A model equation for nonlinear wavelength selection and amplitude evolution of frontal waves |
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Authors: | M Ghil N Paldor |
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Institution: | (1) Institute of Geophysics and Planetary Physics, University of California, 90024-1567 Los Angeles, California, USA;(2) Department of Atmospheric Sciences, UCLA, USA;(3) Present address: Department of Atmospheric Sciences, The Hebrew University of Jerusalem, 91904, Israel |
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Abstract: | Summary We study a model equation describing the temporal evolution of nonlinear finite-amplitude waves on a density front in a rotating
fluid. The linear spectrum includes an unstable interval where exponential growth of the amplitude is expected. It is shown
that the length scale of the waves in the nonlinear situation is determined by the linear instabilities; the effect of the
nonlinearities is to limit the amplitude's growth, leaving the wavelength unchanged. When linearly stable waves are prescribed
as initial data, a short interval of rapid decrease in amplitude is encountered first, followed by a transfer of energy to
the unstable part of the spectrum, where the fastest growing mode starts to dominate. A localized disturbance is broken up
into its Fourier components, the linearly unstable modes grow at the expense of all other modes, and final amplitudes are
determined by the nonlinear term. Periodic evolution of linearly unstable waves in the nonlinear situation is also observed.
Based on the numerical results, the existence of low-order chaos in the partial differential equation governing weakly nonlinear
wave evolution is conjectured. |
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Keywords: | frontal instability weakly nonlinear waves regularization energy integrals reduction to finite dimensions |
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