Magnetic Reversals in a Simple Model of Magnetohydrodynamics
Roberto Benzi1 and Jean-Francois Pinton2
1Dipartimento di Fisica and INFN, Universita Tor Vergata, Via della Ricerca Scientifica 1, I-00133 Roma, Italy2Laboratoire de Physique, CNRS and Ecole Normale Superieure de Lyon, 46 allee dItalie, F69007 Lyon, France
(Received 1 May 2009; published 8 July 2010)
We study a simple magnetohydrodynamical approach in which hydrodynamics and MHD turbulence
are coupled in a shell model, with given dynamo constraints in the large scales. We consider the case of a
low Prandtl number fluid for which the inertial range of the velocity field is much wider than that of the
magnetic field. Random reversals of the magnetic field are observed and it shown that the magnetic field
has a nontrivial evolutionlinked to the nature of the hydrodynamics turbulence.
DOI: 10.1103/PhysRevLett.105.024501 PACS numbers: 47.27.Ak, 91.25.Cw
The question of transitions between statistical solutionsis central to the behavior of many out-of-equilibrium sys-tems in physics and geophysics . As one particularexample addressed here, we note that natural dynamos areintrinsically dynamical. Formally, the coupled set of mo-mentum and induction equations is invariant under thetransform u;B ! u;B so that states with oppositepolarities can be generated from the same velocity field (uand B are, respectively, the velocity and magnetic fields).In the case of the geodynamo, polarity switches are calledreversals  and occur at very irregular time intervals .Such reversals have been observed recently in laboratoryexperiments using liquid metals, in arrangements wherethe dynamo cycle is either favored artificially  or stemsentirely from the fluid motions [4,8]. In these laboratoryexperiments, as also presumably in Earths core, the ratioof the magnetic diffusivity to the viscosity of the fluid(magnetic Prandtl number PM) is quite small. As a result,the kinetic Reynolds number RV of the flow is very highbecause its magnetic Reynolds number RM RVPM needsto be large enough so that the stretching of magnetic fieldlines balances the Joule dissipation. Hence, the dynamoprocess develops over a turbulent background, and in thiscontext it is often considered as a problem of bifurcationin the presence of noise. For the dynamo instability, theeffect of noise enters both additively and multiplicatively, asituation for which a complete theory is not currentlyavailable. Some specific features have been ascribed toits onset (e.g., bifurcation via an on-off scenario ) andto its dynamics . Turbulence also implies that pro-cesses occur over an extended range of scales; however,in a low magnetic Prandtl number fluid the hydrodynamicrange of scales is much wider than the magnetic one. Inlaboratory experiments, the induction processes that par-ticipate in the dynamo cycle involve the action of largescale velocity gradients [4,11,12], with possible contribu-tions of velocity fluctuations at small scales .
Building upon the above observations, we propose herea simple model which incorporates hydromagnetic fluctu-ations (as opposed to noise) in a dynamo instability. Ourgoal here is not to derive a low dimensional model from the
interaction of selected modes (see, for instance,  andreferences therein) but to assume the existence of suchlarge scale symmetry-breaking features and to investigatethe effects of turbulence fluctuations onto the dynamics ofreversals. The models stems from the approach introducedin  for the hydrodynamic studies.We consider an energy cascade model, i.e., a shell
model aimed at reproducing a few of the relevant charac-teristic features of the statistical properties of the Navier-Stokes equations . In a shell model, the basic variablesdescribing the velocity field at scale rn 2nr0 k1nis a complex number un satisfying a suitable set of non-linear equations (here r0 2). There are many versions ofshell models which have been introduced in literature.Here we choose the one referred to as Sabra shell model.Let us remark that the statistical properties of intermittentfluctuations, computed using either shell variables or theinstantaneous rate of energy dissipation, are in close quali-tative and quantitative agreement with those measured inlaboratory experiments, for homogeneous and isotropicturbulence . The MHD shell modelintroduced inallows a description of turbulence at low magneticPrandtl number since the steps of both cascades can befreely adjusted [20,21]. Although geometrical features arelost, this is a clear advantage over 3D simulations [22,23].We consider here a formulation extended from the Sabrahydrodynamic shell model:
i3nu; u nB; B k2nun fn; (1)
i3nu; B nB; u mk2nBn; (2)
where n 1; 2; . . . andnu;wkn11un2wn12un1wn2
for which, following , we chose 0:4. For this
PRL 105, 024501 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending9 JULY 2010
0031-9007=10=105(2)=024501(4) 024501-1 2010 The American Physical Society
value of , the Sabra model is known to show statisticalproperties (i.e., anomalous scaling) close to the ones ob-served in homogenous and isotropic turbulence. Themodel, without forcing and dissipation, conserves the ki-netic energy EV njunj2, the magnetic energy EB njBnj2, and the cross-helicity RenunBn. In the samelimit, the model has a U1 symmetry corresponding to aphase change expi in both complex variables un and Bn.The quantity nv;w is the shell model version of thetransport term ~ur ~w. The forcing term fn is given by fn S1nf0=u
1; i.e., we force injection in the large scale with a
constant power. We want to introduce in Eq. (2) an extra(large scale) term aimed at producing two statisticallystationary equilibrium solutions for the magnetic field.For this purpose, we add to the right-hand side of (2) anextra term M2B2; namely, for n 2, Eq. (2) becomes
F2u; B M2B2 mk22B2; (4)
where F2u; B is a shorthand notation for i=32u; B 2B; u. The term M2B2 is chosen with two require-ments: (1) it must break theU1 symmetry, and (2) it mustintroduce a large scale dissipation needed to equilibrate thelarge scale magnetic field production. There are manypossible ways to satisfy these two requirements. Here wesimply choose M2B2 amB32. We argue, see the discus-sion at the end of this Letter, that the two requirements area necessary condition to observe large scale equilibration.From a physical point of view, symmetry breaking alsooccurs in real dynamos since the magnetic field is directedin one preferential direction which changes sign during areversal. Thus symmetry breaking is a generic featurewhich we introduce in our model by prescribing somelarge scale geometrical constrain. On the other hand, largescale dissipation must be responsible for the equilibrationmechanism of the large scale field. The choice of a non-linear equilibration is made here to highlight the existenceof a nonlinear center manifold for the large scale dynamics. In other words, Eq. (4) with M2B2 amB32 issupposed to describe the normal form dynamics of thelarge scale magnetic field. Note that our assumption onM2does not necessarily imply a time scale separation betweenthe characteristic time scale of B2 and the magnetic turbu-lent field. Finally, since the system has an inverse cascadeof helicity , we set B1 0 as a boundary condition atlarge scale in order to prevent nonstationary behavior.
The free parameters of the model are the power input f0,the magnetic viscosity m, and the saturation parametersam. Our numerical simulations have been computed withn 1; 2; . . . ; 25. Actually, the parameter f0 could be elim-inated by a suitable rescaling of the velocity field. We shallkeep it fixed to f0 1 i. In Fig. 1 we show the ampli-tude of hjB2ji and the magnetic energy EB hnjBnj2i asa function of m for 107, where the symbol h istands for time average. In this system, a possible estimate
of Reynolds numbers is RV ffiffiffiffiffiffiffiffiffiffihEVip =k2 ffiffiffiffiffiffiffiffiffiffihEVip r0=4
and RM ffiffiffiffiffiffiffiffiffiffihEVip r0=4m; in the runs shown, hEVi 0:3,
this yields RV 1:5 106 and RM 0:15=m.For very large m, the magnetic field does not grow.
Then, for m greater than some critical value, hB2i, as wellas EB, increases for decreasing m. Eventually, hjB2jisaturates at a given value while EB still increases, showingthat for m small enough a fully developed spectrum of Bnis achieved. This type of behavior is in agreement withprevious studies of Taylor-Green flows [25,26], s2t2 flowsin a sphere , or MHD shell models . In the top insetof Fig. 1 we show the magnetic and energy spectrum form 103. Finally, in the lower inset we plot the magneticdissipation B mnk2nhjBnj2i and the large scale dissi-pation due to Mn. Note that at the dynamo threshold weobserve a sudden bump in the magnetic dissipation whichdecreases for decreasing n. At relatively small m, themagnetic dissipation becomes constant and quite close tothe large scale dissipation.We can reasonably predict the behavior of hjB2j2i as a
function of m by the following argument. The onset ofdynamo implies that there exists a net flux of energy fromthe velocity field to the magnetic field. At the largest scale,the magnetic field B2 is forced by the velocity field due tothe terms F2u; B. The quantity A RF2u; BB2 is theenergy pumping due to the velocity field which is inde-pendent of B2 and am. Thus, from Eq. (4) we can obtain
A amjB2j2B22r B22i mk22jB2j2; (5)
whereB2r andB2i are the real and imaginary part of B2. Forlarge m, the amplitude of B2 is small and th