We study the system of Euler equations with the so-called Ekman damping in the whole 2D space. The global well-posedness and dissipativity for the weak infinite energy solutions of this problem in the uniformly local spaces is verified based on the further development of the weighted energy theory for the Navier–Stokes and Euler type problems. In addition, the existence of weak locally compact global attractor is proved and some extra compactness of this attractor is obtained. © 2015, The Author(s).
A hydrodynamic model of an oceanic gyre is proposed. The ﬂuid motion is considered in the leading-order shallowwater approximation in the spherical Lagrangian 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.