fluctuations. The first term on the right-hand side of (\[eq:Equations\_Ut\]) is the Reynolds stress, while the second term is specific to the mTHE system. For the form of $\hat{\delta}$ given by (\[eq:Equations\_delta\]), the second term becomes $$-\langle\tilde{v}_{x}{\rm i}\hat{\delta}\tilde{\varphi}\rangle=\delta_{0}^{2}\langle\tilde{v}_{x}^{2}\rangle>0.\label{eq:Equation_T(t)}$$ Therefore, the second term will always increase the local ZF velocity $U$, and meanwhile, the value of $U$ at other locations will be adjusted by the effect of $\mathcal{T}(t)$, which is an integration constant that ensures conservation of the total momentum. Specifically, $\partial_{t}\int U{\rm d}x=0$ implies $$\quad\mathcal{T}(t)=\frac{1}{L_x}\int\langle\tilde{v}_{x}{\rm i}\hat{\delta}\tilde{\varphi}\rangle{\rm d}x.$$ Due to nonzero $\mathcal{T}$, ZFs cannot remain (quasi)stationary in the presence of fluctuations within the mTHE. In other words, either ZFs completely suppress DW turbulence, or both ZFs and DWs keep evolving indefinitely.

Primary, secondary, and tertiary instability {#sec:Linear}
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We have integrated the mTHE numerically using random noise for the initial conditions. Typical simulation results are presented in figures \[fig:THhistory\] and \[fig:PI\_and\_SI\]. It is seen that the primary instability of DWs arises and is followed by ZF generation through the secondary instability. Then, at the fully nonlinear stage, DW turbulence becomes inhomogeneous, exhibiting signatures of the TI. In the following, we study these stages in detail.

![Snapshots from numerical simulations of the 