The test field method
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The Test-field Method. Output so far. Simulations showing large-scale fields. Helical turbulence ( B y ). Helical shear flow turb. Convection with shear. Magneto-rotational Inst. K äpyla et al (2008). Low Pr M dynamos. Sun Pr M = n/h =10 -5. Schekochihin et al (2005).

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The Test-field Method

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The test field method

The Test-field Method

Output so far

Output so far

Simulations showing large scale fields

Simulations showing large-scale fields

Helical turbulence (By)

Helical shear flow turb.

Convection with shear

Magneto-rotational Inst.

Käpyla et al (2008)

Low pr m dynamos

Low PrM dynamos

Sun PrM=n/h=10-5


et al (2005)

Here: non-helically

forced turbulence


Helical turbulence

Upcoming dynamo effort in stockholm

Upcoming dynamo effort in Stockholm

Soon hiring:

  • 4 students

  • 3 post-docs

  • 1 assistant professor

  • Long-term visitors

Calculate full a ij and h ij tensors

Calculate full aij and hij tensors

turbulent emf

  • Correlation method

    • MRI accretion discs (Brandenburg & Sokoloff 2002)

    • Galactic turbulence (Kowal et al. 2005, 2006)

  • Test field method

    • Stationary geodynamo (Schrinner et al. 2005, 2007)

    • Shear flow turbulence (Brandenburg 2005)

a effect and turbulent

aagnetic diffusivity

Calculate full a ij and h ij tensors1

Calculate full aij and hij tensors

Original equation (uncurled)

Mean-field equation


Response to arbitrary mean fields

Test fields

Test fields


Validation roberts flow

Validation: Roberts flow


SOCA result


Kinematic a and h t independent of rm 2 200

Kinematic a and ht independent of Rm (2…200)

Sur et al. (2008, MNRAS)

Non helical with rotation

Non-helical with rotation

Rotational quenching, finite d>0 for W>0

Shear turbulence

Shear turbulence

Growth rate

Use S<0, so need negative h*21 for dynamo

Dependence on sh and rm

Dependence on Sh and Rm

Direct simulations

Direct simulations

Fluctuations of a ij and h ij

Fluctuations of aij and hij

Incoherent a effect

(Vishniac & Brandenburg 1997,

Sokoloff 1997, Silantev 2000,

Proctor 2007)

From linear to nonlinear

From linear to nonlinear

Use vector potential

Mean and fluctuating

U enter separately

Nonlinear a ij and h ij tensors

Nonlinear aij and hij tensors

Consistency check: consider steady state to avoid da/dt terms


l=0 (within error bars)  consistency check!

R m dependence for b b eq

Rm dependence for B~Beq

  • l is small  consistency

  • a1 and a2 tend to cancel

  • making a small

  • h2 small

Earlier results on h t quenching

Earlier results on ht quenching

Yousef et al.

(2003, A&A)

Nonlinear a ij and h ij tensors1

Nonlinear aij and hij tensors

Another consistency check:

passive vector equation

Solar paradigm shifts

Solar paradigm shifts

  • 1980: magnetic buoyancy (Spiegel & Weiss) overshoot layer dynamos

  • 1985: helioseismology: dW/dr > 0  dynamo dilema, flux transport dynamos

  • 1992: catastrophic a-quenching a~Rm-1(Vainshtein & Cattaneo) Parker’s interface dynamo  Backcock-Leighton mechanism

I is magnetic buoyancy a problem

(i) Is magnetic buoyancy a problem?

Stratified dynamo simulation in 1990

Expected strong buoyancy losses,

but no: downward pumping

Tobias et al. (2001)

Ii before helioseismology

(ii) Before helioseismology

  • Angular velocity (at 4o latitude):

    • very young spots: 473 nHz

    • oldest spots: 462 nHz

    • Surface plasma: 452 nHz

  • Conclusion back then:

    • Sun spins faster in deaper convection zone

    • Solar dynamo works with dW/dr<0: equatorward migr

Benevolenskaya et al. (1998)

Thompson et al. (2003)

Yoshimura (1975)

Revisit paradigm shifts

Revisit paradigm shifts

  • 1980: magnetic buoyancy  counteracted by pumping

  • 1985: helioseismology: dW/dr > 0  negative gradient in near-surface shear layer

  • 1992: catastrophic a-quenching  overcome by helicity fluxes  in the Sun: by coronal mass ejections



  • 11 yr cycle

  • Dyamo (SS vs LS)

  • Problems

    • a-quenching

    • slow saturation

  • Solution

    • Modern a-effect theory

    • j.b contribution

    • Magnetic helicity fluxes

  • Location of dynamo

    • Distrubtion, shaped by

    • near-surface shear

1046 Mx2/cycle

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