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Entraining gravity currents

Published online by Cambridge University Press:  19 August 2013

Christopher G. Johnson  and
Andrew J. Hogg
Christopher G. Johnson*
Affiliation:
Department of Mathematics, University of Bristol, Bristol BS8 1TW, UK
Andrew J. Hogg
Affiliation:
Department of Mathematics, University of Bristol, Bristol BS8 1TW, UK
*
†Email address for correspondence: chris.johnson@bristol.ac.uk
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Abstract

Entrainment of ambient fluid into a gravity current, while often negligible in laboratory-scale flows, may become increasingly significant in large-scale natural flows. We present a theoretical study of the effect of this entrainment by augmenting a shallow water model for gravity currents under a deep ambient with a simple empirical model for entrainment, based on experimental measurements of the fluid entrainment rate as a function of the bulk Richardson number. By analysing long-time similarity solutions of the model, we find that the decrease in entrainment coefficient at large Richardson number, due to the suppression of turbulent mixing by stable stratification, qualitatively affects the structure and growth rate of the solutions, compared to currents in which the entrainment is taken to be constant or negligible. In particular, mixing is most significant close to the front of the currents, leading to flows that are more dilute, deeper and slower than their non-entraining counterparts. The long-time solution of an inviscid entraining gravity current generated by a lock-release of dense fluid is a similarity solution of the second kind, in which the current grows as a power of time that is dependent on the form of the entrainment law. With an entrainment law that fits the experimental measurements well, the length of currents in this entraining inviscid regime grows with time approximately as ${t}^{0. 447} $. For currents instigated by a constant buoyancy flux, a different solution structure exists in which the current length grows as ${t}^{4/ 5} $. In both cases, entrainment is most significant close to the current front.

JFM classification

Geophysical and Geological Flows: Gravity currents Geophysical and Geological Flows: Shallow water flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 731 , 25 September 2013 , pp. 477 - 508
Copyright
©2013 Cambridge University Press 

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