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2015
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.

2013
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.

2008
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.

2007
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.

1999
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.

1994
Carnevale, GF, Kloosterziel RC.  1994.  Lobe shedding from propagating vortices. Physica D. 76:147-167.   10.1016/0167-2789(94)90256-9   AbstractWebsite

Associated with intense propagating vortices is a separatrix defining a region of fluid that is transported with the vortex. The distortion of this separatrix, under external perturbations, leads to entrainment and detrainment of fluid. The detrained fluid is shed in lobes in the wake of the vortex. Examples of this phenomenon for a propagating monopole and dipole are provided from rotating-tank experiments and numerical simulations.

Carnevale, GF, Kloosterziel RC.  1994.  Emergence and evolution of triangular vortices. Journal of Fluid Mechanics. 259:305-331.   10.1017/s0022112094000157   AbstractWebsite

Laboratory observations and numerical simulations reveal that, in addition to monopoles, dipoles and tripoles, yet another stable coherent vortex may emerge from unstable isolated circular vortices. This new vortex is the finite-amplitude result of the growth of an azimuthal wavenumber-3 perturbation. It consists of a triangular core of single-signed vorticity surrounded by three semicircular satellites of oppositely signed vorticity. The stability of this triangular vortex is analysed through a series of high-resolution numerical simulations and by an investigation of point-vortex models. This new compound vortex rotates about its centre and is stable to small perturbations. If perturbed strongly enough, it undergoes an instability in which two of the outer satellites merge, resulting in the formation of an axisymmetric tripole, which subsequently breaks down into either a pair of dipoles or a dipole plus a monopole. The growth of higher-azimuthal-wavenumber perturbations leads to the formation of more intricate compound vortices with cores in the shape of squares, pentagons, etc. However, numerical simulations show that these vortices are unstable, which agrees with results from point-vortex models.

1993
Kloosterziel, RC, Carnevale GF, Philippe D.  1993.  Propagation of barotropic dipoles over topography in a rotating tank. Dynamics of Atmospheres and Oceans. 19:65-100.   10.1016/0377-0265(93)90032-3   AbstractWebsite

It is shown how symmetric dipolar vortices can be formed by the action of an impulsive jet in a homogeneous single layer of fluid in a rotating tank. These dipoles are allowed to interact with a constant topographic slope, which can model a beta-plane or a continental shelf. A dipole's trajectory bends toward the right when climbing a slope and to the left when descending, as predicted by numerical simulations and analytical arguments. The maximum penetration of the dipoles over a slope, the adjustment to the slope, and formation of trailing lobes are compared with both numerical simulations and a two-point vortex model. The results suggest that Rossby wave radiation plays an important role in the interaction process.