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Put $P=p/{\rho }_0$ and use the downward buoyancy anomaly $b=g{\rho }^{\prime }/{\rho }_0$. This sign convention is required by the printed potential vorticity. The linear rotating Boussinesq approximation givesLet $D=u_x+v_y$ and $\zeta =v_x-u_y$. Horizontal divergence and curl give $D_t-f\zeta =-{\nabla }_H^{2}P$ and ${\zeta }_t=-fD=f{w}_z$. HenceThus 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 ${\nabla }^2{P}_{tt}=(N^2-f^2)(P_{zz}+b_z)$. Combining terms,The linear pressure equation for rotating stratified flow has source $Q={\rho }_{0}f{N}^{2}q$. “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 $-g{\rho }^{\prime }/{\rho }_0$, both appearances of its sign would change consistently. The pressure source represents the balanced component alongside propagating inertia-gravity waves.