Dipole-quadrupole dynamics during magnetic field reversals
Christophe GissingerDepartment of Astrophysical Sciences, Princeton University, Princeton, New Jersey 08544, USA
Received 10 August 2010; published 4 November 2010
The shape and the dynamics of reversals of the magnetic field in a turbulent dynamo experiment areinvestigated. We report the evolution of the dipolar and the quadrupolar parts of the magnetic field in the VKSexperiment, and show that the experimental results are in good agreement with the predictions of a recentmodel of reversals: when the dipole reverses, part of the magnetic energy is transferred to the quadrupole,reversals begin with a slow decay of the dipole and are followed by a fast recovery, together with an overshootof the dipole. Random reversals are observed at the borderline between stationary and oscillatory dynamos.
DOI: 10.1103/PhysRevE.82.056302 PACS numbers: 47.65.d, 52.65.Kj, 91.25.Cw
Despite a large variability of the internal structure of plan-ets and stars, most of the observed astrophysical bodies pos-sess a coherent large scale magnetic field. It is widely ac-cepted that these natural magnetic fields are self-sustained bydynamo action 1. Although reversals of the magnetic fieldin planetary and stellar dynamos are now considered to be acommon feature, they still remain poorly understood.Whereas the Sun shows periodic oscillations of its magneticfield, the polarity of the Earths dipole field reverses ran-domly. During the last decades, several mechanisms havebeen proposed for geomagnetic reversals, among which wecan mention, the analogy with a bistable oscillator 2, amean-field dynamo model 3, or interaction between dipolarand higher axisymmetric components of the magnetic field4,5. The comprehension of dynamo reversals have alsobenefited from direct numerical simulations of the MHDequations, which have displayed several possible mecha-nisms for reversals 68.
Reversals of a dipolar magnetic field have also been re-ported in the VKS Von Karman Sodium dynamo experi-ment 9. In this experiment, periodic or chaotic flips of po-larity can be observed depending on the magnetic Reynoldsnumber. Based on these results, a model for reversals hasbeen recently proposed by Ptrlis and Fauve 11. It relieson the interaction between the dipolar and the quadrupolarmagnetic components, and describe transitions to periodicoscillations or randomly reversing dynamos. It has beenclaimed that such a mechanism could apply to the reversalsof the Earth magnetic field 12, and temptatively be con-nected to the periodic oscillations of the Solar dynamo. Un-fortunately, as for many other models, the lack of observa-tions of the magnetic field during a reversal limits a directcomparison with the actual geomagnetic reversals. From thispoint of view, the VKS experiment is a unique opportunity totest the validity of different models of turbulent reversingdynamos. In particular, the model 11 makes predictionsabout the dynamics that are easily confrontable to experi-mental results. We propose a simple way to analyze datafrom the VKS experiment in order to test this model. To wit,we extract from the data the dipolar and the quadrupolarcomponents of the magnetic field. We show that the charac-teristics of the reversals in the VKS experiment are in verygood agreement with the predictions, and that the dynamicsof the magnetic field in this turbulent dynamo mainly resultfrom an interaction between dipolar and quadrupolar modes.
In the VKS experiment, a turbulent von Karman flow ofliquid sodium is generated inside a cylinder by two counter-rotating impellers, with independent rotation frequencies F1and F2 see Fig. 1, and 13 for the description of the setup.When F1=F2, the system is invariant about a rotation of anangle around any axis located in the midplane. On thecontrary, if the impellers rotate at different rates, this sym-metry, hereafter referred to as the R symmetry, is broken.The dynamics of the magnetic field observed in the experi-ment strongly depend on this symmetry. When F1=F2, astatistically stationary magnetic field with either polarity isgenerated, with a dominant axial dipolar component. Dy-namical regimes, including periodic oscillations and chaoticreversals of the magnetic field, are observed only when theR symmetry is broken F1F2.
Previous studies have suggested that the evolution of themagnetic field in the VKS experiment results from low di-mensional dynamics, involving only a few modes in interac-tion 10. This can be ascribed to the proximity of the bifur-cation threshold and to the smallness of the magnetic Prandtlnumber in liquid metals Pm105. Indeed, in the low-Pmregime, the magnetic field is strongly damped compared tothe velocity field. Hence, the dynamics are governed only bya small number of magnetic modes. Based on this observa-tion, Ptrlis and Fauve 11 have proposed that close to thedynamo threshold, the magnetic field can be decomposedinto two components,
B = Dtdr + Qtqr , 1
where D respectively, Q represents the amplitude of dipolardr respectively, quadrupolar qr component of the field.
FIG. 1. Color online Sketch of the VKS experiment.
PHYSICAL REVIEW E 82, 056302 2010
1539-3755/2010/825/0563024 2010 The American Physical Society056302-1
As emphasized in 11, these components do not only in-volve a dipole or a quadrupole, but also all the higher com-ponents with the same symmetry in the transformation R. Inother words, dr respectively, qr is the antisymmetricrespectively, symmetric part of the magnetic field.
The evolution equations for Dt and Qt can then beobtained by symmetry arguments see 11 for a detaileddescription of the model. Since dd and qq in thetransformation R, dipolar and quadrupolar modes cannot belinearly coupled when F1=F2. Breaking the R symmetry byrotating the impellers at different speeds allows a linear cou-pling between dipolar and quadrupolar modes. For a suffi-ciently strong symmetry-breaking, this coupling can generatea limit cycle that involves an energy transfer between dipolarand quadrupolar modes. This mechanism has been recentlyvalidated on a numerical model of the VKS experiment 15and also in the case of a mean-field 2 dynamo model 17.
Two scenarios of transition from a stationary dynamo to aperiodically reversing magnetic field can be described in theframework of this dipole-quadrupole model. When the cou-pling is such that the system is close to both a stationary anda Hopf bifurcation, i.e., in the vicinity of a codimension-twobifurcation point, one can have bistability between a station-ary and a time-periodic reversing dynamo 13. We thus get asubcritical transition from a stationary dynamo to a periodicone with a finite frequency at onset. Turbulent fluctuationscan generate random transitions between these two regimes14. Far from this codimension-two point, a reversing mag-netic field can be generated through an Andronov bifurcationwhen the stationary state disappears through a saddle-nodebifurcation 11. Then, the frequency of the limit cycle van-ishes at onset. In the vicinity of this transition, turbulentfluctuations drive random reversals of the magnetic field. Asa consequence, random reversals always occur at the border-line between stationary and oscillatory dynamos. This simplemechanism also yields several predictions about the shape ofthe reversals. First, when the dipole D vanishes, part of themagnetic energy is transferred to the quadrupole Q. An over-shoot of the dipolar amplitude is expected after each reversal.Random reversals are asymmetric. During a first phase, fluc-tuations push the system from the stable solution to the un-stable one, thus acting against the deterministic dynamics.This phase is slow compared to the one beyond the unstablefixed point, where the system is driven to the opposite polar-ity under the action of the deterministic dynamics.
In this paper, we use data of the VKS experiment in orderto reconstruct the dipolar and the quadrupolar parts of themagnetic field and study their behavior in the time dependentregimes. Time-dependent regimes only occur for specificvalues of F1 and F2, inside three delimited regions of theparameter space 14. We will focus on the regimes observedwhen following the line F1 /F2=0.6 in the parameter space.Figure 2 shows that when the rotation rates are increasedalong this line, one first bifurcates to a stationary dynamo,then to time-dependent regimes. Figure 3top shows thetime recordings of the three components of the magneticfield close to the fastest disk, displaying the bifurcation fromstationary to time-dependent dynamo when the frequenciesof the two disks are increased from 14.4/24 Hz F1+F2=38.4 Hz to 15/25 Hz F1+F2=40 Hz. After a short tran-
sient state, the three components of the magnetic field un-dergo a transition to nearly periodic oscillations.
However, it is hard to test the pertinence of the model 11from the time recording of the magnetic field at a singlepoint. Using measurements obtained from two probes 1 and2, symmetric with respect to the midplane, we compute thedipolar part, Dtdi= Bi1, t+Bi2, t /2 and the quadrupo-lar part, Qtqi= Bi1, tBi2, t /2. In order to obtain ob-servables which are independent of the spatial component i,each of these vectors is projected on its value at a given timet0. We thus extract Dt and Qt up to a multiplicative con-stant. In the measurements displayed here, the dipolar andquadrupolar components are projected on their stationaryvalues obtained at F1+F2=38.4 Hz. Note however that dif-ferent methods could be used to reconstruct these ampli-tudes. In particular, plotting the sum and the difference of agiven component does not change the qualitative behavior19. Figure 3bottom shows the evolution of Dt and Qtduring periodic oscillations of the magnetic field. We