Three-dimensional Global Scale Permanent-wave Solutions of the Nonlinear Quasigeostrophic Potential Vorticity Equation and Energy Dispersion

The three-dimensional nonlinear quasi-geostrophic potential vorticity equation is reduced to a linear form in the stream function in spherical coordinates for the permanent wave solutions consisting of zonal wavenumbers from 0 to n and rn vertical components with a given degree n. This equation is solved by treating the coefficient of the Coriolis parameter square in the equation as the eigenvalue both for sinusoidal and hyperbolic variations in vertical direction. It is found that these solutions can represent the observed long term flow patterns at the surface and aloft over the globe closely. In addition, the sinusoidal vertical solutions with large eigenvalue G are trapped in low latitude, and the scales of these trapped modes are longer than 10 deg. lat. even for the top layer of the ocean and hence they are much larger than that given by the equatorial β-plane solutions. Therefore such baroclinic disturb-ances in the ocean can easily interact with those in the atmosphere.Solutions of the shallow water potential vorticity equation are treated in a similar manner but with the effective depth H = RT / g taken as limited within a small range for the atmosphere.The propagation of the flow energy of the wave packet consisting of more than one degree is found to be along the great circle around the globe both for barotropic and for baroclinic flows in the atmosphere.