= Vertical shear instability
{title2=VSI}
Vertical shear instability destabilizes differential rotation that varies with height, when buoyancy restoration is absent or sufficiently weakened. In the homogeneous <incompressible flow> model with background $U_y=-3\Omega x/2+q\Omega z$, an axisymmetric perturbation with <wavevector> $(k_x,0,k_z)\ne0$ has
$$
s^2=\Omega^2\frac{2qk_xk_z-k_z^2}{k_x^2+k_z^2}.
$$
The pressure-free radial/vertical momentum equation and the azimuthal equation give this result after using <incompressibility>. Growth requires $2qk_xk_z>k_z^2$, so a small vertical shear favors nearly radial wavevectors. Here $q$ is the vertical shear amplitude, distinct from the <orbital shear parameter> $3/2$. Stratification, cooling and boundaries can change this simplified criterion.
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