On the transport and concentration of enstrophy in 3D magnetohydrodynamic turbulence (Q2845641)

A mathematical model of the 3D MHD turbulence is studied. This model is the PDE system for the velocity field \(v(x,t)\), magnetic field \(b(x,t)\) and pressure field \(p(x,t)\), NEWLINE\[NEWLINE\begin{aligned} &\frac{\partial v}{\partial t}+v\cdot\nabla v-\Delta v+\nabla p-b\cdot\nabla b=0,\\ &\frac{\partial b}{\partial t}+v\cdot\nabla b-\Delta b-b\cdot\nabla v=0,\\ &\nabla\cdot v=0,\quad \nabla\cdot b=0,\\ &v(x,0)=v_0(x),\quad b(x,0)=b_0(x), \end{aligned}NEWLINE\]NEWLINE where \(x\in \mathbb{R}^3\), \(t\geq 0\).NEWLINENEWLINEThe authors are interested in such a characteristic of turbulence as the combined enstrophy flux, which is the sum of integrals of squares of the vorticity and current. The main result of the paper establishes the positivity and near-constancy of the combined enstrophy flux across a range of physical scales. The ensemble averaging method is used in the proof. The main result has a corollary that the transport of the combined enstrophy is mainly between scales of comparable size.