- Entropy production selects nonequilibrium states in multistable systems
- Presentations and Authors
- Singularities and Black Holes (Stanford Encyclopedia of Philosophy)
- Maximum entropy spectral analysis and autoregressive decomposition
Both Kraichnan and Zakharov also clearly realized the applicability of these notions of dual cascades and inverse cascade to many other systems, including quantum dynamics of Bose condensation. In a J. Kraichnan also studied the 2D cnstrophy cascade and, in particular, worked out the logarithmic correction mentioned in A paper in J. Kraichnan proposed there a new interpretation of the inverse cascade in terms of a negative eddy viscosity, an idea that goes back to V.
Starr and he gave a very simple heuristic explanation for this effect: If a small-scale motion has the form of a compact blob of vorticity, or an assembly of uncorrelated blobs, a steady straining will eventually draw a typical blob out into an elongated shape, with corresponding thinning and increase of typical wavenumber. The typical result will be a decrease of the kinetic energy of the small-scale motion and a corresponding reinforcement of the straining field.
This idea has been particularly influential in the geophysical literature, where it has often been invoked to explain inverse energy cascade. What is the empirical status of Kraichnan's dual cascade theory of 2D turbulence? A complete review would be out of place here, but we shall briefly discuss its verification with an emphasis on the most current work.
Only very recently, in fact, has it become possible to observe both 2D cascades, inverse energy and direct enstrophy, in a single simulation. Earlier numerical simulations and labora- tory experiments which have focussed on a single range have, however, separately confirmed the predictions of the paper.
A number of numerical studies of the enstrophy cascade with "hyperviscosity" powers of the Laplacian replacing the usual dissipative term have reported observing the log-correction to the energy spectrum.
Entropy production selects nonequilibrium states in multistable systems
The quasi-steady inverse cascade predicted by Kraichnan as a transient before energy from pumping reaches the largest scales has also been observed in both experiments and simulations, first by L. Smith Bray, ; Batchelor, Kadomtscv informed Zakharov of Kraichnan's 2D paper sometime around V. Zakharov, private communication, Another notable paper on 2D turbulence is Kraichnan, b. However, contrary to Kraichnan's speculations in his paper, the cascade is not associated to "coalescence" of vortices and, indeed, the statistics of the velocity are quite close to Gaussian and strong, coherent vortices do not appear until the energy begins to accumulate at the largest scales.
Experiments and simulations on the statistical steady-state have instead found considerable evidence for Kraichnan's "vortex-thinning" mechanism of energy transfer even in the local cascade regime. In the situation without large-scale damping, there is "energy conden- sation" at large scales as Kraichnan had supposed, but not confined to the gravest mode. Recent simulations show that condensation in a periodic domain appears as a pair of large, counterrotating vortices with a spectrum.
These vortices are close to what is predicted by an equilibrium, maximum-entropy argument although the system is non-equilibrium, with continuously growing energy and constant negative energy flux. This question is com- plicated by the limited scale ranges that exist in those systems and the greater complexity of the dynamics. However, several recent observational studies have found evidence for both inverse energy and direct cnstrophy cascades in the Earth's atmosphere and oceans.
In the late forties Batchelor and A. Townscnd observed intermittent behavior of low-order velocity deriva- tives; since such derivatives come predominantly from the transition region between the incrtial and dissipation ranges, this intermittcncy cannot be directly taken as evidence that the self-similarity postulated for the K41 inertial-range is breaking down. Dissipation-range intermittency Kraichnan was the first to explain intermittcncy in the far dissipation range or, equiv- alcntly, for high-order velocity derivatives.
This argument can be made more systematic by using singularities of the analytic continuation of the velocity field to complex space-time locations. Chertkov et al. Inertial-range intermittency Much more difficult is tlie issue of intermittency at inertial-range scales and the problem of anomalous scaling, that is scaling for which the exponents cannot be obtained by a dimensional argument, as in K In the early sixties, A.
Obukhov and his advisor Kolmogorov began to suspect that K41 must be somewhat modified because spatial averages Er of the local energy dissipation over balls with a radius r staying within the inertial range appeared to fluctuate more and more when r is decreased; they proposed a lognormal model of intermittency. Novikov and R.
Stewart and then A. Yaglom constructed ad hoc random multiplicative models to capture such intermittency and the corresponding scaling exponents. Mandelbrot showed that, in these models, the dissipation is taking place on a set with non-integer fractal dimension; in general such models are actually multifractal.
Indeed, it was known since that the full hierarchy of moment or cumulant equations derived for statistical solutions of the Navier-Stokes equation is com- patible with the scale-invariant K41 theory in the limit of infinite Reynolds numbers. But Kraichnan was also aware that K41 is equally compatible with the Burgers equation, which definitely has no K41 scaling because of the presence of shocks ; he also noticed that the presence of the pressure in the incompressible Navier-Stokes was likely to reduce the inter- mittency one would otherwise expect from a simple vortex stretching argument.
Closure seemed incapable to say anything about the breaking of the K41 scale invariance one major exception to this statement is discussed in Section liV. Kraichnan pursued some of these ideas further himself in an influential paper in J. This paper is pure Kraichnan. A wealth of intriguing ideas arc tossed out, very original model calculations sketched in brief, and clever counterexamples devised against conventional ideas.
At least two contributions of this paper are now well-known. First, See the contribution by Falkovich in this volume. On Kcompatibility of the Navier-Stokes equations, cf. Kraichnan, a, Kraichnan, p. On using inertial-range quantities, cf. Kraichnan, ; p. In the same paper, Kraichnan gave what is now the standard formulation of the "Landau argument" on intermittency and non-universality of coefhcients in scaling laws.
His argument is considerably clearer and more compelling than the brief remarks originally made by Landau in There is at least as much Kraichnan in this argument as there is Landau. Passive scalar intermittency and the "Kraichnan model" The story of the Kraichnan model and of the birth of the first ab initio derivation of anomalous scaling is rather complex, spanning nearly three decades. Since it is understood that in this book the emphasis should be on what happened before , we shall concentrate on the early developments, that begin in the late sixties.
Examples of passive scalar transported fields are provided by the temperature of a fluid when buoyancy is negligible , the humidity of the atmosphere, the concentration of chemical or biological species. Passive scalar transport has thus an important domain of applications and considerable efforts were made since the forties to gain an understanding at least as good as for turbulence dynamics.
In particular Obukhov and, independently, S. It was thus quite natural for Kraichnan to see how well the closure tools he developed for turbulence in the fifties and the sixties were able to cope with passive scalar dynamics. He applied his LHDIA closure to the passive scalar problem, for example, reproducing Obukhov-Corrsin scaling with precise numerical coefficients. The DIA closure is exact for this special system, reducing to a single equation for the scalar correlation function at two space points and simultaneous times.
The mean Green function reduces to a Dirac delta because of the zero correlation-time assumption. This model is now usually called "the Kraichnan model" [of passive scalar dynamics] and has assumed a paradigmatic status for turbulence theory, comparable to that of the Ising model in statistical mechanics of critical phenomena. Its importance stems from a string of major discoveries by Kraichnan and others on the fundamental mechanism of intermittency, some of which will be described only briefly because they took place in the nineties. Kraichnan showed that even when the velocity field is not at all intermittent, e.
Obukhov, ; Corrsin, ; Yaglom, ; Batchelor, ; cf. Kraichnan, b, b, Al and will not concern us further. Actually doing this in a system- atic way would have required all kinds of heavy-duty theoretical tools: path integrals, large deviation theory, fluctuations of Lyapunov exponents, etc.
Actually, all this was done — and correctly so — by Kraichnan in a remarkable paper published in , just after the paper on Kolmogorov's inertial-range theories. Kraichnan's analysis was carried out for general space dimension d — following a suggestion of M.
Kraichnan's work, which was going to strongly influence subsequent more formally rigorous analyses, showed a thorough understanding of the mechanism of intermittency in the Batchelor regime. The third mechanism identified by Kraichnan was rather close to one of the Holy Grails of turbulence theory, namely understanding incrtial-range anomalous scaling and predicting the scaling exponents. This is a rather amazing proposal: how can a self- similar velocity field act on a transported temperature field to endow it with anomalous scaling and thus with lack of self-similarity?
As we shall see, the qualitative aspects of Kraichnan's conjecture have been fully corroborated by later work. In Kraichnan derived the equation for the two-point temperature correlation functions by this technique and found that the second-order temperature structure functions displayed scaling. The scaling exponent C2 can actually be obtained by simple dimensional analysis. So far no evidence of anomalous scaling had emerged. By a method similar to that used in for the two-point correlations of a passive scalar, Kraichnan derived in an equation for the structure function of order p.
One year later it was shown that there is indeed anomalous scaling, using a zero modes method, borrowed partially from field theory: the equation for the moments of order 2p has a linear operator L2p acting on the 2p-point correlation function and an inhomogencous right hand side involving correlation functions of lower order. The zero modes correspond to certain functions of 2p variables which are killed by L2p. Actually, determining the zero modes turned out to be quite difficult.
Scattering of sound by turbulence In M. Lighthill published a landmark paper on the generation of sound by turbu- lence. The next year Kraichnan observed that the production of noise in this theory depends on a high power of the Mach number and that the scattering [of sound by turbulence] is the most conspicuous acoustical phenomenon associated with very low Mach number tur- bulence. This paper, together with further developments was to be the basis of a nonintrusive ultrasonic technique for the re- mote probing of vorticity.
The same year and independently Lighthill also published a theory of scattering. As done by Lighthill, Kraichnan assumed that density and pressure fluctuations are related by an adiabatic equation of state with a uniform speed of sound. On zero-mode methods, cf. On simulations, cf. The whole story about anomalous scaling for passive scalars is recounted in [www. This is known as a Hodge decomposition in mathematics. He then obtained a wave equation which has four terms. One term is linear in , related to viscous stresses and is mostly negligible.
The T-T term is Lighthill's quadrupolar sound production term. The L-T term gives the scattering of a preexisting sound wave by the turbulence. Kraichnan then worked out the angular distribution and frequency distribution of the scattered wave in terms of the four-dimensional Fourier transform of the shear velocity field. Explicit expressions for cross sections were obtained for the case of a scattering from a region of isotropic turbulence. Some remarks are in order. High-Rayleigh number convection Thermal convection is ubiquitous in technology and is amenable to controlled experiments where a fluid heated from below is placed between two horizontal plates.
Here, g is the acceleration due to gravity, 6T the vertical temperature difference across the fluid of height h, and a, v and K arc the thermal expansion coefficient of the fluid, its kinematic viscosity and its thermal diffusivity. Turbulent thermal convection was and remains a central topic of the Woods Hole Oceanographic Institute Geophysical Fluid Dynamics summer program, with which Kraich- nan had considerable interaction from the late fifties.
Around the same time he also had much interaction with E. Spiegel, who had been trained in astrophysical fluid dynamics: it is usually convective transport which allows the heat generated in the interiors of stars to escape. In the early days the easiest way to model astrophysical convection was through the mixing length theory, which follows ideas of Boussinesq and of Prandtl. In Kraichnan devoted a fairly substantial paper to thermal convection, which we cannot summarize in detail because of lack of space.
We shall thus concentrate on his most orignal contribution, to what is now called "ultimate convection" , at extremely high Rayleigh numbers. One important question in high-Rayleigh number convection is the dependence upon Rayleigh and Prandtl numbers of the Nusselt number N, the heat flux non-dimensionalizcd by the conductive heat flux. In the fifties C. As pointed out by Kraichnan "[In] Priestley's theory Kraichnan also discussed the Prandtl number dependence of the various regimes. This al- His first relativity paper was to be published only two years later Kraichnan, b.
Boussinesq, ; Prandtl, ; Kraichnan, Priestley, and references therein; Kraichnan, p. It may be shown that this argument breaks down at large Prandtl numbers.
It is generally believed that the threshold is significantly lower and depends on the Prandtl number and the boundary conditions. Artefacts masquerading as a Ra ' law cannot be ruled out. In Gottingen a two-meter high convection experiment using sulfur hexafluoride SF6 at 20 times atmo- spheric pressure is under construction to try and capture Kraichnan's ultimate convection regime. Kraichnan and computers Kraichnan, although basically a theoretician, was very far from being allergic to comput- ers.
Actually, not only was he a very talented programmer, but he got occasionally involved in writing system software and even in modifying hardware. Some of his closest collab- orators, foremost S. Orszag, prodded by him, got deeply involved in three-dimensional simulations of Navier-Stokes turbulence.
This was — and still is — called "Direct Numer- ical Simulation" DNS because the original goal was to check on the validity of various closures by going directly to the fluid dynamical equations. Convinced that many features of high-Reynolds number turbulence should be universal, Kraichnan encouraged the use of the simplest type of boundary conditions periodic which allows the simple and efficient use of spectral methods.
He also suggested using Gaussian initial conditions rather than more realistic ones.
Presentations and Authors
Curiously, although the thrust to do DNS started just after Kraichnan's discovery of the 2D inverse cascade, he strongly recommended focusing on 3D flow. Considerable effort — this time often in collaboration with J. Herring — went into the numerical integration of various closure equations. Kraichnan proposed using discrete wavcnumbcrs in geometric rather than arithmetic progression.
This allowed reaching very high Reynolds numbers. Kraichnan himself was actively involved in writing code for these investigations. His punched cards were shipped from New Hampshire to NASA Goddard Institute where the machine computations were performed during the 's and 's. Herring recalls that "Bob's programs very rarely contained any bugs.
Conclusions Our survey has focussed on three of Kraichnan's contributions to turbulence theory: 1 spectral closures and realizability, 2 inverse cascade of energy in 2D turbulence and 3 intermittency of passive scalars advected by turbulence. These are, arguably, his most signif- icant achievements which have had the greatest impact on the field. Spectral closures of the DIA class still have numerous interesting applications when the questions under investiga- tion do not depend crucially on deviations from K Even today an EDQNM calculation, for example, will often be the first line of assault on a difficult new turbulence problem.
Furthermore, Kraichnan's criterion of realizability has become part of the standard toolbox of turbulence closure techniques. Realizability is necessary both for physical meaningfulness and, often, for successful numerical solution of the closure equations. Kraichnan's prediction of inverse cascade has been well verified by experiments and simulations and has relevance in explaining dynamical processes in the Earth's atmosphere and oceans.
The concept of an inverse cascade has proved very fruitful in other systems, too, where similar fluxes of invari- ants to large-scales may occur, such as magnetic helicity in 3D MHD turbulence, magnetic potential in 2D MHD turbulence, and particle number in quantum Bose systems. Finally, Kraichnan's model of a passive scalar advected by a white-in-time Gaussian random veloc- ity has become a paradigm for turbulence intermittency and anomalous scaling — an "Ising model" of turbulence. The theory of passive scalar intermittency has not yet led to a similar Spiegel, On artefacts, Sroenivasan private communication, On Helium gas experiments, cf.
Chavannc ct al. On SF6, cf. However, the Kraichnan model has raised the scientific level of discourse in the field by providing a nontrivial ex- ample of a multifractal field generated by a turbulence dynamics. It is no longer debatable that anomalous scaling is possible for Navier-Stokes. For example we have not been able to discuss the numerous interactions Kraichnan had with many people in the USA and in other countries, particularly in France, Israel and Japan.
Even focusing our discussion to his turbulence research prior to , we have been forced to omit mention of a large number of problems to which Kraichnan made important contributions in that period.
Singularities and Black Holes (Stanford Encyclopedia of Philosophy)
Furthermore, Kraichnan invested substantial time to other theoretical approaches, that came after the cutoff for this paper. We provide below just a few references to this additional work on turbulence by Kraichnan. It may be that later generations will find that our survey has missed some of Kraichnan's most significant accomplishments. The richness of his oeuvre can only be appreciated by poring over his densely written research articles, bristling with original ideas and novel methods, for oneself.
The reader who docs so will be generously rewarded for his effort. It is amusing to wonder what might be Einstein's assessment from the welkins of his for- mer assistant. He would probably have to conclude that Kraichnan had a lot of Sitzfleisch. Turbulence is a dauntingly difficult subject where any significant advance is won by a hard-fought battle; and yet Kraichnan has left his record of victories throughout the field. Several international conferences held in his honor arc testimony to the lasting impact of Robert H.
Many have helped us with their remarks and their own recollec- tions. We are particularly indebted to B. Castaing, C. Connaughton, G. Falkovich, H. Frisch, T. Gotoh, J. Herring, C. Leith, H. Moffatt, S. Nazarenko, S. Orszag, I. Procaccia, H. Rose, E. Spiegel, K. Sreenivasan, B. Villone and V. References Abe, R.
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