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Carnevale, GF, Kloosterziel RC, Orlandi P.  2016.  Equilibration of centrifugally unstable vortices: A review. European Journal of Mechanics B-Fluids. 55:246-258.   10.1016/j.euromechflu.2015.06.007   AbstractWebsite

In three-dimensional flow, a vortex can become turbulent and be destroyed through a variety of instabilities. In rotating flow, however, the result of the breakup of a vortex is usually a state comprising several vortices with their axes aligned along the ambient rotation direction. This article is a review of our recent work on how the combined effect of centrifugal and barotropic instabilities can breakup a vortex and lead to its reformation in a predictable way even though an intermediate stage in the evolution is turbulent. Centrifugal instability tends to force the unstable vortex into a turbulent state that mixes absolute angular momentum in such a way as to precondition the flow for a subsequent barotropic instability. A method for predicting the redistribution of angular momentum and theresulting velocity profile is discussed. The barotropic instability horizontally redistributes the component of vorticity that is aligned along the ambient rotation vector, resulting in the final byproducts of the instability, which are stabilized by the effects of ambient rotation. A prediction scheme that puts the tendencies of these two instabilities together proves to be very reliable. (C) 2015 Elsevier Masson SAS. All rights reserved.

Kloosterziel, RC, Orlandi P, Carnevale GF.  2015.  Saturation of equatorial inertial instability. Journal of Fluid Mechanics. 767:562-594.   10.1017/jfm.2015.63   AbstractWebsite

Inertial instability in parallel shear flows and circular vorwes in a uniformly rotating system (f-plane) redistributes absolute linear momentum or absolute angular momentum in such a way as to neutralize the instability. In previous studies we showed that, in the absence of other instabilities, at high Reynolds numbers the final equilibrium can be predicted with a simple construction based on conservation of total momentum. In this paper we continue this line of research with a study of barotropic shear flows on the equatorial /3-plane, Through numerical simulations the evolution of the instability is studied in select illuminating cases: a westward flowing Gaussian jet With the flow axis exactly on the equator, a uniform shear flow and eastward and Wes t w ard flowing jets that have their flow axis shifted away from the equator. In the numerical simulations it is assumed that there are no along -stream variations. 'fins suppresses equatorial Rossby W a ve s and barotropic shear instabilities and allows only inertial instability to develop. We investigate whether for these flow S the equatorial t-plane the final equilibrated flow can be predicted as was possible for flows on the f-plane. For the Gaussian jet centred on the equator the prediction of the equilibrated flow is obvious by mere inspection of the initial momentum distribution and by assuming that momentum is mixed and homogenized to render the equilibrated flow inertially stable. For the uniform shear flow, however, due to the peculiar nature of the initial momentum distribution and the fact that the Coriolis parameter f varies with latitude, it appears that, unlike in our earlier studies of flows on the f-plane, additional constraints need to be considered to correctly predict the outcome of the highly nonlinear evolution of the instability, The mixing range of the linear shear flow and the value of the mixed momentum is determined numerically and this is used to predict the equilibrated flow that emerges from an eastward flowing jet that is shifted a small distance away from the equator. For shifts large enough to induce no shcar at the equator the equilibrium flow can be well predicted using the simple rccipe used in our carlicr studies of parallel shear flows on the f-plane. For thc westward flowing jet shifted a very small distance from the equator, no prcdiction appears feasible. For modcstly small shifts a prediction is possiblc by combining the empirical prcdiction for thc linear shear flow with a prediction similar to what we used in our previous studies for flows on the f-plane.

Orlandi, P, Pirozzoli S, Bernardini M, Carnevale GF.  2014.  A minimal flow unit for the study of turbulence with passive scalars. Journal of Turbulence. 15:731-751.   10.1080/14685248.2014.927066   AbstractWebsite

The concept of a minimal flow unit (MFU) for the study of the basic physics of turbulent flows is introduced. The MFU is an initial vorticity configuration that consists of a few simple well-defined large-scale vortex structures. The form and position of these structures are chosen so that their interaction produces turbulence capturing many of the essential characteristics of isotropic homogeneous turbulence produced from random-phase initial conditions or that produced by continual random-phase forcing. The advantage of using the MFU is that the evolution of the vortex structures can be followed more clearly and the relationship between the evolving vortex structures and the various ranges in the energy spectrum can be more clearly defined. The addition of passive scalar fields to the MFU permits an investigation of passive scalar mixing that is relevant to the study of combustion. With a particular choice of the MFU, one that produces a trend to a finite-time singularity in the vorticity field, it is demonstrated that passive scalar distributed in the original large-scale vortices will develop intense gradients in the region where the vorticity is tending toward a singularity. In viscous flow, the evolution of the MFU clearly shows how the volume of the regions where originally well-separated passive scalars come into contact increases with increasing Reynolds number.

Carnevale, GF, Kloosterziel RC, Orlandi P.  2013.  Inertial and barotropic instabilities of a free current in three-dimensional rotating flow. Journal of Fluid Mechanics. 725:117-151.   10.1017/jfm.2013.191   AbstractWebsite

A current in a homogeneous rotating fluid is subject to simultaneous inertial and barotropic instabilities. Inertial instability causes rapid mixing of streamwise absolute linear momentum and alters the vertically averaged velocity profile of the current. The resulting profile can be predicted by a construction based on absolute-momentum conservation. The alteration of the mean velocity profile strongly affects how barotropic instability will subsequently change the flow. If a current with a symmetric distribution of cyclonic and anticyclonic vorticity undergoes only barotropic instability, the result will be cyclones and anticyclones of the same shape and amplitude. Inertial instability breaks this symmetry. The combined effect of inertial and barotropic instability produces anticyclones that are broader and weaker than the cyclones. A two-step scheme for predicting the result of the combined inertial and barotropic instabilities is proposed and tested. This scheme uses the construction for the redistribution of streamwise absolute linear momentum to predict the mean current that results from inertial instability and then uses this equilibrated current as the initial condition for a two- dimensional simulation that predicts the result of the subsequent barotropic instability. Predictions are made for the evolution of a Gaussian jet and are compared with three-dimensional simulations for a range of Rossby numbers. It is demonstrated that the actual redistribution of absolute momentum in the three-dimensional simulations is well predicted by the construction used here. Predictions are also made for the final number and size of vortices that result from the combined inertial and barotropic instabilities.

Orlandi, P, Pirozzoli S, Carnevale GF.  2012.  Vortex events in Euler and Navier-Stokes simulations with smooth initial conditions. Journal of Fluid Mechanics. 690:288-320.   10.1017/jfm.2011.430   AbstractWebsite

We present high-resolution numerical simulations of the Euler and Navier-Stokes equations for a pair of colliding dipoles. We study the possible approach to a finite-time singularity for the Euler equations, and contrast it with the formation of developed turbulence for the Navier-Stokes equations. We present numerical evidence that seems to suggest the existence of a blow-up of the inviscid velocity field at a finite time (t(s)) with scaling vertical bar u vertical bar(infinity) similar to (t(s) - t)(-1/2), vertical bar omega vertical bar(infinity) similar to (t(s) - t)(-1). This blow-up is associated with the formation of a k(-3) spectral range, at least for the finite range of wavenumbers that are resolved by our computation. In the evolution toward t(s), the total enstrophy is observed to increase at a slower rate, Omega similar to (t(s) - t)(-3/4), than would naively be expected given the behaviour of the maximum vorticity, omega(infinity) similar to (t(s) - t)(-1). This indicates that the blow-up would be concentrated in narrow regions of the flow field. We show that these regions have sheet-like structure. Viscous simulations, performed at various Re, support the conclusion that any non-zero viscosity prevents blow-up in finite time and results in the formation of a dissipative exponential range in a time interval around the estimated inviscid t(s). In this case the total enstrophy saturates, and the energy spectrum becomes less steep, approaching k(-513). The simulations show that the peak value of the enstrophy scales as Re-3/2, which is in accord with Kolmogorov phenomenology. During the short time interval leading to the formation of an inertial range, the total energy dissipation rate shows a clear tendency to become independent of Re, supporting the validity of Kolmogorov's law of finite energy dissipation. At later times the kinetic energy shows a t(-1.2) decay for all Re, in agreement with experimental results for grid turbulence. Visualization of the vortical structures associated with the stages of vorticity amplification and saturation show that, prior to ts, large-scale and the small-scale vortical structures are well separated. This suggests that, during this stage, the energy transfer mechanism is non-local both in wavenumber and in physical space. On the other hand, as the spectrum becomes shallower and a k(-5/3) range appears, the energy-containing eddies and the small-scale vortices tend to be concentrated in the same regions, and structures with a wide range of sizes are observed, suggesting that the formation of an inertial range is accompanied by transfer of energy that is local in both physical and spectral space.

Carnevale, GF, Kloosterziel RC, Orlandi P, van Sommeren D.  2011.  Predicting the aftermath of vortex breakup in rotating flow. Journal of Fluid Mechanics. 669:90-119.   10.1017/s0022112010004945   AbstractWebsite

A method for predicting the outcome of vortex breakup in a rotating flow is introduced. The vortices dealt with here are subject to both centrifugal and barotropic instabilities. The prediction of the aftermath of the breakup relies on knowing how both centrifugal and barotropic instabilities would equilibrate separately. A theoretical model for non-linear equilibration in centrifugal instability is wedded to two-dimensional simulation of barotropic instability to predict the final vortices that emerge from the debris of the original vortex. This prediction method is tested against three-dimensional Navier-Stokes simulations. For vortices in which a rapid centrifugal instability triggers a slower barotropic instability, the method is successful both qualitatively and quantitatively. The skill of the prediction method decreases as the time scales of the two instabilities become comparable.

Espa, S, Cenedese A, Mariani M, Carnevale GF.  2009.  Quasi-two-dimensional flow on the polar beta-plane: Laboratory experiments. Journal of Marine Systems. 77:502-510.   10.1016/j.jmarsys.2008.10.015   AbstractWebsite

Geophysical turbulence is strongly affected by the variation of the Coriolis parameter with latitude. This variation results in the so-called beta-effect, which forces energy from small-scales to be transferred preferentially into zonal motions. This effect results in the formation of narrow jet-like zonal flows that dominate the dynamics and act as transport barriers. Here, laboratory experiments are used to reproduce this effect in decaying turbulent flows. An electromagnetic cell is used to generate an initial field of vorticity in a rotating tank. Under conditions of quasigeostrophic flow, the beta-effect is produced by depth variation of the flow instead of variation of the Coriolis parameter. The effects of changing the container geometry and the overall fluid depth on the production of jets are investigated. The results suggest that this laboratory configuration can be used to model jet formation in the oceans and that increasing fluid depth is a practical way to decrease viscous effects. (C) 2008 Elsevier B.V. All rights reserved.

Kloosterziel, RC, Carnevale GF.  2008.  Vertical scale selection in inertial instability. Journal of Fluid Mechanics. 594:249-269.   10.1017/s0022112007009007   AbstractWebsite

The linear instability of a barotropic flow with uniform horizontal shear in a stratified rotating fluid is investigated with respect to perturbations invariant in the alongflow direction. The flow can be inertially unstable if there is sufficiently strong anticyclonic shear, but only for sufficiently high Reynolds numbers Re. We determine the critical Reynolds numbers required for amplification of the instability as a function of Prandtl number, strength of the stratification and magnitude of the shear. The vertical scales at the onset of the instability are calculated. For Prandtl number P < 1.44 instability always sets in through stationary overturning motions, for P > 1.44 it may also commence through overstable (oscillatory) motions. For Re exceeding the critical value, we determine the vertical scale of the most rapidly amplifying modes and the corresponding growth rates and how they vary with Re, P, the shear and the strength of stratification.

Espa, S, Carnevale GF, Cenedese A, Mariani M.  2008.  Quasi-two-dimensional decaying turbulence subject to the effect. Journal of Turbulence. 9:1-18.   10.1080/14685240802464417   AbstractWebsite

Freely decaying quasi-2D turbulence under the influence of a meridional variation of the Coriolis parameter f ( effect) is experimentally and numerically modelled. The experimental flow is generated in a rotating electromagnetic cell where the variation of f is approximated by a nearly equivalent topographical effect. In the presence of a high effect, the initial disordered vorticity field evolves to form a weak polar anticyclonic circulation surrounded by a cyclonic zonal jet demonstrating the preferential transfer of energy towards zonal motions. In agreement with theoretical predictions, the energy spectrum becomes peaked near the Rhines wave number with a steep fall-off beyond, indicating the presence of a soft barrier to the energy transfer towards larger scales. DNS substantially confirmed the experimental observations.

Dietrich, D, Carnevale GF, Orlandi P.  2007.  Flow over the Mid Adriatic Pit. Nuovo Cimento Della Societa Italiana Di Fisica C-Geophysics and Space Physics. 30:277-290.   10.1393/ncc/i2007-10242-x   AbstractWebsite

The influence of the Mid Adriatic Pit (MAP) on the general circulation of the Adriatic is explored through numerical simulations. The numerical code used is the DieCAST model specifically modified for application to the Adriatic Sea. A ten-year simulation is performed and the ability of the model to capture important features of the Adriatic circulation is demonstrated. A series of numerical experiments on the importance of the MAP on the general circulation is performed. It is demonstrated that the current over the northern flank of the MAP, which flows from the Croatian toward the Italian coast, is primarily a topographic current and that such a current would reverse direction if the gradient of the bathymetry were reversed.

Orlandi, P, Carnevale GF.  2007.  Nonlinear amplification of vorticity in inviscid interaction of orthogonal Lamb dipoles. Physics of Fluids. 19   10.1063/1.2732438   AbstractWebsite

Dynamical arguments based on the structure of the Euler equations suggest the possibility of rapid amplification of vorticity in which the vorticity and the rate of strain grow proportionately. During such growth, the vorticity is expected to amplify as a (t(s)-t)(-1) power-law in time. This behavior is difficult to demonstrate numerically, in part, because initial transients tend to obscure it. Lamb dipoles are used here to construct the initial vorticity. This helps to avoid these transients and results in a flow exhibiting the expected power-law vorticity amplification for a period of time. The spatial region where the vorticity growth rate is maximal is investigated in detail using a decomposition of the vorticity along the principal axes of the rate-of-strain tensor. It is demonstrated that the vorticity and strain rate in one direction in this decomposition are proportional during the period of rapid vorticity growth. These findings suggest that, during this period, the Euler equations can be reduced to a one-dimensional model equation for vorticity in the rate-of-strain coordinate system. (C) 2007 American Institute of Physics.

Kloosterziel, RC, Carnevale GF, Orlandi P.  2007.  Inertial instability in rotating and stratified fluids: barotropic vortices. Journal of Fluid Mechanics. 583:379-412.   10.1017/s0022112007006325   AbstractWebsite

The unfolding of inertial instability in intially barotropic vortices in a uniformly rotating and stratified fluid is studied through numerical simulations. The vortex dynamics during the instability is examined in detail. We demonstrate that the instability is stabilized via redistribution of angular momentum in a way that produces a new equilibrated barotropic vortex with a stable velocity profile. Based on extrapolations from the results of a series of simulations in which the Reynolds number and strength of stratification are varied, we arrive at a construction based on angular momentum mixing that predicts the infinite-Reynolds-number form of the equilibrated vortex toward which inertial instability drives an unstable vortex. The essential constraint is conservation of total absolute angular momentum. The construction can be used to predict the total energy loss during the equilibration process. It also shows that the equilibration process can result in anticyclones that are more susceptible to horizontal shear instabilities than they were initially, a phenomenon previously observed in laboratory and numerical studies.

Kloosterziel, RC, Orlandi P, Carnevale GF.  2007.  Saturation of inertial instability in rotating planar shear flows. Journal of Fluid Mechanics. 583:413-422.   10.1017/s0022112007006593   AbstractWebsite

Inertial instability in a rotating shear flow redistributes absolute linear momentum in such a way as to neutralize the instability. In the absence of other instabilities, the final equilibrium can be predicted by a simple construction based on conservation of total momentum. Numerical simulations, invariant in the along-stream direction, suppress barotropic instability and allow only inertial instability to develop. Such simulations, at high Reynolds numbers, are used to test the theoretical prediction. Four representative examples are given: a jet, a wall-bounded jet, a mixing layer and a wall-bounded shear layer.

Kloosterziel, RC, Carnevale GF.  2007.  Generalized energetics for inertially stable parallel shear flows. Journal of Fluid Mechanics. 585:117-126.   10.1017/s0022112007006933   AbstractWebsite

For simple parallel shear flows on the f-plane and the equatorial beta-plane we derive an energy norm for zonally invariant perturbations. It is used to derive the linear stability boundary for when these flows are inertially stable in the classical sense but may be destabilized due to unequal rates of diffusion of momentum and heat. The analysis is valid when there are arbitrary, zonally invariant, no-slip boundaries which are perfect thermal conductors.

Zavala Sanson, L, Serravall R, Carnevale GF, vanHeijst GJF.  2005.  Experiments and simulations on coastal flows in the presence of a topographic slope. Journal of Physical Oceanography. 35:2204-2218.   10.1175/jpo2815.1   AbstractWebsite

The evolution of a barotropic coastal current in the presence of a bottom ramp-shaped topography is studied by means of laboratory experiments and numerical simulations. The experiments are performed in a rectangular rotating tank filled with freshwater. The fluid depth is shallow at one side of the domain and deeper at the other side, and both regions are divided by a narrow slope, whose depth contours are perpendicular to the long sides of the tank. A current approaching the slope is produced along one of the vertical walls, having the boundary at its right. Two configurations are analyzed: when the current flows from shallow to deep water and when flowing in the opposite direction. In the first scenario, the current is divided in two parts, one of them following the coastline. The other part of the current pairs with a cell of negative relative vorticity generated at the slope due to squeezing effects, forming a dipolar structure moving offshore, back toward the shallow side. In addition, a weak current moving inshore along the slope is clearly formed. In the second configuration, when the flow goes from deep to shallow water, a part of or even the whole current might be forced to move along the contours of the topography, away from the coast. In this case there is no dipole formation. The experiments are well reproduced by means of quasigeostrophic numerical simulations, which allow a more detailed systematic study of the influence of flow parameters such as the topography height and the width of the slope.

Kloosterziel, RC, Carnevale GF.  2003.  Closed-form linear stability conditions for magneto-convection. Journal of Fluid Mechanics. 490:333-344.   10.1017/s0022112003005329   AbstractWebsite

Chandrasekhar (1961) extensively investigated the linear dynamics of Rayleigh-Benard convection in an electrically conducting fluid exposed to a uniform vertical magnetic field and enclosed by rigid, stress-free, upper and lower boundaries. He determined the marginal stability boundary and critical horizontal wavenumbers for the onset of convection as a function of the Chandrasekhar number Q or Hartmann number squared. No closed-form formulae appeared to exist and the results were tabulated numerically. We have discovered simple expressions that concisely describe the stability properties of the system. When the Prandtl number Pr is greater than or equal to the magnetic Prandtl number Pm the marginal stability boundary is described by the curve Q = pi(-2)[R - (RcR2/3)-R-1/3] where R is the Rayleigh number and R-c = (27/4)pi(4) is Rayleigh's famous critical value for the onset of stationary convection in the absence of a magnetic field (Q = 0). When Pm > Pr the marginal stability boundary is determined by this curve until intersected by the curve Q = 1/pi(2)[Pm-2(1 + Pr)/Pr-2(1 + Pm)R - ((1 + Pr)(Pr + Pm)/Pr-2)(1/3) (Pm-2(1 + Pr)/Pr-2(1 + Pm))(2/3) (RcR2/3)-R-1/3]. An expression for the intersection point is derived and also for the critical horizontal wavenumbers for which instability sets in along the marginal stability boundary either as stationary convection or in an oscillatory fashion. A simple formula is derived for the frequency of the oscillations. Also we show that in the limit of vanishing magnetic diffusivity, or infinite electrical conductivity, the system is unstable for sufficiently large R. Instability in this limit always sets in via overstability.

Kloosterziel, RC, Carnevale GF.  2003.  Closed-form linear stability conditions for rotating Rayleigh-Benard convection with rigid stress-free upper and lower boundaries. Journal of Fluid Mechanics. 480:25-42.   10.1017/s0022112002003294   AbstractWebsite

The linear dynamics of rotating Rayleigh-Benard convection with rigid stress-free boundaries has been thoroughly investigated by Chandrasekhar (1961) who determined the marginal stability boundary and critical horizontal wavenumbers for the onset of convection and overstability as a function of the Taylor number T. No closed-form formulae appeared to exist and the results were tabulated numerically. However, by taking the Rayleigh number R as independent variable we have found remarkably simple expressions. When the Prandtl number P greater than or equal to P-c = 0.67659, the marginal stability boundary is described by the curve T(R) = R[(R/R-c)(1/2) - 1] where R-c = 27/4pi(4) is Rayleigh's famous critical value for the onset of stationary convection in a non-rotating system (T = 0). For P < P-c the marginal stability boundary is determined by this curve until it is intersected by the curve T(R, P) = R [(1+P/2(3) P-4)(1/2) (R/R-c)(1/2) - 1 + P/2P(2)]. A simple expression for the intersection point is derived and also for the critical horizontal wavenumbers for which, along the marginal stability boundary, instability sets in either as stationary convection or in an oscillatory fashion. A simple formula is derived for the frequency of the oscillations. Further, we have analytically determined critical points on the marginal stability boundary above which an increase of either viscosity or diffusivity is destabilizing. Finally, we show that if the fluid has zero viscosity the system is always unstable, in contradiction to Chandrasekhar's conclusion.

Carnevale, GF, Orlandi P, Zhou Y, Kloosterziel RC.  2002.  Rotational suppression of Rayleigh-Taylor instability. Journal of Fluid Mechanics. 457:181-190.   10.1017/s0022112002007772   AbstractWebsite

It is demonstrated that the growth of the mixing zone generated by Rayleigh-Taylor instability can be greatly retarded by the application of rotation, at least for low Atwood number flows for which the Boussinesq approximation is valid. This result is analysed in terms of the effect of the Coriolis force on the vortex rings that propel the bubbles of fluid in the mixing zone.

Orlandi, P, Carnevale GF, Lele SK, Shariff K.  2001.  Thermal perturbation of trailing vortices. European Journal of Mechanics B-Fluids. 20:511-524.   10.1016/s0997-7546(01)01131-1   AbstractWebsite

The possibility of diminishing the danger of trailing vortices through thermal forcing is investigated. It is shown that heating the vortices would have two beneficial effects. First, it would cause the vortices to descend more rapidly thus clearing the flight path more quickly. Second, it would cause the vortices to draw closer together, thus greatly increasing the growth rate of the short-wave instabilities that can ultimately destroy the vortices through cross-diffusion. (C) 2001 Editions scientifiques et medicales Elsevier SAS.

Carnevale, GF, Briscolini M, Orlandi P.  2001.  Buoyancy- to inertial-range transition in forced stratified turbulence. Journal of Fluid Mechanics. 427:205-239.   10.1017/s002211200000241x   AbstractWebsite

The buoyancy range, which represents a transition from large-scale wave-dominated motions to small-scale turbulence in the oceans and the atmosphere, is investigated through large-eddy simulations. The model presented here uses a continual forcing based on large-scale standing internal waves and has a spectral truncation in the isotropic inertial range. Evidence is presented for a break in the energy spectra from the anisotropic k(-3) buoyancy range to the small-scale k(-5/3) isotropic inertial range. Density structures that form during wave breaking and periods of high strain rate are analysed. Elongated vertical structures produced during periods of strong straining motion are found to collapse in the subsequent vertically compressional phase of the strain resulting in a zone or patch of mixed fluid.

Carnevale, GF, Cavallini F, Crisciani F.  2001.  Dynamic boundary conditions revisited. Journal of Physical Oceanography. 31:2489-2497.   10.1175/1520-0485(2001)031<2489:dbcr>;2   AbstractWebsite

The applicability of the super-slip boundary condition in wind-driven quasigeostrophic ocean circulation models is reexamined. The energy and enstrophy characteristics of the super-slip condition are discussed for the equilibrium state. A model is constructed with super-slip on the western boundary and free slip on the other boundaries. Both linear and nonlinear solutions are presented. Compared to the case with all free-slip boundaries, this new model gives a more energetic and narrower western boundary current, but otherwise the differences are not very great.

Carnevale, GE, Smith SGL, Crisciani F, Purini R, Serravall R.  1999.  Bifurcation of a coastal current at an escarpment. Journal of Physical Oceanography. 29:969-985.   10.1175/1520-0485(1999)029<0969:boacca>;2   AbstractWebsite

The evolution of a coastal current as it encounters an escarpment depends strongly on whether the geometry of the coast and escarpment is right or left "handed," independent of the direction of the coastal current. Handedness is defined such that right-handed means that when looking across the escarpment from the deep to the shallow side, the coast is found on the right. The essential aspects of the difference in behavior of the current in the two geometries are captured by a simple quasigeostrophic model of coastal flow over a step. An exact analytic solution to the nonlinear stationary problem is obtained. This solution shows that, when a coastal current crosses an escarpment in the left-handed geometry, the speed of the current will increase independent of whether the flow is from shallow to deep or from deep to shallow. For the right-handed geometry, the speed of the current decreases, also independent of the direction of the coastal flow. In the left (right)-handed geometry, there is associated to the coastal flow an inshore (offshore) current along the escarpment. These results are explained in terms of linear wave theory and vortex dynamics. Numerical simulations are used to examine the evolution of the flow from the initial encounter to the establishment of a stationary flow. The relevance of this research is discussed in light of recent results from laboratory experiments and oceanic observations.

Kloosterziel, RC, Carnevale GF.  1999.  On the evolution and saturation of instabilities of two-dimensional isolated circular vortices. Journal of Fluid Mechanics. 388:217-257.   10.1017/s0022112099004760   AbstractWebsite

Laboratory observations and numerical experiments have shown that a variety of compound vortices can emerge in two-dimensional flow due to the instability of isolated circular vortices. The simple geometrical features of these compound vortices suggest that their description may take a simple form if an appropriately chosen set of functions is used. We employ a set which is complete on the infinite plane for vorticity distributions with finite total enstrophy. Through projection of the vorticity equation (Galerkin method) and subsequent truncation we derive a dynamical system which is used to model the observed behaviour in as simple as possible a fashion. It is found that at relatively low-order truncations the observed behaviour is qualitatively captured by the dynamical system. We determine what the necessary ingredients are for saturation of instabilities at finite amplitude in terms of wave-wave interactions and feedback between various azimuthal components of the vorticity field.

Orlandi, P, Carnevale GF.  1999.  Evolution of isolated vortices in a rotating fluid of finite depth. Journal of Fluid Mechanics. 381:239-269.   10.1017/s0022112098003693   AbstractWebsite

Laboratory experiments have shown that monopolar isolated vortices in a rotating flow undergo instabilities that result in the formation of multipolar vortex states such as dipoles and tripoles. In some cases the instability is entirely two-dimensional, with the vortices taking the form of vortex columns aligned along the direction of the ambient rotation at all times. In other cases, the vortex first passes through a highly turbulent three-dimensional state before eventually reorganizing into vortex columns. Through a series of three-dimensional numerical simulations, the roles that centrifugal instability, barotropic instability, and the bottom Ekman boundary layer play in these instabilities are investigated. Evidence is presented that the centrifugal instability can trigger the barotropic instabilities by the enhancement of vorticity gradients. It is shown that the bottom Ekman layer is not essential to these instabilities but can strongly modify their evolution.

Carnevale, GF, Fuentes OUV, Orlandi P.  1997.  Inviscid dipole-vortex rebound from a wall or coast. Journal of Fluid Mechanics. 351:75-103.   10.1017/s0022112097007155   AbstractWebsite

A vortex approaching a no-slip wall 'rebounds' due to the creation of vorticity at the wall in a viscous boundary layer. Here it is demonstrated that a purely inviscid mechanism can also produce vortex rebound from a slip wall. In inviscid vortex rebound, vortex tube stretching generates the necessary vorticity to allow rebound, eliminating the need for viscous vorticity generation. This vortex stretching mechanism is demonstrated through numerical simulations and laboratory experiments on dipole-vortex rebound from a boundary. In an application to oceanography, numerical simulations of both quasi-geostrophic and shallow water dynamics are used to show that the beta-effect at an eastern boundary can produce this inviscid rebound. Through a series of numerical experiments in which the strength of the beta-effect is varied, a formula for predicting the point of separation of the vortices from the boundary in a dipole-coast collision is deduced. Through simulations, the flux of vorticity and fluid away from the boundary is measured as a function of beta and initial angle of incidence. It is found that, in contrast to viscous vortex rebound, which typically does not produce a flux of material away from the boundary farther than a distance comparable to the initial vortex radius, the beta-induced rebound does carry fluid far from the coast. Laboratory experiments in a rotating tank are used to show that a sloping bottom can also provide an inviscid mechanism for dipole-vortex rebound from the wall of the tank under certain conditions. A relation determining the conditions under which inviscid or viscous processes will dominate in the rebound of the dipole from a boundary is obtained.