The spin-up of fluid in a rectangular container with sloping bottom

Abstract

The spin-up from rest of a homogeneous free-surface fluid contained in a rectangular tank with an inclined bottom has been studied in the laboratory. As in the case of a tank without bottom topography, it is found that in the spin-up process leading to the ultimate state of rigid-body rotation a number of stages can be distinguished, these being (i) the starting flow, characterized by zero absolute vorticity, (ii) the viscous generation of cyclonic vorticity at the lateral tank walls, leading to flow separation, and (iii) the formation of cyclonic and anticyclonic flow cells, which show a complicated interaction. When the topography steepness is small, these cells become organized in a regular array similar to what is observed in the non-sloping bottom case. For steeper topography, however, no organization into a regular cellular pattern is observed, and the relative fluid motion remains unsteady and irregular until eventually it has decayed owing to the spin-ip/spin-down mechanism provided by the Ekman layer at the tank bottom. During the first stage of the adjustment process the starting flow takes on the appearance of a large anticyclonic cell that fills the fluid domain entirely. Depending on the ratio of the horizontal and vertical lengthscales of the tank this cell is either symmetric or asymmetric, with a higher density of streamlines in the deeper part of the tank. The coupled vorticity equation, governing the depth-independent part of the starting flow, and the potential equation describing its depth-dependent part have been solved analytically, and the comparison between these results and observational data is generally good.