= Solution
Put $P=p/\rho_0$ and use the downward buoyancy anomaly $b=g\rho'/\rho_0$. This sign convention is required by the printed <potential vorticity>. The linear rotating <Boussinesq approximation> gives
$$
u_t-fv=-P_x,\quad v_t+fu=-P_y,\quad w_t=-P_z-b,\quad b_t=N^2w,\quad u_x+v_y+w_z=0.
$$
Let $D=u_x+v_y$ and $\zeta=v_x-u_y$. Horizontal divergence and curl give $D_t-f\zeta=-\nabla_H^2P$ and $\zeta_t=-fD=fw_z$. Hence
$$
\nabla^2P=f\zeta-b_z,\qquad
q_t=\zeta_t-\frac f{N^2}b_{zt}=0.
$$
Thus the linear <potential vorticity> is a time-independent field fixed by the initial data. Differentiate the pressure identity twice. Using $\zeta_{tt}=-f(P_{zz}+b_z)$ and $b_{ztt}=-N^2(P_{zz}+b_z)$ gives $\nabla^2P_{tt}=(N^2-f^2)(P_{zz}+b_z)$. Combining terms,
$$
\boxed{\nabla^2p_{tt}+f^2p_{zz}+N^2\nabla_H^2p
=\rho_0fN^2\left(\zeta-\frac f{N^2}b_z\right)=Q(\mathbf x).}
$$
The <linear pressure equation for rotating stratified flow> has source \b[$Q=\rho_0fN^2q$]. “Arbitrary” means that different initial potential-vorticity fields give different time-independent sources; it is not independent forcing that may vary in time. If buoyancy were instead defined upward as $-g\rho'/\rho_0$, both appearances of its sign would change consistently. The pressure source represents the balanced component alongside propagating <inertia-gravity waves>.
Back to article page
