Self-consistent definition for the variable depth of unsteady, turbulent gravity currents

We used the data from two-dimensional direct numerical simulations of Boussinesq gravity currents to define a self-consistent depth, $h$, and reduced gravity, $g'=\frac{g}{\rho_0}(\rho-\rho_0)$, for the current in terms of moments of the current density field. We demonstrate that using these definitions to calculate the Froude number, Fr=$u/\sqrt{g'h}$, gives a constant Froude number in constant-velocity and self-similar regime. At sufficiently high Reynolds number, our results are consistent with previous experimental and theoretical models (Shin \emph{et al}. 2004, Nokes \emph{et al}. 2008). We also develop a simple model to quantify the loss of mass from the gravity current head due to shear-induced vortices which propagate away from and behind the head. \\[4pt] Shin, J. O., Dalziel, S. B., Linden, P. F. (2004). Gravity currents produced by lock exchange. \emph{J. Fluid Mech.} \textbf{521},1-34. \\[0pt] Nokes, R.I., Davidson, M.J., Stepien, C.A., Veale, W. B., Oliver, R.L. (2008). The front condition for intrusive gravity currents. \emph{J. Hydraul. Res.} \textbf{46} (6) 788-801.