Dynamics of Eddies on a Rotating Sphere

A hydrodynamic model of an oceanic gyre is proposed. The fluid motion is considered in the leading-order shallow- water approximation in the spherical Euler coordinates. Motion of liquid particles at the spherical surfaces is studied versus latitude and longitude as unknown variables. The boundary condition at the edge of the gyre is not formulated. An approximation of the “averaged latitude” is introduced when the coefficients of the momentum equation are replaced by constant values corresponding to the latitude of the gyre’s center. It is shown that the resulting set of equations is similar to the equations of plane hydrodynamics. Its analytical solutions containing two arbitrary functions and two arbitrary constants (time frequencies) are obtained. The trajectories of liquid particles represent a superposition of two rotational motions, and their general properties are discussed. A family of the gyres with invariable shape in time is selected. Their outer boundaries either remain motionless or rotate uniformly. An example of the unsteady gyre both rotating and deforming in its shape is studied numerically.