Journal of the Meteorological Society of Japan. Ser. II
Online ISSN : 2186-9057
Print ISSN : 0026-1165
ISSN-L : 0026-1165
Amplitude of Rossby Wavetrains on a Sphere
Yoshi-Yuki HayashiTaroh Matsuno
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1984 Volume 62 Issue 3 Pages 377-387


The amplitudes of stationary Rossby wave packets propagating in a super-rotational zonal flow are studied with the non-divergent linear barotropic vorticity equation. Discussions are made concerning various pitfalls in determining wave amplitudes on the basis of ray theory. It is recalled that the change of wave amplitude along a ray in a slowly varying medium can be calculated by ray theory only by specifying the ray configuration neighbouring the ray considered (owing to the ∇•Cg term). The poleward increase of stream function amplitude of stationary Rossby wavetrains as emphasized by Hoskins and Karoly (1981) is a consequence of a particular ray configuration implicitly assumed (all longitudinally neighbouring rays pointing the same direction). In their problem, a caustic forms at each turning latitude where northward-directed rays turn southward or vice versa.
Although ∇•Cg is important for amplitude calculation in general, a spherical harmonic solution cosυθ'eiυφ', which is often referred to as a wavetrain along a great circle (θ'=0 in a tilted spherical coordinate system), is shown to be an exception. It is suggested that this solution should be interpreted to consist of an isolated single great circle ray. The direct application.of ray theory is impossible in this case so that a slight modification of the method is suggested.
It is also noted that attention must be paid to zonally symmetric components (k=0) of solutions, especially when the total wave number of stationary wave is small. This is because steady state solutions of the linearized vorticity equation with basic zonal flows contain an ambiguity of arbitrary function of latitude. A physical consideration that the zonal mean flows do not change at the first order of wave amplitude leads to a modification of spherical harmonic solution.

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