Hall-MHD dynamos and turbulence

Hall-MHD dynamos and turbulence

P.D. MininniDepartamento de Física, Facultad de Ciencias Exactas y Naturales,

Universidad de Buenos Aires, Argentina,

and

National Center for Atmospheric Research, Boulder, Colorado, USA

A. Alexakis, D.O. Gómez, S.M. Mahajan, and A. Pouquet

[email protected], [email protected]

Momentum equation:

Induction equation:

• P is the pressure, J is the current, F an external force, ν the viscosity, η the resistivity, v the velocity, and B the induction (in Alfven velocity units); incompressibility is assumed.

• Non-relativisticflow, isotropicOhm’s law.

The MHD equations

( ) UBJUUU 2∇+×+−∇=∇⋅+

∂∂ νP

t

BUBBUB 2∇+∇⋅=∇⋅+

∂∂ η

t( )

EB

BUEJ

BJ

×−∇=∂∂

×+=

×∇=

tc

c

c

1

/4σ

π

• The MHD equations follow from an isotropic Ohm’s law.• If there are not enough collisions, the current is not parallel to the electric field;

an extra term appears associated with the trajectories of the electrons in the presence of a magnetic field.

• The amplitude of this term is proportional to the ratio of the two associated timescales.

The Hall effect from the fluid perspective

'EB ×∇−=

∂∂

ct

Momentum equation:

Induction equation:

( ) UBJUUU 2∇+×+−∇=∇⋅+

∂∂ νP

t

c

BUEE

×+=´nec

BJEEJ

×−≈= ´´ σσ

• Solar wind and Earth’s bow shock (Cluster)

• Accretion discs: H/O ≈ 102, H/I ≈ 10-4

Balbus & Terquem, ApJ 552, 253 (2001); Sano & Stone, ApJ 570, 314 (2002)

• Neutron stars and white dwarfs: H/O ≈ 103

Shalybkov & Urpin, A&A 321, 687 (1997)


c

BUEE

×+=´

( )[ ] BBJUB 2∇+×−×∇=

∂∂ ηε

t

nec

BJEEJ

×−≈= ´´ σσ

We can define:

04 UV

nm

ec

L AH π

=L

LH=ε

Induction equation:

H: Hall

O: Ohmic

I: Induction

• Coulomb’s gauge used for the vector potential (enhances stability).• The terms on the right are associated with the electric field, and U-εJ can

be interpreted as the velocity of the electrons.• The scalar potential can also be associated with the electron pressure


( ) UBJUUU 2∇+×+−∇=∇⋅+

∂∂ νP

t

( ) '2 EABJUA −=∇+×−=∇+

∂∂ ηεφ

t

( )[ ] BBJUB 2∇+×−×∇=

∂∂ ηε

t

0=⋅∇ A

ep∇−=∇ εφ

Hall-MHD and two-fluid descriptionMomentum equation for the ions:

Momentum equation for the electrons:

ii pc

endt

dnm ∇−

×+= BUE

U

ee pc

en ∇−

×+−= BUE0

Mass conservation:

( ) ( ) 0=⋅∇+∂∂

Unmnmt ii

Ampere’s law:

( ) BUUJ ×∇=−=π4

cen e

The Hall-MHD equations are obtained adding and subtracting the momentum equations.

Incompressible, dimensionless Hall-MHD:

The Hall-MHD equations

ε = LHall /L0 where L0 is a characteristic scaleIf L0 = 2π (the size of the box) ⇒ kHall = 1/ ε

Lhall = cUA / ωpiU0

For UA = U0 ⇒ Lhall is the ion skin-depthWe will study the case L0 > Lhall > Lη

Invariants

( ) BJUUU ×+−∇=∇⋅+

∂∂

Pt

( )[ ]BJUB ×−×∇=

∂∂ ε

t

∫

⋅+⋅= 3

22

1xωUBU dH g

ε

Turner (1986), Servidio et al. (2008)

( ) ( )∫ +⋅+= 3

2

1xωBBA dH h εε

Energy

Magnetic helicity

Generalized helicity

Hybrid helicity

Multiple motivations• In astrophysics and space physics, several cases are known where extensions

of the MHD fluid description are needed.

• Apparently conflicting results have been reported in Hall-MHD:– Generation of small scales and filamentation Laveder, Passot & Sulem, Phys.

Plasmas 9, 293 (2002); 9, 305 (2002)

– Fast reconnection Gosh, Siregar, Roberts & Goldstein, JGR 101, 2493 (1996);Bhattacharjee, Ma & Wang, JGR 104, 14543 (1999); Smith, Ghosh, Dmitruk &Matthaeus, JGR 31, L02805 (2004)

– Large scale self-organization Numata, Yoshida & Hayashi, Comp. Phys. Comm.

164, 291 (2004); Ohsaki, Phys. Plasmas 12, 032306 (2005)

• Can small scale effects have an impact in the evolution of large scale fields? (e.g., through inverse cascade processes)

• We analyze results from 2563 dynamo simulations (mechanical forcing)

• An analysis of scale interactions and transport coefficients in 3D Hall-MHD simulations can help us understand the basic processes in Hall-MHD turbulence.

• What is the effect of resolution? Structures in MHD and Hall-MHD.

Evidence of turbulent fluctuations are known since Mariner 2.

Helios 2 allowed studies with heliocentric distance.

Clear power laws are observed, with at least three different indices.

An example: solar wind observations

1/f: Dmitruk & Matthaeus, PRE 76, 036305 (2007)

• Although the large-scales can be described by MHD turbulence, at sufficiently large frequencies a knee is observed, followed by a third slope in the spectrum.

• The frequency corresponds to the Hall-frequency, although it is unclear if the range that follows is inertial or dissipative.

Solar wind observations

Dynamo action in Hall-MHD:Small-scale dynamos

Numerical simulations

• GHOST: Geophysical High Order Suite for Turbulence.• Hybrid (MPI/OpenMP) pseudospectral code that solves Hydrodynamic/Quasigeostrophic/Magnetohydrodynamics/HallMHD/ tracer equations in periodic domains (2D/3D).Mininni, Rosenberg, Reddy,

Pouquet (2010); extensions for GPU support (2011).

• DNS plus multiple parametrizations, and tools for turbulence analysis.

• 3D MHD/Hall-MHD pseudospectral/explicit code.

• 2563 simulations with time stepping satisfying CFL for Hall-MHD.

• Non-helicalmechanical forcing at large scales Lf = L0.

• Non-helicalmagnetic seed at small scales.

• Magnetic fields are amplified and sustained against Ohmic dissipation by a dynamo process.

• Without the Hall current, the induction equation is linearin B.

• Stretching by the electron velocity (=U when the magnetic field is weak) must overcome Ohmic dissipation to obtain growing solutions.

• A turbulent steady state is reached at late times (non-helical Hall-MHD turbulence).

Direct numerical simulations

U

B

( )[ ] BBJUB 2∇+×−×∇=

∂∂ ηε

t

Magnetic fields in astrophysics

• Generation of magnetic fields occurs in media for which the viscosityand the magnetic diffusivityare vastly different.

• A large separation of scales exists between the energy containing scales (convection, rotation), and the magnetic and kinetic dissipation scales.

• The flows are in a turbulent state, and we are interested in thestatistically stable properties of the magnetic and velocity fields.

10610120,415,3Jupiter

108101527104Sun

101010110,110-2Disks

7·1010

1

T [days]

106

109

RV

109

102

RM

10-6Galaxy

1,9Earth

B [Gauss]

V

MM

M

V

R

RP

LUR

LUR

=

=

=

η

ν

• In MHD, given v in the induction equation, and solving for B (kinematicproblem)

exponentially growing solutions are found when η is small enough.

• The growth rate can be computed from simulations

for each value of RV and RM. Near the threshold, the growth rate depends on both.

Dynamo growth rate

t

EM

dlogd

21=σ

BvBBvB 2∇+∇⋅=∇⋅+

∂∂ η

t

Ponty, Mininni, Montgomery, Pinton, Politano, & Pouquet, PRL 94, 164502 (2005)

• We consider cases with the Hall scale close to the dissipation scale.

• For small magnetic energy, the Hall term is negligible.

• In MHD, during the kinematic regime the magnetic field grows exponentially.

• In Hall-MHD, as the magnetic field grows a non-linear regimedevelops (before Lorentz saturation takes place).

• In this regime, the magnetic field grows more than in the MHD case.

Time evolution

U

BKinematicdynamo

Steady state

Gomez, Mininni, & Dmitruk, PRE 82, 036406 (2010)

• A Kazantsev small-scale dynamo spectrum develops in the magnetic energyKleeorin & Rogachevskii 1994

• Note the Hall effect is negligible at small wavenumbers.• Similar solutions are obtained for different magnetic Prandtl numbers.• However, compared with MHD, there is slightly more energyat large scales,

and the spectrum decays faster at smaller scales.

Energy spectrum

U


-5/3 -5/3

3/23/2

ε=0.1Pm=1

• The peak in the current density spectrum moves towards larger scales as the Hall effect is increased.

• The position of the peak in the Hall-MHD case corresponds roughly to the Hall scale.

• The thinnest currents sheets are at the Hall scale!

• The energy dissipation rate also decreases in the presence of the Hall effect.

Current density spectrum

U

B


• The energy dissipation rate substantially decreases with the Hall effect.

• Also, its time evolution becomes smoother.

Energy dissipation rate

U

B


Dynamo action in Hall-MHD:Large-scale dynamos

• Many astrophysical objects have magnetic fields with correlations at scales much larger than the flow energy-containing scale.

• Solar activity has a cyclic behavior with superimposed fluctuations.

• Magnetic cycles are observed in other astrophysical objects. Many planets (including the earth) go through magnetic field reversals.

• This quasi-regular behavior is accompanied by (and coupled to) complex small-scale / fast dynamics.

Large-scale / slow behavior

• Helical mechanic forcing at large scales Lf = L0/3 or smaller.• Non-helicalmagnetic seed at small scales.• In the presence of helicity, magnetic fields grow at scales larger

than the energy containing scale in the flow (large-scaledynamo).

• Shear is also known to give (under certain conditions) large-scale dynamo action.

• A turbulent steady state is reached at late times (helicalHall-MHD turbulence).

• The Hall effect is relevant at small scales. Can it affect the growth of magnetic fields at large scales?

Direct numerical simulations

U

B

• In 1958 Woltjerintroduces the magnetic helicity(later studied by Chandrasekhar and Kendall):

• In 1967, Moffattfinds its hydrodynamic equivalent:

• Helicity measures the structural complexity of the field: it is proportional to the number of linksin the field lines.

• Helicity is zero for 2D flows.

• When maximal,

Magnetic and kinetic helicity

dVHm ∫ ⋅= AB

dVH ∫ ⋅= uω

AB ×∇=

uω ×∇=

ω

u

0=×uω

• Helical magnetic fields are observed in the solar corona, in the solar wind, and the magnetosphere.

• First indirect observations date back to 1925(“sunspot whirls” by Hale).

• However, helicity was not know at that time.

• More measurements since the 90’s:Abramenko et al. (1996), Canfield et al. (1999).

• Helicity is known to be preserved in interplanetary magnetic clouds.

Helicity and magnetic fields

Bleyvel et al (2002)

Time evolution

Kinematicdynamo

Kinetic energy

Magnetic energy

• Kinetic and magnetic energy as a function of time in MHD and Hall-MHD• The saturation level in the turbulent steady state is a function of the Hall lengthscale

Steady state

Energy spectrumε = 0, Re = 560

ε = 0.1, Re = 560

• 2563 pseudospectralsimulations, 3D periodic boundary conditions.η = ν = 2×10-3.

• In the steady state, the ratio of magnetic to kinetic energy at small scales depends on the value of ε.

Should we find k -7/3 or k -11/3 for k > kHall? Galtier, J. Plasma Phys. 72, 721 (2006); Galtier & Buchlin, ApJ 656, 560 (2007).

The Hall length scale

( )0=εME

( )εME

• Hall-suppressed regime: inhibits dynamo activity• Hall-enhanced regime: enhances dynamo activity• As ε → 0 we obtain the classical MHD dynamo

Mininni, Gómez & Mahajan, ApJ 619, 1019 (2005)

Turbulent transport coefficients

• Can we understand these processes from turbulent transport coefficients? We split the fields into the mean and fluctuating components

• Mean Field Theory (Krause & Raedler 1980), and Reduced Smooth Approximation (Blackman and Field 1999, also Gruzinov & Diamond 1995).

• RSA allows for the recovery of higher order terms (backreaction) not present in MFT and first obtained using EDQNM (Pouquet, Frisch, & Leorat 1976).


Isotropic, fluctuating solution

Large scale fields Anisotropic

correction to the fluctuating fields

• Replacing into the induction equation and averaging:

where the mean field electromotive force is

• We want to close this equation. From an argument of symmetry

and we also have and equations for the fluctuations (neglecting terms quadratic in the fluctuations):


• Finally, to close the equations,d/dt in the equations for the fluctuations are replaced by the inverse of a correlation time1/τ :

• The alpha-effectis responsible for the growth of large-scale magnetic fields in MHD. In Hall-MHD, it is associated with the helicity of the electron velocity field.

• The third term is new (and important) in Hall-MHD• The beta-effect(magnetic diffusivity) can now be negative!• The gamma-effectis also new in Hall-MHD.


Mininni, Gómez, & Mahajan, ApJ 567, L81 (2002); Mininni, Alexakis, & Pouquet, JPP 73, 377 (2007)

• In the MHD case, α=0 for Alfven waves u0= ±b0.

• In the Hall-MHD case, α=0 should be zero for the normal modes:

• The third term is needed to satisfy this condition.

• While the second term is associated with non-linear saturation through the Lorentz force, the third term is associated with the non-linearHall effect.

• This term can give faster saturation or enhance dynamo action.

The alpha-effect

ee k 00 uu ±=×∇ 00 bb k±=×∇

• In the MHD case, the alpha effect is also responsible for the generation of large-scale magnetic helicity.

• The accumulation of large-scale magnetic helicity results in the slow saturation of the dynamo.

The alpha-effect

• In Hall-MHD, less magnetic helicity is generated, resulting in a faster growth (and saturation) of the dynamo.

• Even for small values of the Hall-effect, a faster (and larger) growth of the large-scale magnetic field is observed.

Revisiting the alpha-effect

In Hall-MHD, the kinetic helicity (or the kinetic helicity containing scale) increases after the dynamo saturation, an indication of “handedness”associated with the Hall currents.


• As scale separation is increase, the large-scale dynamo becomes more clear.• In the Hall-MHD case, the large scale magnetic field is accompanied by

large-scale flows.


• In the MHD case, β>0, and doesn’t change during non-linear saturation.

• In the Hall-MHD case, when β<0 we can have non-local transfer of energy from small-scale turbulent fields to large scale magnetic fields.

• For large values of ε and if the system is magnetically dominated, then β<0.

• For the normal modes, one branch gives β>0 (right handed polarized) while the other gives β<0 (left handed polarized).

The beta-effect

• In a magnetically forced run, large scale magnetic fields are observed to grow.

• The magnetic field is non-helical.

• The amplitude of the magnetic field at k = 1 grows as ε is increased.

ε = 0 (MHD, solid line), ε = 0.05 (dotted line),

ε = 0.1 (dashed line).

Backscatter of magnetic energy

Cascades and energy transfers in Hall-MHD

The Hall-MHD energy transfer

• Incompressible Hall-MHD:

The Hall-MHD energy transfer

Mininni, Alexakis & Pouquet, J. Plasma Phys. 73, 377 (2007)

The Tuu and TbbMHD transfers (advection) give a local and direct cascade of

energy.

Hall-MHD: just the same?

• From the shell-to-shell transfer functions we can get information of the direction of the cascades, as details of interactions between fields and scales.

• From the transfer functions we can also get the individual contribution to the energy flux summing over wave numbers.

The Tbu transfer is different: in the turbulent regime there is a local backscatterof magnetic to kinetic energy.

The TbbHall term gives a

local direct transferof magnetic energy to small scales, and a nonlocalinverse transferto large scales.

Hall-MHD: nonlocal effects

Mininni, Alexakis & Pouquet, J. Plasma Phys. 73, 377 (2007)

The Tbu transfer is different: in the turbulent regime there is a local backscatterof magnetic to kinetic energy.

The TbbHall term gives a

local direct transferof magnetic energy to small scales, and a nonlocalinverse transferto large scales.

Hall-MHD: nonlocal effects

Tuu, Tbb

TbbHall


• The magnetic field at wavenumber K (=20 in the figure) receives energy from wavenumbers K slightly smaller and gives energy to wavenumbers K slightly larger (direct local transfer).

• But the magnetic field at K also gives energy to scales much smaller (inverse nonlocal transfer).

• When all the terms are considered, what direction wins?

Hall-MHD: TbbHall transfer

Turbulentsteady state

• The net flux is positive, giving a direct cascade of energy to small scales.

• However, the flux of magnetic energyis positive at scales smaller than Lhall, but negative at scales larger.

• The Hall effect gives enhancement of both small and large scales.

• In a non-helical magnetically dominated run, can large scale magnetic fields grow?

Hall-MHD: Energy flux

∏(k)

∏bbHall(k)

∏bbMHD(k)

∏uu(k)

∏ub(k)

• The combined effect of advection by the velocity field, and of the Hall current, still gives a net upscaling of magnetic energy in a range of scales.

• Also, the downscaling of magnetic energy is decreased as the strength of the Hall effect is increased.

Hall-MHD: total magnetic flux

Generation of large-scale flows can also be observed, as the result of the interaction of the magnetic field with the velocity field in the Hall-MHD case.

Hall-MHD: magnetic-kinetic flux

• The transfer of magnetic helicity in the Hall-MHD case is strongly quenched.

• Since magnetic helicity is transferred to larger (and smaller) scales at a lower rate, helicity of opposite signs piles up near the forcing scale.

• This results in a net value of magnetic helicity closer to zero, and prevents “catastrophic quenching” of the dynamo.

Hall-MHD: TH transfer

Conclusions

• Dynamo is enhanced whenLHall is at intermediate scales and is inhibited whenLHall is at large scales

• The maximun magnetic field generated depends non monotonicallywith ε. The efficiency of the dynamo is increased as the scale separation is increased

• The effect of the Hall term in 3D Hall-MHD turbulence is more complex than what was expected.

• The transfer of magnetic energy is modified, and the new term gives both a direct transfer (and an enhancement of small scales) as well as an inverse nonlocal transfer to large scales.

• The coupling between velocity and magnetic field also changes, and the magnetic field can transfer energy to flows at large scales.

• Can we expect changes in the power law due to this dual transfer? Simulations at higher resolution are needed.

Hall-MHD dynamos and turbulence

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