Euler equations in geophysics and astrophysics

F. H. Busse, University of Bayreuth

Euler equations used in geophysics and astrophysics typically include the Coriolis force. This property extends the range for applications of inviscid descriptions of flows since dissipative processes are often confined to thin Ekman layers. Features such as geostrophic flows, Rossby waves, inertial oscillations and thermal winds are induced by the Coriolis force. The modifications of these basic flows by effects of buoyancy and viscous friction permit the solution of numerous problems through perturbation techniques. The non-uniqueness of vorticity distributions may sometimes cause difficulties. Examples from convection in rotating spheres to flows in precessing cavities will be discussed. Finally the problem of the absence of turbulence in Taylor-Couette configurations in the range -Re_E ≤ Re ≤ τ+(Re_E)²/4τ (with τ ≥ Re_E/2) of Reynolds numbers Re (which includes Keplerian flows) will be addressed. Here Re_E = 2(1708)^1/2 denotes the energy Reynolds number for the small gap limit, and the rotation rate τ = 2Ωd²/ν is made dimensionless with the gap width d and the kinematic viscosity ν.