= Cubic-exponent decay of a nonaxisymmetric shearing wave
{title2=$|\widetilde{\mathbf v}(t)|^2\le |\widetilde{\mathbf v}(0)|^2e^{2|A|t-2\nu\int_0^t k(s)^2\,ds}$}
For a <zero-mean magnetic shearing wave>, the real <wavevector> satisfies $k_x=k_{x0}+2Ak_yt$, with $k_y,k_z$ constant. Dotting the velocity-amplitude equation with its <complex conjugate> eliminates <Coriolis force> and pressure work, giving $d|\widetilde{\mathbf v}|^2/dt=4A\operatorname{Re}(\widetilde v_x\widetilde v_y^*)-2\nu k^2|\widetilde{\mathbf v}|^2$. The inequality $2|v_xv_y|\le|\mathbf v|^2$ proves the displayed bound; the magnetic bound has the same form with $\eta$ replacing $\nu$. Now
$$
\int_0^t k(s)^2\,ds=(k_{x0}^2+k_y^2+k_z^2)t+2Ak_{x0}k_yt^2+\frac43A^2k_y^2t^3.
$$
For $Ak_y\ne0$ and positive <kinematic viscosity> and <magnetic diffusivity>, the negative cubic exponent dominates the linear shear-work bound, so both amplitudes tend to zero. Transient growth is allowed. Positivity matters: an inviscid vertical velocity mode with $k_z=0$ can remain constant, and an ideal vertical magnetic mode with $k_z=0$ can also remain constant.
Back to article page
