## Instability of nonlinear wave-current interactions in a modified equatorial $$\beta$$-plane approximation.(English)Zbl 1411.76015

Summary: We present an instability analysis of some exact and explicit solutions to the geophysical equatorial $$\beta$$-plane equations incorporating a gravitational-correction term. A criterion for the instability is given by means of the short-wavelength perturbation method. Thresholds for both, a solution with a zonal current under constant density and a solution admitting stratification, are derived and expressed in terms of the steepness of the waves.

### MSC:

 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76E20 Stability and instability of geophysical and astrophysical flows 37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents
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### References:

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