The linear stability of a wavetrain propagating on water of variable depth

Girish K. Rajan, Diane Marie Henderson

Research output: Contribution to journalArticle

Abstract

A narrow-banded spectrum of gravity waves propagating in one horizontal direction at the surface of water with finite depth may be governed by a nonlinear Schrödinger equation (NLSE). If the depth is nonuniform, then the coefficients in the NLSE are variable and an additional linear term due to conservation of wave action flux is present, which may act as a dissipative term that causes the wave envelope to broaden and decrease in amplitude or which may act as a growth term that causes the wave envelope to focus and increase in amplitude. The addition of linear, viscous dissipation either enhances the dissipative process or competes with the growth effect. Here we consider such a variable-coefficient, dissipative NLSE. We find a solution that has uniform amplitude in the appropriate reference frame and examine its linear stability. We find that for waves propagating into shallower water, both bathymetry and viscous dissipation act to stabilize the modulational instability. For waves propagating into deeper water, bathymetry acts to enhance instability, but viscous dissipation eventually causes stabilization for small enough initial perturbation amplitudes.

Original languageEnglish (US)
Pages (from-to)2030-2041
Number of pages12
JournalSIAM Journal on Applied Mathematics
Volume76
Issue number5
DOIs
StatePublished - Jan 1 2016

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Linear Stability
Viscous Dissipation
Water
Nonlinear equations
Bathymetry
Nonlinear Equations
Envelope
Term
Modulational Instability
Dissipative Equations
Gravity Waves
Gravity waves
Shallow Water
Variable Coefficients
Conservation
Stabilization
Horizontal
Fluxes
Perturbation
Decrease

All Science Journal Classification (ASJC) codes

  • Applied Mathematics

Cite this

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abstract = "A narrow-banded spectrum of gravity waves propagating in one horizontal direction at the surface of water with finite depth may be governed by a nonlinear Schr{\"o}dinger equation (NLSE). If the depth is nonuniform, then the coefficients in the NLSE are variable and an additional linear term due to conservation of wave action flux is present, which may act as a dissipative term that causes the wave envelope to broaden and decrease in amplitude or which may act as a growth term that causes the wave envelope to focus and increase in amplitude. The addition of linear, viscous dissipation either enhances the dissipative process or competes with the growth effect. Here we consider such a variable-coefficient, dissipative NLSE. We find a solution that has uniform amplitude in the appropriate reference frame and examine its linear stability. We find that for waves propagating into shallower water, both bathymetry and viscous dissipation act to stabilize the modulational instability. For waves propagating into deeper water, bathymetry acts to enhance instability, but viscous dissipation eventually causes stabilization for small enough initial perturbation amplitudes.",
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The linear stability of a wavetrain propagating on water of variable depth. / Rajan, Girish K.; Henderson, Diane Marie.

In: SIAM Journal on Applied Mathematics, Vol. 76, No. 5, 01.01.2016, p. 2030-2041.

Research output: Contribution to journalArticle

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