Table Of ContentThe small-scale dynamo: Breaking universality at high Mach numbers
Dominik R.G. Schleicher∗
Institut fu¨r Astrophysik, Georg-August-Universita¨t G¨ottingen,
Friedrich-Hund-Platz 1, D-37077 G¨ottingen, Germany
Jennifer Schober†
Universita¨t Heidelberg, Zentrum fu¨r Astronomie, Institut fu¨r Theoretische Astrophysik,
Albert-U¨berle-Strasse 2, D-69120 Heidelberg, Germany
Christoph Federrath‡
3 Monash Centre for Astrophysics, School of Mathematical Sciences, Monash University, Vic 3800, Australia
1 Universita¨t Heidelberg, Zentrum fu¨r Astronomie, Institut fu¨r Theoretische Astrophysik,
0 Albert-U¨berle-Strasse 2, D-69120 Heidelberg, Germany
2
n Stefano Bovino§
a Institut fu¨r Astrophysik, Georg-August-Universita¨t G¨ottingen,
J Friedrich-Hund-Platz 1, D-37077 G¨ottingen, Germany
8
1 Wolfram Schmidt¶
Institut fu¨r Astrophysik, Georg-August-Universita¨t G¨ottingen,
]
O Friedrich-Hund-Platz 1, D-37077 G¨ottingen, Germany
(Dated: January 21, 2013)
C
. The small-scale dynamo may play a substantial role in magnetizing the Universe under a large
h
range of conditions, including subsonic turbulence at low Mach numbers, highly supersonic turbu-
p
lence at high Mach numbers and a large range of magnetic Prandtl numbers Pm, i.e. the ratio
-
o of kinetic viscosity to magnetic resistivity. Low Mach numbers may in particular lead to the well-
r known, incompressible Kolmogorov turbulence, while for high Mach numbers, we are in the highly
t compressible regime, thus close to Burgers turbulence. In this study, we explore whether in this
s
a large range of conditions, a universal behavior can be expected. Our starting point are previous
[ investigations in the kinematic regime. Here, analytic studies based on the Kazantsev model have
shown that the behavior of the dynamo depends significantly on Pm and the type of turbulence,
1
andnumerical simulations indicateastrong dependenceof thegrowth rateon theMach numberof
v
the flow. Once the magnetic field saturates on the current amplification scale, backreactions occur
1
and the growth is shifted to the next-larger scale. We employ a Fokker-Planck model to calculate
7
themagneticfieldamplificationduringthenon-linearregime,andfindaresultingpower-lawgrowth
3
that depends on the type of turbulence invoked. For Kolmogorov turbulence, we confirm previous
4
resultssuggestingalineargrowthofmagneticenergy. Formoregeneralturbulentspectra,wherethe
.
1 turbulentvelocityv scaleswiththecharacteristiclengthscaleasu ∝ℓϑ,wefindthatthemagnetic
t ℓ
0 energy grows as (t/T )2ϑ/(1−ϑ), with t the time-coordinate and T theeddy-turnovertime on the
ed ed
3
forcing scaleof turbulence. ForBurgersturbulence,ϑ=1/2, aquadraticratherthanlinear growth
1
may thus be expected, as the spectral energy increases from smaller to larger scales more rapidly.
:
v The quadratic growth is due to the initially smaller growth rates obtained for Burgers turbulence,
i and thus implies longer timescales until saturation is reached. Similarly, we show that the char-
X acteristic length scale of the magnetic field grows as t1/(1−ϑ) in the general case, implying t3/2 for
r Kolmogorov and t2 for Burgers turbulence. Overall, we find that high Mach numbers, as typically
a
associated with steep spectra of turbulence, may break the previously postulated universality, and
introducea dependenceon theenvironment also in the non-linearregime.
I. INTRODUCTION solarsurface[1,2],galaxiesandgalaxyclusters[3–8],the
intergalactic medium [9] and the formation of the first
Thesmall-scaledynamohasbeensuggestedtooperate stars and galaxies [10–17]. It thus operates on a large
under a large range of different conditions, including the range of different conditions, concerning for instance the
magneticPrandtlnumberPm,i.e. theratioofkinematic
viscosityν tomagneticresistivityη,theMachnumberof
the turbulence , i.e. the ratio of turbulent velocities
∗ [email protected] M
to the soundspeed, and, mostlikely relatedto the Mach
† [email protected]
‡ [email protected] number, the expected type of turbulence in the system.
§ [email protected] Most studies of the small-scale dynamo performed so
¶ [email protected] far have focused on incompressible Kolmogorov turbu-
2
lence [18], assuming a scaling relation u ℓ1/3 between marize our main results in section 5.
ℓ
∝
turbulentvelocityu andlengthscaleℓ. ForKolmogorov
ℓ
turbulence, it was previously concluded that the mag-
neticenergygrowsexponentiallyinthekinematicregime II. NON-UNIVERSALITY IN THE KINEMATIC
[e.g. 19–22] and linearly once the backreactionsfrom the REGIME
magnetic field become important (e.g. [23–25]). The
latter was interpreted by Beresnyak [25] as evidence for The small-scale dynamo is well-studied in the kine-
universalityofthe small-scaledynamo,suggestingthata matic regime, where an exponential growth of the mag-
fixed fraction of the global turbulence dissipation rate is netic field is expected on the viscous scale. The growth
converted into magnetic energy. rate of the magnetic field can be calculated in the
However, observations of turbulence in molecular framework of the Kazantsev model, assuming homoge-
clouds [e.g. 26, 27] and numerical simulations of super- neous turbulence that is δ-correlated in time, or with
sonic turbulence [e.g. 28–31] often reveal steeper turbu- 3-dimensional magneto-hydrodynamical simulations. In
lentspectra,typically inbetweenthe incompressible Kol- this section, we discuss hints and evidence for a non-
mogorovturbulenceandthehighlycompressibleBurgers universal behavior in the kinematic regime.
turbulence [32]. So far, only a small amount of studies
have investigated the turbulent dynamo in this regime.
For instance, Haugen et al. [33] provided the first study A. Indications for non-universality in the
Kazantsev model
exploring the dependence of the dynamo on the Mach
number in simulations of driven turbulence, and Balsara
et al. [34, 35], Balsara and Kim [36] explored the ampli- Theamplificationofmagneticfieldsisgovernedbythe
fication of magnetic fields in turbulence produced from induction equation, which is given as
strongsupernovashocks. Thefirstsystematicstudycov-
∂ B~ = ~v B~ +η∆B~. (1)
eringturbulentMachnumbersfrom0.02to20anddiffer- t
∇× ×
enttypesofturbulencedrivinghasbeenpursuedbyFed-
We assume in the following B~ = 0, although we note
errath et al. [37], while the effect of a large range of dif- h i
that scenarios considering B~ = 0 have been recently
ferent turbulence spectra has been explored by Schober
h i 6
et al. [22] based on the Kazantsev model [19]. exploredbyBoldyrevetal.[54],MalyshkinandBoldyrev
[55, 56, 57]. In the Kazantsev model, the velocity field
We note that the small-scale dynamo has also been
andthe magneticfieldaredecomposedintoameanfield,
studied in the context of so-called shell models [38–41].
denoted with brackets , and a fluctuating component
The latter originate from shell models of hydrodynam- hi
denoted with δ:
ical turbulence, which originally considered turbulence
in 2D [42–44], but were extended to 3D once a descrip- ~v = ~v +δ~v, B~ = B~ +δB~. (2)
tion of kinetic helicity was obtained [45]. The first 2D h i h i
MHD shell model has been derived by Frik [46], while A central input is the correlation function of the turbu-
3D models have been developed by Brandenburg et al. lent velocity, which is δ-correlated in time and (in the
[47],Basuetal.[48],FrickandSokoloff[49]. Moresophis- absence of helicity) can be decomposed as
ticated processes such as non-local interactions [50, 51],
δv (~r ,t)δv (~r ,s) =T (r)δ(t s), (3)
anisotropies[52]andthe Halleffect [53]havebeen incor- h i 1 j 2 i ij −
r r r r
porated in more recent studies. These approaches allow T (r)= δ i j T (r)+ i jT (r),
tostudyboththeevolutionofthepowerspectrumaswell ij ij − r2 N r2 L
(cid:16) (cid:17)
as the saturated regime, and are highly complementary withr = ~r ~r andT ,T arethetransverseandlon-
1 2 N L
to the methods presented here. gitudinal|pa−rts o|f the correlation function, respectively
In the following, we will consider the small-scale dy- [58]. Thesamedefinitionscanbeappliedtothemagnetic
namointhekinematicandnon-linearregime,andpresent field, yielding a two-point correlation function M (r,t)
ij
evidence from existing and new calculations suggesting with transverse and longitudinal components M (r,t)
N
a strong dependence on the magnetic Prandtl number, andM (r,t). Unlikethevelocityfield,themagneticfield
L
as well as the Mach number of the flow. In section 2, is always divergence-free, leading to the additional con-
we summarize the evidence and indications for a non- straint
universal behavior in the kinematic regime, which has
1 d
beenderivedinpreviousstudies. Insection3,wepresent M = r2M . (4)
N L
2rdr
the first exploration concerning different types of turbu-
(cid:0) (cid:1)
lence during the non-linear phase of the dynamo, where As the Kazantsev model assumes that the flow is δ-
the backreaction of the magnetic field becomes impor- correlatedintime,conceptssuchasviscosityorthemag-
tant. We show that a linear growth is only obtained in netic Prandtl number cannot be directly incorporated
the case of Kolmogorov turbulence, while steeper power into the flow, as the turbulent velocity field is destroyed
laws result from turbulent spectra with ϑ > 1/3. We and regenerated at each instant, leaving no time for vis-
discuss the physical implications in section 4, and sum- cosity to act. However, it can be indirectly included by
3
adopting turbulent velocity spectra that become steeper with Rm=VL/η the magnetic Reynolds number, and η
below a given viscous scale ℓ . This is the approach em- the magnetic diffusivity. We thus observe a fundamen-
ν
ployed here. For a given relation of type tal difference between the limiting cases Pm 1 and
≪
Pm 1inthekinematicregime: ForPm 1,magnetic
u ℓϑ (5) ≫ ≫
ℓ field amplification occurs predominantly on the viscous
∝
scale, corresponding to the most negative range of the
intheinertialrange,thelongitudinalcorrelationfunction
potential. For Pm 1, on the other hand, the resis-
of turbulence can be parametrized as [22] ≪
tive scale becomes larger than the visous scale. Amplifi-
VL 1 Re(1−ϑ)/(1+ϑ) r 2 0<r <ℓ cation on the viscous scale is thus not possible, and the
3 − L ν strongestcontributionisclosetotheresistivescaledueto
TL(r)=0V3L(cid:16)(cid:16)1−(cid:0)Lr(cid:1)ϑ+1(cid:17) (cid:0) (cid:1) (cid:17) ℓLν<<rr,<L tohfethsehomrtaegdndetyi-ctifimeelds.dCeporernedsspoonnditnhgelyR,etyhneogldroswntuhmrbaeter
Re for Pm 1, and on the magnetic Reynolds number
with ℓν the viscous scale, L the driving scale of turb(u6)- RmThfoerrPesmu≫l≪ts 1ca[sneebaelsgoen60er].alized further for different
lence, V the turbulent velocity on scale L, Re = VL/ν
types of turbulence. In the limit Pm 1, one obtains
the Reynoldsnumberofthe gasandν the kinetic viscos- ≫
[22]
ity. Similarly, we have
(163 304ϑ)V
V3L 1−θ(ϑ)Re(1−ϑ)/(1+ϑ) Lr 2 0<r <ℓν Γϑ,Pm≫1 = −60 LRe(1−ϑ)/(1+ϑ). (13)
TN(r)=0V3L(cid:16)(cid:16)1−θ(ϑ)(cid:0)Lr(cid:1)ϑ+1(cid:17) (cid:0) (cid:1) (cid:17) ℓLν<<rr, <L In the regime PmΓ≪=1α,VonRemfi(n1−dsϑ)a/(1s+imϑ)ilar relation [5(91]4,)
with θ(ϑ)= (21 38ϑ)/5. As we expect an exponent(i7a)l L
− with the prefactor α defined through the quantities
growth of the magnetic energy as a function of time, we
make the following ansatz for the kinematic regime: a(ϑ)=ϑ(56 103ϑ), (15)
−
b(ϑ)=ϑ(79 157ϑ), (16)
M (r,t) 1 ψ(r)e2Γt. (8) −
L ≡ r2√κdiff 25+ 135a(ϑ)+(b(ϑ) 25)2 b(ϑ)
c(ϑ)= − − . (17)
Inserting (8) in the induction equation (1), one obtains q a(ϑ)
theKazantsevequation,whichisofthesameformasthe as
quantum-mechanical Schr¨odinger equation:
a(ϑ) ϑ−1 5
α= c(ϑ)1+ϑ exp π (ϑ 1) 2 . (18)
d2ψ(r) 5 s3a(ϑ) − − !
κ (r) +U(r)ψ(r) = Γψ(r). (9)
− diff d2r −
A numerical evaluation shows that these coefficients are
In this framework, the amplification depends on the ef- smaller by about two orders of magnitude in the limit
fective potential U(r) in Eq. (9), which depends on the Pm 1,assumingthesametypeofturbulence. Thiscan
≪
properties of turbulence via be expected, as the amplification then occurs on larger
scales, with larger eddy-turnover times.
κ′′ (κ′ )2 2κ 2T′ 2(T T )
U(r) diff diff + diff + N + L− N , Similarly, also the type of turbulence reflected in the
≡ 2 − 4κ r2 r r2
diff parameterϑmaychangethe amplificationratebyabout
κdiff =η+TL(0) TL(r). (10) an order of magnitude, in case of the same value of
−
Rm. ThemostefficientamplificationrateoccursforKol-
As recently shown by Schober et al. [22], this form of
mogorov turbulence, ϑ = 1/3, while it is less efficient
the potential also accounts for the effect of compressibil-
for highly compressible Burgers turbulence, ϑ=1/2, for
itybykeepingtermsrelatedto ~vduringthederivation.
∇· whichtheturbulentvelocitiesdecreasemorerapidlywith
The equation can be solved using the WKB approxima-
length scale.
tion in the limit of Pm [19–22]. For Kolmogorov
→ ∞ The Kazantsev model thus indicates that the behav-
turbulence, one obtains
ior of the dynamo depends both on the type of turbu-
V lence and the magnetic Prandtl number. A potential
Γ =1.028 Re1/2. (11)
K,Pm≫1 L restriction of the underlying model is the assumption
of δ-correlated turbulence, although the characteristic
Inarecentstudy,analyticalsolutionsbasedontheWKB
timescales are certainly small compared to the dynam-
approximationhavebeenderivedinthelimitPm 1by
≪ ical time. To investigate the resulting uncertainties, we
Schober et al. [59]. For Kolmogorov,they yield
refer the reader to Schekochihin and Kulsrud [61]. The
V main results from these considerations are thus the fol-
ΓK,Pm≪1 =0.0268 Rm1/2, (12) lowing:
L
4
The behavior of the small-scale dynamo depends
•
sensitively on the value of Pm, and in particular
whether Pm 1 or Pm 1. We note that there
≪ ≫
is a continuous transition at Pm 1, as detailed
∼
by Bovino et al. [62].
The adopted type of turbulence has a significant
•
influence on the efficiency of magnetic field ampli-
fication, as turbulent spectra with ϑ > 1/3 cor-
respond to larger eddy-turnover times and smaller
amplification rates.
B. Results from numerical simulations
Due to the numericalviscosityandresistivity,itis dif-
ficult to perform magneto-hydrodynamical simulations
FIG.1. AfittothegrowthratesobtainedbyFederrathet al.
with Pm significantly different from 1. However, a lim-
[37] for compressive and solenoidal forcing as a function of
ited rangeof Pmhas neverthelessbeen explored. Forin-
Mach number. These simulations correspond to Pm∼2.
stance, Haugen et al. [63] investigated magnetic Prandtl
numbers between 0.1 and 30. For Pm < 1, they report
that the miminum magnetic Reynolds number required
for dynamo action, Rmc, scales as They performed a fit to the growth rate and saturation
levels as a function of Mach number, using the function
Rm 35πPm−1/2. (19)
c ∼ f( )= p Mp1 +p2 +p p6. (20)
Wenotethatthefactorπintheaboveisduetotheirdef- M (cid:18) 0Mp3 +p4 5(cid:19)M
inition of the magnetic Reynolds number. They further The fit coefficients for the different cases are given in
reportdifferencesinthe obtainedpowerspectra,indicat- Table I, and the normalized growth rates are given in
ing a steeper decrease on small scales for small values of Fig.1. Inthesubsonicregime,theirresultsindicatethat
Pm. the growth rate (normalized by the eddy-turnover time
Schekochihin et al. [64] and Iskakov et al. [65] report T ontheforcingscale)stronglydecreaseswithdecreas-
ed
numerical simulations exploring the small-scale dynamo ing Mach number for compressive driving, while it is al-
from Pm 0.017 up to Pm = 1. In this regime, they most constant at solenoidal driving. At > 1, there
findthat∼evenforconstantvaluesofRm,thegrowthrate is an initial drop due to the appearance Mof shocks, but
decreaseswith decreasingPm. In particular,for Pm 1 increasesas 1/3 atlargervalues. Asimilardependence
and Rm 830, they report a normalized growth rate∼of is reported oMn the saturation level, which is particularly
∼
1.8, which decreases to 0.9 for Pm 0.2 and the same high for solenoidal driving, and decreases in the regime
∼
magnetic Reynolds number. The simulations further of large, supersonic Mach numbers.
indicate thatthe value ofRm settles to aconstantlimit
c
forPm 1,Re 1 andRm 1,eventhoughthis case Γ [T−1] Γ [T−1] (E /E ) (E /E )
≪ ≫ ≫ sol ed comp ed m k sol m k comp
ishardtonumericallyexplore. Thescalingofthegrowth p −18.71 2.251 0.020 0.037
0
rateonRm,ontheotherhand,hasnotbeenconclusively
p 0.051 0.119 2.340 1.982
1
explored. p −1.059 −0.802 23.33 −0.027
2
All in all, simulations thus show that the growth rate
p 2.921 25.53 2.340 3.601
depends on the magnetic Prandtl number even in the 3
p 1.350 1.686 1 0.395
range Pm . 1. Another quantity which was shown to 4
p 0.313 0.139 0 0.003
influence the dynamo is the Mach number of the gas. 5
Haugen et al. [33] explored Mach numbers in the range p6 1/3 1/3 0 0
of 0.1 2.1 and reported a clear dependence of the criti-
−
calmagneticReynoldsnumberfordynamoactiononthe
TABLE I. Fit coefficients reported by Federrath et al. [37].
Mach number . For Pm 5, they report Rm 25π
c
M ∼ ∼
for <1andarapidincreasetoRm 45πfor >1.
M c ∼ M We thus summarize the results from numerical simu-
A similar behavior was found for Pm 1, with critical
∼ lations as follows:
values of 40π and 80π, respectively.
∼ ∼
AlargerseriesofsimulationshasbeenreportedbyFed- ThecriticalmagneticReynoldsnumberfordynamo
•
errathetal.[37],exploringMachnumbersfrom0.02upto action as well as the resulting spectra for the mag-
20,withcompressiveandsolenoidalforcing,respectively. neticfielddependonthemagneticPrandtlnumber.
5
Both the growth rates and the saturation levels of ofthe magnetic energy,assumingthatthe density distri-
•
the dynamo depend significantly on the turbulent bution function will not change significantly over time.
Mach number and the type of forcing that is em- In the case of well-developed driven turbulence, one in-
ployed. deed expects a characteristic log-normal density prob-
ability distribution function, which naturally complies
with these requirements [66–68]. Strictly speaking, the
III. NON-UNIVERSALITY IN THE following considerations apply to the quantity B˜ = B/ρ
NON-LINEAR REGIME and W˜ = W/ρ2, with W the magnetic energy. In the
following, the˜is however dropped for simplicity.
The exponential growth phase will come to an end
when the tension forceof the magnetic field, B~ B~, be-
·∇
comescomparabletotheinertialtermoftheflow,~u ~u. A. First considerations based on a toy model
·∇
At this point, magnetic field amplification will stop on
the scalesthat fulfill this condition, andcontinue to pro- In the toy model previouslyproposedby Schekochihin
ceedonlargerscales. AsdiscussedbySchekochihinet al. et al. [23], the dominant fraction of the magnetic energy
[23], this condition translates to residesonthescaleℓ ,thesmallestscalewheremagnetic
a
field amplification still occurs (thus yielding the shortest
B2 u2
ℓa ℓa, (21) amplificationtimescale). Onthatscale,themagneticen-
ℓa ∼ ℓa ergy is expected to be already close to saturation. The
magnetic energy W(t) can thus be related to the ampli-
where ℓ denotes the smallest scale where amplification
a
fication scale ℓ by the approximate relation
still occurs. In this regime, a linear growth of the mag- a
neticenergyhasbeenreportedinpreviousstudies,based
1
ontheassumptionofKolmogorovturbulence[e.g.23–25]. W(t) ρ u2 . (26)
∼ 2h i ℓa(t)
In the following, we will generalize these investigations
by employing a simplified toy model as well as a more The magnetic energy is evaluated here at the mean den-
sophisticated Fokker-Planck model previously suggested sity ρ ofthe turbulentbox,asweareinterestedonlyin
h i
by Schekochihin et al. [23]. As a result, we will show the magnetic field amplification by shear. Adopting the
thatdifferenttypesofpower-lawgrowthcanbeexpected eddy-turnoverrate onthe scale ℓ as the growthrate for
a
depending on the adopted type of turbulence. the magnetic field, i.e.
We further point out that in the non-linear regime,
u
we expect the magnetic Prandtl number to play a less Γ(t) ℓa(t), (27)
∼ ℓ (t)
critical role, as the amplification scale of the magnetic a
field is now expected to be larger than both the viscous
the magnetic energy evolves as
and the resistive scale, such that no strong dependence
on Re or Rm can be expected. d
W =Γ(t)W(t) 2ηk2 W(t) (28)
Wenotethatthemodelsconsideredinthissectionhave dt − rms
previously been motivated in the context of the incom-
with
pressible induction equation, given as
1 ∞
∂ B~ +~v B~ =B~ ~v+η∆B~. (22) k2 (t)= dkk2M(t,k) (29)
t ·∇ ·∇ rms W Z0
However, they can be naturally extended into the com-
and
pressible regime with the replacement
1
B~ B~. (23) M(t,k)= 2Z dΩ~kh|B~(t,~k)|2i. (30)
→ ρ
Now, we have Γ(t)W(t) ρ u3 /ℓ (t)=:ǫ(t). Insert-
Insertingthisreplacementaswellasthecontinuityequa- ∼h i ℓa(t) a
ing in Eq. (28) yields
tion,
d
ρ˙ = (ρ~v), (24) W =χǫ(t) 2ηk2 (t)W(t), (31)
−∇· dt − rms
it is straightforward to show that one obtains the com-
whereχisaconstantoforderunity. ForKolmogorovtur-
pressible form of the induction equation, bulence, the quantity ǫ(t)= ρ u3 /ℓ (t) is a constant
h i ℓa(t) a
∂ B~ +~v B~ =B~ ~v B~ ( ~v)+η∆B~, (25) [18]. In this case, and as long as magnetic energy dis-
t
·∇ ·∇ − ∇· sipation is negligible, dW/dt = const, implying a phase
equivalent to Eq. (1). As long as the mean density ρ of linear growth. In this limit, we obtain the result of
h i
in the box is constant, a significant growth of the quan- Beresnyak [25], where a constant fraction of the turbu-
tity B/ρ nevertheless implies a corresponding growth lence dissipation rate is converted into magnetic energy.
h i
6
In the general case with u ℓϑ, ǫ(t) is however not To describe the evolution in the nonlinear regime,
constant, but varies as ℓ3ϑ−ℓ1a.∝Inathe case of Burgers Schekochihin et al. [23] postulated the following expres-
a
turbulence, we thus obtain ǫ ℓ0.5. In this case, the sions:
∝ a
growthof the magnetic energy is no longer linear, as the
1/2
turbulent energy dissipation rate is not independent of ks(t)
Γ(t)=c dkk2E(k) , (37)
scale! 1
"Z0 #
For comparison, we note that the quantity ǫ˜ =
∞
ρe3/2 /ℓ,withe thespecificenergydensityofsubgrid- W(t)=c dkE(k). (38)
SGS SGS 2
scale turbulence, is practically independent of ℓ. It how- Zks(t)
ever has a weak dependence on the Mach number, and
The constants c and c are of order unity, E(k) is the
1 2
a strong dependence on the type of forcing [69]. As the
hydrodynamic energy spectrum neglecting the influence
density fluctuations will however not contribute to the
of the magnetic field, and the wave vector k (t) is de-
s
shearing,wewilladoptǫasthe quantityofinteresthere.
fined via Eq. (38). It corresponds to the smallest scale
To quantify the expected behavior, we need to solve
where amplification efficiently occurs. As input for the
Eq. (26) for ℓ . For this purpose, we recall that u is
a ℓa Fokker-Planck model, we require an energy spectrum of
relatedtotheturbulencedrivingscaleLandthevelocity
the turbulence. As before, we assume that the velocity
V on that scale via
in the inertial range scales as
ϑ
uℓa =V ℓLa . (32) uℓ ∝ℓϑ. (39)
(cid:18) (cid:19)
The hydrodynamic energy spectrum is then approxi-
From (26), we thus obtain
mately given as
1/(2ϑ)
ℓa =L ρ2WV2 . (33) E(k)= Ctǫ2/3k−2ϑ−1 for k ∈[kf,kν] (40)
(cid:18)h i (cid:19) (0 elsewhere,
We can now evaluate (27) and (28), yielding
with C a constant which depends on the type of tur-
t
d 2W 1/(2ϑ) ϑ−1 bulence, kf and kν the wave vectors describing the in-
W W L W1+(ϑ−1)/(2ϑ). jection scale of turbulence and the viscous scale, respec-
dt ∼ " (cid:18)hρiV2(cid:19) # ∝ tively. The value of kν is set to enforce the condition
(34) ǫ = 2ν ∞dkk2E(k). Unlikely in (6) and (7), we do
0
For Kolmogorov turbulence (ϑ = 1/3), we confirm that not explicitly model the turbulent spectra in the viscous
dW/dt = const, while in the more general case, this regime, Ras these no longer contribute during the non-
quantity will increase with increasing W. This can be linear stage. With these input data, Eq. (38) can be
intuitively understood, as the steep spectra for ϑ > 1/3 evaluated as
implyamoremodestincreaseoftheeddy-timescalewith
length scale, suggesting that the amplification rate re- W(t)= c2Ctǫ2/3 k−2ϑ k−2ϑ . (41)
mains larger when increasing the scale. We re-assess 2ϑ s − ν
these results with the Fokker-Planck model below and (cid:2) (cid:3)
We further introduce the quantities
explore the physical implications in more detail.
∞ c C ǫ2/3
W =c dkE(k)= 2 t k−2ϑ k−2ϑ ,(42)
0 2 2ϑ f − ν
B. Implications of the Fokker-Planck model for Z0 h i
universality W = c2Ctǫ2/3k−2ϑ. (43)
ν 2ϑ ν
ThestartingpointforourinvestigationsistheFokker- Wenotethatintheaboveexpressions,theintegral ∞dk
PlanckmodelofSchekochihinet al. [23]. Here,the time- corresponds to an integration from k to k , as the0tur-
f ν
evolution of the magnetic-energy spectrum is given as bulent energy is non-zero only in this regime (see R40).
Usingthesedefinitions,the wavevectorsk , k andk
∂ ∂M ν s f
∂ M = D(k) V(k)M +2Γ(t)M 2ηk2M, can be expressed as
t
∂k ∂k − −
(cid:20) (cid:21)
(35) 2ϑ −1/(2ϑ)
with the diffusion coefficient D(k) = Γ(t)k2/5 and the ks = c C ǫ2/3 [W(t)+Wν]−1/(2ϑ), (44)
drift velocity in k-space V(k) = 4Γ(t)k/5. We recall (cid:18) 2 t (cid:19)
−1/(2ϑ)
that the magnetic-energy spectrum M is related to the k = 2ϑ [W +W ]−1/(2ϑ), (45)
magnetic energy W via f c C ǫ2/3 0 ν
(cid:18) 2 t (cid:19)
∞ 2ϑ −1/(2ϑ)
W(t)= dkM(t,k). (36) k = W−1/(2ϑ). (46)
ν c C ǫ2/3 ν
Z0 (cid:18) 2 t (cid:19)
7
Integrating Eq. (37) now yields the following:
C ǫ2/3 1/2
Γ(t)=c t k2−2ϑ(t) k2−2ϑ . (47)
1 2 2ϑ s − f
(cid:20)(cid:18) − (cid:19)(cid:16) (cid:17)(cid:21)
SubstitutingEqs.(44)-(46)into(47)yieldstheexpression
C ǫ2/3 1/2 c C ǫ2/3 12−ϑϑ
t 2 t
Γ(t)=c (48)
1
2 2ϑ 2ϑ
(cid:18) − (cid:19) (cid:18) (cid:19)
(W(t)+W )1−ϑ1 (W +W )1−ϑ1 1/2.(49)
ν 0 ν
× −
h i
ConsideringturbulencemodelsbetweenKolmogorovand
Burgers, we have 1/3 ϑ 1/2. We further assume
≤ ≤
that W(t) W , implying that the magnetic field is far
0
≪
from saturation on the current amplification scale. In
this case, we can neglect the second term in the square
FIG. 2. The power-law growth of magnetic energy for differ-
brackets. As we focus here on the non-linear regime, we
enttypesofturbulenceinthenon-linearregime,followingthe
can further neglect W compared to W(t), and obtain
ν evolution Eq.(56) for Re=104.
the expression
Γ(t)=c Ctǫ2/3 1/2 c2Ctǫ2/3 12−ϑϑ W(ϑ−1)/(2ϑ)(t). Model and reference ϑ W ∝ ℓa ∝
1
(cid:18)2−2ϑ(cid:19) (cid:18) 2ϑ (cid:19) Kolmogorov [18] 1/3 t1 t3/2
(50)
Intermittency of Kolmogorov turbulence[70] 0.35 t1.077 t1.54
As in our toy model, the growth of the magnetic energy
Driven supersonic MHD turbulence[28] 0.37 t1.17 t1.59
thus scales as
Observation in molecular clouds [26] 0.38 t1.23 t1.61
dW W(t)Γ(t) W1+(ϑ−1)/(2ϑ). (51) Solenoidal forcing of turbulence[31] 0.43 t1.51 t1.75
dt ∝ ∝ Compressive forcing of turbulence[31] 0.47 t1.77 t1.89
Observation in molecular clouds [27] 0.47 t1.77 t1.89
For Kolmogorov turbulence, the growth is thus linear,
while it growsfaster thanlinear forϑ>1/3. Integrating Burgers turbulence[32] 1/2 t2 t2
Eq. (51), we obtain
TABLEII.Thepower-lawbehaviorofthesmall-scaledynamo
W(t)=C˜t2ϑ/(1−ϑ), (52)
for different types of turbulencein thenon-linear regime.
with
If we normalize Eq. (52) in terms of T (k √W )−1,
C˜ = Ctǫ2/3 1/2 c2Ctǫ2/3 (1−ϑ)/(2ϑ) 5 1 −1. we thus obtain ed ∼ f 0
(cid:18)2−2ϑ(cid:19) (cid:18) 2ϑ (cid:19) (cid:18)2 − 2ϑ(cid:19) t 2ϑ/(1−ϑ)
(53) W(t)=C , (55)
T
From this expression, we already see that the energy (cid:18) ed(cid:19)
grows linearly in t for Kolmogorov, while it grows as t2 C ǫ2/3 1/2 5 1 −1
t
forBurgersturbulence. Foraphysicalinterpretation,the C =
2 2ϑ 2 − 2ϑ
normalizationin terms of the eddy-turnovertime T on (cid:18) − (cid:19) (cid:18) (cid:19)
ed
theforcingscaleisstillrequired,whichweperformbelow. Re2ϑ(1−2ϑ)/(1+ϑ)k2ϑ−2ϑ/(1−ϑ)W2ϑ−ϑ/(1−ϑ).
× f 0
Adopting a system of units with W = 1 and k = 1,
0 f
it is evident that E 1, v(k ) 1 and thus T 1.
f f ed
IV. PHYSICAL IMPLICATIONS ∼ ∼ ∼
From Eq. (40), we also expect ǫ 1. In these units, our
∼
evolution equations simplifies as
To explore the physical implications of the above-
mentioned results, we now perform a normalization in W(t) t 2ϑ/(1−ϑ)
=C , (56)
terms of the eddy-turnover time T on the forcing scale W T
kf−1. Forthispurpose,wenotethatedtheexpressionwithin 0 (cid:18) 1ed(cid:19) 1/2 5 1 −1
the centralbracketsofEq.(53)isidenticaltoW k , and C = Re2ϑ(1−2ϑ)/(1−ϑ).
ν ν 2 2ϑ 2 − 2ϑ
it is straightforwardto show that (cid:18) − (cid:19) (cid:18) (cid:19)
We illustrate the behavior for the different types of tur-
W k =W k Re(1−2ϑ)/(1+ϑ). (54) bulence in Fig. 2 for Re = 104, and summarize the
ν ν 0 f
8
power-law behavior in Table II. The solution suggests −2ϑ−1
E(k)~k
that turbulence spectra closer to Kolmogorov saturate
earlier (in terms of the eddy-turnover time on the forc-
ing scale k ), and initially start at a higher value. The
f
latter is fully consistent with our expectations for the
kinematic regime, where the growth rates are higher for Kolmogorov
Kolmogorov turbulence, and a larger amount of mag- ~k−5/3
neticenergymaybuildupbeforethenon-linearregimeis
reached(duetotheincreasedamountofturbulentenergy Burgers
that is available on the same scale). We note that in the −2
~k
finalstageclosetosaturation,theevolutionmaystartto
deviate from the power-law behavior reported here, pro-
viding a transition to the regime where W(t)=const.
cf. Eq. (36)
From the relation derived above, we further calcu-
latethecharacteristicscalingofthecurrentamplification
scale l as a function of time t. Adopting Eq. (26), we
s
have W(t) ρ u2 ℓ2ϑ, thus k
∼h i ℓa(t) ∝ a
ℓ (t) W1/(2ϑ)(t) t1/(1−ϑ). (57) FIG.3. AsketchofKolmogorovvsBurgersturbulence. While
a ∝ ∝ theturbulentenergyisconsiderablysmallerforBurgersspec-
For Kolmogorov turbulence, the characteristic length tra (ϑ = 1/2) on small scales, it approaches the values for
scale of the magnetic field thus grows as t3/2, while it Kolmogorov turbulence (ϑ = 1/3) on larger scales. As a re-
sult,themagneticenergygrowsfasterthanlinearforBurgers
growsas t2 for Burgersturbulence. The results are sum-
turbulence, as the growth rates gradually approach the Kol-
marized for all types of turbulence in Table (II).
mogorov values at later times.
Thepower-lawsderivedheredependonthetypeoftur-
bulenceduetothedifferenteddy-turnovertimescalesasa
functionofscale,aswesketchinFig.(3). Wesummarize
the main ingredients based on the toy model developed
V. DISCUSSION AND CONCLUSIONS
in section IIIA:
Considering a driving scale L with a turbulence veloc-
In this paper, we have explored both the kinematic
ity V on that scale, the ratio of the eddy-turnover times
regime of the small-scale dynamo, where an exponential
on scale l L for Kolmogorov and Burgers turbulence
≪ growthofthemagneticenergyisgenerallyobserved,and
is given as
thenon-linearregime,wherebackreactionsstartoccuring
t (ℓ/L)1−1/3 l 1/6 on small scales and shift the amplification scale of the
K
= = . (58) magnetic field to larger scales.
t (ℓ/L)1−1/2 L
B (cid:18) (cid:19) In the kinematic regime, analytical studies based on
During the growth of the magnetic energy, the relevant the Kazantsevmodel suggesta fundamental dependence
length scale however shifts to larger scales. According on the magnetic Prandtl number. In particular, for
to Eq. (58), the ratio of the eddy timescales approaches Pm 1, the growth rate of the dynamo is a function of
≪
unity for ℓ L. For Burgers turbulence, the magnetic the magnetic Reynolds number Rm, while for Pm 1,
→ ≫
field amplification is thus initially delayed with respect it depends on the kinematic Reynolds number Re. In
to Kolmogorov, and catches up later, resulting into the addition, the amplification rates significantly depend on
non-linear behavior and the power-lawgrowth described theadoptedtypeofturbulence. ForPm 1,itscalesas
here. Re1/2 for Kolmogorovturbulence andas≫Re1/3 forBurg-
Due to these results, it is clearthat the growthrateof ers turbulence. The same scaling relations, with a dif-
the dynamo is not a fixed fraction of the global turbu- ferent normalization, were found for Pm 1, with the
≪
lence dissipation rate, as previously proposed by Beres- replacement Re Rm.
→
nyak[25]. Due to the dependence on the turbulent spec- Numerical simulations confirm the dependence on Pm
trum, such a considerationmay only hold locally, i.e. on also in the range Pm 1, and find a strong dependence
∼
a given scale, where the growth rate of the field is in- ofthe growthrateandthe saturationlevelonthe turbu-
deed related to the local eddy timescale. From a more lent Mach number and the type of turbulence forc-
M
global perspective, however, the turbulence dissipation ing. Magnetic field amplification is particularly efficient
rate changes as a function of scale for models different forsolenoidalforcingandlowMachnumbers,butalsooc-
from Kolmogorov, such that the previously postulated cursforhighMachnumbersandsolenoidal/compressive
universal behavior cannot be expected. From Eq. (56), forcing. IftheMachnumbersareverysmall,compressive
it is further evident that the evolution depends on the forcing is hardly able to trigger magnetic field amplifica-
Reynolds number of the gas, and that larger Reynolds tion,asthepresenceofdensitygradientsarerequiredfor
numbers imply stronger magnetic fields at earlier times. the production of solenoidal turbulence in this case.
9
To investigate the non-linear regime of the dynamo, occur in the highly compressible regime. Universality in
we employed the Fokker-Planck model of Schekochihin the sense of a uniform behavior under all conditions can
et al. [23] and explored the effect of different turbulent thus not be expected. Nevertheless, we note that there
spectraonthe magneticfieldamplificationrate. We find are still universal laws governingthe behavior of the dy-
that the previously known linear growth only occurs for namo, which relate the growth of the magnetic energy
Kolmogorov turbulence, while in the general case with to the eddy-turnover time on the current amplification
u ℓϑ, we expect the magnetic energy to scale as scale. Thisquantityingeneraldoesdepend onthe Mach
ℓ
t2ϑ/∝(1−ϑ). The energy growth is thus faster than linear, number and the type of turbulence involved, such that
and may even become quadratic for Burgers turbulence the breaking of universality is a result of the properties
(ϑ = 1/2). However, we note that the growth rate is of different environments. We propose to explore such
initially smaller for Burgers turbulence, as the turbu- effectsinfurtherdetailwithnumericalsimulationstoim-
lent energy available for amplification is initially much prove our understanding of such non-universalbehavior.
smaller on small scales. While magnetic field amplifica-
tionisshiftedtolargerscales,thedifferenceintheturbu-
lent energy decreases, implying the reported power-law
behavior as a function of time.
ACKNOWLEDGMENTS
We have further shown that also the scaling of the
characteristic length scale ℓ for magnetic field amplifi-
a
cation depends on the turbulent slope. Specifically, we We thank Robi Banerjee and Ralf Klessen for stim-
find a scaling as t1/(1−ϑ), corresponding to t3/2 for Kol- ulating discussions on the topic. D.R.G.S., J.S. and
mogorov and t2 for Burgers turbulence. The change of S.B.acknowledgefundingfromtheDeutscheForschungs-
length scales proceeds thus in a fashion analogousto the gemeinschaft (DFG) in the Schwerpunktprogramm SPP
inverse-cascade in case of helicity [e.g. 71, 72]. The evo- 1573 “Physics of the Interstellar Medium” under grant
lutionofthisquantitymaythusprovideanotherrelevant KL 1358/14-1 and SCHL 1964/1-1. D.R.G.S. and
diagnostic for a comparison with numerical simulations. W.S. thank for funding via the SFB 963/1 on “Astro-
Due to the above considerations, we point out that physicalflowinstabilitiesandturbulence”. J.S.acknowl-
the non-linear stage of the small-scale dynamo does not edges the support by IMPRS HD, the HGSFP and the
generallycorrespondtoconvertingafixedfractionofthe SFB 881 ”The Milky Way System”. C.F. thanks for
turbulence dissipation rate into magnetic energy, as pre- fundingprovidedbytheAustralianResearchCouncilun-
viously suggested by Beresnyak [25]. While their results dertheDiscoveryProjectsscheme(grantDP110102191).
agree with our model for the case of Kolmogorov tur- We thank the anonymous referees for valuable sugges-
bulence (low Mach numbers), steeper power laws may tions that improved the manuscript.
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