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Hugoniot-Maslov chains for solitary vortices of the shallow water equations. II: The analysis of solutions of the truncated chain and an approximate description of possible trajectories of mesoscale vortices (typhoons). (English) Zbl 1060.76533
Summary: This is the second part of the paper [Part I, cf. Russ. J. Math. Phys. 6, No. 2, 137–173 (1999; Zbl 1059.76506)]. Here we analytically and numerically study some special families of solutions to truncated chains of Hugoniot-Maslov equations for solitary vortices of the shallow wave equations. The solutions belonging to these families are in a sense critical and describe sufficiently smooth approximate trajectories of solitary vortices. The existence of such solutions is closely related to the so-called $$b$$-effect (the slow dependence of Coriolis frequency on latitude). The asymptotic analysis of such equations, which is essentially based on the (partial) averaging method, quite unexpectedly results in a simple system of second-order equations that is integrable by quadratures and coincides in some approximation with the physical pendulum equations. We compare the approximate trajectories of solitary vortices thus obtained with the trajectories of several actual typhoons.

MSC:
 76B25 Solitary waves for incompressible inviscid fluids 76B47 Vortex flows for incompressible inviscid fluids 86A10 Meteorology and atmospheric physics