Mathematical models with exact renormalization for turbulent transport,II: Fractal interfaces,non-Gaussian statistics and the sweeping effect

This paper continues the study of a model for turbulent transport with an exact renormalization theory which has recently been proposed and developed by the authors. Three important topics are analyzed with complete mathematical rigor for this model: (1) Renormalized higher order statistics of a passively advected scalar such as the pair distance distribution and the fractal dimension of interfaces, (2) the effect of non-Gaussian turbulent velocity statistics on renormalization theory, (3) the sweeping effect of additional large scale mean velocities. A special emphasis is placed on renormalization theory in the vicinity of the value of the analogue of the Kolmogorov-spectrum in the model. In the authors' earlier paper, it was established that the Kolmogorov value is at a phase transition boundary in the exact renormalization theory. It is found here that the qualitative model, despite its simplicity contains, in the vicinity of the Kolmogorov value, a remarkable amount of the qualitative behavior of turbulent transport which has been uncovered in recent experiments and proposed in phenomenological theories. In particular, the Richardson 4/3-law for pair dispersion and interfaces with fractal dimension defect of 2/3 occur in the model rigorously as limits when the Kolmogorov spectrum is approached as a limit from one side of the phase transition boundary; alternative corrections to the Richardson law with the same form as those proposed heuristically in the recent literature and interfaces with fractal dimension defect 1/3, occur in the model when the Kolmogorov spectrum is approached from the other side of the phase transition. It is very interesting that fractal dimension defects of roughly the value either 1/3 or 2/3 for level sets and interfaces of passive scalars have been ubiquitous in recent turbulence experiments. As regards non-Gaussian the asymptotic normality of normalized integrals (B.56) corresponding to compactly supported blobs with mean zero. The proof of this latter fact is done in the same way as Step 2, Proposition B.3, using the fact that the corresponding random processes have finite domain of dependence. This concludes the proof of Proposition B.4.Research partially supported by NSF-DMS-9005799, ARO-DAAL03-89-K-0039 and AFOSR 90-0090Research partially supported by grants NSF-DMS-90-01805, ARO-DAAL03-89-K-0013, and ONR-N00014-89-J-1014     

A renormalization group analysis of turbulent transport

The field-theoretic renormalization group is used to derive scaling relations for the transport of passive scalars by an incompressible velocity field with a specified energy spectrum. Results are obtained with the analog of the expansion of critical phenomena and compared to exact results which are available for shear flows in two dimensions.A 1/N expansion is proposed for the regions in which the expansion fails.     

BRS symmetry for Yang-Mills theory with exact renormalization group

In the exact renormalization-group (RG) flow in the infrared cutoff Λ one needs boundary conditions. In a previous paper on SU(2) Yang-Mills theory we proposed to use the nine physical relevant couplings of the effective action as boundary conditions at the physical point Λ= 0 (these couplings are defined at some non-vanishing subtraction point μ≠ 0). In this paper we show perturbatively that it is possible to appropriately fix these couplings in such a way that the full set of Slavnov-Taylor (ST) identities are satisfied. Three couplings are given by the vector and ghost wave-function normalization and the three-vector coupling at the subtraction point; three of the remaining six are vanishing (e.g. the vector mass) and the others are expressed by irrelevant vertices evaluated at the subtraction point. We follow the method used by Becchi to prove ST identities in the RG framework. There the boundary conditions are given at a non-physical point Λ = Λ′ ≠ 0, so that one avoids the need of a non-vanishing subtraction point.     

Analytical solutions of exact renormalization group equations

We study exact renormalization group equations in the framework of the effective average action. We present analytical solutions for the scale dependence of the potential in a variety of models. These solutions display a rich spectrum of physical behaviour such as fixed points governing the universal behaviour near second-order phase transitions, critical exponents, first-order transitions (some of which are radiatively induced) and tricritical behaviour.     

Strategies for fluctuation renormalization in nonlinear transport theory

We are concerned here with the problems encountered in the derivation of nonlinear transport equations from a correspondingly nonlinear Langevin equation. A dynamical coupling between the time-dependent averages and the fluctuations must be accounted for by a procedure which leads to a renormalization of the nonlinear transport equation. Generalizing the familiar phenomenological approach to Brownian motion to nonlinear dynamics, we illustrate how the problem arises and show how the fluctuation renormalization can be obtained exactly by a formal procedure or approximately by more tractable methods.     

Explicit inertial range renormalization theory in a model for turbulent diffusion

The inertial range for a statistical turbulent velocity field consists of those scales that are larger than the dissipation scale but smaller than the integral scale. Here the complete scale-invariant explicit inertial range renormalization theory for all the higher-order statistics of a diffusing passive scalar is developed in a model which, despite its simplicity, involves turbulent diffusion by statistical velocity fields with arbitrarily many scales, infrared divergence, long-range spatial correlations, and rapid fluctuations in time-such velocity fields retain several characteristic features of those in fully developed turbulence. The main tool in the development of this explicit renormalization theory for the model is an exact quantum mechanical analogy which relates higher-order statistics of the diffusing scalar to the properties of solutions of a family ofN- body parabolic quantum problems. The canonical inertial range renormalized statistical fixed point is developed explicitly here as a function of the velocity spectral parameter, which measures the strength of the infrared divergence: for<2, mean-field behavior in the inertial range occurs with Gaussian statistical behavior for the scalar and standard diffusive scaling laws; for>2 a phase transition occurs to a fixed point with anomalous inertial range scaling laws and a non-Gaussian renormalized statistical fixed point. Several explicit connections between the renormalization theory in the model and intermediate asymptotics are developed explicitly as well as links between anomalous turbulent decay and explicit spectral properties of Schrödinger operators. The differences between this inertial range renormalization theory and the earlier theories for large-scale eddy diffusivity developed by Avellaneda and the author in such models are also discussed here.     

Preturbulent states and renormalization group for simple models

Some one-parameter first-order difference-equations of the form X_i+₁=f(R,X_i), arising in theoretical biology and now used as simple models for turbulent behavior, are studied. We show that they have qualitative features quite similar to those found in critical phenomena.Such concepts as universality, scale invariance, critical temperature, scaling law find their counterpart here using a formalism directly inspired from the Renormalization Group Theory. Then, standard techniques permit us to correlate various different numerical results. On the other hand, these considerations serve to illustrate the renormalization group approach on simple models exhibiting chaotic behavior.     

Micromixing models for turbulent flows

In transported probability density function and filtered density function methods, micromixing models are required to close the molecular mixing term. The accuracy and computational efficiency of improved versions of the parameterized scalar profile (PSP) model are assessed and compared with commonly used mixing models such as Curl, modified Curl, interaction by exchange with the mean and Euclidean minimum spanning tree. Different generalizations of the PSP mixing model for spatially inhomogeneous flow configurations are presented. The selected test cases focus on molecular mixing and avoid interference with other models. Simulation results for a three-stream problem, involving two inert scalars, and a multi-scalar test case with mean-scalar-gradients are presented.     

Analytical approximation schemes for solving exact renormalization group equations. II Conformal mappings

We present a new efficient analytical approximation scheme to two-point boundary value problems of ordinary differential equations (ODEs) adapted to the study of the derivative expansion of the exact renormalization group equations. It is based on a compactification of the complex plane of the independent variable using a mapping of an angular sector onto a unit disc. We explicitly treat, for the scalar field, the local potential approximations of the Wegner–Houghton equation in the dimension d=3$d=3$ and of the Wilson–Polchinski equation for some values of d∈]2,3]$d\in ]2,3]$. We then consider, for d=3$d=3$, the coupled ODEs obtained by Morris at the second order of the derivative expansion. In both cases the fixed points and the eigenvalues attached to them are estimated. Comparisons of the results obtained are made with the shooting method and with the other analytical methods available. The best accuracy is reached with our new method which presents also the advantage of being very fast. Thus, it is well adapted to the study of more complicated systems of equations.     

The running gauge coupling in the exact renormalization group approach

We discuss the perturbative running Yang-Mills coupling constant in the Wilsonian exact renormalization group approach, and compare it to the running coupling in the more conventional MS scheme. The exact renormalization group approach corresponds to a particular renormalization scheme, and we relate explicitly the corresponding Λ parameters. The unambiguous definition of an exact renormalization group scheme requires, however, the use of a one-loop improved high energy effective action.     

Barriers for transport in turbulent plasmas

The control of transport due to electrostatic turbulence is investigated using test-particle simulations. We show that a barrier for the transport, that is, a region where transport is reduced, can be generated through the randomization of phases of the turbulent field. This corresponds to the annihilation of coherent structures which are present at all scales, without actually suppressing turbulence. When the barrier is active, a flux of particles towards the center of the simulation box is present inside the region where the barrier is located.     

On the renormalization group transformation for scalar hierarchical models

We give a new proof for the existence of a non-Gaussian hierarchical renormalization group fixed point, using what could be called a beta-function for this problem. We also discuss the asymptotic behavior of this fixed point, and the connection between the hierarchical models of Dyson and Gallavotti.Supported in Part by the National Science Foundation under Grant No. DMS-8802590Supported in Part by the Swiss National Science Foundation     

Tests for Monte Carlo renormalization studies on Ising models

In this expanded version of an earlier letter, we consider many computational details that were omitted for want of space. Ford = 2 Ising spins with up to 13 different short-range interactions, we construct the critical surface in the vicinity of (Onsager's) nearest-neighbor (nn) critical point by using the body of the available information on the solvable nn case. We then see if the Monte Carlo renormalization group flows generated from this point do indeed lie on this surface and quantify the errors if they do not.     

Perturbative renormalization of multi-channel Kondo-type models

The poor man's scaling is extended to higher order by the use of the open-shell Rayleigh-Schr?dinger perturbation theory. A generalized Kondo-type model with the SU(n)SU(m) symmetry is proposed and renormalized to the third order. It is shown that the model has both local Fermi-liquid and non-Fermi-liquid fixed points, and that the latter becomes unstable in the special case of n=m=2. Possible relevance of the model to the newly found phase IV in Ce_xLa_1-xB₆ is discussed. Received: 24 February 1998 / Accepted: 17 April 1998     

Analytical approximation schemes for solving exact renormalization group equations in the local potential approximation

The relation between the Wilson–Polchinski and the Litim optimized ERGEs in the local potential approximation is studied with high accuracy using two different analytical approaches based on a field expansion: a recently proposed genuine analytical approximation scheme to two-point boundary value problems of ordinary differential equations, and a new one based on approximating the solution by generalized hypergeometric functions. A comparison with the numerical results obtained with the shooting method is made. A similar accuracy is reached in each case. Both two methods appear to be more efficient than the usual field expansions frequently used in the current studies of ERGEs (in particular for the Wilson–Polchinski case in the study of which they fail).     

Revisiting the local potential approximation of the exact renormalization group equation

The conventional absence of field renormalization in the local potential approximation (LPA) — implying a zero value of the critical exponent η   — is shown to be incompatible with the logic of the derivative expansion of the exact renormalization group (RG) equation. We present a LPA with η≠0$\eta \ne 0$ that strictly does not make reference to any momentum dependence. Emphasis is made on the perfect breaking of the reparametrization invariance in that pure LPA (absence of any vestige of invariance) which is compatible with the observation of a progressive smooth restoration of that invariance on implementing the two first orders of the derivative expansion whereas the conventional requirement (η=0$\eta =0$ in the LPA) precluded that observation.