Magnetoelastic wave in an isotropic magnetic tangle (${\omega }^2=⟨B^2⟩k^2/3$)

A long-wavelength transverse disturbance of an initially isotropic magnetic second-moment tensor behaves as an elastic shear wave when small-scale rearrangements of the tangle are omitted. For $v_A^2=⟨B^2⟩/3$, its fluid displacement obeys ${\partial }_t^2\mathbf{\xi }=v_A^2{\nabla }^2\mathbf{\xi }$ and $\nabla \cdot \mathbf{\xi }=0$ on the oscillatory branch. The tensor perturbation is $\delta M_{ij}=v_A^2({\partial }_i{\xi }_j+{\partial }_j{\xi }_i)$. A zero mean magnetic field does not imply zero restoring magnetic tension.

The linear system ${\partial }_t{v}_i=-{\partial }_i\pi +{\partial }_j{m}_{ij}$, ${\partial }_t{m}_{ij}=v_A^2({\partial }_i{v}_j+{\partial }_j{v}_i)$ and $\nabla \cdot \mathbf{v}=0$ admits stationary stress as well as propagating waves. For instance, $\mathbf{v}=0$, $m_{ij}=f(\mathbf{r}){\delta }_{ij}$ and $\pi =f(\mathbf{r})$ solve it for any sufficiently small smooth $f$. Thus arbitrary initial magnetic second-moment tensor perturbations need not be oscillatory.