Internal waves in a sheeted thermocline

Abstract

The eigenvalue problem for internal waves in a thermocline for which the density profile may exhibit finite discontinuities is formulated as a homogeneous Fredholm integral equation. The corresponding quadratic functional yields both upper and lower bounds for the dominant-mode eigenvalue and lower bounds for the remaining eigenvalues (which are proportional to the square of the wave speed) for any value of the wavenumber. The lower bounds based on simple trial functions appear to provide adequate approximations for typical density profiles and all wavenumbers. The upper bound (which is based on a Schwarz inequality and does not require the choice of a trial function) is sharp only for relatively long waves. A simple approximation is developed for the effect of the free surface on the dominant mode. Two algebraic formulations are given for a thermocline of homogeneous layers separated by a finite number of sheets, across each of which the density is discontinuous. The various approximations are compared with the exact results for a thermocline with a hyperbolic-tangent density profile, a three-sheeted thermocline, and a five-sheeted model of the summer thermocline in the Mediterranean.