Keplerian shearing sheet

A Keplerian shearing sheet has background velocity ${\mathbf{u}}_0=-(3/2)\Omega x{\mathbf{e}}_y$, radial shear rate $S=3\Omega /2$, and radial epicyclic frequency ${\kappa }_r=\Omega$.

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 $(k_x,0,k_z)\ne 0$ hasThe pressure-free radial/vertical momentum equation and the azimuthal equation give this result after using incompressibility. Growth requires $2q{k}_x{k}_z>k_z^2$, 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.

For the unstratified local vertical shear instability, maximize $s^2/{\Omega }^2=(2qR-1)/(1+R^2)$ over $R=k_x/k_z$. Its derivative vanishes when $q{R}^2-R-q=0$. For $q>0$ the maximum uses $R=(1+\sqrt{1+4q^2})/(2q)$, and substitution gives the displayed expression. For small $q$, $R\sim q^{-1}$ and $s_{\max }=\Omega q(1-q^2/2+O(q^4))$. Changing the sign of the shear changes the preferred sign of $R$ but gives the same maximum magnitude; there is no exponential growth at $q=0$.

For a perturbation ${\mathbf{u}}^{\prime }=\tilde{\mathbf{u}}f(\mathbf{k}\cdot \mathbf{x})e^{st}$ with constant $\tilde{\mathbf{u}},\mathbf{k}$ and nonconstant differentiable $f$, incompressibility forces $\mathbf{k}\cdot \tilde{\mathbf{u}}=0$. Consequently ${\mathbf{u}}^{\prime }\cdot \nabla {\mathbf{u}}^{\prime }=f{f}^{\prime }{e}^{2st}(\mathbf{k}\cdot \tilde{\mathbf{u}})\tilde{\mathbf{u}}=0$. About a shear depending only on $x,z$ and directed along $y$, an axisymmetric such perturbation therefore obeys its linear amplitude equations exactly, including at finite amplitude in this homogeneous model. For pressure $P^{\prime }=\tilde{P}g(\xi )e^{st}$, nonzero pressure requires $g^{\prime }=Cf$, and $C$ can be absorbed into $\tilde{P}$; choose $g^{\prime }=f$. This relation does not imply $g=f$ for an arbitrary profile.

For constant density ${\rho }_0$ and body force ${\Omega }^2(3x-2qz){\mathbf{e}}_x$, dotting the incompressible flow momentum equation with ${\rho }_0\mathbf{u}$ gives the kinetic energy flux $(K+P)\mathbf{u}$ and source ${\rho }_0{\Omega }^2(3x-2qz)u_x$, where $K={\rho }_0|\mathbf{u}{|}^2/2$. Coriolis acceleration does no work. Absorbing ${\Phi }_t=-3{\Omega }^2{x}^2/2$ gives the displayed balance. For nonzero $q$, the remaining force has curl $-2{\Omega }^{2}q{\mathbf{e}}_y$, so it cannot be incorporated into a scalar mechanical potential. The model can exchange energy with its maintained vertical-shear background; at $q=0$ ordinary Jacobi energy in a shearing sheet conservation is recovered under zero boundary flux.

An adverse radial specific entropy gradient can drive an oscillatory instability through the finite lag supplied by thermal conduction or relaxation. This differs from a viscous-convective instability, in which viscosity changes rotational stabilization.

For $|N^2|\ll {\Omega }^2$, this leading amplitude-growth rate is largest at thermal diffusion rate of a Fourier mode $\beta =\Omega$, giving ${\sigma }_{\max }=-N^2/(4\Omega )$ when $N^2<0$.

A horizontal-velocity perturbation depending only on height has zero vertical velocity for nonzero wavenumber under incompressibility. Its pressure amplitude then vanishes in the unstratified vertical momentum equation.

The nonoscillatory thermal branch of the radially stratified shearing sheet spectrum. Its decoupled small-buoyancy limit is a diffusing scalar perturbation.