On the dispersion relation for trapped internal waves

Abstract

An analysis is constructed in order to estimate the dispersion relation for internal waves trapped in a layer and propagating linearly in a fluid of infinite depth with a rigid surface. The main interest is in predicting the structure of internal wave wakes, but the results are applicable to any internal waves. It is demonstrated that, in general 1/cp= 1/CpO+k/ωmax+ ∈(k) wherecpis the wave phase speed for a particular mode,CpOis the phase speed atk= 0, ωmaxis the maximum possible wave angular frequency and ωmax≤NmaxwhereNmaxis the maximum buoyancy frequency. Also, ∈(0) = 0, ∈(k) =o(k) forklarge, and is bounded for finitek.In particular, when ∈(k) can be neglected, the dispersion relation for a lowest mode wave is approximately 1/cp≈ (∫∞0N2(y)ydy)-½+k/ωmax. The eigenvalue problem is analysed for a class of buoyancy frequency squared functionsN2(x) which is taken to be a class of realvalued functions of a real variablexwhere O ≤x∞ such thatN2(x) =O(e-βx) asx→ ∞ and 1/β is an arbitrary length scale. It is demonstrated thatN2(x) can be represented by a power series in e-βx. The eigenfunction equation is constructed for such a function and it is shown that there are two cases of the equation which have solutions in terms of known functions (Bessel functions and confluent hypergeometric functions). For these two cases it is shown that ∈(k) can be neglected and that, in addition, ωmax=Nmax. More generally, it is demonstrated that whenk→ ∞ it is possible to approximate the equation uniformly in such a way that it can be compared with the confluent hypergeometric equation. The eigenvalues are then, approximately, zeros of the Whittaker functions. The main result which follows from this approach is that ifN2(x) isO(e-βx) asx→ ∞ and has a maximum valueN2maxthen a sufficient condition for 1/cp∼k/Nmaxto hold for largekfor the lowest mode is thatN2(t)/tis convex for O ≤t≤ 1 wheret= e-βx.