Envelope solitary Rossby waves and modulational instabilities of uniform Rossby wave trains in two space dimensions

Abstract

Envelope solitary Rossby waves and modulational instability of a uniform Rossby wave train in two space dimensions are investigated. It is found that the condition for an envelope solitary Rossby wave to exist is [ (3m 2 − k 2) cos 2 θ + (5m 2 + k 2) sin 2 θ] [3m 2(m 2 − 2k 2) − k 4 ] > 0 , in which k and m are the zonal and meridional wave numbers, respectively, and θ is a fixed angle of orientation representing the modulation direction in two space dimensions. Moreover, under a moderate condition the envelope solitary Rossby wave possesses quasi-stationary dipole structure that can tilt horizontally either westward or eastward due to the change of modulation direction, which resembles observed vortex pair blocking. If there is a set of infinitesimal sideband perturbations imposing on a uniform Rossby wave train, then the condition for instability to occur is 0 < [ λ( pk) 2 + Q( qm) 2] λ < 2 δb 0 2, where b 0 is the amplitude of the uniform Rossby wave train, and the restrictions that p ⪡ 1 and q ⪡ 1 are required. A noteworthy property is that the instability region of p will become narrower if q increases in the small value limit. On the other hand, it will become wider if the latitude increases for a fixed q.