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Linear stability of fully developed flows of air over water is carried out in order to study nonlinear effects in the generation of solitons by wind. A linear stability analysis of the basic flow is made and the conditions at which solitons first begin to grow is determined. Then, following [10], the non-linear stability of the flow is examined and the quintic non-linear Schr¨odinger equation is derived for the amplitude of disturbances. The coefficients of the non-linear Schr¨odinger equation are calculated from the eigenvalue problem which determines the stability of air-water interface.

An asymptotic and a numerical stability analysis is carried out to determine the neutrally stable flow conditions for air-sea interface. Four different profiles are considered for the airflow blowing over the surface of the sea, namely, plane Couette flow (pCf), plane Poiseuille flow (pPf), laminar and turbulent boundary layer (L,TBL) profiles. For each of the above cases the shear flow counterpart in the water is assumed to be a pPf.

A nonlinear stability analysis results in the nonlinear Schr¨odinger equation

∂A − a22A = d1 A + κA|A|2 + ϖA|A|4 ,
∂τ ∂ξ2 d1r

where ξ and τ are local variables, and the amplitude of the surface wave is proportional to A. The complex constants a2, d1, κ and ϖ are evaluated from the linear stability of the two-fluid interface. The profile for the initial condition considered here is that of the Stokes wave

η = a cos kx + 1/2 a2k cos 2kx,

where a is the amplitude and k is the wavenumber of the surface Stokes wave.

It is shown that the above amplitude equation produces ‘snake’ solitons [9] for pCf, pPf and LBL profiles, with striking similarities. On the other hand, for TBL we observe a very violent surface motion. For cases of pCf and LBL remarkable similarity is observed with observations made at sea.

We conclude that the effect of nonlinearity in the airflow over the sea surface is much larger than nonlinear interactions in the water, and hence it is not possible to decouple the motion in the air and the water for finite amplitude wind-wave interactions, particularly in the case of wind-generated solitons in shallow waters.

Publication Title

Advances and Applications in Fluid Dynamics


Pushpa Publishing House