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Rotating planar gravity currents at moderate Rossby numbers: fully resolved simulations and shallow-water modelling

Published online by Cambridge University Press:  20 March 2019

Jorge S. Salinas ,
Thomas Bonometti Open the ORCID record for Thomas Bonometti [Opens in a new window] ,
Marius Ungarish Open the ORCID record for Marius Ungarish [Opens in a new window]  and
Mariano I. Cantero
Jorge S. Salinas*
Affiliation:
Comisión Nacional de Energía Atómica e Instituto Balseiro, Centro Atómico Bariloche, 8400 San Carlos de Bariloche, Argentina
Thomas Bonometti
Affiliation:
Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS-INPT-UPS, 31400 Toulouse, France
Marius Ungarish
Affiliation:
Department of Computer Science, Technion, Haifa 32000, Israel
Mariano I. Cantero
Affiliation:
Comisión Nacional de Energía Atómica e Instituto Balseiro, Centro Atómico Bariloche, 8400 San Carlos de Bariloche, Argentina Consejo Nacional de Investigaciones Científicas y Técnicas, Centro Atómico Bariloche, 8400 San Carlos de Bariloche, Argentina
*
†Email address for correspondence: [email protected]
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Abstract

The flow of a gravity current of finite volume and density $\unicode[STIX]{x1D70C}_{1}$ released from rest from a rectangular lock (of height $h_{0}$) into an ambient fluid of density $\unicode[STIX]{x1D70C}_{0}$ (${<}\unicode[STIX]{x1D70C}_{1}$) in a system rotating with $\unicode[STIX]{x1D6FA}$ about the vertical $z$ is investigated by means of fully resolved direct numerical simulations (DNS) and a theoretical model (based on shallow-water and Ekman layer spin-up theories, including mixing). The motion of the dense fluid includes several stages: propagation in the $x$-direction accompanied by Coriolis acceleration/deflection in the $-y$-direction, which produces a quasi-steady wedge-shaped structure with significant anticyclonic velocity $v$, followed by a spin-up reduction of $v$ accompanied by a slow $x$ drift, and oscillation. The theoretical model aims to provide useful insights and approximations concerning the formation time and shape of wedge, and the subsequent spin-up effect. The main parameter is the Coriolis number, ${\mathcal{C}}=\unicode[STIX]{x1D6FA}h_{0}/(g^{\prime }h_{0})^{1/2}$, where $g^{\prime }=(\unicode[STIX]{x1D70C}_{1}/\unicode[STIX]{x1D70C}_{0}-1)g$ is the reduced gravity. The DNS results are focused on a range of relatively small Coriolis numbers, $0.1\leqslant {\mathcal{C}}\leqslant 0.25$ (i.e. Rossby number $Ro=1/(2{\mathcal{C}})$ in the range $2\leqslant Ro\leqslant 5$), and a large range of Schmidt numbers $1\leqslant Sc<\infty$; the Reynolds number is large in all cases. The current spreads out in the $x$ direction until it is arrested by the Coriolis effect (in ${\sim}1/4$ revolution of the system). A complex motion develops about this state. First, we record oscillations on the inertial time scale $1/\unicode[STIX]{x1D6FA}$ (which are a part of the geostrophic adjustment), accompanied by vortices at the interface. Second, we note the spread of the wedge on a significantly longer time scale; this is an indirect spin-up effect – mixing and entrainment reduce the lateral (angular) velocity, which in turn decreases the Coriolis support to the $\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}x$ slope of the wedge shape. Contrary to non-rotating gravity currents, the front does not remain sharp as it is subject to (i) local stretching along the streamwise direction and (ii) convective mixing due to Kelvin–Helmholtz vortices generated by shear along the spanwise direction and stemming from Coriolis effects. The theoretical model predicts that the length of the wedge scales as ${\mathcal{C}}^{-2/3}$ (in contrast to the Rossby radius $\propto 1/{\mathcal{C}}$ which is relevant for large ${\mathcal{C}}$; and in contrast to ${\mathcal{C}}^{-1/2}$ for the axisymmetric lens).

JFM classification

Geophysical and Geological Flows: Gravity currents Geophysical and Geological Flows: Rotating flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 867 , 25 May 2019 , pp. 114 - 145
Copyright
© 2019 Cambridge University Press 

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Salinas et al. supplementary movie 1

Effect of the Coriolis number - cases (from top to bottom) S1-C25-F, S1-C15-F, S1-C10-F - side view of spanwise averaged density (Re=4000, Sc=1, free-slip)

Download Salinas et al. supplementary movie 1(Video)
Video 16.2 MB

Salinas et al. supplementary movie 2

Case S1-C15-F - top view of bottom density distribution (C=0.15, Ro=3.3, Re=4000, Sc=1, free-slip)

Download Salinas et al. supplementary movie 2(Video)
Video 5.9 MB

Salinas et al. supplementary movie 3

Case S5-C15-N - general view of the density surface rho=0.05 (C=0.15, Ro=3.3, Re=4000, Sc=5, no-slip)

Download Salinas et al. supplementary movie 3(Video)
Video 11.4 MB

Salinas et al. supplementary movie 4

Effect of the top-bottom boundary conditions - cases (from top to bottom) S1-C15-F, S1-C15-N - side view of spanwise averaged density (C=0.15, Ro=3.3, Re=4000, Sc=1)

Download Salinas et al. supplementary movie 4(Video)
Video 14.9 MB

Salinas et al. supplementary movie 5

Case SI-C15-N - general view of the density surfaces rho=0.1 and rho=0.9 (C=0.15, Ro=3.3, Re=4000, Sc=infinite, no-slip)

Download Salinas et al. supplementary movie 5(Video)
Video 7.1 MB

Salinas et al. supplementary movie 6

Case SI-C15-N - bottom view of the density surfaces rho=0.1 and rho=0.9 (C=0.15, Ro=3.3, Re=4000, Sc=infinite, no-slip)

Download Salinas et al. supplementary movie 6(Video)
Video 11.4 MB

Salinas et al. supplementary movie 7

Effect of the Schmidt number - cases (from top to bottom) S5-C15-N, S1-C15-N - side view of spanwise averaged density (C=0.15, Ro=3.3, Re=4000, no-slip)

Download Salinas et al. supplementary movie 7(Video)
Video 13.1 MB