Eddy-damped Quasi-normal Markovian Approximation Research Articles

Role of helicities for the dynamics of turbulent magnetic fields

Investigations of the inverse cascade of magnetic helicity are conducted with pseudospectral, three-dimensional direct numerical simulations of forced and decaying incompressible MHD turbulence. The high-resolution simulations which allow for the necessary scale separation show that the observed self-similar scaling behavior of magnetic helicity and related quantities can only be understood by taking the full nonlinear interplay of velocity and magnetic fluctuations into account. With the help of the eddy-damped quasi-normal Markovian approximation a probably universal relation between kinetic and magnetic helicities is derived which closely resembles the extended definition of the prominent dynamo pseudoscalar α. This unexpected similarity suggests an additional nonlinear quenching mechanism of the current-helicity contribution to α.

Spectral modeling of magnetohydrodynamic turbulent flows

We present a dynamical spectral model for large-eddy simulation of the incompressible magnetohydrodynamic (MHD) equations based on the eddy damped quasinormal Markovian approximation. This model extends classical spectral large-eddy simulations for the Navier-Stokes equations to incorporate general (non-Kolmogorovian) spectra as well as eddy noise. We derive the model for MHD flows and show that the introduction of an eddy damping time for the dynamics of spectral tensors, in the absence of equipartition between the velocity and magnetic fields, leads to better agreement with direct numerical simulations, an important point for dynamo computations.

Spectral modeling of turbulent flows and the role of helicity

We present a version of a dynamical spectral model for large eddy simulation based on the eddy damped quasinormal Markovian approximation [S. A. Orszag, in, edited by R. Balian, Proceedings of Les Houches Summer School, 1973 (Gordon and Breach, New York, 1977), p. 237; J. P. Chollet and M. Lesievr, J. Atmos. Sci. 38, 2747 (1981)]. Three distinct modifications are implemented and tested. On the one hand, whereas in current approaches, a Kolmogorov-like energy spectrum is usually assumed in order to evaluate the non-local transfer, in our method the energy spectrum of the subgrid scales adapts itself dynamically to the large-scale resolved spectrum; this first modification allows in particular for a better treatment of transient phases and instabilities, as shown on one specific example. Moreover, the model takes into account the phase relationships of the small scales, embodied, for example, in strongly localized structures such as vortex filaments. To that effect, phase information is implemented in the treatment of the so-called eddy noise in the closure model. Finally, we also consider the role that helical small scales may play in the evaluation of the transfer of energy and helicity, the two invariants of the primitive equations in the inviscid case; this leads as well to intrinsic variations in the development of helicity spectra. Therefore, our model allows for simulations of flows for a variety of circumstances and a priori at any given Reynolds number. Comparisons with direct numerical simulations of the three-dimensional Navier-Stokes equation are performed on fluids driven by an Arnold-Beltrami-Childress (ABC) flow which is a prototype of fully helical flows (velocity and vorticity fields are parallel). Good agreements are obtained for physical and spectral behavior of the large scales.

The residual energy in freely decaying magnetohydrodynamic turbulence

Within the framework of the eddy-damped quasi-normal Markovian approximation for incompressible isotropic magnetohydrodynamic (MHD) turbulence a prediction for the inertial range scaling of the residual energy spectrum, , is obtained. This scaling, while in contradiction to earlier theoretical results, is shown to be in agreement with high-resolution direct numerical simulations of nonhelical decaying MHD turbulence. The underlying phenomenology states a dynamic quasi-equilibrium of the small-scale turbulent dynamo and the Alfvén effect.

Free decay of turbulence and breakdown of self-similarity

It has been generally assumed, since the work of von Kármán and Howarth in 1938, that free decay of fully-developed turbulence is self-similar. Here we present a simple phenomenological model of the decay of three-dimensional incompressible turbulence, which predicts breakdown of self-similarity for low-wavenumber spectral exponents n in the range nc<n<4, where nc is some threshold wavenumber. Calculations with the eddy-damped quasi-normal Markovian approximation give the value as nc≈3.45. The energy spectrum for this range of exponents develops two length-scales, separating three distinct wavenumber ranges.

The separation of particle pairs in the eddy-damped quasinormal Markovian approximation

The implications of the eddy-damped quasinormal Markovian (EDQNM) approximation for the separation of particle pairs in (three-dimensional) inertial subrange turbulence are explored. In particular the EDQNM equations are solved for the particle separation probability distribution, without making the usual approximation of replacing the damping time scales by their large time asymptotic form. A number of properties of the separation probability distribution that are of significance for the prediction of concentration fluctuations are calculated. The values show quite large departures from the values obtained with the large time form of the damping time scales.

Analytical theories of turbulence and the ε expansion

The fixed-point form of hydrodynamic equations emerging from renormalization group analysis of strong turbulence is analyzed using perturbation expansion in powers of the renormalized coupling constant (Reynolds number) Re*∝ε1/2. In the second order of ε expansion, transport equations are derived for energy and variance of a passive scalar which are identical to the inertial-range form of the eddy-damped quasinormal Markovian approximation (EDQNM), but without adjustable parameters which must be determined from experiment. The energy and scalar variance spectra are predicted to be Ek=1.617ε̄2/3k−5/3, and Θk=1.146N̄ ε̄−1/3k−5/3, where ε̄ and N̄ are, respectively, the dissipation rate of energy and scalar variance. This level of agreement with experiment has in the past proved difficult to obtain from renormalized perturbation theory applied to the bare hydrodynamic equations, without introduction of parameters that must be adjusted to different values for different dynamical fields. The fidelity of the present method, which emphasizes in a particular way the distant interactions in k space, may be related to the degree of nonlocality of the bare dynamics as manifested by possible generic Beltramization tendencies in strong turbulence.

The Anisotropic Temperature Spectrum in Isotropic Turbulence with a Constant Temperature Gradient

The anisotropic temperature spectrum is numerically determined within the eddy-damped quasi-normal Markovian approximation for the case that a constant temperature gradient is imposed on isotropic and incompressible turbulence with the Kolmogorov velocity spectrum. The temperature spectrum is conveniently expanded in terms of Legendre polynominals. The isotropic part of the spectrum is calculated as k -1.701 , very close to the Kolmogorov spectrum, while the anisotropic spectrum which corresponds to the coefficient of a second Legendre function is shown to be k -3 . A model is examined which exhibits a crossover of the anisotropic spectrum from coupling-constant-dependent scaling to the independent scaling k -3 .

D-dimensional turbulence

$d$-dimensional homogeneous isotropic incompressible turbulence is defined, for arbitrary nonintegral $d$, by analytically continuing the Taylor expansion in time of the energy spectrum ${E}_{k}(t)$, assuming Gaussian initial conditions. If $dl2$, the positivity of the energy spectrum is not necessarily preserved in time. For $d\ensuremath{\ge}2$ all steady-state and initial-value calculations have been made with a realizable second-order closure, the eddy-damped quasi normal Markovian approximation. Near two dimensions the enstrophy (mean square vorticity) conservation law is weakly broken, enough to allow ultraviolet singularities to develop in a finite time but not enough to prevent energy from cascading in the infrared direction. A systematic investigation is made of zero-transfer (inertial) steady-state scaling solutions ${E}_{k}\ensuremath{\propto}{k}^{\ensuremath{-}m}$ and of their stability. Energy-inertial solutions with $m=\frac{5}{3}$ exist for arbitrary $d$; the direction of the energy cascade reverses at $d={d}_{c}\ensuremath{\simeq}2.05$. For $dl{{d}^{\ensuremath{'}}}_{c}\ensuremath{\simeq}2.06$ there are in addition, as in the cascade studied by Bell and Nelkin, inertial solutions with zero energy flux; their exponents $m(d)$ are given by a roughly parabolic curve in the ($m, d$) plane, linking enstrophy cascade ($m=3$, $d=2$) to enstrophy equipartition ($m=1$, $d=2$) For any point in the ($m, d$) plane such that the transfer integral is finite and negative, a steady-state scaling solution ${E}_{k}\ensuremath{\propto}{k}^{\ensuremath{-}\mathrm{m}}$ is obtained when the fluid is subject to random forces with spectrum ${F}_{k}\ensuremath{\propto}{k}^{\frac{3(m\ensuremath{-}1)}{2}}$. A special case is the model B [$m=\ensuremath{-}1=\frac{2}{3}\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})$, $d=4\ensuremath{-}\ensuremath{\epsilon}$] obtained by Forster, Nelson, and Stephen using a dynamical renormalization-group procedure. Forced steady-state solutions are actually not resticted to the neighborhood of $m=\ensuremath{-}1$, $d=4$; they are amenable to renormalization-group calculations on the primitive equations for arbitrary $dg2$ when $m$ is close to the crossover -1 and, perhaps, also near the crossover +3.

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 a)’ 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 ‘Alfvén effect’; the small-scale ‘residual’ helicity, which is the difference between a purely kinetic and a purely magnetic helical term, induces growth of large-scale 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 −3/2 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 will grow 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 Frischet al.(1975), results from a competition between the helicity and Alféh 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 linjand the rate$\tilde{\epsilon}^V$(per unit mass), the time of build-up of magnetic energy with scaleL[Gt ] linjis$t \approx L(|\tilde{\epsilon}^V|l^2_{\rm inj})^{-1/3}.$