 Zero-mean magnetic shearing wave

In an incompressible shearing sheet with background velocity $-2Ax{\mathbf{e}}_y$, take both the perturbation velocity and the entire magnetic field to be real single Fourier modes with a common real wavevector and transverse amplitudes. Then every quadratic directional derivative vanishes, including terms involving complex conjugates. There is no uniform background magnetic field and consequently no linear magnetic-tension coupling between the amplitudes. Advecting the phase requires ${\dot{k}}_x=2A{k}_y$, ${\dot{k}}_y={\dot{k}}_z=0$. The velocity amplitude obeys $\dot{\tilde{\mathbf{v}}}-2A{\tilde{v}}_x{\mathbf{e}}_y+2\mathbf{\Omega }\times \tilde{\mathbf{v}}=-i\mathbf{k}\tilde{\psi }-\nu k^2\tilde{\mathbf{v}}$; the magnetic amplitude obeys the displayed equation. The modified pressure includes magnetic pressure; the physical pressure can contain a mean and second harmonic even when the modified pressure is a single mode.