Vertically propagating Rossby waves

Returning to (1.1) and (1.2) and forming the vorticity equation using (1.4) gives

But, from (1.5) and (1.3)

Finally, using the geostrophic relation for the u, v terms in (5.1) gives the quasi-geostrophic vorticity equation for p(x,y,z,t)

of which (4.10) is the modal counterpart for the nth mode function .

• Simple plane wave solutions for N constant