H.K. Moffatt- Turbulence in Conducting Fluids

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  • 8/3/2019 H.K. Moffatt- Turbulence in Conducting Fluids




    No 108

    M E C A N I Q U ED E L AU L E N C EMarseille

    28 aoiit . 2 septembre 1961



    1 9 6 2


  • 8/3/2019 H.K. Moffatt- Turbulence in Conducting Fluids



    S0MMAIREMaintenant que l extrbme complexi tb de la thbor ie de la turbulence dans les f luidesordinaires a btb rbvblbe, i l peut apparaltre I beaucoup une extravagance tbmbraire daborderlexamen des fluides conducteurs de lblectricitb. La situation est dbjI assez mauvaise,pourquoi la rendre encore pire en autor isant les blectrons , auss i bien que les molbcules ,21 se mouvoir sans entraves ? A premibre vue, btendre toute thbor ie bien connue des f luidesord ina i r es I quelques-uns possbdant cet te dernikre propr ibtb semble ref lbter une autreexplos ion de cet te panique contagieuse. Su r quelques points, laccusation est justif ibe. Unetendance s es t rbvblbe de prbsenter des extens ions directes de quelques-unes parmi les plusconnues des theor ies mathbmatiques de la turbulence, nbcessairement lourdes de formal ismemathbmatique, soutenues par des hypothkses dune val idi tb discutable et impliquant unesbr ie de conclus ions dont la s ignif icat ion nes t comprise que par t iel lement . Mais l aspectphys ique du sujet nes t pas encore suff isamment clar i f ib pour jus t i f ier une approche exclu-s ivement mathbmatique.I1 es t impor tant I cet te btape dessayer de dbf inir les sor tes de s i tuat ions phys iquessuscept ibles de se produire, et ces t en par t ie mon but dans cet te conversat ion.I1 y a relat ivement peu de publ icat ions I ce sujet et aucun t ravai l expbr iniental naprat iquement Btb rbal isb. Nbanmoins , i l y a deux raisons encourageantes de poursuivre ce

    sujet I fond.- abord, da ns les recherc hes c oncern ant l as t rophys ique, et 1a phys ique des plasmas ,la prbsence de la turbulence es t souvent supposbe lorsquon ne peut pas expl iquer les obser-vat ions par une thbor ie a bien carbnbe B.I1 est cependant trop facile duser, ou plutBt dabuser, du mot a tu rbu lence B, co m m edune baguet te magique, pour fai re disparal t re ce qui ne peut Ctre interprbtb autrement .I1 est important darriver I des conclus ions prbcises , quant I savoir quels phbnomknes ,dans des f luides conducteurs , peuvent btre v raime nt at t r ibubs a la turbulence, et quels phbno-mknes ne le peuven t pas ,- a seco nde raison es t peut-btre plus acadbm ique. Lact ion de la turbulence sur u negrandeur scalai re, tel le que la tempbrature, qui es t a la fois t ransmise et di f fusbe dans lef luide, es t maintenant bien connue. Pour complbter le tableau, i l serai t intbressant de biencom prendre l act ion de la turbulence s ur une gra ndeu r vector ielle , qui es t de mOmetransmise et di f fusbe. Le champ rotat ionnel es t un exemple, mais i l es t t rop par t icul ier , carint imement l ib au ch am p, des vi tesses .Le champ magnbt ique dans un f luide conducteur es t le parfai t exemple de sujet detravail . Les l ignes de, force dun cham p m agnbt ique, dans un f luide de conduct ivi tb inf inie,sont t ranspor tbes avec le f luide. Dans les f luides de conduct ivi tb f inie, el les se dif fusent itun t aux dependant de l a g randeur de ce t t e conduct iv i tb .La s i tuat ion es t compliqube du fai t que le champ magnbt ique exerce une force sur lef luide; i l nest gbnbralement pas passi f dyna miqu em ent; mais da ns cer tain es ci rcons ta ncesi l sera poss ible de nbgl iger c et te force, et de se conc entrer su r l ef fet combine de la convec-t ion et de la dif fus ion, dan s un f luide turbulent , de propr ibtbs s tat is t iques connues .

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    Now that the extreme complexity of the theory of turbulence in ordi nary fluids hasbeen revealed, i t may seem to many a rash extravagance to admit to consideration fluidswhich conduct electricity. The situation is bad enough already - hy make it worse byallowing electrons as well as molecules to move unfettered ? At first sight it seems to reflectanother outburst of that infectious stampede to extend every known theory of ordinaryfluids to those few with this " latest " property. To some extent, the accusation is justified.A tendency has revealed itself to present direct extens ions of some of the better-knownmathematical theories of turbulence, necessarily heavy with mathematical formalism,bolstered with assumptions of debatable validity, and ca rrying a trail of conclusions ofpartial ly understood significance. But the physics of the subject is not yet sufficientlyclarified to justify an all-out mathematical approach. It is important at this stage to attemptto define the types of physical situation that may arise, and this is partly my aim in thistalk.There are relatively few published papers on the subject and practically no experimentalwork has been done. Nevertheless two compelling reasons can be given for pursuing thesubject to its limit. Firstly, in astrophysics and in plasma physics research, the presence ofturbulence is often inferred when observations cannot be explained by a streamlined theory.However it is too facile to use, or rather abuse, the word " turbulence ", like a magic wand,to dispel what cannot otherwise be understood. It is important to arrive at some preciseconclusions as to what phenoniena in conducting fluids can truly be attributed to thepresence of turbulence, and what cannot. The second reason is perhaps more academic.The action of turbulence on a scalar quantity, such as temperature, which is both convectedand diffused in the fluid is now fairly well understood. To complete the picture it wouldbe satisfying to understand fully the action of turbulence on a vector quantity which islikewise convected and diffused. The vorticity field is an example, but it is too special,being closely related to the velocity field. The magnetic field in a conducting fluid is theperfec t working example. The lines of fo rce of a magnetic field in a fluid of infiniteconductivity are convected with the fluid. In fluids of finite conductivity, they diffuse ata rate determined by the magnitude of this conductivity. The situation is complicated bythe fact that the magnetic field exe rts a force on the fluid- t is not in general dynamicallypassive; but in certain circumstances it will be possible to neglect this force, and concen-trate on the combined effect of convection and diffusion in a turbulent fluid with knownstatistical properties.

    2. The turbulent dynamoThe standard equations of magnetohydrodynamics call be conveniently written in

    terms of the fluid velocity U (r, t ) and the Alfvhn velocity at each point h (r , t) , whichis simply p roportional to the magnetic field H ( r , t ) :

    where p. and p ar e the constant magnetic permeability nlld density of the fluid. I n this1 12 2notation the kinetic energy and the magnetic energy per uni t mass are - 2 and - Zrespectively. The total pressure x (r , ) is the sum of the fluid pressure p (r , t ) and the

    magnetic pressure ph2,< 112x = p +- h2.

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    397The equations fo r U (r, t ) and h ( I , t ) are then

    a h- u * V h= h*Vu+ h V hattogether with V * u = V * h = O . ( 5 )Two diffusive constants appear, the kinematic viscosity v, and the magnetic diffu-sivity A. When h = 0, one can deduce from equations (4)and ( 5 ) the well-known resultthat the flux of magnetic field through any circuit moving with the fluid remainsconstant, or equivalently t ha t the lines of fo rce move with th e fiuid and t he s trengthof the magnetic field at any point moving with the fluid is proportional to the lengthof an element of the line of force through t ha t point. I shall, for simplicity, restrictattention t o homogeneous turbulence, and shall use the spectrum tensors @,(k) an dI',(k) of velocity and magnetic fields to describe th e energy distribu tions in any steadystate. Let it be our first aim to describe the development and steady s ta te form of thesespectra, given certain gross conditions defining the various situations that may arise.

    In 1950, tw o irrecoricilal~lr heories w r e proposed, thci O I ~ Pby T~AT(WFX,OR [11 , theother by BIERMANNnd SCHLIJTER2], o predict the development of a n initi ally weakrandom magnetic field in a fiuid in tnrbulent notion. To be fair, it must be stated thatno fully convincing argument has yet been given to prove or disprove either theory.The matter is of fundamental importance and it seems highly appropriate that thetheories should be reviewed at this meeting at any rate to clarify the points at whichthey diverge, and perh aps t o suggest some critical problem whose solution might finallydistinguish between the two Rtandpoints. Let me therefore recall the main points ofthese theories.BATCHELORsploited the analogy between equation (4) or the magnetic field and

    that for vortieity w (= VA ) in a non-conducting fluid, viz.,at (6 )

    V . 0 = 0. (7 )am- u*vm= w.vu + v v 2 w

    Vorticity is generated by the stretching of vortex lines as they are convected by thefluid motion and it is destroyed by viscous diffusion a t high wave-numbers. These twoprocesses are approximately in equilibrium. In the same way, magnetic energy isgenerated by the stretching of magnetic lines of force in so far as they are convectedby the turbulent motion. It is to be expected therefore that those statistical propertiesof the magnetic field that depend only upon this stretching mechanism will in timeapprox imate to the corresponding statist ical properties of the vorticity field. If h= ,the conductive diffusion'of lines of force is then just rapid enough for the magneticfield spectrum (like the vorticity spectrum) to remain approximately steady. If h > v,conduction wins over stretching and the field decays to zero, while if h < v, conductionis less important and the field increases in intensity. When h is only slightly less than v,it is not clear whether an all- round decrease of scale together with increased Ohmicdissipation limits the growth of the field, or whether it is the Lorentz force whichmodifies the straining motion and.so l imi ts the growth. Buth when A

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    398argued th at conduction alone would be of small importance and th at the magneticenergy level must increase until magnetic stresses are comparable with the dynamicstresses governing the smallest turbulent eddies in which most of the vorticity isconcentrated, that is until the mean magnetic energy per unit mass - h 2 is comparablewith the kinetic energy per unit mass of the small-scale motion, ( e v ) ' l z (E being the usualra te of di ssipation of energy per unit mass).

    Now BIERMANNnd SCHLIITERid not explicitly discuss the criterion for growth,but in any case they were considering a fluid, the interstellar gas, to which the conditionh 1 (8)UI 8hor R , ( E ) = - -2I 12where uz (= (eZ)lI3) is the velocity in a n eddy of size I and R, ( I ) is the magneticReynolds number for that length-scale. Substituting for ut this condition givesI > ( y ) / * ,

    \ Ihasituation that can arise only if L > > y .The t,est case on which the two theories really collide is therefore when


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    the second (tantamount to h >> v ) the BAT(-HELORittack predicts that ;ill randommagnetic field fluctuations ultimately decay to zero. If 1 use the semi-empirical formulaE=- (12)

    for 8 , zL

    where U is the r. m. 8. velocity, ant1 define the I ieynolds number R and magnetic Heynoldsnumber R, of the turbulence l y

    then the case (11)is defined in more fundamental terms by the inequalities1

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    3. Fluctuations at low magnetic Reynolds numberwhen a uniform field is appliedIt is noteworthy that the fluctuations will not be small compared with the appliedmagnetic field when the magnetic Reynolds number is large compared with unity. Henceany perturbation method which assumes that the fluctuiitioiis are small compared withthe applied field can be valid only when the magnetic Reynolds number is smaller, andpreferably much smaller, than unity. The perturbation approach was used by L I E P M A N N [8 ]in 1052 and by GOLITSYN9] in 19G0, and although the condition B,

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    401The star indicates a complex conjugate, and the overbar an ensemble average. @ { j (k, )here represents the Fourier transform of t he space-time velocity correlat ion and its timedependence is not in general known. Howver, in the case considered here (R,

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    402was to inc rease the mean square vor t i c ity wi th ou t l imi t . The sam e mathemat ics app l ied tothe p resen t p rob lem shows th a t un i fo rm s t r a in inc reases the m agne t ic energy wi thou t li-mit . This is cons i s ten t wi th the conclusion th a t when h v ) is in a sense complementaryto tha t cons ide red by Ha tche lo r (&I= 0, v >> A). Kovasznays work was motivated byobservat ions of sponta neous turbulence in the presence of appl ied f ie lds ; Batchelors bythe w idespread as tropl iysical phenomenon of spon taneous magne tic f i e lds in the p resenceof background turbulence. The kinet ic and magnet ic spectra for Batchelors case and forthe isotropic analo gue of Kovasznays case ar e sketched in f igures 1 an d 2, in which th i scomplementarity i8 pronounced.

    To mak e the foregoing pic tu re of magnetohyd rodynamic turb ulence less impressionis t ic ,8ome experim ental resul t8 are very much required. For example the determinat ion of theampl i f i ca t ion fa c to r of a weak appl ied f ie ld in the case R >> R, >> 1would be suff ic ientto d i s t ingu ish be tween the Ba tche lo r s tandpo in t and tha t o f B ie rmann and Sch l i i t e r .The condi t ion R, >> I i s unfor tuna te ly ha rd to rea l i se in l abora to ry cond i t ions , bu t it



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    kViscousdissipat ion

    I"nterchange Jouledissipation

    F I GLI RE

    Jou led iss ipa t ion

    Viscousdissipat ion


    F I G U R E 2

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    404may not remain hopelessly beyond experimental technique. The less interesting situationR, > 1, a condition that did indeed apply in the type ofturbulent plasma that he considered.Let me conclude by summarising the above observations in the following roughclawwiflcittion of types of stationary rnagiic~toliytlrocty~iiii~iicuiabulenct. together with thechief situations in which each type may arise.( a ) Kinetic source dominant, weak applied field, K >> M.turbulent mercury, liquid sodium etc.)interiorw, regions of the ionosphere)of interstellar gas)( b ) Magnetic source dominant : strong applied fields, 11 >> K

    (i) R, > 1 : Magnetic driven turbulence (hot plasma, stellar interiors)This is only a tentative scheme of limiting cases. A more thorough examination of theparticular case8 K = M and R, = 1 might also throw light on the general situation.

    (i ) R,

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    Une session spbciale btait consacrb e aux effets de la coiiipressibili tb et de la p resen cedun milieu possbdant une conductibili tb blectrique.I1 semble Ctre un peu prbtentieux de soccuper de ces com plications q uan d la tur bu-lence simple dun fluide incompressible et non-conducteur prbsente elle-mCme des difficultbspresque insurmontables .De nombreuses raisons conduisent A exbcutcr d cs recl ierches dans ces domaincs quelquepeu bsotbriques. Le bruit produit par lbcoulement turbulent gst un problbmc pratique ence qui con cerne les avions A rbact ion. La turbulence magnbto-hydrodyn amique semble devenirun obstacle important au dbveloppement des rbacteurs thermo-nuclbaires contrdlbs. Mais,mCme du point de vue de la recherche de base, cet te ques t ion es t intkressante parce que lecas du f luide incompress ible et non-conducteur peut Ctre mieux c omp ris en tan t que casl imite du f luide compress ible et conducteur .Monsieur MORKOVIN a discutb les rksul tats obtenus dans la couche l imite turbulentesupersonique.Les mesures de turbulence fai tes A lanbmombtre h f i l chaud dans l a couche l imi tenous ont surpr is . M6me A un nombre de Mach de 1.75 a 2.00, nous avons cons tatb que lemecanisme interne de la turbulence dif fbre peu de celui de la couche l imite incompress ible.Bien en tendu, i l y a des f luctuations dentropie, et m& me des f luctuat ions de press ion (desonde s acoustiques), ma is la veritable turbu lence qu i possbde une diverge nce nulle, cest-A-dire la par t ie incompress ible du champ de vi tesse, change t rbs peu.Une des ques t ions essent iel les es t le comportement des tens ions de Reynolds en mil ieucompress ible, et un c hoix convenable des l ignes de co urant m oyennes la ranibne au casincompress ible.Les spectres des f luctuat ions ressemblent auss i for tement B ceux des couches l imitesincoinpress ibles .Ces fluctuations peuvent Ctre dbcomposkes en trois modes : le mode rotat ionnel , lemode dentropie et le mode acous t ique. Deux de ces modes sont parabol iques , autrementdit , obbissent A des Bquat ions du type conduct ion de la chaleur . Par contre le mode acous-tique est hyperbolique, et obbit A une bquation de propagation dondes.Dans un bcoulement oh la rbgion turbulente es t bornbe, comme par exemple une couchelimite turbulente, ou un jet, ou un s i l lage, les ondes acous t iques engendrbes au sein de laportion turbulente se propagent et peuvent Ctre observbes dans lbcoulement extbrieur nonturbulent .M onsieur L A U F E ~ous a prbsentb les rbsul tats de mesures des f luctuat ions acous t iquesobtenues P l extbr ieur de la couche l imite supersonique, et a fai t auss i la cr i t ique des theor iesexis tantes sur la prot luct ion de brui t par la couche l imite supersonique. La thbor ie asymp-tot ique de Ph i l l ips (valable Q un nombre de Mach inf ini) se t rouve approximat ivement con-firmbe. Dailleurs Ibnergie rayonnbe est trks faible par rapport a la dissipation visqueuse,m6me A un nombre de Mach trbs Clevb, et par exemple B M = 5, elle est de lordre de 1 %,ce qui cons t i tue un rbsul tat surprenant .

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    40GDiverses considerations sur la turbulence magnbto-hydrodynamique ont btb prbsentbespar Monsieur MOFFATT t jai apportb personnellement quelques preuves expbrimentales delexistence de la turbulence dans un plasma. Quand le milieu posskde une conductibilitbblectrique, les bquations dynamiques (de Navier Stokes) comprennent un terme supplbmen-

    taire traduisant la force de Lorentz, qui est une fonction quadratique du champ magnbtique.Par contre, lbquation qui gouverne le champ magnbtique est linbaire.Le problbme essentiel de laugmentation de lbnergie magnbtique totale par lagitationde la turbulence cinbtique nest pas rbsolu dune facon dbfinitive. Dautre part, un progrksconsidbrable a btb apportb dans le cas oh le Nombre de Reynolds magnbtique est trksinfbrieur au Nombre de Reynolds cinbtique. Dans ce cas particulier, le champ magnbtiquepeut Ctre traitb par une mbthode analogue ii celle utilisbe pour la diffusion turbulenle, B cettedifference prks que le champ magnbtique est une quantitb vectorielle transportbe dune faconpassive, tandis que la chaleur, ou la concentration dune matikre qui diffuse sont des quan-titbs scalaires. La question expbrimentale qui savbre la plus importante est de trouver desmoyens pour rbaliser un bcoulement turbulent de plasma qui soit simple et bien dbfini.