Erratum to: Shallow-water solutions for gravity currents in non-rectangular cross-area channels with stratified ambient

Abstract

We consider the propagation of a high-Reynolds-number gravity current in a horizontal channel along the horizontal coordinate $$x$$ . The current is of constant density, $$\rho _c$$ , and the ambient has a linear stable stratification, from $$\rho _{b}$$ at the bottom $$z=0$$ to $$\rho _{o}$$ at the top $$z=H$$ . The cross-section of the channel is given by the quite general $$-f_1(z)\le y \le f_2(z)$$ for $$0 \le z \le H$$ . We develop a one-layer shallow-water (SW) formulation, and we implement it for the solution of a gravity current of fixed volume released from a lock (we focus on a “heavy” bottom current with $$\rho _{c}\ge \rho _{b}$$ ). The dependent variables are the position of the interface, $$h(x,t)$$ , and the speed (averaged over the area of the current), $$u(x,t)$$ , where $$t$$ is the time. The non-rectangular cross-section geometry enters the formulation via $$f(h)$$ and integrals of $$f(z), z f(z)$$ and $$z^2 f(z)$$ , where $$f(z) = f_1(z) + f_2(z)$$ is the width of the channel. For a given geometry $$f(z)$$ , the free input parameters are (1) the height ratio $$H/h_0$$ of ambient to lock; and (2) the stratification parameter $$S = (\rho _{b}- \rho _{o})/(\rho _{c}- \rho _{o})$$ . In general, the SW equations of motion are a hyperbolic PDE system which we solve by a finite-difference method. However, we show that the initial motion displays a “slumping” stage with constant speed of propagation; this can be calculated exactly by the method of characteristics. An analytical solution for the long-time self-similar propagation is also presented for $$S=1$$ and the power-law $$f(z) = bz^\alpha $$ cross section profile. Solutions are presented for various stratification and typical geometries: power-law ( $$f(z) = b z^\alpha $$ , where $$b, \alpha $$ are positive constants), power-law B ( $$f(z) = b( H-z)^\alpha $$ ), trapezoidal, and circle. In general, the increase of the stratification parameter $$S$$ causes a reduction of the speed of propagation, but the details depend on the geometry of the cross section. We also show that, upon a simple transformation, the solutions for the bottom (“heavy”) current can be used for the prediction of top (“light”) currents The present solution is a significant generalization of the classical analysis of the gravity current in a stratified ambient problem (including the “intrusion” situation when $$S=1$$ ). The classical formulation for a rectangular (or laterally unbounded) channel is now just a particular case, $$f(z) = $$ const., in the wide domain of cross-sections covered by the new model. The present solution is reduced to the non-stratified (homogeneous) ambient counterpart by setting $$S=0$$ . The flow under consideration is relevant to environmental and geophysical applications like ventilation of tunnels and discharges in ducts and rivers.