Inertial waves and an initial-value problem for a thin spherical rotating fluid shell

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Abstract
Natural modes of oscillation of a vanishingly thin spherical rotating fluid shell, with frequencies σ less than twice the angular velocity Ω, were found by Haurwitz (1940). Their validity is, however, put in question by the presence of a singularity at critical co-latitudes θc: 2Ω cosθc = σ in the O(ε) term of an expansion in the relative shell thickness ε (Stewartson & Rickard 1969). The problem is investigated here by considering the evolution of flow from a specified initial distribution. The principal features are as follows:(i) The O(1) natural modes of Haurwitz, decaying on a time scale Ω−1ε−2.(ii) Corrections to (i), regular and of magnitude O(ε) except near critical latitudes.(iii) Essentially transient inertial waves of magnitude O(ε).(iv) Inertial waves of magnitude O(ε½) with the natural-mode frequencies σ and generated by (i) at critical latitudes.On a time scale Ω−1ε−1, (iii) and (iv) develop vertical and horizontal length scales ε and propagate throughout the ocean. The continuing energy transfer from (i) to (iv), at a rate O(ε2Ω), appears to be the principal respect in which (i) and (ii) fail to constitute a conventional normal mode.

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