Stabilization of a linearly unstable wave participating in a three-wave resonant interaction |
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| Authors: | N N Romanova I G Yakushkin |
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| Institution: | 1.Obukhov Institute of Atmospheric Physics,Russian Academy of Sciences,Moscow,Russia |
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| Abstract: | An analytical study of the influence of three-wave resonant interactions on the evolution of unstable wave disturbances is
presented in the Kelvin-Helmholtz model. These results may be of interest in analyzing the dynamics of disturbances at the
ocean-atmosphere interface and in two-layer flows which arise in the ocean and are characterized by large gradients of flow
velocity at the boundary of layers. In the case under consideration, the instability arises when eigenfrequencies coincide
in the framework of a single mode and the instability is algebraic. The amplitudes of the two other interacting stable waves
are assumed to be small compared to the amplitude of the third, unstable, mode. The system of amplitude equations for this
case is investigated using the WKB method. As a result, we obtain the formulas coupling the solutions for the time before
and after a transition through a singular point, where the amplitude of the linearly unstable wave has a local minimum. These
formulas give the rule of transformation of the parameter that characterizes a phase shift between fast and slow modes and
determines the behavior of the system. It is shown that, in a transition through a singular point, this parameter changes
randomly. As long as the parameter is positive, the amplitude of the linearly unstable wave remains limited and oscillates
stochastically. In a transition of the parameter through zero, we exit the stabilization region and have an infinite growth
of amplitude. The transition into the instability region is random. However, if the time interval where the amplitude remains
limited is large enough, the scenario of the behavior of the system we have obtained can be treated as the partial stabilization
of instability. The results make it possible for us to investigate the stochasticity caused by the nonlinear interaction of
gravity-capillary waves in a two-layer model of a shear flow. These results are also of interest in analyzing secondary flows
in laboratory facilities modeling the ocean and atmospheric processes. |
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