Poincaré equation for inertial waves (${\partial }_t^2{\nabla }^{2}w+4{\Omega }^2{\partial }_z^{2}w=0$)

Linear incompressible motion about rest in a frame rotating with $\mathbf{\Omega }=\Omega \mathbf{k}$ obeys ${\mathbf{u}}_t+2\mathbf{\Omega }\times \mathbf{u}=-\nabla P/\rho$. If $\zeta =v_x-u_y$ is its vertical vorticity, taking curl gives ${\zeta }_t=2\Omega w_z$, while eliminating pressure using continuity gives ${\partial }_t{\nabla }^{2}w+2\Omega {\zeta }_z=0$. Differentiating in time yields the displayed scalar equation. All three velocity components must be retained, even when the fields depend on only two spatial coordinates. The Coriolis acceleration generally generates a transverse velocity that is absent in a strictly two-component description.