= Solution
The variable $\psi$ is a <rotating-flow pressure potential> including the conservative body-force and centrifugal potentials. For constant <mass density>,
$$
\boxed{\psi=\frac p\rho+\Phi-\frac12|\boldsymbol\Omega\times\mathbf r|^2,}
$$
up to an additive function of time, where $\Phi$ is any imposed conservative potential. The last term is the <centrifugal potential>; its negative gradient is the centrifugal acceleration. The <Coriolis force> remains explicit in the equation because it is velocity-dependent and cannot be absorbed into a scalar position potential.
The further constraint is <incompressible flow>:
$$
\boxed{\nabla\cdot\mathbf u=0.}
$$
For the specified shear, $\mathbf u_0\cdot\nabla\mathbf u_0=0$ and $\nabla^2\mathbf u_0=0$. Its <Coriolis acceleration> is $4\Omega Ax\mathbf e_x$, so the corresponding total basic <pressure> potential can be chosen as $\psi_0=-2\Omega Ax^2$. This supplies the force balance required for the basic state used below.
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