Approximate solution for Euler equations ofstratified water via numerical solution of coupled KdV system

A A Halim,S B Leble,S P Kshevetskii

doi:10.1155/s0161171203212242

A A Halim, S B Leble + Show 1 more

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https://doi.org/10.1155/s0161171203212242

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Abstract

We consider Euler equations with stratified background state that is valid for internal water waves. The solution of the initial‐boundary problem for Boussinesq approximation in the waveguide mode is presented in terms of the stream function. The orthogonal eigenfunctions describe a vertical shape of the internal wave modes and satisfy a Sturm‐Liouville problem. The horizontal profile is defined by a coupled KdV system which is numerically solved via a finite‐difference scheme for which we prove the convergence and stability. Together with the solution of the Sturm‐Liouville problem, the stream functions give the internal waves profile.

Highlights

The basic system of Euler equations for internal water waves in two dimensions, with a stable stratified ambient state and the buoyancy frequency N(z), is ux + wz = 0, ρout = −ρo →v, → ∇u − px, ρowt = −ρo w − pz − ρ g, Tt + wTz =− T (1.1)where u, v are velocity components, ρo is the density, p is the pressure, ρ g is the body force due to stratification, Tz is the vertical background temperature gradient, and T is the temperature variable [3]
The internal water waves are described by system (1.1). The solution of this system is constructed as the representation for the stream function (1.5), where Zn(z) are solutions of the correspondent Sturm-Liouville problem (1.6), Zzz +(N2/cn2 )Z = 0, Z(0) = Z(h) = 0, and describe a vertical shape of the wave modes
We can select the initial perturbation for the stream function (1.5) which has the general form ψ(z, x, 0) = Zn(z)θn(x, 0) = φ(x, z) = φ1(x)φ2(z)

Summary

Introduction

Substitute in (1.4) by the stream function of the form ψ(z, x, t) = Zm(z)θm(x, t), m multiply by Zn, integrate with respect to z, and use the separation of variables that give The equations of the separated propagated modes are obtained by substituting θtn = un, cnθxn = vn, so (1.7) becomes unt − cnvxn cn3 β2 N2 vxnxx Operating P+, P− on (1.9) and using (1.12), we obtain the equations for the separated modes φn+, φn− as φtn+

Results

Conclusion