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(NH 0 ), where N = √ g /H 0 is the buoyancy frequency, and g = g ρ/ρ 0 is the reduced acceleration due to gravity. Here ρ 0 is the density and ρ 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 σ on the bottom boundary shows that, without rotation, U F has a maximum value for σ = π/6, while with rotation, U F tends to zero on any slope. For the asymptotic stage when f t F 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 √ Bu = L R /R 0 , where L R = NH 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 ∼ 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 (f t F , Bu). Thus the front only tends to the geostrophic asymptotic state of zero radial velocity very slowly (i.e. as f t F → ∞) for finite values of L R /R 0 . However, as L R /R 0 → 0, it reaches this state when f t F ∼ 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 f t F > 1, are consistent with recent experiments.

Rotating planar gravity currents at moderate Rossby numbers: fully resolved simulations and shallow-water modelling

“…The present work extends Hunt et al's (2005) results in many ways. Firstly, the present 3-D DNS include viscous effects and thus fully capture the Ekman layers.…”

“…A forerunner to the present investigation was done by Hunt et al (2005). Briefly, these authors performed 2-D numerical simulations of the Euler equations to investigate the effect of rotation on the dynamics of the front, up to a time corresponding to the first oscillation of the front.…”

An Introduction to Ocean Turbulence

“…Gill (1981) gives a mathematical treatment of the spread of a mixed patch in stratified surroundings, and Griffiths and Linden (1981) demonstrate through laboratory experiments how patches become baroclinically unstable. Analysis of the spread and adjustment to the effects of the Earth's rotation in a related problem, that of a spreading body of water in the form of a density or gravity current, is given by Hunt et al (2005).…”

Gravity currents: a personal perspective

“…The frequency of these pulses was observed to be slightly higher than inertial, independent of h o and g , with an amplitude of the order of magnitude of half the maximum radius, as illustrated in figure 9. Analysis of the subsequent motion (Ωt 1) and the influence of a slope is discussed in Hunt et al (2005), along with some interesting applications to atmospheric fronts.…”