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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.

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.

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.

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.  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.