EXISTENCE AND NUMERICAL APPROXIMATION OF SOLUTIONS OF AN IMPROVED INTERNAL WAVE MODEL

Juan Carlos Munoz Grajales

doi:10.3846/13926292.2014.924039

Abstract

In this paper we establish local existence of solutions for a new model to describe the propagation of an internal wave propagating at the interface of two immiscible fluids with constant densities, contained at rest in a long channel with a horizontal rigid top and bottom. We also introduce a spectral-type numerical scheme to approximate the solutions of the corresponding Cauchy problem and perform a complete error analysis of the semidiscrete scheme.

Highlights

IntroductionTo describe the propagation of a weakly nonlinear internal wave propagating at the interface of two immiscible fluids with constant densities ρ1, ρ2 with ρ2/ρ1 > 1 (for stable stratification), contained at rest in a long channel with a horizontal rigid top and bottom, and the thickness of the lower layer is assumed to be effectively infinite (deep water limit)
In this work we derive the following system written in dimensionless variables: ζt − (1 − αζ)u x = 6 ζxxt, ut + αuux + 1 − ρ2 ρ1 ζx = ρ2 ρ1 H(uxt) +6 uxxt, ζ(x, 0) = ζ0(x), u(x, 0) = u0(x) (1.1)to describe the propagation of a weakly nonlinear internal wave propagating at the interface of two immiscible fluids with constant densities ρ1, ρ2 with ρ2/ρ1 > 1, contained at rest in a long channel with a horizontal rigid top and bottom, and the thickness of the lower layer is assumed to be effectively infinite
6 uxxt, ζ(x, 0) = ζ0(x), u(x, 0) = u0(x) to describe the propagation of a weakly nonlinear internal wave propagating at the interface of two immiscible fluids with constant densities ρ1, ρ2 with ρ2/ρ1 > 1, contained at rest in a long channel with a horizontal rigid top and bottom, and the thickness of the lower layer is assumed to be effectively infinite

Summary

Introduction

To describe the propagation of a weakly nonlinear internal wave propagating at the interface of two immiscible fluids with constant densities ρ1, ρ2 with ρ2/ρ1 > 1 (for stable stratification), contained at rest in a long channel with a horizontal rigid top and bottom, and the thickness of the lower layer is assumed to be effectively infinite (deep water limit). The constants α and are small positive real numbers that measure the intensity of nonlinear and dispersive effects, respectively. J.C. Munoz Grajales depth z = 1 − 2/3, the function ζ = ζ(x, t) is the wave amplitude at the point x and time t, measured with respect to the rest level of the two-fluid interface, and Hf (x) denotes the Hilbert transform defined by ∞ f (y) Hf (x) = p.v.

Objectives

Results

Conclusion