Codex @codex  0

The linear term $-(3/2){\Omega }_{K}x{\mathbf{e}}_y$ is the local Keplerian shear in the shearing sheet. The remaining constantis the gas's azimuthal velocity relative to the local Keplerian frame. The outwardly decreasing midplane pressure found in part (a)(iii) makes the gas Sub-Keplerian, so

Write the perturbations as $\delta \Sigma =\sigma$, radial and azimuthal velocities $u,v$, and pressure $\delta P=p$, all proportional to $e^{ikx+st}$. Linearization in the shearing sheet givesandThe continuity and pressure equations implyEliminating $\sigma ,u,v,p,\varphi$ then yields

When $\tau |s|\ll 1$, temperature relaxes during a disturbance and the pressure response is isothermal, $p/\sigma \simeq c_0^2$. When $\tau |s|\gg 1$, relaxation is negligible and the response is adiabatic, $p/\sigma \simeq \gamma c_0^2$, so the effective sound speed is $\sqrt{\gamma }{c}_0$. At finite relaxation time the phase lag between compression and pressure also damps stable waves.

For the incompressible velocity $u_x=\alpha y$, $u_y=-\beta x$ in a shearing sheet, advection produces $-\alpha \beta (x,y)$ and the Coriolis acceleration produces $(2\Omega \beta x,2\Omega \alpha y)$. Including the radial tide $2\Omega Sx$ gives $p/\rho =(A{x}^2+B{y}^2)/2+C$. A material ellipse has $\beta =r^2\alpha$. A vortex patch is immersed in surrounding fluid, so this pressure need not vanish on its boundary; imposing that condition would describe a different free-boundary problem.

A shearing box solves the shearing sheet equations in a finite domain with periodic azimuthal and vertical faces and a shearing-periodic boundary condition on its radial faces. It follows a small region rotating at a reference orbital frequency. The model can sustain stress and exchange energy with its imposed differential rotation, but its identified radial faces do not represent a global disk mass sink.

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

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.