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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.

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.

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, 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, Orlandi P.  2017.  Equatorial inertial instability with full Coriolis force. Journal of Fluid Mechanics. 825:69-108.   10.1017/jfm.2017.377   AbstractWebsite

The zonally symmetric inertial instability of oceanic near-equatorial flows is studied through high-resolution numerical simulations. In homogeneous upper layers, the instability of surface-confined westward currents implies potentially fast downward mixing of momentum with a predictable final equilibrium. With increasing Reynolds number, latitudinal scales along the surface associated with the instability become ever smaller and initially the motions are ever more concentrated underneath the surface. The results suggest that even if the upper layer is stratified, it may still be necessary to include the full Coriolis force in the dynamics rather than use the traditional beta-plane approximation.

Kloosterziel, RC, Carnevale GF.  1992.  Formal stability of circular vortices. Journal of Fluid Mechanics. 242:249-278.   10.1017/s0022112092002362   AbstractWebsite

The second variation of a linear combination of energy and angular momentum is used to investigate the formal stability of circular vortices. The analysis proceeds entirely in terms of Lagrangian displacements to overcome problems that otherwise arise when one attempts to use Arnol'd's Eulerian formalism. Specific attention is paid to the simplest possible model of an isolated vortex consisting of a core of constant vorticity surrounded by a ring of oppositely signed vorticity. We prove that the linear stability regime for this vortex coincides with the formal stability regime. The fact that there are formally stable isolated vortices could imply that there are provable nonlinearly stable isolated vortices. The method can be applied to more complicated vortices consisting of many nested rings of piecewise-constant vorticity. The equivalent expressions for continuous vorticity distributions are also derived.

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.  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.

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.

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.

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.

Koniges, AE, Crotinger JA, Dannevik WP, Carnevale GF, Diamond PH.  1991.  Equilibrium spectra and implications for a two‐field turbulence model. Physics of Plasmas. 3:1297-1299.   10.1063/1.859822   Abstract

Analytic expressions are given for statistical mechanical equilibrium solutions of two‐field turbulence model equations that are used in describing plasma drift waves and for passive scalar advection in a neutral fluid. These are compared with those previously proposed [Phys. Fluids B 1, 1331 (1989)], in particular regarding the role of the cross correlations between fields. Implications for two‐point closure calculations are discussed.