= Solution
With zero <velocity> and no time dependence, the <magnetic second-moment tensor> equation holds automatically, but the <magnetohydrodynamic momentum equation> still requires
$$
\boxed{\partial_jM_{ij}^{(0)}=\partial_i p_0.}
$$
The <magnetic tension> must therefore be a <pressure gradient>. Locally, its necessary integrability condition is
$$
\nabla\times\bigl(\nabla\cdot M^{(0)}\bigr)=0;
$$
on a simply connected region this is also sufficient for a scalar $p_0$, subject to the global <boundary conditions>. In physical units the same balance is $\nabla P_0=\mathbf J_0\times\mathbf B_{{\rm phys},0}$: the <Lorentz force density> is balanced by mechanical <pressure>, rather than necessarily vanishing. A zero mean <magnetic field> does not make the local <magnetic tension> zero, so an arbitrary tangle is not automatically a static <equilibrium>.
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