2 Two-dimensional flow over topography on a generalized $\unicode[STIX]{x1D6FD}$ -plane

For this study of the development and performance of three versions of the MIC model we employ the same equations for two-dimensional flow over topography on a generalized $\unicode[STIX]{x1D6FD}$ -plane as introduced by FO05. Again the streamfunction is written in the form $\unicode[STIX]{x1D6F9}=\unicode[STIX]{x1D713}-Uy$ where the ‘small scales’ are determined by $\unicode[STIX]{x1D713}$ and the large-scale westerly flow by $U$ .

2.1 Barotropic vorticity equation for the small scales

The ‘small scales’ evolve according to the barotropic vorticity equation in the presence of the large-scale flow and topography,

(2.1a ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}{\unicode[STIX]{x2202}t}=-J(\unicode[STIX]{x1D713}-Uy,\unicode[STIX]{x1D701}+h+\unicode[STIX]{x1D6FD}y+k_{0}^{2}Uy)+\hat{\unicode[STIX]{x1D708}}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D701}+f^{0}.\end{eqnarray}$$

Here, $\unicode[STIX]{x1D701}$ is the vorticity, $h$ is the scaled topography, $\hat{\unicode[STIX]{x1D708}}$ is the viscosity, $f^{0}$ is a forcing function, $\unicode[STIX]{x1D6FD}$ is the beta effect and $k_{0}$ is a wavenumber that, on a sphere, would determine the strength of the vorticity of the solid-body rotation. This wavenumber can be made as small as desirable to recover the standard $\unicode[STIX]{x1D6FD}$ -plane equations. However, the advantage of retaining a small $k_{0}$ is that it allows us to combine the spectral equations for the small and large scales into a single elegant form. As shown in FO05, there is then a one-to-one correspondence between the spherical geometry and $\unicode[STIX]{x1D6FD}$ -plane equations that extends to the statistical mechanics equilibrium solutions in the two geometries. The Jacobian is

(2.1b ) $$\begin{eqnarray}J(\unicode[STIX]{x1D713},\unicode[STIX]{x1D701})=\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}x}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}{\unicode[STIX]{x2202}y}-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}y}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}{\unicode[STIX]{x2202}x}\end{eqnarray}$$

and the vorticity is related to the streamfunction through

(2.1c ) $$\begin{eqnarray}\unicode[STIX]{x1D701}=\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D713}\equiv \left(\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}x^{2}}+\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}y^{2}}\right)\unicode[STIX]{x1D713}.\end{eqnarray}$$

2.2 Large-scale flow equation

The large-scale flow $U$ evolves according to the form-drag equation

(2.2) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}U}{\unicode[STIX]{x2202}t}=\frac{1}{(2\unicode[STIX]{x03C0})^{2}}\int _{0}^{2\unicode[STIX]{x03C0}}\text{d}^{2}\boldsymbol{x}h(\boldsymbol{x})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}(\boldsymbol{x})}{\unicode[STIX]{x2202}x}+\unicode[STIX]{x1D6FC}_{U}(\bar{U}-U).\end{eqnarray}$$

The integrations of equations (2.1) and (2.2) are carried out for flows on the doubly periodic plane $0\leqslant x\leqslant 2\unicode[STIX]{x03C0},0\leqslant y\leqslant 2\unicode[STIX]{x03C0}$ . Here $\boldsymbol{x}=(x,y)$ and the flow $U$ is forced by relaxing it towards $\bar{U}$ with relaxation coefficient $\unicode[STIX]{x1D6FC}_{U}$ .

3 Spectral equations

Spectral equations corresponding to the system in § 2 are derived by expanding each of the ‘small-scale’ functions in a Fourier series. For example,

(3.1a ) $$\begin{eqnarray}\unicode[STIX]{x1D701}(\boldsymbol{x},t)=\mathop{\sum }_{\boldsymbol{k}\in \boldsymbol{R}}\unicode[STIX]{x1D701}_{\boldsymbol{k}}(t)\exp (\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}),\end{eqnarray}$$

where

(3.1b ) $$\begin{eqnarray}\unicode[STIX]{x1D701}_{\boldsymbol{k}}(t)=\frac{1}{(2\unicode[STIX]{x03C0})^{2}}\int _{o}^{2\unicode[STIX]{x03C0}}\text{d}^{2}\boldsymbol{x}\unicode[STIX]{x1D701}(\boldsymbol{x},t)\exp (-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}),\end{eqnarray}$$

$\boldsymbol{x}=(x,y),\boldsymbol{k}=(k_{x},k_{y}),k=(k_{x}^{2}+k_{y}^{2})^{1/2},\unicode[STIX]{x1D701}_{-\boldsymbol{k}}=\unicode[STIX]{x1D701}_{\boldsymbol{k}}^{\ast }$ and $\boldsymbol{R}$ is a circular domain in wavenumber space excluding the origin $\mathbf{0}$ . As noted in FO05, it is possible to combine the spectral equations for the ‘small scales’ with the spectral representation of the form drag equation by defining

(3.2a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D701}_{-\mathbf{0}}=\text{i}k_{0}U,\quad \unicode[STIX]{x1D701}_{\mathbf{0}}=\unicode[STIX]{x1D701}_{-\mathbf{0}}^{\ast },\end{eqnarray}$$

as the zero wavenumber spectral component. In addition, the interaction coefficients $A(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})$ and $K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})$ need to be generalized as summarized in appendix A. Thus, the spectral form of the vorticity equation may be written as in the same form as for flows on a non-rotating domain (F99; OF04),

(3.3) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D708}_{0}(\boldsymbol{k})k^{2}\right)\unicode[STIX]{x1D701}_{\boldsymbol{k}}(t)=\mathop{\sum }_{\boldsymbol{p}\in \boldsymbol{T}}\mathop{\sum }_{\boldsymbol{q}\in \boldsymbol{T}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})[K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})\unicode[STIX]{x1D701}_{-\boldsymbol{p}}\unicode[STIX]{x1D701}_{-\boldsymbol{q}}+A(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})\unicode[STIX]{x1D701}_{-\boldsymbol{p}}h_{-\boldsymbol{q}}]+f_{\boldsymbol{k}}^{0},\end{eqnarray}$$

where $\boldsymbol{T}=\boldsymbol{R}\cup \mathbf{0}$ . From (A 1c ), $\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})=1$ if $\boldsymbol{k}+\boldsymbol{p}+\boldsymbol{q}=0$ and otherwise is zero. In addition

(3.4) $$\begin{eqnarray}\unicode[STIX]{x1D708}_{0}(\boldsymbol{k})k^{2}=\hat{\unicode[STIX]{x1D708}}k^{2}+\text{i}\unicode[STIX]{x1D714}_{\boldsymbol{ k}}\end{eqnarray}$$

and the Rossby wave frequency

(3.5) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{\boldsymbol{k}}=-\frac{\unicode[STIX]{x1D6FD}k_{x}}{k^{2}}.\end{eqnarray}$$

The $\boldsymbol{k}=\mathbf{0}$ components of $f_{\boldsymbol{k}}^{0}$ and $\unicode[STIX]{x1D708}_{0}(\boldsymbol{k})$ are defined by

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle f_{\mathbf{0}}^{0}=\unicode[STIX]{x1D6FC}_{U}\bar{\unicode[STIX]{x1D701}}_{\mathbf{0}}, & \displaystyle\end{eqnarray}$$

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D708}_{0}(\mathbf{0})k_{0}^{2}=\unicode[STIX]{x1D6FC}_{U}. & \displaystyle\end{eqnarray}$$

4 QDIA closure equations

We consider next an ensemble of flows satisfying (3.3) where the ensemble mean is denoted by $\langle \unicode[STIX]{x1D701}_{\boldsymbol{k}}\rangle$ and angle brackets denote expectation value. Then we can express the vorticity component for a given realization by

(4.1) $$\begin{eqnarray}\unicode[STIX]{x1D701}_{\boldsymbol{k}}=\langle \unicode[STIX]{x1D701}_{\boldsymbol{k}}\rangle +\tilde{\unicode[STIX]{x1D701}}_{\boldsymbol{k}},\end{eqnarray}$$

where $\tilde{\unicode[STIX]{x1D701}}_{\boldsymbol{k}}$ denotes the deviation from the ensemble mean. The spectral equations (3.3) can then be expressed in terms of $\langle \unicode[STIX]{x1D701}_{\boldsymbol{k}}\rangle$ and $\tilde{\unicode[STIX]{x1D701}}_{\boldsymbol{k}}$ as follows:

(4.2a ) $$\begin{eqnarray}\displaystyle \left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D708}_{0}(\boldsymbol{k})k^{2}\right)\langle \unicode[STIX]{x1D701}_{\boldsymbol{ k}}\rangle & = & \displaystyle \mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})[\!K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})\{\langle \unicode[STIX]{x1D701}_{-\boldsymbol{p}}\rangle \langle \unicode[STIX]{x1D701}_{-\boldsymbol{q}}\rangle +C_{-\boldsymbol{p},-\boldsymbol{q}}(t,t)\}\nonumber\\ \displaystyle & & \displaystyle +\,A(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})\langle \unicode[STIX]{x1D701}_{-\boldsymbol{p}}\rangle h_{-\boldsymbol{q}}\! ]+\,\bar{f}_{\boldsymbol{k}}^{0},\end{eqnarray}$$

and

(4.2b ) $$\begin{eqnarray}\displaystyle \left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D708}_{0}(\boldsymbol{k})k^{2}\right)\tilde{\unicode[STIX]{x1D701}}_{\boldsymbol{ k}} & = & \displaystyle \mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})[\!K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})\{\langle \unicode[STIX]{x1D701}_{-\boldsymbol{p}}\rangle \tilde{\unicode[STIX]{x1D701}}_{-\boldsymbol{q}}+\tilde{\unicode[STIX]{x1D701}}_{-\boldsymbol{p}}\langle \unicode[STIX]{x1D701}_{-\boldsymbol{q}}\rangle \nonumber\\ \displaystyle & & \displaystyle +\,\tilde{\unicode[STIX]{x1D701}}_{-\boldsymbol{p}}\tilde{\unicode[STIX]{x1D701}}_{-\boldsymbol{q}}-C_{-\boldsymbol{p},-\boldsymbol{q}}(t,t)\!\}+A(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})\tilde{\unicode[STIX]{x1D701}}_{-\boldsymbol{p}}h_{-\boldsymbol{q}}\! ]+\tilde{f}_{\boldsymbol{k}}^{0}.\end{eqnarray}$$

Here, and in subsequent equations, the wave vectors lie in the larger $\boldsymbol{T}$ domain and

(4.3a ) $$\begin{eqnarray}\displaystyle f_{\boldsymbol{k}}^{0}=\bar{f}_{\boldsymbol{k}}^{0}+\tilde{f}_{\boldsymbol{k}}^{0}, & & \displaystyle\end{eqnarray}$$

with

(4.3b ) $$\begin{eqnarray}\displaystyle \bar{f}_{\boldsymbol{k}}^{0}=\langle f_{\boldsymbol{k}}^{0}\rangle & & \displaystyle\end{eqnarray}$$

and

(4.3c ) $$\begin{eqnarray}\displaystyle C_{-\boldsymbol{p},-\boldsymbol{q}}(t,s)=\langle \tilde{\unicode[STIX]{x1D701}}_{-\boldsymbol{p}}(t)\tilde{\unicode[STIX]{x1D701}}_{-\boldsymbol{q}}(s)\rangle , & & \displaystyle\end{eqnarray}$$

are two-time covariance (and also two-point cumulant) matrix elements.

We briefly outline the derivation of the QDIA closure equations for barotropic flow over topography by employing the expressions for the off-diagonal elements of the two- and three-point cumulants and response functions, in terms of the diagonal elements for the QDIA closure equations, as described in appendix B. First, to close (4.2a ) we need the expression for the two-point cumulant $C_{-\boldsymbol{p},-\boldsymbol{q}}(t,t)$ in (B 1). This results in the expression

(4.4) $$\begin{eqnarray}\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})C_{-\boldsymbol{p},-\boldsymbol{q}}(t,t)=-\!\int _{t_{o}}^{t}\text{d}s\unicode[STIX]{x1D702}_{\boldsymbol{ k}}(t,s)\langle \unicode[STIX]{x1D701}_{\boldsymbol{k}}(s)\rangle +f_{\boldsymbol{k}}^{h}(t),\end{eqnarray}$$

where

(4.5a ) $$\begin{eqnarray}\unicode[STIX]{x1D702}_{\boldsymbol{k}}(t,s)=-4\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(-\boldsymbol{p},-\boldsymbol{q},-\boldsymbol{k})R_{-\boldsymbol{p}}(t,s)C_{-\boldsymbol{q}}(t,s),\end{eqnarray}$$

and

(4.5b ) $$\begin{eqnarray}f_{\boldsymbol{k}}^{h}(t)\equiv f_{\boldsymbol{ k}}^{\unicode[STIX]{x1D712}}(t)=h_{\boldsymbol{ k}}\int _{t_{o}}^{t}\text{d}s\unicode[STIX]{x1D712}_{\boldsymbol{ k}}(t,s),\end{eqnarray}$$

with

(4.5c ) $$\begin{eqnarray}\unicode[STIX]{x1D712}_{\boldsymbol{k}}(t,s)=2\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})A(-\boldsymbol{p},-\boldsymbol{q},-\boldsymbol{k})R_{-\boldsymbol{p}}(t,s)C_{-\boldsymbol{q}}(t,s).\end{eqnarray}$$

Here, $\unicode[STIX]{x1D702}_{\boldsymbol{k}}(t,s)$ is the nonlinear damping, $\unicode[STIX]{x1D712}_{\boldsymbol{k}}(t,s)$ is a measure of the strength of the interaction of the transient eddies with the topography and $f_{\boldsymbol{k}}^{h}(t)$ is the eddy–topographic force. The response function is defined for $t\geqslant t^{\prime }$ through the functional derivative

(4.6a ) $$\begin{eqnarray}\displaystyle \tilde{R}_{\boldsymbol{k},\boldsymbol{l}}(t,t^{\prime }) & = & \displaystyle \frac{\unicode[STIX]{x1D6FF}\tilde{\unicode[STIX]{x1D701}}_{\boldsymbol{k}}(t)}{\unicode[STIX]{x1D6FF}\tilde{f}_{\boldsymbol{l}}^{0}(t^{\prime })},\end{eqnarray}$$

(4.6b ) $$\begin{eqnarray}\displaystyle R_{\boldsymbol{k},\boldsymbol{l}}(t,t^{\prime }) & = & \displaystyle \langle \tilde{R}_{\boldsymbol{k},\boldsymbol{l}}(t,t^{\prime })\rangle .\end{eqnarray}$$

In (4.5) we have also used the shortened notation

(4.6c ) $$\begin{eqnarray}\displaystyle R_{\boldsymbol{k}}(t,t^{\prime }) & \equiv & \displaystyle R_{\boldsymbol{k},\boldsymbol{k}}(t,t^{\prime }),\end{eqnarray}$$

(4.6d ) $$\begin{eqnarray}\displaystyle C_{\boldsymbol{k}}(t,t^{\prime }) & \equiv & \displaystyle C_{\boldsymbol{k},-\boldsymbol{k}}(t,t^{\prime }).\end{eqnarray}$$

Substituting (4.4) into (4.2a ) yields

(4.7) $$\begin{eqnarray}\displaystyle \left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D708}_{0}(\boldsymbol{k})k^{2}\right)\langle \unicode[STIX]{x1D701}_{\boldsymbol{ k}}(t)\rangle & = & \displaystyle \mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})A(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})\langle \unicode[STIX]{x1D701}_{-\boldsymbol{p}}(t)\rangle h_{-\boldsymbol{q}}\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})\langle \unicode[STIX]{x1D701}_{-\boldsymbol{p}}(t)\rangle \langle \unicode[STIX]{x1D701}_{-\boldsymbol{q}}(t)\rangle \nonumber\\ \displaystyle & & \displaystyle -\,\int _{t_{o}}^{t}\text{d}s\unicode[STIX]{x1D702}_{\boldsymbol{k}}(t,s)\langle \unicode[STIX]{x1D701}_{\boldsymbol{k}}(s)\rangle +f_{\boldsymbol{k}}^{h}(t)+\bar{f}_{\boldsymbol{k}}^{0}(t).\end{eqnarray}$$

From (4.2b ), we can obtain an equation for the diagonal two-time cumulant, needed in (4.5), by multiplying by $\tilde{\unicode[STIX]{x1D701}}_{-\boldsymbol{k}}(t^{\prime })$ . Then taking the expectation value we have

(4.8a ) $$\begin{eqnarray}\displaystyle & & \displaystyle \left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D708}_{0}(\boldsymbol{k})k^{2}\right)C_{\boldsymbol{ k}}(t,t^{\prime })\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})A(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})C_{-\boldsymbol{p},-\boldsymbol{k}}(t,t^{\prime })h_{-\boldsymbol{q}}+\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,[\langle \unicode[STIX]{x1D701}_{-\boldsymbol{p}}(t)\rangle C_{-\boldsymbol{q},-\boldsymbol{k}}(t,t^{\prime })+C_{-\boldsymbol{p},-\boldsymbol{k}}(t,t^{\prime })\langle \unicode[STIX]{x1D701}_{-\boldsymbol{q}}(t)\rangle +\langle \tilde{\unicode[STIX]{x1D701}}_{-\boldsymbol{p}}(t)\tilde{\unicode[STIX]{x1D701}}_{-\boldsymbol{q}}(t)\tilde{\unicode[STIX]{x1D701}}_{-\boldsymbol{k}}(t^{\prime })\rangle ]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\int _{t_{0}}^{t^{\prime }}\text{d}sF_{\boldsymbol{k}}^{0}(t,s)R_{-\boldsymbol{k}}(t^{\prime },s),\end{eqnarray}$$

where $t>t^{\prime }$ and $C_{\boldsymbol{k}}(t,t^{\prime })=C_{-\boldsymbol{k}}(t^{\prime },t)$ for $t^{\prime }>t$ and

(4.8b ) $$\begin{eqnarray}\displaystyle F_{\boldsymbol{k}}^{0}(t,s)=\langle \,\tilde{f}_{\boldsymbol{k}}^{0}(t)\tilde{f}_{\boldsymbol{k}}^{0\ast }(s)\rangle . & & \displaystyle\end{eqnarray}$$

Again, equation (B 1) can be used to express the off-diagonal elements of the two-time cumulant in terms of the diagonal elements. In the quasi-diagonal approximation, the treatment of the three-point cumulant in (4.8) is the same as in the standard DIA closure for homogeneous turbulence (Kraichnan Reference Kraichnan1959a ; Frederiksen et al. Reference Frederiksen, Davies and Bell1994; Frederiksen (Reference Frederiksen, Ball and Akhmediev2003) contains a simple derivation) and is given in (B 3). Thus,

(4.9) $$\begin{eqnarray}\displaystyle \left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D708}_{0}(\boldsymbol{k})k^{2}\right)C_{\boldsymbol{ k}}(t,t^{\prime }) & = & \displaystyle \int _{t_{0}}^{t^{\prime }}\text{d}s(S_{\boldsymbol{k}}(t,s)+P_{\boldsymbol{k}}(t,s)+F_{\boldsymbol{k}}^{0}(t,s))R_{-\boldsymbol{k}}(t^{\prime },s)\nonumber\\ \displaystyle & & \displaystyle -\,\int _{t_{0}}^{t}\text{d}s(\unicode[STIX]{x1D702}_{\boldsymbol{k}}(t,s)+\unicode[STIX]{x03C0}_{\boldsymbol{k}}(t,s))C_{-\boldsymbol{k}}(t^{\prime },s).\end{eqnarray}$$

In (4.9),

(4.10a ) $$\begin{eqnarray}\displaystyle S_{\boldsymbol{k}}(t,s) & = & \displaystyle 2\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})C_{-\boldsymbol{p}}(t,s)C_{-\boldsymbol{q}}(t,s),\qquad\end{eqnarray}$$

(4.10b ) $$\begin{eqnarray}\displaystyle P_{\boldsymbol{k}}(t,s) & = & \displaystyle \mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})C_{-\boldsymbol{p}}(t,s)[2K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})\langle \unicode[STIX]{x1D701}_{-\boldsymbol{q}}(t)\rangle +A(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})h_{-\boldsymbol{q}}]\nonumber\\ \displaystyle & & \displaystyle \times \,[2K(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})\langle \unicode[STIX]{x1D701}_{\boldsymbol{q}}(s)\rangle +A(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})h_{\boldsymbol{q}}],\end{eqnarray}$$

(4.10c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x03C0}_{\boldsymbol{k}}(t,s) & = & \displaystyle -\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})R_{-\boldsymbol{p}}(t,s)[2K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})\langle \unicode[STIX]{x1D701}_{-\boldsymbol{q}}(t)\rangle +A(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})h_{-\boldsymbol{q}}]\nonumber\\ \displaystyle & & \displaystyle \times \,[2K(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})\langle \unicode[STIX]{x1D701}_{\boldsymbol{q}}(s)\rangle +A(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})h_{\boldsymbol{q}}],\end{eqnarray}$$

$\unicode[STIX]{x1D702}_{\boldsymbol{k}}(t,s)$

$F_{\boldsymbol{k}}^{0}(t,s)$

$\unicode[STIX]{x03C0}_{\boldsymbol{k}}(t,s)$

$S_{\boldsymbol{k}}(t,s)$

$P_{\boldsymbol{k}}(t,s)$

The equation for the diagonal response function is derived in a similar way using (B 2) and (B 4). We find (F99; FO05)

(4.11) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D708}_{0}(\boldsymbol{k})k^{2}\right)R_{\boldsymbol{ k}}(t,t^{\prime })+\int _{t^{\prime }}^{t}\text{d}s(\unicode[STIX]{x1D702}_{\boldsymbol{ k}}(t,s)+\unicode[STIX]{x03C0}_{\boldsymbol{k}}(t,s))R_{\boldsymbol{k}}(s,t^{\prime })=\unicode[STIX]{x1D6FF}(t-t^{\prime })\end{eqnarray}$$

for $t\geqslant t^{\prime }$ , where the delta function implies that $R_{\boldsymbol{k}}(t,t)=1$ . Finally, the single-time cumulant equation takes the form

(4.12) $$\begin{eqnarray}\displaystyle \left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+2Re\unicode[STIX]{x1D708}_{0}(\boldsymbol{k})k^{2}\right)C_{\boldsymbol{ k}}(t,t) & = & \displaystyle 2Re\int _{t_{0}}^{t}\text{d}s(S_{\boldsymbol{ k}}(t,s)+P_{\boldsymbol{k}}(t,s)+F_{\boldsymbol{k}}^{0}(t,s))R_{-\boldsymbol{k}}(t,s)\nonumber\\ \displaystyle & & \displaystyle -\,2Re\int _{t_{0}}^{t}\text{d}s(\unicode[STIX]{x1D702}_{\boldsymbol{k}}(t,s)+\unicode[STIX]{x03C0}_{\boldsymbol{k}}(t,s))C_{-\boldsymbol{k}}(t,s).\end{eqnarray}$$

The CUQDIA closure, the cumulant update version of the QDIA, uses non-Gaussian and inhomogeneous cumulants, accumulated in the integrations, in the initial conditions of the periodic restarts, as described in the Appendix of FO05. For the sake of brevity these terms are not included here since they are not needed for the development of our three MIC models that follows next.

5 MIC equations

In this section, we formulate three slightly different versions of the MIC equations. We employ three commonly used versions of the FDT together with a Markovian version of the response function equation in the derivation. We begin by reformulating the single-time covariance equation (4.9), which we first write in the form

(5.1) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}C_{\boldsymbol{k}}(t,t) & = & \displaystyle 2Re[{\mathcal{F}}_{\boldsymbol{k}}^{S}(t)+{\mathcal{F}}_{\boldsymbol{ k}}^{PA}(t)+{\mathcal{F}}_{\boldsymbol{ k}}^{PB}(t)+{\mathcal{F}}_{\boldsymbol{ k}}^{PC}(t)+{\mathcal{F}}_{\boldsymbol{ k}}^{PD}(t)+{\mathcal{F}}_{\boldsymbol{ k}}^{0}(t)]\nonumber\\ \displaystyle & & \displaystyle -\,2Re[{\mathcal{N}}_{\boldsymbol{k}}^{\unicode[STIX]{x1D702}}(t)+{\mathcal{N}}_{\boldsymbol{k}}^{\unicode[STIX]{x03C0}A}(t)+{\mathcal{N}}_{\boldsymbol{k}}^{\unicode[STIX]{x03C0}B}(t)+{\mathcal{N}}_{\boldsymbol{k}}^{\unicode[STIX]{x03C0}C}(t)+{\mathcal{N}}_{\boldsymbol{k}}^{\unicode[STIX]{x03C0}D}(t)+{\mathcal{N}}_{\boldsymbol{k}}^{0}].\end{eqnarray}$$

Here, we have first split up the $P_{\boldsymbol{k}}(t,s)$ and $\unicode[STIX]{x03C0}_{\boldsymbol{k}}(t,s)$ terms into their mean vorticity, topographic and cross terms. Simple algebra shows that the ${\mathcal{F}}_{\boldsymbol{k}}(t)$ and ${\mathcal{N}}_{\boldsymbol{k}}(t)$ functions have the following expressions:

(5.2a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{F}}_{\boldsymbol{k}}^{S}(t)=2\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})\unicode[STIX]{x1D6E5}^{S}(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})(t),\qquad & \displaystyle\end{eqnarray}$$

(5.2b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E5}^{S}(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})(t)=\int _{t_{0}}^{t}\text{d}sR_{-\boldsymbol{k}}(t,s)C_{-\boldsymbol{p}}(t,s)C_{-\boldsymbol{q}}(t,s), & \displaystyle\end{eqnarray}$$

(5.2c ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{F}}_{\boldsymbol{k}}^{PA}(t)=4\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})\unicode[STIX]{x1D6E5}^{PA}(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})(t),\quad & \displaystyle\end{eqnarray}$$

(5.2d ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E5}^{PA}(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})(t)=\langle \unicode[STIX]{x1D701}_{-\boldsymbol{q}}(t)\rangle \int _{t_{0}}^{t}\text{d}sR_{-\boldsymbol{k}}(t,s)C_{-\boldsymbol{p}}(t,s)\langle \unicode[STIX]{x1D701}_{\boldsymbol{q}}(s)\rangle , & \displaystyle\end{eqnarray}$$

(5.2e ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{F}}_{\boldsymbol{k}}^{PB}(t)=\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})A(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})A(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})\unicode[STIX]{x1D6E5}^{PB}(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})(t),\qquad & \displaystyle\end{eqnarray}$$

(5.2f ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E5}^{PB}(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})(t)=h_{-\boldsymbol{q}}h_{\boldsymbol{q}}\int _{t_{0}}^{t}\text{d}sR_{-\boldsymbol{k}}(t,s)C_{-\boldsymbol{p}}(t,s), & \displaystyle\end{eqnarray}$$

(5.2g ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{F}}_{\boldsymbol{k}}^{PC}(t)=2\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})A(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})\unicode[STIX]{x1D6E5}^{PC}(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})(t),\qquad & \displaystyle\end{eqnarray}$$

(5.2h ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E5}^{PC}(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})(t)=\langle \unicode[STIX]{x1D701}_{-\boldsymbol{q}}(t)\rangle h_{\boldsymbol{q}}\int _{t_{0}}^{t}\text{d}sR_{-\boldsymbol{k}}(t,s)C_{-\boldsymbol{p}}(t,s), & \displaystyle\end{eqnarray}$$

(5.2i ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{F}}_{\boldsymbol{k}}^{PD}(t)=2\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})A(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})\unicode[STIX]{x1D6E5}^{PD}(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})(t),\qquad & \displaystyle\end{eqnarray}$$

(5.2j ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E5}^{PD}(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})(t)=h_{-\boldsymbol{q}}\int _{t_{0}}^{t}\text{d}sR_{-\boldsymbol{k}}(t,s)C_{-\boldsymbol{p}}(t,s)\langle \unicode[STIX]{x1D701}_{\boldsymbol{q}}(s)\rangle , & \displaystyle\end{eqnarray}$$

(5.2k ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{F}}_{\boldsymbol{k}}^{0}(t)=\int _{t_{0}}^{t}\text{d}sF_{\boldsymbol{k}}^{0}(t,s)R_{-\boldsymbol{k}}(t,s). & \displaystyle\end{eqnarray}$$

(5.3a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{N}}_{\boldsymbol{k}}^{\unicode[STIX]{x1D702}}(t)=-4\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(-\boldsymbol{p},-\boldsymbol{q},-\boldsymbol{k})\unicode[STIX]{x1D6E5}^{\unicode[STIX]{x1D702}}(-\boldsymbol{p},-\boldsymbol{q},-\boldsymbol{k})(t),\qquad & \displaystyle\end{eqnarray}$$

(5.3b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E5}^{\unicode[STIX]{x1D702}}(-\boldsymbol{p},-\boldsymbol{q},-\boldsymbol{k})(t)=\int _{t_{0}}^{t}\text{d}sR_{-\boldsymbol{p}}(t,s)C_{-\boldsymbol{q}}(t,s)C_{-\boldsymbol{k}}(t,s)\equiv \unicode[STIX]{x1D6E5}^{S}(-\boldsymbol{p},-\boldsymbol{q},-\boldsymbol{k})(t),\qquad & \displaystyle\end{eqnarray}$$

(5.3c ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{N}}_{\boldsymbol{k}}^{\unicode[STIX]{x03C0}A}(t)=-4\mathop{\sum }_{p}\mathop{\sum }_{q}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})\unicode[STIX]{x1D6E5}^{\unicode[STIX]{x03C0}A}(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})(t),\qquad & \displaystyle\end{eqnarray}$$

(5.3d ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E5}^{\unicode[STIX]{x03C0}A}(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})(t)=\langle \unicode[STIX]{x1D701}_{-\boldsymbol{q}}(t)\rangle \int _{t_{0}}^{t}\text{d}sR_{-\boldsymbol{p}}(t,s)C_{-\boldsymbol{k}}(t,s)\langle \unicode[STIX]{x1D701}_{\boldsymbol{q}}(s)\rangle & \displaystyle \nonumber\\ \displaystyle & \displaystyle \hphantom{\unicode[STIX]{x1D6E5}^{\unicode[STIX]{x03C0}A},}\equiv \unicode[STIX]{x1D6E5}^{PA}(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})(t), & \displaystyle\end{eqnarray}$$

(5.3e ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{N}}_{\boldsymbol{k}}^{\unicode[STIX]{x03C0}B}(t)=-\mathop{\sum }_{p}\mathop{\sum }_{q}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})A(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})A(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})\unicode[STIX]{x1D6E5}^{\unicode[STIX]{x03C0}B}(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})(t),\qquad & \displaystyle\end{eqnarray}$$

(5.3f ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E5}^{\unicode[STIX]{x03C0}B}(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})(t)=h_{-\boldsymbol{q}}h_{\boldsymbol{q}}\int _{t_{0}}^{t}\text{d}sC_{-\boldsymbol{k}}(t,s)R_{-\boldsymbol{p}}(t,s)\equiv \unicode[STIX]{x1D6E5}^{PB}(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})(t),\qquad & \displaystyle\end{eqnarray}$$

(5.3g ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{N}}_{\boldsymbol{k}}^{\unicode[STIX]{x03C0}C}(t)=-2\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})A(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})\unicode[STIX]{x1D6E5}^{\unicode[STIX]{x03C0}C}(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})(t),\qquad & \displaystyle\end{eqnarray}$$

(5.3h ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E5}^{\unicode[STIX]{x03C0}C}(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})(t)=\langle \unicode[STIX]{x1D701}_{-\boldsymbol{q}}(t)\rangle h_{\boldsymbol{q}}\int _{t_{0}}^{t}\text{d}sR_{-\boldsymbol{p}}(t,s)C_{-\boldsymbol{k}}(t,s) & \displaystyle \nonumber\\ \displaystyle & \displaystyle \hphantom{\unicode[STIX]{x1D6E5}^{\unicode[STIX]{x03C0}C}(t)=}\equiv \unicode[STIX]{x1D6E5}^{PC}(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})(t), & \displaystyle\end{eqnarray}$$

(5.3i ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{N}}_{\boldsymbol{k}}^{\unicode[STIX]{x03C0}D}(t)=-2\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})A(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})\unicode[STIX]{x1D6E5}^{\unicode[STIX]{x03C0}D}(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})(t),\qquad & \displaystyle\end{eqnarray}$$

(5.3j ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E5}^{\unicode[STIX]{x03C0}D}(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})(t)=h_{-\boldsymbol{q}}\int _{t_{0}}^{t}\text{d}sR_{-\boldsymbol{p}}(t,s)C_{-\boldsymbol{k}}(t,s)\langle \unicode[STIX]{x1D701}_{\boldsymbol{q}}(s)\rangle & \displaystyle \nonumber\\ \displaystyle & \displaystyle \hphantom{\unicode[STIX]{x1D6E5}^{\unicode[STIX]{x03C0}D}(t)=,}\equiv \unicode[STIX]{x1D6E5}^{PD}(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})(t), & \displaystyle\end{eqnarray}$$

(5.3k ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{N}}_{\boldsymbol{k}}^{0}(t)=\unicode[STIX]{x1D708}_{0}(\boldsymbol{k})k^{2}C_{\boldsymbol{k}}(t,t).\qquad & \displaystyle\end{eqnarray}$$

(5.4) $$\begin{eqnarray}\displaystyle C_{\boldsymbol{k}}(t,t^{\prime })\equiv [C_{\boldsymbol{k}}(t,t)]^{1-X}R_{\boldsymbol{k}}(t,t^{\prime })[C_{\boldsymbol{k}}(t^{\prime },t^{\prime })]^{X} & & \displaystyle\end{eqnarray}$$

for $t\geqslant t^{\prime }$ and $C_{\boldsymbol{k}}(t,t^{\prime })=C_{-\boldsymbol{k}}(t^{\prime },t)$ for $t^{\prime }>t$ . Here $X=0$ corresponds to the current-time FDT (see Frederiksen & Davies (Reference Frederiksen and Davies1997, equation (A.16)) and references therein) usually used in the EDQNM, $X=1/2$ is the correlation FDT in Bowman (Reference Bowman, Krommes and Ottaviani1993, equation (61)) and $X=1$ is the prior-time FDT (see Carnevale & Frederiksen (Reference Carnevale and Frederiksen1983a , equation (3.5)) and references therein). Bowman (Reference Bowman, Krommes and Ottaviani1993) pointed out that in the presence of wave phenomena it is possible for the forms with $X=0$ and $X=1$ to lead to unphysical results whereas the form with $X=1/2$ (together with Markovian response functions with positive damping) will always be realizable. Their demonstration showing that $C_{\boldsymbol{k}}(t,t)$ is real and non-negative when $X=1/2$ , in their equation (65), applies equally in the QDIA inhomogeneous formalism. This follows by replacing their eddy–eddy nonlinear noise $\mathscr{F}_{\boldsymbol{k}}$ by our total nonlinear noise $[S_{\boldsymbol{k}}+P_{\boldsymbol{k}}]$ . This is an important point in general, but whether unphysical results ensue will of course also depend on the parameter regime of the flow. For that reason we examine all three cases in this study.

With (5.4) substituted into equations (5.2) and (5.3) it is then possible to express the nonlinear noises and dampings in forms that involve triad relaxation functions $\unicode[STIX]{x1D6E9}^{X}$ , $\unicode[STIX]{x1D6F7}^{X}$ and $\unicode[STIX]{x1D6F9}^{X}$ . Thus,

(5.5a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{F}}_{\boldsymbol{k}}^{S}(t)=2\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q}) & \displaystyle \nonumber\\ \displaystyle & \displaystyle \hphantom{2\sum }\times \,\mathop{C}_{-\boldsymbol{p}}^{1-X}(t,t)C_{-\boldsymbol{q}}^{X}(t,t)\unicode[STIX]{x1D6E9}^{X}(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})(t), & \displaystyle\end{eqnarray}$$

(5.5b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E9}^{X}(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})(t)=\int _{t_{0}}^{t}\text{d}sR_{-\boldsymbol{k}}(t,s)R_{-\boldsymbol{p}}(t,s)R_{-\boldsymbol{q}}(t,s)C_{-\boldsymbol{p}}^{X}(s,s)C_{-\boldsymbol{q}}^{X}(s,s), & \displaystyle\end{eqnarray}$$

(5.5c ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{F}}_{\boldsymbol{k}}^{PA}(t)=4\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q}) & \displaystyle \nonumber\\ \displaystyle & \displaystyle \hphantom{2\sum }\times \,\mathop{C}_{-\boldsymbol{p}}^{1-X}(t,t)\langle \unicode[STIX]{x1D701}_{-\boldsymbol{q}}(t)\rangle \unicode[STIX]{x1D6F7}^{X}(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})(t), & \displaystyle\end{eqnarray}$$

(5.5d ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F7}^{X}(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})(t)=\int _{t_{0}}^{t}\text{d}sR_{-\boldsymbol{k}}(t,s)R_{-\boldsymbol{p}}(t,s)C_{-\boldsymbol{p}}^{X}(s,s)\langle \unicode[STIX]{x1D701}_{\boldsymbol{q}}(s)\rangle , & \displaystyle\end{eqnarray}$$

(5.5e ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{F}}_{\boldsymbol{k}}^{PB}(t)=\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})A(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})A(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q}) & \displaystyle \nonumber\\ \displaystyle & \displaystyle \hphantom{2\sum }\times \,\mathop{C}_{-\boldsymbol{p}}^{1-X}(t,t)h_{-\boldsymbol{q}}h_{\boldsymbol{q}}\unicode[STIX]{x1D6F9}^{X}(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})(t), & \displaystyle\end{eqnarray}$$

(5.5f ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F9}^{X}(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})(t)=\int _{t_{0}}^{t}\text{d}sR_{-\boldsymbol{k}}(t,s)R_{-\boldsymbol{p}}(t,s)C_{-\boldsymbol{p}}^{X}(s,s), & \displaystyle\end{eqnarray}$$

(5.5g ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{F}}_{\boldsymbol{k}}^{PC}(t)=2\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})A(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q}) & \displaystyle \nonumber\\ \displaystyle & \displaystyle \hphantom{\sum \sum }\times \,\mathop{C}_{-\boldsymbol{p}}^{1-X}(t,t)\langle \unicode[STIX]{x1D701}_{-\boldsymbol{q}}(t)\rangle h_{\boldsymbol{q}}\unicode[STIX]{x1D6F9}^{X}(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})(t), & \displaystyle\end{eqnarray}$$

(5.5h ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{F}}_{\boldsymbol{k}}^{PD}(t)=2\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})A(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q}) & \displaystyle \nonumber\\ \displaystyle & \displaystyle \hphantom{2}\times \,C_{-\boldsymbol{p}}^{1-X}(t,t)h_{-\boldsymbol{q}}\unicode[STIX]{x1D6F7}^{X}(-\boldsymbol{k},-\boldsymbol{p},-\boldsymbol{q})(t), & \displaystyle\end{eqnarray}$$

(5.5i ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{F}}_{\boldsymbol{k}}^{0}(t)=\int _{t_{0}}^{t}\text{d}sF_{\boldsymbol{k}}^{0}(t,s)R_{-\boldsymbol{k}}(t,s). & \displaystyle\end{eqnarray}$$

(5.6a ) $$\begin{eqnarray}{\mathcal{N}}_{\boldsymbol{k}}^{\unicode[STIX]{x1D702}}(t)=D_{\boldsymbol{ k}}^{\unicode[STIX]{x1D702}}(t)C_{\boldsymbol{ k}}(t,t),\end{eqnarray}$$

(5.6b ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{D}}_{\boldsymbol{k}}^{\unicode[STIX]{x1D702}}(t)=-4\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(-\boldsymbol{p},-\boldsymbol{q},-\boldsymbol{k}) & \displaystyle \nonumber\\ \displaystyle & \displaystyle \hphantom{\sum }\times \,\mathop{C}_{-\boldsymbol{q}}^{1-X}(t,t)C_{-\boldsymbol{k}}^{-X}(t,t)\unicode[STIX]{x1D6E9}^{X}(-\boldsymbol{p},-\boldsymbol{q},-\boldsymbol{k})(t), & \displaystyle\end{eqnarray}$$

(5.6c ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{N}}_{\boldsymbol{k}}^{\unicode[STIX]{x03C0}A}(t)={\mathcal{D}}_{\boldsymbol{k}}^{\unicode[STIX]{x03C0}A}(t)C_{\boldsymbol{k}}(t,t), & \displaystyle\end{eqnarray}$$

(5.6d ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{D}}_{\boldsymbol{k}}^{\unicode[STIX]{x03C0}A}(t)=-4\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q}) & \displaystyle \nonumber\\ \displaystyle & \displaystyle \hphantom{2-}\times \,C_{-\boldsymbol{k}}^{-X}(t,t)\langle \unicode[STIX]{x1D701}_{-\boldsymbol{q}}(t)\rangle \unicode[STIX]{x1D6F7}^{X}(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})(t), & \displaystyle\end{eqnarray}$$

(5.6e ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{N}}_{\boldsymbol{k}}^{\unicode[STIX]{x03C0}B}(t)=D_{\boldsymbol{k}}^{\unicode[STIX]{x03C0}B}(t)C_{\boldsymbol{k}}(t,t), & \displaystyle\end{eqnarray}$$

(5.6f ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{D}}_{\boldsymbol{k}}^{\unicode[STIX]{x03C0}B}(t)=-\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})A(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})A(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})h_{-\boldsymbol{q}}h_{\boldsymbol{q}} & \displaystyle \nonumber\\ \displaystyle & \displaystyle \times \,C_{-\boldsymbol{k}}^{-X}(t,t)\unicode[STIX]{x1D6F9}^{X}(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})(t),\hphantom{\sum \sum ,} & \displaystyle\end{eqnarray}$$

(5.6g ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{N}}_{\boldsymbol{k}}^{\unicode[STIX]{x03C0}C}(t)={\mathcal{D}}_{\boldsymbol{k}}^{\unicode[STIX]{x03C0}C}(t)C_{\boldsymbol{k}}(t,t), & \displaystyle\end{eqnarray}$$

(5.6h ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{D}}_{\boldsymbol{k}}^{\unicode[STIX]{x03C0}C}(t)=-2\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})A(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})\langle \unicode[STIX]{x1D701}_{-\boldsymbol{q}}(t)\rangle h_{\boldsymbol{q}} & \displaystyle \nonumber\\ \displaystyle & \displaystyle \times \,C_{-\boldsymbol{k}}^{-X}(t,t)\unicode[STIX]{x1D6F9}^{X}(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})(t),\hphantom{2\sum \sum \sum ,} & \displaystyle\end{eqnarray}$$

(5.6i ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{N}}_{\boldsymbol{k}}^{\unicode[STIX]{x03C0}D}(t)={\mathcal{D}}_{\boldsymbol{k}}^{\unicode[STIX]{x03C0}D}(t)C_{\boldsymbol{k}}(t,t), & \displaystyle\end{eqnarray}$$

(5.6j ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{D}}_{\boldsymbol{k}}^{\unicode[STIX]{x03C0}D}(t)=-2\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})A(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})h_{-\boldsymbol{q}} & \displaystyle \nonumber\\ \displaystyle & \displaystyle \times \,C_{-\boldsymbol{k}}^{-X}(t,t)\unicode[STIX]{x1D6F7}^{X}(-\boldsymbol{p},-\boldsymbol{k},-\boldsymbol{q})(t),\hphantom{2\sum \sum } & \displaystyle\end{eqnarray}$$

(5.6k ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{N}}_{\boldsymbol{k}}^{0}(t)={\mathcal{D}}_{\boldsymbol{k}}^{0}(t)C_{\boldsymbol{k}}(t,t), & \displaystyle\end{eqnarray}$$

(5.6l ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{D}}_{\boldsymbol{k}}^{0}(t)=\unicode[STIX]{x1D708}_{0}(\boldsymbol{k})k^{2}. & \displaystyle\end{eqnarray}$$

Note that $C_{\boldsymbol{k}}(t,t)=C_{-\boldsymbol{k}}(t,t)$ since $C_{\boldsymbol{k}}(t,t)$ is real. It convenient then to define

(5.7a-e ) $$\begin{eqnarray}{\mathcal{F}}_{\boldsymbol{k}}^{1}(t)={\mathcal{F}}_{\boldsymbol{ k}}^{S}(t),\quad \hspace{-1.0pt}{\mathcal{F}}_{\boldsymbol{ k}}^{2}(t)={\mathcal{F}}_{\boldsymbol{ k}}^{PA}(t),\quad \hspace{-1.0pt}{\mathcal{F}}_{\boldsymbol{ k}}^{3}(t)={\mathcal{F}}_{\boldsymbol{ k}}^{PB}(t),\quad \hspace{-1.0pt}{\mathcal{F}}_{\boldsymbol{ k}}^{4}(t)={\mathcal{F}}_{\boldsymbol{ k}}^{PC}(t),\quad \hspace{-1.0pt}{\mathcal{F}}_{\boldsymbol{ k}}^{5}(t)={\mathcal{F}}_{\boldsymbol{ k}}^{PD}(t),\end{eqnarray}$$

(5.7f-j ) $$\begin{eqnarray}{\mathcal{D}}_{\boldsymbol{k}}^{1}(t)={\mathcal{D}}_{\boldsymbol{ k}}^{\unicode[STIX]{x1D702}}(t),\quad \hspace{-1.0pt}{\mathcal{D}}_{\boldsymbol{ k}}^{2}(t)={\mathcal{D}}_{\boldsymbol{ k}}^{\unicode[STIX]{x03C0}A}(t),\quad \hspace{-1.0pt}{\mathcal{D}}_{\boldsymbol{ k}}^{3}(t)={\mathcal{D}}_{\boldsymbol{ k}}^{\unicode[STIX]{x03C0}B}(t),\quad \hspace{-1.0pt}{\mathcal{D}}_{\boldsymbol{ k}}^{4}(t)={\mathcal{D}}_{\boldsymbol{ k}}^{\unicode[STIX]{x03C0}C}(t),\quad \hspace{-1.0pt}{\mathcal{D}}_{\boldsymbol{ k}}^{5}(t)={\mathcal{D}}_{\boldsymbol{ k}}^{\unicode[STIX]{x03C0}D}(t).\end{eqnarray}$$

From equations (4.12) and (5.1) we see that

(5.7k ) $$\begin{eqnarray}\mathop{\sum }_{j=0}^{5}{\mathcal{D}}_{\boldsymbol{k}}^{j}(t)=\unicode[STIX]{x1D708}_{0}(\boldsymbol{k})k^{2}+\int _{t_{0}}^{t}\text{d}s\{(\unicode[STIX]{x1D702}_{\boldsymbol{ k}}(t,s)+\unicode[STIX]{x03C0}_{\boldsymbol{k}}(t,s))C_{-\boldsymbol{k}}(t,s)[C_{\boldsymbol{k}}(t,t)]^{-1}\}.\end{eqnarray}$$

Thus,

(5.8) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}C_{\boldsymbol{k}}(t,t)=2Re\left[\mathop{\sum }_{j=0}^{5}\{{\mathcal{F}}_{\boldsymbol{ k}}^{j}(t)-{\mathcal{D}}_{\boldsymbol{ k}}^{j}(t)C_{\boldsymbol{ k}}(t,t)\}\right]\!.\end{eqnarray}$$

Now to complete the Markovianization, the response function equation (4.11) must also be replaced by

(5.9) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}R_{\boldsymbol{k}}(t,t^{\prime })+\mathop{\sum }_{j=0}^{5}{\mathcal{D}}_{\boldsymbol{k}}^{j}(t)R_{\boldsymbol{ k}}(t,t^{\prime })=\unicode[STIX]{x1D6FF}(t-t^{\prime }).\end{eqnarray}$$

From equation (5.7k ) we see that (5.9) is equivalent to

(5.10) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}R_{\boldsymbol{k}}(t,t^{\prime })+\left[\unicode[STIX]{x1D708}_{0}(\boldsymbol{k})k^{2}+\int _{t_{0}}^{t}\text{d}s\{(\unicode[STIX]{x1D702}_{\boldsymbol{ k}}(t,s)+\unicode[STIX]{x03C0}_{\boldsymbol{k}}(t,s))C_{-\boldsymbol{k}}(t,s)[C_{\boldsymbol{k}}(t,t)]^{-1}\}\right]R_{\boldsymbol{ k}}(t,t^{\prime })\nonumber\\ \displaystyle & & \displaystyle =\unicode[STIX]{x1D6FF}(t-t^{\prime }).\end{eqnarray}$$

The advantage of the forms (5.8) and (5.9) is however that the integral terms can be replaced by manifestly Markovian equations for the triad relaxation times that also involve the damping terms ${\mathcal{D}}_{\boldsymbol{k}}(t)$ . This can be seen as follows. The integral expression for the relaxation time,

(5.11a ) $$\begin{eqnarray}\unicode[STIX]{x1D6E9}^{X}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})(t)=\int _{t_{0}}^{t}\text{d}sR_{\boldsymbol{ k}}(t,s)R_{\boldsymbol{p}}(t,s)R_{\boldsymbol{q}}(t,s)C_{\boldsymbol{p}}^{X}(s,s)C_{\boldsymbol{ q}}^{X}(s,s),\end{eqnarray}$$

is equivalent to the differential equation

(5.11b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D6E9}^{X}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})(t)+\mathop{\sum }_{j=0}^{5}[{\mathcal{D}}_{\boldsymbol{ k}}^{j}(t)+{\mathcal{D}}_{\boldsymbol{ p}}^{j}(t)+{\mathcal{D}}_{\boldsymbol{ q}}^{j}(t)]\unicode[STIX]{x1D6E9}^{X}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})(t)=C_{\boldsymbol{ p}}^{X}(t,t)C_{\boldsymbol{ q}}^{X}(t,t),\end{eqnarray}$$

with $\unicode[STIX]{x1D6E9}^{X}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})(0)=0$ . In addition,

(5.11c ) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}^{X}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})(t)=\int _{t_{0}}^{t}\text{d}sR_{\boldsymbol{ k}}(t,s)R_{\boldsymbol{p}}(t,s)C_{\boldsymbol{p}}^{X}(s,s)\langle \unicode[STIX]{x1D701}_{-\boldsymbol{q}}(s)\rangle\end{eqnarray}$$

is equivalent to

(5.11d ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D6F7}^{X}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})(t)+\mathop{\sum }_{j=0}^{5}[{\mathcal{D}}_{\boldsymbol{ k}}^{j}(t)+{\mathcal{D}}_{\boldsymbol{ p}}^{j}(t)]\unicode[STIX]{x1D6F7}^{X}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})(t)=C_{\boldsymbol{ p}}^{X}(t,t)\langle \unicode[STIX]{x1D701}_{-\boldsymbol{q}}(t)\rangle ,\end{eqnarray}$$

with $\unicode[STIX]{x1D6F7}^{X}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})(0)=0$ . Finally,

(5.11e ) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}^{X}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})(t)=\int _{t_{0}}^{t}\text{d}sR_{\boldsymbol{ k}}(t,s)R_{\boldsymbol{p}}(t,s)C_{\boldsymbol{p}}^{X}(s,s)\end{eqnarray}$$

is equivalent to

(5.11f ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D6F9}^{X}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})(t)+\mathop{\sum }_{j=0}^{5}[{\mathcal{D}}_{\boldsymbol{ k}}^{j}(t)+{\mathcal{D}}_{\boldsymbol{ p}}^{j}(t)]\unicode[STIX]{x1D6F9}^{X}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})(t)=C_{\boldsymbol{ p}}^{X}(t,t),\end{eqnarray}$$

with $\unicode[STIX]{x1D6F9}^{X}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})(0)=0$ .

To close the Markovian equations for the single-time diagonal cumulant and response function, with auxiliary equations for the triad relaxation times, we need to formulate a Markov version of the mean field equation. From (4.7) we have

(5.12) $$\begin{eqnarray}\displaystyle \left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D708}_{0}(\boldsymbol{k})k^{2}\right)\langle \unicode[STIX]{x1D701}_{\boldsymbol{ k}}\rangle & = & \displaystyle \mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})[\!K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})\langle \unicode[STIX]{x1D701}_{-\boldsymbol{p}}\rangle \langle \unicode[STIX]{x1D701}_{-\boldsymbol{q}}\rangle \nonumber\\ \displaystyle & & \displaystyle +\,A(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})\langle \unicode[STIX]{x1D701}_{-\boldsymbol{p}}\rangle h_{-\boldsymbol{q}}\! ]-{\mathcal{N}}_{\boldsymbol{k}}^{M}(t)+f_{\boldsymbol{k}}^{h}(t)+\bar{f}_{\boldsymbol{k}}^{0}(t).\end{eqnarray}$$

Here, the nonlinear damping and eddy–topographic force are given by

(5.13a ) $$\begin{eqnarray}\displaystyle {\mathcal{N}}_{\boldsymbol{k}}^{M}(t) & = & \displaystyle -4\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(-\boldsymbol{p},-\boldsymbol{q},-\boldsymbol{k})\nonumber\\ \displaystyle & & \displaystyle \times \,\int _{t_{0}}^{t}\text{d}sR_{-\boldsymbol{p}}(t,s)C_{-\boldsymbol{q}}(t,s)\langle \unicode[STIX]{x1D701}_{\boldsymbol{k}}(s)\rangle ,\end{eqnarray}$$

(5.13b ) $$\begin{eqnarray}\displaystyle f_{\boldsymbol{k}}^{h}(t) & = & \displaystyle 2\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})A(-\boldsymbol{p},-\boldsymbol{q},-\boldsymbol{k})\nonumber\\ \displaystyle & & \displaystyle \times \,h_{\boldsymbol{k}}\int _{t_{0}}^{t}\text{d}sR_{-\boldsymbol{p}}(t,s)C_{-\boldsymbol{q}}(t,s).\end{eqnarray}$$

(5.14a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{N}}_{\boldsymbol{k}}^{M}(t)=-4\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(-\boldsymbol{p},-\boldsymbol{q},-\boldsymbol{k})\unicode[STIX]{x1D6E5}^{M}(-\boldsymbol{p},-\boldsymbol{q},-\boldsymbol{k})(t),\quad \qquad & \displaystyle\end{eqnarray}$$

(5.14b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E5}^{M}(-\boldsymbol{p},-\boldsymbol{q},-\boldsymbol{k})(t)=\int _{t_{0}}^{t}\text{d}sR_{-\boldsymbol{p}}(t,s)C_{-\boldsymbol{q}}(t,s)\langle \unicode[STIX]{x1D701}_{\boldsymbol{k}}(s)\rangle , & \displaystyle\end{eqnarray}$$

and

(5.14c ) $$\begin{eqnarray}\displaystyle & \displaystyle f_{\boldsymbol{k}}^{h}(t)=2\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})A(-\boldsymbol{p},-\boldsymbol{q},-\boldsymbol{k})\unicode[STIX]{x1D6E5}^{h}(-\boldsymbol{p},-\boldsymbol{q},-\boldsymbol{k})(t),\quad & \displaystyle\end{eqnarray}$$

(5.14d ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E5}^{h}(-\boldsymbol{p},-\boldsymbol{q},-\boldsymbol{k})(t)=h_{\boldsymbol{k}}\int _{t_{0}}^{t}\text{d}sR_{-\boldsymbol{p}}(t,s)C_{-\boldsymbol{q}}(t,s). & \displaystyle\end{eqnarray}$$

Next, we again use the FDT (5.4) for $t\geqslant t^{\prime }$ . Then we have

(5.15a ) $$\begin{eqnarray}{\mathcal{N}}_{\boldsymbol{k}}^{M}(t)={\mathcal{D}}_{\boldsymbol{ k}}^{M}(t)\langle \unicode[STIX]{x1D701}_{\boldsymbol{ k}}(t)\rangle ,\end{eqnarray}$$

where

(5.15b ) $$\begin{eqnarray}\displaystyle {\mathcal{D}}_{\boldsymbol{k}}^{M}(t) & = & \displaystyle -4\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(-\boldsymbol{p},-\boldsymbol{q},-\boldsymbol{k})\nonumber\\ \displaystyle & & \displaystyle \times \,C_{-\boldsymbol{q}}^{1-X}(t,t)\langle \unicode[STIX]{x1D701}_{\boldsymbol{k}}(t)\rangle ^{-1}\unicode[STIX]{x1D6F7}^{X}(-\boldsymbol{p},-\boldsymbol{q},-\boldsymbol{k})(t),\end{eqnarray}$$

and

(5.15c ) $$\begin{eqnarray}f_{\boldsymbol{k}}^{h}(t)=2\mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})A(-\boldsymbol{p},-\boldsymbol{q},-\boldsymbol{k})C_{-\boldsymbol{q}}^{1-X}(t,t)h_{\boldsymbol{ k}}\unicode[STIX]{x1D6F9}^{X}(-\boldsymbol{p},-\boldsymbol{q},-\boldsymbol{k})(t).\end{eqnarray}$$

Here, the expressions for $\unicode[STIX]{x1D6F7}^{X}$ and $\unicode[STIX]{x1D6F9}^{X}$ are the same as in (5.5). Similarly, we have the manifestly Markovian equations for the triad relaxation times associated with the mean field as in (5.11).

Now, equation (5.12) can be written in the form

(5.16) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\langle \unicode[STIX]{x1D701}_{\boldsymbol{k}}\rangle & = & \displaystyle \mathop{\sum }_{\boldsymbol{p}}\mathop{\sum }_{\boldsymbol{q}}\unicode[STIX]{x1D6FF}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})[K(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})\langle \unicode[STIX]{x1D701}_{-\boldsymbol{p}}\rangle \langle \unicode[STIX]{x1D701}_{-\boldsymbol{q}}\rangle +A(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})\langle \unicode[STIX]{x1D701}_{-\boldsymbol{p}}\rangle h_{-\boldsymbol{q}}]\nonumber\\ \displaystyle & & \displaystyle -\,[{\mathcal{D}}_{\boldsymbol{k}}^{0}+{\mathcal{D}}_{\boldsymbol{k}}^{M}(t)]\langle \unicode[STIX]{x1D701}_{\boldsymbol{k}}(t)\rangle +f_{\boldsymbol{k}}^{h}(t)+\bar{f}_{\boldsymbol{k}}^{0}.\end{eqnarray}$$

This then closes the equations for the three versions of the MIC with the unique triad relaxation times being $\unicode[STIX]{x1D6E9}^{X}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})(t),\unicode[STIX]{x1D6F7}^{X}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})(t)$ and $\unicode[STIX]{x1D6F9}^{X}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})(t)$ . Of course, rather than performing the integrals over time, the partial differential equations for the triad relaxation times must be solved. The relative efficiency of the methods will depend on how long the integration needs to be done before a cumulant update restart can be performed (OF04; FO05) and the dimensionality of the problem. However, the MIC also offers the possibility of analytic forms of the triad relaxation times, such as is typically used in EDQNM calculations, and this of course would speed up the calculations enormously.

6 Performance of MIC models for turbulent flow and Rossby waves over topography

We test the performance of the three versions of the MIC model, described in § 5, against two variants of the non-Markovian QDIA and against a large ensemble of direct numerical simulations (DNS) for the case of inhomogeneous turbulent flows over an isolated topographic feature. The numerical simulation and closure calculation setup is similar to that described by FO05. We consider the turbulent interaction of an initial two-dimensional eastward mean flow $U$ impinging on isolated topography on a $\unicode[STIX]{x1D6FD}$ -plane. As noted by FO05, where many references are given, this is an iconic problem in mean flow–topographic interaction going back to the work of Kasahara (Reference Kasahara1966). This is a far from equilibrium problem that is a severe test of the closures with the Rossby wavetrains rapidly spun up in the presence of turbulence. The topography is a circular conical mountain centred at $30~^{\circ }\text{N}$ , $180~^{\circ }\text{W}$ , with a height of 2.5 km and a diameter of $45^{\circ }$ latitude (figure 1 of FO05) and is also similar to that used in the linear and nonlinear study on the sphere by Frederiksen (Reference Frederiksen1982).

Figure 1. (a) The evolved day 10 CUQDIA, (b) QDIA, (c) MIC with $X=0$ , (d) MIC with $X=1/2$ , (e) MIC with $X=1$ and (f) ensemble of DNS Rossby wave streamfunction in non-dimensional units. Multiply values by $(1/4)a_{e}^{2}\unicode[STIX]{x1D6FA}^{-1}=740~\text{km}^{2}~\text{s}^{-1}$ to convert into units of $\text{km}^{2}~\text{s}^{-1}$ .

The focus here is on a case similar to case 1 of FO05 where the initial mean flow $U$ is eastward at $7.5~\text{m}~\text{s}^{-1}$ and the $\unicode[STIX]{x1D6FD}$ -effect is $1.15\times 10^{-11}~\text{m}^{-1}~\text{s}^{-1}$ . Throughout we use a length scale of one-half of the Earth’s radius, $a_{e}/2$ , and a time scale of the inverse of the Earth’s rotation rate, $\unicode[STIX]{x1D6FA}^{-1}$ , which means the non-dimensional values are $U=0.0325$ , $\unicode[STIX]{x1D6FD}=1/2$ and $k_{0}^{2}=1/2$ , and the $\unicode[STIX]{x1D6FD}$ -effect is typical of that at $60^{\circ }$ latitude. The viscosity is $2.5\times 10^{4}~\text{m}^{2}~\text{s}^{-1}$ or $\hat{\unicode[STIX]{x1D708}}=3.378\times 10^{-5}$ in non-dimensional units and $f^{0}=0=\unicode[STIX]{x1D6FC}_{U}$ . Other details of the initial setup are specified in table 1. As well as the large-scale flow $U$ the initial conditions include a small-amplitude mean flow, which is localized over the isolated topography, and an isotropic transient spectrum, both of which are specified in table 1. The calculations are started from this mean field, to which is added Gaussian isotropic perturbations with the spectrum given in table 1. The calculations use a circular C16 resolution, with $k\leqslant 16$ , which is adequate for studying Rossby wave dispersion in a turbulent environment.

Table 1. Parameters and initial conditions used in the calculations and for figures.

Some care is needed in setting up the 1800 initial conditions for the ensemble of DNS fields that the closure calculations are compared with, as noted by FO05. First a field is constructed as a Gaussian sample with zero mean and unit variance. Then from this field further initial members are obtained by moving its origin by a given increment in the $x$ -direction and then in the $y$ -direction. A total of $n$ realizations are obtained by successively moving the initial field by $2\unicode[STIX]{x03C0}/n$ in the $x$ -direction. By subsequently moving each of these $n$ realizations by $2\unicode[STIX]{x03C0}/n$ in the $y$ -direction we obtain a total of $n^{2}$ realizations, and also taking the negative values of each gives $2n^{2}$ realizations. The process can be repeated until the required number of realizations are arrived at and the fields then scaled as required and added to the mean vorticity in table 1. The method ensures that the DNS covariance matrix is nearly isotropic for sufficiently large $n$ .

The time evolutions with the DNS, with two variants of the non-Markovian QDIA and with the three versions of the MIC, are carried out with a time step of $1/30$ day (non-dimensional $\unicode[STIX]{x0394}t=0.21$ ) for a total of 10 days each. The time stepping employs a predictor–corrector method and the integrals are performed using the trapezoidal rule (Frederiksen et al. Reference Frederiksen, Davies and Bell1994; OF04; FO05). The CUQDIA calculations use a restart every day as in FO05, but we also integrate the non-Markovian QDIA equations with the full 10-day integrals evaluated without restart. As noted, the CUQDIA uses the calculated off-diagonal elements of the two- and three-point cumulants in the new initial conditions of the periodic restart procedure. This makes the code more efficient at the expense of a judicious choice of the restart time. However, here we have also performed the much larger computational task with the full non-Markovian QDIA to make a very clean comparison with the Markovian closures.

Figure 1 shows the Rossby wavetrains in the zonally asymmetric component of the mean streamfunction on day 10 for the non-Markovian CUQDIA and QDIA closures, for the three variants of the MIC model and for the ensemble of DNS. The wave patterns show the typical Rossby wave dispersion found in both linearized models and in nonlinear simulations (see FO05 and references therein). The results in all six panels are extremely similar. The numbers in brackets above each figure panel are the pattern correlations between each of the closure results and those from the ensemble of DNS. The agreement between the DNS and the three MICs and the non-Markovian QDIA and CUQDIA closures is excellent, with pattern correlations greater than 0.9998 in all cases.

Next we examine the evolution of the statistics through the band-averaged kinetic energy spectra where the mean and transient components are defined as

(6.1a ) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{E}(k_{i},t)=\frac{1}{2}\mathop{\sum }_{\boldsymbol{k}\in {\mathcal{S}}}\langle \unicode[STIX]{x1D701}_{\boldsymbol{k}}(t)\rangle \langle \unicode[STIX]{x1D701}_{-\boldsymbol{k}}(t)\rangle k^{-2}, & \displaystyle\end{eqnarray}$$

(6.1b ) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{E}(k_{i},t)=\frac{1}{2}\mathop{\sum }_{\boldsymbol{k}\in {\mathcal{S}}}C_{\boldsymbol{k}}(t,t)k^{-2}, & \displaystyle\end{eqnarray}$$

$E(k_{i},t)=\bar{E}(k_{i},t)+\tilde{E}(k_{i},t)$

${\mathcal{S}}$

(6.2) $$\begin{eqnarray}{\mathcal{S}}=[\boldsymbol{k}\mid k_{i}=\text{Int.}(k+{\textstyle \frac{1}{2}})].\end{eqnarray}$$

Here, the average is taken over all $\boldsymbol{k}$ that lie within a band of unit width at a radius $k_{i}$ and the energy of the large-scale flow is plotted at zero wavenumber.

Figure 2. The evolved non-dimensional kinetic energy spectra for the five inhomogeneous closures (a) CUQDIA, (b) QDIA, (c) MIC with $X=0$ , (d) MIC with $X=1/2$ , (e) MIC with $X=1$ , and the ensemble of DNS in each panel. Shown are initial mean energy (dotted), initial transient energy (thin solid), evolved DNS transient energy (thick solid), evolved DNS mean energy (short wide dashed), evolved closure transient energy (thin dashed) and evolved closure mean energy (thick dashed). Multiply by $(1/4)a_{e}^{2}\unicode[STIX]{x1D6FA}^{-2}=5.4\times 10^{4}~\text{m}^{2}~\text{s}^{-2}$ to convert into units of $\text{m}^{2}~\text{s}^{-2}$ .

Figure 2 shows the initial and 10-day evolved mean and transient kinetic energy spectra for the non-Markovian CUQDIA and QDIA closures and each of the three variants of the MIC model, and spectra for the ensemble of DNS are also depicted in each panel for comparison. There are only slight differences in the six sets of results on day 10. The CUQDIA kinetic spectra, both mean and transient, are virtually identical to those of the ensemble of DNS, as are the QDIA spectra with just the slightest increase in the transients between wavenumbers 2 and 6. The realizable MIC that employs the correlation FDT ( $X=1/2$ ) and the MIC that uses the prior-time FDT ( $X=1$ ) again perform exceptionally. For both, the evolved mean kinetic energy spectra are virtually indistinguishable from the ensemble of DNS, as are the transient spectra but with some slight increases between wavenumbers 2 and 6. This range of wavenumbers is of course where the changes in the mean kinetic energy are largest as the Rossby waves amplify, and it is also where there is some increase in the turbulent kinetic energy. The MIC that uses the current-time FDT ( $X=0$ ) again has evolved mean and transient kinetic energy spectra in close agreement with the ensemble of DNS but with somewhat weaker transients between wavenumbers 2 and 6 than the DNS and other closures. The method of Markovianization for this MIC (with $X=0$ ) is of course closest to that employed for the EDQNM where the current-time FDT in (1.2) is also used. Interestingly, the large-amplitude Rossby waves between wavenumbers 2 and 6 will tend to reduce the kinetic energy transfer into those wavenumbers as is the case for the EDQNM (see Frederiksen, Dix & Kepert Reference Frederiksen, Dix and Kepert1996 and references therein). The magnitude of the 10-day evolved relative kinetic energy difference between the closures and DNS, $|E(k)-E^{DNS}(k)|/E^{DNS}(k)$ , has a root mean square (r.m.s.) value, over $k$ , that varies little between the closures. The r.m.s. difference is less than 1.8 % for the QDIA, CUQDIA and MIC closure with $X=1$ , for the MIC closure with $X=1/2$ it is 1.9 % whereas for $X=0$ it is 2.1 %. For the QDIA, CUQDIA and MIC closures with $X=1/2$ and $X=1$ the maximum difference in $|E(k)-E^{DNS}(k)|/E^{DNS}(k)$ is ${\sim}3\,\%$ between wavenumbers 7 and 9 whereas for the MIC with $X=0$ the peak magnitudes of ${\sim}4\,\%$ occur at wavenumbers 11 and 16.

Altogether, the performance of the three versions of the MIC model, and of the non-Markovian CUQDIA and QDIA closures, is quite remarkable in this parameter regime of large-scale Rossby wave dispersion over topography in a turbulent environment. As noted, the performance of the MIC with $X=0$ is slightly poorer in terms of the kinetic energy spectra than the other closures. For higher resolution and higher Reynolds numbers we expect that the MICs, like the non-Markovian DIA (Frederiksen & Davies Reference Frederiksen and Davies2004) and QDIA (OF04), will need to incorporate a regularization, or empirical vertex renormalization, to yield the correct small-scale spectra. The regularized MICs are documented in appendix C.

The results here show the potential of the MICs to reproduce the mean and transient statistics of ensembles of DNS in a strongly inhomogeneous context. If broadly universal and analytic expressions for the triad relaxation times, or the underlying response functions, can be established in the presence of waves and inhomogeneities, then very efficient closures, like the EDQNM, can be developed to tackle the general problem of inhomogeneous turbulent flows.

7 Discussion and conclusions

We have formulated manifestly MIC models for turbulent flows and Rossby waves over topography on a generalized $\unicode[STIX]{x1D6FD}$ -plane. These have been derived as modifications of the non-Markovian QDIA closure (F99; OF04; FO05) in which the diagonal two-time covariance is replaced by one of three versions of the FDT and the diagonal response function equation is modified to a form that is Markovian. The three different MICs employ the current-time FDT (quasi-stationary), the prior-time FDT (non-stationary) and the correlation FDT, respectively. Bowman (Reference Bowman, Krommes and Ottaviani1993) pointed out that the correlation FDT for homogeneous turbulence, in the presence of waves, together with positive damping in Markovian response functions, resulted in an RMC that guaranteed real and non-negative cumulants $C_{\boldsymbol{k}}(t,t)$ . We have noted that the cumulants $C_{\boldsymbol{k}}(t,t)$ are again realizable in the more complex inhomogeneous MIC formalism with the correlation FDT, but, as in the homogeneous case, not necessarily with the current-time FDT or the prior-time FDT. Of course, whether unphysical results arise will depend on the parameter regimes of the flows, and in this article we have documented results for each of the three Markovian closures in the same regimes.

The MICs differ from the non-Markovian QDIA closure in that the response function has been modified to a form that is Markovian and the time history integrals have also been modified by the FDTs in such a way that their information can be characterized by three triad relaxation functions (for each variant) that satisfy auxiliary Markovian tendency equations. Thus, the MICs contain much the same information as the non-Markovian QDIA, but the time history integrals can be replaced by differential equations that become relatively more efficient for long integrations. In addition, there is the prospect of developing analytical forms of the triad relaxation functions, or underpinning response functions, as is the case for isotropic EDQNM closures, that would increase the computational efficiency enormously.

The performance of the three MICs has been compared with results from an 1800 member ensemble of DNS and with the non-Markovian QDIA and CUQDIA closures for turbulent flow over isolated topography on a generalized $\unicode[STIX]{x1D6FD}$ -plane. For each of these calculations, the initial flow consists of an eastward mean flow $U$ , together with smaller-amplitude mean and transient ‘small scales’. The impact of the mean flow on the conical mountain topography results in the rapid generation of large-amplitude Rossby waves in a turbulent environment in 10-day integrations. The calculations are performed at a C16 resolution with wavenumbers $k\leqslant 16$ , which is sufficient for this large-scale process. This is a far from equilibrium process (Frederiksen Reference Frederiksen1982; FO05), which severely tests the closures. The performance of each of the three MICs, and the non-Markovian QDIA integrated without a restart over the 10 days, is excellent, like the CUQDIA closure calculations with periodic 1-day restarts (FO05). In all cases, the pattern correlations of the day 10 mean Rossby wave streamfunction for the closures with the ensemble result for 1800 DNS are greater than 0.9998. Over the 10 days, there is significant evolution of the mean and transient energy spectra particularly between wavenumbers 2 and 6 where the Rossby waves amplify by orders of magnitude in the mean; in this range, there are some slight differences in the transient kinetic energy between the MIC calculations and the ensemble of 1800 DNS. The MIC with $X=0$ is slightly poorer in terms of the kinetic energy spectra than the other closures.

We have also formulated a regularized version of the inhomogeneous closures, along the lines of the regularized DIA closure of Frederiksen & Davies (Reference Frederiksen and Davies2004) and the regularized QDIA closure of OF04, which are needed for high-resolution and high-Reynolds-number calculations. The regularized closures have a wavenumber cut-off parameter $\unicode[STIX]{x1D6FC}$ which localizes the interactions, and corresponds to an empirical vertex renormalization (C.1). It has been found that the value of $\unicode[STIX]{x1D6FC}$ that is consistent with inertial range behaviour in the DIA (Frederiksen & Davies Reference Frederiksen and Davies2004) and QDIA (OF04) is essentially universal.

In future studies we aim to experiment with different analytical expressions of the triad relaxation functions, or underpinning Markovian response functions, to see whether it is possible to establish Markovian closures for the inhomogeneous problem in the presence of waves that have similar computational efficiency to the EDQNM closure for isotropic turbulence. We also aim to examine the performance of regularized versions of the Markovian closures at high resolution and Reynolds numbers. We further seek to generalize the LET closure of McComb (Reference McComb1974, Reference McComb1990, Reference McComb2014) to inhomogeneous turbulent flows and compare the performance of non-Markovian and Markovian variants.