scholarly journals On the baroclinic instability of axisymmetric rotating gravity currents with bottom slope

2000 ◽  
Vol 408 ◽  
pp. 149-177 ◽  
Author(s):  
PAUL F. CHOBOTER ◽  
GORDON E. SWATERS
Keyword(s):  
Numerical Simulations ◽  
Linear Stability ◽  
Lower Layer ◽  
Baroclinic Instability ◽  
Laboratory Data ◽  
Gravity Current ◽  
Gravity Currents ◽  
Finite Amplitude ◽  
Rotating System ◽  
A Current

The baroclinic stability characteristics of axisymmetric gravity currents in a rotating system with a sloping bottom are determined. Laboratory studies have shown that a relatively dense fluid released under an ambient fluid in a rotating system will quickly respond to Coriolis effects and settle to a state of geostrophic balance. Here we employ a subinertial two-layer model derived from the shallow-water equations to study the stability characteristics of such a current after the stage at which geostrophy is attained. In the model, the dynamics of the lower layer are geostrophic to leading order, but not quasi-geostrophic, since the height deflections of that layer are not small with respect to its scale height. The upper-layer dynamics are quasi-geostrophic, with the Eulerian velocity field principally driven by baroclinic stretching and a background topographic vorticity gradient.Necessary conditions for instability, a semicircle-like theorem for unstable modes, bounds on the growth rate and phase velocity, and a sufficient condition for the existence of a high-wavenumber cutoff are presented. The linear stability equations are solved exactly for the case where the gravity current initially corresponds to an annulus flow with parabolic height profile with two incroppings, i.e. a coupled front. The dispersion relation for such a current is solved numerically, and the characteristics of the unstable modes are described. A distinguishing feature of the spatial structure of the perturbations is that the perturbations to the downslope incropping are preferentially amplified compared to the upslope incropping. Predictions of the model are compared with recent laboratory data, and good agreement is seen in the parameter regime for which the model is valid. Direct numerical simulations of the full model are employed to investigate the nonlinear regime. In the initial stage, the numerical simulations agree closely with the linear stability characteristics. As the instability develops into the finite-amplitude regime, the perturbations to the downslope incropping continue to preferentially amplify and eventually evolve into downslope propagating plumes. These finally reach the deepest part of the topography, at which point no more potential energy can be released.

Gravity currents over fractured substrates in a porous medium

2007 ◽  
Vol 584 ◽  
pp. 415-431 ◽  
Author(s):  
DAVID PRITCHARD
Keyword(s):  
Porous Medium ◽  
Steady State ◽  
Gravity Current ◽  
Gravity Currents ◽  
Limited Extent ◽  
Form Part ◽  
Long Time ◽  
Self Similar ◽  
Steady State Solutions ◽  
A Current

We consider the behaviour of a gravity current in a porous medium when the horizontal surface along which it spreads is punctuated either by narrow fractures or by permeable regions of limited extent. We derive steady-state solutions for the current, and show that these form part of a long-time asymptotic description which may also include a self-similar ‘leakage current’ propagating beyond the fractured region with a length proportional to t1/2. We discuss the conditions under which a current can be completely trapped by a permeable region or a series of fractures.

Nonlinear effects in two-layer large-amplitude geostrophic dynamics. Part 2. The weak-beta case

2000 ◽  
Vol 412 ◽  
pp. 161-196 ◽  
Author(s):  
RICHARD H. KARSTEN ◽  
GORDON E. SWATERS
Keyword(s):  
Numerical Simulations ◽  
Nonlinear Stability ◽  
Large Amplitude ◽  
Baroclinic Instability ◽  
Finite Amplitude ◽  
Linear Instability ◽  
Steady Flows ◽  
Leading Order ◽  
Unstable Flow ◽  
Amplitude Analysis

This paper is a continuation of our study on nonlinear processes in large-amplitude geostrophic (LAG) dynamics. Here, we examine the so-called weak-β models. These models arise when the intrinsic length scale is large enough so that the dynamics is geostrophic to leading order but not so large that the β-effect enters into the dynamics at leading order (but remains, nevertheless, dynamically non-negligible). In contrast to our previous analysis of strong-β LAG models in Part 1, we show that the weak-β models allow for vigorous linear baroclinic instability.For two-layer weak-β LAG models in which the mean depths of both layers are approximately equal, the linear instability problem can exhibit an ultraviolet catastrophe. We argue that it is not possible to establish conditions for the nonlinear stability in the sense of Liapunov for a steady flow. We also show that the finite-amplitude evolution of a marginally unstable flow possesses explosively unstable modes, i.e. modes for which the amplitude becomes unbounded in finite time. Numerical simulations suggest that the development of large-amplitude meanders, squirts and eddies is correlated with the presence of these explosively unstable modes.For two-layer weak-β LAG models in which one of the two layers is substantially thinner than the other, the linear stability problem does not exhibit an ultraviolet catastrophe and it is possible to establish conditions for the nonlinear stability in the sense of Liapunov for steady flows. A finite-amplitude analysis for a marginally unstable flow suggests that nonlinearities act to stabilize eastward and enhance the instability of westward flows. Numerical simulations are presented to illustrate these processes.

The front speed of intrusions into a continuously stratified medium

2007 ◽  
Vol 594 ◽  
pp. 369-377 ◽  
Author(s):  
DIOGO BOLSTER ◽  
ALICE HANG ◽  
P. F. LINDEN
Keyword(s):  
Numerical Simulations ◽  
Gravity Current ◽  
Gravity Currents ◽  
Stratified Fluid ◽  
Stratified Medium ◽  
Fluid Density ◽  
Neutral Buoyancy ◽  
Front Speed ◽  
Predicted Values ◽  
Good Agreement

This paper examines intrusive Boussinesq gravity currents, propagating into a continuously stratified fluid. We develop a model, based on energy arguments, to predict the front speed of such an intrusive gravity current from a lock release. We find that the depth at which the intrusion occurs, which corresponds to the level of neutral buoyancy (i.e. the depth where the intrusion density equals the stratified fluid density), affects the front speed. The maximum speeds occur when the intrusion travels along the top and bottom boundaries and the minimum speed occurs at mid-depth. Experiments and numerical simulations were conducted to compare to the theoretically predicted values, and good agreement was found.

On the propagation of internal bores

1997 ◽  
Vol 331 ◽  
pp. 81-106 ◽  
Author(s):  
JOSEPH B. KLEMP ◽  
RICHARD ROTUNNO ◽  
WILLIAM C. SKAMAROCK
Keyword(s):  
Numerical Simulations ◽  
Laboratory Data ◽  
Gravity Current ◽  
Leading Edge ◽  
Strong Shear ◽  
Hydraulic Theory ◽  
Front Condition ◽  
Internal Bores ◽  
Boussinesq Fluid ◽  
Internal Bore

According to classical hydraulic theory, the energy losses within an external bore must occur within the expanding layer. However, the application of this theory to describe the propagation of internal bores leads to contradiction with accepted gravity-current behaviour in the limit as the depth of the expanding layer ahead of the bore becomes small. In seeking an improved expression for the propagation of internal bores, we have rederived the steady front condition for a bore in a two-layer Boussinesq fluid in a channel under the assumption that the energy loss occurs within the contracting layer. The resulting front condition is in good agreement with available laboratory data and numerical simulations, and has the appropriate behaviour in both the linear long-wave and gravity-current limits. Analysis of an idealized internal bore assuming localized turbulent stresses suggests that the energy within the expanding layer should, in fact, increase. Numerical simulations with a two-dimensional non-hydrostatic model also reveal a slight increase of energy within the expanding layer and suggest that the structure of internal bores is fundamentally different from classical external bores, having the opposite circulation and little turbulence in the vicinity of the leading edge. However, if there is strong shear near the interface between layers, the structure and propagation of internal jumps may become similar to their counterparts in classical hydraulic theory. The modified jump conditions for internal bores produce some significant alterations in the traditional Froude-number dependence of Boussinesq shallow-water flow over an obstacle owing to the altered behaviour of the upstream-propagating internal bore.

Lock-exchange gravity currents propagating in a channel containing an array of obstacles

2015 ◽  
Vol 765 ◽  
pp. 544-575 ◽  
Author(s):  
Ayse Yuksel Ozan ◽  
George Constantinescu ◽  
Andrew J. Hogg
Keyword(s):  
Reynolds Number ◽  
Numerical Simulations ◽  
Drag Coefficient ◽  
Gravity Current ◽  
Gravity Currents ◽  
Front Velocity ◽  
Release Time ◽  
Exchange Flow ◽  
Layer Model ◽  
Shallow Layer

AbstractLarge eddy simulation (LES) is used to investigate the evolution of Boussinesq gravity currents propagating through a channel of height $H$ containing a staggered array of identical cylinders of square cross-section and edge length $D$. The cylinders are positioned with their axes horizontal and perpendicular to the (streamwise) direction along which the lock-exchange flow develops. The effects of the volume fraction of solids, ${\it\phi}$, the Reynolds number and geometrical parameters describing the array of obstacles on the structure of the lock-exchange flow, total drag force acting on the gravity current, front velocity and global energy budget are analysed. Simulation results show that the currents rapidly transition to a state in which the extra resistance provided by the cylinders strongly retards the motion and dominates the dissipative processes. A shallow layer model is also formulated and similarity solutions for the motion are found in the regime where the driving buoyancy forces are balanced by the drag arising from the interaction with the cylinders. The numerical simulations and this shallow layer model show that low-Reynolds-number currents transition to a drag-dominated regime in which the resistance is linearly proportional to the flow speed and, consequently, the front velocity, $U_{f}$, is proportional to $t^{-1/2}$, where $t$ is the time measured starting at the gate release time. By contrast, high-Reynolds-number currents, for which the cylinder Reynolds number is sufficiently high that the drag coefficient for most of the cylinders can be considered constant, transition first to a quadratic drag-dominated regime in which the front speed determined from the simulations is given by $U_{f}\sim t^{-0.25}$, before undergoing a subsequent transition to the aforementioned linear drag regime in which $U_{f}\sim t^{-1/2}$. Meanwhile, away from the front, the depth-averaged gravity current velocity is proportional to $t^{-1/3}$, a result that is in agreement with the shallow water model. It is suggested that the difference between these two is due to mixing processes, which are shown to be significant in the numerical simulations, especially close to the front of the motion. Direct estimation of the drag coefficient $C_{D}$ from the numerical simulations shows that the combined drag parameter for the porous medium, ${\it\Gamma}_{D}=C_{D}{\it\phi}(H/D)/(1-{\it\phi})$, is the key dimensionless grouping of variables that determines the speed of propagation of the current within arrays with different $C_{D},{\it\phi}$ and $D/H$.

The dynamics of sedimenting surface gravity currents

1999 ◽  
Vol 392 ◽  
pp. 27-44 ◽  
Author(s):  
T. MAXWORTHY
Keyword(s):  
Interstitial Fluid ◽  
Lower Layer ◽  
Gravity Current ◽  
Gravity Currents ◽  
Surface Gravity ◽  
Physical Situation ◽  
Total Density ◽  
Heavy Particles ◽  
Two Fluids ◽  
Series Of Experiments

We have performed a series of experiments on the dynamics of sedimenting, surface gravity currents. The physical situation concerns a current, with total density ρC, evolving at the surface of a fluid of greater density, ρA. In turn ρC is made up of interstitial fluid of density ρI and heavy particles with a concentration by weight c and a density ρP. Only the case of the release of a constant volume of particles and interstitial fluid has been considered in detail. It has been found that the sedimentation of the particles, plus some of the interstitial fluid, through the interface between the two fluids has a profound effect upon the motion of the current. When the rejected mixture of particles and upper- and lower-layer fluids reaches the bottom of the experimental tank it generates a secondary gravity current which in turn interacts with the primary current to further modify its behaviour. Using simple models we have been able to rationalize the observations and reveal the dynamical balances which appear to be important. A subsidiary experiment and analysis on the flux characteristics of the interface have been performed in order to further clarify the important effects of the particle motion through that region.

scholarly journals Gravity current propagation up a valley

2014 ◽  
Vol 762 ◽  
pp. 417-434 ◽  
Author(s):  
Catherine S. Jones ◽  
Claudia Cenedese ◽  
Eric P. Chassignet ◽  
P. F. Linden ◽  
Bruce R. Sutherland
Keyword(s):  
Numerical Simulations ◽  
Laboratory Experiments ◽  
Rectangular Channel ◽  
Gravity Current ◽  
Gravity Currents ◽  
Momentum Transport ◽  
Front Speed

AbstractThe advance of the front of a dense gravity current propagating in a rectangular channel and V-shaped valley both horizontally and up a shallow slope is examined through theory, full-depth lock–release laboratory experiments and hydrostatic numerical simulations. Consistent with theory, experiments and simulations show that the front speed is relatively faster in the valley than in the channel. The front speed measured shortly after release from the lock is 5–22 % smaller than theory, with greater discrepancy found in upsloping V-shaped valleys. By contrast, the simulated speed is approximately 6 % larger than theory, showing no dependence on slope for rise angles up to ${\it\theta}=8^{\circ }$. Unlike gravity currents in a channel, the current head is observed in experiments to be more turbulent when propagating in a V-shaped valley. The turbulence is presumably enhanced due to the lateral flows down the sloping sides of the valley. As a consequence, lateral momentum transport contributes to the observed lower initial speeds. A Wentzel–Kramers–Brillouin like theory predicting the deceleration of the current as it runs upslope agrees remarkably well with simulations and with most experiments, within errors.

scholarly journals Instability of Lenticular Vortices: Results from Laboratory Experiments, Linear Stability Analysis and Numerical Simulations

Fluids ◽  
2021 ◽  
Vol 6 (11) ◽  
pp. 380
Author(s):  
Noé Lahaye ◽  
Alexandre Paci ◽  
Stefan G. Llewellyn Smith
Keyword(s):  
Stability Analysis ◽  
Numerical Simulations ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Laboratory Experiments ◽  
Lower Layer ◽  
Water Model ◽  
Layer Potential ◽  
Rotating Shallow Water ◽  
Layer Flow

The instability of surface lenticular vortices is investigated using a comprehensive suite of laboratory experiments combined with numerical linear stability analysis as well as nonlinear numerical simulations in a two-layer Rotating Shallow Water model. The development of instabilities is discussed and compared between the different methods. The linear stability analysis allows for a clear description of the origin of the instability observed in both the laboratory experiments and numerical simulations. While global qualitative agreement is found, some discrepancies are observed and discussed. Our study highlights that the sensitivity of the instability outcome is related to the initial condition and the lower-layer flow. The inhibition or even suppression of some unstable modes may be explained in terms of the lower-layer potential vorticity profile.

Gravity currents descending a ramp in a stratified tank

1999 ◽  
Vol 379 ◽  
pp. 39-69 ◽  
Author(s):  
J. J. MONAGHAN ◽  
R. A. F. CAS ◽  
A. M. KOS ◽  
M. HALLWORTH
Keyword(s):  
Numerical Simulations ◽  
Large Amplitude ◽  
Gravity Current ◽  
Gravity Currents

This paper describes experiments and numerical simulations of a gravity current flowing down a ramp in a tank stratified in two layers. We study the dynamics of the configuration for different densities of the gravity current and different ramp angles. The experiments show that waves of large amplitude can be generated easily and that, depending on the density of the gravity current, the initial gravity current splits into a gravity current along the interface of the stratified layers, and a gravity current along the bottom of the tank. We also describe numerical simulations which give results in agreement with the results of the experiments and enable us to study three-fluid configurations with wider ranges of density than is possible in the laboratory.

Rotational flow in gravity current heads

2005 ◽  
Vol 363 (1832) ◽  
pp. 1603-1623 ◽  
Author(s):  
J.N McElwaine
Keyword(s):  
Froude Number ◽  
Gravity Current ◽  
Gravity Currents ◽  
Rotational Flow ◽  
Arbitrary Angle ◽  
Von Karman ◽  
Von Kármán ◽  
A Current

The structure of gravity currents and plumes, in an unbounded ambient, on a slope of arbitrary angle is analysed. Inviscid, rotational flow solutions in a wedge are used to study the flow near the front of a current, and used to show that the Froude number is and the angle of the front to the slope is 60°. This extends the result of von Kármán (1940) to arbitrary slope angles and large internal current velocities. The predictions of the theory are briefly compared with experiments and used to explain the large negative (relative to ambient) pressures involved in avalanches.

Sign in / Sign up

Export Citation Format

Share Document