E.M. Drobyshevski and V.S. Yuferev- Topological pumping of magnetic flux by three-dimensional convection

8/3/2019 E.M. Drobyshevski and V.S. Yuferev- Topological pumping of magnetic flux by three-dimensional convection

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J . Fluid Mech. (1974), vol. 65, part 1 , p p . 33-44Printed in. Great Britain.

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Topological pumping of magnetic flux bythree-dimensional convection

By E. M. DRO BYSH EVSK I A N D V. S . YUFE RE VIoEe Physical Technical Institute, Academy of Sciences of theUSSR, Leningrad

(Received 17 September 1973)Attention is drawn to a principal difference between the transfer of a horizontalmagnetic field by turbulence and by three-dimensional cell convection.

If the motion of the conducting medium in the cells is such that the heatedmaterial ascends at the centre while descending along the sides of the cells, thenthe magnetic tubes of force will be carried downwards by peripheral flows.Discrete ascending flows separated from one another by descending materialcarry only closed magnetic field loops. Such loops do not transfer net magneticflux. As a result, the magnetic flux becomes blocked at the base of the convectivelayer.

1. IntroductionA crucial point for any theory trying to explain the origin of the magnetic fieldof celestial bodies is the time scale, extent and manner of penetration (or decay)of the magnetic field into the bulk of the objects considered. If the conductingmaterial is fixed, this time scaler is determined by the diffusion of the field due toohmic dissipation:

7 = 4nv1yc2, (1)where v s the conductivity of the material, 1 the length scale and c the velocity oflight (the Gaussian system of units being used).For the earth this time is about104yr, for the sun and the stars 2 0lOyr. The hypotheses on the primordialnature of the magnetic fields of the sun and of the stars (Cowling1945) are basedon the latter estimate.

If the material can move, the field can change as a result of either the develop-ment of hydromagnetic instabilities when the field itself initiates motion of thematerial (Kipper 1958), or simply the transfer of field by the material when thefield is so weak tha t it does not affect the pattern of motion and plays the role ofa passive admixture.

The latter situation is pertinent to the generation of magnetic field by materialmotion. Intensive motion of material in celestial bodies originates, as a rule, fromthermal convection. Convection in the stars is of a turbulent nature. A largenumber of papers (see reviews of Parker 1970; Vainshtein & Zeldovich 1971)are devoted t o a consideration of field generation by the turbulent dynamomechanism. In these papers, one either does not consider the question of field

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34 E . M. Drobyshevski and V .S. Yuferevpenetration into the body or assumes th at in the presence of turbulent convectionthe field is transferred as if it were a scalar admixture on a characteristic timescale

T~= 12/Dt= 12 /K& l t ,where Dt is the turbulent diffusion coefficient, V, and It being the characteristicvelocity a,nd ength scales of the turbulence. The dimensionless factor K is of orderunity. Calculations of Parker ( 1 9 7 1 ) yield K = 0.15. When using ( 2 ) one assumesV, and lt to be equal to the velocity and length scale of convective motion, thusmaking no distinction between the transport properties of convection andturbulence.

The purpose of the present work is to draw attention to a principal differencebetween the transfer of magnetic field by convective motions and by randomturbulence.We deal here with three-dimensional thermal convection. Its velocity fieldpossesses an ordered structure in that the entire convective layer is divided intocells a t whose centres the material ascends while at th e peripheries it descends(the direction of motion can be of either sense depending on the actual physicalproperties of the material, etc.; see Krishnamurti 1968).Convective motion withsuch structure is observed both under laboratory conditions (BBnard convection)and in nature (granulation and supergranulation on the sun).

In an appendix to the paper (by H. K. Moffatt) the phenomenon of topologicalpumping is analysed by the methods of 'mean field electrodynamics ' under theapproximation of low magnetic Reynolds number, and the main conclusion ofthe paper is confirmed.

c,

2. The mechanism of flux pumpingLet three-dimensional BBnard convection take place in an infinite horizontal

liquid layer. The flow in the entire layer has a cellular structure (hexagonal in theideal case). The heated material ascends along the axis of each cell. Near theupper layer boundary it spreads towards th e periphery of the cell and, on cooling,descends along the sides of the cell. On reaching the lower layer boundary, thematerial is heated, moves towards the cell axis and ascends again, and so on.

The velocity field of such three-dimensional convection has a topologicalfeature essential to us. This is t ha t regions with a downward velocity form acontinuous network within which one can draw a continuous line connecting cellsarbitrarily far away from one another. Regions with upward velocities do notpossess this property. They are embedded in th e network of descending materialand are isolated one from another.

Imagine now an infinite ideally flexible filament to be placed on th e top surfaceof the layer consisting of such cells. Near the top surface of the layer the materialin each cell will spread from the centre towards the periphery. The flow of materialwill bend the filament, dragging it towards the sides of the cells. As a result, theentire filament will move into the region of downward velocities and eventuallywill be carried down, to the base of the layer. The filament as a whole cannot

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Topological pumping of magnetic fzux 35return to the top surface of the layer because of the discrete nature of theascending flows.

The behaviour of this filament simulates that of an individual tube of magneticfield in a highly conducting medium. Such a tube will also be carried by con-vective flows to the base of the convective layer. The ascending flows cannotdrag the tube as a whole back up. They can only tear from it closed loops. Theseloops do not carry any net magnetic flux. As a result, the original magnetic fluxof the tube will be confined near the lower boundary of the layer.

Thus three-dimensional convection acts as a valve with respect to the magneticfield, producing, on the average, a vertical gradient of the horizontal magneticfield component. This may be called a magnetic flux pumping effect.

The transfer of a scalar field does not depend on the topology of convection.

@ 3. Formulation of the problem and its solutionTo check the above reasoning, we have considered a concrete case of the effectof three-dimensional convection on magnetic field.The problem was formulated in a rather simplified form and solved numerically.

The magnetic field was assumed to be weak and not to affect the velocity field.The convective cells were taken t o be of rectangular (square) shape with thevelocity field in them given by

( 3 )I=V ,= -sinnx(l++cosny)cosnz,v = V = -(l+&cosnz)sinnycosnx,w =V,= [(1+cosnx)(1+cosny)- 11sinnz.The z axis is vertical, the x and y axes being normal to the cell sides. The celloccupies 0 < x < 1, - 1 < x < + 1, - < y < + 1.

This field satisfies the continuity equation for an incompressible fluid,v .v = 0, (4)

@ and has the desired topology, the fluid ascending a t the centre and descendingalong the sides of the cells. The net flux of fluid through any plane x = constantis zero.

It is noteworthy t hat the three-dimensional velocity field given for rectangular(square) cells by Chandrasekhar (1961, chap. 2 ) ,

I= - sin nx cos ny cosnz,v = - cosnx sin ny cosnz,w = cosnx cosny sinnx,does not possess the required structure since here the regions of ascending anddescending fluid alternate like the black-and-white chess-board pattern. One candraw arbitrarily long continuous lines through both ascending and descendingregions. Therefore from the topological viewpoint, field (5) is two-dimensional.

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36 E . M . Drobyshevski and V.S. YzcferevThe behaviour of a magnetic field H in the given velocity field is completely

described by the following equations:V . H = 0, (6)

(7 )H/at = rV2H- V V ) H + (H V ) V,where 7 = c2/4nc is the coefficient of magnetic field diffusion due to ohmicdissipation.

0). Because of theperiodicity and symmetry, computation was carried out for one quarter of theconvective cell (0 < x < 1 , 0 < y < 1 , 0 < z < 1 ) . The total magnetic field fluxwas assumed to be non-zero only in the direction of the x axis:

We sought a stationary solution of the problem (a/at


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i0000111112222333

\ j\0123012340123012

7+ 1.000+0.020+0.263+0.048

0.000- .484- .037-0.041+0.004

0.000+0*018- .018- 0.019

0.000- .006- .006

H,"jk H:9kv Ar >1 2 3 4 5 0 1 2 3 4+0.654 + 0.540 +0.120 + 0.080 +0.008 0.000 0.000 0.000 0~000 0.000+1.051 +0.591 +0.135 + 0.085 +0.007 0.000 0.000 0.000 0.000 0.000+0.295 +0.167 +0.068 + 0.025 +0.007 0.000 0.000 0.000 0.000 0.000+0.128 +0.060 +0.022 +0.008 +0.003 0.000 0.000 0.000 0.000 0.0000.000 0.000 0.000 0.000 0.000.891 - .437 - 0.264 - .061 - 43fj

- .538 - .435 - .270 - .071 - .040 - 0.484 +0.590 +0.246 + 0.029 +0.022-0.261 - .186 - .085 - .038 - .012 - 0.019 + 0.088 +0.085 - .008 +0.007-0.028 - .044 - .027 - .012 - .003 - .014 +0.036 + 0.034 + 0.002 +0.003- .013 - .006 - .006 - .002 - .001 + 0.001 + 0.006 + 0.010 + 0.003 +O*OOO- .107 - .039 +0.062 +0.023 + 0.010 0.000 0.000 0.000 0.000 0.000- .212 - .052 + 0.087 + 0.031 + 0.015 +0.036 - .030 - .000 - 0.026 - 0.002- .097 - .027 +0.028 +0.011 +0.007 - .018 +0,009 +0.012 - .018 - .001-0.026 - .015 +0.003 +0.003 +0.002 - .013 +0.006 +0.009 - .004 - .001-0.011 + 0.010 +0.008 - .005 - .001 0.000 0.000 0.000 0.000 0.000- .008 +0.010 +0.012 - .007 - .002 - .019 +0'017 - .003 - '003 +0.001- '004 - .000 +0.004 - .002 - .001 - .009 +0.003 - .001 - .002 +0.001

TABLE . Amplitudes of first harmonics of magnetic field components, R, = 12 (H;j,, = 0).

70.0000.0000.0000.000- .891- .129- .438- .137- .035-0.214- 0.395-0.212- .070- .032- .040- .019

H f j kA -72 3

0.000 0.000 y0.000 0.000 @0.000 0.000 80.000 0.000 9.Ec)-0.218 -0.088-0.341 -0.099 S-0.178 -0.023-0.073 -0.011 3.-0.024 -0.006


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E. M . Drobyshevski and V .S. Yuferev

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'I X

H = 1-FIGURE. Magnetic field lines in planes with zero norrnd field components. The fieldstrength is proportional to the distance between marks on curves. R, = 12. ( a )y = 0.( b )2 = 0. ( c )y = 1. ( d ) 2 = 1.Substitiiting (15)-(18) into (6) and into the x and y projections of (7) ields aninfinite set of linear algebraic equations for the coefficients. The set was solvednumerically on a computer by iteration. For solution, the chain of equations wastruncated by setting all harmonics with i, ,k > m to zero. The magnitude of mwas chosen so that the higher harmonics would be sufficiently small. For R, < 4 @it is sufficient to take m = 4; for R, < 16, m = 6 (the number of algebraicequations here exceeds 1000).

4. DiscussionWe carried out computations for R, = 4 r r d V / c 2< 16.Table 1 lists the first-order coefficients for the magnetic field components forR, = 12. Higher-order coefficients do not exceed 0.023 for H,, 0.006 for H, and

0.023 for H,.It is difficult to imagine and display the spatial distribution of the magneticlines of force. Therefore figure I presents the lines of force only in the planes withzero normal components. Note that areas where the lines of force come closer toone another do not, generally speaking, necessarily correspond to stronger fields.


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Topological pum pin g oj- magnetic Jlu x 39

X YH = 1FIGURE. Spatial magnetic field lines. The field strength is proportional to the distancebetween marks on curves. -, R, = 12; ---, R, = 16. Only a few of the computed linesare shown. For the sake of simplicity, projections of the lines onto the planes y = constantand z = constant are given (yielding nformation required for the construction of the spatialcurve as a whole). x , points where the lines of force intersect the plane x = 0; 0,ointswhere the lines of force intersect the plane 12: = 1 .

w

This is caused by the three-dimensional field structure. On the figures, it is thedistance between marks on lines th at is proportional to the field strength.Figure 2 provides some idea of the spatial field distribution.

The moving medium is seen to interact very strongly with the magnetic field.I n the absence of motion, or a t cr = 0, one would have only a uniform fieldH =H, = 1 , while H, =H, = 0. The motion of the medium results in deforma-tion and stretching of the magnetic tubes. This will cause generation of H , and H,components comparable in magnitude t o the original field H,, while at R, 2 8the field H, proper will reverse the sign in a part of the volume thus even formingclosed magnetic loops somewhere (figures 1 and 2).

We are interested in the behaviour of the magnetic field averaged over many@ cells. We have conditions (8)and ( 9 ) . From (16) it follows also thatH,dxdy = 0./'I/'1

As for the magnetic flux pumping process discussed in $2, its existence wouldproduce asymmetry in the distribution of the averaged magnetic field

(H,) =/'/' H,dxdy0 0with respect to the plane x = 4.As follows from (15), the magnitude of (H,) is determined unambiguously bythe coefficients HgOk,o that (H,) will be a function of z only if these coefficientswith k =+ 0 are non-zero. The asymmetry in question occurs if H & are non-zero atodd k . It turns out t hat these odd harmonics are zero for two-dimensional roll


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40 E . M . Drobyshevski and V . S. Yuferev

0 8 16R7n

FIGURE. Amplitudes of first three harmonics of z fieldcomponent 08.magnetic Reynolds number R,.convection and this is confirmed by the results of Weiss (1966), who also foundthat horizontal flux is redistributed symmetrically about th e centre-plane by alayer of two-dimensional eddies.The same holds for the three-dimensional convection with Chandrasekharsvelocity field (5).Here the three-dimensionality may be thought of as producedby deformation of rolls conserving the topological structure of th e ascending anddescending regions.? For motions with isolated ascending flows odd-k harmonicsare non-zero.

The values of the first three coefficientsH&,, for the velocity field (3 )are shownin figure 3 as functions of R,. Both the first and the second harmonics increasemonotonically with growingR,; at first (R,5 6) the second one dominates overthe rest. The last circumstance masks somewhat a t smallR, the field asymmetryof interest to us. However the growth of the second harmonic gradually slowsdown while from R, w 4 the first harmonic grows continuously in direct pro-portion to R,. As for the th ird coefficient, it is very small and even negative a t0 < R, 5 6, but afterwards it begins t o increase.

The distribution in height of the (H,) field averaged in the horizontal planeover many cells in the layer is presented in figure 4 for various values of R,.

The effect of asymmetry in the net magnetic flux distribution across the con-vective layer is obvious. It should be stressed that the redistribution occursdespite the absence of a net material flow across the layer ((w)= 0 ) , he effectbeing totally due to the structural properties of the three-dimensional convection.Both the absolute and relative differences in the field strengths (H,) between the

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t Generally speaking, one can expect that any difference in the geometry or parameters(e.g. conductivity) of the ascending and descending flows will result in an asymmetricredistribution of magnetic flux.


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Topological pumping of magnetic Jlux 41

0 1.0 2.0 3.0( H a >

FIGURE. Distribution of averaged horizontal magnetic fieldcomponent across convective layer.

@ upper ( z = 1)and lower ( z = 0 ) urfaces increase monotonically with growing E,.Thus the original magnetic flux becomes more and more confined to the lowerboundary of the convective layer.

I n conclusion, it should be stressed that, in contrast to isotropic turbulence,thermal convection does not necessarily result in magnetic field dissipation. Ifthe motion possesses a certain structure and direction which is often observed innature, the field will be transferred into the bulk of the liquid, becoming blockeda t the base of the convective layer.

AppendixBy H. K.MOFFATT,University of Cambridge

The pumping effect described in this paper may be demonstrated explicitly(without recourse to numerical methods), under the assumption that the magneticReynolds number is small.? The equation for the steady field H(x) s then

-V2H = c V A ( U A H ) , (A 1)where s -g 1. We are primarily interested in the mean field H,(z) = (H(x))' averaged over horizontal sections of the flow; this satisfies the equation

- 2H,/dz2= SV A 6, (A 2)where &(z ) = (U A H) i s the mean electromotive force generated. Under theboundary conditions imposed in $ 3 , we have the further constraint that thehorizontal flux is trapped between the planes z = 0 , l :

(A 3)with t he normalization adopted above. It is evident from symmetry considera-tions that

t The methods used are essentially the methods of 'mean field electrodynamics' asapplied t o space-periodicvelocity fields by Childress (1970) and Roberts (1970).


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42and so (A2)becomes

E.M. Drobyshevski and V . S. YuferevdHo/dz=dy,

the constant of integration being zero, sinceboundaries z = 0 , l [equations ( 1 1 ) and (12)].

= 0 and dH,ldz = 0 on theWhen 4 1 , we expand H as a power series

where -and H(0)= (1,0,0). When u(x) is periodic, these equations may be solvedsuccessively by elementary methods, making repeated use of the identity

- v-2 cos(Zx+ p )cos (my+q )cos (nz+r )= Z 2 +m2+n2)-lcos (Zx+ p )cos (my+q )cos ( n z+r ) . (A 8) 0

If 8%)(U A Hen)) and Hh")= (H@)), hen (A 5) givesd H F + l ) / d z= &g).

Clearly f)= 0 and Hh1)= 0. The constraint (A 3) implies thatP l

With n = 0, (A 7 ) gives-V2Hc1)= V A (U A H(O))= H(O).VU = au/&, (A 11)

and when U is given by? [equation (3)]U = (-sinx(l+icosy)cosz,

- 1+gcosx) sin y cosz, (cosx+ cosy +cos%cosy) sinz) (A 12)it follows thatH(l)= &(- o s x ( 3+cosy) cosz, sinx siny cosz, - 3 + 2 cosy) sinx sinz).Hence

(A 13)(A14) @t )= ( u e H g )- ,HL1)) = - &sin 22,and, from (A 9) and (A l O ) , HL2)=&COS 22 .

Hence, to order e2 ,th e mean field is symmetrically perturbed about th e centre-plane z = 3n; t is slightly intensified for z < fn and for z > $71 owing to th ehorizontal divergence of the flow in these regions. I n order t o obtain the pumping(or valve) effect discovered (see $2) ,it is necessary t o continue to the nex t orderin 6 .

For this purpose, we need only calculate6'F' = (U , H g )- , Hi2)) , (A 16)

f We put x*= TIX and drop the asterisk for convenience of notation; the magneticReynolds number used in $ 9 3 and 4 above is then R, = m.


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Topological pumping of magnetic flux 43and we need therefore retain only those terms of the horizontal Fourier expansionof H(2)nvolving sinx, osx, in y and cosy. For example, we have

(U A Hcl)), = uY L - U ~ H ~ =&(12sinx siny+ siny sin 2x+2sinx sin2y)sin2z7 (A 17)and we need only retain

(U H(l))z= $ sinx sin y sin 22+ . , (A 18)in calculating H(2) rom (A 7). Similarly, we have

(U H(l)),= -&sin 2.414+2 cos x+ 15:cos y+ 12 cos x cosy+ . , (A 19)and (U H(l))Z -&(1+ cos 22) (5sin y+ 12cos x siny+ . ). (A 20)We can immediately calculate the components of V A (U HS), and from (A 7 ) ,a, sing (A S), we then haveHi2= - g CO S t~(5+ 6 COS X)

+3&cos 2435+4cosx+25 cosy+ 10cos x cos:y + . ), (A 21)and HL2)= i&sin2z(sinx+...). (A 22)

h2) = - & sin z+& cosz sin 22, (A231Bi3)(z)= hCoSZ(37- 12C 0S 22) = &C OS Z-& ~~S 3~ . (A 24)

Hence from (A 16),

and so, from (A 9) and (A O ) ,

This contribution to the field is antisymmetric about z = Qrr; in fact t he flux ofHi3) n the lower half 0 < z < Qrr is

and the flux in the upper half in < z < rr is - 0.121. Continuation of the processshows clearly that HLn)(z) s symmetric or antisymmetric about z = &rr accordingas n is even or odd. The mean field H,(z) is now given t o order e3by

(A 26)7e2 348 240, ( ~ )= i +- os2z+- 2800s - cos 32)+o(e4) .It seems likely that the first three terms give a reasonable approximation for 5 .Comparison of (A 26) with (15)shows th at

7e3 7e2 - 3H:,~=go+ o(e5 ) , ~k~ 48 + 0(e4), H;,,~=8o 0 ( 5 ) , (A 27)results that are not inconsistent with figure 3 (with E = R&).


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44 E . M . Drobyshevski and V .S. YuferevR E F E R E N C E S

CHANDRASEKHAR,. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon Press.CHILDRESS,S. 1970 New solutions of the kinematic dynamo problem.J. Math. Phys. 11,

3063.COWLING,T. G. 1945 Mon. Not. Roy. Astr. Soc. 105, 166.KIPPER,A. J. 1958 Electromagnetic Phenomena in Cosmical Physics, 6th I A U Symp . (ed.KRISKNAMURTI,. 1968 J . Fluid Mech. 33, 445, 457.LANDAU,L. D. & LIFSHITZ,E. M.PARKER, . N. 1970 Astrophys. J. 160, 383.PARKER,E. N. 1971 Astrophys. J . 163, 279.ROBERTS, . 0 . 1970 Spatially periodic dynamos. Phil. Trans. A 266, 535.VAINSHTEIN,.I. & ZELDOVICH,YA.B. 1971 Usp. Fiz . Nauk. 106, 431.WEISS,N. 0. 1966 Proc. R o y . Soc. A293, 310.

B. Lehnert), p. 33. Cambridge University Press.1957 Elektrodinamika Xploshnykh Sred. Moscow:GITTL. (Trans. Electrodynamics of Continuous Media. Pergamon.)

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E.M. Drobyshevski and V.S. Yuferev- Topological pumping of magnetic flux by three-dimensional convection