Kinematic Dynamo Problem Research Articles

Relaxation of the kinematic dynamo equations

We compute the exact relaxation and Λ \Lambda -convex hull of the kinematic dynamo equations and show that they coincide. We also find the relaxation in the stationary case and present applications to the infinite-time limit in the kinematic dynamo problem.

Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. III. Asymptotics of branching

The previous paper of this series presented the results of a numerical investigation of the dependence of the dominant growth rates of Bloch eigenmodes on the diffusivity parameters (the molecular viscosity ν and molecular magnetic diffusivity η) in three linear stability problems: the kinematic dynamo problem, and the hydrodynamic and MHD stability problems for steady spaceperiodic flows and MHD states. The dominant eigenmodes (i.e., the stability modes, whose growth rates are maximum over the wave vector q of the planar wave involved in the Bloch modes) comprise branches. In some branches, the dominant growth rates are attained for constant half-integer q. In all the three stability problems for parity-invariant steady states, offshoot branches, stemming from the branches of this type, were found, in which the dominant growth rates are attained for q depending on ν and/or η. We consider now such a branching of the dominant magnetic modes in the kinematic dynamo problem, where an offshoot stems from a branch of neutral eigenmodes for q = 0, and construct power series expansions for the offshoots and the associated eigenvalues of the magnetic induction operator near the point of bifurcation. We show that the branching occurs for the molecular magnetic diffusivities, for which the two eigenvalues of the eddy diffusivity operator become imaginary, and magnetic field generation by the mechanism of the negative eddy diffusivity ceases. The details of branching in the other linear stability problems under consideration are distinct.

Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. II. Numerical results

We consider Bloch eigenmodes of three linear stability problems: the kinematic dynamo problem, the hydrodynamic and MHD stability problem for steady space-periodic flows and MHD states comprised of randomly generated Fourier coefficients and having energy spectra of three types: exponentially decaying, Kolmogorov with a cut off, or involving a small number of harmonics (“big eddies”). A Bloch mode is a product of a field of the same periodicity as the perturbed state and a planar harmonic wave, exp(iq · x). Such a mode is characterized by the ratio of spatial scales which, for simplicity, we identify with the length |q| < 1 of the Bloch wave vector q. Computations have revealed that the Bloch modes, whose growth rates are maximum over q, feature the scale ratio that decreases on increasing the nondimensionalized molecular diffusivity and/or viscosity from 0.03 to 0.3, and the scale separation is high (i.e., |q| is small) only for large molecular diffusivities. Largely this conclusion holds for all the three stability problems and all the three energy spectra types under consideration. Thus, in a natural MHD system not affected by strong diffusion, a given scale range gives rise to perturbations involving only moderately larger spatial scales (i.e., |q| only moderately small), and the MHD evolution consists of a cascade of processes, each generating a slightly larger spatial scale; flows or magnetic fields characterized by a high scale separation are not produced. This cascade is unlikely to be amenable to a linear description. Consequently, our results question the allegedly high role of the α-effect and eddy diffusivity that are based on spatial scale separation, as the primary instability or magnetic field generating mechanisms in astrophysical applications. The Braginskii magnetic α-effect in a weakly non-axisymmetric flow, often used for explanation of the solar and geodynamo, is advantageous not being upset by a similar deficiency.

Линейные возмущения Блоховского типа пространственно-периодических магнитогидодинамических стационарных состояний. I. Математические вопросы

We consider Bloch eigenmodes in three linear stability problems: the kinematic dynamo problem, the hydrodynamic and MHD stability problem for steady space-periodic flows and MHD states. A Bloch mode is a product of a field of the same periodicity, as the state subjected to perturbation, and a planar harmonic wave, exp(iqx). The complex exponential cancels out from the equations of the respective eigenvalue problem, and the wave vector q remains in the equations as a numeric parameter. The resultant problem has a significant advantage from the numerical viewpoint: while the Bloch mode involves two independent spatial scales, its growth rate can be computed in the periodicity box of the perturbed state. The three-dimensional space, where q resides, splits into a number of regions, inside which the growth rate is a smooth function of q. In preparation for a numerical study of the dominant (i.e., the largest over q) growth rates, we have derived expressions for the gradient of the growth rate in q and proven that, for parity-invariant flows and MHD steady states or when the respective eigenvalue of the stability operator is real, half-integer q (whose all components are integer or half-integer) are stationary points of the growth rate. In prior works it was established by asymptotic methods that high spatial scale separation (small q) gives rise to the phenomena of the α-effect or, for parity-invariant steady states, of the eddy diffusivity. We review these findings tailoring them to the prospective numerical applications.

A set of codes for numerical convection and geodynamo calculations

Abstract We present a set of codes for calculating and displaying solutions to diverse problems within thermal convection and magnetic field generation in rotating fluid-filled spheres and spherical shells. There are diverse programs for the kinematic dynamo problem, the onset of thermal convection, and boundary-locked thermal convection, and time-stepping codes for non-magnetic convection and the dynamo with either homogeneous or spatially varying thermal boundary conditions. Where possible, all programs have been benchmarked against other codes and tested by reproducing previously published results. Each program comes with the complete source code, a pdf instruction manual, and at least one example run with a sample input file and all necessary files for describing an initial condition. The only prerequisite for running most of the codes is a FORTRAN compiler. The plotting programs require in addition the PGPLOT graphics library. All source code, examples, input files, solutions, and instructions are available for download from github and Zenodo.

Nonlinear laminar `dynamos' linear in one coordinate

We consider two self-similar flow fields, for which the Navier--Stokes equations reduce to ODEs. If the magnetic field has a similar structure, then the induction equation also reduces to ODEs. The coupled system can be regarded as a kinematic dynamo problem, but also the fields grow until they saturate as exact solutions of the nonlinear system. The extent to which these solutions can be regarded as genuine dynamos is discussed. Figs 6, Refs 13.

Kinematic dynamos in triaxial ellipsoids

Planetary magnetic fields are generated by motions of electrically conducting fluids in their interiors. The dynamo problem has thus received much attention in spherical geometries, even though planetary bodies are non-spherical. To go beyond the spherical assumption, we develop an algorithm that exploits a fully spectral description of the magnetic field in triaxial ellipsoids to solve the induction equation with local boundary conditions (i.e. pseudo-vacuum or perfectly conducting boundaries). We use the method to compute the free-decay magnetic modes and to solve the kinematic dynamo problem for prescribed flows. The new method is thoroughly compared with analytical solutions and standard finite-element computations, which are also used to model an insulating exterior. We obtain dynamo magnetic fields at low magnetic Reynolds numbers in ellipsoids, which could be used as simple benchmarks for future dynamo studies in such geometries. We finally discuss how the magnetic boundary conditions can modify the dynamo onset, showing that a perfectly conducting boundary can strongly weaken dynamo action, whereas pseudo-vacuum and insulating boundaries often give similar results.

Computation of Kinematic and Magnetic α-Effect and Eddy Diffusivity Tensors by Padé Approximation

We present examples of Padé approximations of the α -effect and eddy viscosity/diffusivity tensors in various flows. Expressions for the tensors derived in the framework of the standard multiscale formalism are employed. Algebraically, the simplest case is that of a two-dimensional parity-invariant six-fold rotation-symmetric flow where eddy viscosity is negative, indicating intervals of large-scale instability of the flow. Turning to the kinematic dynamo problem for three-dimensional flows of an incompressible fluid, we explore the application of Padé approximants for the computation of tensors of magnetic α -effect and, for parity-invariant flows, of magnetic eddy diffusivity. We construct Padé approximants of the tensors expanded in power series in the inverse molecular diffusivity 1 / η around 1 / η = 0 . This yields the values of the dominant growth rate to satisfactory accuracy for η , several dozen times smaller than the threshold, above which the power series is convergent. We do computations in Fortran in the standard “double” (real*8) and extended “quadruple” (real*16) precision, and perform symbolic calculations in Mathematica.

Kazantsev dynamo in turbulent compressible flows.

We consider the kinematic fluctuation dynamo problem in a flow that is random, white-in-time, with both solenoidal and potential components. This model is a generalization of the well-studied Kazantsev model. If both the solenoidal and potential parts have the same scaling exponent, then, as the compressibility of the flow increases, the growth rate decreases but remains positive. If the scaling exponents for the solenoidal and potential parts differ, in particular if they correspond to typical Kolmogorov and Burgers values, we again find that an increase in compressibility slows down the growth rate but does not turn it off. The slow down is, however, weaker and the critical magnetic Reynolds number is lower than when both the solenoidal and potential components display the Kolmogorov scaling. Intriguingly, we find that there exist cases, when the potential part is smoother than the solenoidal part, for which an increase in compressibility increases the growth rate. We also find that the critical value of the scaling exponent above which a dynamo is seen is unity irrespective of the compressibility. Finally, we realize that the dimension d = 3 is special, as for all other values of d the critical exponent is higher and depends on the compressibility.

Growth rate distribution and intermittency in kinematic turbulent dynamos: Which moment predicts the dynamo onset?

We consider the generation of magnetic field by a turbulent flow. For the linear induction equation (i.e., the kinematic dynamo problem), we show that the statistical moments of the magnetic field display multiscaling and in particular moments of different order turn unstable for different values of the control parameter. On a canonical example, we map the problem onto the calculation of the injected power by a time correlated fluctuating force acting on a Brownian particle. We are then able to calculate analytically the growth rate of the moments of the magnetic field and explain the origin of this intermittency. We finally show that the onset for the nonlinear problem is predicted by the linear onset of the moment of order (i.e., the logarithm of the magnetic field).

The kinematic dynamo problem, part I: analytical treatment with the Bullard–Gellman formalism

This paper is dedicated to the description of kinematic dynamo action in a sphere and its analytical treatment with the Bullard–Gellman formalism. One goal of dynamo theory is to answer the question: Can magnetic fields of stellar objects be generated or sustained due to (fluid) motion in the interior? Bullard and Gellman were among the first to study this question, leading the way for many subsequent studies, cf. Bullard (Philos Trans R Soc A 247(928):213–278, 1954). In their publication the differential equations resulting from a toroidal–poloidal decomposition of the velocity and magnetic field are stated without an in-depth discussion of the employed methods and computation steps. This study derives the necessary formalism in a compact and concise manner by using an operator-based approach. The focus lies on the mathematical steps and necessary properties of the considered formalism. Prior to that a derivation of the induction equation is presented based on rational continuum electrodynamics. As an example of the formalism the decay of two magnetic fields is analyzed.

Kinematic dynamos in spheroidal geometries.

The kinematic dynamo problem is solved numerically for a spheroidal conducting fluid of possibly large aspect ratio with an insulating exterior. The solution method uses solenoidal representations of the magnetic field and the velocity by spheroidal toroidal and poloidal fields in a non-orthogonal coordinate system. Scaling of coordinates and fields to a spherical geometry leads to a modified form of the kinematic dynamo problem with a geometric anisotropic diffusion and an anisotropic current-free condition in the exterior, which is solved explicitly. The scaling allows the use of well-developed spherical harmonic techniques in angle. Dynamo solutions are found for three axisymmetric flows in oblate spheroids with semi-axis ratios 1≤a/c≤25. For larger aspect ratios strong magnetic fields may occur in any region of the spheroid, depending on the flow, but the external fields for all three flows are weak and concentrated near the axis or periphery of the spheroid.

Dynamo action in rapidly rotating Rayleigh–Bénard convection at infinite Prandtl number

In order better to understand the processes that lead to the generation of magnetic fields of finite amplitude, we study dynamo action driven by turbulent Boussinesq convection in a rapidly rotating system. In the limit of infinite Prandtl number (the ratio of viscous to thermal diffusion) the inertia term drops out of the momentum equation, which becomes linear in the velocity. This simplification allows a decomposition of the velocity into a thermal part driven by buoyancy, and a magnetic part driven by the Lorentz force. While the former velocity defines the kinematic dynamo problem responsible for the exponential growth of the magnetic field, the latter encodes the magnetic back reaction that leads to the eventual nonlinear saturation of the dynamo. We argue that two different types of solution should exist: weak solutions in which the saturated velocity remains close to the kinematic one, and strong solutions in which magnetic forces drive the system into a new strongly magnetised state that is radically different from the kinematic one. Indeed, we find both types of solutions numerically. Interestingly, we also find that, in our inertialess system, both types of solutions exist on the same subcritical branch of solutions bifurcating from the non-magnetic convective state, in contrast with the more traditional situation for systems with finite inertia in which weak and strong solutions are thought to exist on different branches. We find that for weak solutions, the force balance is the same as in the non-magnetic case, with the horizontal size of the convection varying as the one-third power of the Ekman number (the ratio of viscous to Coriolis forces), which gives rise to very small cells at small Ekman numbers (i.e. high rotation rates). In the strong solutions, magnetic forces become important and the convection develops on much larger horizontal scales. However, we note that even in the strong cases the solutions never properly satisfy Taylor’s constraint, and that viscous stresses continue to play a role. Finally, we discuss the relevance of our findings to the study of planetary dynamos in rapidly rotating systems such as the Earth.

Kinematic dynamo in a tetrahedron composed of helical Fourier modes

It is generally believed that helicity can play a significant role in turbulent systems, e.g. supporting the generation of large-scale magnetic fields, but its exact contribution is not clearly understood. For example there are well-known examples of large scale dynamos produced by a flow which is pointwise non-helical. In any case a break of mirror symmetry seems to be always at the heart of the dynamo mechanism. A fruitful framework to analyze such processes is the use of helical mode decomposition. In pure hydrodynamics such framework has proved its availability in study of the processes responsible for helicity cascades. It has also been used in the analysis of MHD helical mode interactions. The present work deals with the kinematic dynamo problem, solving the induction equation within the framework of helical Fourier modes decomposition. We show that the simplest modes configuration leading to an unstable solution has the form of a tetrahedron. Then the dynamo is produced by only two scales flow. We find necessary conditions for such dynamo action, not certainly related to flow helicity. The results help to understand generic dynamo flows like the one studied by G.O. Roberts (1972).

The Axisymmetric Antidynamo Theorem Revisited

The axisymmetric kinematic dynamo problem is reconsidered, and a number of open questions are answered. Apart from axisymmetry and smoothness of data and solution we deal with this problem under quite general conditions; i.e., we assume a compressible fluid of variable (in space and time) conductivity moving in an arbitrary (axisymmetric) domain. We prove unconditional, pointwise, and exponential decay of the magnetic field and electric current to zero. The decay rate of the external (meridional) magnetic field can become very small (compared to free decay) for special flow fields and large magnetic Reynolds numbers, as we demonstrate with an example. On the other hand, we show for fluids with weak variation of mass density and conductivity that the meridional and azimuthal decay rates do not drop significantly below those of free decay.

Space–time behaviour of magnetic anomalies induced by tsunami waves in open ocean

The magnetic anomaly induced by an inhomogeneous velocity field under tsunami waves in open ocean is investigated. With asymptotical analysis, an explicit series solution of the kinematic dynamo problem is established for weak dispersive water waves. The magnetic field induced by typical tsunami models, including single wave and -wave, can be directly obtained using the proposed series solution. The characteristics of the magnetic field induced by two realistic tsunami events are investigated. By analysis, the magnetic magnitude induced by a 1 m high tsunami is estimated as of the order of 10 nT at the sea surface, which depends on the wave parameters as well as the Earth's magnetic field. The space and time behaviour of the magnetic field shows fair similarity with the field data at Easter Island during the 2010 Chile tsunami.

Kinematic dynamo by large scale tsunami waves in open ocean

Kinematic dynamo problem is studied with tsunami motion in open oceans. Using long wave approximation, a series solution of the dynamo problem is established with fast convergent rate based on a small parameter relating water wave dispersive effects. Taking solitary wave and single wave as typical tsunami wave models, the magnitude of tsunami induced magnetic field is estimated at the order of 10 nano Tesla (nT) just over sea level and 1 nT at altitudes of several hundreds kilometers, respectively, depending on the wave parameters as well as earth magnetic field. The space and time behavior of the magnetic field predicted by present model shows fairly similarity with the field data at Easter Island during 2010 Chile tsunami.

Well‐posedness of the kinematic dynamo problem

In the framework of magnetohydrodynamics, the generation of magnetic fields by the prescribed motion of a liquid conductor in a bounded region is described by the induction equation, a linear system of parabolic equations for the magnetic field components. Outside G, the solution matches continuously to some harmonic field that vanishes at spatial infinity. The kinematic dynamo problem seeks to identify those motions, which lead to nondecaying (in time) solutions of this evolution problem. In this paper, the existence problem of classical (decaying or not) solutions of the evolution problem is considered for the case that G is a ball and for sufficiently regular data. The existence proof is based on the poloidal/toroidal representation of solenoidal fields in spherical domains and on the construction of appropriate basis functions for a Galerkin procedure. Copyright © 2012 John Wiley & Sons, Ltd.

Electromagnetic induction in non-uniform domains

Kinematic simulations of the induction equation are carried out for different setups suitable for the von-Kármán-Sodium (VKS) dynamo experiment. The material properties of the flow driving impellers are modeled by means of high-conducting and high-permeability disks in a cylindrical volume filled with a conducting fluid. Two entirely different numerical codes are mutually validated by showing quantitative agreement on Ohmic decay and kinematic dynamo problems using various configurations and physical parameters. Field geometry and growth rates are strongly modified by the material properties of the disks even if the disks are thin. In contrast the influence of external boundary conditions remains small. Utilizing a VKS like mean fluid flow and high-permeability disks yield a reduction of the critical magnetic Reynolds number Rmc for the onset of dynamo action of the simplest non-axisymmetric field mode. However, this threshold reduction is not sufficient to fully explain the VKS experiment. We show that this reduction of Rmc is influenced by small variations in the flow configuration so that the observed reduction may be changed with respect to small modifications of setup and properties of turbulence.

Stabilized finite element formulation applied to the kinematic Ponomarenko dynamo problem

A stabilized finite element (B, q) formulation is developed to solve the kinematic dynamo problem. As a test case, we solve the induction equation for a given solid body helical flow, embedded in a cylindrical conducting shell. This problem corresponds to the well-known Ponomarenko dynamo. It has the interesting property to have an exact dispersion relation giving the magnetic growth rate as a function of the flow properties. Therefore, it is a good benchmark to test our kinematic dynamo code. We calculated the dynamo threshold and plotted the geometry of the generated magnetic field. We also evaluated the residual error due to our stabilized formulation.