On a similarity solution for lock-release gravity currents affected by slope, drag and entrainment

We consider the long-time propagation of a Boussinesq inertia–buoyancy (large-Reynolds- number) gravity current released from a lock over a downslope of angle γ , affected by entrainment and drag. We show that the shallow-water (depth-averaged) equations with a Benjamin-type front-jump condition admit a similarity solution xN(t) = K t2/3 while h, φ, u change like t to the power of $2/3, -4/3, -1/3$ , respectively; here xN, h, φ, u and t are the position of the nose (distance from backwall), thickness, concentration of dense fluid, velocity and time, respectively, and K is a constant. Assuming that γ and the coefficients of entrainment and drag are constant, we derive an analytical exact solution for the similarity profiles and show that K ∝ (tan γ )1/3 ; the driving of the slope is balanced by entrainment and/or drag. The predicted t2/3 propagation is in agreement with previously published experimental data but a conclusive quantitative assessment of the present theory cannot be performed due to various uncertainties (discussed in the paper) that must be resolved by future work.