H.K. Moffatt- Magnetic eddies in an incompressible viscous fluid of high electrical conductivity

8/3/2019 H.K. Moffatt- Magnetic eddies in an incompressible viscous fluid of high electrical conductivity

1/15

Reprinted without change of p agination fr o m theJournal of Fluid Mechanics, volume 17, p a r t 2, pp. 226-239

Magnetic eddies in an incompressible viscous fluid ofhigh electrical conductivityBy H. K. MOFFATT

Trinity College, Cambridge(Received 11 April 1963)

It is shown th a t in a n incompressible fluid in which th e magnetic diffusivity h ismuch less th an th e kinematic viscosity v, certain m agnetic field distributions oflimited spa tial ex te nt (conveniently described as m agnetic eddies) can ex ist ona length scale such th a t the associated Reynolds num ber a nd m agnetic Reynoldsnumber are respectively small and large compared with unity. The Lorentzforces are in equilibrium with the dynamic forces associated with the fluidmotion. The b oundary condition imposed on this motion is th a t a t a large distancefrom a magnetic edd y th e velocity field should be a uniform axisym me tric irrota-tional straining motion. The eddies are stead y in th e limit h + 0, b u t decay slowlyin a fluid of finite con duc tivity, Two particu lar eddies ar e considered in d etail:a disk-shaped eddy in a compressive straining motion, a nd a spherical eddy in a nextensive straining motion. Possible applications t o turbulence in interstellargas clouds are qualitatively considered.1. Introduction

I n th is paper, a fairly general motion of a n incompressible fluid of ver y highconductivity is considered. It will be supposed that the magnetic diffusivityh is very m uch smaller tha n th e kinem atic viscosity v , or more strongly, th a t(It m ay be necessary to impose a m ore stringent condition in special cases; seeequation ( 2 . 3 5 ) . ) I n th e absence of moving boundaries it is useful to characterizethe m otion by a typical ra te of strain a rather th an by a typical velocity. Therestriction (1.1) s strong enough to ensure th a t there ex ists a range of lengthscales I , < I < 1, within which the Reynolds numbe r R = a12/vand the magnet icReynolds number R, = d 2 / h atisfy the inequ alitiesThis com bination of circumstances means t h a t th e lines of force of any am bientmagnetic field are convected with the fluid and can diffuse only very slowlyrelative to it, whereas vortic ity diffuses rapid ly so th a t the m otion of the fluid iscontrolled by viscous rather th a n b y inertial forces.The restriction on the Reynolds number R is no t a n essential pa rt .of th ephysical theory to be presented, but it does allow considerable mathematicalsimplifications. Moreover, the types of solution available in this extreme casemay indicate method s of approaching similar problems when R as well as R, islarge compared w ith u nity, The only application th a t is ma de of the results of

(hlv))< 1. (1 .1)

Rh < 1, Ri % 1. (1.2)

http://moffatt.tc


2/15

226 H . K . Moflattthis paper is to those small-scale features of a turbulent velocity field for whichthe associated Reynolds number is small.

In the absence of any electric current or magnetic field distributions, thevelocity relative to any point moving with the fluid is approximately a linearfunction of position throughout any region of dimension 1, i.e.

ui U i j X j . (1 .3 )This is an accepted approximation in turbulent flows where 1 is any lengthsmall compared with the scale of the smallest eddies of the turbulence (i.e.1< (v3/e)4where e is the rate of dissipation of energy per unit mass of fluid). I n asteady laminar flow, the velocity distribution ( 1 . 3 )may be a reasonable approxi-mation for a large variety of flows, even if the length scale of the region underconsideration is such that the associated Reynolds number is of order, or largerthan, unity. I n general the tensor aijwill not be symmetric and will be time-dependent. Here, for simplicity, attention is restricted to steady flows, irrota-tional in the absence of electric currents and symmetrical about an axis, so that,referred to the principal axes of strain, equation ( 1 . 3 ) becomes (using the in-compressibility condition, V ,U = 0 )

U = ( a x , a y , -2az) . (1.41This will be taken as the outer boundary condition (asxixi+ CO) in what follows.If a > 0, material volumes tend to be flattened and oriented parallel to the(x, )-plane, while if a < 0, they tend to be drawn out into filaments parallel tothe z-axis. The strain field may be conveniently described as compressive orextensive in these respective cases.

The main purpose of this paper is t o enquire what happens when an axisym-metric blob of magnetic field is present at the centre of such a strain field.Specifically, we suppose that a t time t = 0 the velocity field is given by equation(1.4) (which is subsequently maintained as an outer boundary condition), andthat there is a magnetic field H(r,z) (in cylindrical polar co-ordinates ( r ,8 , x ) )which is non-zero in the volume V,inside a closed surface 1x1 = f r ) , nd identicallyzero outside this surface (on which there may flow a surface current). We supposein the first instance that the conductivity is infinite ( A= 0),so that the magneticfield is subsequently confined to the material volume V ( t ) hat was initially V,.

The Lorentz force associated with the magnetic blob acts as a perturbing forceon the uniform strain field. If the Lorentz force is negligible, the strain distorts thevolume V ( t )nto a disk whose dimensions increase as eatifa > 0,or into a filamentwhose length increases as e-2at if CL < 0 , and in either case the magnetic lines offorce in V ( t ) re stretched without limit. It seems likely that eventually it willnot be possible to ignore the Lorentz forces and that these will ultimately sodistort the strain field that the magnetic energy of the blob will not continue togrow without limit. It is not easy toprove this assertion; there is an infinite energyavailable in a uniform strain field that extends to infinity, and it is not impossiblethat an infinite energy is transferred from the velocity field to the magnetic fielddespite the back-reaction of the Lorentz force. I n this paper, only a search forpossible steady states in which the magnetic energy is finite is attempted; whether


3/15

Magnetic eddies in a n incompressible viscous JEuid 22 7these states are the inevitable asymptotic states consequent upon the aboveinitial conditions is left an open question.

The problem posed above arose naturally in a detailed study of the turbulentdynamo problem. The condition h < v was first put forward by Batchelor (1950)as the condition that an initially weak random magnetic field should be intensifiedby a turbulent fluid motion. The intensification is chiefly associated with thesmall-scale straining, and the approximation ( 1 . 3 ) t o the velocity field, thoughnot ideal, is a t least a reasonable assumption over regions of small enough dimen-sion. I n the limit h -+ 0 , the intensification must ultimately be checked by theLorentz forces which resist the stretching. (This may still be true when h is smallbut non-zero, although Saffman 1963 has recently stressed that the increase inthis case may be checked by the increased Ohmic dissipation associated with thedecrease in scale of the strained magnetic field.) There is still considerable doubtas to the level attained by the magnetic field even in the limit h -+0. Batchelorestimated that approximate equipartition of energy would be established betweenthe magnetic energy and the kinetic energy of the smallest turbulent eddies.Biermann & Schliiter ( 1 9 5 0 , 1 9 5 1 ) on the other hand estimated tha t equipartitionbetween magnetic and kinetic modes would ultimately be established at alllength scales. These estimates differ by a factor of order the square-root of theoverall Reynolds number of the turbulence, which may be huge in astrophysicalapplications. Hence the importance of examining in more detail the types ofbalance that can persist between magnetic and velocity fields.

Interest in the problem is not, however, derived solely from the turbulencecontext. An enormous effort has been devoted in recent years t o the problem ofdetermining magnetostatic equilibrium configurations in a perfect conductor,that is, magnetic field configurations that can be supported by pressure gradientsin a fluid a t rest (see, for example, Grad & Rubin 1959;Kruskal & Kulsrud 1956) .The chief stimulus for these investigations comes from thermonuclear fusionresearch, although the results are also of great interest in astrophysics. If theconductivity is large but finite, localized magnetic field distributions diffuseoutwards in a fluid a t rest. The dynamo problem is usually understood t o signifythe problem of determining a fluid motion which can maintain a steady magneticfield against persistent Ohmic decay. Cowling (1934) showed that no axisym-metric motion could reproduce this type of dynamo effect. Bullard (1949) ,Herzenberg (1958) and others have aimed at determining suitable non-axisym-metric velocity fields, bu t no a ttempt (other than qualitative) has been made tosatisfy the dynamical equation of motion. In this contribution, the dynamicalequation is satisfied in an extreme situation (R 1)) ut the motion is axisym-metric and the field decays slowly if the conductivity is finite. By approachingthe dynamo problem from this complementary angle, it is hoped that a littlelight may be thrown on some of the inherent difficulties involved.2. The disk eddy in a region of compressive strain

I n a region of compressive strain (a> 0 ) he volume V ( t ) , s observed above,tends to be flattened into a disk-shaped region near the plane z = 0. The tensionin the azimuthal lines of force provides an inward force resisting the outward


4/15

228 H. K. Moffattstraining motion. As this tension increases, the field may eventually succeed inreversing the straining motion within V and thus in generating a disk-shaped eddyin which the streamlines are closed and (as t -+ CO) the motion becomes steady assuggested schematically in figure 1 . This possibility is now investigated in detail.

It is supposed that the magnetic field B is purely toroidal

wherep is the fluid density. If the velocity fieldU = (U ,0, w) is expressed in termsof a stream function $ ( r , z ) by the usual relationsU. = -r-I&,, w = r-1 7, (2.2)

FIGURE. Schematic representation of the disk eddy. The shaded region IS the magneticboundary layer and the magnetic wake. The flux of magnetic field through the materialcircuit r is constant.then it is readily verified that the solution of the steady induction equation(withh = 0), V h ( U A h ) = 0, (2.3)is h = (0,4$),) , (2.4)where H ( $ ) is an arbitrary function of $.V = lim V ( t ) ) nd a velocity distribution U satisfyingThe dynamic problem that remains is to find a closed surfaceS (the surfaceoft+m

(2.5)(2.6)1and u.Vu = - - V X + h . V h + v V 2 u inside S,P

where x = p ++ph2 s the sum of fluid and magnetic pressures inside S. Theboundary conditions are thatU N (ar,0, - az) (2.7)

1U. u = - V p+vV2uP

outside S ,


5/15

Magne tic eddies in an incompressible viscous $uid 229at large distances, and th at ud nd the total force on a surface elemen t of S (withunit normal ni), - p+ 4ph2)ni+ vp(aai/8xj 8aj/axi)nj,should be continuous across S. (Th e mag netic field has no com ponent across S,so th at the contribution phi hjnj o th e force on a surface element of S vanishesidentically.)Since the eddy is flattened by th e external strain, it is reasonable t o look for asolution in which ( at least in a first approxim ation) the volume V is bounded b yth e planes z = & b an d th e cylinder r = a , where a $ b. For large z the streamfunction is $ = -ar2z, (2.8)an d hence in th e region r < a sufficiently far from th e edge of the disk for edgeeffects to be negligible, it may be possible to express the stream function in theform - r 2 f ( z ) fo r Iz( > b ,-r2g(z) fo r 1x1 < b. (2.9)(2.10)Sub stitution of equation (2.9) in equations (2.2) an d (2.5) leads to the equation,well known in the context of axisymmetric stagnation flow (Homann 1936;Schlichting 1960), f'2-2"' = a 2 + v f ' r l . (2.11)

The pressure p is independent of z hroughout th e region z > b. There is asudden drop in fluid pressure, however, on passing into the eddy regionz > b, since the total pressure is continuous. (The normal viscous stress iscontinuous on z = b because th e velocity an d tangen tial stress are continuousthere-see equation (2.20).) Substitution of equation (2.10) in equations (2.2)an d (2.6))using the linking pressure co ndition,

x = p on [zI= b (all r )yields the equation for g 9'2 - 2gg" = a 2-H2(11p) + vg"(2 .12)(2 .13)

Clearly the assumed form (2.10) of 4 is self-consistent on ly if H ( $ ) is equal to aconstant, K say, independent of 11p (otherwise equation (2 .13) still involves thevariable r ) . I n this case, th e magnetic field within V assumes th e simple form

h = (0 , r , 0 )and t he equation for g is g'2- gg" = E2- 2+vg",(2.14)(2.15)

Other magnetic field distributions may also yield steady states, but this iscertainly the simplest possibility, and it is the one to which attention is nowrestricted. The electric current distribution t h a t gives rise to th e field (2 .14) isa uniform current through V parallel to the z-axis, the current loops beingcompleted by a surface current on S.It can be shown th a t th e effect of slow Ohmic diffusion within th e edd y is (asmight be expected) to smooth ou t current variations within th e eddy so t h a t t heuniform current distribution chosen above is in a sense the preferred one.A relevant resu lt (in which vorticity, rath er th an magnetic field, was the slowly


6/15

230 H . K . Moffattdiffused quantity) was proved by Batchelor (1956). By integrating the equationof motion round a closed streamline in a region of closed streamlines in axisym-metric flow, he showed that the azimuthal component of vorticity must be pro-portional (in the steady sta te) to the distance r from the axis. The same reasoningapplies here to magnetic field, so that the distribution (2.14) is the one whichOhmic diffusion slowly selects. Unfortunately this same diffusion is also respon-sible for a slow leakage of flux from the eddy (to be estimated later in this section)and it seems likely that, starting from an arbitrary initial distribution of theform (2.4))the eddy would disappear before the particular distribution (2.14)could be established by Ohmic diffusion alone. It is, nevertheless, possible toimagine a situation in which magnetic flux is continuously supplied to the eddyin which case the solution preferred by Ohmic diffusion is likely to be the relevantone. (In the situations envisaged by Batchelor, some vorticity would likewiseleak from the region of closed streamlines, to be replaced by an input of vorticitygenerated at a solid boundary.)

Now the Reynolds number of the flow within V is assumed small so that thenon-linear terms of equation (2.15) may be neglected. The simplified equationmay then be integrated with the boundary conditions that the velocity profileis symmetrical about the plane x = 0, and that the plane z = b is a stream-surface,

(2.16).e.to give (2.17)The velocity inside V therefore has components

g(0) = g ( b ) = 0 ) g(0)= 0)g(Z) = ( K ~ 2) (x2- b2)/6v.

U = T ~ ( Z )= ( K ~ - C L ~ ) T ( ~ Z - ~ ~ ) / ~ V , (2.18)W = - 9 = - K 2- 2) ( Z 2 - b2)/3V. (2.19)

The profile within the disk is thus parabolic, and, since the stress must be con-tinuous a t the surface of the disk, it can be matched to the solution outside onlyif K~ > a2,when the profile is convex t o the origin; this is a necessary conditionon the strength of the magnetic field if a disk eddy of this type with closedstreamlines is t o be possible.

Equation (2.11)must now be solvedforf(z) subject to the boundary conditions

(2.20)f ( b )= g(b)= ( K - a 2 ) b/v. I

The last two follow from the continuity of velocity and tangential stress on thesurface x = b. Three of these conditions determine a solution of equation (2.11);the fourth will determine the thickness b of the disk. The non-linear terms can-not be neglected in this equation, because they dominate for large x .

Now Honiann determined the particular solution fi(z)of equation (2.11)thatsatisfies the stagnation flow boundary conditions

f;(.o)= a, fi(0) = 0, f;(o) = 0,


7/15

Magnetic eddies in an incompressible viscous $uid 231by expanding the solution in powers of x (for small x ) and 2- l (for large z ) andmatching the two solutions. The first few coefficients in the following expansionfor small x COfl(4=(Wt % ( a / w Z n , ( 2 . 2 1 )

a0= a1= 0 , u2 = 0 . 6 6 , a3 = 0 . 1 6 , a4= as = 0 , .... ( 2 . 2 2 )n=Owere found to be

A more general solution satisfying the condition f ( m) = a isf (4 = c +fI@ 0 ) ) ( 2 . 2 3 )

where c and zoare constants; these can be chosen so that the remaining conditionsof the set ( 2 . 2 0 )are satisfied. Clearly the conditionf(b) = 0 determines c. In theremaining two conditions only the first non-vanishing term of the series in theexpression ( 2 . 2 3 )need be retained, since when z is of order b (and provided zo isalso of order b, as the subsequent analysis confirms (see equation 2 . 2 6 ) ) , he ratioof each term t o it s successor is of order ( v /a ) : -l, and this is a t least of order R-3,i.e. much greater than unity by the assumption ( 1 . 2 )(assuming th at b lies in therange of length scales for which ( 1 . 2 ) s valid).

The conditions are therefore( 2 . 2 4 )1a,(b - o ) (a3/v)4= ( K ~ a2) 2/3v2 a , ( a 3 / v ) 3= ( K ~ a2) 2/v,nd

which may be solved immediately t o giveb = 1 * 3 2 ( a 3 v ) 4 K ~ 2)-I ( 2 . 2 5 )and z0 = gb. ( 2 . 2 6 )It is now apparent that the solution is self-consistent only if K~ % a2, or onlythen is b ( v ia )+ s the model requires. I n fact from ( 2 . 2 5 ) , b is then givenapproximately byor equivalently b z 1 * 3 2 ( v / a ) *$ / K ~ ) ,K 2 / a 2 M 1.32Rb-4,

( 2 . 2 7 )( 2 . 2 7 ~ )

where Rb is the Reynolds number based on the length scale b.If the to tal volume of the eddy is r , ts radius a s then given by

nu2b= r. ( 2 . 2 8 )If K, is not very much greater than a2 n eddy with closed streamlines may stillexist but clearly the flat disk approximation is not then appropriate.

The simplest interpretation of the ratio K / a is that ~ ~ / a ~s the ratio of themagnetic energy in the disk

to the kinetic energy of the fundamental strain field


8/15

232 H . K . Moflatt(neglectingb2 compared with a2) hat would exist in the same volume if the eddywere not present to distort it. Alternatively, since the electric current insidethe eddy is

J = - - (")" v Ah = (O,O,($)t"),4n Pthe ratio K ICX may be interpreted as giving a measure of the strength of this

current compared with the rate of strain const,anta.

FIGURE . Streamlines and velocity profile for a disk eddy of infinite radius.The streamlines and radial velocity profile determined by the above solution

are sketched in figure 2. The solution of course gives no information about theflow near the rim of the disk, where, as suggested in figure 1 , the streamlines mustclose. The solution strictly describes only a disk of infinite radius; it is not certainthat a similar steady solution describing a disk of finite radius exists a t all. Thedifficulties encountered in searching for such a solution are similiar to, if notgreater than, those encountered in investigations of the flow due to a rotatingdisk of finite radius, and they have not been overcome in either problem.

The real significance of the above solution is that it demonstrates the possibilityof a balance between magnetic and dynamic forces. If Lorentz forces are negli-gible, the intensity of any magnetic field in a perfect conductor increasesexponentially under uniform strain. Here, in one case, albeit specialized, it hasbeen demonstrated th at the effect of Lorentz forces is to admit the possibility ofa steady state; and it seems likely (although this has not been proved) that anymagnetic field initially confined near the (x, )-plane in a region of compressivestrain will either approach a steady state similar to tha t of the disk eddy, or a tleast will ultimately fluctuate about such a state.


9/15

Magnetic eddies in an incompressible viscous JEuidEffects of f ini te conductivi ty

If the conductivity is large but finite, the discontinuity in the magnetic fieldacross the surfaces z = k of the disk will be smoothed out in some form ofmagnetic boundary layer, and, if the disk is of finite radius, some magnetic fluxwill be convected radially outwards into a magnetic wake. These features maybe analysed simply as follows.Near both sides of the surface z = b of the disk, the velocity distribution ( 2 . 1 8 )and ( 2 . 1 9 ) (continuous across the surface), with neglect of a2 compared with

233

~ 2 ,s K 2 b 231,=- r ,0, - ( ~ ) ) .

In the magnetic boundary layer, the field h satisfies the equation

with boundary conditions V h ( U h h ) + h V 2 h= 0,h = 0 and h = (0,~ r ,)

( 2 . 2 9 )

( 2 . 3 0 )

at the outer and inner edges of the layer respectively. A trial solution of the formh = (0, T ( z ) ,)gives an equation for F(x)

with relevant solution

( 2 . 3 1 )

( 2 . 3 2 )

( 2 . 3 3 )an error function tending t o a step-function as h +. 0. The layer on z = - is ofcourse the mirror image in the plane x = 0 of that on z = +b . The thickness of

( 2 . 3 4 )he layer is of order 8 = ( h / K 2 b 2 ) *M (h/a)*K / c z ) ,on using the expression ( 2 . 2 7 ) for the thickness b . The condition S4 is then

( 2 . 3 5 )atisfied providedor equivalently, using ( 2 . 2 7 ~ )nd the fact that (A/v)* = (R/R,)*,

( h / V ) * 4 3 / K 3 ,

R i 8 R-f . (2.35a)It is worth remarking that the contribution h.Vh to the Lorentz force, that

cannot be compensated by the steep pressure gradient that exists in the layer,is of the form - [ T ( z ) ] 2U, (whereU, is a unit vector in the radial direction). Thisincreases smoothly from zero outside the eddy to its value - K%, inside, so thatno wild departures from the velocity distribution ( 2 . 2 9 )are to be expected withinthe layer.

For a disk eddy of finite radius, the inner part of this layer is presumably sweptround into the interior of the disk giving rise to some kind of magnetic shear layeron the plane x = 0. The outer part is swept radially outwards by the external strain


10/15

234 H . K . Moflattinto a magnetic wake. Fa r downstream; equation (2.30) with U = (ET, 0 , - ax)admits the similarity solution

h = (0, ,ar-lexp ( - a x 2 / h ) , )) (2.36)where h, is a measure of the maximum field strength that leaks from the eddy. Ithas already been remarked that an eddy can form only if K > a ; t therefore seemspossible tha t the outer part of the boundary layer (2 .31) satisfying F ( x)< awillescape into the wake and that the inner part where F ( x )> a will be drawn backinto the eddy. If this is true, then h, = O(aa) and the field in the wake

h = (0, au2r-lexp (-ax2/h), )) (2 .37)where C is a constant of order unity. This will hold only for r $ a.

The eddy decays slowly as the toroidal magnetic flux escapes into the wake.It is possible to estimate the rate of decay knowing the form of the magnetic wake(2 .37) . Consider the flux @ of the field h (proportional to the usual magnetic flux)through the material circuit F in the plane 0 = 0, sketched in figure 1,consistingof the axis of the disk, two lines 1x1 = x1 $ b, and a line r = rl $ a, cutting thewake. The diffusive transport of flux across I? is -h V A h .dl and this is zerosince V A h is radial in the wake and zero elsewhere on r. Hence @ is constant as I?moves with the fluid. The contribution to Q through the eddy itself is of ordera2bK which may decrease through a decrease in a2or b or K . I n a time dt, the partof r crossing the wake moves a distance arod t , so that, using equation (2 .37) , thecontribution to @ from the wake increases by an amount

k

This must be compensated by the leakage of flux from the eddy, so that- (U2bK) = - 2C 2a2$at

where C is another constant of order unity. Since the magnetic field is convectedaway from the rim of the eddy, it seems likely that only a2decreases in time andthat K (and consequently b from equation (2 .27)) remain constant. Then

a2aa2at b K_ - 2C(hol)+ ,so that the radius of the disk decreases according to the law

a(t)= a(0)exp [-C(ha)*a~- lb - l t ] . (2 .38 )The volume of the eddy therefore decreases, which is not surprising, because asthe magnetic field leaks away from the eddy its power to stop the strainingmotion becomes less and less effective. The half-life of the eddy, again using therelation ( 2 . 2 7 ) ) s of order (v/h)) - ~ .


11/15

Hag netic eddies in an incompressible viscous jluid 2353. Regions of extensive strain; the spherical eddy

I n regions of extensive strain magnetic loops will be drawn out parallel to thez-axis. A search for steady cigar-shaped magnetic eddies is unsuccessful becausethe magnetic lines of force, which must coincide in the steady state with thestreamlines within the eddy, are not suitably arranged to hold the eddy togetheragainst the disruptive stress of the external strain.

IFIGURE. Streamlines for the spherical eddy in a perfectly conducting fluid.

There is one interesting solution, however, in regions of axisymmetric extensivestrain, which describes a spherical eddy in which the electric current is againuniform and parallel to the axis of symmetry (figure 3). The magnetic field insuch an eddy (of radius a ) in spherical polar co-ordinates ( r ,8,$) is

h = (0,0,KrsinO ) , (3 .1 )and the equations of motion, neglecting non-linear terms, are

(3 .2 )(3 .3 )

1P

and - - V X + h . V h + v V 2 u = 0 ( r < a ) .The Stokes approximation is justified here, as for the problem of low Reynoldsnumber streaming past a sphere, by the fact that a solution is found whichsatisfies the outer boundary condition sufficiently accurately within the region inwhich inertia forces are negligible.

- - V p + v V 2 u = 0 ( r > a )1P


12/15

236 H . K . M o f a t tUsing the expression ( 3 . 1 ) ,equation (3 .3) may be written in the form

1P- -vq+ vv2u = 0, ( 3 . 4 )

where q = x ++pK2r2sin28. ( 3 . 5 )The velocity U = (U, U , 0) may be derived from a Stokes stream function $( r , 8)by the relations U = (r2sin ,g)-11~rg, = - rsin 8)-11~.,, (3 .6 )and the outer boundary condition is

@ N -ar3sin28cos8 as r + GO. ( 3 . 7 )A solution is therefore sought of the form

f(r)sin28cosB ( r > a )g(r)sin28cos8 ( r < a )

with boundary conditions (on r = a )f (4= g (a ) = 0, f ( a ) = g(a), f ( a )= g ( a ) , ( 3 . 9 )

the last reflecting continuity of tangential stress across the surface of the sphere.Equations ( 3 . 2 )and (3 .5 ) mply that $everywhere satisfies the Stokes equation

Hence

with general solutions f = A,r5+B1r3+Cl+D1r-2 ,g = A , r 5 + B 2 r 3 + C 2 + D 2 r - 2 .

Conditions at r = 0 and r = 00 imply thatthen the four conditions ( 3 . 9 )determine C,, D,, A, and B,, giving

A , = C, =D, 0 , B, = -a ;f = - ar3(4- a3/r3+ 3a5/r5),g = $ar3(i -rz/a2).

(3 .10)

(3 .11)

(3.12)

(3.13)We have yet to satisfy the condition that the normal stress should be continuous

x = p or r = a (all8). (3 .14)cross r = a , i.e.From equations ( 3 . 2 )and (3 .13) ,the pressure is

p / p = ( 21vaa3 /2r3 ) i n2 8 - 21v aa3 / 4 r 3+ pa / p ,and, from equation ( 3 . 4 ) ,

q / p = - 63var2 /4a2)in28+ 21vccr2/2a2+ g,/p,so that , using the relation ( 3 4 , he condition (3 .14)gives

2 a 2 ~ 2 - 105va, (3 .15)


13/15

Magnetic eddies in an incompressibleviscous$uid 237once again relating the electric current density (proportional to K) to the dimen-sion a of the eddy. Only if a < 0, i.e. only in regions of extensive strain, can thisflow exist. The gross effect of the magnetic forces is to reverse the rates of strainin the neighbourhood of the origin (as for the disk eddy), so that a steady statecan be realized. The streamlines for the flow described by the stream function(3.8) and (3.13) are sketched in figure 3.4. Application to turbulence in interstellar gas clouds

Although the magnetic eddies described above are certainly of mathematicalinterest, the conditions (1.2) under which they may materialize are ratherextreme, and the physical relevance of the solutions requires some discussion.The condition (1.1)is satisfied in hot HI1 interstellar gas clouds, fo r which theusual estimates of temperature T, number density of hydrogen ionsn, and lengthscaleL areMolecular transport theory for a totally ionized hydrogen plasma gives thefollowing expressions for h and v (Spitzer 1956; Kantrowitz & Petschek 1957):

(4.2)(4.3)

where A is a collision parameter, depending only on T and n, and tabulated bySpitzer. Substituting the particular values (4.1) gives

(4.4)(4.5)

so that the condition (1 .1) is indeed satisfied. The velocity of sound, corre-sponding to the conditions (4.1), is of order 10km/sec. An irregular componentof fluidvelocity U (say)of order 5 to 7 km/sec is observedin these clouds (Burbidge1959) so that the motion is subsonic, though perhaps not incompressible. At thesame time i t is likely to be highly turbulent; the overall Reynolds number basedon the above estimates is of order 108. In this case the kinetic energy cascadesthrough the turbulent spectrum and is dissipated a t a rate (per unit mass)

E w U3L-l x 10 cm2 ec-3.

T x lO4"K, n w 1~ m - ~ , x 1019cm. (4.1)

h = 5 .19 x 10l1T-&InA,v = 1.81x 10gTft(nInA)-l,

h w 1.2 x 107cm2/sec, v x 7 .8 x 1017cm2/sec,h/v% 1.5x 10-9,

(4.6)

(4.7)The length scale of the smallest turbulent eddies is

zv x v 3 / 4 1 x 1013~~1.Throughout any region of fluid small compared with 1, the strain field is uniformand the associated Reynolds number small. Moreover, the small-scale velocitydifferencesoccurring in these regions are an order of magnitude smaller than thelarge scale velocity U , so that on the small scale the motion may certainly betreated as incompressible. These conditions are then conducive to the formationof magnetic eddies of the type described in $0 2 and 3. The dimension of theseeddies would be so small (ona cosmical scale) that they could not be individuallydetected. If a large number of eddies were distributed throughout the fluid,


14/15

2 3 8 H . K . Moffatthowever, one eddy (say) in each region of fluid of dimension 1, in which the strainis approximately uniform, then they would all contribute to a stationary randommagnetic field whose statistical properties and effects might be observable.

Suppose, for the sake of argument, that there is one disk eddy of radiusa = O(1,) (the maximum possible) within every sphere of compressive strain ofradius I,, and one spherical eddy of radius a = O(1,) in every sphere of extensivestrain of radius I,. The eddies are then packed as closely as possible a t any instantwithout mutual interference. The magnetic energy within the disk eddy is

M pa4(a3v))using equations (2.14) and ( 2 . 2 7 ) , nd similarly the magnetic energy in a sphericaleddy of radius a , M, say, is of order pvaa3. Under the close packing assumption,the magnetic energy density is

M z M , or M')/$nl~O[p(sv )*] , ( 4 . 8 )using a = O(Z,), Z = O(v3/s)*,a = O ( e / v ) i .Thus the magnetic energy density is ofthe same order of magnitude as the kinetic energy density of the small-scalemotion; as pointed out by Batchelor (1950)) this is the only dimensionalpossibility if i t is the average strain which controls the magnetic energy densitylevel.

The particular values for 6, v and p (x 10-24n)given above imply a magneticenergy density. - M x 2 x 10-15erg/c.c.,2-=8nPcorresponding to a root-mean-square magnetic field (with ,U M 1)

(P)i 3 x lO- ' Gauss. (4.9)This is weaker than currect estimates of the general galactic magnetic field( N 5 x 10-5 Gauss) based on polarization measurements. However, the evidencealso suggests that this general field is uniform over length-scales much greaterthan 1013 cm and is roughly aligned along the spiral arms of the galaxy, suggestingthat it is not generated by the internal turbulence of the galaxy, but that it iseither generated by a regular large-scale motion or is of extra-galactic origin.There may, nevertheless, exist certain hot clouds of high conductivity whichcannot be penetrated by a general uniform field, but within which a genuineturbulent magnetic field of the type described above, whose r.m.s. intensity is oforder lO-' Gauss, may be supported.

The ratio h/v, rom the expressions (4.2) and ( 4 . 3 ) ) s proportional to T-4(apart from the weakly varying logarithmic factor), and therefore decreasesfairly rapidly with temperature. It may, therefore, be much less than uni ty in ahot laboratory plasma, but it is hardly realistic to apply the ideas of this paper insuch circumstances, because, first, such a plasma is usually permeated by a strongapplied magnetic field which controls (rather than is controlled by) the dynamicsof any plasma motion, steady or turbulent, and secondly, a t the high tempera-


15/15

Magnetic eddies in a n incompressible viscous fluid 239tures a t which hlv < 1 (based on the expressions (4.2) and (4.3))he mean freepath of ions and electrons is much greater than the dimensions of the apparatus(a criticism tha t cannot be levelled a t the cosmic laboratory) and the conven-tional equations of magnetohydrodynamics are of dubious applicability.

It is a pleasure to record my gratitude to Dr G. K. Batchelor for the guidanceand encouragement that he has given throughout the course of this research.

R E F E R E N C E SBATCHELOR,. K. 1950 Proc. Roy. Soc. A, 201, 405.BATCHELOR,. K. 1956 J. Fluid Mech. 1, 177.BIERMANN,. & SCHLUTER,A. 1950 2 . Naturforsch. 5 a , 237.BIERMANN,. & SCHLUTER,A. 1951 Phys. Rev. 82, 863.BULLARD,. C. 1949 Proc. Roy. Soc. A, 197, 433.BURBIDGE,. 1959 SymposiumonPlasmaDynamics (ed.F. H. Clauser),p.267. Pergamon.COWLING,T. G. 1934 Mon. Not. R. str. Soc. 94 , 39.GRAD,H. & RUBIN,H. 1959 Proc. 2nd Geneva Conf. 31, 190.HERZENBERU,. 1958 Phil. Trans. A, 250, 543.HOMANN,'. 1936 2. angew. Math. Mech. 16, 153.KAUTROWITZ,. R. & PETSCHEX,. E. 1957 Mag~etohydrodynamics Ed. by R. K. M.KRUSKAL,. D. & KULSRUD,. M. 1986 Phys. Fluids, 1, 265.SAFFMAN,. G. 1963 J. Fluid Mech. 16 , 545.SCHLICHTING, . 1960 Boundary Layer Theory, 4th ed., p. 81. MoGraw-Hill.SPITZER,L. 1956 Physics of Fully Ionized Gases, p. 83. Intersoienoe.

Landshoff), pp. 7 an d 8. Stanford University Press.

PRINTED I N GREAT BRITAIN AT TH E Uh-IVERSITY PRIKTING HOUSE, CAMBRIDGE(BROOKE CRUTCHLEY, UNIVERSITY PRINTER)

H.K. Moffatt- Magnetic eddies in an incompressible viscous fluid of high electrical conductivity

Documents

Transcript of H.K. Moffatt- Magnetic eddies in an incompressible viscous fluid of high electrical conductivity