Fourth-order nonlinear evolution equations for surface gravity waves in the presence of a thin thermocline

Abstract

Two coupled nonlinear evolution equations correct to fourth order in wave steepness are derived for a three-dimensional wave packet in the presence of a thin thermocline. These two coupled equations are reduced to a single equation on the assumption that the space variation of the amplitudes takes place along a line making an arbitrary fixed angle with the direction of propagation of the wave. This single equation is used to study the stability of a uniform wave train. Expressions for maximum growth rate of instability and wave number at marginal stability are obtained. Some of the results are shown graphically. It is found that a thin thermocline has a stabilizing influence and the maximum growth rate of instability decreases with the increase of thermocline depth.

The effect of randomness on the stability of deep water surface gravity waves in the presence of a thin thermocline

“…A fourth order nonlinear evolution equation for a surface gravity wave packet in the presence of a thin thermocline has been obtained by the present authors in a previous paper [2]. By the use of this equation a deterministic growth rate of the instability has also been obtained in that paper.…”

“…A similar study is made in the present paper for surface gravity waves in the presence of a thin thermocline and we start from a fourth order nonlinear evolution equation obtained in our previous paper [2]. A spectral transport equation is derived from this equation.…”

“…£2, is the growth rate of the instability obtained from the third order nonlinear evolution equation and this decreases with the increase in bandwidth a. The deterministic growth rate of the instability obtained in our previous paper [2] is recovered from (52) by setting a = 0. Since by setting a = 0 in (52) we get…”

“…The evolution of the complex amplitude £ ( is governed by the following fourth order nonlinear evolution equation, which has recently been derived by the present 2 , £n and £ 12 being the first two terms in the expansion of £1 in powers of e and where it has been assumed that the space variation of the complex amplitude £ takes place along a line making an arbitrary fixed angle 6 with the direction of propagation of the wave and £ is the co-ordinate along this line. The quantities wave amplitude £, space co-ordinate £ and time r have been made dimensionless by the factors k, k and co respectively.…”

“…The effect of randomness on the stability of surface gravity waves has been investigated by Alber [1] and Crawford, Saffman and Yuen [3], and they started respectively [2] The effect of randomness on the stability of deep water surface gravity waves 191 from the third order nonlinear evolution equation and Zakharov's integral equation [7]. A similar study is made in the present paper for surface gravity waves in the presence of a thin thermocline and we start from a fourth order nonlinear evolution equation obtained in our previous paper [2].…”