,|\unicode[STIX]{x1D639}\rangle \langle \unicode[STIX]{x1D63A}|\text{Tr}[\widehat{\mathscr{K}}(\unicode[STIX]{x1D639})\,\widehat{\mathscr{W}}\,\widehat{\mathscr{K}}^{\dagger }(\unicode[STIX]{x1D63A})\,\widehat{\mathscr{W}}]. & \displaystyle\end{eqnarray}$$

In summary, equations (3.2), (3.19), and (3.20) form a complete set of equations for the zonal flow $U(t,y)$ and for the correlation operator $\widehat{\mathscr{W}}$ . Multiplying (3.19) by $\langle t,\boldsymbol{x}|$ and $|t^{\prime },\boldsymbol{x}^{\prime }\rangle$ leads to the ‘quasinormal’ equation for the two-point correlation function written in the double-physical coordinate space $(\unicode[STIX]{x1D639},\unicode[STIX]{x1D639}^{\prime })$ . However, we shall instead project these equations into the phase space by using the Weyl transform. Then, by adopting approximations that are consistent with the GO description of eikonal fields, we shall obtain a WKE model describing the vorticity fluctuations. (Readers who are not familiar with the Weyl calculus are encouraged to read § A.1 before continuing further.)

4.1 Wigner–Moyal equation

The Weyl transform is a mapping from operators in a Hilbert space into functions of phase space (Tracy et al. Reference Tracy, Brizard, Richardson and Kaufman2014). In this work, the Weyl transform is defined as

(4.1) $$\begin{eqnarray}\displaystyle A(\unicode[STIX]{x1D639},\unicode[STIX]{x1D62C})\doteq \unicode[STIX]{x1D61E}[\widehat{\mathscr{A}}]=\int \text{d}^{3}\unicode[STIX]{x1D634}\,\text{e}^{\text{i}\unicode[STIX]{x1D62C}\cdot \unicode[STIX]{x1D634}/\unicode[STIX]{x1D716}}\langle \unicode[STIX]{x1D639}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D634}\mid \widehat{\mathscr{A}}\mid \unicode[STIX]{x1D639}-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D634}\rangle , & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D62C}\doteq (\unicode[STIX]{x1D714},\boldsymbol{k})$ , $\unicode[STIX]{x1D634}\doteq (\unicode[STIX]{x1D70F},\boldsymbol{s})$ , $\unicode[STIX]{x1D62C}\cdot \unicode[STIX]{x1D634}=\unicode[STIX]{x1D714}\unicode[STIX]{x1D70F}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{s}$ and $\text{d}^{3}\unicode[STIX]{x1D634}\doteq \text{d}\unicode[STIX]{x1D70F}\,\text{d}^{2}\boldsymbol{s}$ . Here the integrals span over $\mathbb{R}^{3}$ . The Weyl symbol $A(\unicode[STIX]{x1D639},\unicode[STIX]{x1D62C})$ of an operator $\widehat{\mathscr{A}}$ is a function on the extended six-dimensional phase space $(\unicode[STIX]{x1D639},\unicode[STIX]{x1D62C})=(t,\boldsymbol{x},\unicode[STIX]{x1D714},\boldsymbol{k})$ ; i.e. $A=A(t,\boldsymbol{x},\unicode[STIX]{x1D714},\boldsymbol{k})$ . Physically, $A(\unicode[STIX]{x1D639},\unicode[STIX]{x1D62C})$ can be interpreted as a local Fourier transform of the space–time representation of the operator $\widehat{\mathscr{A}}$ (see, for instance, equation (A 3)).

The Weyl symbol $W(t,y,\unicode[STIX]{x1D714},\boldsymbol{k})$ corresponding to $\widehat{\mathscr{W}}$ is referred to as the Wigner function of the vorticity fluctuations (Wigner Reference Wigner1932). Since $W(t,y,\unicode[STIX]{x1D714},\boldsymbol{k})$ is a Weyl symbol of the zonal-averaged operator (3.1), it does not depend on the $x$ coordinate. Upon following § A.2, one can write $W(t,y,\unicode[STIX]{x1D714},\boldsymbol{k})$ explicitly as

(4.2) $$\begin{eqnarray}\displaystyle W(t,y,\unicode[STIX]{x1D714},\boldsymbol{k}) & = & \displaystyle \int \text{d}^{3}\unicode[STIX]{x1D634}\,\text{e}^{\text{i}\unicode[STIX]{x1D62C}\cdot \unicode[STIX]{x1D634}/\unicode[STIX]{x1D716}}\langle \unicode[STIX]{x1D639}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D634}\mid \widehat{\mathscr{W}}\mid \unicode[STIX]{x1D639}-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D634}\rangle \nonumber\\ \displaystyle & = & \displaystyle \int \text{d}^{3}\unicode[STIX]{x1D634}\,\text{d}x\,\text{e}^{\text{i}\unicode[STIX]{x1D62C}\cdot \unicode[STIX]{x1D634}/\unicode[STIX]{x1D716}}\boldsymbol{\langle }\!\langle \widetilde{w}(\unicode[STIX]{x1D639}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D634})\,\widetilde{w}(\unicode[STIX]{x1D639}-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D634})\rangle \!\boldsymbol{\rangle }/L_{x}.\end{eqnarray}$$

Since $\widetilde{w}$ is real, then $W(t,y,\unicode[STIX]{x1D714},\boldsymbol{k})=W(t,y,-\unicode[STIX]{x1D714},-\boldsymbol{k})$ . Also, $W(t,y,\unicode[STIX]{x1D714},\boldsymbol{k})$ is a real function because $\widehat{\mathscr{W}}$ is Hermitian. Similar arguments apply to the Weyl symbol $S(t,y,\unicode[STIX]{x1D714},\boldsymbol{k})$ corresponding to the operator $\widehat{\mathscr{S}}$ .

Applying the Weyl transform to (3.19) leads to the Wigner–Moyal formulation of DW–ZF dynamics:

(4.3) $$\begin{eqnarray}\displaystyle & & \displaystyle \mathbf{\{}\!\{D_{\text{H}},W\}\!\mathbf{\}}+\text{i}\,\mathbf{[}[D_{\text{A}},W]\mathbf{]}\nonumber\\ \displaystyle & & \displaystyle \quad =2\text{i}\unicode[STIX]{x1D716}_{\text{nl}}^{2}\,\text{Im}\{F\star [D^{-1}]^{\ast }\}-2\text{i}\unicode[STIX]{x1D716}_{\text{nl}}^{2}\,\text{Im}(\unicode[STIX]{x1D702}\star W)+2\text{i}\unicode[STIX]{x1D716}^{2}\,\text{Im}\{S\star [D^{-1}]^{\ast }\}.\end{eqnarray}$$

Here $D(t,y,\unicode[STIX]{x1D714},\boldsymbol{k})$ is the Weyl symbol corresponding to $\widehat{\mathscr{D}}$ . From the properties of the Weyl transform, we observe that $D_{\text{H}}=\text{Re}\,D$ and $D_{\text{A}}=\text{Im}\,D$ are the (real) Weyl symbols corresponding to the operators $\widehat{\mathscr{D}}_{\text{H}}$ and $\widehat{\mathscr{D}}_{\text{A}}$ , respectively. (‘ $\text{Re}$ ’ and ‘ $\text{Im}$ ’ denote the real and imaginary parts, respectively.) These are given by

(4.4a ) $$\begin{eqnarray}\displaystyle & \displaystyle D_{\text{H}}(t,y,\unicode[STIX]{x1D714},\boldsymbol{k})=\unicode[STIX]{x1D714}-k_{x}U+{\textstyle \frac{1}{2}}\mathbf{[}[(\unicode[STIX]{x1D6FD}-U^{\prime \prime }),k_{x}/k_{\text{D}}^{2}]\mathbf{]}, & \displaystyle\end{eqnarray}$$

(4.4b ) $$\begin{eqnarray}\displaystyle & \displaystyle D_{\text{A}}(t,y,\boldsymbol{k})=\unicode[STIX]{x1D716}\unicode[STIX]{x1D707}_{\text{dw}}-{\textstyle \frac{1}{2}}\mathbf{\{}\!\{U^{\prime \prime },k_{x}/k_{\text{D}}^{2}\}\!\mathbf{\}}. & \displaystyle\end{eqnarray}$$

$F(t,y,\unicode[STIX]{x1D714},\boldsymbol{k})$

$\unicode[STIX]{x1D702}(t,y,\unicode[STIX]{x1D714},\boldsymbol{k})$

$[D^{-1}]^{\ast }(t,y,\unicode[STIX]{x1D714},\boldsymbol{k})$

$\widehat{\mathscr{F}}$

$\widehat{\mathscr{\{}}$

$(\widehat{\mathscr{D}}^{-1})^{\dagger }$

$\star$

$\mathbf{\{}\!\{\cdot ,\cdot \}\!\mathbf{\}}$

$\mathbf{[}[\cdot ,\cdot ]\mathbf{]}$

Modulo the statistical closure introduced in § 3, equation (4.3) is an exact equation for the dynamics of the Wigner function. However, equation (4.3) is difficult to solve as is, both analytically and numerically. Thus, we shall reduce (4.3) to a first-order partial-differential equation (PDE) in phase space by using the GO approximation.

4.2 Collisional wave kinetic equation

In order to simplify (4.3), we shall expand the Moyal products and brackets in terms of the ordering parameter $\unicode[STIX]{x1D716}$ . As a reminder, the parameter $\unicode[STIX]{x1D716}$ was introduced in (2.7) in order to denote the spatio-temporal scale separation between the DW and ZF dynamics. From the definition of the Moyal product (A 6), one has

(4.5) $$\begin{eqnarray}\displaystyle A\star B=AB+\frac{\text{i}\unicode[STIX]{x1D716}}{2}(A\overleftrightarrow{{\mathcal{L}}}B)+O(\unicode[STIX]{x1D716}^{2}), & & \displaystyle\end{eqnarray}$$

where $\overleftrightarrow{{\mathcal{L}}}$ is the Janus operator (A 7), which basically serves as the canonical Poisson bracket in the extended six-dimensional phase space $(t,\boldsymbol{x},\unicode[STIX]{x1D714},\boldsymbol{k})$ . Likewise, the Moyal brackets in (A 9) and (A 10) are approximated by

(4.6a,b ) $$\begin{eqnarray}\displaystyle \mathbf{\{}\!\{A,B\}\!\mathbf{\}}=\text{i}\unicode[STIX]{x1D716}(A\overleftrightarrow{{\mathcal{L}}}B)+O(\unicode[STIX]{x1D716}^{3}),\quad \mathbf{[}[A,B]\mathbf{]}=2AB+O(\unicode[STIX]{x1D716}^{2}). & & \displaystyle\end{eqnarray}$$

For the purposes of this paper, higher-order corrections to (4.5) and (4.6) will not be needed. It is to be noted that the asymptotic expansions above are valid as long as the functions involved are smooth.

In addition to asymptotically expanding the Moyal products and brackets, let us find a proper ansatz for the Wigner function. Note that (3.3) can be written as $\widehat{\mathscr{D}}_{\text{H}}\widehat{\mathscr{W}}=O(\unicode[STIX]{x1D716},\unicode[STIX]{x1D716}_{\text{nl}})$ , where we assumed small dissipation $[D_{\text{A}}\sim O(\unicode[STIX]{x1D716})]$ . In the Weyl representation, this equation becomes $D_{\text{H}}\star W=O(\unicode[STIX]{x1D716},\unicode[STIX]{x1D716}_{\text{nl}})$ . Using (4.5) gives

(4.7) $$\begin{eqnarray}\displaystyle D_{\text{H}}(t,y,\unicode[STIX]{x1D714},\boldsymbol{k})\,W(t,y,\unicode[STIX]{x1D714},\boldsymbol{k})\simeq O(\unicode[STIX]{x1D716},\unicode[STIX]{x1D716}_{\text{nl}}). & & \displaystyle\end{eqnarray}$$

To satisfy this equation, we adopt the GO ansatz (McDonald & Kaufman Reference McDonald and Kaufman1985; McDonald Reference McDonald1991)

(4.8) $$\begin{eqnarray}\displaystyle W(t,y,\unicode[STIX]{x1D714},\boldsymbol{k})=2\unicode[STIX]{x03C0}\unicode[STIX]{x1D716}\,\unicode[STIX]{x1D6FF}\boldsymbol{(}D_{\text{H}}(t,y,\unicode[STIX]{x1D714},\boldsymbol{k})\boldsymbol{)}\,J(t,y,\boldsymbol{k}). & & \displaystyle\end{eqnarray}$$

Here $J(t,y,\boldsymbol{k})$ is interpreted as the wave-action density for the vorticity fluctuations, and $2\unicode[STIX]{x03C0}\unicode[STIX]{x1D716}$ is added to ensure the proper normalization.

To remain consistent with the asymptotic expansion in (4.5) and (4.6), the symbols $D_{\text{H}}$ and $D_{\text{A}}$ in (4.4) are approximated as well to lowest order in $\unicode[STIX]{x1D716}$ :

(4.9a ) $$\begin{eqnarray}\displaystyle & \displaystyle D_{\text{H}}(t,y,\unicode[STIX]{x1D714},\boldsymbol{k})\simeq \unicode[STIX]{x1D714}-k_{x}U+(\unicode[STIX]{x1D6FD}-U^{\prime \prime })k_{x}/k_{\text{D}}^{2}, & \displaystyle\end{eqnarray}$$

(4.9b ) $$\begin{eqnarray}\displaystyle & \displaystyle D_{\text{A}}(t,y,\boldsymbol{k})\simeq \unicode[STIX]{x1D716}\unicode[STIX]{x1D707}_{\text{dw}}+\unicode[STIX]{x1D716}k_{x}k_{y}U^{\prime \prime \prime }/k_{\text{D}}^{4}, & \displaystyle\end{eqnarray}$$

Let us now obtain an approximate expression for $[D^{-1}]^{\ast }$ in (4.3). As a reminder, $[D^{-1}]$ and $D^{-1}$ are not equivalent; $[D^{-1}]$ is the Weyl symbol of $\widehat{\mathscr{D}}^{-1}$ while $D^{-1}$ is simply the inverse of the Weyl symbol $D$ corresponding to $\widehat{\mathscr{D}}$ . However, note that the Weyl representation of $\widehat{\mathscr{D}}\widehat{\mathscr{D}}^{-1}=\widehat{1}$ is $D\star [D^{-1}]=1$ . By replacing the Moyal product with an ordinary product, we obtain $[D^{-1}]\simeq D^{-1}$ . Then, by using the Sokhotski–Plemelj theorem, we replace $[D^{-1}]^{\ast }$ with its limiting form as $\unicode[STIX]{x1D716}$ tends to zero:

(4.10) $$\begin{eqnarray}\displaystyle [D^{-1}]^{\ast }\simeq \lim _{\unicode[STIX]{x1D716}\rightarrow 0}\,\frac{1}{D_{\text{H}}-\text{i}D_{\text{A}}}=\text{i}\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}(D_{\text{H}})+{\mathcal{P}}\,\frac{1}{D_{\text{H}}}, & & \displaystyle\end{eqnarray}$$

where ‘ ${\mathcal{P}}$ ’ denotes the Cauchy principal value.

We then insert (4.6)–(4.10) into (4.3) and integrate over the frequency variable $\unicode[STIX]{x1D714}$ . Afterwards, we expand the Moyal products and brackets using (4.5) and (4.6). Although the GO ansatz (4.8) and the expression for $[D^{-1}]$ in (4.10) involve Dirac delta functions whose derivatives in phase space are not smooth, in § B.1 we show that these singularities can be removed via integration by parts. After some calculations, we obtain the following to lowest order in $\unicode[STIX]{x1D716}$ :

(4.11) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D716}\int \text{d}\unicode[STIX]{x1D714}\,[\unicode[STIX]{x1D716}(D_{\text{H}}\overleftrightarrow{{\mathcal{L}}}J)\,\unicode[STIX]{x1D6FF}(D_{\text{H}})+2D_{\text{A}}\unicode[STIX]{x1D6FF}(D_{\text{H}})J][1+O(\unicode[STIX]{x1D716})]\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D716}_{\text{nl}}^{2}\int \text{d}\unicode[STIX]{x1D714}\,[F\unicode[STIX]{x1D6FF}(D_{\text{H}})-2\text{Im}(\unicode[STIX]{x1D702})J\unicode[STIX]{x1D6FF}(D_{\text{H}})][1+O(\unicode[STIX]{x1D716})]+\unicode[STIX]{x1D716}^{2}\int \text{d}\unicode[STIX]{x1D714}\,S\unicode[STIX]{x1D6FF}(D_{\text{H}})[1+O(\unicode[STIX]{x1D716})].\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Note that in (4.11) the Weyl symbols $W$ , $F$ and $S$ are real because their associated operators are Hermitian.

For simplicity, we consider that the nonlinear scaling parameter $\unicode[STIX]{x1D716}_{\text{nl}}$ and the GO parameter $\unicode[STIX]{x1D716}$ scale as $\unicode[STIX]{x1D716}\sim \unicode[STIX]{x1D716}_{\text{nl}}$ . Upon integrating (4.11) over the frequency and neglecting higher-order terms in $\unicode[STIX]{x1D716}$ , we obtain the collisional wave kinetic equation (cWKE)

(4.12) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}J+\{J,\unicode[STIX]{x1D6FA}\}=-2\unicode[STIX]{x1D707}_{\text{dw}}J+2\unicode[STIX]{x1D6E4}J+S_{\text{ext}}+\unicode[STIX]{x1D716}\,C[J,J], & & \displaystyle\end{eqnarray}$$

where $\{\cdot ,\cdot \}=\overleftarrow{\unicode[STIX]{x2202}_{\boldsymbol{x}}}\cdot \overrightarrow{\unicode[STIX]{x2202}_{\boldsymbol{k}}}-\overleftarrow{\unicode[STIX]{x2202}_{\boldsymbol{k}}}\cdot \overrightarrow{\unicode[STIX]{x2202}_{\boldsymbol{x}}}$ . The wave frequency $\unicode[STIX]{x1D6FA}$ and dissipation term $\unicode[STIX]{x1D6E4}$ are respectively obtained from the lowest-order expansion in $\unicode[STIX]{x1D716}$ of $D_{\text{H}}$ and $D_{\text{A}}$ in (4.9):

(4.13a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FA}(t,y,\boldsymbol{k})\doteq k_{x}U-(\unicode[STIX]{x1D6FD}-U^{\prime \prime })k_{x}/k_{\text{D}}^{2}, & \displaystyle\end{eqnarray}$$

(4.13b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}(t,y,\boldsymbol{k})\doteq -U^{\prime \prime \prime }k_{x}k_{y}/k_{\text{D}}^{4}. & \displaystyle\end{eqnarray}$$

The external source term $S_{\text{ext}}(t,y,\boldsymbol{k})$ appearing in (4.12) is given by

(4.14) $$\begin{eqnarray}\displaystyle S_{\text{ext}}(t,y,\boldsymbol{k})=\int \text{d}^{3}\unicode[STIX]{x1D634}\,\text{d}x\,\text{e}^{\text{i}\unicode[STIX]{x1D62C}\cdot \unicode[STIX]{x1D634}/\unicode[STIX]{x1D716}}\boldsymbol{\langle }\!\langle \widetilde{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D639}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D634})\,\widetilde{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D639}-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D634})\rangle \!\boldsymbol{\rangle }/L_{x}. & & \displaystyle\end{eqnarray}$$

Assuming that $\widetilde{\unicode[STIX]{x1D709}}(t,\boldsymbol{x})$ is white noise leads to

(4.15) $$\begin{eqnarray}\displaystyle \int \text{d}x\,\boldsymbol{\langle }\!\langle \widetilde{\unicode[STIX]{x1D709}}(t,\boldsymbol{x})\widetilde{\unicode[STIX]{x1D709}}(t^{\prime },\boldsymbol{x}^{\prime })\rangle \!\boldsymbol{\rangle }/L_{x}=\unicode[STIX]{x1D6FF}(t-t^{\prime })\,\unicode[STIX]{x1D6EF}\boldsymbol{(}{\textstyle \frac{1}{2}}(y+y^{\prime }),\boldsymbol{x}-\boldsymbol{x}^{\prime }\boldsymbol{)}. & & \displaystyle\end{eqnarray}$$

Thus, the Weyl symbol of the zonal-averaged operator for the stochastic forcing is

(4.16) $$\begin{eqnarray}\displaystyle S_{\text{ext}}(t,y,\boldsymbol{k})=2\int \text{d}^{2}\boldsymbol{s}\,\unicode[STIX]{x1D6EF}(y,\boldsymbol{s})\cos (\boldsymbol{p}\cdot \boldsymbol{s}/\unicode[STIX]{x1D716}). & & \displaystyle\end{eqnarray}$$

Here we used $\unicode[STIX]{x1D6EF}(y,\boldsymbol{s})=\unicode[STIX]{x1D6EF}(y,-\boldsymbol{s})$ , which is due to the fact that $\widetilde{\unicode[STIX]{x1D709}}$ is real by definition.

The term $C[J,J](t,y,\boldsymbol{k})$ in (4.12) represents wave–wave collisions and is given by

(4.17) $$\begin{eqnarray}\displaystyle C[J,J](t,y,\boldsymbol{k})\doteq S_{\text{nl}}[J,J]-2\unicode[STIX]{x1D6FE}_{\text{nl}}[J]J. & & \displaystyle\end{eqnarray}$$

As shown in § B.2, to the leading order in $\unicode[STIX]{x1D716}$ , $\unicode[STIX]{x1D6FE}_{\text{nl}}[J]$ and $S_{\text{nl}}[J,J]$ are given by

(4.18a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FE}_{\text{nl}}[J](t,y,\boldsymbol{k})\doteq \int \frac{\text{d}^{2}\boldsymbol{p}\,\text{d}^{2}\boldsymbol{q}}{(2\unicode[STIX]{x03C0}\unicode[STIX]{x1D716})^{2}}\,\unicode[STIX]{x1D6FF}^{2}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\unicode[STIX]{x1D6E9}(t,y,\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})\,M(\boldsymbol{p},\boldsymbol{q})M(\boldsymbol{p},\boldsymbol{k})\,J(t,y,\boldsymbol{p}), & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(4.18b ) $$\begin{eqnarray}\displaystyle & \displaystyle S_{\text{nl}}[J,J](t,y,\boldsymbol{k})\doteq \int \frac{\text{d}^{2}\boldsymbol{p}\,\text{d}^{2}\boldsymbol{q}}{(2\unicode[STIX]{x03C0}\unicode[STIX]{x1D716})^{2}}\,\unicode[STIX]{x1D6FF}^{2}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\,\unicode[STIX]{x1D6E9}(t,y,\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})\,|M(\boldsymbol{p},\boldsymbol{q})|^{2}\,J(t,y,\boldsymbol{p})J(t,y,\boldsymbol{q}). & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

$\unicode[STIX]{x1D6E9}(t,y,\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})\doteq \unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}(\unicode[STIX]{x0394}\unicode[STIX]{x1D6FA})$

(4.19) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0394}\unicode[STIX]{x1D6FA}(t,y,\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})\doteq \unicode[STIX]{x1D6FA}(t,y,\boldsymbol{k})-\unicode[STIX]{x1D6FA}(t,y,\boldsymbol{p})-\unicode[STIX]{x1D6FA}(t,y,\boldsymbol{q}). & & \displaystyle\end{eqnarray}$$

One can identify $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FA}=0$ as the frequency-resonance condition. Finally, the kernel $M(\boldsymbol{p},\boldsymbol{q})$ in (4.18) is

(4.20) $$\begin{eqnarray}\displaystyle M(\boldsymbol{p},\boldsymbol{q})\doteq \boldsymbol{e}_{z}\cdot (\boldsymbol{p}\times \boldsymbol{q})(q_{\text{D}}^{-2}-p_{\text{D}}^{-2}). & & \displaystyle\end{eqnarray}$$

4.3 Dynamics of the ZF velocity $U$

Returning to (2.19) for the zonal flow velocity, the term on the right-hand side can be rewritten in terms of the Wigner function $W$ by using (A 4). One has

(4.21) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}U+\unicode[STIX]{x1D707}_{\text{zf}}U & = & \displaystyle \unicode[STIX]{x1D716}_{\text{nl}}^{2}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\int \frac{\text{d}\unicode[STIX]{x1D714}\,\text{d}^{2}\boldsymbol{k}}{(2\unicode[STIX]{x03C0}\unicode[STIX]{x1D716})^{3}}\,\frac{k_{x}}{k_{\text{D}}^{2}}\star W(t,y,\unicode[STIX]{x1D714},\boldsymbol{k})\star \frac{k_{y}}{k_{\text{D}}^{2}},\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D716}_{\text{nl}}^{2}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\int \frac{\text{d}^{2}\boldsymbol{k}}{(2\unicode[STIX]{x03C0}\unicode[STIX]{x1D716})^{2}}\,\frac{k_{x}}{k_{\text{D}}^{2}}\star J(t,y,\boldsymbol{k})\star \frac{k_{y}}{k_{\text{D}}^{2}},\end{eqnarray}$$

where we used the Moyal product and substituted (4.8). Upon following Ruiz et al. (Reference Ruiz, Parker, Shi and Dodin2016), we substitute $\unicode[STIX]{x1D716}\sim \unicode[STIX]{x1D716}_{\text{nl}}$ and obtain to lowest order in $\unicode[STIX]{x1D716}$ :

(4.22) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}U+\unicode[STIX]{x1D707}_{\text{zf}}U=\unicode[STIX]{x1D716}^{2}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\int \frac{\text{d}^{2}\boldsymbol{k}}{(2\unicode[STIX]{x03C0}\unicode[STIX]{x1D716})^{2}}\,\frac{k_{x}k_{y}}{k_{\text{D}}^{4}}J(t,y,\boldsymbol{k}). & & \displaystyle\end{eqnarray}$$