Propagation of hydormagnetic planetary waves on a beta-plane through magnetic and velocity shear

The propagation of hydromagnetic planetary waves in a rotating thin shell of fluid in regions of magnetic and velocity shears is studied using the β-plane approximation. In slowly varying shear the use of the WKBJ approximation makes it possible to construct the various types of ray trajectory that can occur and consequently the conditions that give rise to critical-latitude phenomena and trapping are deduced. The opposite extreme to the WKBJ limit, namely reflexion and refraction of waves by a current-vortex sheet, is also analysed. In this case the conditions that lead to wave amplification (or over-reflexion) are investigated. Qualitatively, it is found that reflected Alfvén modes are amplified if the jump in the flow speed across the sheet lies between two speeds which are respectively greater and less than the sum of the Alfvén speeds on either side of the sheet. Also, Rossby waves incident upon a sufficiently strong easterly flow can suffer over-reflexion. The general case of reflexion and refraction at a finite double (magnetic and velocity) shear layer is discussed. In analogy with the invariance of ‘wave action’ of gravity waves in a shear flow we construct a quantity A which is invariant except at critical latitudes, where it is discontinuous. By using the asymptotic solutions near these critical latitudes and by adopting the proper matching procedure for the solutions on either side of these latitudes it is possible to relate the two constant values of A on either side of each critical latitude. These general results are then applied to various profiles of shear flow and magnetic field so as to elucidate the manner in which an incident wave is reflected from and transmitted through a double layer.