Free surface equatorial two-layer flows in the $\beta$-plane approximation with discontinuous stratification

An exact solution of the nonlinear governing equations in the $\beta$-plane approximation is established and analysed in this paper. Such solution describes a purely azimuthal equatorial two-layer flows with free surface and discontinuous stratification. We present the explicit solution for the velocity field and the pressure. Moreover, we derive the implicit formulas for the shape of the free surface and the interface. Finally, the monotonicity properties between the free surface and the pressure on the free surface, as well as the regularity result for the interface are proved, respectively.

2010 Mathematics Subject Classification

Primary 35Q31. Secondary 35Q35, 35R35.

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Received 29 December 2023

Published 21 May 2024