= Solution
For the <Keplerian shearing sheet> background $\boldsymbol U=-3\Omega x\boldsymbol e_y/2$, the only spatial derivative is $\partial_xU_y=-3\Omega/2$. Since $U_x=0$ and all fields are independent of $y$, $\boldsymbol U\cdot\nabla\boldsymbol U=0$ and $\nabla\cdot\boldsymbol U=0$.
The <Coriolis acceleration> exactly balances the tidal term:
$$
-2\Omega\boldsymbol e_z\times\boldsymbol U=-3\Omega^2x\boldsymbol e_x.
$$
With constant <pressure> and $\theta=0$, the <buoyancy displacement variable> equation also has zero advection, zero source $U_x$ and zero diffusion. Therefore
$$
\boxed{\boldsymbol u=-\frac32\Omega x\boldsymbol e_y,\qquad\theta=0,\qquad P=\text{constant}}
$$
is a steady solution. Its <absolute vorticity> is $\boldsymbol\omega_0=(\Omega/2)\boldsymbol e_z$, since the background <relative vorticity> is $-3\Omega\boldsymbol e_z/2$.
Back to article page
