Power-law for gravity currents on slopes in the deceleration phase

The power-law for gravity currents on slopes is essentially an asymptotic form of the solution of thermal theory developed in Beghin, Hopfinger, and Britter (J. Fluid Mech. 107 (1981) 407-422), when the gravity current is sufficiently far into the deceleration phase. The power-law not only describes the long-term front location versus time relationship but also provides a useful means to estimate the buoyancy contained in the gravity current head. However, the hypothesis that gravity current is sufficiently far into the deceleration phase is hardly satisfied in experiments. In this paper, we re-formulated the power-law, considering the influence of bottom friction, and supplement the formulation by proposing a correct version of the power-law. When the gravity current is not sufficiently far into the deceleration phase, we showed that the power-law still robustly describes the front location versus time relationship, but the amount of heavy fluid in the head can be easily underestimated. The underestimation of heavy fluid in the head also depends on where the gravity current is in the deceleration phase. Therefore, a correction factor is suggested according to the location of gravity current. The amount of heavy fluid in the head, when estimated by the power-law, should be understood as the 'effective' buoyancy in driving the gravitational convection and is deemed as a lower limit for the estimation of buoyancy contained in the head.