Interplay between experimental and numerical approaches in the fluid dynamo problem

After years of purely analytical and numerical investigations, the dynamo fluid problem has advanced to a phase of experimental study. We present an outline of the numerical steps that have accompanied the Von Kármán Sodium (VKS) experiment at Cadarache for the past ten years. We show how various numerical studies contributed progressively to the optimization of the experimental facility. The recent success of the VKS2 experiment of September 2006 in achieving dynamo action has prompted an extension of the numerical code. Modeling of the electromotive force induced in the volume of the impellers shows that an axial dipole is excited, as observed in the experiment. We infer from these results that the observed value of the critical magnetic Reynolds number may be linked to the soft iron of the impellers and not to turbulence which occurs for any choice of materials. We conclude with proposals for further lines of numerical development. To cite this article: J. Léorat, C. Nore, C. R. Physique 9 (2008).     

An optimal Galerkin scheme to solve the kinematic dynamo eigenvalue problem in a full sphere

The kinematic dynamo approximation describes the generation of magnetic field in a prescribed flow of electrically-conducting liquid. One of its main uses is as a proof-of-concept tool to test hypotheses about self-exciting dynamo action. Indeed, it provided the very first quantitative evidence for the possibility of the geodynamo. Despite its utility, due to the requirement of resolving fine structures, historically, numerical work has proven difficult and reported solutions were often plagued by poor convergence. In this paper, we demonstrate the numerical superiority of a Galerkin scheme in solving the kinematic dynamo eigenvalue problem in a full sphere. After adopting a poloidal–toroidal decomposition and expanding in spherical harmonics, we express the radial dependence in terms of a basis of exponentially convergent orthogonal polynomials. Each basis function is constructed from a terse sum of one-sided Jacobi polynomials that not only satisfies the boundary conditions of matching to an electrically insulating exterior, but is everywhere infinitely differentiable, including at the origin. This Galerkin method exhibits more rapid convergence, for a given problem size, than any other scheme hitherto reported, as demonstrated by a benchmark of the magnetic diffusion problem and by comparison to numerous kinematic dynamos from the literature. In the axisymmetric flows we consider in this paper, at a magnetic Reynolds number of O(100), a convergence of 9 significant figures in the most unstable eigenvalue requires only 40 radial basis functions; alternatively, 4 significant figures requires 20 radial functions. The terse radial discretization becomes particularly advantageous when considering flows whose associated numerical solution requires a large number of coupled spherical harmonics. We exploit this new method to confirm the tentatively proposed positive growth rate of the planar flow of Bachtiar et al. [4], thereby verifying a counter-example to the Zel’dovich anti-dynamo theorem in a spherical geometry.     

Structure of the electric field in the skin-effect problem

The structure of the electric field in a metal has been elucidated for the skin-effect problem. It is demonstrated that the electric field is the sum of the integral term and two (or one) exponentially decreasing particular solutions to the initial system and that one particular solution disappears depending on the anomality parameter. An expression for the distribution function profile in the half-space and at the metal boundary is obtained in the explicit form. The absolute value, the real part, and the imaginary part of the electric filed are analyzed in the case of the anomalous skin effect.     

Self-similar turbulent dynamo

The amplification of magnetic fields in a highly conducting fluid is studied numerically. During growth, the magnetic field is spatially intermittent: it does not uniformly fill the volume, but is concentrated in long thin folded structures. Contrary to a commonly held view, intermittency of the folded field does not increase indefinitely throughout the growth stage if diffusion is present. Instead, as we show, the probability-density function (PDF) of the field-strength becomes self-similar. The normalized moments increase with magnetic Prandtl number in a powerlike fashion. We argue that the self-similarity is to be expected with a finite flow scale and system size. In the nonlinear saturated state, intermittency is reduced and the PDF is exponential. Parallels are noted with self-similar behavior recently observed for passive-scalar mixing and for map dynamos.     

Solitary dynamo waves

Long dynamo waves are a characteristic feature of interface dynamo models with spatially localized α and Ω effects. The evolution of such waves is described by the modified Korteweg–de Vries equation. Solutions to this equation take the form of solitary waves, breathers, and snoidal and cnoidal waves, and represent nonlinear waves of magnetic activity that migrate towards the equator, as observed on the Sun. Averaging techniques extend the theory to longer times and relate the amplitude of these waves to the dynamo number.     

The solar dynamo

We shortly review the basic observational facts concerning the solar dynamo. Then, a brief overview of our current understanding of the large-scale evolution of the magnetic field of the Sun is proposed, showing, in particular, some successes and difficulties of the mean-field models. We illustrate the complications of this problem with recent work on stellar dynamos. We also compare the solar situation to that of the core of the Earth as well as those of laboratory and numerical experiments. To cite this article: M. Rieutord, C. R. Physique 9 (2008).     

Chaos in the segmented disc dynamo

It is shown that the equations describing the segmented disc dynamo in the presence of friction are identical to the Lorenz equations.     

Universal nonlinear small-scale dynamo

We consider astrophysically relevant nonlinear MHD dynamo at large Reynolds numbers (Re). We argue that it is universal in a sense that magnetic energy grows at a rate which is a constant fraction C(E) of the total turbulent dissipation rate. On the basis of locality bounds we claim that this "efficiency of the small-scale dynamo", C(E), is a true constant for large Re and is determined only by strongly nonlinear dynamics at the equipartition scale. We measured C(E) in numerical simulations and observed a value around 0.05 in the highest resolution simulations. We address the issue of C(E) being small, unlike the Kolmogorov constant which is of order unity.     

On the lyman problem

This paper attempts to answer Lyman's question (1990) on the non-uniqueness in defining the 3D measure of the boundary vorticity-creation rate. Firstly, a straightforward analysis of the vorticity equation introduces a definition of a general vorticity flux-density tensor and its ‘effective’ part. The approach is strictly based on classical field theory and is independent of the constitutive structure of continuous medium. Secondly, the fundamental question posed by Lyman dealing with the ambiguity of the 3D measure of the boundary vorticity-creation rate for incompressible flow is discussed. It is shown that the original 3D measure (for an incompressible Newtonian fluid defined by Panton 1984), which is reminiscent of an analogy to Fourier's law, is in its character ‘effective’ and plays a crucial role in the prognostic vorticity transport equation. The alternative 3D measure proposed by Lyman includes, on the other hand, a ‘non-effective’ part, which plays a role in the local determination of the ‘effective’ measure as well as in a certain diagnostic integral boundary condition.     

Landau problem with a general time-dependent electric field

The time evolution is studied for the Landau level problem with a general time dependent electric field E(t) in a plane perpendicular to the magnetic field. A general and explicit factorization of the time evolution operator is obtained with each factor having a clear physical interpretation. The factorization consists of a geometric factor (path-ordered magnetic translation), a dynamical factor generated by the usual time-independent Landau Hamiltonian, and a nonadiabatic factor that determines the transition probabilities among the Landau levels. Since the path-ordered magnetic translation and the nonadiabatic factor are, up to completely determined numerical phase factors, just ordinary exponentials whose exponents are explicitly expressible in terms of the canonical variables, all of the factors in the factorization are explicitly constructed. New quantum interference effects are implied by this result. The factorization is unique from the point of view of the quantum adiabatic theorem and provides a seemingly first rigorous demonstration of how the quantum adiabatic theorem (incorporating the Berry phase phenomenon) is realized when infinitely degenerate energy levels are involved. Since the factorization separates the effect caused by the electric field into a geometric factor and a nonadiabatic factor, it makes possible to calculate the nonadiabatic transition probabilities near the adiabatic limit. A formula for matrix elements that determines the mixing of the Landau levels for a general, nonadiabatic evolution is also provided by the factorization.     

Infrared problem in QED and electric charge renormalization

The electric charge renormalization in quantum electrodynamics is discussed, by taking into account the fact that an “infrared dressing transformation” is needed to go from the local states occurring in the Green's functions, to the (physical) charged states which obey Gauss' law. Apparent difficulties discussed in the literature are resolved. The construction of physical multicharged states is discussed explicitly.     

Saturated state of the nonlinear small-scale dynamo

We consider the problem of incompressible, forced, nonhelical, homogeneous, and isotropic MHD turbulence with no mean magnetic field and large magnetic Prandtl number. This type of MHD turbulence is the end state of the turbulent dynamo, which generates folded fields with small-scale direction reversals. We propose a model in which saturation is achieved as a result of the velocity statistics becoming anisotropic with respect to the local direction of the magnetic folds. The model combines the effects of weakened stretching and quasi-two-dimensional mixing and produces magnetic-energy spectra in remarkable agreement with numerical results at least in the case of a one-scale flow. We conjecture that the statistics seen in numerical simulations could be explained as a superposition of these folded fields and Alfvén-like waves that propagate along the folds.     

Magnetic structures produced by the small-scale dynamo

Small-scale dynamo action has been obtained for a flow previously used to model fluid turbulence, where the sensitivity of the magnetic field parameters to the kinetic energy spectrum can be explored. We apply quantitative morphology diagnostics, based on the Minkowski functionals, to magnetic fields produced by the kinematic small-scale dynamo to show that magnetic structures are predominantly filamentary rather than sheetlike. Our results suggest that the thickness, width, and length of the structures scale differently with magnetic Reynolds number as R(m)(-2/(1-s)) and R(m)(-0.55} for the former two, whereas the latter is independent of R(m), with s the slope of the energy spectrum.     

Subcritical dynamo bifurcation in the Taylor-Green flow

We report direct numerical simulations of dynamo generation for flow generated using a Taylor-Green forcing. We find that the bifurcation is subcritical and show its bifurcation diagram. We connect the associated hysteretic behavior with hydrodynamics changes induced by the action of the Lorentz force. We show the geometry of the dynamo magnetic field and discuss how the dynamo transition can be induced when an external field is applied to the flow.     

Influence of turbulence on the dynamo threshold

We use direct and stochastic numerical simulations of the magnetohydrodynamic equations to explore the influence of turbulence on the dynamo threshold. In the spirit of the Kraichnan-Kazantsev model, we model the turbulence by a noise, with given amplitude, injection scale, and correlation time. The addition of a stochastic noise to the mean velocity significantly alters the dynamo threshold and increases it for any noise at large scale. For small-scale noise, the result depends on its correlation time and on the magnetic Prandtl number.