Non-Stationary Flows of a Rotating Fluid with Three Coriolis Parameters

The paper presents a class of exact solutions of the Navier–Stokes equations that allows the original system of partial differential equations to be reduced analytically to a system of ordinary differential equations. The resulting ODE system is integrated numerically with high accuracy for three representative sets of Coriolis parameters. Stratified flows of a viscous incompressible fluid in a rotating coordinate system with an arbitrary angular velocity vector are considered. The solutions are constructed in the class of velocity fields linear in horizontal coordinates and pressure quadratic in horizontal coordinates. Three problems are considered: equatorial (vertical Coriolis parameter is zero), mid-latitude (three non-zero parameters; the Ekman boundary layer with a no‑slip bottom is included), and fast rotation (doubled angular velocity at the equator). A numerical analysis of the velocity and pressure fields in time is carried out. It is found that at the equator the maximum velocity reaches 1.02 m/s (Reynolds number 1023), while at mid-latitudes due to Ekman damping the velocity drops to 0.039 m/s (Re = 39). Under fast rotation (doubled angular velocity) the velocity increases to 0.075 m/s but remains significantly lower than the equatorial one. The obtained solutions can be used for verifying numerical models of geophysical hydrodynamics and for planning laboratory experiments. The evolution of the expansion coefficients is studied; pressure contours and velocity vector fields are constructed at different times.