Effect of Magnetic Helicity on Non-Helical Turbulent Dynamos

Effect of Effect of Magnetic Helicity on Non-Helical Magnetic Helicity on Non-Helical

Turbulent DynamosTurbulent Dynamos

N. KLEEORIN and I. ROGACHEVSKII

Ben-Gurion University of the Negev, Beer Sheva, ISRAEL

OutlineOutline

IntroductionIntroduction

Physics of Physics of “shear-current” effect “shear-current” effect and and

comparison with alpha effect (hydrodynamic comparison with alpha effect (hydrodynamic

helicity)helicity)

Generation ofGeneration of large-scale magnetic field large-scale magnetic field due due

to the to the “shear-current” effect“shear-current” effect (kinematic and (kinematic and

nonlinear dynamos with magnetic helicity)nonlinear dynamos with magnetic helicity)

Alpha-Omega DynamoAlpha-Omega Dynamo(Mean-Field Approach)(Mean-Field Approach)

Induction equation for Induction equation for mean magnetic fieldmean magnetic field::

Electromotive forceElectromotive force::




12

tpr

+ ×h rot h




12

tpr

+ ×a b

h rot h: g

144424443

Generation of the mean magnetic Generation of the mean magnetic field due to the dynamofield due to the dynamo

Dynamo number:

Mean magnetic field:

BpB

)(rΩ

Physics of the alpha-effectPhysics of the alpha-effect The -effectThe -effect is related to the is related to the

hydrodynamic helicityhydrodynamic helicity in an in an

inhomogeneous turbulenceinhomogeneous turbulence. .

The The deformations of the magnetic fielddeformations of the magnetic field

lines are caused by lines are caused by upwardupward and and

downwarddownward rotating turbulent eddies. rotating turbulent eddies.

The The inhomogeneity of turbulenceinhomogeneity of turbulence breaks breaks

a symmetry between the a symmetry between the upwardupward and and

downwarddownward eddies. eddies.

Therefore, the Therefore, the total effect of the upward total effect of the upward

and downwardand downward eddies on the mean eddies on the mean

magnetic field magnetic field does not vanishdoes not vanish and it and it

creates the creates the mean electric currentmean electric current parallel parallel

to the to the original mean magnetic fieldoriginal mean magnetic field..

J

B

Mean-Field DynamoMean-Field Dynamo Is it possible to generate a large-scale Is it possible to generate a large-scale

magnetic field in a magnetic field in a non-helicalnon-helical and and non-non-

rotatingrotating homogeneoushomogeneous turbulence ? turbulence ?

The answer is YESThe answer is YES(I. Rogachevskii and N. Kleeorin(I. Rogachevskii and N. Kleeorin, , Phys. Rev. EPhys. Rev. E 6868, ,

036301 (2003).)036301 (2003).)

The answer is YESThe answer is YES(I. Rogachevskii and N. Kleeorin(I. Rogachevskii and N. Kleeorin, , Phys. Rev. EPhys. Rev. E 6868, ,

036301 (2003).)036301 (2003).)

turbulenceturbulenceturbulenceturbulence

0

0

Sx

æ ö÷ç ÷ç ÷ç ÷+ =- ç ÷ç ÷ç ÷÷çè ø

U

Direct Numerical SimulationsDirect Numerical Simulations A. BrandenburgA. Brandenburg, , Astrophys. J. Astrophys. J. 625625, 539-547 (2005)., 539-547 (2005). A. Brandenburg, N.E.L. Haugen, P.J. Käpylä, C. SandinA. Brandenburg, N.E.L. Haugen, P.J. Käpylä, C. Sandin,,

Astron. Nachr. Astron. Nachr. 326326, 174-185 (2005)., 174-185 (2005).

1. 1. Non-helical forcingNon-helical forcing

2. 2. Imposed mean velocity Imposed mean velocity shearshear

3. 3. Open boundary conditionsOpen boundary conditions (non-zero flux of magnetic (non-zero flux of magnetic

helicity)helicity)

Direct Numerical Simulations Direct Numerical Simulations (linear shear velocity)(linear shear velocity)

T. A. Yousef, T. Heinemann, A.A. Schekochihin, T. A. Yousef, T. Heinemann, A.A. Schekochihin, N. Kleeorin, N. Kleeorin, I. Rogachevskii, A.B. I. Rogachevskii, A.B. Iskakov, S.C. Cowley, J.C. Iskakov, S.C. Cowley, J.C. McWilliams,McWilliams, Phys. Rev. Lett.Phys. Rev. Lett., , v.100, 184501 v.100, 184501 (2008) (2008)

1. 1. A white noise non-A white noise non-helical homogeneous helical homogeneous and isotropic random and isotropic random forcingforcing

2.2. Imposed mean linear Imposed mean linear shear flowshear flow

3. 3. Sheared box (shear-periodic Sheared box (shear-periodic boundary conditions)boundary conditions)

30/l uRmRe 0 yxz LLL 3/ 0 lLx

Numerical set upIncompressible MHD equations with background shear

(Units: )

ParametersTurbulence:

Weak shear:

)0),(),(( zBzB yxB

S

l

S

uL rms

B0

Magnetic field grows

30/l uRmRe 0 1 yxz LLL 3/ 0 lLx

Generated field is large scale

Growth rate S

Mean-Field ApproachMean-Field Approach



Generation of the mean magnetic Generation of the mean magnetic field due to the shear-current effectfield due to the shear-current effect

Mean velocity shear:

The growth rate of B

Comparison of the alpha-effect Comparison of the alpha-effect with the ''shear-current" effectwith the ''shear-current" effect

The effectThe effect is caused by a is caused by a uniform rotationuniform rotation and and

inhomogeneity of turbulenceinhomogeneity of turbulence::

, where, where

The ''shear-current" effectThe ''shear-current" effect is related to is related to the termthe term and and

is caused by is caused by mean shearmean shear and and nonuniform mean nonuniform mean

magnetic fieldmagnetic field,,

, , wherewhere

Therefore,Therefore,

The effectThe effect of large-scale shear of large-scale shear The The large-scale shear motionslarge-scale shear motions cause the cause the

stretchingstretching of the magnetic field of the magnetic field

generating the field componentgenerating the field component

The The interactioninteraction of the non-uniform of the non-uniform

magnetic field with the background magnetic field with the background

vorticity produces vorticity produces electric currentelectric current along along

the fieldthe field

Physics of ''shear-current" effectPhysics of ''shear-current" effect In a turbulent flow with the mean In a turbulent flow with the mean

velocity shearvelocity shear, the , the inhomogeneity of inhomogeneity of

the original mean magnetic fieldthe original mean magnetic field breaks breaks

a symmetrya symmetry between the influence of between the influence of

the the upwardupward and and downwarddownward turbulent turbulent

eddies on the mean magnetic field.eddies on the mean magnetic field.

The deformations of the magnetic field The deformations of the magnetic field

lines lines in the ''shear-current"in the ''shear-current" dynamo are dynamo are

caused by the caused by the upwardupward and and downwarddownward

turbulent eddies which result in the turbulent eddies which result in the

mean electric currentmean electric current parallel to the parallel to the

mean magnetic field and mean magnetic field and produce the produce the

magnetic dynamomagnetic dynamo..

J

B

Generation of the mean magnetic Generation of the mean magnetic field (kinematic dynamo)field (kinematic dynamo)

Solution for the symmetric mode:

The growth rate of B:

Critical dynamo number:

The shear-current nonlinear The shear-current nonlinear dynamo (algebraic nonlinearity)dynamo (algebraic nonlinearity)

Dynamo number:

Nonlinear shear-current effect:


Shear number:

Nonlinear shear-current effectNonlinear shear-current effect

Weak magnetic field:

Strong mean magnetic field:

There is no quenching of the nonlinear "shear-current" effect contrary to the quenching of the nonlinear alpha effect, the nonlinear turbulent magnetic diffusion, etc.

Nonlinear “shear-current” dynamoNonlinear “shear-current” dynamo(algebraic nonlinearity)(algebraic nonlinearity)

Magnetic HelicityMagnetic Helicity

Magnetic part of alpha effect:

Total magnetic helicityTotal magnetic helicity is conservedis conserved for very large for very large magnetic Reynolds numbersmagnetic Reynolds numbers

Dynamics of small-scale magnetic helicity:

The nonlinear function:

Dynamics of magnetic helicityDynamics of magnetic helicity

In the absence of the magnetic helicity flux,

In the presence of the flux of magnetic helicity:

i.e., catastrophic quenching (Vainshtein and Cattaneo, 1992)

Kleeorin and Ruzmaikin (1982); Gruzinov and Diamond (1994); Kleeorin and Rogachevskii (1999); Kleeorin, Moss, Rogachevskii and Sokoloff (2000); Blackman and Field (2000); Brandenburg and Subramanian (2005); etc.

The shear-current nonlinear dynamo The shear-current nonlinear dynamo (algebraic and dynamic nonlinearities)(algebraic and dynamic nonlinearities)


ReferencesReferences I. Rogachevskii and N. KleeorinI. Rogachevskii and N. Kleeorin, , Phys. Rev. EPhys. Rev. E 6868, 036301 (2003)., 036301 (2003).

I. Rogachevskii and N. KleeorinI. Rogachevskii and N. Kleeorin, Phys. Rev. E , Phys. Rev. E 7070, 046310 (2004)., 046310 (2004).

I. Rogachevskii, N. Kleeorin, A. D. Chernin and E. LivertsI. Rogachevskii, N. Kleeorin, A. D. Chernin and E. Liverts, Astron. , Astron. Nachr. Nachr. 327, 327, 591-594 (2006).591-594 (2006).

I. RogachevskiiI. Rogachevskii, , N. Kleeorin and E. Liverts, N. Kleeorin and E. Liverts, Geophys. Astroph. Fluid Geophys. Astroph. Fluid Dyn. Dyn. 100, 100, 537-557537-557 (2006). (2006).

I. Rogachevskii andI. Rogachevskii and N. Kleeorin, N. Kleeorin, Phys. Rev. E Phys. Rev. E 75, 75, 046305046305 (2007). (2007).

N. Kleeorin and I. Rogachevskii, N. Kleeorin and I. Rogachevskii, Planet. Space Sci.Planet. Space Sci. 5555, 2315-2318 , 2315-2318 (2007).(2007).

N. Kleeorin and I. Rogachevskii, N. Kleeorin and I. Rogachevskii, Phys. Rev. E Phys. Rev. E 77, 77, 036307 036307 (2008).(2008).

T. A. Yousef, T. Heinemann, A.A. Schekochihin, T. A. Yousef, T. Heinemann, A.A. Schekochihin, N. Kleeorin, N. Kleeorin, I. Rogachevskii, A.B. I. Rogachevskii, A.B. Iskakov, S.C. Cowley and J.C. McWilliams,Iskakov, S.C. Cowley and J.C. McWilliams, Phys. Rev. Lett.Phys. Rev. Lett. 100 100, 184501 , 184501 (2008). (2008).

ConclusionsConclusions

Generation of large-scale magnetic fieldGeneration of large-scale magnetic field is caused by is caused by a a new ''shear-current" effectnew ''shear-current" effect which acts even in a which acts even in a nonrotating and nonhelical homogeneous nonrotating and nonhelical homogeneous turbulence. turbulence.

The shear-current dynamo The shear-current dynamo and and generation of large-generation of large-scale vorticity scale vorticity can occur only when can occur only when

Re>1 and Rm > 1 Re>1 and Rm > 1 During the growth of the mean magnetic field, During the growth of the mean magnetic field, the the

nonlinear nonlinear ''shear-current" effect''shear-current" effect is not quenched and is not quenched and it only it only changes its signchanges its sign at some value of the mean at some value of the mean magnetic field which can determine the magnetic field which can determine the level of the level of the saturated mean magnetic field.saturated mean magnetic field.

ConclusionsConclusions

We have taken into account the We have taken into account the transport of transport of magnetic helicitymagnetic helicity as dynamical nonlinearity. The as dynamical nonlinearity. The magnetic helicity fluxmagnetic helicity flux strongly affects the magnetic strongly affects the magnetic field dynamics during the field dynamics during the nonlinear shear-currentnonlinear shear-current dynamo. The dynamo. The level of the saturated mean level of the saturated mean magnetic fieldmagnetic field is of the order of the is of the order of the equipartition equipartition fieldfield..

The shear current dynamoThe shear current dynamo can occur in can occur in laboratory dynamo laboratory dynamo

experimentsexperiments..

The estimated saturated large-scale magnetic field for The estimated saturated large-scale magnetic field for merging protogalactic cloudsmerging protogalactic clouds and and colliding giant galaxy colliding giant galaxy clustersclusters is about is about several microgaussseveral microgauss, and for , and for merging merging protostellar cloudsprotostellar clouds is of the order of is of the order of several tenth of several tenth of microgaussmicrogauss..

THE ENDTHE END

Shear-current dynamo with Shear-current dynamo with generated mean vorticitygenerated mean vorticity

Growing perturbations of vorticity:

Generation of the mean magnetic Generation of the mean magnetic field (kinematic dynamo)field (kinematic dynamo)

Solution for the antisymmetric mode:

The growth rate of B:

Critical dynamo number:

The magnetic scale at maximum :

Anisotropic Turbulent Magnetic Anisotropic Turbulent Magnetic DiffusionDiffusion

Anisotropic Turbulent ViscosityAnisotropic Turbulent Viscosity

Method of DerivationMethod of Derivation

The spectral -approximation (the third-order closure procedure)

Equations for the correlation functions for:

The velocity fluctuations

The magnetic fluctuations

The cross-helicity tensor

The shear-current nonlinear dynamo The shear-current nonlinear dynamo (algebraic and dynamic nonlinearities)(algebraic and dynamic nonlinearities)

Magnetic part of alpha effect:

Dynamical nonlinearity: magnetic helicity evolution


The nonlinear function:

Astrophysical cloudsAstrophysical clouds We apply We apply the universal mechanismthe universal mechanism of generation of of generation of

large-scale magnetic fields due to shear-current effect to large-scale magnetic fields due to shear-current effect to

several astrophysical objects:several astrophysical objects:

merging protostellar cloudsmerging protostellar clouds

merging protogalactic cloudsmerging protogalactic clouds

colliding giant galaxy clusterscolliding giant galaxy clusters

Interactions of protostellar clouds, or colliding Interactions of protostellar clouds, or colliding

protogalactic clouds or giant galaxy clusters produce protogalactic clouds or giant galaxy clusters produce

large-scale shear motionslarge-scale shear motions which are superimposed on which are superimposed on

small-scale turbulence.small-scale turbulence.

Chernin (1993). Non-central collisionChernin (1993). Non-central collision

Different cloud sizes, Chernin (1993)Different cloud sizes, Chernin (1993)

ParametersParameters

ParametersParameters Protostellar Protostellar CloudsClouds

Protogalactic Protogalactic CloudsClouds

Giant Galaxy Giant Galaxy ClustersClusters

MassMass

R (pc)R (pc)

V (cm/s)V (cm/s)

ParametersParametersParametersParameters Protostellar Protostellar

CloudsCloudsProtogalactic Protogalactic

CloudsCloudsGiant Giant Galaxy Galaxy ClustersClusters

(cm/s)(cm/s)

(cm)(cm)

u (cm/s)u (cm/s)

(cm)(cm)

(years) (years)

ParametersParametersParametersParameters Protostellar Protostellar

CloudsCloudsProtogalactic Protogalactic

CloudsCloudsGiant Giant

Galaxy Galaxy ClustersClusters

(cm/s)(cm/s)

(cm)(cm)

(years)(years)

Necessary condition for Necessary condition for the shear-current dynamothe shear-current dynamo

The parameter :The growth rate of B:

The Kolmogorov Scaling (large Re and Rm):

Small Re and Rm (random flow):

Rogachevskii and Kleeorin (2003): there is shear-current dynamo

In a good agreement with: there is no dynamo for:Rädler and Stepanov (2006) (SOCA)Rüdiger and Kitchatinov (2006)


Large Re and arbitrary Rm

Generation of the mean vorticity in Generation of the mean vorticity in turbulence with mean velocity shearturbulence with mean velocity shear


The growth rate of the mean vorticity

Elperin, Kleeorin and Rogachevskii, PRE, 68, 016311 (2003)


Re > 1 and arbitrary Rm


There is no shear-current dynamo

Kraichnan - Kazantsev model:

- correlated in time random velocity field

The shear-current dynamo (as well as effect of shear) requires finite correlation time of turbulent velocity field

Magnetic Field and VorticityMagnetic Field and Vorticity

Induction equation for Induction equation for magnetic fieldmagnetic field::

Equation forEquation for vorticity vorticity::

Generation of the mean vorticity and Generation of the mean vorticity and magnetic field in sheared turbulencemagnetic field in sheared turbulence


The growth rate of

The mean vorticityThe mean magnetic field

The growth rate of B

Effect of Magnetic Helicity on Non-Helical Turbulent Dynamos

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