On some homogenization problems from shallow water theory

This note is devoted to the effect of topography on geophysical flows. We consider two models derived from shallow water theory: the quasigeostrophic equation and the lake equation. Small scale variations of topography appear in these models through a periodic function, of small wavelength ε. The asymptotic limit as ε goes to zero reveals homogenization problems in which the cell and averaged equations are both nonlinear. In the spirit of article [P.-L. Lions and N. Masmoudi, J. Math. Pures Appl. (9), 84(1):1–20, (2005)], we derive rigorously the limit systems, through the notion of two-scale convergence.