On internal solitary waves

Abstract

Earlier investigations of internal solitary waves in stable, stratified shear flows are generalized on the hypothesis that cubic and quadratic nonlinearity may be of comparable significance—or, equivalent, that β = O(α²), where α = a/h ≪ 1 and β = (h/l)² ≪ 1 for a wave of amplitude a (which may be positive or negative) and length l in a fluid of depth h. An infinite, discrete set of internal solitary waves is possible for prescribed density and horizontal velocity in the primary flow, and to each of these modes there corresponds a relation β = β(α),0 |β| |β_n|. Cubic nonlinearity is significant vis-à-vis quadratic nonlinearity if and only if β_n = O(β), and h|β_n| then appears as a maximum achievable amplitude. The limit β_n → 0 (which corresponds to a critical combination of the basic flow parameters) implies a → 0 and l → ∞ if the mass carried by the wave is fixed.