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A non-dissipative solution of Benjamin-type gravity current for a wide range of depth ratios

We consider the steady-state propagation of a high-Reynolds-number gravity current of height h and density ρc on the bottom of a horizontal channel of height H filled with ambient fluid of density ρa(<ρc), usually known as Benjamin's current problem. The objective is to derive an analytical result for the speed of propagation, U, in the form of the dimensionless Froude number, Fr(a) = U/(g′h)1/2). Here g′ = (ρc/ρa − 1)g is the reduced gravity-driving effect (g being the gravity acceleration) and a = h/H is the depth (thickness) ratio of the layer of the current to that of the ambient fluid into which the current propagates. The analysis is performed in a frame of reference attached to the current; in this frame the current is a motionless slug. The original analysis of Benjamin assumes that the speed of the ambient in the domain above the parallel-horizontal main part of the current (behind the head) is independent of the vertical coordinate z, but here we assume that a small u′(z) fluctuation about the depth-averaged speed u exists. Then, we impose the balances of volume flux, flow-force (momentum flux) and global energy conservation, for a control volume attached to the current. We show that this gives a unique analytical result for Fr as a function of a = h/H. We recall that the original counterpart solution FrB(a) of Benjamin does not satisfy the above-mentioned energy conservation condition, i.e. the system displays energy dissipation (except for the half-depth current case a = 1/2). The present dissipationless-flow Fr(a) result is valid for any a ≤ 1/2, i.e. currents of at most half-depth of the channel height. On the other hand, in agreement with Benjamin's solution, gravity currents of more than half-depth of the channel height require an energy source and are impossible in normal conditions. The new Fr(a) is slightly smaller than Benjamin's FrB(a) result for 0 < a < 1/2, and the difference vanishes at a = 1/2 and a → 0 (a current of finite height in a very deep ambient).