= Solution
For the <polytropic equation of state> $p=K\rho^{1+1/n}$, take $K>0$ and $n>0$ for the physical model. The <specific enthalpy> relative to vacuum is
$$
Q=\int_0^\rho\frac{dp(\tilde\rho)}{\tilde\rho}=(n+1)K\rho^{1/n}=\frac{(n+1)p}{\rho}.
$$
Its gradient obeys $\nabla Q=\rho^{-1}\nabla p$, so the <Euler equations> become
$$
\boxed{\partial_t\mathbf u+\mathbf u\cdot\nabla\mathbf u=-\nabla Q-2\Omega\mathbf e_z\times\mathbf u-\nabla\Phi_t.}
$$
For the material derivative $D_t=\partial_t+\mathbf u\cdot\nabla$, $D_tQ=(Q/n\rho)D_t\rho$. The <continuity equation> thus becomes
$$
\boxed{nD_tQ=-Q\nabla\cdot\mathbf u.}
$$
The choice $Q=0$ at zero <mass density> fixes the additive convention in this equation. Although the momentum equation alone is unchanged by adding a constant to $Q$, the polytropic relation and enthalpy evolution must use the same vacuum reference.
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