The flow field and bare-spot formation in spin-up from rest of a two-layer fluid about a vertical axis

The spin-up from rest of a two-layer fluid with a free surface in a cylindrical container rotating about a vertical axis is investigated for small Ekman numbers. Numerical results from the axisymmetric Navier–Stokes equations, supported by comparisons with improved boundary-layer approximations, show that the Ekman-type layer on the bottom pushes the dense fluid of the lower layer to the periphery, and consequently the interface between the layers curves upward near the sidewall and descends near the centre. When the lower layer of fluid is sufficiently thin a bare spot appears at the bottom, i.e. a region where the light fluid is in direct contact with the horizontal boundary. The lower-layer fluid is spun-up quickly by the bottom Ekman layer, but the angular motion in the upper layer is provided by the much weaker detached Ekman layer on the interface between the two fluids, and hence the global spin-up process is prolonged compared with the homogeneous fluid case. The influence of the various dimensionless parameters and the connection with the continuous stratified case are discussed.