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Carnevale, GF, Frisch U, Salmon R.  1981.  H theorems in statistical fluid dynamics. Journal of Physics a-Mathematical and General. 14:1701-1718.   10.1088/0305-4470/14/7/026   AbstractWebsite

It is demonstrated that the second-order Markovian closures frequently used in turbulence theory imply an H theorem for inviscid flow with an ultraviolet spectral cut-off. That is, from the inviscid closure equations, it follows that a certain functional of the energy spectrum (namely entropy) increases monotonically in time to a maximum value at absolute equilibrium. This is shown explicitly for isotropic homogeneous flow in dimensions d>or=2, and then a generalised theorem which covers a wide class of systems of current interest is presented. It is shown that the H theorem for closure can be derived from a Gibbs-type H theorem for the exact non-dissipative dynamics.

Carnevale, GF, Martin PC.  1982.  Field theoretical techniques in statistical fluid dynamics: With application to nonlinear wave dynamics. Geophysical and Astrophysical Fluid Dynamics. 20:131-164.   10.1080/03091928208209002   AbstractWebsite

A derivation of two-point Markovian closure is presented in classical statistical field theory formalism. It is emphasized that the procedures used in this derivation are equivalent to those employed in the quantum statistical field theory derivation of the Boltzmann equation. Application of these techniques to the study of two-dimensional flow on a β-plane yields a quasi-homogeneous, quasi-stationary transport equation and a renormalized dispersion relation for Rossby waves

Carnevale, GF, Holloway G.  1982.  Information decay and the predictability of turbulent flows. Journal of Fluid Mechanics. 116:115-121.   10.1017/s0022112082000391   AbstractWebsite

A measure of predictability that has many superior features compared to currently used measures is introduced. Through statistical theory it is demonstrated that in inviscid truncated flow this new predictability measure increases monotonically in time while all initial information about the system decays. Under the influence of forcing and viscosity the behaviour of this measure is shown always to satisfy intuitive expectations.

Carnevale, GF.  1982.  A nonstationary solution to Liouville’s equation for a randomly forced two‐dimensional flow. Physics of Fluids. 25:1547-1549.   10.1063/1.863942   AbstractWebsite

Liouville’s equation for randomly forced two‐dimensional flow with Rayleigh friction is examined. An exact nonstationary solution is presented for a special form of the forcing and zero energy initial condition. This solution is such that the fluctuation‐dissipation relation is valid at all times.

Carnevale, GF.  1982.  Statistical features of the evolution of two-dimensional turbulence. Journal of Fluid Mechanics. 122:143-153.   10.1017/s0022112082002134   AbstractWebsite

Statistical fluid dynamics identifies a functional of the fluid energy spectrum that plays the role of Boltzmann's entropy for fluids. Through a series of two-dimensional flow simulations we confirm the theoretical predictions for the behaviour of this entropy functional. This includes a demonstration of Loschmidt's paradox and an examination of the effects of Rossby waves and viscosity on the behaviour of the entropy.

Weiss, J, Tabor M, Carnevale G.  1983.  The Painlevé property for partial differential equations. Journal of Mathematical Physics. 24:522-526.   10.1063/1.525721   AbstractWebsite

In this paper we define the Painlevé property for partial differential equations and show how it determines, in a remarkably simple manner, the integrability, the Bäcklund transforms, the linearizing transforms, and the Lax pairs of three well‐known partial differential equations (Burgers’ equation, KdV equation, and the modified KdV equation). This indicates that the Painlevé property may provide a unified description of integrable behavior in dynamical systems (ordinary and partial differential equations), while, at the same time, providing an efficient method for determining the integrability of particular systems.

Carnevale, GF, Frederiksen JS.  1983.  A statistical dynamical theory of strongly nonlinear internal gravity waves. Geophysical and Astrophysical Fluid Dynamics. 23:175-207.   10.1080/03091928308209042   AbstractWebsite

A statistical dynamical closure theory describing the interaction of strongly (and weakly) nonlinear two-dimensional internal waves in the presence of viscous dissipation and thermal conduction is derived. By applying renormalization methods originally formulated for quantum and classical statistical field theory, closures similar to the Direct Interaction and eddy-damped quasi-normal procedures of turbulence are derived. These methods are applied directly to the strongly nonlinear primitive field equations in Eulerian variables, thus avoiding the small amplitude assumptions inherent in the resonant interaction formalism. Propagator renormalization techniques provide formulas for the nonlinear internal wave frequency and spectral width in terms of the energy spectrum. The commonly used multiple time and space scale analysis is replaced by an analysis of the two-point correlation functions in terms of sum and difference variables. This permits the systematic development of a Landau equation. This generalization of the Boltzmann equation incorporates spatial variation of the group velocity and scattering due to spatial inhomogeneity. In the limit of weakly interacting waves and zero viscosity, the closures reduce to the resonant interaction approximation formalism. It is shown that the inviscid resonant interaction limit is singular in the sense that the quilibrium spectrum differs from that of the general inviscid nonlinear off-resonant case. This is due to the fact that in the resonant interaction limit there is an additional constant of motion, viz. “z-momentum”. The implications of these results are discussed.

Carnevale, GF, Frederiksen JS.  1983.  Viscosity renormalization based on direct-interaction closure. Journal of Fluid Mechanics. 131:289-303.   10.1017/s0022112083001330   AbstractWebsite

Approximations in statistical turbulence theory often rely on modelling the decay in time of velocity correlations with a simple exponential decay. The decay rate is viewed as a renormalized viscosity. The three simplest implementations of this approximation scheme were originally given independently by Kraichnan, Edwards and Leslie. Each of these investigators used a different formalism and each achieved different renormalization prescriptions. These three different results are reexamined here entirely in terms of direct-interaction theory. The difference in the prescriptions of Kraichnan and Leslie is shown to be the product of different definitions of renormalized viscosity. Edwards’ prescription is shown to result from an inconsistent identification of the non-stationary energy-spectrum relaxation rate with the viscosity. An assessment of the validity of the Markovian closure approximation, and a prescription for non-stationary renormalized viscosity are provided.

Frederiksen, JS, Carnevale GF.  1986.  Stability properties of exact nonzonal solutions for flow over topography. Geophysical and Astrophysical Fluid Dynamics. 35:173-207.   10.1080/03091928608245892   AbstractWebsite

The nonlinear stability properties of stationary exact nonzonal solutions for inviscid flow over topography are examined within a barotropic model in spherical geometry. For stationary solutions, such that the potential vorticity is proportional to the streamfunction, necessary and sufficient conditions for nonlinear stability are established. For a truncated system with rhomboidal truncation wave number J these are that the solid body rotation component of the zonal wind u(i) be negative, corresponding to westward flow, as J ->infinity. These results are established by using the methods of statistical mechanics. The sufficient condition for stability is also established by applying Arnol'd's method. The results are illustrated by numerical calculations. The stationary solutions are perturbed by disturbances in the streamfunction fields or by small changes in the topographic height; the climatic states for the system are obtained directly using statistical mechanics methods. The nonlinear stability properties of the stationary solutions are obtained by comparing the stationary solution with the climate, which for inviscid flow is shown to be unique. Stationary flows for which u(i) is eastward, are found to be unstable even in the limit as the streamfunction perturbation or change in the topographic height vanish. Large amplitude transient waves are generated which break the time invariance symmetry of the initial stationary flows. In contrast, for stationary flows with westward u(i), the climate is identical to the initial flow in the limit as the initial streamfuncton perturbation or the change in the topographic height vanishes. The linear instability characteristics of the stationary solutions are also obtained by solving a linear eigenvalue problem. The difficulties in establishing the stability properties of more general exact solutions, where the streamfunction is a general differentiable function of the potential vorticity, within numerical spectral models are discussed.

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.

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, 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.

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.

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.

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.

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, 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.

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, 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.

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, 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, 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.