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

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).

Internal Bore Evolution across the Shelf near Pt. Sal, California, Interpreted as a Gravity Current

“…Gravity currents are also modified by rotation (e.g., Griffiths 1986). In regions without boundaries gravity currents on flat bottoms initially propagate at speed independent of Coriolis parameter and arrest due to geostrophic adjustment after many inertial periods (e.g., Salinas et al 2019).…”

“…Gravity currents under the effect of rotation, particularly flowing along boundaries, have been extensively investigated (e.g., Griffiths 1986;Lentz and Helfrich 2002). Numerically modeled lock-release gravity currents with rotation and periodic along-front boundary conditions show that gravity currents eventually geostrophically adjust over many inertial periods (Salinas et al 2019).…”

A 3D Numerical Model for Turbidity Currents

“…In eqns. (8), (9) and hereinafter, * + and * − are the contour surfaces of the volume * on which α is constant and which are located at the larger and the smaller value of α , respectively (the coefficients , , γ = 1,2,3 are cyclic). In a generalized curvilinear coordinate system, eqns.…”

“…Hogg et al [8] investigated theoretically on gravitationally driven motion arising from a sustained constant source of dense fluid in a horizontal channel, using shallow-layer models and direct numerical simulations of the Navier-Stokes equations. Salinas et al [9] used a 3D DNS model to investigate on the flow of a gravity current of a dense fluid, released from rest from a rectangular lock, into an ambient fluid with a density lower than the one of the dense fluid. See [10] for a detailed review about gravity currents.…”

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

“…First, we wish to clarify that in equations (3.14)-(3.15) of Salinas et al (2019) all variables are dimensional except for the angular velocity ω and the coefficient k. After (3.15), we switch to dimensionless variables for the streamwise direction x, the front location x N , the Ekman layer δ E , the local height h (scaled by the initial height h 0 ) and time t (scaled by 1/Ω, with Ω the angular velocity of the rotating system).…”