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February 28, 2001

The Madison Dynamo Experiment: magnetic instabilities driven by sheared flow in a sphere

Cary ForestDepartment of PhysicsUniversity of Wisconsin

February 28, 2001 2

Planets, stars and perhaps the galaxy all have magnetic fields produced by dynamos

Solar prominences and x-rays from sunspots

Magnetic fields observed inM83 spiral galaxy

Glatzmaier Roberts simulation of geodynamo

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Outline• What are dynamos?• Some models of astrophysical and geophysical dynamos• The Madison dynamo experiment• Questions for experiments to study

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Physics of magnetic field generation

1. Charged particles moving in a magnetic field experience the Lorentz force

2. The motional EMF generates a current

3. Currents produce a magnetic field

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Dynamos generate magnetic energy from mechanical energy• this is easy if you allow yourself

the luxury of using insulators and solid conductors

• in a conductor, currents are generated by motion across a magnetic field

feedback

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• plasmas or liquid metals • Magnetohydrodynamics: systems are describe by two

vector fields:– the magnetic field, is generated by electrical currents in the conducting fluid

In astrophysical dynamos the conductors are simply connected (no insulators) and can flow

Induction Resistivediffusion

-The velocity field evolves according to Navier-Stokes + electromagnetic forces

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• in a fast moving, or highly conducting fluid, magnetic field lines are frozen into the moving fluid

• transverse component of field is generated and amplified• finite resistance leads to diffusion of field lines

Fluid flow can amplify and distort magnetic fields

0

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Cartoon of a laminar geodynamo driven by convection and differential rotation

Differential rotationgenerates Bφfrom dipole

Thermal convection in rotatingsphere generates cyclonic

eddies which produce dipole field from Bφ

Dipole field reinforcesinitial magnetic field

pB RmBφ ∼

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The kinematic dynamo problem• Start with a sphere filled with a uniformly conducting fluid of

conductivity σ, radius a, surrounded by an insulating region

• Find a velocity field V(r) inside the sphere, which leads to growing B(r,t)

• Ignore the back-reaction of magnetic field on flow

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The kinematic dynamo problem (continued)• Transform to dimensionless variables:

• Since its linear in B, use separation of variables:

• Solve eigenvalue equation for given V(r) profile

( ), ) λ=∑ B rB(r i

it

it e

m0 axRm aVσ=µ

2

02

1

1ˆ ˆ ˆˆ

→

∂ =∇× × + ∇µ σ∂∂ =∇× × + ∇∂

B V B B

B V B B

t

Rmt

21λ =∇× × + ∇B V B Bi i iRm


Solution technique involves solving induction equation using spherical harmonic basis functions

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

m m m mn n n n

m m m mn n n n

ˆ ˆt r Y , s r Y ,

ˆ ˆT r Y , S r Y ,

θ ϕ θ ϕ

θ ϕ θ ϕ

= ∇ × + ∇ × ∇ ×

= ∇ × + ∇ × ∇ ×

V r r

B r r

Since fields are incompressible, use vector potential formulation, based on spherical harmonics (Bullard and Gellman, 1954)

Integration over angles and selection rules produce a coupled set of differential equations, which are solved numerically, using finite differences in the radial direction

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

2

2 2 2

2

2 2 2

1

1

,

,

n nS RmS S t S S s T S s S S

r r r

n nT RmT T t T T t S T s T T s S T

r r r

γ γγγ γ α β γ α β γ α β γ

α β

γ γγγ γ α β γ α β γ α β γ α β γ

α β

λ

λ

+∂ − − = + + ∂

+∂ − − = + + + ∂

∑

∑Solutions for complex eigenvalues are found using inverse iteration


Growing magnetic fields are predicted for simple flow topologies in a sphere

• induction equation is solved numerically

• flow fields are axisymmetric • growth rate depends upon

m0 axRm aVσ=µconductivity X size X velocity


Numerical search is used to find optimized velocity profiles forgiven flow topology• Radial function is

parameterized and search algorithm is used to find low Rmcrit

( )

( )

220

2 22

220

2 22

1

1

tt t

t

ss s

s

r rl lt ( r ) exp( )

r r w

r rl ls ( r ) exp( )

r r wε

−= − + −

−

−= − + −

−


Optimized solutions have low critical Rm


Can these flows be produced in a laboratory?• How big and how much power is required?• Is this simply an linear eigenmode problem in a

turbulent/mechanically homogeneous system?• Estimates suggest 100 kW, and a=0.5 m with sodium

should provide Rm=100


The Dynamo Experiment• 1 m diameter• 200 Hp • sodium• Rm=200


Why sodium?• the control parameter is the magnetic Reynolds number

– quantifies relative importance of generation of B field by velocity and diffusion (decay) of B due to resistive decay of electrical currents

– must exceed critical value for system to self-excite• Sodium is more conducting than any other liquid metal

(melts at 100 C)– Rm = 120 for a=0.5 m, Vmax=15 m/s

m0 axRm aVσ=µ conductivity X size X velocity


Dimensionally identical water experiment is used to create flows and test technology• Laser Doppler velocimetry is

used to measure vector velocity field

• Measured flows are used as input to MHD calculation

• Full scale, half power


Measurements at discrete positions are used to find radial profile functions for input to eigenvalue code


Data are fit by profile functions


Simple baffles and Kort nozzles produce T2S2 flows


Measured water flows extrapolate to an experimental dynamo in the sodium experiment


Flows are turbulent and large scale velocity field varies on resistive time scales

• Large scale velocity field varies with time– Mechanically homogeneous

• Magnetic eigenmode analysis is only valid on correlation time of velocity fluctuations

• Turbulence from small scales may play important role


Flow should be linearly unstable some fraction of the time• Assume mean flow is bounded

by rms fluctuation levels• Specific geometry corresponds

to a family of flow profiles• Some fraction of flow profiles

self-excite• Monte-Carlo analysis shows

estimates fraction of time in growing phase

• Temporal intermittency for dynamo?


Sodium laboratory is 20 miles from Water laboratory (remember high school chemistry)

Physics Department

Liquid sodium laboratory


Molten sodium laboratory is operational• A new laboratory has been constructed for housing the

dynamo experiment


The Dynamo Experiment will address several dynamo issues experimentally• Do dynamically consistent flows exist for kinematic

dynamos?• How does a dynamo saturate?

– role of Lorentz force on fluid velocity• What role does turbulence play in a real dynamo?

– energy equipartition of velocity fields and magnetic fields– enhanced electrical resistivity– current generation


Back-reaction and saturation is due to non-linearities in MHD Equations • Induction equation by itself is linear in B

– If V is given, linear solutions can be found (kinematic dynamo problem)

• Induction equation is non-linear if V is affected by B• Navier-stokes is nonlinear in V and JxB

– V is naturally turbulent– JxB force modifying V we can call the back-reaction


What happens as the magnetic field energy grows?• Simulations show a strong back-reaction when b=v• Turbulent flows can be strongly modified

2 2

0

1 12 2

v bρµ

= %%


Nonlinear effects in mean-field electrodynamics• α-effect as derived is a kinematic treatment• Treatment did not include back-reaction of induced

magnetic field on velocity fluctuations

( )( )

( )( )

( )

0

0

23 1

t

corr

corr

t

d

t dt

B

t t

δ

δ

τα

στρ

δ

δ−∞

−∞

×

′ ′= ∇ × ×

⋅∇×≈ ⋅∇× − ⋅∇×

×

′ ′= ∇ ×

=

×

+

∫

∫

E = + v b

b v B

v vv v b b

v b

v b B

%%

%

% %% %

%

%

%

%

%

See Gruzinov and Diamond’s quasilinearTreatment of backreaction term


MHD turbulence introduces effects beyond scope of laminar DynamoTheory for Experiments to Study

– equipartition between magnetic and kinetic energy is predicted for small scales

– resistivity enhancement due to mixing of magnetic fields on small scales is predicted

– Currents can be generated by helical velocity fluctuations on small scales


The β-effect: eddies move flux

( )( ) ( )0

2

0

1

1 3corr

T ,

σ α β

µ σβ σ α

τσσ β

µ σβ

= × − ∇×

⇒ + = ×

⇒ = ≈+

J E + V B + B B

J E + V B + B

v

Will there be a turbulent modification to Rm-crit in experiments?

01T

RmRm

µ σβ⇒ =

+


Measured turbulence levels already indicate resistivity enhancement may be important• Turbulence levels are measured

on small experiment

0

2

1

3

T

corr

,σ

σµ σβ

τβ

=+

≈ v


Experiment to measure β-effect• Use transient techniques to

estimate resistivity (measure electrical skin depth of system)

• Use ω-effect to measure Rm with applied poloidal field

zB Rm Bϕ =


Experiment to measure turbulent α-effect• Apply toroidal field. If toroidal

current is observed, symmetry breaking fluctuations must be responsible

• Increase B and look for modifications to a-effect

• Study role of large scale B on small scale fluctuations (saturation through Alfven effect)

21 corr Bα

στρ

⋅∇×≈

+

v v% %


Summary• water experiments have demonstrated flows which may

lead to dynamo action• sodium laboratory facility is completed

– sodium experiment is under construction – sodium operation is imminent

• initial experiments will search for growing eigenmodes and evaluation of role of turbulence


Thanks to:

• The David and Lucille Packard Foundation

• The Sloan Foundation• NSF• The Research Corporation• The DoE

Roch Kendrick, Jim Truit Rob O’Connell, Cary ForestErik Spence, Mark Nornberg