An exact solution for two-dimensional frictionless motion in the atmosphere

A solution of the nonlinear problem for determining the wind velocity in frictionless atmosphere (the gradient wind) under given geopotential (pressure) field is proposed. The approach is analytical and is based on quadratic polynomial approximation of the geopotential field and linear approximation of the wind velocity field with respect to x and y, the coefficients of the expansions being functions of the time t. The derived system of ordinary nonlinear differential equations is analyzed as a dynamical system. Exact analytical solutions are found for some par-ticular cases, Some of their properties bear a resemblance to those or really existing atmospheric vortices (cyclones and anticyclones).