Gravity Current Front Research Articles

Porous media gravity current flow over an interbed layer: the impact of dispersion and distributed drainage

Motivated by buoyancy-driven flows within geological formations, we study the evolution of a (dense) gravity current in a porous medium bisected by a thin interbed layer. The gravity current experiences distributed drainage along this low-permeability boundary. Our theoretical description of this flow takes into account dispersive mass exchange with the surrounding ambient fluid by considering the evolution of the bulk and dispersed phases of the gravity current. In turn, we model basal draining by considering two bookend limits, i.e. no mixing versus perfect mixing in the lower layer. Our formulations are assessed by comparing model predictions against the output of complementary numerical simulations run using COMSOL. Numerical output is essential both for determining the value of the entrainment coefficient used within our theory and for assessing the reasonableness of key modelling assumptions. Our results suggest that the degree of dispersion depends on the dip angle and the depth and permeability of the interbed layer. We further find that the nose position predictions made by our theoretical models are reasonably accurate up to the point where the no mixing model predicts a retraction of the gravity current front. Thereafter, the no mixing model significantly under-predicts, and the perfect mixing model moderately over-predicts, numerical data. Reasons for the failure of the no mixing model are provided, highlighting the importance of convective instabilities in the lower layer. A regime diagram is presented that defines the parametric region where our theoretical models do versus do not yield predictions in good agreement with numerical simulations.

Energy balances for the collision of gravity currents of equal strengths

Collision of two counterflowing gravity currents of equal densities and heights was investigated by means of three-dimensional high-resolution simulations with the goal of understanding the flow structures and energetics in the collision region in more detail. The lifetime of collision is approximately $3 \tilde {H}/\tilde {u}_f$ , where $\tilde {H}$ is the depth of heavy and ambient fluids, and $\tilde {u}_f$ is the front velocity of the approaching gravity currents, and the lifetime of collision can be divided into three phases. During Phase I, $-0.2 \leqslant (\tilde {t}-\tilde {t}_c) \tilde {u}_f/\tilde {H} \leqslant 0.5$ , where $\tilde {t}$ is the time, and $\tilde {t}_c$ is the time instance at which the two colliding gravity currents have fully osculated, geometric distortions of the gravity current fronts result in stretching of pre-existing vorticity in the wall-normal direction inside the fronts, and an array of vertical vortices extending throughout the updraught fluid column develop along the interface separating the two colliding gravity currents. The array of vertical vortices is responsible for the mixing between the heavy fluids of the two colliding gravity currents and for the production of turbulent kinetic energy in the collision region. The presence of the top boundary deflects the updraughts into the horizontal direction, and a number of horizontal streamwise vortices are generated close to the top boundary. During Phase II, $0.5 \leqslant (\tilde {t}-\tilde {t}_c) \tilde {u}_f/\tilde {H} \leqslant 1.2$ , the horizontal streamwise vortices close to the top boundary induce turbulent buoyancy flux and break up into smaller structures. While the production of turbulent kinetic energy weakens, the rate of transfer of energy to turbulent flow due to turbulent buoyancy flux reaches its maximum and becomes the primary supply in the turbulent kinetic energy in Phase II. During Phase III, $1.2 \leqslant (\tilde {t}-\tilde {t}_c) \tilde {u}_f/\tilde {H} \leqslant 2.8$ , the collided fluid slumps away from the collision region, while the production of turbulent kinetic energy, turbulent buoyancy flux and dissipation of energy attenuate. From the point of view of energetics, the production of turbulent kinetic energy and turbulent buoyancy flux transfers energy away from the mean flow to the turbulent flow during the collision. Our study complements previous experimental investigations on the collision of gravity currents in that the flow structures, spatial distribution and temporal evolution of the mean flow and turbulent flow characteristics in the collision region are presented clearly. It is our understanding that such complete information on the energy budgets in the collision region can be difficult to attain in laboratory experiments.

Lifecycle of a Submesoscale Front Birthed from a Nearshore Internal Bore

AbstractThe ocean is home to many different submesoscale phenomena, including internal waves, fronts, and gravity currents. Each of these processes entails complex nonlinear dynamics, even in isolation. Here we present shipboard, moored, and remote observations of a submesoscale gravity current front created by a shoaling internal tidal bore in the coastal ocean. The internal bore is observed to flatten as it shoals, leaving behind a gravity current front that propagates significantly slower than the bore. We posit that the generation and separation of the front from the bore is related to particular stratification ahead of the bore, which allows the bore to reach the maximum possible internal wave speed. After the front is calved from the bore, it is observed to propagate as a gravity current for approximately 4 h, with associated elevated turbulent dissipation rates. A strong cross-shore gradient of alongshore velocity creates enhanced vertical vorticity (Rossby number ≈ 40) that remains locked with the front. Lateral shear instabilities develop along the front and may hasten its demise.

Removal of a dense bottom layer by a gravity current

Abstract

Origin of lobe and cleft at the gravity current front.

In this Rapid Communication, the Rayleigh-Taylor instability (RTI) along the density interfaces of gravity current fronts is analyzed. Both the location and the spanwise wave number of the most unstable mode determined by the local dispersion relation agree with those of the strongest perturbation obtained from numerical simulations, suggesting that the original formation mechanism of lobes and clefts at the current front is RTI. Furthermore, the predictions of the semi-infinite RTI model, i.e., the original dominating spanwise wave number of the Boussinesq current substantially depends on the Prandtl number and has a 1/3 scaling law with the Grashof number, are confirmed by the three-dimensional numerical simulations.

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

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.

Lobe‐cleft instability in the buoyant gravity current generated by estuarine outflow

AbstractGravity currents represent a broad class of geophysical flows including turbidity currents, powder avalanches, pyroclastic flows, sea breeze fronts, haboobs, and river plumes. A defining feature in many gravity currents is the formation of three‐dimensional lobes and clefts along the front and researchers have sought to understand these ubiquitous geophysical structures for decades. The prevailing explanation is based largely on early laboratory and numerical model experiments at much smaller scales, which concluded that lobes and clefts are generated due to hydrostatic instability exclusively in currents propagating over a nonslip boundary. Recent studies suggest that frontal dynamics change as the flow scale increases, but no measurements have been made that sufficiently resolve the flow structure in full‐scale geophysical flows. Here we use thermal infrared and acoustic imaging of a river plume to reveal the three‐dimensional structure of lobes and clefts formed in a geophysical gravity current front. The observed lobes and clefts are generated at the front in the absence of a nonslip boundary, contradicting the prevailing explanation. The observed flow structure is consistent with an alternative formation mechanism, which predicts that the lobe scale is inherited from subsurface vortex structures.

Characteristics of colliding sea breeze gravity current fronts: a laboratory study

Sea and land breeze circulations driven by surface temperature differences between land and sea often evolve into gravity currents with sharp fronts. Along narrow peninsulas, islands and enclosed seas, sea/land breeze fronts from opposing shorelines may converge and collide and may initiate deep convection and heavy precipitation. Here we investigate the collision of two sea breeze gravity current fronts in an analogue laboratory setting. We examine these collisions by means of ‘lock‐exchange’ experiments in a rectangular channel. The effects of differences in gravity current density and height are studied. Upon collision, a sharp front separating the two currents develops. For symmetric collisions (the same current densities and heights) this front is vertical and stationary. For asymmetric collisions (density differences, similar heights) the front is tilted, changes shape in time and propagates in the same direction as the heavier current before the collision. Both symmetric and asymmetric collisions lead to upward displacement of fluid from the gravity currents and mixing along the plane of contact. The amount of mixing along the collision front decreases with asymmetry. Height differences impact post‐collision horizontal propagation: there is significant propagation in the same direction as the higher current before collision, independent of density differences. Collisions of two gravity current fronts force sustained ascending motions which increase the potential for deep convection. From our experiments we conclude that this potential is larger in stationary collision fronts from symmetric sea breeze collisions than in propagating collision fronts from asymmetric sea breeze collisions.

Filling box flows in porous media

We report upon a theoretical and experimental investigation of a porous medium ‘filling box’ flow by specifically examining the details of the laminar descending plume and its outflow in a control volume having an impermeable bottom boundary and sidewalls. The plume outflow initially comprises a pair of oppositely directed gravity currents. The gravity currents propagate horizontally until they reach the lateral sidewalls at $y=\pm L$. The flow then becomes of filling box type, with a vertically ascending ‘first front’ separating discharged plume fluid below from ambient fluid above. The flow details are described analytically by first deriving a new similarity solution for Darcy plumes with $\mathit{Pe}>O(1)$, where $\mathit{Pe}$ is the Péclet number. From the similarity solution so obtained, we then derive expressions for the plume volume flux and mean reduced gravity as functions of the vertical distance from the source. Regarding the plume outflow, a similarity solution adopted from Huppert & Woods (J. Fluid Mech., vol. 292, 1995, pp. 55–69) describes the height and front speed of the gravity currents, whereas a semi-implicit finite difference scheme is used to predict the first front elevation versus time and horizontal distance. As with high-Reynolds-number filling box flows, that studied here is an example of a coupled problem: the gravity current source conditions are prescribed by the plume volume flux and mean reduced gravity. Conversely, discharged plume fluid may be re-entrained into the plume, be it soon or long after reaching the bottom impermeable boundary. To corroborate our model predictions, analogue laboratory experiments are performed with fresh water and salt water as the working fluids. Our experiments consider as independent variables the porous medium bead diameter and the plume source volume flux and reduced gravity. Predictions for the gravity current front position and height compare favourably against analogue measured data. Good agreement is likewise noted when considering either the mean elevation or the profile of the first front. Results from this study may be adopted in modelling geological plumes. For example, our equations can be used to predict the time required for discharged plume fluid to return to the point of injection in the case of aquifers closed on the sides and below by impermeable boundaries.

Gravity current flow over sinusoidal topography in a two-layer ambient

We report upon a laboratory experimental study of dense full- and partial-depth locks released gravity currents propagating through a two-layer ambient and over sinusoidal topography of amplitude A and wavelength λ. Attention is restricted to a Boussinesq flow regime and the initial (or slumping) stage of motion. Because of the presence of the topography, the lower layer depth and channel depth vary in the downstream direction by as much as ±52% and ±40%, respectively. Despite such large variations, the instantaneous gravity current front speed typically changes by only a small amount from its average value, Ū. We conclude that the reason for this unexpected observation is a counterbalancing between the contraction/expansion of the channel on one hand and along-slope variations of the buoyancy force on the other hand. Compared to the case of gravity current flow over a horizontal surface, the topography has a retarding effect on Ū; we characterize the variation of Ū with the following parameters: A, A/λ, the interface height, the height of gravity current fluid inside the lock, and the layer densities as characterized by S ≡ (ρ1 − ρ2)/(ρ0 − ρ2) where ρ0 is the gravity current density, ρ1 is the lower ambient layer density, and ρ2 is the upper ambient layer density. The topography also alters the structure of the gravity current head by inducing large-scale Kelvin-Helmholtz (K-H) vortices directly downstream of regions of significant shear. These vortices are transient flow features; after formation and saturation, they relax at which point gravity current fluid sloshes back and forth in the topographic troughs. A simple geometric model is presented that prescribes the minimum number of topographic peaks a gravity current fluid will overcome. As in the horizontal bottom case, the forward advance of the gravity current can excite a downstream-propagating interfacial disturbance, which is more prominent for larger S. We identify when the gravity current front will travel faster (supercritical flow) or slower (subcritical flow) than the interfacial disturbance based on S and A/λ. Unlike in the horizontal-bottom case, however, we typically also observe interfacial oscillations far ahead of the gravity current front. These have wavelength approximately equal to λ and are due to spatial variations in the speed of the ambient return flow.

Gravity currents propagating up a slope in a two-layer fluid

Gravity currents produced by partial-depth lock-release and travelling along the base of a two-layer stratified ambient are investigated as they propagate along a rising slope. The initial gravity current front speed is found to be consistent with a theory adapted from Shin et al. [“Gravity currents produced by lock exchange,” J. Fluid Mech. 521, 1-34 (2004)] to the case of partial-depth currents in two-layer ambients. The subsequent evolution depends on the gravity current speed relative to the speed of the interfacial disturbance it creates. The deceleration of supercritical gravity currents, which travel faster than the interfacial disturbance, is gradual and agrees well with the relationship developed by Marleau et al. [“Gravity currents propagating up a slope,” Phys. Fluids 26, 046605 (2014)] for upslope gravity currents in uniform density ambients. In several subcritical cases, the gravity current suddenly came to rest as a consequence of interactions with the interfacial disturbance. The disturbance amplitude, speed, and width are found to be nearly constant during its evolution. In cases for which the ambient interface intersected the bottom slope, the amplitude, speed, and width were nearly constant up to the point where the lower layer shallowed and the disturbance transformed into an upslope-propagating gravity current.

Axisymmetric gravity currents in two-layer density-stratified media

Numerous studies have considered the flow of a rectilinear, high Reynolds number, Boussinesq gravity current through a two-layer stratified ambient, however, far less is known concerning the analogue axisymmetric problem. Whereas in both instances there is the possibility of a dynamic coupling between the gravity current front and the waves that are excited by its forward advance, axisymmetric gravity currents entail the added complexity of a radially-diverging flow. Because a steady-state formulation cannot then be developed, we instead present a one-layer shallow water model that describes the flow evolution for various initial conditions and ambient stratifications. We also report upon $$>\!\!30$$ full- and partial-depth lock release laboratory experiments that span a densitometric range $$0 \le S < 0.8868$$ where $$S=(\rho _1-\rho _2)/(\rho _c-\rho _2)$$ in which $$\rho _c$$ , $$\rho _1$$ and $$\rho _2$$ denote, respectively, the densities of the gravity current and lower and upper ambient layers. Of principal interest is the initial front speed of the gravity current for which good agreement is observed between laboratory measurement and shallow water numerical simulation, despite the limiting assumptions of the latter. The horizontal distance over which the initial front speed is maintained may span several lock-lengths, however, this depends on whether or not the gravity current is substantially impacted by the interfacial wave(s). For example, when the lower ambient layer is moderate and $$S$$ is large, the transfer of momentum from the gravity current front to the wave may lead to a deceleration so severe that gravity current fluid is swept in the $$-r$$ direction. The connection between our analysis and problems of pollution dispersion is briefly outlined.

Thermal Theory for Non-Boussinesq Gravity Currents Propagating on Inclined Boundaries

In this study the author derived the thermal theory for non-Boussinesq gravity currents produced from an instantaneous buoyancy source propagating on an inclined boundary. For Boussinesq gravity currents on a slope, it is known that the gravity current front location history follows an asymptotic relationship, xf3/2∼t, where xf is the front location, and t is the time, when the gravity current is sufficiently far into the deceleration phase. For non-Boussinesq gravity currents, the distance for the acceleration phase is extended attributable to the non-Boussinesq effects. When the gravity current is sufficiently far into the deceleration phase, this paper shows that the non-Boussinesq gravity currents tend to approach similar asymptote in the deceleration phase as Boussinesq gravity currents, but the approach is less rapid in non-Boussinesq gravity currents.

Gravity currents propagating up a slope

Bottom propagating gravity currents resulting from full- and partial-depth lock-release experiments are investigated as they approach and then propagate up a rising slope. Consistent with the prediction of a WKB-like theory, the gravity current front decelerates in a nearly uniform manner along the slope as 0.112g′s(D/H)(2 − D/H), in which g′ is the reduced gravity, s is the slope, D is the initial lock-fluid height, and H is the ambient fluid height. The shape of the gravity current as it decelerates over relatively steep slopes is found to be self similar with a nearly linear decrease of the head height between the start of the slope and up to 80% of the distance to the nose. Some deviation from self-similar behaviour is found in cases with small s because of the comparatively large volume of fluid in the gravity current tail that flows downslope while the front continues to advance upwards.

Dense gravity currents moving beneath progressive free-surface water waves

AbstractThe characteristics of dense gravity currents in coastal regions, where free-surface gravity waves are dominant, have yet to be studied in the laboratory. This paper provides a first insight into the dynamics of dense saline gravity currents moving beneath regular progressive free-surface water waves. The gravity currents were generated by releasing a finite volume of saline into a large wave tank with an established periodic wave field. After the initial collapse, the gravity currents propagated horizontally with two fronts, one propagating in the wave direction and the other against the wave direction. The fronts of the gravity currents oscillated with an amplitude and phase that correlated with the orbital velocities within a region close to the bed. To leading order, the overall length of the gravity current was found to be weakly affected by the wave action and the dynamics of the current could be approximated by simply considering the buoyancy of the released fluid. Other characteristics such as the position of the gravity current centre and the shape of the two leading profiles were found to be significantly affected by the wave action. The centre was displaced at constant speed dependent on the second-order wave-induced mean Lagrangian velocity. For long waves, the centre was advected downstream in the direction of wave propagation owing to the dominance of Stokes drift. For short waves, the gravity current centre moved upstream against the wave direction, as under these wave conditions Stokes drift is negligible at the bed. An asymmetry in the shape of the upstream and downstream current heads was observed, with the gravity current front moving against the waves being much thicker and the front steeper, similar to the case of a current moving in a stream.

A general description of a gravity current front propagating in a two-layer stratified fluid

AbstractThe behaviour of a gravity current propagating into a two-layer stratified ambient fluid is described in detail. A comprehensive description is given of the different flow regimes, with particular emphasis on the front condition linking the thickness of the gravity current to its speed of propagation and the transfer of energy to upstream disturbances in the form of internal bores and nonlinear solitary waves. Hydraulic theory analogous to that of two-layer flow over topography (Baines, J. Fluid Mech., vol. 146, 1984, pp. 127–167) is extended to the gravity current problem to classify frontal behaviour into the following regimes: Type I, subcritical currents; Type II, currents that generate upstream undular bores; Type III, currents that generate an upstream monotonic bore connected by a rarefaction; Type IV, supercritical fronts with a large-amplitude trapped solitary-wave-like disturbance; and Type V, supercritical gravity currents. Over 200 two-dimensional Boussinesq–Euler simulations spanning a range of gravity current properties demonstrate good agreement, for both the behavioural regime and the front condition ${U}_{o} (h)$, with hydraulic theory that extends original work by Rottman &amp; Simpson (Q. J. R. Meteorol. Soc., vol. 115, 1989, pp. 941–963) to arbitrary ambient layer thickness, and uses an improved closure for the upstream bore that correctly predicts the behaviour in the limit of large bore amplitude. In addition, the energy balance is analysed, and it is shown that the energy transfer from the gravity current to upstream disturbances is significant, and consistent with the hydraulic theory. The results demonstrate a clear connection to the problem of upstream resonance in two-layer flow over topography, and have significant implications for interpreting field observations of nonlinear internal waves generated by atmospheric density currents and coastal river plumes.

Gravity currents in a two-layer stratified ambient: The theory for the steady-state (front condition) and lock-released flows, and experimental confirmations

We consider the propagation of a gravity current of density ρc at the bottom of a two-layer stratified ambient in a horizontal channel of height H, in the high-Reynolds number Boussinesq domain. The study emphasizes theoretical-analytical modeling, however, experimental and Navier-Stokes simulation data are also presented and their comparison with theory is discussed. The stratification parameters are S = (ρ1 − ρ2)/(ρc − ρ2) where ρ is the fluid density, and φ = h1R/H where h1R is the (unperturbed) ambient interface height. Here, 1 and 2 denote, respectively, the lower and upper layer and c denotes the gravity current. The reduced gravity is defined as g′ = (ρc/ρ2 − 1)g. Rigorous results are obtained for the steady-state analogue of the classical problem of Benjamin [J. Fluid Mech. 31, 209 (1968)]10.1017/S0022112068000133, in which the half-infinite gravity current has thickness h and speed U. We thereby demonstrate that the Froude number \documentclass[12pt]{minimal}\begin{document}${\cal F}= U/(g^{\prime } h)^{1/2}$\end{document}F=U/(g′h)1/2 is a function of a = h/H, S, and φ. In general, two solutions (or modes) may be realized. Issues of energy dissipation, sub- vs. supercriticality with respect to long internal waves and, more generally, the influence of upstream-propagating disturbances are discussed. For a gravity current released from a lock of height h0 and length x0, we derive an approximate shallow-water model and show that the motion is in this case governed by Ξ = H/h0, S, and φ. Although the shallow-water model neglects motion in the ambient layers and ignores the impact of propagation on stratification, the gravity current front speed in the slumping stage is in excellent agreement with measured data. Our theoretical solutions are consistent with previous results (in particular, Holyer and Huppert [J. Fluid Mech. 100, 739 (1980)] and Tan et al. [Environ. Fluid Mech. 11, 203 (2011)]), but have the advantages of being (i) derived without reliance on adjustable constants and ad hoc closures; (ii) applicable to a significantly broader range of dimensionless parameters; and (iii) better assessed by comparison against measured data. The present one-layer shallow-water approximation turns out to be a simple and versatile extension of existing models for homogeneous and linearly stratified ambients, and can be straightforwardly incorporated into the available prediction tools for gravity currents.

The impact of mesh adaptivity on the gravity current front speed in a two-dimensional lock-exchange

Numerical simulations of the two-dimensional lock-exchange flow are used to evaluate the performance of adaptive meshes as implemented in the non-hydrostatic, finite-element model Fluidity-ICOM. The lock-exchange is a widely studied laboratory-scale set-up that produces two horizontally propagating gravity currents and incorporates key physical processes associated with gravity currents over many scales, including ocean overflows. The Froude number (non-dimensional front speed) is used to assess simulations performed on structured-fixed, unstructured-fixed and unstructured-adaptive meshes and different adaptive mesh configurations are compared. Fluidity-ICOM successfully captures the flow dynamics, including the development of Kelvin–Helmholtz billows. Mesh adapts are guided by a metric which is key to the ability of an adaptive mesh to represent the flow. The metric employed in Fluidity-ICOM is simple, based on the curvature of the solution fields and user-defined solution field weights. Good representation of the gravity current front region is essential to the quality of the solution and for the adaptive meshes this is achieved by reducing the horizontal velocity field weight near the boundaries. Adaptive meshes that are configured in this way are seen to perform as well as high-resolution fixed meshes whilst using at least one order of magnitude fewer nodes. The Froude numbers also compare well with previously published values determined from experimental, numerical and theoretical approaches. The substantial reduction in the number of nodes used by the adaptive meshes is particularly encouraging as it suggests that even greater gains may be achieved in three-dimensional simulations and larger-scale problems. Results show that successful use of the adaptive mesh approach employed requires a clear understanding of the physics of the system and the metric. These considerations will be vital to the effective application of adaptive mesh approaches in numerical modelling of more complex ocean flows.

Turbulent structures in planar gravity currents and their influence on the flow dynamics

Direct numerical simulations (DNS) of planar gravity current in the Boussinesq limit have been conducted with the objective of identifying, visualizing, and describing turbulent structures and their influence on the flow dynamics. The simulations are performed for Reynolds numbers of Re = 8950 and Re = 15,000 with 31‐ and 131‐million grid point resolutions, respectively. This range of Reynolds numbers ensures fully developed turbulent gravity currents, which have never been simulated before using DNS. The flow develops zones with different turbulence characteristics, which eventually interact with each other. The near‐wall bottom flow resembles boundary layer flow with several hairpin‐like vortices oriented in the direction of the flow and preferential patterns of low‐ and high‐speed streaks. The separation between low‐speed streaks at the front scales with the lobe size, which is about 200 wall units for Re = 15,000. Upstream of the front, the separation between low‐speed streaks scales with the well‐accepted value of 100 wall units for turbulent boundary layers. These patterns have associated regions of low and high bottom shear stresses with implications for sediment erosion and bed load transport. Most of the erosive power of the flow is found in the gravity current front. The interface between heavy and light fluids rolls up by baroclinic generation of Kelvin‐Helmholtz vortices, which undergo sudden breakup and decay to small‐scale turbulence. The effect of turbulence and three‐dimensionality on the flow dynamics is addressed by comparing two‐ and three‐dimensional simulations. Three‐dimensional simulations present active mechanisms that undermine the strong flow coherence, comparing well with experimental observations.

Effects of rotation and sloping terrain on the fronts of density currents

The initial stage of the adjustment of a gravity current to the effects of rotation with angular velocity $f/2$ is analysed using a short time analysis where Coriolis forces are initiated in an inviscid von Karman–Benjamin gravity current front at $t_F\,{=}\,0$ . It is shown how, on a time-scale of order $1/f$ , as a result of ageostrophic dynamics, the slope and front speed $U_F$ are much reduced from their initial values, while the transverse anticyclonic velocity parallel to the front increases from zero to $O(N H_0)$ , where $N\,{=}\,\sqrt{g'/H_0}$ is the buoyancy frequency, and $g'\,{=}\,g \Delta \rho /\rho_0$ is the reduced acceleration due to gravity. Here $\rho_0$ is the density and $\Delta \rho$ and $H_0$ are the density difference and initial height of the current. Extending the steady-state theory to account for the effect of the slope $\sigma$ on the bottom boundary shows that, without rotation, $U_{F}$ has a maximum value for $\sigma \,{=}\, \upi/6$ , while with rotation, $U_{F}$ tends to zero on any slope. For the asymptotic stage when $ft_F \,{\gg}\, 1$ , the theory of unsteady waves on the current is reviewed using nonlinear shallow-water equations and the van der Pol averaging method. Their motions naturally split into a ‘balanced’ component satisfying the Margules geostrophic relation and an equally large ‘unbalanced’ component, in which there is horizontal divergence and ageostrophic vorticity. The latter is responsible for nonlinear oscillations in the current on a time scale $f^{-1}$ , which have been observed in the atmosphere and field experiments. Their magnitude is mainly determined by the initial potential energy in relation to that of the current and is proportional to the ratio $\sqrt{\hbox{\it Bu}} \,{=}\, L_R/R_0$ , where $L_R\,{=}\,N H_0/f$ is the Rossby deformation radius and $R_0$ is the initial radius. The effect of slope friction also prevents the formation of a steady front. From the analysis it is concluded that a weak mean radial flow must be driven by the ageostrophic oscillations, preventing the mean front speed $U_F$ from halting sharply at $f t_F \,{\sim}\, 1$ . Depending on the initial value of $L_R/R_0$ , physical arguments show that $U_F$ decreases slowly in proportion to $(f t_F)^{-1/2}$ , i.e. $U_F/U_{F_0}\,{=}\,F(ft_F,\hbox{\it Bu})$ . Thus the front only tends to the geostrophic asymptotic state of zero radial velocity very slowly (i.e. as $f t_F \,{\rightarrow}\, \infty $ ) for finite values of $L_R/R_0$ . However, as $L_R/R_0 \,{\rightarrow}\, 0$ , it reaches this state when $f t_F \,{\sim}\, 1$ . This analysis of the overall nonlinear behaviour of the gravity current is consistent with two two-dimensional non-hydrostatic (Navier–Stokes) and axisymmetric hydrostatic (shallow-water) Eulerian numerical simulations of the varying form of the rotating gravity current. When the effect of surface friction is considered, it is found that the mean movement of the front is significantly slowed. Furthermore, the oscillations with angular frequency $f$ and the slow growth of the radius, when $ft_F \,{\ge}\, 1$ , are consistent with recent experiments.