Dynamo Experiment Research Articles

Nonlinear MHD dynamo operating at equipartition

Context. We present results from non linear MHD dynamo experiments with a three-dimensional steady and smooth flow that drives fast dynamo action in the kinematic regime. In the saturation regime, the system yields strong magnetic fields, which undergo transitions between an energy-equipartition and a turbulent state. The generation and evolution of such strong magnetic fields is relevant for the understanding of dynamo action that occurs in stars and other astrophysical objects. Aims. We study the mode of operation of this dynamo, in the linear and non-linear saturation regimes. We also consider the effect of varying the magnetic and fluid Reymolds number on the non-linear behaviour of the system. Methods. We perform three-dimensional non-linear MHD simulations and visualization using a high resolution numerical scheme. Results. We find that this dynamo has a high growth rate in the linear regime, and that it can saturate at a level significantly higher than intermittent turbulent dynamos, namely at energy equipartition, for high values of the magnetic and fluid Reynolds numbers. The equipartition solution however does not remain time-independent during the simulation but exhibits a much more intricate behaviour than previously thought. There are periods in time where the solution is smooth and close to energy-equipartition and others where it becomes turbulent. Similarities and differences in the way the magnetic field is amplified and sustained for experiments with varying Reynolds numbers are discussed. Conclusions. Strong magnetic fields, in near equipartition, can be generated also by a non-turbulent dynamo. A striking result is that the saturation state of this dynamo reveals interesting transitions between turbulent and laminar states.

On the magnetic fields generated by experimental dynamos

We review the results obtained by three successful fluid dynamo experiments and discuss what has been learnt from them about the effect of turbulence on the dynamo threshold and saturation. We then discuss several questions that are still open and propose experiments that could be performed to answer some of them.

Forward and inverse problems in MHD: Numerical and experimental results

AbstractWhen a conducting fluid comes under the influence of a magnetic field, electrical currents are induced that give rise to a modification of this magnetic field. The ratio of induced magnetic field to applied magnetic field is characterized by the magnetic Reynolds number Rm of the flow. For large Rm, even self‐excitation of a magnetic field can occur. This hydromagnetic dynamo effect is responsible for the maintenance of the magnetic fields of planets, stars and galaxies. In the present paper, we delineate some recent developments in the numerical treatment of induction effects in arbitrary geometry, and their application for dynamo experiments as well as for a “Contactless Inductive Flow Tomography (CIFT)”. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Numerical simulations of current generation and dynamo excitation in a mechanically forced turbulent flow

The role of turbulence in current generation and self-excitation of magnetic fields has been studied in the geometry of a mechanically driven, spherical dynamo experiment, using a three-dimensional numerical computation. A simple impeller model drives a flow that can generate a growing magnetic field, depending on the magnetic Reynolds number Rm=micro0sigmaVa and the fluid Reynolds number Re=Vanu of the flow. For Re<420, the flow is laminar and the dynamo transition is governed by a threshold of Rmcrit=100, above which a growing magnetic eigenmode is observed that is primarily a dipole field transverse to the axis of symmetry of the flow. In saturation, the Lorentz force slows the flow such that the magnetic eigenmode becomes marginally stable. For Re>420 and Rm approximately 100 the flow becomes turbulent and the dynamo eigenmode is suppressed. The mechanism of suppression is a combination of a time varying large-scale field and the presence of fluctuation driven currents (such as those predicted by the mean-field theory), which effectively enhance the magnetic diffusivity. For higher Rm, a dynamo reappears; however, the structure of the magnetic field is often different from the laminar dynamo. It is dominated by a dipolar magnetic field aligned with the axis of symmetry of the mean-flow, which is apparently generated by fluctuation-driven currents. The magnitude and structure of the fluctuation-driven currents have been studied by applying a weak, axisymmetric seed magnetic field to laminar and turbulent flows. An Ohm's law analysis of the axisymmetric currents allows the fluctuation-driven currents to be identified. The magnetic fields generated by the fluctuations are significant: a dipole moment aligned with the symmetry axis of the mean-flow is generated similar to those observed in the experiment, and both toroidal and poloidal flux expulsion are observed.

Coupled fluid-flow and magnetic-field simulation of the Riga dynamo experiment

Magnetic fields of planets, stars, and galaxies result from self-excitation in moving electroconducting fluids, also known as the dynamo effect. This phenomenon was recently experimentally confirmed in the Riga dynamo experiment [A. Gailitis et al., Phys. Rev. Lett. 84, 4365 (2000); A. Gailitis et al., Physics of Plasmas 11, 2838 (2004)], consisting of a helical motion of sodium in a long pipe followed by a straight backflow in a surrounding annular passage, which provided adequate conditions for magnetic-field self-excitation. In this paper, a first attempt to simulate computationally the Riga experiment is reported. The velocity and turbulence fields are modeled by a finite-volume Navier-Stokes solver using a Reynolds-averaged-Navier-Stokes turbulence model. The magnetic field is computed by an Adams-Bashforth finite-difference solver. The coupling of the two computational codes, although performed sequentially, provides an improved understanding of the interaction between the fluid velocity and magnetic fields in the saturation regime of the Riga dynamo experiment under realistic working conditions.

Low-dimensional dynamo modelling and symmetry-breaking bifurcations

Motivated by the successful Karlsruhe dynamo experiment, a relatively low-dimensional dynamo model is proposed. It is based on a strong truncation of the magnetohydrodynamic (MHD) equations with an external forcing of the Roberts type and the requirement that the model system satisfies the symmetries of the full MHD system, so that the first symmetry-breaking bifurcations can be captured. The backbone of the Roberts dynamo is formed by the Roberts flow, a helical mean magnetic field and another part of the magnetic field coupled to these two by triadic mode interactions. A minimum truncation model (MTM) containing only these energetically dominating primary mode triads is fully equivalent to the widely used first-order smoothing approximation. However, it is shown that this approach works only in the limit of small wave numbers of the excited magnetic field or small magnetic Reynolds numbers ( R m ≪ 1 ). To obtain dynamo action under more general conditions, secondary mode triads must be taken into account. Altogether a set of six primary and secondary mode types is found to be necessary for an optimum truncation model (OTM), corresponding to a system of 152 ordinary differential equations. In a second step, the OTM is used to study symmetry-breaking bifurcations on its route to chaos, with the Reynolds number or strength of the driving force as the control parameter. A decisive role in this scenario is played by a symmetry of the form of Z 2 × S 1 resulting from the Z 2 reflection symmetry of the magnetic field in the MHD equations in conjunction with a circle symmetry S 1 of the Roberts flow. Under its influence, in a secondary Hopf bifurcation from a circle of steady reflection-symmetric states a time-periodic solution branch of oscillating waves (OW) is generated retaining the reflection symmetry, however in a spatio-temporal manner. Finally, the subsequent bifurcations on the route to chaos are examined.

A paradigmatic model of Earth's magnetic field reversals

Introduction. There is an ample paleomagnetic evidence that the Earth’s magnetic field has reversed its polarity many times. The last reversal occurred approximately 780000 years ago.The mean rate of reversals varies from nearly zero during the Permian and Cretaceous superchrons to approximately 5 per Myr in the present. Some observations suggest that the decay of the dipole of one polarity might be much slower than the following recreation of the dipole with opposite polarity [1]. Observational data also indicate a possible correlation of the dipole moment with the persistence time of the field in one polarity [2]. In a recent paper, a bimodal distribution of the dipole moment has been hypothesized [3]. Although some of the above mentioned features are controversially discussed in the literature, it is worthwhile to ask if and how they could be modeled by theory. With view on the successful dynamo experiments in Riga and Karlsruhe [4], one could extend this question and ask what would be the most essential ingredient for a dynamo experiment to exhibit irregular reversals in a similar way as the Earth’s dynamo does. Thus motivated, we focus in this paper on a very generic mechanism of field reversals and study it in detail by means of an extremely simple dynamo model. It is well known that the non-self adjoint dynamo operator can provide transitions between non-oscillatory and oscillatory eigenmodes. Typically, this transition occurs at the so-called ”exceptional points” [5] of the spectrum, where the eigenvalues and the eigenfunctions of two non-oscillatory modes coalesce and continue as a pair of oscillatory modes with complex conjugate eigenvalues. It is the goal of the present paper to show that even within a very simple mean-field dynamo model the main characteristics of reversals can be attributed to the magnetic field dynamics in the vicinity of such exceptional points.

Fluctuation of magnetic induction in von Kármán swirling flows

Studies of magnetic induction in von Kármán swirling flows have so far linked the time-averaged induced magnetic field to the structure of the mean flow. They have evidenced the Omega and Parker mechanism generated, respectively, by the flow differential rotation and helicity, which underly the Dudley and James [N. L. Dudley and R. W. James, Proc. R. Soc. London, Ser. A 425, 407 (1989)] dynamos. Using an array of Hall probes we study here the dynamical regime. In the experimental flow, turbulence is fully developed and large fluctuations are observed in the magnetic induction processes. We find that the large scale turbulent fluctuations have different characteristics when induction results from the differential rotation or from the dynamics of the stagnation point in the midplane of the von Kármán flow. Symmetry considerations indicate that the dynamical flow spends half of its time away from the time-averaged structure. The consequences of these observations for dynamo experiments are discussed.

Marginal stability of almost adiabatic planetary convection

Almost adiabatic states are typical for the deep convective interiors of all known planets and their moons, e.g., the deviations from the adiabatic state in the Earth's outer core and in the MHD dynamo region of Jupiter are about or less than 10−5%. We approximated the equations governing convection in the deep interiors of planets and their moons to obtain a system, which is more accurate than the traditional Boussinesq equations. Fortunately, our system, which adopts almost uniform entropy instead of the temperature as the basic thermodynamic states, can still be investigated by standard methods. We considered the marginal stability of well-mixed almost adiabatic states in rapidly rotating thick spherical shells, whose inner to outer radius ratio does not exceed that of the modern Earth. The critical Rayleigh-type numbers, frequencies and solution structures of the marginal states were determined by both analytical and numerical methods. Our new estimates differ from those obtained previously using the Boussinesq equations, suggesting that the earlier Boussinesq results for convection in the deep planetary interiors should be re-assessed. The small molecular Prandtl number limit was adopted to model the marginal stability of thermal planetary convection. It was found that the critical Rayleigh number for convection sharply diminished as the radius of the inner rigid core is increased. We modelled the instability of the combined compositional–thermal turbulent geo-convection for Prandtl number unity. When thermal convection is in opposition to compositional convection, extremely large critical Rayleigh numbers are possible. This might happen for a terrestrial planet during its later stage of evolution. Pure compositional convection has been investigated in the large compositional Prandtl number limit, for which the critical Rayleigh number is rather large and the variations of all critical parameters are small. The large size of the critical Rayleigh number ensures that the actual values used in numerical dynamo experiments are only moderately supercritical.

Intermittent Magnetic Field Excitation by a Turbulent Flow of Liquid Sodium

The magnetic field measured in the Madison dynamo experiment shows intermittent periods of growth when an axial magnetic field is applied. The geometry of the intermittent field is consistent with the fastest-growing magnetic eigenmode predicted by kinematic dynamo theory using a laminar model of the mean flow. Though the eigenmodes of the mean flow are decaying, it is postulated that turbulent fluctuations of the velocity field change the flow geometry such that the eigenmode growth rate is temporarily positive. Therefore, it is expected that a characteristic of the onset of a turbulent dynamo is magnetic intermittency.

Dynamo and hydromagnetic liquid metal experiments

AbstractMagnetic dynamo experiments are being performed in many laboratories throughout the world. The most successful have been at Karlsruhe and Riga, where liquid sodium has been forced in constrained flows and has produced the predicted positive dynamo gain. The unconstrained flows are being investigated at Wisconsin, Cadarache, and Maryland. These experiments have not reached the critical values of the magnetic Reynolds number, Rm , necessary for dynamo gain, despite the positive predictions of laminar flow theory. Experiments in understanding the relationship of turbulence to MHD are being performed in addition to these at Perm, Princeton, and Swarthmore. A naturally constrained dynamo, using angular momentum gradient, Couette flow, to suppress turbulence is being attempted at New Mexico Tech. Flow constrained by ridged walls is likely to have less turbulence than unconstrained flow where the criterion of constraint is the degree of suppression of the Helmholtz shear flow instability. The study of MHD in conjunction with fluid turbulence has become of major importance in dynamo experiments as well as to future dynamo theory. (© 2006 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Do mean-field dynamos in nonrotating turbulent shear-flows exist?

A plane-shear flow in a fluid with forced turbulence is considered. If the fluid is electrically-conducting then a mean electromotive force (EMF) results even without basic rotation and the magnetic diffusivity becomes a highly anisotropic tensor. It is checked whether in this case self-excitation of a large-scale magnetic field is possible (so-called W × J-dynamo) and the answer is NO. The calculations reveal the cross-stream components of the EMF perpendicular to the mean current having the wrong signs, at least for small magnetic Prandtl numbers. After our results numerical simulations with magnetic Prandtl number of about unity have only a restricted meaning as the Prandtl number dependence of the diffusivity tensor is rather strong. If, on the other hand, the turbulence field is strati.ed in the vertical direction then a dynamo-active α -effect is produced. The critical magnetic Reynolds number for such a self-excitation in a simple shear flow is slightly above 10 like for the other – but much more complicated – flow patterns used in existing dynamo experiments with liquid sodium or gallium. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Measurements of the magnetic field induced by a turbulent flow of liquid metal

Initial results from the Madison Dynamo Experiment provide details of the inductive response of a turbulent flow of liquid sodium to an applied magnetic field. The magnetic field structure is reconstructed from both internal and external measurements. A mean toroidal magnetic field is induced by the flow when an axial field is applied, thereby demonstrating the omega effect. Poloidal magnetic flux is expelled from the fluid by the poloidal flow. Small-scale magnetic field structures are generated by turbulence in the flow. The resulting magnetic power spectrum exhibits a power-law scaling consistent with the equipartition of the magnetic field with a turbulent velocity field. The magnetic power spectrum has an apparent knee at the resistive dissipation scale. Large-scale eddies in the flow cause significant changes to the instantaneous flow profile resulting in intermittent bursts of nonaxisymmetric magnetic fields, demonstrating that the transition to a dynamo is not smooth for a turbulent flow.

On the effects of turbulence on a screw dynamo

In an experiment in the Institute of Continuous Media Mechanics in Perm (Russia) a non-stationary screw dynamo is intended to be realized with a helical flow of liquid sodium in a torus. The flow is necessarily turbulent, that is, may be considered as a mean flow and a superimposed turbulence. In this study, the induction processes of the turbulence are investigated within the framework of mean-field electrodynamics. They imply, of course, a part which leads to an enhanced dissipation of the mean magnetic field. As a consequence of the helical mean flow there are also helical structures in the turbulence. They lead to some kind of α-effect, which might basically support the screw dynamo. The peculiarity of this α-effect explains measurements made at a smaller version of the device envisaged for the dynamo experiment. The helical structures of the turbulence lead also to other effects, which in combination with a rotational shear are potentially capable of dynamo action. A part of them can basically support the screw dynamo. Under the conditions of the experiment all induction effects of the turbulence prove to be rather weak in comparison to that of the main flow. Numerical solutions of the mean-field induction equation show that all the induction effects of the turbulence together let the screw dynamo threshold slightly, at most by a few percent, rise. The numerical results also give some insights into the action of the individual induction effects of the turbulence.

Solar activity, Earth’s rotation rate and climate variations in the secular and semi-secular time scales

By applying the wavelet formalism to sudden storm commencements and aa geomagnetic indices and solar total irradiation, as a proxy data for solar sources of climate-forcing, we have searched the signatures of those variables on the Northern Hemisphere surface temperature. We have found that cyclical behaviour in surface temperature is not clearly related to none of these variables, so we have suggested that besides them surface temperature might be related to Earth’s rotation rate variations. Also it has been suggested that in the long-term Earth’s rotation rate variations might be excited by geomagnetic storm time variations which, in turn, depends on solar activity. The study of these phenomena and its relationships is addressed in the present paper. With this purpose we perform an analysis of the evolution during the last 350 years of the signals that conform the 11-year sunspot cycle maxima envelope for different time scales. The result is applied to analyze the relationship between long-term variations in sunspot number and excess of length of day; and to unearth the signature in surface temperature of this last from that of sudden storm commencement and aa geomagnetic indices and total solar irradiation. As solar dynamo experiments transient processes in all the time scales from seconds to centuries, Fourier analysis and its requirement for the process to be stationary is likely to produce spurious periodicities. We resort instead to wavelet formalism. By this representation we found that sunspot maxima envelope for the last 350 years may be described by means of the superposition of two cycles – a decadal and a semi-secular one – and a secular trend. And that the changing amplitude and phase of the cycles are well reconstructed using the superposition of two wavelets with nearby periods. The strong temporal changes in amplitude of the cycles facilitate to detect its phase in a given time series. It is found that a strong semi-secular cycle in the sunspot maxima envelope that started during the 1705 chaotic transition having its maximum amplitude at the Dalton minimum, was mapped 94 years later in Earth’s rotation rate and simultaneously in surface temperature. Also, there was a decrease (increase) of 0.022 °C for each millisecond of decrease (increase) in the Earth’s rotation period for the semi-secular cycle as well as the secular trend.

An optimal scale separation for a dynamo experiment

Scale separation between the flow and the magnetic field is a common feature of natural dynamos. It has also been used in the Karlsruhe sodium experiment in which the scale of the magnetic field is roughly seven times larger than the scale of the flow [R. Stieglitz and U. Müller, “Experimental demonstration of the homogeneous two-scale dynamo,” Phys. Fluids 13, 561 (2001)]. Recently, Fauve and Pétrélis [Peyresq Lectures on Nonlinear Phenomena, edited by J. Sepulchre (World Scientific, Singapore, 2003), p. 1] have shown that the power needed to reach the dynamo threshold in a dynamo experiment increases with the scale separation in the limit of large scale separation. With a more elaborate method based on subharmonic solutions [F. Plunian and K.-H. Rädler, “Subharmonic dynamo action in the Roberts flow,” Geophys. Astrophys. Fluid Dyn. 96, 115 (2002)], we show, for the Roberts flow, the existence of an optimal scale separation for which this power is minimum. Previous results obtained by Tilgner [“A kinematic dynamo with a small scale velocity field,” Phys. Lett. A 226, 75 (1997)] with a completely different numerical method are also reconsidered here. Again, we find an optimal scale separation in terms of minimum power for dynamo action. In addition we find that this scale separation compares very well with the one derived from the subharmonic solutions method.

Magnetic field oscillation in simulations of the Karlsruhe Dynamo

AbstractThe dynamo effect is numerically simulated for a model of the Karlsruhe dynamo. A flow periodic in two directions and independent of the third coordinate is considered. Each periodicity cell contains four helical vortices. The novelty of the model is that it includes cylindrical walls enclosing each vortex. These walls mimic the mechanical constraints inside the Karlsruhe experiment. Periodic oscillations of the magnetic field appear in the course of which the magnetic field pattern in each vortex rotates. The oscillations already exist in the kinematic regime. It is shown that the oscillations found in our simulations and oscillations observed in the Karlsruhe dynamo experiment must be of different origin. (© 2005 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

On the sensitivity of dynamo action to the system’s magnetic diffusivity

An experiment at the Karlsruhe dynamo test facility has shown that the intensity of the dynamo magnetic field decreases significantly with increasing temperature of the test fluid, liquid sodium, while keeping the other control parameters constant. This is explained by the increasing magnetic diffusivity with increasing temperatures utilizing results from a weakly nonlinear analysis of Tilgner and Busse [“Saturation mechanism in a model of the Karlsruhe dynamo,” in Dynamo and Dynamics, a Mathematical Challenge, NATO Science Series, II. Mathematics, Physics and Chemistry, Vol. 26, edited by P. Chossat, D. Armbruster, and I. Oprea (Kluwer, Dordrecht, 2001)] for the transcritical regime of the Karlsruhe dynamo experiment. The observations are also interpreted by dimensional reasoning utilizing previous experimental results.

Power requirement of the geodynamo from ohmic losses in numerical and laboratory dynamos

In the Earth's fluid outer core, a dynamo process converts thermal and gravitational energy into magnetic energy. The power needed to sustain the geomagnetic field is set by the ohmic losses (dissipation due to electrical resistance). Recent estimates of ohmic losses cover a wide range, from 0.1 to 3.5 TW, or roughly 0.3-10% of the Earth's surface heat flow. The energy requirement of the dynamo puts constraints on the thermal budget and evolution of the core through Earth's history. Here we use a set of numerical dynamo models to derive scaling relations between the core's characteristic dissipation time and the core's magnetic and hydrodynamic Reynolds numbers--dimensionless numbers that measure the ratio of advective transport to magnetic and viscous diffusion, respectively. The ohmic dissipation of the Karlsruhe dynamo experiment supports a simple dependence on the magnetic Reynolds number alone, indicating that flow turbulence in the experiment and in the Earth's core has little influence on its characteristic dissipation time. We use these results to predict moderate ohmic dissipation in the range of 0.2-0.5 TW, which removes the need for strong radioactive heating in the core and allows the age of the solid inner core to exceed 2.5 billion years.

Riga dynamo experiment and its theoretical background

It is widely believed that almost all magnetic fields in a natural environment are the result of the dynamo process—the field generation in moving nearly homogeneous electro-conducting fluids. This dynamo process occurs in the depths of celestial bodies such as the Earth, most of the planets, the Sun, other stars, and even galaxies. The Riga dynamo experiment is not intended as a model of any particular celestial body. It aims at demonstrating the basic dynamo mechanism—that the intense motion in a large volume of a good electro-conducting liquid creates a magnetic field. In the present paper, the set-up and the main results of this experiment are presented, with some focus on the theoretical interpretation of the data.