Table Of ContentAstron.Nachr./AN000,No.00,000–3(2010)/DOI draftversionofJanuary25,2010
A Self-Consistent Treatment of the Electromotive Force
in Magnetohydrodynamics for Large Diffusivities
A.Courvoisier1,D.W.Hughes1,⋆,andM.R.E.Proctor2
1 DepartmentofAppliedMathematics,UniversityofLeeds,LeedsLS29JT,UK
2 DAMTP,CentreforMathematicalSciences,UniversityofCambridge,CambridgeCB30WA,UK
0
1 Thedatesofreceiptandacceptanceshouldbeinsertedlater
0
2
Keywords magnetohydrodynamics(MHD)–magneticfields–turbulence
n
The coupled equations that describe the effect of large-scale magnetic and velocity fields on forced high-diffusivity
a
magnetohydrodynamicflowsareinvestigatedthroughanextensionofmeanfieldelectrodynamics.Ourresultsgeneralise
J
thoseofRa¨dler&Brandenburg(2010),whoconsiderasimilarsituationbutassumethattheeffectoftheLorentzforceon
5 themomentumequationcanbeneglected.Newmeancouplingtermsareshowntoappear,whichcanleadtolarge-scale
2 growthofmagneticandvelocityfieldsevenwhentheusualα-effectsareabsent.
] (cid:13)c 2010WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
R
S
h. 1 Introduction Equation(3)canbeexpressedas
p
L(b)=∇×(u×B ), (4)
- Magnetohydrodynamicturbulence,particularlyin the con- 0
o
r text of astrophysics,is often studied within the framework whereLisalinearoperator.Ifthesmall-scalefieldcangrow
st of mean field electrodynamics, an elegant closure theory intheabsenceofameanfield,i.e.iftheequationL(b)=0
a that describes the evolution of mean quantities in termsof hasexponentiallygrowingsolutions,thenasmall-scaledy-
[
transportcoefficientsdeterminedfromaveragedsmall-scale namoisoperative,andtheinterpretationofanymeanfield
1 propertiesoftheflowandmagneticfield(see,forexample, equationsbecomesproblematic(Cattaneo&Hughes2009).
v Moffatt1978,Krause&Ra¨dler1980). If,however,non-decayingsolutionsforb dependcrucially
8 onanon-zeromeanfieldB ,thenb,andhenceE,depend
9 Thesimplestformofthetheoryconsidersthecasewhen 0
linearlyon B, and itis customaryto expressE as a series
3 the magneticfield B can be regardedas kinematic,evolv-
inBanditsspatialderivatives(thoughseeHughes&Proc-
4 ingundertheinfluenceofavelocityU,butexertingnoin-
1. fluencebackonthevelocity.Onlytheinductionequationis tor(2010)foradiscussionofthesignificanceoftheomitted
temporalderivatives),usuallywrittenas
0 then of significance.The theoryproceedsby decomposing
0 thevelocityandthemagneticfieldintomeanandfluctuating ∂B
1 parts, Ei =αijB0j +βijk ∂x0j +··· , (5)
: k
v U =U +u, B =B +b, (1)
0 0 whereitisanticipatedthatthelargespatialscaleofB will
i 0
X leadtorapidconvergence.Thetensorsα andβ depend
under an averagingprocedureobeyingthe Reynoldsrules. ij ijk
r onthestatisticalpropertiesofthevelocityfieldandonη.
Themostcommonpracticeistoadoptspatialaveraging,as-
a
The above kinematic formulation can be regarded as
sumingadistinctscaleseparationbetweenthatofthelarge-
the study of the evolution of weak, long wavelength mag-
scale (mean)fields and the small-scale (fluctuating)fields.
neticfieldperturbationstoapre-existingnon-magneticflow.
If,forsimplicity,weassumethatthereisnomeanflow(i.e.
Onemaythenextendthisideasoastoconsiderlongwave-
U =0),thenaveragingtheinductionequationleadstothe
0
lengthperturbations,inboththemagneticfieldandtheve-
followingequationsforthemeanandfluctuatingmagnetic
locity, of a magnetohydrodynamic [MHD] state involving
fields,
∂B both small-scale field and flow; here the perturbations are
0 =∇×E +η∇2B0, (2) againinfinitesimal,butthe backgroundstate is, in general,
∂t
fully nonlinear. This problem has been tackled recently in
∂b a variety of ways. Courvoisier, Hughes& Proctor (2010a)
=∇×(u×B )+∇×G+η∇2b, (3)
0
∂t considered the evolution of linear, three-dimensional per-
whereE = hu×biisthemeanelectromotiveforce(emf), turbations to two-dimensional MHD basic states resulting
fromprescribedforcingsinthemomentumequationinthe
G=(u×b)−hu×biandηisthemagneticdiffusivity.
presenceofabackgroundmagneticfieldwithnon-zeroflux.
⋆ Correspondingauthor:e-mail:[email protected] Courvoisieretal.(2010a)stressedtheimportanceoftreat-
ingthemagneticandvelocityfieldsonanequalfooting,and
(cid:13)c 2010WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
Astron.Nachr./AN(2010) 1
∂B
showedhowthelinearevolutionofthemeanfieldandmean R +U ·∇B
m
flow are both dependenton terms proportionalto both the (cid:18) ∂t (cid:19) (7)
meanfieldandthemeanflow.InanAppendix,Courvoisier =R (B·∇U)+∇2B+h,
m
etal.(2010a)outlinedtheformalextensionofthetheoryto
whereΠrepresentsthetotalpressure.Thevelocityisscaled
describe perturbationsof a three-dimensional,MHD back-
withU =FL2/ηandthemagneticfieldwithB =HL2/η,
ground state. This work is extended in Courvoisier et al.
whereF,HandLarerepresentativescalesfortheapplied
(2010b), which considers in more detail the evolution of
body force, the applied electromagnetic force and length.
long wavelength perturbations to a fully nonlinear three-
The dimensionless parameters are the magnetic Reynolds
dimensional,small-scaleturbulentstate,asmayresultfrom
number R = UL/η = FL3/η2; the magnetic Prandtl
thenonlinearsaturationofasmall-scaledynamo.Ra¨dler& m
number,P =ν/η;andχ=(H2/ρµ )/F2,whereρisthe
Brandenburg (2010) have also recently considered the na- m 0
(constant) density and µ the magnetic permeability. The
tureoftheemfinMHDturbulence.Theymadethesimpli- parameter χ, which may0be written as M−2, where M
fyingassumptionthatboththebasicstateandperturbations A A
istheAlfve´nicMachnumber,describestherelativeimpor-
toitwerekinematic,omittingtheLorentzforcethroughout.
tanceofthemagneticfieldandthevelocityinthebasicstate.
However,they allowed a basic MHD state to arise by pre-
ThesituationinvestigatedbyRa¨dler&Brandenburg(2010)
scribinganexternalelectromotiveforceintheinductioneq-
isrecoveredbylettingχ→0.
uation,inadditiontoanexternalbodyforceinthemomen-
We now introduce applied uniform velocity and mag-
tumequation.Theythenconsideredtheinfluenceofaweak
netic fields, U and B , with the aim of evaluating the
meanvelocity(butnomeanmagneticfield),theirmainre- 0 0
essential mean quantities in the momentum and induction
sultbeingthattheensuingmeanemfhasacomponentpro-
equations. We assume that the energies in the mean flow
portionaltothemeanflow.Thisresulthadalsobeennoted
andfieldareofthesameorder,orsmaller,thanthoseofthe
byCourvoisieretal.(2010a:equations(A.17)and(A.18)).
basicstate.Ifweexpressthetotalvelocityasu+U andthe
It is interesting to note that in the model studied by 0
totalmagneticfieldasb+B ,thenthegoverningequations
Ra¨dler&Brandenburg(2010),itisnotactuallynecessaryto 0
taketheform,
makethesimplifyingassumptionofneglectingtheLorentz
force. In this paper we therefore extend their analysis by ∂u
retainingtheLorentzforcethroughout,thusallowingusto Rm +u·∇u+U0·∇u
∂t
studytheevolutionofmoregeneralMHDstates.However, (cid:18) (cid:19) (8)
=R (−∇π+χb·∇b+χB ·∇b)
weignorerotationandanyeffectsofmeanfieldgradients,in m 0
ordertofocusonthemostbasiceffects:themeanemfsand +Pm∇2u+f,
ReynoldsandMaxwellstressesthatariseowingtoimposed
meanvelocityandmagneticfields.Weadoptaslightlydif- ∂b
R +u·∇b+U ·∇b
m 0
ferent approach to that of Ra¨dler & Brandenburg (2010), ∂t (9)
(cid:18) (cid:19)
makingtheassumptionthatthemagneticReynoldsnumber =R (b·∇u+B ·∇u)+∇2b+h.
m 0
Rmissmall,soastomakeanalyticprogress.Inorderthat
We supposethatbothR andtheReynoldsnumberR ≡
we can consider a basic MHD state for small Rm, we re- m e
P R aresmall,withP oforderunity.Wemaythusex-
taintheexternalemfintheinductionequation,since,with- m m m
panduandbinpowersofR ,namely
outthis,anysmall-scalefieldwouldsimplydecay.Weshow m
thatwhenconsideringabackgroundMHDstate(asopposed
u=u(0)+R u(1)+··· , b=b(0)+R b(1)+··· .
toapurelyhydrodynamicstate),themagneticandvelocity m m
(10)
perturbationsmust,forconsistency,betreatedonthesame
Similarly, we may expand the fields u′ = u−U and
footing.Thus, one must considernotjust the mean emf in
′
b = b −B, induced by the imposed mean velocity and
the averaged induction equation, but also the mean stress
magneticfields,as
tensorsintheaveragedmomentumequation.
u′ =R u′(1)+··· , b′ =R b′(1)+··· , (11)
m m
2 The mean emfandthe meanstress tensor
where
Our treatment starts with a basic MHD state consisting of
P ∇2u′(1) =(U ·∇u(0)−χB ·∇b(0)), (12)
fluctuating,small-scalevelocityandmagneticfields,U and m 0 0
B,thataredrivenbyanappliedbodyforceandanapplied ∇2b′(1) =(U ·∇b(0)−B ·∇u(0)). (13)
0 0
electromagnetic forcing. The fluid has kinematic viscosity
νandmagneticdiffusivityη.Thebasicstateisdescribedby Therighthandsideof(12)issolenoidal,andhencethereis
thefollowingnon-dimensionalequations, nopressuregradientterm.Weassumethatthereisnomean
emf in the basic state, described by equations (6) and (7);
∂U
R +U ·∇U hencetheleadingordercontributiontotheemfisgivenby
m
∂t (6)
(cid:18) (cid:19)
=Rm(−∇Π+χB·∇B)+Pm∇2U +f, E =Rmhu(0)×b′(1)+u′(1)×b(0)i. (14)
www.an-journal.org (cid:13)c 2010WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
2 A.Courvoisier,D.W.Hughes&M.R.E.Proctor:Self-consistenttreatmentoftheelectromotiveforce
Toaidcalculation,wenowfurthersupposethatf andhare forthemeanemfandthemeanstresstensorcanbeusedto
independentoft,spatiallyperiodicandmonochromatic,so determinetheevolutionofvelocityandmagneticfieldsde-
that pendingslowlyonspaceandtime.EvenforsmallRm,ifthe
∇2f =−λ2f, ∇2h=−λ2h, (15) lengthscales are sufficientlylongthenthe termsinvolving
with|λ|∼L−1.Itcanbeshownthatu(0)andb(0) arethen first derivativesdominateoverdiffusionterms, which may
therefore be neglected. In this case, the mean momentum
alsomonochromatic.Hencewecanwrite
andinductionequationstaketheform
hu(0)×b′(1)i=−λ−2h∇2u(0)×b′(1)i (16)
∂ U +∂ R =−∂ P , (22)
=−λ−2hu(0)×∇2b′(1)i, (17) T 0i Xj ij Xi 0
∂ B =ǫ ∂ E , (23)
andsimilarlyfortheothertermin(14).Allsurfacetermsin- T 0i ijk Xj k
volvedintheintegrationsbypartsneededtotransform(16) whereT andX arelongtimeandspacevariables.Wenow
to (17) vanish, by assumption of spatial periodicity. Then ignore the familiar α-effect terms, and look for solutions
usingequations(12)and(13)weobtainexactly, proportionaltoeiK·X+sT.Afterappropriatescalingofthe
time and space variables, to absorb the factors of λ2R−1,
λ2Rm−1Ei =U0·h(1−Pm−1)ǫijk∇u(j0)b(k0)i andeliminationofthepressureusing∇·U0 =0,weobtmain
±
+B0·hǫijk(u(j0)∇u(k0)+Pm−1χ∇b(j0)b(k0))i. (writingWijk =Qikj ±Qkij)
(18) sB = 1−P−1 W− ·iK U , (24)
0i m ikl k 0l
Tmhaegnteertmicsαin-eBffe0cmts,awyhbielerethcoegtenrimsedinaUsthehaussuthaelksianmeteicfoarnmd sU0i =(cid:0)1+Pm−1(cid:1)χSijWj+kl ·iKkB0l, (25)
0
asthatdiscussedbyRa¨dler&Brandenburg(2010)inamore whereS =δ −(cid:0)K K /|K(cid:1)|2.WhileW+,beingsymmet-
ij ij i j
generalcontext. ricinitsfirsttwoarguments,willvanishinisotropicsitua-
Thusfarthereislittlethatisnew.However,crucially,we tions,thereisnoreasonforitto vanishingeneral.Indeed,
canseethatbecausethemomentumequationnowcontains for a simplified probleminvolving a 2D basic state, Cour-
B0, through the Lorentz force, we also have a non-trivial voisier et al. (2010a)have exhibitedcases wherethis term
expressionforthemeanstresstensor,Rij,atleadingorder, isnon-zero.Becausetheequationsarecoupled,exponential
namely growthispossibleevenwithouttheα-effect.Thesquareof
R−1R =hu′(1)u(0)+u(0)u′(1)i thegrowthratesisgivenastheeigenvalueofthematrix
m ij i j i j
(19)
−χhb′(1)b(0)+b(0)b′(1)i. −χ 1− 1 W−K K S W+ , (26)
i j i j P2 ikl k p lm mpq
(cid:18) m(cid:19)
Under the assumptions (15), and via manipulationsanalo-
and it appears that solutions with a positive real part of s
goustothoseleadingto(18),weobtain
can be found in a wide variety of situations, providedthat
λ2R−1R =χB ·h(1+P−1)(∇b(0)u(0)+∇b(0)u(0))i. P 6=1.
m ij 0 m i j j i m
(20)
Itmighthavebeenexpectedthattherewouldalsobeaterm
3 Discussion
proportionalto U in thisexpression.Were suchatermto
0
exist, it would be analogous to the AKA (anisotropic ki-
Starting from a small-scale, MHD basic state, forced via
netic α) effectintroducedby Frisch, She & Sulem (1987).
a body force and an externally applied emf, we have con-
There are such contributions present here, but they cancel
sideredthemeanemf,E,andthemeanReynolds-Maxwell
out; in fact it can be shown, in general and not just for
stresstensor,R ,resultingfromtheimpositionofspatially
monochromatic velocity fields, that the AKA effect van- ij
uniform,steady,velocityandmagneticfields.Undertheas-
ishesinthefirstordersmoothingapproximation,forsteady
sumption that the fluids and magnetic Reynolds numbers
forcingasassumed.However,anon-zerocontributionmay
aresmall,wehaveobtainedanalyticexpressionsforE and
beexpectedformoregeneralforcings.Inspiteofthis,itis
R ,(18)and(20).Thesearelinearinthemeanfields,with
plainfromtheformofthetermsinexpressions(18)and(20) ij
transport coefficients dependent on the means of various
thattheevolutionequationsforthelarge-scalevelocityand
quadraticfluctuatingquantitiesofthebasicstate.
magneticfieldswill genericallybe coupledtogether.Inter-
ThemainpointofourpaperisthatinatrueMHDstate,
estingly,bothtypesofcouplingtermdependonthetensor
in which neither the velocity nor the magnetic field can
∂u(0) ∂b(0) be regarded as dominant, it is imperative that the velocity
Q = i b(0) =− k u(0) . (21)
ijk ∂x k ∂x i and magnetic fields are treated on an equal footing. This
* j + * j +
isinsharpcontrasttotraditionalmeanfieldelectrodynam-
The above calculations have been performed for uni- ics, which is essentially a kinematic theory,describingthe
form, steady U and B . However, and this is the under- evolution of a (formally weak) large-scale magnetic field
0 0
lying principle of mean field electrodynamics, the results undertheinfluenceofa prescribedvelocity.Itisthenonly
(cid:13)c 2010WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim www.an-journal.org
Astron.Nachr./AN(2010) 3
themeanemfthatisofinterest;furthermore,themeanemf
depends only on the fluctuating velocity field. In the case
considered in this paper, however, E and R are of equal
ij
importance and, moreover, they both depend on both the
fluctuating velocity and the fluctuating magnetic field. We
believethatthisisanimportantmessage,inthatithighlights
theshortcomingsin,forexample,consideringonlytheemf,
and then only the part proportional to the mean magnetic
field(theα-effect),forMHDturbulence.Althoughitisof-
tenclaimedthattheα-effectinthenonlinear(MHD)regime
canbeexpressedasthedifferencebetweentheflowhelicity
andthecurrenthelicity(essentiallytheB termin(18)),our
0
analysis shows clearly that this tells only partof the story.
Inthegeneralcase,thecouplingbetweenthemeanvelocity
andmagneticfieldsissuchthatthegrowthofalarge-scale
disturbancecanbeexpectedevenwhentheusualkinematic
and magnetic α-effects vanish. Such a mode of instability
isnotaccessibleiftheinfluenceoftheLorentzforceonthe
fluidmomentumisdropped.
Acknowledgements. ThisworkwassupportedbytheUKScience
andTechnologyFacilitiesCouncil.
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www.an-journal.org (cid:13)c 2010WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
