U. Frisch

A simple dynamical model of intermittent fully developed turbulence

We present a phenomenological model of intermittency called the P-model and related to the Novikov-Stewart (1964) model. The key assumption is that in scales N &2-” only a fraction /3n of the total space has an appreciable excitation. The model, the idea of which owes much to Kraichnan (1972, 1974)’ is dynamical in the sense that we work entirely with inertial-range quantities such as velocity amplitudes, eddy turnover times and energy transfer. This gives more physical insight than the traditional approach based on probabilistic models of the dissipation. The P-model leads in an elementary way to the concept of the self-similarity dimension D, a special case of Mandelbrot’s (1974, 1976) ‘fractal dimension’. For threedimensional turbulence, the correction B to the Q exponent of the energy spectrum is equal to +( 3 - D) and is related to the exponent p of the dissipation correlation function by B = Qp (0.17 for the currently accepted value). This is a borderline case of the Mandelbrot inequality B < Qp. It is shown in the appendix that this inequality may be derived from the Navier-Stokes equation under the strong, but plausible, assumption that the inertial-range scaling laws for second- and fourth-order moments have the same viscous cut-off. The predictions of the P-model for the spectrum and for higher-order statistics are in agreement with recent conjectures based on analogies with critical phenomena (Nelkin 1975) but generally diasgree with the 1962 Kolmogorov lognormal model. However, the sixth-order structure function (8v6(Z)) and the dissipation correlation function (e(r) e(r + 1)) are related by

Strong MHD helical turbulence and the nonlinear dynamo effect

To understand the turbulent generation of large-scale magnetic fields and to advance beyond purely kinematic approaches to the dynamo effect like that introduced by Steenbeck, Krause & Radler (1966)’ a new nonlinear theory is developed for three-dimensional, homogeneous, isotropic, incompressible MHD turbulence with helicity, i.e. not statistically invariant under plane reflexions. For this, techniques introduced for ordinary turbulence in recent years by Kraichnan (1971~~)’ Orszag (1970, 1976) and others are generalized to MHD; in particular we make use of the eddy-damped quasi-normal Markovian approximation. The resulting closed equations for the evolution of the kinetic and magnetic energy and helicity spectra are studied both theoretically and numerically in situations with high Reynolds number and unit magnetic Prandtl number. Interactions between widely separated scales are much more important than for non-magnetic turbulence. Large-scale magnetic energy brings to equipartition small-scale kinetic and magnetic excitation (energy or helicity) by the ‘AlfvBn effect ’; the small-scale ‘residual’ helicity, which is the difference between a purely kinetic and a purely magnetic helical term, induces growth of largescale magnetic energy and helicity by the ‘helicity effect’. In the absence of helicity an inertial range occurs with a cascade of energy to small scales; to lowest order it is a - power law with equipartition of kinetic and magnetic energy spectra as in Kraichnan (1965) but there are - 2 corrections (and possibly higher ones) leading to a slight excess of magnetic energy. When kinetic energy is continuously injected, an initial seed of magnetic field willgrow to approximate equipartition, at least in the small scales. If in addition kinetic helicity is injected, an inverse cascade of magnetic helicity is obtained leading to the appearance of magnetic energy and helicity in ever-increasing scales (in fact, limited by the size of the system). This inverse cascade, predicted by Frisch et aZ. (1975), results from a competition between the helicity and Alfvh effects and yields an inertial range with approximately - 1 and - 2 power laws for magnetic energy and helicity. When kinetic helicity is injected at the scale Zinj and the rate k (per unit mass), the time of build-up of magnetic energy with scale L 9 Zinl is t % L( prp;nj)-k 21 FLM 77

Small-scale structure of the Taylor–Green vortex

The dynamics of both the inviscid and viscous Taylor–Green (TG) three-dimensional vortex flows are investigated. This flow is perhaps the simplest system in which one can study the generation of small scales by three-dimensional vortex stretching and the resulting turbulence. The problem is studied by both direct spectral numerical solution of the Navier–Stokes equations (with up to 256 3 modes) and by power-series analysis in time. The inviscid dynamics are strongly influenced by symmetries which confine the flow to an impermeable box with stress-free boundaries. There is an early stage during which the flow is strongly anisotropic with well-organized (laminar) small-scale excitation in the form of vortex sheets located near the walls of this box. The flow is smooth but has complex-space singularities within a distance

Possibility of an inverse cascade of magnetic helicity in magnetohydrodynamic turbulence

\hat{\delta}(t)

Lattice Gas Models for 3D Hydrodynamics

of real (physical) space which give rise to an exponential tail in the energy spectrum. It is found that

FULLY DEVELOPED TURBULENCE AND INTERMITTENCY

\hat{\delta}(t)

Helicity cascades in fully developed isotropic turbulence

decreases exponentially in time to the limit of our resolution. Indirect evidence is presented that more violent vortex stretching takes place at later times, possibly leading to a real singularity (

Dynamo action in a family of flows with chaotic streamlines

\hat{\delta}(t) = 0

Brownian motion of harmonic oscillator with stochastic frequency

) at a finite time. These direct integration results are consistent with new temporal power-series results that extend the Morf, Orszag & Frisch (1980) analysis from order t 44 to order t 80 . Still, convincing evidence for or against the existence of a real singularity will require even more sophisticated analysis. The viscous dynamics (decay) have been studied for Reynolds numbers R (based on an integral scale) up to 3000 and beyond the time t max at which the maximum energy dissipation is achieved. Early-time, high- R dynamics are essentially inviscid and laminar. The inviscidly formed vortex sheets are observed to roll up and are then subject to instabilities accompanied by reconnection processes which make the flow increasingly chaotic (turbulent) with extended high-vorticity patches appearing away from the impermeable walls. Near t max the small scales of the flow are nearly isotropic provided that R [gsim ] 1000. Various features characteristic of fully developed turbulence are observed near t max when R = 3000 and R λ = 110: a k − n inertial range in the energy spectrum is obtained with n ≈ 1.6–2.2 (in contrast with a much steeper spectrum at earlier times); th energy dissipation has considerable spatial intermittency; its spectrum has a k −1+μ inertial range with the codimension μ ≈ 0.3−0.7. Skewness and flatness results are also presented.

Turbulence: Challenges for Theory and Experiment

Some of the consequences of the conservation of magnetic helicity