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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.

Carnevale, GF, Briscolini M, Kloosterziel RC, Vallis GK.  1997.  Three-dimensionally perturbed vortex tubes in a rotating flow. Journal of Fluid Mechanics. 341:127-163.   10.1017/s0022112097005430   AbstractWebsite

Numerical experiments are used to study the evolution of perturbed vortex tubes in a rotating environment in order to better understand the process of two-dimensionalization of unsteady rotating flows. We specifically consider non-axisymmetric perturbations to columnar vortices aligned along the axis of rotation. The basic unperturbed vortex is chosen to have a Gaussian cross-sectional vorticity distribution. The experiments cover a parameter space in which both the strength of the initial perturbation and the Rossby number are varied. The Rossby number is defined here as the ratio of the maximum amplitude of vorticity in the Gaussian vorticity profile to twice the ambient rotation rate. For small perturbations and small Rossby numbers, both cyclones and anticyclones behave similarly, relaxing rapidly back toward two-dimensional columnar vortices. For large perturbations and small Rossby numbers, a rapid instability occurs for both cyclones and anticyclones in which antiparallel vorticity is created. The tubes break up and then re-form again into columnar vortices parallel to the rotation axis (i.e. into a quasi-two-dimensional flow) through nonlinear processes. For Rossby numbers greater than 1, even small perturbations result in the complete breakdown of the anticyclonic vortex through centrifugal instability, while cyclones remain stable. For a range of Rossby numbers greater than 1, after the breakdown of the anticyclone, a new weaker anticyclone forms, with a small-scale background vorticity of spectral shape given approximately by the -5/3 energy spectral law.

Verzicco, R, Orlandi P, Eisenga AHM, vanHeijst GJF, Carnevale GF.  1996.  Dynamics of a vortex ring in a rotating fluid. Journal of Fluid Mechanics. 317:215-239.   10.1017/s0022112096000730   AbstractWebsite

The formation and the evolution of axisymmetric vortex rings in a uniformly rotating fluid, with the rotation axis orthogonal to the ring vorticity, have been investigated by numerical and laboratory experiments. The flow dynamics turned out to be strongly affected by the presence of the rotation. In particular, as the background rotation increases, the translation velocity of the ring decreases, a structure with opposite circulation forms ahead of the ring and an intense axial vortex is generated on the axis of symmetry in the tail of the ring. The occurrence of these structures has been explained by the presence of a self-induced swirl flow and by inspection of the extra terms in the Navier-Stokes equations due to rotation. Although in the present case the swirl was generated by the vortex ring itself, these results are in agreement with those of Virk et al. (1994) for polarized vortex rings, in which the swirl flow was initially assigned as a 'degree of polarization'. If the rotation rate is further increased beyond a certain value, the flow starts to be dominated by Coriolis forces. In this flow regime, the impulse imparted to the fluid no longer generates a vortex ring, but rather it excites inertial waves allowing the flow to radiate energy. Evidence of this phenomenon is shown. Finally, some three-dimensional numerical results are discussed in order to justify some asymmetries observed in flow visualizations.

Carnevale, GF, Purini R, Orlandi P, Cavazza P.  1995.  Barotropic quasi-geostrophic f-plane flow over anisotropic topography. Journal of Fluid Mechanics. 285:329-347.   10.1017/s0022112095000565   AbstractWebsite

For an anisotropic topographic feature in a large-scale flow, the orientation of the topography with respect to the flow will affect the vorticity production that results from the topography-flow interaction. This in turn affects the amount of form drag that the ambient flow experiences. Numerical simulations and perturbation theory are used to explore these effects of change in topographic orientation. The flow is modelled as a quasi-geostrophic homogeneous fluid on anf-plane. The topography is taken to be a hill of limited extent, with an elliptical cross-section in the horizontal. It is shown that, as a result of a basic asymmetry of the quasi-geostrophic flow, the strength of the form drag depends not only on the magnitude of the angle that the topographic axis makes with the oncoming stream, but also on the sign of this angle. For sufficiently low topography, it is found that a positive angle of attack leads to a stronger form drag than that for the corresponding negative angle. For strong topography, this relation is reversed, with the negative angle then resulting in the stronger form drag.

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.

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.

Bates, E, Carnevale GF.  1993.  New directions in research on language development. Developmental Review. 13:436-470.   10.1006/drev.1993.1020   AbstractWebsite

In this paper, we will describe what are (in our view) the newest and most exciting trends in current research on language development; trends that are likely to predominate in the few years that remain until the millennium. The paper is organized into six sections: (1) advances in data sharing (including the Child Language Data Exchange System), (2) improved description and quantification of the linguistic data to which children are exposed and the data that they produce (with implications for theories of language learning); (3) new theories of learning in neural networks that challenge old assumptions about the "learnability" (or unlearnability) of language, (4) increased understanding of the nonlinear dynamics that may underlie behavioral change, (5) research on the neural correlates of language learning, and (6) an increased understanding of the social factors that influence normal and abnormal language development.

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.

Carnevale, GF, Pierrehumbert RT.  1992.  Nonlinear phenomena in atmospheric and oceanic sciences. IMA volumes in mathematics and its applications. 40:259., New York: Springer-Verlag Abstract
Carnevale, GF, McWilliams JC, Pomeau Y, Weiss JB, Young WR.  1992.  Rates, pathways, and end states of nonlinear evolution in decaying two‐dimensional turbulence: Scaling theory versus selective decay. Physics of Fluids. 4:1314-1316.   10.1063/1.858251   Abstract

A recently proposed scaling theory of two‐dimensional turbulent decay, based on the evolutionary pathway of successive mergers of coherent vortices, is used to predict the rate and end state of the evolution. These predictions differ from those based on the selective‐decay hypothesis and traditional ideas of spectrum evolution, and they are in substantially better agreement with numerical solutions at large Reynolds number.

Carnevale, GF, Falcioni M, Isola S, Purini R, Vulpiani A.  1991.  Fluctuation‐response relations in systems with chaotic behavior. Physics of Fluids a-Fluid Dynamics. 3:2247-2254.   10.1063/1.857905   AbstractWebsite

The statistics of systems with good chaotic properties obey a formal fluctuation-response relation which gives the average linear response of a dynamical system to an external perturbation in terms of two-time correlation functions. Unfortunately, except for particularly simple cases, the appropriate form of correlation function is unknown because an analytic expression for the invariant density is lacking. The simplest situation is that in which the probability distribution is Gaussian. In that case, the fluctuation-response relation is a linear relation between the response matrix and the two-time two-point correlation matrix. Some numerical computations have been carried out in low-dimensional models of hydrodynamic systems. The results show that fluctuation-response relation for Gaussian distributions is not a useful approximation. Nevertheless, these calculations show that, even for non-Gaussian statistics, the response function and the two-time correlations can have similar qualitative features, which may be attributed to the existence of the more general fluctuation-response relation.

Carnevale, GF, McWilliams JC, Pomeau Y, Weiss JB, Young WR.  1991.  Evolution of vortex statistics in two-dimensional turbulence. Physical Review Letters. 66:2735-2737.   10.1103/PhysRevLett.66.2735   AbstractWebsite

Freely evolving two-dimensional turbulence is dominated by coherent vortices. The density of these vortices decays in time as rho approximately t^-ɛ with ɛ almost-equal-to 0.75. A new scaling theory is proposed which expresses all statistical properties in terms of ɛ. Thus the average circulation of the vortices increases as t^ɛ/2 and their average radius as t^ɛ/4. The total energy is constant, the enstrophy decreases as t^ɛ/2, and the vorticity kurtosis increases as t^ɛ/2. These results are supported both by numerical simulations of the fluid equations and by solutions of a modified point-vortex model.

Carnevale, GF, Cavazza P, Orlandi P, Purini R.  1991.  An explanation for anomalous vortex merger in rotating‐tank experiments. Physics of Fluids a-Fluid Dynamics. 3:1411-1415.   10.1063/1.858019   AbstractWebsite

Theory and simulations based on the two-dimensional Euler equation predict a critical distance of separation for the merger of two like-signed vortices. By the symmetry of the equation, this separation must be the same for both cyclone and anticyclone pairs. In rotating-tank experiments, the observed critical separation distance for anticyclone merger is in accord with predictions; however, pairs of cyclones have been found to merge in all cases examined, even with separations substantially greater than the predicted critical separation. The hypothesis that this discrepancy is due to the presence of Ekman volume fluxes, which are not incorporated in the two-dimensional theory, is examined and found not quantitatively supportable. A second hypothesis is that the parabolic curvature of the free upper surface of the fluid in the rotating tank induces motion of the cyclones toward the center of the tank and hence promotes the cyclone pair merger. Quasigeostrophic simulations which capture this "topography effect" show good agreement with the rotating-tank experiments.

Carnevale, GF, Kloosterziel RC, vanHeijst GJF.  1991.  Propagation of barotropic vortices over topography in a rotating tank. Journal of Fluid Mechanics. 233:119-139.   10.1017/s0022112091000411   AbstractWebsite

A small-scale cyclonic vortex in a relatively broad valley tends to climb up and out of the valley in a cyclonic spiral about the centre, and when over a relatively broad hill it tends to climb toward the top in an anticyclonic spiral around the peak. This phenomenon is examined here through two-dimensional numerical simulations and rotating-tank experiments. The basic mechanism involved is shown to be the same as that which accounts for the northwest propagation of cyclones on a beta-plane. This inviscid nonlinear effect is also shown to be responsible for the observed translationary motion of barotropic vortices in a free-surface rotating tank. The behaviour of isolated vortices is contrasted with that of vortices with non-vanishing circulation.

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.

Carnevale, GF, Pomeau Y, Young WR.  1990.  Statistics of ballistic agglomeration. Physical Review Letters. 64:2913-2916.   10.1103/PhysRevLett.64.2913   AbstractWebsite

We consider a ‘‘sticky gas’’ in which collisions between spherical particles are perfectly inelastic. Thus the two colliding particles conserve mass and momentum, but merge to form a single more massive sphere. A scaling argument suggests that the average mass of a particle grows as t^2D/(2+D), where D is the spatial dimension. In the case D=1 this result is confirmed by numerical simulation.

Carnevale, GF, Vallis GK.  1990.  Pseudo-advective relaxation to stable states of inviscid two-dimensional fluids. Journal of Fluid Mechanics. 213:549-571.   10.1017/s0022112090002440   AbstractWebsite

The continuous transformation of one flow into another of higher or lower energy while preserving the potential vorticity of all particles can be accomplished by advection with an artificial velocity field. Since isolated extremal energy states are stable states, this method can be used to find stable stationary flows on a prescribed isovortical sheet. A series of numerical simulations of this method for two-dimensional fluids that demonstrates its feasibility and utility is presented. Additionally, a corollary to Arnol'd's nonlinear stability theorems is discussed, which shows that there can be at most two Arnol'd stable states per isovortical sheet.

Carnevale, GF, Shepherd TG.  1990.  On the interpretation of Andrews’ theorem. Geophysical and Astrophysical Fluid Dynamics. 51:1-17.   10.1080/03091929008219847   AbstractWebsite

Andrews (1984) has shown that any flow satisfying Arnol'd's (1965, 1966) sufficient conditions for stability must be zonally-symmetric if the boundary conditions on the flow are zonally-symmetric. This result appears to place very strong restrictions on the kinds of flows that can be proved to be stable by Arnol'd's theorems. In this paper, Andrews’ theorem is re-examined, paying special attention to the case of an unbounded domain. It is shown that, in that case, Andrews’ theorem generally fails to apply, and Arnol'd-stable flows do exist that are not zonally-symmetric. The example of a circular vortex with a monotonic vorticity profile is a case in point. A proof of the finite-amplitude version of the Rayleigh stability theorem for circular vortices is also established; despite its similarity to the Arnol'd theorems it seems not to have been put on record before.

Vallis, GK, Carnevale GF, Young WR.  1989.  Extremal energy properties and construction of stable solutions of the Euler equations. Journal of Fluid Mechanics. 207:133-152.   10.1017/s0022112089002533   AbstractWebsite

Certain modifications of the Euler equations of fluid motion lead to systems in which the energy decays or grows monotonically, yet which preserve other dynamically important characteristics of the field. In particular, all topological invariants associated with the vorticity field are preserved. In cases where isolated energy extrema exist, a stable steady flow can be found. In two dimensions, highly constrained by vorticity invariants, it is shown that the modified dynamics will lead to at least one non-trivial stationary, generally stable, solution of the equations of motion from any initial conditions. Numerical implementation of the altered dynamics is straightforward, and thus provides a practical method for finding stable flows. The method is sufficiently general to be of use in other dynamical systems.Insofar as three-dimensional turbulence is characterized by a cascade of energy, but not topological invariants, from large to small scales, the procedure has direct physical significance. It may be useful as a parameterization of the effects of small unresolved scales on those explicitly resolved in a calculation of turbulent flow.

Benzi, R, Carnevale GF.  1989.  A possible measure of local predictability. Journal of the Atmospheric Sciences. 46:3595-3598.   10.1175/1520-0469(1989)046<3595:apmolp>;2   AbstractWebsite

In this paper we suggest that the longevity of the enhanced predictability periods often observed in the weather and general circulation models can he quantified by a study of the statistical moments of error growth rates as has been demonstrated for dynamical systems. As an illustration, it is shown how this approach can he pursued in simple cases. For the Lorenz model, the probability density distribution of error growth is close to log-normal and the average growth rate is two times shorter than the most probable. In general, we argue that the ratio of the average growth rate to the most probable is a measure of enhanced predictability.

Carnevale, GF, Briscolini M, Purini R, Vallis GK.  1988.  Numerical experiments on modon stability to topographic perturbations. Physics of Fluids. 31:2562-2566.   10.1063/1.866533   AbstractWebsite

A summary of a numerical study of the stability of modons to topographic perturbation is presented. Previous studies have suggested a monotonic relationship between the horizontal scale of the perturbation and the amplitude needed to destroy a modon—as the scale of the perturbation increases the strength needed for destruction decreases. The results presented here show that this relationship does not hold for scales larger than the modon radius. For large‐scale perturbations, the strength needed for destruction again increases. The modon is most stable to perturbations of horizontal scale either much larger or much smaller than the modon radius. Stability graphs are presented for three types of perturbations; ridges, hills, and irregular terrain.

Carnevale, GF, Vallis GK, Purini R, Briscolini M.  1988.  The role of initial conditions in flow stability with an application to modons. Physics of Fluids. 31:2567-2572.   10.1063/1.866534   AbstractWebsite

Lyapunov stability arguments may be used to show that an otherwise unstable flow can be stabilized by restriction of the class of possible perturbations. It is shown that, in general, such a restriction applied only to the initial perturbation does not imply stability for dynamics on the entire phase space nor does it necessarily imply a delay of the onset of instability. As a result, proofs of linear stability based on a restriction of the initial perturbation actually are not valid. In particular, certain criteria for the stability of modons given by Pierini [Dyn. Atmos. Oceans 9, 273 (1985)] and Swaters [Phys. Fluids 29, 1419 (1986)] and synthesized by Flierl [Annu. Rev. Fluid Mech. 19, 493 (1987)] do not, in fact, ensure stability. A model is used to demonstrate that these stability criteria do not preclude instantaneous onset of linear instability. The model also demonstrates that, although conservation of energy and enstrophy implies that the transfer of energy in an instability must be to scales both larger and smaller than the modon scale, the principal direction of transfer remains undetermined.

Carnevale, GF, Vallis GK, Purini R, Briscolini M.  1988.  Propagation of barotropic modons over topography. Geophysical and Astrophysical Fluid Dynamics. 41:45-101.   10.1080/03091928808208831   AbstractWebsite

This is a broad survey of the interaction of modons with topography in a one-layer, quasigeostrophic model. Numerical simulations of modons interacting with ridges, hills, random topography and other obstacles are presented. The behavior of the modon is compared to numerical simulations of a two-point-vortex model, which proves a useful guide to the basic trajectory deflection mechanism. Under sufficiently strong but quasigeostrophically valid topographic perturbations, the modon is shown to fission into two essentially independent, oppositely-signed vortices. In the breakup of a modon near a hill it is found that the positive vortex migrates to the top of the hill. The resulting correlation between the positive vorticity trapped over the hill and the topography is in sharp contrast with the theories of turbulent flow over topography and generation of flow over topography by large scale forcing, both of which describe the development of vorticity anticorrelated with topography. A heuristic explanation of this new behavior is provided in terms of the dynamics of β bT-plane vortices. Further, it is found that a modon travelling over rough topography homogenizes the field of potential vorticity in its vicinity. This behavior is explained in terms of the induced eddy activity near the modon.

Carnevale, GF, Frederiksen JS.  1987.  Nonlinear stability and statistical mechanics of flow over topography. Journal of Fluid Mechanics. 175:157-181.   10.1017/s002211208700034x   AbstractWebsite

The stability properties and stationary statistics of inviscid barotropic flow over topography are examined. Minimum enstrophy states have potential vorticity proportional to the streamfunction and are nonlinearly stable; correspondingly, canonical equilibrium based on energy and enstrophy conservation predicts mean potential vorticity is proportional to the mean streamfunction. It is demonstrated that in the limit of infinite resolution the canonical mean state is statistically sharp, that is, without any eddy energy on any scale, and is identical to the nonlinearly stable minimum enstrophy state. Special attention is given to the interaction between small scales and a dynamically evolving large-scale flow. On the β-plane, these stable flows have a westward large-scale component. Possibilities for a general relation between inviscid statistical equilibrium and nonlinear stability theory are examined.