Buoyant gravity currents along a sloping bottom in a rotating fluid

The dynamics of buoyant gravity currents in a rotating reference frame is a classical problem relevant to geophysical applications such as river water entering the ocean. However, existing scaling theories are limited to currents propagating along a vertical wall, a situation almost never realized in the ocean. A scaling theory is proposed for the structure (width and depth), nose speed and flow field characteristics of buoyant gravity currents over a sloping bottom as functions of the gravity current transport Q, density anomaly g′, Coriolis frequency f, and bottom slope α. The nose propagation speed is cp ∼ cw/ (1 + cw/cα) and the width of the buoyant gravity current is Wp ∼ cw/ f(1 + cw/cα), where cw = (2Qg′ f)1/4 is the nose propagation speed in the vertical wall limit (steep bottom slope) and cα = αg/f is the nose propagation speed in the slope-controlled limit (small bottom slope). The key non-dimensional parameter is cw/cα, which indicates whether the bottom slope is steep enough to be considered a vertical wall (cw/cα → 0) or approaches the slope-controlled limit (cw/cα → ∞). The scaling theory compares well against a new set of laboratory experiments which span steep to gentle bottom slopes (cw/cα = 0.11–13.1). Additionally, previous laboratory and numerical model results are reanalysed and shown to support the proposed scaling theory.

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Self-similar viscous gravity currents: phase-plane formalism

Journal of Fluid Mechanics ◽

10.1017/s0022112090001240 ◽

1990 ◽

Vol 210 ◽

pp. 155-182 ◽

Cited By ~ 58

Author(s):

Julio Gratton ◽

Fernando Minotti

Keyword(s):

Scaling Laws ◽

Phase Plane ◽

Gravity Current ◽

Classical Problem ◽

Gravity Currents ◽

Steady Flows ◽

Two Phase ◽

Integral Curves ◽

Self Similar ◽

Gravity currents in rotating channels. Part 1. Steady-state theory

Journal of Fluid Mechanics ◽

10.1017/s0022112001007662 ◽

2002 ◽

Vol 457 ◽

pp. 295-324 ◽

Cited By ~ 8

Author(s):

J. N. HACKER ◽

P. F. LINDEN

Keyword(s):

Steady State ◽

Perfect Fluid ◽

Control Point ◽

Gravity Current ◽

Propagation Speed ◽

Gravity Currents ◽

Force Balance ◽

Fluid Approximation ◽

Left Hand ◽

Layer Solution

A theory is developed for the speed and structure of steady-state non-dissipative gravity currents in rotating channels. The theory is an extension of that of Benjamin (1968) for non-rotating gravity currents, and in a similar way makes use of the steady-state and perfect-fluid (incompressible, inviscid and immiscible) approximations, and supposes the existence of a hydrostatic ‘control point’ in the current some distance away from the nose. The model allows for fully non-hydrostatic and ageostrophic motion in a control volume V ahead of the control point, with the solution being determined by the requirements, consistent with the perfect-fluid approximation, of energy and momentum conservation in V, as expressed by Bernoulli's theorem and a generalized flow-force balance. The governing parameter in the problem, which expresses the strength of the background rotation, is the ratio W = B/R, where B is the channel width and R = (g′H)1/2/f is the internal Rossby radius of deformation based on the total depth of the ambient fluid H. Analytic solutions are determined for the particular case of zero front-relative flow within the gravity current. For each value of W there is a unique non-dissipative two-layer solution, and a non-dissipative one-layer solution which is specified by the value of the wall-depth h0. In the two-layer case, the non-dimensional propagation speed c = cf(g′H)−1/2 increases smoothly from the non-rotating value of 0.5 as W increases, asymptoting to unity for W → ∞. The gravity current separates from the left-hand wall of the channel at W = 0.67 and thereafter has decreasing width. The depth of the current at the right-hand wall, h0, increases, reaching the full depth at W = 1.90, after which point the interface outcrops on both the upper and lower boundaries, with the distance over which the interface slopes being 0.881R. In the one-layer case, the wall-depth based propagation speed Froude number c0 = cf(g′h0)−1/2 = 21/2, as in the non-rotating one-layer case. The current separates from the left-hand wall of the channel at W0 ≡ B/R0 = 2−1/2, and thereafter has width 2−1/2R0, where R0 = (g′h0)1/2/f is the wall-depth based deformation radius.

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On the propagation of gravity currents over and through a submerged array of circular cylinders

Journal of Fluid Mechanics ◽

10.1017/jfm.2017.604 ◽

2017 ◽

Vol 831 ◽

pp. 394-417 ◽

Cited By ~ 10

Author(s):

Jian Zhou ◽

Claudia Cenedese ◽

Tim Williams ◽

Megan Ball ◽

Subhas K. Venayagamoorthy ◽

...

Keyword(s):

Volume Fraction ◽

Gravity Current ◽

Gravity Currents ◽

Front Velocity ◽

Circular Cylinders ◽

Dimensional Parameter ◽

Flow Interruption ◽

Line Array ◽

Through Flow ◽

Array Density

The propagation of full-depth lock-exchange bottom gravity currents past a submerged array of circular cylinders is investigated using laboratory experiments and large eddy simulations. Firstly, to investigate the front velocity of gravity currents across the whole range of array density $\unicode[STIX]{x1D719}$ (i.e. the volume fraction of solids), the array is densified from a flat bed ($\unicode[STIX]{x1D719}=0$) towards a solid slab ($\unicode[STIX]{x1D719}=1$) under a particular submergence ratio $H/h$, where $H$ is the flow depth and $h$ is the array height. The time-averaged front velocity in the slumping phase of the gravity current is found to first decrease and then increase with increasing $\unicode[STIX]{x1D719}$. Next, a new geometrical framework consisting of a streamwise array density $\unicode[STIX]{x1D707}_{x}=d/s_{x}$ and a spanwise array density $\unicode[STIX]{x1D707}_{y}=d/s_{y}$ is proposed to account for organized but non-equidistant arrays ($\unicode[STIX]{x1D707}_{x}\neq \unicode[STIX]{x1D707}_{y}$), where $s_{x}$ and $s_{y}$ are the streamwise and spanwise cylinder spacings, respectively, and $d$ is the cylinder diameter. It is argued that this two-dimensional parameter space can provide a more quantitative and unambiguous description of the current–array interaction compared with the array density given by $\unicode[STIX]{x1D719}=(\unicode[STIX]{x03C0}/4)\unicode[STIX]{x1D707}_{x}\unicode[STIX]{x1D707}_{y}$. Both in-line and staggered arrays are investigated. Four dynamically different flow regimes are identified: (i) through-flow propagating in the array interior subject to individual cylinder wakes ($\unicode[STIX]{x1D707}_{x}$: small for in-line array and arbitrary for staggered array; $\unicode[STIX]{x1D707}_{y}$: small); (ii) over-flow propagating on the top of the array subject to vertical convective instability ($\unicode[STIX]{x1D707}_{x}$: large; $\unicode[STIX]{x1D707}_{y}$: large); (iii) plunging-flow climbing sparse close-to-impermeable rows of cylinders with minor streamwise intrusion ($\unicode[STIX]{x1D707}_{x}$: small; $\unicode[STIX]{x1D707}_{y}$: large); and (iv) skimming-flow channelized by an in-line array into several subcurrents with strong wake sheltering ($\unicode[STIX]{x1D707}_{x}$: large; $\unicode[STIX]{x1D707}_{y}$: small). The most remarkable difference between in-line and staggered arrays is the non-existence of skimming-flow in the latter due to the flow interruption by the offset rows. Our analysis reveals that as $\unicode[STIX]{x1D719}$ increases, the change of flow regime from through-flow towards over- or skimming-flow is responsible for increasing the gravity current front velocity.

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Flow Entrainment of Gravity Currents

ASME-JSME-KSME 2011 Joint Fluids Engineering Conference: Volume 1, Symposia – Parts A, B, C, and D ◽

10.1115/ajk2011-16022 ◽

2011 ◽

Author(s):

K. M. Mok ◽

Harry H. Yeh ◽

K. K. Ieong ◽

K. I. Hoi

Keyword(s):

Gravity Current ◽

Three Dimensional ◽

Propagation Speed ◽

Bottom Layer ◽

Gravity Currents ◽

Flow Structures ◽

Visualization Technique ◽

Ambient Fluid ◽

Actual Flow ◽

Eddy Generation

The entrainment of gravity currents advancing over a horizontal bed was studied. A two-dimensional rigid-lid flow model was derived assuming ambient-fluid entrainment to the mixing region being supplied only from the bottom layer of the approaching flow. Two sets of laboratory experiments were carried out using the laser-induced fluorescence (LIF) flow visualization technique. With given parameters such as the total fluid depth, densities of the fluids, height of the gravity current head and its propagation speed, and the denser-fluid flow depth behind the head under the mixing region, our model predicts that the thickness of the front flow layer to be entrained is about 35 percents of the height of the gravity current head. Qualitative examination of the flow structures along various planes in the developed fronts suggests that the actual flow structures at the foremost part of the current head are complex and three dimensional. Entrainment of ambient fluid to the current is through various directions starting at its front, which creates an unstable stratification condition there favorable for the subsequent complex three-dimensional eddy generation and growth leading to the formation of the short-crested billows exhibiting the lobe-and-cleft features in the following flow.

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A note on the propagation speed of a weakly dissipative gravity current

Journal of Fluid Mechanics ◽

10.1017/s0022112006004198 ◽

2007 ◽

Vol 574 ◽

pp. 393-403 ◽

Cited By ~ 11

Author(s):

EUGENY V. ERMANYUK ◽

NIKOLAI V. GAVRILOV

Keyword(s):

Gravity Current ◽

Propagation Speed ◽

Front Propagation ◽

Gravity Currents ◽

Minor Role ◽

Rigid Boundary ◽

Inviscid Flows ◽

First Case ◽

Energy Conserving ◽

A Minor

This paper presents an experimental study on the propagation speed of gravity currents at moderate values of a gravity Reynolds number. Two cases are considered: gravity currents propagating along a rigid boundary and intrusive gravity currents. For the first case, a semi-empirical formula for the front propagation speed derived from simple energy arguments is shown to capture well the effect of flow deceleration because of viscous dissipation. In the second case, the propagation speed is shown to agree with the one predicted for energy-conserving virtually inviscid flows (Shin, Dalziel & Linden, J. Fluid Mech. vol. 521, 2004, p. 1), which implies that the losses due to vorticity generation and mixing at the liquid–liquid interface play only a minor role in the total balance of energy.

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On the dynamics of gravity currents in a channel

Journal of Fluid Mechanics ◽

10.1017/s0022112094001527 ◽

1994 ◽

Vol 269 ◽

pp. 169-198 ◽

Cited By ~ 77

Author(s):

Joseph B. Klemp ◽

Richard Rotunno ◽

William C. Skamarock

Keyword(s):

Shallow Water ◽

Good Description ◽

Water Solution ◽

Gravity Current ◽

Leading Edge ◽

Propagation Speed ◽

Gravity Currents ◽

Force Balance ◽

Cold Pool ◽

Two Dimensional

We attempt to clarify the factors that regulate the propagation and structure of gravity currents through evaluation of idealized theoretical models along with two-dimensional numerical model simulations. In particular, we seek to reconcile research based on hydraulic theory for gravity currents evolving from a known initial state with analyses of gravity currents that are assumed to be at steady state, and to compare these approaches with both numerical simulations and laboratory experiments. The time-dependent shallow-water solution for a gravity current propagating in a channel of finite depth reveals that the flow must remain subcritical behind the leading edge of the current (in a framework relative to the head). This constraint requires that hf/d ≤ 0.347, where hf is the height of the front and d is the channel depth. Thus, in the lock-exchange problem, inviscid solutions corresponding to hf/d = 0.5 are unphysical, and the actual currents have depth ratios of less than one half near their leading edge and require dissipation or are not steady. We evaluate the relevance of Benjamin's (1968) well-known formula for the propagation of steady gravity currents and clarify discrepancies with other theoretical and observed results. From two-dimensional simulations with a frictionless lower surface, we find that Benjamin's idealized flow-force balance provides a good description of the gravity-current propagation. Including surface friction reduces the propagation speed because it produces dissipation within the cold pool. Although shallow-water theory over-estimates the propagation speed of the leading edge of cold fluid in the ‘dam-break’ problem, this discrepancy appears to arise from the lack of mixing across the current interface rather than from deficiencies in Benjamin's front condition. If an opposing flow restricts the propagation of a gravity current away from its source, we show that the propagation of the current relative to the free stream may be faster than predicted by Benjamin's formula. However, in these situations the front propagation remains dependent upon the specific source conditions and cannot be generalized.

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Gravity currents and internal waves in a stratified fluid

Journal of Fluid Mechanics ◽

10.1017/s0022112008003984 ◽

2008 ◽

Vol 616 ◽

pp. 327-356 ◽

Cited By ~ 34

Author(s):

BRIAN L. WHITE ◽

KARL R. HELFRICH

Keyword(s):

Internal Waves ◽

Wave Speed ◽

Numerical Solutions ◽

Gravity Current ◽

Gravity Currents ◽

Stratified Fluid ◽

Front Speed ◽

Long Wave ◽

Conjugate State ◽

Energy Conserving

A steady theory is presented for gravity currents propagating with constant speed into a stratified fluid with a general density profile. Solution curves for front speed versus height have an energy-conserving upper bound (the conjugate state) and a lower bound marked by the onset of upstream influence. The conjugate state is the largest-amplitude nonlinear internal wave supported by the ambient stratification, and in the limit of weak stratification approaches Benjamin's energy-conserving gravity current solution. When the front speed becomes critical with respect to linear long waves generated above the current, steady solutions cannot be calculated, implying upstream influence. For non-uniform stratification, the critical long-wave speed exceeds the ambient long-wave speed, and the critical-Froude-number condition appropriate for uniform stratification must be generalized. The theoretical results demonstrate a clear connection between internal waves and gravity currents. The steady theory is also compared with non-hydrostatic numerical solutions of the full lock release initial-value problem. Some solutions resemble classic gravity currents with no upstream disturbance, but others show long internal waves propagating ahead of the gravity current. Wave generation generally occurs when the stratification and current speed are such that the steady gravity current theory fails. Thus the steady theory is consistent with the occurrence of either wave-generating or steady gravity solutions to the dam-break problem. When the available potential energy of the dam is large enough, the numerical simulations approach the energy-conserving conjugate state. Existing laboratory experiments for intrusions and gravity currents produced by full-depth lock exchange flows over a range of stratification profiles show excellent agreement with the conjugate state solutions.

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Symmetric and Baroclinic Instability in Dense Shelf Overflows

Journal of Physical Oceanography ◽

10.1175/jpo-d-18-0072.1 ◽

2019 ◽

Vol 49 (1) ◽

pp. 39-61 ◽

Cited By ~ 3

Author(s):

Elizabeth Yankovsky ◽

Sonya Legg

Keyword(s):

Baroclinic Instability ◽

Gravity Current ◽

The Arctic ◽

Neutral Buoyancy ◽

Buoyancy Forcing ◽

Dense Water ◽

Density Anomaly ◽

Eurasian Basin ◽

Thermohaline Stratification ◽

2D And 3D

AbstractIn this study, we revisit the problem of rotating dense overflow dynamics by performing nonhydrostatic numerical simulations, resolving submesoscale variability. Thermohaline stratification and buoyancy forcing are based on data from the Eurasian basin of the Arctic Ocean, where overflows are particularly crucial to the exchange of dense water between shelves and deep basins, yet have been studied relatively little. A nonlinear equation of state is used, allowing proper representation of thermohaline structure and mixing. We examine three increasingly complex scenarios: nonrotating 2D, rotating 2D, and rotating 3D. The nonrotating 2D case behaves according to known theory: the gravity current descends alongslope until reaching a relatively shallow neutral buoyancy level. However, in the rotating cases, we have identified novel dynamics: in both 2D and 3D, the submesoscale range is dominated by symmetric instability (SI). Rotation leads to geostrophic adjustment, causing dense water to be confined within the forcing region longer and attain a greater density anomaly. In the 2D case, Ekman drainage leads to descent of the geostrophic jet, forming a highly dense alongslope front. Beams of negative Ertel potential vorticity develop parallel to the slope, initiating SI and vigorous mixing in the overflow. In 3D, baroclinic eddies are responsible for cross-isobath dense water transport, but SI again develops along the slope and at eddy edges. Remarkably, through two different dynamics, the 2D SI-dominated case and 3D eddy-dominated case attain roughly the same final water mass distribution, highlighting the potential role of SI in driving mixing within certain regimes of dense overflows.

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Gravity currents and related phenomena

Journal of Fluid Mechanics ◽

10.1017/s0022112068000133 ◽

1968 ◽

Vol 31 (2) ◽

pp. 209-248 ◽

Cited By ~ 893

Author(s):

T. Brooke Benjamin

Keyword(s):

Salt Water ◽

Gravity Current ◽

Circular Cross Section ◽

Gravity Currents ◽

Head Wave ◽

Physical Problems ◽

Energy Conserving ◽

Fluid Theory ◽

Hydrodynamical Problem ◽

Upper Boundary

This paper presents a broad investigation into the properties of steady gravity currents, in so far as they can be represented by perfect-fluid theory and simple extensions of it (like the classical theory of hydraulic jumps) that give a rudimentary account of dissipation. As usually understood, a gravity current consists of a wedge of heavy fluid (e.g. salt water, cold air) intruding into an expanse of lighter fluid (fresh water, warm air); but it is pointed out in § 1 that, if the effects of viscosity and mixing of the fluids at the interface are ignored, the hydrodynamical problem is formally the same as that for an empty cavity advancing along the upper boundary of a liquid. Being simplest in detail, the latter problem is treated as a prototype for the class of physical problems under study: most of the analysis is related to it specifically, but the results thus obtained are immediately applicable to gravity currents by scaling the gravitational constant according to a simple rule.In § 2 the possible states of steady flow in the present category between fixed horizontal boundaries are examined on the assumption that the interface becomes horizontal far downstream. A certain range of flows appears to be possible when energy is dissipated; but in the absence of dissipation only one flow is possible, in which the asymptotic level of the interface is midway between the plane boundaries. The corresponding flow in a tube of circular cross-section is found in § 3, and the theory is shown to be in excellent agreement with the results of recent experiments by Zukoski. A discussion of the effects of surface tension is included in § 3. The two-dimensional energy-conserving flow is investigated further in § 4, and finally a close approximation to the shape of the interface is obtained. In § 5 the discussion turns to the question whether flows characterized by periodic wavetrains are realizable, and it appears that none is possible without a large loss of energy occurring. In § 6 the case of infinite total depth is considered, relating to deeply submerged gravity currents. It is shown that the flow must always feature a breaking ‘head wave’, and various properties of the resulting wake are demonstrated. Reasonable agreement is established with experimental results obtained by Keulegan and others.

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Numerical simulations of lock-exchange compositional gravity current

Journal of Fluid Mechanics ◽

10.1017/s0022112009007599 ◽

2009 ◽

Vol 635 ◽

pp. 361-388 ◽

Cited By ~ 77

Author(s):

SENG KEAT OOI ◽

GEORGE CONSTANTINESCU ◽

LARRY WEBER

Keyword(s):

High Resolution ◽

Dissipation Rate ◽

Friction Velocity ◽

Gravity Current ◽

Gravity Currents ◽

Inviscid Limit ◽

Grashof Number ◽

Front Speed ◽

Initial Volume ◽

Bed Friction

Compositional gravity current flows produced by the instantaneous release of a finite-volume, heavier lock fluid in a rectangular horizontal plane channel are investigated using large eddy simulation. The first part of the paper focuses on the evolution of Boussinesq lock-exchange gravity currents with a large initial volume of the release during the slumping phase in which the front of the gravity current propagates with constant speed. High-resolution simulations are conducted for Grashof numbers $\sqrt {Gr}$ = 3150 (LGR simulation) and $\sqrt {Gr}$ = 126000 (HGR simulation). The Grashof number is defined with the channel depth h and the buoyancy velocity ub = $\sqrt {g'h}$ (g′ is the reduced gravity). In the HGR simulation the flow is turbulent in the regions behind the two fronts. Compared to the LGR simulation, the interfacial billows lose their coherence much more rapidly (over less than 2.5h behind the front), which results in a much faster decay of the large-scale content and turbulence intensity in the trailing regions of the flow. A slightly tilted, stably stratified interface layer develops away from the two fronts. The concentration profiles across this layer can be approximated by a hyperbolic tangent function. In the HGR simulation the energy budget shows that for t > 18h/ub the flow reaches a regime in which the total dissipation rate and the rates of change of the total potential and kinetic energies are constant in time. The second part of the paper focuses on the study of the transition of Boussinesq gravity currents with a small initial volume of the release to the buoyancy–inertia self-similar phase. When the existence of the back wall is communicated to the front, the front speed starts to decrease, and the current transitions to the buoyancy–inertia phase. Three high-resolution simulations are performed at Grashof numbers between $\sqrt {Gr}$ = 3 × 104 and $\sqrt {Gr}$ = 9 × 104. Additionally, a calculation at a much higher Grashof number ($\sqrt {Gr}$ = 106) is performed to understand the behaviour of a bottom-propagating current closer to the inviscid limit. The three-dimensional simulations correctly predict a front speed decrease proportional to t−α (the time t is measured from the release time) over the buoyancy–inertia phase, with the constant α approaching the theoretical value of 1/3 as the current approaches the inviscid limit. At Grashof numbers for which $\sqrt {Gr}$ > 3 × 104, the intensity of the turbulence in the near-wall region behind the front is large enough to induce the formation of a region containing streaks of low and high streamwise velocities. The streaks are present well into the buoyancy–inertia phase before the speed of the front decays below values at which the streaks can be sustained. The formation of the velocity streaks induces a streaky distribution of the bed friction velocity in the region immediately behind the front. This distribution becomes finer as the Grashof number increases. For simulations in which the only difference was the value of the Grashof number ($\sqrt {Gr}$ = 4.7 × 104 versus $\sqrt {Gr}$ = 106), analysis of the non-dimensional bed friction velocity distributions shows that the capacity of the gravity current to entrain sediment from the bed increases with the Grashof number. Past the later stages of the transition to the buoyancy–inertia phase, the temporal variations of the potential energy, the kinetic energy and the integral of the total dissipation rate are logarithmic.

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