Linear
stability of fully developed flows of air over water is carried out in order to
study non-linear 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 nonlinear Schrödinger
equation is derived for the amplitude of disturbances. The coefficients of the
non-linear Schrödinger 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 Schrödinger equation
where
and
are local variables, and the amplitude of the surface wave is
proportional to A. The complex constants
and
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
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.
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