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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, 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, GE, Smith SGL, Crisciani F, Purini R, Serravall R.  1999.  Bifurcation of a coastal current at an escarpment. Journal of Physical Oceanography. 29:969-985.   10.1175/1520-0485(1999)029<0969:boacca>;2   AbstractWebsite

The evolution of a coastal current as it encounters an escarpment depends strongly on whether the geometry of the coast and escarpment is right or left "handed," independent of the direction of the coastal current. Handedness is defined such that right-handed means that when looking across the escarpment from the deep to the shallow side, the coast is found on the right. The essential aspects of the difference in behavior of the current in the two geometries are captured by a simple quasigeostrophic model of coastal flow over a step. An exact analytic solution to the nonlinear stationary problem is obtained. This solution shows that, when a coastal current crosses an escarpment in the left-handed geometry, the speed of the current will increase independent of whether the flow is from shallow to deep or from deep to shallow. For the right-handed geometry, the speed of the current decreases, also independent of the direction of the coastal flow. In the left (right)-handed geometry, there is associated to the coastal flow an inshore (offshore) current along the escarpment. These results are explained in terms of linear wave theory and vortex dynamics. Numerical simulations are used to examine the evolution of the flow from the initial encounter to the establishment of a stationary flow. The relevance of this research is discussed in light of recent results from laboratory experiments and oceanic observations.

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, 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, 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, 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.  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.

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.

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, 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, Orlandi P.  2001.  Buoyancy- to inertial-range transition in forced stratified turbulence. Journal of Fluid Mechanics. 427:205-239.   10.1017/s002211200000241x   AbstractWebsite

The buoyancy range, which represents a transition from large-scale wave-dominated motions to small-scale turbulence in the oceans and the atmosphere, is investigated through large-eddy simulations. The model presented here uses a continual forcing based on large-scale standing internal waves and has a spectral truncation in the isotropic inertial range. Evidence is presented for a break in the energy spectra from the anisotropic k(-3) buoyancy range to the small-scale k(-5/3) isotropic inertial range. Density structures that form during wave breaking and periods of high strain rate are analysed. Elongated vertical structures produced during periods of strong straining motion are found to collapse in the subsequent vertically compressional phase of the strain resulting in a zone or patch of mixed fluid.

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, 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.

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.

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, 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, 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.  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.

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, 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, Cavallini F, Crisciani F.  2001.  Dynamic boundary conditions revisited. Journal of Physical Oceanography. 31:2489-2497.   10.1175/1520-0485(2001)031<2489:dbcr>;2   AbstractWebsite

The applicability of the super-slip boundary condition in wind-driven quasigeostrophic ocean circulation models is reexamined. The energy and enstrophy characteristics of the super-slip condition are discussed for the equilibrium state. A model is constructed with super-slip on the western boundary and free slip on the other boundaries. Both linear and nonlinear solutions are presented. Compared to the case with all free-slip boundaries, this new model gives a more energetic and narrower western boundary current, but otherwise the differences are not very great.

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, 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, 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.