The methods developed in this book can be applied to various physical systems. We will not detail all the possible applications and will only quote three systems arising in magnetohydrodynamics (MHD) and meteorology, namely conducting fluids in a strong external “large scale” magnetic field, a classical MHD system with high rotation, and the quasigeostrophic limit. The main theorems of this book can be extended to these situations. The theory of rotating fluids is very close to the theory of conducting fluids in a strong magnetic field. Namely the Lorenz force and the Coriolis force have almost the same form, up to Ohm’s law. The common feature is that these phenomena appear as singular perturbation skew-symmetric operators. The simplest equations in MHD are Navier–Stokes equations coupled with Ohm’s law and the Lorenz force where ∇φ is the electric field, j the current, and e the direction of the imposed magnetic field. In this case ε is called the Hartmann number. In physical situations, like the geodynamo (study of the magnetic field of the Earth), it is really small, of order 10−5–10−10, much smaller than the Rossby number. These equations are the simplest model in geomagnetism and in particular in the geodynamo. As ε→0 the flow tends to become independent of x3. This is not valid near boundaries. For horizontal boundaries, Hartmann layers play the role of Ekman layers and in the layer the velocity is given by The critical Reynolds number for linear instability is very high, of order Rec ∼ 104. The main reason is that there is no inflexion point in the boundary layer profile (10.1.2), therefore it is harder to destabilize than the Ekman layer since the Hartmann profile is linearly stable for the inviscid model associated with (10.1.1). As for Ekman layers, Hartmann layers are stable for Re<Rec and unstable for Re>Rec. There is also something similar to Ekman pumping, which is responsible for friction and energy dissipation. Vertical layers are simpler than for rotating fluids since there is only one layer, of size (εν)1/4. We refer to for physical studies.
An algorithm to generate anisotropic rotating fluids with vanishing viscosity