Numerical calculation of planar geophysical flows of an inviscid incompressible fluid by a meshfree-spectral method

In this article, a meshfree-spectral method for numerical investigation of dynamics of planar geophysical flows is proposed. We investigate inviscid incompressible fluid flows with the presence of planetary rotation.Mathematically this problem is described by the non-steady system of two partial differential equations in terms of stream and vorticity functions with different boundary conditions (closed flow region and periodic conditions). The proposed method is based on several assumptions. First of all, the vorticity field is given by its values on the set of particles. The function of vorticity distribution is approximated by piecewise cubic polynomials.Coefficients of polynomials are found by least squares method. The stream function is calculated by using the spectral global Bubnov–Galerkin method at each time step. The dynamics of fluid particles is calculated by pseudo-symplectic Runge–Kutta method. A detailed version of the method for periodic boundary conditions is described in this article for the first time. The adequacy of numerical scheme was examined on test examples.
The dynamics of the configuration of four identical circular vortex patches with constant vorticity located at the vertices of a square with a center at the pole is investigated by numerical experiments. The effect of planetary rotation and the radius of patches on the dynamics and formation of vortex structures is studied. It is shown that, depending on the direction of rotation, the Coriolis force can enhance or slow down the processes of interaction and mixing of the distributed vortices. At large radii the vortex structure does not stabilize.