The Barotropic Normal Modes in Certain Shear Flows and the Traveling Waves in the Atmosphere

Abstract

It is shown analytically and numerically that in certain shear flows the linearized nondivergent barotropic vorticity equation has a limited number of neutral normal modes. The latitudinal structures of these shear flows can be expressed as polynomials of the sine of latitude. The first few such shear flows resemble the gross features of the zonal winds in the atmosphere of the earth at different tines and altitudes. The spatial structures of the neutral normal modes in these shear flows are spherical harmonics, and, as a consequence, these modes are also the exact solutions of the fully nonlinear equation because the nonlinear interaction term vanishes identically. The spatial structures of the observed 5-, 4-, 2-, and 16-day free traveling waves in the atmosphere are often identified with the spherical harmonics with indices of (m, n) = (1, 2), (2, 3), (3, 3), and (1, 4), which are known previously as the neutral normal equation in a motionless background state. Our results could explain why these free traveling waves can survive the shearing effects of zonal flows that are far different from rest because these spherical harmonies are also normal modes in certain shear flows that resemble the observations of the atmosphere.