= Solution
Expand the axisymmetric <Euler equations> about the rotating equilibrium, retaining only terms linear in $u_r,u_\phi,u_z$ and in the <pressure> perturbation. The symbol in this question is the <kinematic pressure> perturbation,
$$
\boxed{\pi(r,z,t)=\frac{p(r,z,t)-p_0(r)}\rho,\qquad[\pi]=\mathrm{m^2\,s^{-2}}.}
$$
The linearized centripetal term in the radial equation is $-2\Omega u_\phi$, after subtracting the equilibrium term $-r\Omega^2$. Radial advection of the background azimuthal <velocity>, together with the cylindrical-coordinate term $u_rv_\phi/r$, gives
$$
\left[\frac d{dr}(r\Omega)+\Omega\right]u_r
=(2\Omega+r\Omega')u_r
=\frac1r\frac d{dr}(r^2\Omega)u_r.
$$
This is the azimuthal coupling in the second equation; it also expresses conservation of <specific angular momentum> $j=r^2\Omega$ for a radially displaced element. The axial equation is acceleration balanced by $-\partial_z\pi$. Finally, the axisymmetric <incompressible flow> condition in <cylindrical coordinates> is
$$
\frac1r\partial_r(ru_r)+\partial_zu_z=0,
$$
which expands to the last supplied equation. Background azimuthal advection contributes no $\Omega\partial_\phi$ term because the perturbations are axisymmetric.
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