Gravity currents from a dam-break in a rotating channel

The generation of a gravity current by the release of a semi-infinite region of buoyant fluid of depth $H$ overlying a deeper, denser and quiescent lower layer in a rotating channel of width $w$ is considered. Previous studies have focused on the characteristics of the gravity current head region and produced relations for the gravity current speed $c_{b}$ and width $w_b$ as a functions of the local current depth along the wall $h_b$ , reduced gravity $g^\prime$ , and Coriolis frequency $f$ . Here, the dam-break problem is solved analytically by the method of characteristics assuming reduced-gravity flow, uniform potential vorticity and a semigeostrophic balance. The solution makes use of a local gravity current speed relation $c_{b} \,{=}\, c_b(h_b,\ldots)$ and a continuity constraint at the head to close the problem. The initial value solution links the local gravity current properties to the initiating dam-break conditions. The flow downstream of the dam consists of a rarefaction joined to a uniform gravity current with width $w_b$ ( ${\le}\, w$ ) and depth on the right-hand wall of $h_b$ , terminated at the head moving at speed $c_b$ . The solution gives $h_b$ , $c_b$ , $w_b$ and the transport of the boundary current as functions of $w/L_R$ , where $L_R \,{=}\, \sqrt{g^\prime H}/f$ is the deformation radius. The semigeostrophic solution compares favourably with numerical solutions of a single-layer shallow-water model that internally develops a leading bore. Existing laboratory experiments are re-analysed and some new experiments are undertaken. Comparisons are also made with a three-dimensional shallow-water model. These show that lateral boundary friction is the primary reason for differences between the experiments and the semigeostrophic theory. The wall no-slip condition is identified as the primary cause of the experimentally observed decrease in gravity current speed with time. A model for the viscous decay is developed and shown to agree with both experimental and numerical model data.

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