Internal Travelling Waves in the Limit of a Discontinuously Stratified Fluid |
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Authors: | Guillaume James |
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Institution: | Institut National des Sciences Appliquées?Département GMM?135 avenue de Rangueil?31077 Toulouse cedex 4, France, FR
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Abstract: | We consider internal travelling waves in a perfect stratified fluid, in the singular limit case when smooth stratifications approach a discontinuous two-layer profile. Our analysis concerns two-dimensional waves of small amplitude, propagating in an infinite horizontal strip of finite depth. The problems with smooth or discontinuous stratification are formulated as a unifying spatial evolution problem, where the stratification ρ plays the role of a functional parameter. The vector field is not smooth with respect to ρ, but has some weak continuity. When the Froude number is close to a critical value, we reduce the problem to one on a center manifold in a neighborhood of the trivial state independent of ρ (for the usual topology). Considering a weaker topology, we prove the continuity in ρ of the center manifold. Then the small solutions are described by an ordinary differential equation in ?2, which depends continuously on ρ in the C k norm. |
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