An analytical model of gravity currents in a stable atmosphere

Abstract

An analytical solution to the nonlinear equations of motion and thermodynamic energy for gravity currents propagating in stable atmosphere is found. This solution differs from the previous analytical studies in several aspects. In our solution the head of the gravity current is a strong vortex and the dynamics are non-hydrostatic. The solution has two regimes: (i) a supercritical regime when the Froude number Fr = (c – U)/Na is larger than 1 – in this case the cold front is local; (ii) a subcritical regime when Fr is smaller than 1. Here, ahead of the front there is a disturbance of nonlinear gravity waves. The scale of the wave and its amplitude increase as the Froude number decreases.

We found that the square of the speed of the gravity current (relative to the synoptic wind) is proportional to the mean drop of potential temperature over the front area times the front height a. The constant of proportionality is function of the environmental conditions. The thermal, velocity and vorticity fields can be described by non-dimensional structure functions of two numbers: pa = 1/Fr and ka. The amplitude of the structure functions is proportional to (c – U) ²/a for the thermal field, to (c – U) for the velocity field, and to (c – U)/a for the vorticity field.

The propagation is studied in terms of the vorticity equation. The horizontal gradient of the buoyancy term always tends to propagate the cold front. The nonlinear advection term in most of the cases investigated here tends to slow the propagation of the gravity current. The propagation of the disturbance of nonlinear gravity waves ahead of the front in regime (ii) in most of the cases is due to the buoyancy term. The nonlinear advection term tends to slow the propagation when the synoptic wind blows in the direction opposite to that of the front propagation, and increase the propagation when the synoptic wind blows in the direction of propagation.