Anisotropic Navier-Stokes equations in a bounded cylindrical domain

Navier-Stokes equations with anisotropic viscosity are classical in geophysical fluid dynamics. Instead of choosing a classical viscosity −ν(∂ 1 + ∂ 2 2 + ∂ 2 3) in the case of three-dimensional fluids, meteorologists often modelize turbulent flows by putting a viscosity of the form −νh(∂ 2 1 + ∂ 2 2) − νv∂ 2 3 , where νv is usually much smaller than νh and thus can be neglected (see Chapter 4 of the book of Pedlovsky [14] for a detailed discussion). More precisely, in geophysical fluids, the rotation of the earth plays a primordial role. This Coriolis force introduces a penalized skew-symmetric term εu × e3 into the equations, where ε > 0 is the Rossby number and e3 = (0, 0, 1) is the unit vertical vector. This leads to an asymmetry between the horizontal and vertical motions. By the Taylor-Proudman theorem (see [14] and [16]), the fluid tends to have a two-dimensional behavior, far from the boundary of the domain. When the fluid evolves between two parallel plates with homogeneous Dirichlet boundary conditions, Ekman boundary layers of the form UBL(x1, x2, ε x3) appear near the boundary. In order to compensate the term εUBL × e3 by the term −νv∂ 2 3UBL, we need to impose that νv = βε, for β > 0 (see [6] and also [2]).