Development of a Rossby Wave Critical Level

Abstract

We discuss the time-dependent behavior of a Rossby wave on a latitudinally varying flow, near the point where the steady-state wave equation is singular. The wave is forced by the switch-on of a steady forcing. Analytic solutions are obtained for the latitudinal propagation of nondivergent Rossby waves in a linear shear flow and for a large longitudinal wavelength. It is shown that the north–south eddy velocity v′ approaches the steady-state solution everywhere when nondimensional time >1, this time being a few days or less for atmospheric planetary waves. The east–west eddy velocity u′ takes much longer to approach a steady state near the singularity. One-half the steady-state amplitude of u′ is approached in a time inversely proportional to the square root of the distance from the singularity. The solution for u′ near the singularity settles down to the steady solution only after a time large compared to the inverse of the distance from the singularity. The steady-state solution for u′ is logarithmically singular at the critical level, violating the assumption of small-amplitude motions. However, the initial-value solution indicates that large amplitudes near a critical level probably will not actually occur in the atmosphere since they require a longer time scale to be set up than the time scale for seasonal changes of the zonal winds. At times ≥O(1), the momentum divergence in the initial value calculation is smeared out over the region where the u′ component of velocity has not yet settled down. The width of this region is inversely proportional to time.