On the barotropic instability using wave ray theory

Abstract

Based on the spherical Rossby wave ray theory, dynamic characteristics of the barotropic disturbances associated with the westerly jet are discussed, and the barotropic instability conditions are reconsidered. The well-known Rayleigh-Kuo instability criterion indicates that the gradient of potential vorticity must change sign in the unstable area. Kuo uses the method of eigenvalues by assuming disturbances to be simple harmonic waves; however, in this work we assume the atmosphere to be a slowly varying medium, and calculate the amplitude of the Rossby wave by ray tracing. We find a band where the quasi-geostrophic potential vorticity gradient ( β M) is negative, and the amplitude of the northward perturbation grows substantially in this area, making it an unstable region for northward perturbations. In addition the perturbations cannot cross the line of β M=0, and therefore, this is referred to as the “trapped line”. Characteristics of the nonstationary Rossby wave are first analyzed using an ideal zonal mean wind profile u = 30 sin 8 ( 2 φ ) + 0.2 , with the “trapped line” located at 58°N and 70°N, and β M l = 8, but with different zonal wave numbers k . The results suggest that the behavior of the disturbances depends on the initial wave numbers k and l . Northward disturbances with smaller k can propagate further, while disturbances with larger k cannot reach high latitudes and turn back towards the south at turning points β M / ( u M − σ k ) − k 2 = 0. As such, only long Rossby waves with small k values can reach the “trapped line”, while short Rossby waves with larger k values cannot reach it. Long Rossby waves with k = 1, l = 8, first propagate northwest, and then turn eastwards at the “trapped line”. Their energy increases while propagating northward, reaching its maximum in the northern part of the jet stream, and then decays rapidly to zero near the “trapped line”. The energy increases during northward propagation, which means the disturbance obtains energy from the base flow in the southern part of the jet stream, and gives it back to the base flow near the “trapped line”, which completes energy transport from south to north. Further, we also calculated the energy evolution with realistic January and June wind profiles using NCEP data. We conclude that long waves with k <3 are easily trapped by the “trapped line” in both winter and summer, while the propagation of short waves differs substantially between winter and summer. The non-stationary disturbances at low latitudes are easily blocked by easterly winds during winter; however, they can cross easterlies and then propagate to the southern hemisphere in summer, forming a westward wave in the easterlies. It is known that stationary Rossby waves cannot propagate in the easterlies, because only non-stationary waves that meet many conditions can propagate in the easterlies; therefore, this scenario will not be detailed in this paper.