A.Y.K. Chui and H.K. Moffatt- A Thermally Driven Disc Dynamo

8/3/2019 A.Y.K. Chui and H.K. Moffatt- A Thermally Driven Disc Dynamo

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A Therm ally Driven Disc DynamoA.Y.K. CHUI & H.K. MOFFATTDepartment of Applied Mathematics and Theoretical PhysicsUniversity of Cambridge, Silver St., Cambridge, CB3 9EW UK

A new simple idealised model for the geodynamo is proposed. Inthis model, the energy required for the self-exciting dynamo ac-tion is provided by thermal convection in a Welander loop; th is iscoupled with two Bullard disc dynamos. It is found that two discswith opposite twists are required in order to achieve reversalsof the magnetic field, and the model can then be described by adynamical system of 5th order. This system is found to be iden-tical to the 5th order system of Kennett (1976) resulting from atrunc ation of th e full magnetohydrodynamic equations. T h e bifur-cation structure of the system is stud ied, and numerical solutionsare presented.

1 INTRODUCTIONA simple mechanical model for self-exci ing dy nam o action was proposedby Bullard (1955). In th is model, a conducting disc (driven by a constant

1 external torque G, say) rotates in t h e presence of a magnetic field at angularvelocity 2nL(t); he current I ( t ) induced by Faradays law flows in a wireconnecting th e rim of t he disc (thro ug h a sliding con tact) and t h e axle. Th issystem is traditionally described by a second order dynamical system

where L and R are the self-inductance and resistance of the total currentcircuit respectively, M is the mutual inductance between the wire loop andthe rim of the disc, C is the moment of inertia of the disc about the axle,and k is a frictional resistance parameter. When G is positive and largeenough, stead y dy nam o sta tes with non-zero current can be found. Note th at

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M . R . E . Proctor, P.C.Matthews d A. M. Rucklidge (e ds . )Theory of Solar and Planetary Dynamos, 51-5801993 Cambridge University Press. Printed in England

www.moffatt.tc


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52 Chui tY Moffalt: A Thermally Driven Disc Dynamo

this model has a preferred parity (or handedness) in th e sense th at dyn am oaction cannot lie found for negative values of G.

Idealised syste m s with a few degrees of freedom such as th e one above areoften studied i n the hope that they may illuminate the essential physics forth e real problem. However, th e Bullard disc mode l is so simplistic tha t them ag ne tic field does not reverse; whereas m agnetic field reversals a re com-monly found in astronomical bodies such as th e Sun and t he E ar th . T he firstidealised model exhibiting reversals was proposed by Rikitake (1958); thisconsisted of two coupled discs resulting in a third order system. Other thirdorder mode ls (M alkus 1972; Moffatt 1979) derived from a single Bullard disccan also exhibit reversals; they are either identical with, or closely relatedto, the Lorenz system. Again, all such models have preferred parities so tha tthere is no d yna m o action i f the direction of applied torque is in th e oppositesense relative t o th e twist in th e wire.

Th ere is however an argu m ent suggesting th at a good model for geodynam oshould have no preferred parity. Helicity (and so the a-effect) results froman interaction between buoyancy and Coriolis effects and may therefore beexpected t o be proportional to th e pseudo-scalar g . 0 , (where g is th e gravita-tional acceleration, and 0 is th e angular velocity of th e E ar th ); it is thereforeantis ym m etric ab ou t th e equatorial plane (Steenbeck, Krause & Radler 1966).We therefore build an analogue model (Figure 1) with two coupled Bullarddiscs mou nted on t h e sam e horizontal axle bu t with wires twisted in oppositedirections in order to provide antisymmetry about the vertical centre-plane;and the discs are thermally driven by a fluid loop hea ted from below. Th ismodel has no global preferred parity, as for the geodynamo.

It turns out that the 5th order system governing this model is identicalwith that found by Kennett (1976) in a truncation of the full magneto-hydrodynamic equations for a convecting layer.2 THE M O D E L2.1 The mechanical deviceT he mo del consists of two ingredients (F igu re 1); h e first is a coupled B ullarddisc syste m i n which two discs are mounted on the same axle and allowed torot ate at th e sa m e angular velocity. T he wires are twisted in opposite sensesso that when the discs rotate in one sense, the right-hand disc acts as a dy-namo and the left-hand disc acts as an anti-d yna m o; if t he discs rot ate inth e opposite sense, then th e right-hand disc becomes an an ti-dyn am o and t heleft-hand disc becomes a dy nam o. T he second model ingredient is a circu-lar fluid loop in a vertical plane, heated from below and cooled from above(W elander 19G7); the rm al convection m us t occur when th e the rm al forcingexceeds some critical value. Suppose now w e combine these two ingredients


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lgChui 4 Moffait: A Thermally Driven Disc Dynamo

t t f cQQ '11 4

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Figure 1. In this model, the coupled discs are driven by a fluid loop heated from below.T he fluid loop must be imagined to be dynamically coupled to the rims of the discs, sothat both fluid and discs have the same angular velocity Q ( t ) .by identifying the ir ang ular velocities, which can b e done by placing t h e fluidloop around the rim of the discs so that the thermal forcing is communi-cated to t h e discs; th e composite system can now b e described by a 5th ordernonlinear dynamical system. A similar construction was first proposed byMoffatt (1992).2.2 The model equationsLet 1;(i = 1,2) be the currents in the wires, positive when flowing outwardfrom the axle; and Qj e the m agn etic fluxes across th e wire loops. T h edirections of fluxes are chosen (conventionally) so that the self inductancesLi of th e wire loops are positive . Similarly we define Ai to be the fluxesacross t h e discs, with th e directions chosen so tha t the m utual inductance Mibetween the wire loop and the disc in each circuit is positive (see Figure 1).For simplicity, we shall assume that the two circuits are identical except forthe 'twists', hence M , = M z %if4 and L1 = Lz % AM ( A > 1). If we furtherle t -c1Ad be th e mu tual ind uctanc e between th e wire loop and t he disc on th edifferent circuit, and - c2M be the m utual inductance between th e two wireloops, the n th e fluxes and th e c urre nts are related by t h e following equations:

Note tha t th e m utual inductances -&,Ad and - c2 M are negative because th ewires are twisted in opposite directions, hence c1 and c2 are positive. T h einduction equations for the two circuits are (cf. Moffatt 1979)

61 = -RI1 -I-R A , , 62 = - R I , - aAZ, (3)


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54 Chui d Moffatt: A Thermally Driven Disc Dynamoand the equation of motion is

C h = -kL' +G - IIA1+ IZAZ,where R,C, k and G are defined as in (1).Denoting

we eliminate I i and Ai using (2) and obtainR 1 + E l

Now let us derive equations for the fluid loop. Let T ( 8 , ) be th e temper-ature perturbation along the fluid loop; then it satisfies the one dimensionalheat diffusion equation

dT dT P Tat 60 6022 ~ 0 - = D- + s ( q , (7)

where D is an effective thermal diffusivity in the &direction and S ( 6 ) s thedifferential heating, which for simplicity we may take to beS(6) = --DsinQ, U > 0 . (8)

T h e solution of (7) then has the formT(6, t )= u ( t ) c o s 6 + v ( t ) s i n 6 , (9)

whereU = -Du - T V Z , i, = -Dv - c -t ~ T U Z . (10)

T he density perturba tion is -aT where a is the coefficient of the rm al expan-sion, and th e gravita.tiona1 torq ue actin g on t he fluid is then(11)12

2sG = agTa cos O(V/2r) O = -agaVu(t) y u ( t ) ,where V is the volume of th e fluid.

For the composite system, we now identify G in (6c) and ( l l ) , nd reinter-pret C as the to tal mo me nt of in ertia and k a.s the total frictional resistance


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Chui d M o f f a l t : A Thermally Driven Disc Dynamo 55coefficient. By rescaling the variables suitably and shifting the origin, weobtain the model equations:

j.= (-7 + w y z ) ,i = - ? j - y t w z z ,E = K(U - - y ) ,ti = -U t z - z ,ir = - v + u z ,

are measures of thermal forcing, frictional resistance and magnetic diffusionrespectively, and

X t E 2 , U = - - 1 -a = -X - 2 27r X + E 2 (14)are geometrical parameters. Note that (Y > 1 since X > > 0. We recallalso the meaning of the dependent variables: z is the flux difference, y is thetotal flux, z is the angular velocity, and U, give the temperature in the fluidloop, via (9).2.3 Basic propertiesThe system has two symmetries: if (z ,y ,z ,u ,v ) is a solution, then so are(-z, -y, z , U, ) and (-x, y , -2, -U, v). There are at most 7 steady stateswhich can be divided into 3 classes: (i) the trivial state, pure heat conduction(one solution, no dynamo):

(14)= y = z = u = v = 0 ;(ii) pure thermal convection (2 solutions, no dynamo):

z = y = o , U = % = & J- - l = f z r J ,ef v = t - l ; (15)I(iii) dynamo solutions (4 solutions, steady dynamo):

1+ zl- 1 = 6x1.\ - \ A 1 ) def

Consider the case when the thermal forcing parameter is increased slowlyfrom zero. Figure 2 shows a typical scenario (with a = 3/2 , w = 1, 17 = 4and tc = 1 ) for hifurca,tions from the trivial state. The first bifurcation occurs 1


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56 Ch,ui d Moffat t : A Thermally Driven Disc Dynamo2

I'

- - - . nstable ton

50 X

- 5-10

- 4 - 2 0 2 4 6

50 Y

-5-10

- 4 - 2 0 2 4 6Figure 2. (left) Bifurcations through increasing thermal forcingtrai ts of the s t range at t ractor . and (right) phase por-

at 5 = 1, when t h e trivial sta te becomes unstable; s table therm al convectionsolutions are created in a pitchfork bifurcation and become preferred states.T h e second pitchfork bifurcation occurs a t < = 1+ ( ~ / u ) ' ,t which point sta -ble dynamo solutions are created. Note that Hopf bifurcations from thermalconvection branches are also possible (when K > 2 ) , but we shall not considerthem in this shor t account .

is increased further, the fixed points corresponding to dy-namo solutions become unstable in a supercritical Hopf bifurcation; 4 stableperiodic solutions (l im it cycles) ar e create d, one from each fixed poin t. Th eyare non-reversing periodic dynamos because y does not change sign, wherey 0: - Qz s a measure of the total flux of the system. The amplitude ofeach periodic d y n a mo solution grows as ( s increased, up to a point when asubcritical bifurcation to invariant torus takes place, th e periodic solution be-comes unstable and th e system now has a strange attracto r wandering about2 unstable f ixed points (steady dynamos). This strange attractor (Lorenz 'typ e) is believed to b e associated with a homoclinic bifurcation a t t h e origin,and always exhibits reversals in 2, but not in y. A closer look at the rolestaken by z and y in the model equations (12) reveals that it is necessary tomake a! < 1 to allow reversals in y; this implies E~ < 0, which is not possiblefor th e configuration of Figure 1.

Typically, as

2 .4 A modified model for magnetic field reversalsThe above dynamo model does not permit field reversals. Before we givea modified version that does exhibit reversals, let us first examine why theabove mo del does no t. Suppose th e discs ar e rota ting in an anti-clockwisedirection, and the right-hand circuit, which acts as a dynamo, maintains an


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Chui & Moffail: A Thermally Driven Disc Dynamo 57

Figure 3. A modif ied mod el th at exhibi ts reversals (gra vi ty now acts hor izontal ly) .positive flux through its disc. Some of th e flux diffuses away, but som e,however tiny it is, must pass through the left-hand disc. Now if th e directionof rotation is reversed, the flux through the right-hand disc decays since theright-hand circuit now ac ts as an anti-d yna m o; th e flux throu gh th e left-handdisc increases b ut , because it remem bers its direction, it can never changesign. In order t o achieve reversals, it is necessary therefore t o modify th emodel so as to change the sign of the mutual inductance between two wireloops. A possible solution is illustrated in Figure 3: the fluid loop is nowembedded in a conducting disc in such a way th at it also acts as an insulatorwhich separates th e inner disc and th e oute r annulus. W ires with slidingcontacts are added to complete two separated circuits. It is not difficult toshow th at the model equations remain unchanged, a nd th e m utua l inductanc ebetween th e two wire loops - c2 M is now positive (i.e.c2< 0). Computat ionswith c2< 0 confirm that y does now show random reversals.

3 CONCLUSIONSWe have shown how a Welander thermal loop may be dynamically coupledwith a pair of B ullard disc dynamo s with opposite twists t o yield a self-exciting dynamo system in which the power supplied is entirely of thermalorigin. T he bifurcations of the resulting 5th order nonlinear dynamical s yst emhave been studied, and the conditions for dynamo action are determined.Flux reversals do not occur i f the mutual inductance between the twistedwire loops is negative; but if the geometrical arrangement is such that thism utua l indu ctan ce is positive (as for exam ple in th e configuration of Figu re 3 ) ,then flux reversals do occur. An analogous constraint on the distribution ofhelical eddies responsible for th e a-effect may be necessary to explain t h ereversals of the Earths magnetic field.


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58 Chui U Moffa i i : A Thermally Driven Disc DynamoREFERENCESBullard, E.C. 1955 Th e stabili ty of a homopolar dynamo. Proc. Camb. Phil.Kennett, R.G. 1976 A model for magnetohydrodynamic convection relevantMalkus, W.V.R . 1972 Reversing Bullards dynamo. EOS, Truns. A m . Geo-Moffatt, H.K. 1979 A self-consistent treatment of simple dynamo systems.Moffatt, H.K. 1992 The Earths magnetism: past achievements and futureRikitake, T. 1958 Oscillations of a system of disk dynam os. Proc. Camb. Phil.Steenbeclc, M., Krause, F. & Radler, I

A.Y.K. Chui and H.K. Moffatt- A Thermally Driven Disc Dynamo

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