= Solution
Let $D/Dt=\partial_t-Sx\partial_y$ be the <material derivative> along the background <linear shear flow> $\mathbf u_0=-Sx\mathbf e_y$. Linearizing about constant $\Sigma$ gives
$$
\frac{D\Sigma'}{Dt}
=-\Sigma(\partial_xv_x+\partial_yv_y),
$$
$$
\frac{Dv_x}{Dt}-2\Omega v_y
=-\partial_x\Psi-\frac{v_s^2}{\Sigma}\partial_x\Sigma',
$$
$$
\frac{Dv_y}{Dt}+(2\Omega-S)v_x
=-\partial_y\Psi-\frac{v_s^2}{\Sigma}\partial_y\Sigma'.
$$
The coefficient $2\Omega-S$ includes the contribution $\mathbf v\mathbin\cdot\nabla\mathbf u_0=-Sv_x\mathbf e_y$ from perturbation advection of the background shear.
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