On the spinup and spreadout of a Cartesian gravity current on a slope in a rotating system

Abstract

Ocean gravity currents flow along the inclined ocean floor for long times compared to the planet's rotation period. Their shape and motion is governed by the gravity buoyancy, Coriolis acceleration and friction-induced Ekman-layer spinup circulation. In order to understand this process, we consider the flow of a dense-fluid Boussinesq gravity current of fixed volume over an inclined bottom in a rotating system, in the framework of Cartesian 2.5-dimensional geometry (no dependency on the lateral direction $y$ , but with a non-trivial $y$ -component velocity $v$ due to Coriolis coupling with the main $u$ along the bottom $x$ ). After release from rest in a lock (co-rotating, with two gates creating propagation in ${\pm }x$ -directions), the current forms a quasi-steady geostrophic ‘vein’ of parabolic height profile with a significant lateral velocity $v$ . Subsequently, a spinup process, driven by the Ekman layers on the bottom and interface, appears and prevails for many revolutions, during which $v$ decays and the shape of the interface changes dramatically. We investigate the spinup motion, using an approximate model, for the case of large Rossby number, small Ekman number and small slope $\gamma$ (relevant to oceanic currents). We show that the initial shape of the natural geostrophic vein can be calculated rigorously (not an arbitrary parabola), and the initial lateral velocity $v(x,t=0)$ is counter-rotation about a fixed point (pivot) $x_{\rm \pi}$ at which $v(x_{\rm \pi},t) =0$ (at the beginning and during spinup). This point is placed excentrically, in the upper part, and this excentre, $\propto \gamma$ , plays a significant role in the process. The spinup in a rigid container is developed as the prototype process; an essential component is the edge (outer wall) where the flux of the Ekman layer is arrested (and then returned to the centre via the inviscid core). While the upper part of the vein adopts this spinup pattern, the lower part (most of the vein, $x< x_{\rm \pi}$ ) develops a leak (drainage) at the edge that (a) modifies the spinup of the vein, and (b) generates a thin tail extension downslope. The tail consists of two merged non-divergent Ekman layers, which chokes the drainage flow rate. The present model provides clear-cut insights and some quantitative predictions of the major spinup stage by analytical algebraic solutions. A comparison with a previously published simple model (Wirth, Ocean Dyn., vol. 59, 2009, pp. 551–563) is presented. We also discuss briefly stability of the initial vein.