On the stability of the Ekman boundary layer

Abstract

We present a result on well-posedness and stability of the Ekman boundary layer problem in the space FM(R 2 , L 2 (R+) 3 ), i.e., in the space of L 2 (R+) 3 -valued Fourier transformed finite Radon measures. In particular we obtain stability in the angle velocity of rotation, which is important in the analysis of fast oscillating singular limits. Description and main resultThe Ekman boundary layer problem is a meteorological model for the motion of a rotating fluid (atmosphere) inside a boundary layer, appearing in between a uniform geostrophic flow (wind) and a solid boundary (earth) at which the no slip condition applies. Mathematically this situation is described by the Navier-Stokes equations with Coriolis forceHere the unknowns u and p denote velocity and pressure of the fluid respectively, whereas e 3 = (0, 0, 1) and the parameters ν and Ω correspond to viscosity and angle velocity of the rotation around the x 3 -axis.There is a famous stationary and exact solution to (1) called Ekman spiral and which is given by the vectorHere δ = ν/|Ω| denotes the layer thickness and U ∞ the velocity of the geostrophic flow away from the boundary pointing in x 1 direction. The corresponding pressure to U E is given by p E (x 2 ) = −ΩU ∞ x 2 . Remarkable persistent stability of the Ekman spiral in atmospheric and oceanic boundary layers has been noticed in geophysical literature. Here we are interested in stability in the parameters t, Ω, ν, and δ. In particular in the existence of solutions with norms uniformly bounded in Ω in spaces including functions nondecaying at infinity. Results of this type are essential in studies of statistical properties of turbulence, see e.g. [6,7], and in the analysis of fast oscillating singular limits for system (1), see e.g. [1,5].The observation that U E depends on the x 3 variable only, i.e., that it has infinite energy, and thatleads to two natural requirements on a potential class E of initial data:The class E should include functions nondecreasing at infinity in tangential direction.(A first result on well-posedness for system (1) is obtained in [4] for u 0 in the classNote that the spaceḂ3 -valued almost periodic functions. Thus, E 1 satisfies (i) and (ii). However, the class E 1 seems to be inappropriate for stability investigations. This relies essentially on the fact that the Poincaré-Riesz semigroup (e −tBΩ ) t≥0 generated by the Coriolis operator B Ω u := 2Ωe 3 × u proved to be unbounded in t *