Effects of rotation and sloping terrain on the fronts of density currents

Abstract

The initial stage of the adjustment of a gravity current to the effects of rotation with angular velocity $f/2$ is analysed using a short time analysis where Coriolis forces are initiated in an inviscid von Kármán–Benjamin gravity current front at $t_F\,{=}\,0$. It is shown how, on a time-scale of order $1/f$, as a result of ageostrophic dynamics, the slope and front speed $U_F$ are much reduced from their initial values, while the transverse anticyclonic velocity parallel to the front increases from zero to $O(N H_0)$, where $N\,{=}\,\sqrt{g'/H_0}$ is the buoyancy frequency, and $g'\,{=}\,g \Delta \rho /\rho_0$ is the reduced acceleration due to gravity. Here $\rho_0$ is the density and $\Delta \rho$ and $H_0$ are the density difference and initial height of the current. Extending the steady-state theory to account for the effect of the slope $\sigma$ on the bottom boundary shows that, without rotation, $U_{F}$ has a maximum value for $\sigma \,{=}\, \upi/6$, while with rotation, $U_{F}$ tends to zero on any slope. For the asymptotic stage when $ft_F \,{\gg}\, 1$, the theory of unsteady waves on the current is reviewed using nonlinear shallow-water equations and the van der Pol averaging method. Their motions naturally split into a ‘balanced’ component satisfying the Margules geostrophic relation and an equally large ‘unbalanced’ component, in which there is horizontal divergence and ageostrophic vorticity. The latter is responsible for nonlinear oscillations in the current on a time scale $f^{-1}$, which have been observed in the atmosphere and field experiments. Their magnitude is mainly determined by the initial potential energy in relation to that of the current and is proportional to the ratio $\sqrt{\hbox{\it Bu}} \,{=}\, L_R/R_0$, where $L_R\,{=}\,N H_0/f$ is the Rossby deformation radius and $R_0$ is the initial radius. The effect of slope friction also prevents the formation of a steady front. From the analysis it is concluded that a weak mean radial flow must be driven by the ageostrophic oscillations, preventing the mean front speed $U_F$ from halting sharply at $f t_F \,{\sim}\, 1$. Depending on the initial value of $L_R/R_0$, physical arguments show that $U_F$ decreases slowly in proportion to $(f t_F)^{-1/2}$, i.e. $U_F/U_{F_0}\,{=}\,F(ft_F,\hbox{\it Bu})$. Thus the front only tends to the geostrophic asymptotic state of zero radial velocity very slowly (i.e. as $f t_F \,{\rightarrow}\, \infty $) for finite values of $L_R/R_0$. However, as $L_R/R_0 \,{\rightarrow}\, 0$, it reaches this state when $f t_F \,{\sim}\, 1$. This analysis of the overall nonlinear behaviour of the gravity current is consistent with two two-dimensional non-hydrostatic (Navier–Stokes) and axisymmetric hydrostatic (shallow-water) Eulerian numerical simulations of the varying form of the rotating gravity current. When the effect of surface friction is considered, it is found that the mean movement of the front is significantly slowed. Furthermore, the oscillations with angular frequency $f$ and the slow growth of the radius, when $ft_F \,{\ge}\, 1$, are consistent with recent experiments.