An Introduction to Magnetohydrodynamics

Self-Organized Criticality

Few areas of geophysics are today progressing as rapidly as basic geomagnetism, which seeks to understand the origin of the Earth's magnetism. Data about the present geomagnetic field pours in from orbiting satellites, and supplements the ever growing body of information about the field in the remote past, derived from the magnetism of rocks. The first of the three parts of this review summarizes the available geomagnetic data and makes significant inferences about the large scale structure of the geomagnetic field at the surface of the Earth's electrically conducting fluid core, within which the field originates. In it, we recognize the first major obstacle to progress: because of the Earth's mantle, only the broad, slowly varying features of the magnetic field within the core can be directly observed. The second (and main) part of the review commences with the geodynamo hypothesis: the geomagnetic field is induced by core flow as a self-excited dynamo. Its electrodynamics define 'kinematic dynamo theory'. Key processes involving the motion of magnetic field lines, their diffusion through the conducting fluid, and their reconnection are described in detail. Four kinematic models are presented that are basic to a later section on successful dynamo experiments. The fluid dynamics of the core is considered next, the fluid being driven into motion by buoyancy created by the cooling of the Earth from its primordial state. The resulting flow is strongly affected by the rotation of the Earth and by the Lorentz force, which alters fluid motion by the interaction of the electric current and magnetic field. A section on 'magnetohydrodynamic (MHD) dynamo theory' is devoted to this rotating magnetoconvection. Theoretical treatment of the MHD responsible for geomagnetism culminates with numerical solutions of its governing equations. These simulations help overcome the first major obstacle to progress, but quickly meet the second: the dynamics of Earth's core are too complex, and operate across time and length scales too broad to be captured by any single laboratory experiment, or resolved on present-day computers. The geophysical relevance of the experiments and simulations is therefore called into question. Speculation about what may happen when computational power is eventually able to resolve core dynamics is given considerable attention. The final part of the review is a postscript to the earlier sections. It reflects on the problems that geodynamo theory will have to solve in the future, particularly those that core turbulence presents.

https://iopscience.iop.org/article/10.1088/0034-4885/76/9/096801/pdf

On the genesis of the Earth's magnetism

Turbulent rotating convection is an important dynamical process occurring on nearly all planetary and stellar bodies, influencing many observed features such as magnetic fields, atmospheric jets and emitted heat flux patterns. For decades, it has been thought that the importance of rotation's influence on convection depends on the competition between the two relevant forces in the system: buoyancy (non-rotating) and Coriolis (rotating). The force balance argument does not, however, accurately predict the transition from rotationally controlled to non-rotating heat transfer behaviour. New results from laboratory and numerical experiments suggest that the transition is in fact controlled by the relative thicknesses of the thermal (non-rotating) and Ekman (rotating) boundary layers. Turbulent rotating convection controls many observed features in stars and planets, such as magnetic fields. It has been argued that the influence of rotation on turbulent convection dynamics is governed by the ratio of the relevant global-scale forces: the Coriolis force and the buoyancy force. This paper presents results from laboratory and numerical experiments which exhibit transitions between rotationally dominated and non-rotating behaviour that are not determined by this global force balance. Instead, the transition is controlled by the relative thicknesses of the thermal (non-rotating) and Ekman (rotating) boundary layers. Turbulent rotating convection controls many observed features of stars and planets, such as magnetic fields, atmospheric jets and emitted heat flux patterns1,2,3,4,5,6. It has long been argued that the influence of rotation on turbulent convection dynamics is governed by the ratio of the relevant global-scale forces: the Coriolis force and the buoyancy force7,8,9,10,11,12. Here, however, we present results from laboratory and numerical experiments which exhibit transitions between rotationally dominated and non-rotating behaviour that are not determined by this global force balance. Instead, the transition is controlled by the relative thicknesses of the thermal (non-rotating) and Ekman (rotating) boundary layers. We formulate a predictive description of the transition between the two regimes on the basis of the competition between these two boundary layers. This transition scaling theory unifies the disparate results of an extensive array of previous experiments8,9,10,11,12,13,14,15, and is broadly applicable to natural convection systems.

Boundary Layer Control of Rotating Convection Systems

These lecture notes are based on a tutorial given in 2017 at a plasma physics winter school in Les Houches. Their aim is to provide a self-contained graduate-student level introduction to the theory and modelling of the dynamo effect in turbulent fluids and plasmas, blended with a review of current research in the field. The primary focus is on the physical and mathematical concepts underlying different (turbulent) branches of dynamo theory, with some astrophysical, geophysical and experimental context disseminated throughout the document. The text begins with an introduction to the rationale, observational and historical roots of the subject, and to the basic concepts of magnetohydrodynamics relevant to dynamo theory. The next two sections discuss the fundamental phenomenological and mathematical aspects of (linear and nonlinear) small- and large-scale MHD dynamos. These sections are complemented by an overview of a selection of current active research topics in the field, including the numerical modelling of the geo- and solar dynamos, shear dynamos driven by turbulence with zero net helicity, and MHD-instability-driven dynamos such as the magnetorotational dynamo. The difficult problem of a unified, self-consistent statistical treatment of small and large-scale dynamos at large magnetic Reynolds numbers is also discussed throughout the text. Finally, an excursion is made into the relatively new but increasingly popular realm of magnetic-field generation in weakly-collisional plasmas. A short discussion of the outlook and challenges for the future of the field concludes the presentation.

Dynamo theories.

Recent years have seen an increased interest in the question of whether the gravitational action of planets could have an influence on the solar dynamo. Without discussing the observational validity of the claimed correlations, we examine which possible physical mechanism might link the weak planetary forces with solar dynamo action. We focus on the helicity oscillations that were recently found in simulations of the current-driven, kink-type Tayler instability, which is characterized by an $m=1$ azimuthal dependence. We show how these helicity oscillations may be resonantly excited by some $m=2$ perturbations that reflect a tidal oscillation. Specifically, we speculate that the tidal oscillation of 11.07 years induced by the Venus–Earth–Jupiter system may lead to a 1:1 resonant excitation of the oscillation of the $\alpha$-effect. Finally, we recover a 22.14-year cycle of the solar dynamo in the framework of a reduced zero-dimensional $\alpha$–$\Omega$ dynamo model.

Synchronized Helicity Oscillations: A Link Between Planetary Tides and the Solar Cycle?

Recent years have seen an increased interest in the question of whether the gravitational action of planets could have an influence on the solar dynamo. Without discussing the observational validity of the claimed correlations, we ask for a possible physical mechanism that might link the weak planetary forces with solar dynamo action. We focus on the helicity oscillations that were recently found in simulations of the current-driven, kink-type Tayler instability, which is characterized by an m=1 azimuthal dependence. We show how these helicity oscillations can be resonantly excited by some m=2 perturbation that reflects a tidal oscillation. Specifically, we speculate that the 11.07 years tidal oscillation induced by the Venus--Earth--Jupiter system may lead to a 1:1 resonant excitation of the oscillation of the alpha-effect. Finally, in the framework of a reduced, zero-dimensional alpha--Omega dynamo model we recover a 22.14-year cycle of the solar dynamo.

Synchronized helicity oscillations: a link between planetary tides and the solar cycle?

The DREsden Sodium facility for DYNamo and thermohydraulic studies (DRESDYN) is intended as a platform both for large scale experiments related to geo- and astrophysics as well as for experiments related to thermohydraulic and safety aspects of liquid metal batteries and liquid metal fast reactors. The most ambitious projects in the framework of DRESDYN are a homogeneous hydromagnetic dynamo driven solely by precession and a large Taylor-Couette type experiment for the combined investigation of the magnetorotational instability and the Tayler instability. In this paper we give a short summary about the ongoing preparations and delineate the next steps for the realization of DRESDYN.

DRESDYN -- a new facility for MHD experiments with liquid sodium

We discuss a solar dynamo model of Tayler-Spruit type whose Omega-effect is conventionally produced by a solar-like differential rotation but whose alpha-effect is assumed to be periodically modulated by planetary tidal forcing. This resonance-like effect has its rationale in the tendency of the current-driven Tayler instability to undergo intrinsic helicity oscillations which, in turn, can be synchronized by periodic tidal perturbations. Specifically, we focus on the 11.07 years alignment periodicity of the tidally dominant planets Venus, Earth, and Jupiter, whose persistent synchronization with the solar dynamo is briefly touched upon. The typically emerging dynamo modes are dipolar fields, oscillating with a 22.14 years period or pulsating with a 11.07 years period, but also quadrupolar fields with corresponding periodicities. In the absence of any constant part of alpha, we prove the subcritical nature of this Tayler-Spruit type dynamo. The resulting amplitude of the alpha oscillation that is required for dynamo action turns out to lie in the order of 1 m/s, which seems not implausible for the sun. When starting with a more classical, non-periodic part of alpha, even less of the oscillatory alpha part is needed to synchronize the entire dynamo. Typically, the dipole solutions show butterfly diagrams, although their shapes are not convincing yet. Phase coherent transitions between dipoles and quadrupoles, which are reminiscent of the observed behaviour during the Maunder minimum, can be easily triggered by long-term variations of dynamo parameters, but may also occur spontaneously even for fixed parameters. Further interesting features of the model are the typical second intensity peak and the intermittent appearance of reversed helicities in both hemispheres.

A model of a tidally synchronized solar dynamo

A crucial point for the understanding of the von K\'arm\'an--sodium (VKS) dynamo experiment is the influence of soft-iron impellers. We present numerical simulations of a VKS-like dynamo with a localized permeability distribution that resembles the shape of the flow driving impellers. It is shown that the presence of soft-iron material essentially determines the dynamo process in the VKS experiment. An axisymmetric magnetic field mode can be explained by the combined action of the soft-iron disk and a rather small $\ensuremath{\alpha}$ effect parametrizing the induction effects of unresolved small scale flow fluctuations.

André Giesecke

Papers

Synchronized Helicity Oscillations: A Link Between Planetary Tides and the Solar Cycle?

Synchronized helicity oscillations: a link between planetary tides and the solar cycle?

DRESDYN -- a new facility for MHD experiments with liquid sodium

A model of a tidally synchronized solar dynamo

Role of soft-iron impellers on the mode selection in the von kármán-sodium dynamo experiment.