Growing Vortex Rossby Waves with Azimuthal Wavenumber One in Quasigeostrophic System

In this study, we show analytically that vortex Rossby waves (VRWs) with azimuthal wavenumber m = 1 in a basic axisymmetric vortex can grow exponentially in a quasi-geostrophic system, although they cannot do so in a barotropic system. VRWs grow exponentially if Rayleigh's condition and Fjortoft's condition are satisfied. Satisfying Rayleigh's condition means that two horizontally aligned VRWs at two different radii propagate (here and hereafter "propagate" refers to propagation relative to the fluid) azimuthally counter to each other. Satisfying Fjortoft's condition means that the cyclonic advective angular velocity of the basic vortex is distributed radially so as to enable the VRWs to be phase-locked with each other. Under these conditions, a strong mutual interaction between the VRWs becomes possible, and thus they grow exponentially. In a barotropic system, even if Rayleigh's condition is satisfied, the azimuthal counter propagation of VRWs with azimuthal wavenumber m = 1 is so strong that phase-locking between them cannot occur, and thus they cannot grow exponentially. In a quasi-geostrophic system, however, the upper and lower VRWs of the first baroclinic vertical mode are equal in magnitude and have opposite signs. Because of this baroclinic structure, the azimuthal counter propagation of the horizontally aligned VRWs is suppressed by the vertical interactions between the upper and lower VRWs. Consequently, horizontally aligned VRWs with azimuthal wavenumber m = 1 may become phaselocked, and hence they may grow exponentially. By analytically solving the linear problem of VRWs in a quasi-geostrophic system, we show that this is indeed the case.