= Stratified incompressible shearing sheet
{title2=$\omega^2=(\kappa_r^2k_z^2+N^2k_x^2)/(k_x^2+k_z^2)$}
A stratified incompressible <shearing sheet> retains vertical <buoyancy> while treating <mass density> as constant in inertial terms, as in the <Boussinesq approximation>. Let $\theta$ be a vertical displacement, with $D\theta/Dt=u_z$, so stable stratification supplies <acceleration> $-N^2\theta\mathbf e_z$ for positive squared <buoyancy frequency> $N^2$.
For uniform local coefficients, no vertical shear, and axisymmetric <plane waves> with <wavevector> $(k_x,0,k_z)\ne0$, the linear incompressible equations give
$$
\omega^2=\frac{\kappa_r^2k_z^2+N^2k_x^2}{k_x^2+k_z^2},\qquad \kappa_r^2=2(2-q)\Omega^2.
$$
Here $q$ is the <orbital shear parameter> and $\kappa_r$ the <radial epicyclic frequency>. In a <Keplerian shearing sheet>, $q=3/2$ and $\kappa_r^2=\Omega^2$, recovering the previously displayed form. The <dispersion relation> interpolates between epicyclic restoration for a vertical <wavevector> and buoyancy restoration for a radial <wavevector>. If both squared restoring frequencies are nonnegative, this branch has no exponential growth; negative $N^2$ permits convective orientations.
Back to article page
