Gravity Currents in a Deep Anelastic Atmosphere

Abstract

Abstract This study presents analytic results for steady gravity currents in a channel using the deep anelastic equations. Results are cast in terms of a nondimensional parameter H/H0 that relates the channel depth H to a scale depth H0 (the depth at which density goes to zero in an isentropic atmosphere). The classic results based on the incompressible equations correspond to H/H0 = 0. For cold gravity currents (at the bottom of a channel), assuming energy-conserving flow, the nondimensional current depth h/H is much smaller, and nondimensional propagation speed C/(gH)1/2 is slightly smaller as H/H0 increases. For flows with energy dissipation, C/(gH)1/2 decreases as H/H0 increases, even for fixed h/H. The authors conclude that as H/H0 increases the normalized hydrostatic pressure rise in the cold pool increases near the bottom of the channel, whereas drag decreases near the top of the channel; these changes require gravity currents to propagate slower for steady flow to be maintained. From these results, the authors find that steady cold pools have a likely maximum depth of 4 km in the atmosphere (in the absence of shear). For warm gravity currents (at the top of a channel), h/H is slightly larger and C/(gH)1/2 is much larger as H/H0 increases. The authors also conduct two-dimensional numerical simulations of “lock-exchange flow” to provide an independent evaluation of the analytic results. For cold gravity currents the simulations support the analytic results. However, for warm gravity currents the simulations show unsteady behavior that cannot be captured by the analytic theory and which appears to have no analog in incompressible flow.