Mean zonal flow generated by azimuthal harmonic forcing in a rotating cylinder

In this paper, we study analytically the flow in a rotating container subjected to an azimuthal forcing. We show that this mechanical forcing generates a correction to the solid body rotation called mean zonal flow, similar to the time oscillation of the rotation rate of an axisymmetric container. This axisymmetric correction induced by nonlinear effects in the Ekman layers modifies the solid body rotation of the fluid in the container. At the leading order, the contribution in the bulk is shown to be an azimuthal flow which scales as the square of the amplitude of the multipolar deformation and is independent of the Ekman number. We also show that the mean zonal flow depends on the symmetry of the angular forcing n and the ratio of the angular rate of the deformation to the angular rate of the cylinder Omega(R) = Omega(orb)/Omega(spin). We found that for an elliptical forcing, n = 2, the rotation rate of the zonal flow does not depend on the radial position. In addition, the angular rate is found to be asymmetric with respect to Omega(R). These scalings are similar to the time harmonic forcing in a cylinder. The particular case of a tidal forcing is also considered.