Table Of ContentThe alpha-effect in a turbulent liquid-metal plane Couette flow
G. Ru¨diger
Leibniz-Institut fu¨r Astrophysik Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany and
Helmholtz-Zentrum Dresden-Rossendorf, P.O. Box 510119, D-01314 Dresden, Germany∗
A. Brandenburg
Nordita, AlbaNova University Center, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden and
Department of Astronomy, Stockholm University, SE-10691 Stockholm, Sweden
(Dated: January 4, 2012)
TheelectromotiveforceinaplaneCouetteflowofconductingfluidsundertheinfluenceofalarge-
scale magnetic field is calculated for driven turbulence. In case of a vertically stratified intensity
2
of the turbulence the resulting α-tensor in presence of a horizontal shear also possesses diagonal
1
elements known as the so-called α-effect. The possibility to measure these components is probed
0
2 bybothquasilinear theory andnonlinear numericalsimulations. Forall magnetic Prandtlnumbers
<∼1 we find the α-component along the flow in the high-conductivity limit exceeding the other
n components. In this limit the α’s run linear with the magnetic Reynolds number Rm while in the
a low-conductivitylimittheyrunwiththeproductRm·Re. ForthesmallmagneticPrandtlnumbers
J
of liquid sodium a value for the electromotive force across the channel of order 0.05...0.5 mV for
3 magnetic fields of about 1 kGauss and velocity fluctuations of about 1 m/s is found for a channel
of(say) 50cm heightindependentofitswidth. Suchvaluesshouldbeappropriatetoberealized in
]
R a futurelaboratory experiment.
S
PACSnumbers: 47.27.Pa,47.65.+a,95.30.Qd
.
h
p
- I. INTRODUCTION Here, α is a pseudo-scalar formed by the rotation vector
o Ω and the anisotropy direction g with α∝ g·Ω. Often
r
t The electrodynamics in turbulent conducting fluids as the anisotropy direction is the gradient of the density
s
a provides the standard explanation of the existence of stratification in the fluid is used, but it is also possible
[ magnetic fields of the cosmic objects. The excitation of thatanintensitygradientclosetorigidboundariesforms
fields results from the interplay of two different elemen- the preferred direction.
1
v tary processes. It is known that the turbulent motions The resulting dynamo equation in rotating and strati-
2 are reducing large-scale electrical currents by inducing fied plasma is
5 an electromotive force (EMF) opposite to the direction ¯
6 of the current. One can write ∂B =rot αB¯−(η+η )rotB¯ , (3)
0 ∂t T
. (cid:18) (cid:19)
1
0 E¯=u×b=−µ η J¯, (1) which indeed has non-decaying solutions if the α-effect
0 T
2 exceeds a critical value (‘α2-dynamo’).
1 where η is the turbulent magnetic diffusivity. In stel- In many of papers the presented concept of a turbu-
T
v: lar convection zones its value exceeds the molecular (or lentdynamohasbeenappliedtoplanets,stars,accretion
i ‘microscopic’) value of the magnetic resistivity by many disks,galaxiesandgalaxyclusters(manyreferencesgiven
X
orders of magnitudes. by [1]. Only very few papers, however, deal with an ex-
r InthispapertheEMFisderivedforaturbulentfluidin perimental confirmation of the validity of relations (1)
a
thepresenceofashearflowU andauniformbackground and (2) in the laboratory. This is surprising insofar as
fieldB. Thefluctuatingflowcomponentsaredenotedby Eq.(3)asasimpleconsequenceofEqs.(1)and(2)seems
uandthefluctuatingfieldcomponentsaredenotedbyb. tosummarizeallourknowledgeabouttheelectrodynam-
Foraturbulentdynamotheenhanceddissipationmust ics in rotatingturbulent fluid conductors. Generally, the
be overcomeby aninduction process whichdoes not run validityofEq.(1)isnotseriouslydoubted. However,the
with the electriccurrent. One alsoknowsthat under the existinglaboratoryexperimentsreportanincreaseofthe
influenceofglobalrotationandauniformmagneticfield, molecular value of η by only a few percent. This is a
anisotropic turbulence produces an EMF parallel to the surprising situation as the eddy-concept of the effective
field, i.e. dissipation in turbulent media governs much of the cos-
micalphysicsfromtheclimateresearch,geophysicsupto
E¯=αB¯ −.... (2) the theory of star formation and quasars.
Anevenmoredramaticsituationholdswithrespectto
the α-effect. There are one or two existing experiments
on the basis of the idea that the α-effect is essentially
∗Electronicaddress: [email protected] a measure of the swirl of the fluid. It has indeed been
2
shown that a fluid with imposed helicity (imposed by Theshearflowmayexistinaturbulencefieldwhichdoes
rigid, swirling channels), produces an EMF in the direc- not possess any other anisotropy than that induced by
tion of an imposed field; see [2–4]. It is not yet shown, the shear (5) itself. The one-point correlation tensor is
however, that a rotating fluid with helicity which is not
imposed(butresultsfromtheglobalrotationofthefluid) Qij =ui(x,t)uj(x,t). (6)
leadstoanobservableα-effect. Innature,helicityisusu-
ally due to the interaction of rotating turbulence with Thecorrelationtensormaybeconstructedbyaperturba-
density stratification. The present proposal also aims in tionmethod. Thefluctuatingvelocityfieldisrepresented
the direction to probe the possibilities of such a more by a series expansion,
rigorous α-effect experiment. The many difficulties for
u=u(0)+u(1)+u(2)+... , (7)
such an experiment which we shall demonstrate make it
understandable that an experiment without prescribed
where the upper index shows the order of the contribu-
helicity notyetexists inthe theoryandinthe MHDlab-
tions in terms of the mean shear flow.
oratory.
The zero-order term represents the ‘original’ isotropic
It is easy to formulate the overall problem of all α-
turbulencewhichisassumedasnotyetinfluencedbythe
effect experiments. With Eq. (2) the potentialdifference
shear. The spectral tensor for the original turbulence is
betweentheendplatesofthecontainerindirectionofthe
mean magnetic fields is E(k,ω) k k
Qˆ(0) = δ − i j , (8)
ij 16πk2 ij k2
∆Φ=α BH (4) (cid:18) (cid:19)
dyn
wherethepositive-definitespectrumEgivestheintensity
with H as the distance between the endplates. Hence of isotropic fluctuations, i.e.
for (say) H ≃ 100 cm and (say) B ≃ 1000 Gauss the
∞∞
potential difference is ∆Φ = |α | in mV and cm/s, re-
spectively. The maximum α-vdaylune is of the order u u(0)2 = E(k,ω) dk dω. (9)
rms
hence ∆Φ<∼urms in mV. For urms ≃1 cm/s the induced Z0 Z0
potentialdifference,therefore,ismaximally1mV.Acon-
For analytical calculations the one-parametric spectrum
tainer of 5 cm radius rotating with 1 Hz has a linear
outer velocity of more than 30 cm/s so that u ≃ 1
rms 2 w
cm/smightbeconsideredasaconservativeestimate. We E(k,ω)= Eˆ(k), (10)
πω2+w2
find as a necessary condition for any α-experiment that
onemustbeabletomeasurepotentialdifferencessmaller canbeusedwhichprovidesE ∝δ(ω)inthelimitofw →
than a few mV. The α-experiment in Riga worked with 0 and leads to white noise for large w. The correlation
B ≃1kGaussandvelocitiesoforderm/ssothatthe∆Φ time of the turbulence is defined by τ =1/w.
corr
exceeded 10 mV. This experiment, however, used a pre- By definition, the magnetic-diffusivity tensor relates
scribedhelicalgeometrytomimicthesymmetrybreaking the mean electromotive force (1) to the gradients of the
between left and right helicity [2]. mean magnetic field via the relation E¯ = η B¯ . This
i ijk j,k
If the basic rotation is not uniform then its shear tensor for an isotropic turbulence influenced by a mean
induces toroidal magnetic fields so that for sufficiently field shear flow (5) has been constructed up to the first
strong shear even the α-effect can be rather small (‘αΩ- order in the shear [5], [6]. It has been shown that the
dynamo’). Also turbulence subject to a plane shear flow combination of the shear and the shear-induced parts of
(without rotation!) of a liquid metal is able to work as the magnetic-diffusion tensor are not able to operate as
a dynamo if the turbulence intensity is stratified in the a dynamo.
direction(orthogonaltotheshearflowplane,see[5].Aba- It has also been shown that the shear in combination
sic rotation may thus not be the only flow system whose with a stratified turbulence provides a helical turbulence
influenceenablestheturbulencetogenerateglobalfields. whichleadsinEq.(2)toanα-effect[5]. Hereαmustbea
InthepresentpaperaplaneCouetteflowisconsidered pseudotensor so that an ǫ-tensor has to appear in the α-
to analyze the characteristic issues of the corresponding coefficients. The construction of the EMF E¯i = ǫijkujbk
α effect and to design a possible experiment to measure is the only possibility for the ǫ-tensor to appear. There-
its amplitude. fore, the subscript of E is always a subscript of the ǫ-
i
tensor. Astheǫ-tensorisof3rdrankaninhomogeneityof
turbulence with the stratification vector, g =∇logu2
rms
II. THE MEAN ELECTROMOTIVE FORCE with u = u2, must also be present for the α-effect
rms
to exist. If the shear flow is included to its first order,
p
Consider a plane shear flow with uniform vorticity in the general structure of the α-tensor is
the vertical z direction, i.e.
α = γǫ g + α ǫ U¯ +α ǫ U¯ g +
ij ijk k 1 ikl j,k 2 ikl k,j l
U¯ =Sx. (5) + α ǫ g U¯ +α ǫ U¯ g +α ǫ U¯ g .(11)
y 3 ikl j l(cid:0),k 4 ikj l,k l 5 ij(cid:1)k k,l l
3
If the stratification is along the vertical z-axis it follows The first expression on the RHS for ν → 0 forms a δ-
from (11) that function, so that
∞
α = α g S =α S,
xx 2 z x 2π E(k,0)
αyy = −α1gzS =αyS, α1 = 15w ηk2 dk+...
Z
α = α g S =α S 0
zz 3 z z
2π
αxy = −αyx =γgz =Γ. (12) ≃ 15Rm ℓ2corr+... (18)
The turbulent pumping is described by α . The
xy with
anisotropy of the α-tensor is described by the difference
between αx and αy. In the used geometry the αyy (i.e. Rm= u2rmsτcorr ≃ 3ηT. (19)
thecoefficientα1definedbelow)playsthemainroleinall η η
cosmicalapplicationswhileintheproposedexperimentin
a turbulent shear flow the coefficient α is probed which The missingtermsin(18) resultto be ofthe sameorder.
2
produces the EMF perpendicular to the flow. The white-noise approximation mimics the high-
The coefficients of (11) read conductivity limit which holds for cosmical applications.
In this approach the spectrum does not depend on the
∞∞
1 ηk2E(k,ω) frequencyωuptoamaximumvalueωmaxfromwhichone
γ = dk dω (13) the power spectrum vanishes. It is a turbulence model
6 ω2+η2k4
ZZ withveryshortcorrelationtime,i.e.τ ≃1/ω . One
0 0 corr max
finds from (17) after integration
for the pumping term and
∞
∞∞
π E(k,0)
α = dk. (20)
αn = AnE(k,ω)dk dω, n=1,2,...5, (14) 1 6η k2
Z
ZZ 0
0 0
for the α-effect with The resulting integral can be written as τcorrℓ2corru2rms so
that
4νη3k8+2ω2η(ν+η)k4
A = + π
1 15(ω2+ν2k4)(ω2+η2k4)2 α1 = 6Rm ℓ2corr, (21)
η2k4 η2k4−3ω2
+ , which hardly differs from the result (18). As it must the
15((cid:0)ω2+η2k4)3(cid:1) factorπ/6alsoappearsinFig.1(middle, bottom)asthe
η2ν3(4η−5ν)k12 value of I1 for τ = 0. For Pm = 1 the corresponding
A = − −
2 60(ω2+ν2k4)2(ω2+η2k4)2 expression is
ω2ν 28η3−4η2ν+12ην2+5ν3 k8 α = π Rm ℓ2 . (22)
− − 1 15 corr
60(ω2+ν2k4)2(ω2+η2k4)2
(cid:0) (cid:1)
Note that the value for small Pm exceeds the value for
ω4η(η+36ν)k4−5ω6
− Pm = 1. For given η smaller ν lead to higher values of
60(ω2+ν2k4)2(ω2+η2k4)2
the EMF.
8ω2k4(ω2+νηk4)η(ν +η) The full expression might be written in the form
A = − −
3 30(ω2+ν2k4)2(ω2+η2k4)2
α ≃I Rm ℓ2 , (23)
3η4k8−24ω2η2k4+5ω4 1 1 corr
− (15)
30(ω2+η2k4)3 whereI1isnumericallygiveninFig.1. Itistheresultofa
numerical integrationunder use of a spectral function of
for the kernels. Only the terms occurring in (12) have the form exp(−τ2ω2) with the dimensionless parameter
been given. Note that A for ν →0 exists
3
τ =ηk2τ ≃1/Rm. (24)
11η4k8−16ω2η2k4+5ω4 corr
A =− . (16)
3 30(ω2+η2k4)3 While small τ represent the high-conductivity limit the
larger τ represent the low-conductivity limit. Obviously,
For small Pm the expression for A simplifies to
1 thehigh-conductivitylimitcanbemodeledwithawhite-
noise spectrum (Eq. (10) with w → ∞) while the low-
1 1 4νη3k8
A = + conductivitylimitcanbetreatedbyuseoftheδ-function
1 15(ω2+η2k4)2(ω2+ν2k4 (Eq. (10) with w→0). It follows from (15)
1
η2k4 3η2k4−ω2
1 4
+ . (17) α = 1+ Rm2 ℓ2 , (25)
ω(cid:0)2+η2k4 (cid:1)) 1 15 Pm corr
(cid:18) (cid:19)
4
FIG.2: I1 forvariousPm. Notethatforgivenmagneticdif-
fusivityη fluidswithmuchsmallerviscosity(likesodiumand
gallium)areevenbetterqualifiedforlaboratoryexperiments.
rotation (Ru¨diger 1978).
Figure 2 demonstrates the behavior of the numerical
integralsfordecreasingPmandgrowingτ. Theintegrals
arerunningwith(1+Pmτ)−1 sothatforPmτ ≫1 the
relation α ∝ Re Rm indeed results. Again, for given
1
magnetic diffusivity smaller values of the viscosity en-
hance (!) the resulting α-effect.
In experiments with liquid metals the Rm is of order
0.1...1 so that τ ≃ 1...10. In this domain and for small
Pm the coefficient I hardly changes with τ (see Fig. 1).
1
Its approximate value is 0.5 which is very close to the
value π/6 valid in the high-conductivity limit. The rea-
son is that for small Pm the transition of I to the low-
1
conductivity limit only happens at rather high values of
τ (Fig. 2).
Similar calculations for α lead to α ≃−I Rm ℓ2 ,
2 2 2 corr
with I also plotted in Fig. 1. For medium τ and small
2
Pm we find I ≃I , i.e.
2 1
α ≃−0.5 Rm ℓ2 . (26)
2 corr
In the low-conductivity limit we have
1 4
α =− −5 Rm2 ℓ2 , (27)
2 60 Pm corr
FIG. 1: The numerical values of the coefficients I1, I2 and (cid:18) (cid:19)
I3. Top: Pm=1, middle: Pm=10−5, bottom: Pm=10−6. whichchangesits signatPm=0.8. Forsmall Pmit can
Note that in the high-conductivity limit (τ →0, white noise also be written as
spectrum) the coefficients I1 and I2 for Pm ≪ 1 exceed the
values for Pm = 1. Laboratory experiments with turbulent α =− 1 RmRe ℓ2 , (28)
flows of liquid metals work with τ ≃1...10. 2 15 corr
In summary, the plots in Fig. 1 reveal an important in-
fluence of the viscosity value for given diffusivity value.
so that Pm = 1 leads to α1 = 13Rm2 ℓ2corr and for small ForsmallPminthelow-conductivitylimittheratioofI2
magneticPrandtlnumber one finds α = 4 Re Rmℓ2 , to I approachesunity while it is very small for Pm=1.
1 15 corr 1
with Re = Rm/Pm as the Reynolds number. Hence, in The magnetic Prandtl number is highly different for liq-
the low-conductivity (small Rm) limit the α effect runs uid metal and for numericalsimulations. Figure 1 shows
with Re·Rm while in the high-conductivity limit (large that in numerical simulations the α should be much
xx
Rm)theαeffectrunswithRm. Verysimilarexpressions smallerthantheα whatisnottrueforlaboratorycon-
yy
alsooccuriftheshearflowisformallyreplacedbyabasic ditions with their small Pm.
5
The α yields α ≃−I Rm ℓ2 with I of about 0.1 where D/Dt = ∂/∂t+Sx ∂/∂y is the advective deriva-
3 3 3 corr 3
for small Pm (see Fig. 1). The horizontal components tive with respect to the imposed shear flow only, and
′
of the α-tensor are thus of the same order of magnitude u×bT =u×bT−u×bT is the fluctuating part of
but the vertical component appears to be smaller. Note u×bT. We compute the correspondingmeanelectromo-
that for Pm = 1 the α1 strongly exceeds the α2 but it (cid:0)tiveforc(cid:1)e,ET =u×bT,whichisthenrelatedtoBT and
does not for small Pm. The signs of all components are
its curl, µ JT =∇×BT, via
identical. 0
Thepumpingtermdoesnotdependontheshear. After
T T T
(13) for small η it does not run with 1/η, so that it is Ei =αijBj −ηijµ0Jj. (33)
simply
We use 4 different test fields with x or y components
γ ≃hu′2iτcorr ≃η Rm. (29) being proportionalto sinkz or coskz. The x andy com-
ponents ofEq. (33) constitute then 8 equations for the 4
It can only be measured if the external field B lies in
0 relevant components of α (z,t) and η (z,t).
the shear plane. ij ij
We adoptperiodic boundaryconditionsinthe y direc-
tion, shearing–periodic boundary conditions in the x di-
III. NUMERICAL SIMULATIONS rection, and stress-free perfect conductor boundary con-
ditions in the z direction, i.e.,
It is straightforward to verify the existence of an ∂ u =∂ u =u =aT =aT =∂ aT =0. (34)
z x z y z x y z z
α-effect using numerical simulations of non-uniformly
forced turbulent shear flows in Cartesian coordinates. A numerical resolution of 643 mesh points was found to
We perform simulations in a cubic domain of size L3, be sufficient. The Pencil Code[18] has been used for
so the minimal wavenumber is k ≡k =2π/L. We solve
1 all calculations.
the governing equations of compressible hydrodynamics
Simulations are performed for a number of different
withanisothermalequationofstatewithconstantsound
parameter combinations. The quantities S and g are
z
speed c . The set of governing equations is
s positive in the calculations presented here, i.e. the ba-
DU sic velocity (5) grows in the positive x direction while
=SU yˆ−c2∇lnρ+f +ρ−1∇·2ρνS, (30)
Dt x s the turbulence intensity grows in the positive z direc-
tion. To make contact with laboratory experiments, we
Dlnρ focus here on the case of low conductivity and choose
=−∇·U, (31)
Dt Rm≡ urms/ηkf = 0.2, which is consistent with our defi-
nition of Eq. (19) with a Strouhal number of unity, i.e.,
where D/Dt=∂/∂t+U ·∇+Sx∂/∂y is the advective
τ u k =1.
derivative with respect to the full velocity field (includ- corr rms f
As in earlier work using fully helical turbulence, we
ing the shear flow), U is the departure from the mean
presentthecomponentsofα andη innormalizedform
shear flow, and S = 1(∂ U +∂ U )− 1δ ∇·U is the ij ij
ij 2 i j j i 3 ij in terms of α = u /3 and η = u /3k , respec-
trace-less rate of strain matrix. The flow is driven by 0 rms T0 rms f
tively. The shear of the background flow is normalized
a randomforcing function consisting of nonhelicalwaves
with the speed of sound, i.e.
with wavenumbers whose modulus lies in a narrowband
aroundanaveragewavenumberkf =5k1[8]. Wearrange 2πsurms
the amplitude of the forcing function such that the rms S =scack1 = (35)
MaD
velocity increases with height while the maximum Mach
numberremainsbelow0.1,sothe effectsofcompressibil- with the Mach number Ma = u /c . Here, D (= L)
rms ac
ity are negligible. is the distance between the walls, i.e. the width of the
Weusethekinematictest-fieldmethod[9]intheCarte- domain in the x direction. In the simulations we work
sian implementation [10] to compute from the simula- with s=0.2 and Ma=0.05. Hence,
tions simultaneously the relevant components of the α
effect and turbulent diffusivity tensors, αij and ηij, We αyy =−12 I sℓ2corr Rm. (36)
do this by solving an additional set of equations govern- α 1 D2 Ma
0
ingthedepartureofthemagneticfieldfromasetofgiven
Following Eqs. (12), both the streamwise and the span-
meanfields. Thismeanfieldisreferredtoasthetestfield
wise α-tensor components, α and α , should be neg-
and is marked by the superscript T. For each test field yy xx
BT,wefindacorrespondingfluctuationbT =∇×aTby ative. As the simulations are done for Pm= 1 the |αyy|
should strongly exceed the value of |α |.
solving the inhomogeneous equation for the correspond- xx
Figure 3a shows these predictions as fulfilled. They
ing vector potential aT,
holdfor Rm=0.2 andk /k =5 so that 10 cells exist in
f 1
DaT =−SaTxˆ+U×bT+u×BT+ u×bT ′+η∇2aT, the verticaldirection. FromEq.(36) one finds αyy/α0 ≃
Dt y −0.01. Both the α-tensor components are negative and
(32) |α |dominates. Itsamplitude is indeed0.01. ForPm=
(cid:0) (cid:1) yy
6
FIG. 3: Simulations with positive shear (s=0.2): The numerical values for α (top) and the two shear–current terms ηxy and
ηyx (bottom).
1theI isstronglyreducedrelativetoI sothatthesmall earliermeasurementsofturbulentdiffusivityinturbulent
2 1
amplitudeofα inFig.3abecomesunderstandable. For shearflows[10,11]. Themaximumofη =(η +η )/2
xx T xx yy
small magnetic Prandtl number this reduction does not is about 20% of the reference value η , which agrees
T0
exist (see Fig. 1). The simulations also confirm that the with the fact that for Rm≪1, η /η ≈Rm [12].
T T0
signs of the diagonal components of the α tensor only Let us turn to the off-diagonal components. For non-
depend on the sign of the product g S, as formulated in stratified shear flows one finds
z
the relations (12).
η =η S, η =η S, (37)
Next, we look at the off-diagonal components of α ; xy x yx y
ij
see Fig. 3b. Note that αyx ≈ −αxy, which corresponds sothattheη-termsdependonthesignandtheamplitude
to a turbulent pumping velocity in the z direction. It is of S. In a quasilinear theory (SOCA) the η proves to
y
negativefork1z <2.5,correspondingtodownwardtrans- be negative-definite[5]. Forpositiveshearthe coefficient
port,i.e.,downthegradientoftheturbulentintensity,as η ,therefore,isexpectedtobenegative. Thisresulthas
yx
expected [7]. Near the top of the domain, the gradient alsobeenconfirmedforRm≤200[10]. Thepossibilityof
ofthe turbulentintensity isreversedandsoisthe signof dynamo action in these nonhelical shear flows [10, 13] is
the pumping velocity, αyx. therefore not explained by the shear–current effect [14],
but instead by the incoherent alpha–shear dynamo [15].
The same dependence of the sign of η on that of S
yx
IV. SHEAR–CURRENT EFFECTS
holds also for η , but only for Pm of order unity and
xy
in the low conductivity limit (small Rm). Both the con-
Let us now discuss the components of the η tensor. ditions are fulfilled for our simulations. Experiments of
ij
Before we turn to the shear–current effect, we note that liquid metals, however, concern small magnetic Prandtl
the two relevant diagonal components are nearly equal, numbers for which the η results as positive. As simu-
x
i.e., η ≈ η (Fig. 3c). This high degree if isotropy of lations for such small Pm are not positive, only experi-
xx yy
turbulent diffusion in the xy plane has already been in ments can finally provide the sign of η . The numerical
x
7
simulations with positive shear and Pm= 1 indeed pro- same. Note that the statement bases on the assumption
duce the expected negative values for both η and η ℓ /H ≃const.inbothcases. Notealsothatturbulence
xy yx corr
(Fig. 3d). whichisstratifiedinthez direction,thereisnopumping
ifalsothemagneticfieldpointsinthespanwisedirection.
V. SHEAR-FLOW ELECTRODYNAMICS
The electromotive force across a channel after (12) is
VI. CONCLUSIONS
E =α g SB (38)
x 2 z 0
Laboratory studies of homogeneous dynamos are still
so that the potential difference δΦ between the walls in their infancy. The only working dynamo where the
x
with their distance D is flowpatternisnotstronhglyconstrainedbypipesorcon-
tainer walls is the experiment in Cadarache [16] where,
δΦx = α2gzSDB0 ≃−0.5Rmℓ2corrgzUB0 however, the effects of soft iron play an important and
≃ −Rm λ2 HUB (39) not well understood role. No contact with the α-effect
0
in mean-field theory has been possible in that case. The
with g = −2/H as the vertical scale height of the tur- present proposal of measuring the α-effect in an uncon-
z
bulence stratification and the ratio λ = ℓ /H. The strained turbulent flow would therefore be a major step
corr
amplitude of the mean shear flow is U. Note that the forward. Insuchanexperiment,thepseudo-scalarneces-
width D of the channel does not appear in (39). Hence, sary for producing helicity comes from the stratification
ofturbulentintensitygivingrisetoapolarvectorandthe
δΦ [mV]≃10 Rmλ2 H U B0 (40) vorticity associated with the shear flow giving rise to an
x 10cm m/s kGauss axialvector. Ithasbeenshown,therefore,thatthe basic
(cid:20) (cid:21)(cid:20) (cid:21)(cid:20) (cid:21)
effects of the theory of the turbulent dynamo which are
sothatwith(say)λ≃0.1andachannelof50cmheight, usuallyconsideredas specialpropertiesofrotatingfluids
a shear flow of 1 m/s subject to a magnetic field of 1000 can also be found for the plane-shear flow.
Gauss leads to a potential difference of The present work yields a detailed prediction about
the sign and magnitude of the tensor components of α
ij
δΦ ≃0.5 Rm [mV]. (41)
x and η . It may motivate the construction of a suitable
ij
experiment using liquid metals to achieve a measurable
For the (maximal) value of Rm ≃ 0.1...1 (u ≃ 1m/s
rms α-effect. The needed vertical stratification of the turbu-
and ℓ ≃ 5 cm) the measurements may provide a po-
corr lence intensity must experimentally imitated using grids
tentialdifferenceof0.05...0.5mV.Thenumbersarequite
with nonuniform mesh sizes and/or by walls of increas-
similar to those of the Riga experiment [2, 7]. The basic
ing/decreasing roughness of the walls in vertical direc-
difference is that in a shear flow experiment, the helicity
tion.
isnotprescribedbutitisnaturallyproducedbytheinter-
Wehaveshownthatthevaluesofαaresmallcompared
actionofthe (stratified)turbulence withthebackground
withnaiveestimates. Infact,itmaynotbepossiblethat
shear.
such flows could produce a supercritical dynamo in the
Iftheexternalfieldisperpendicular totheshearplane,
conceivable future. Nevertheless, even in the subcritical
thenthe resultforthe potentialdifference inthe z direc-
case,anα-effectshouldbemeasurable,whichwouldthus
tion results as
openthepossibilityofdetailedcomparisonsbetweenthe-
H I ory, simulations, and experiments. Once such a compar-
3
δΦ ≃δΦ . (42)
z x isonispossible,there willbe moredetails thatshouldbe
DI
2
investigated. One of them concerns the modifications of
After Fig. 1 the ratio I /I is of the order of 0.1 so that the results in the presence of imperfect scale separation
3 2
for D ≃ H the spanwise potential difference is smaller in space and time. For oscillatory dynamos, this effect
than the same value in the perpendicular x direction. can significantly lower the excitation conditions for the
For narrow channels with D ≃0.1H both values are the dynamocomparedtostandardmean-fieldestimates[17].
[1] G. Ru¨diger and R. Hollerbach, The Magnetic Universe: [3] R. Stepanov, R. Volk, S. Denisov, P. Frick, V. Noskov,
Geophysical and Astrophysical Dynamo Theory (Wiley- and J.-F. Pinton, Phys. Rev.E 73, 046310 (2006).
VCH,Berlin, 2004). [4] P.Frick,S.Denisov,V.Noskov,andR.Stepanov,Astron.
[2] M.Steenbeck,I.M.Kirko,A.Gailitis, A.P.Klawina,F. Nachr. 329, 706 (2009).
Krause,I.J.Laumanis,andO.A.Lielausis,Monats.Dt. [5] G. Ru¨diger and L. L. Kitchatinov, Astron. Nachr. 327,
Akad.Wiss. 9, 714 (1967). 298 (2006).
8
[6] K.-H.Ra¨dlerand R.Stepanov,Phys.Rev.E73, 056311 [13] T. A. Yousef, T. Heinemann, A. A. Schekochihin, N.
(2006). Kleeorin, I. Rogachevskii, A.B. Iskakov, S. C. Cowley,
[7] F. Krauseand K.-H.Ra¨dler, Mean-Field Magnetohydro- and J. C. McWilliams, Phys. Rev. Lett. 100, 184501
dynamicsandDynamoTheory(PergamonPress,Oxford, (2008).
1980). [14] I.RogachevskiiandN.Kleeorin,Phys.Rev.E68,036301
[8] N. E. L. Haugen, A. Brandenburg, and W. Dobler, (2003).
Phys.Rev.E 70, 016308 (2004). [15] E. T. Vishniac and A. Brandenburg, Astrophys. J. 475,
[9] M.Schrinner,K.-H.Ra¨dler,D.Schmitt,M.Rheinhardt, 263 (1997).
and U. R. Christensen, Geophys. Astrophys. Fluid Dy- [16] R.Monchaux,M.Berhanu,M.Bourgoin,M.Moulin,Ph.
nam. 101, 81 (2007). Odier, J. -F. Pinton, et al., Phys. Rev. Lett. 98, 044502
[10] A.Brandenburg,K.-H.Ra¨dler, M. Rheinhardt,and P.J. (2007).
Ka¨pyla¨, Astrophys.J. 676, 740 (2008). [17] M.RheinhardtandA.Brandenburg,Astron.Nachr.333,
[11] A.Brandenburg, Astron.Nachr. 326, 787 (2005). 80 (2012).
[12] S. Sur, A. Brandenburg, and K. Subramanian, [18] http://pencil-code.googlecode.com
Mon. Not. R.Astron. Soc. 385, L15 (2008).
