On the propagation, reflexion, transmission and stability of atmospheric Rossby-gravity waves on a beta-plane in the presence of latitudinally sheared zonal flows

Abstract

The propagation properties of Rossby-gravity waves in an isothermal atmosphere on a beta-plane are investigated in the presence of a latitudinally sheared zonal now. The perturbation equation is found to possess seven regular singularities provided the fluid is non -Boussinesq, and only five for Boussinesq fluids. In slowly varying shear a local dispersion relation is derived and used to study the wave normal surfaces and ray trajectories. The cross sections of the wave normal surfaces in horizontal planes possess three critical latitudes occurring where the intrinsic frequency w takes the values 0, + N ,where N is the Brunt—Väisällä frequency. The former is the usual Ross by wave critical latitude (R.w.c.l.) and the latter are essentially gravity wave critical latitudes (g.w.c.l.). Waves can propagate only on one side of a R.w.c.l. while propagation is possible on both sides of a g.w.c.l. provided the vertical wavenumber, m , there is real and non-zero. Also for real values of m and provided the atmosphere is non -Boussinesq the g.w.c.l. exhibits valve-like behaviour. Such valve behaviour is shown to be responsible for aiding high frequency waves (i.e. gravity waves) to penetrate jet-like wind streams and may facilitate the transfer of energy and momentum across latitudes. The full wave treatment shows that the system possesses a wave-invariant which has a simple physical interpretation only when m is real in which case it represents the conservation of the total northward wave energy flux. The invariant is used, together with the legitimate solutions near the critical latitudes, to study the influence of each of the critical latitudes on the intensity of the wave. It is found that the R.w.c.l. can be associated with energy absorption or emission, depending on certain specified conditions, but the g.w.c.l. is always associated with energy absorption although the amount of energy absorbed depends crucially on whether m is real or imaginary. The reflexion, transmission and stability of atmospheric waves by a finite shear, thickness L , are also studied, by using a full wave treatment in the presence of general flow profiles. For smoothly varying shear flows the use of the wave-invariant yields a relation between the reflexion and transmission coefficients. It is then deduced that in the absence of critical latitudes within the shear over-reflexion (i.e. the amplitude of the reflected wave exceeds that of the incident one) is possible only for real values of m in which case planetary waves incident on the shear are transmitted as gravity waves on the far side of the shear. Such over-reflecting régimes are, however, due to the presence of natural modes of the system. Moreover, it is found possible to isolate certain situations in which the R.w.c.l. within the shear enhances over-reflexion. The situation when the shear is linear and thin is studied analytically. Explicit expressions for the reflexion and transmission coefficients are obtained. It is then shown that over-reflexion is present in a stable shear in which case the energy is extracted from the mean flow primarily at the R.w.c.l. The influence of a general flow on the R.w.c.l. is also studied in a special case to show that the reflectivity and stability properties of the thin shear are strongly dependent on the type of shear present. Comparison of these results with those obtained for the corresponding vortex sheet show wide disagreement. A general criterion for determining whether a vortex sheet will adequately represent the corresponding thin shear layer is offered in the final section.