The Madison Dynamo Experiment: magnetic instabilities driven by sheared flow in a sphere Cary Forest Department of Physics University of Wisconsin February 28, 2001
Planets, stars and perhaps the galaxy all have magnetic fields produced by dynamos Glatzmaier Roberts simulation of geodynamo Solar prominences and x-rays from sunspots Magnetic fields observed in M83 spiral galaxy February 28, 2001 2
Outline What are dynamos? Some models of astrophysical and geophysical dynamos The Madison dynamo experiment Questions for experiments to study February 28, 2001 3
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 February 28, 2001 4
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 February 28, 2001 5
In astrophysical dynamos the conductors are simply connected (no insulators) and can flow plasmas or liquid metals Magnetohydrodynamics: systems are describe by two vector fields: the magnetic field, is generated by electrical currents in the conducting fluid Induction Resistive diffusion -The velocity field evolves according to Navier-Stokes + electromagnetic forces February 28, 2001 6
Fluid flow can amplify and distort magnetic fields in a fast moving, or highly conducting fluid, magnetic field lines are frozen into the moving fluid 0 transverse component of field is generated and amplified finite resistance leads to diffusion of field lines February 28, 2001 7
Cartoon of a laminar geodynamo driven by convection and differential rotation Differential rotation generates B f from dipole Bφ RmB p Dipole field reinforces initial magnetic field Thermal convection in rotating sphere generates cyclonic eddies which produce dipole field from B f February 28, 2001 8
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 February 28, 2001 9
The kinematic dynamo problem (continued) Transform to dimensionless variables: B = V B + 1 2 µσ B t 0 B ˆ ˆ 1 = V B + ˆ 2B tˆ Rm Since its linear in B, use separation of variables: = B () r i e t B(r,) t λ i Solve eigenvalue equation for given V(r) profile λ = + 1 Rm B V B 2B i i i i Rm=µ σav 0 max February 28, 2001 10
Solution technique involves solving induction equation using spherical harmonic basis functions Since fields are incompressible, use vector potential formulation, based on spherical harmonics (Bullard and Gellman, 1954) ( ) ( θϕ) ˆ + ( ) ( θϕ) ˆ ( ) ( θϕ) ˆ + ( ) ( θϕ) V = t r Y, r s r Y, r m m m m n n n n B= T r Y, r S ry, ˆr m m m m n n n n 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 S n ( nγ + 1) ( nγ + 1) Rm ( ) ( ) ( ) γ γ S S tss sts s S S 2 2 γ λ γ = 2 α β γ + α β γ + α β γ αβ, r r r T n Rm ( ) ( ) ( ) ( ) γ γ T T ttt tst stt sst 2 2 γ λ γ = 2 α β γ + α β γ + α β γ + α β γ αβ, r r r Solutions for complex eigenvalues are found using inverse iteration February 28, 2001 11
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 Rm=µ σav 0 max conductivity X size X velocity February 28, 2001 12
Numerical search is used to find optimized velocity profiles for given flow topology Radial function is parameterized and search algorithm is used to find low Rmcrit ( r r ) l l t(r) = exp( + ) 0 t t t 2 2 2 r 1 r wt 2 2 ( r r ) l l s(r) = ε exp( + ) 0 s s s 2 2 2 r 1 r ws 2 2 February 28, 2001 13
Optimized solutions have low critical Rm February 28, 2001 14
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 February 28, 2001 15
The Dynamo Experiment 1 m diameter 200 Hp sodium Rm=200 February 28, 2001 16
Why sodium? the control parameter is the magnetic Reynolds number Rm =µ σav 0 max conductivity X size X velocity 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, V max =15 m/s February 28, 2001 17
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 February 28, 2001 18
Measurements at discrete positions are used to find radial profile functions for input to eigenvalue code February 28, 2001 19
Data are fit by profile functions February 28, 2001 20
Simple baffles and Kort nozzles produce T2S2 flows February 28, 2001 21
Measured water flows extrapolate to an experimental dynamo in the sodium experiment February 28, 2001 22
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 February 28, 2001 23
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? February 28, 2001 24
Sodium laboratory is 20 miles from Water laboratory (remember high school chemistry) Physics Department Liquid sodium laboratory February 28, 2001 25
Molten sodium laboratory is operational A new laboratory has been constructed for housing the dynamo experiment February 28, 2001 26
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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 February 28, 2001 28
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 February 28, 2001 29
What happens as the magnetic field energy grows? Simulations show a strong back-reaction when b=v Turbulent flows can be strongly modified 1 1 ρv = b 2 2µ 2 2 0 February 28, 2001 30
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 E= δ v b + v δ b t t ( ( t ) ) δb = v B dt α ( t ) δ v= b B dt ( ) τ corr 3 ( v v b b ) 0 0 See Gruzinov and Diamond s quasilinear Treatment of backreaction term v v = στcorr 1+ B ρ February 28, 2001 31 2
MHD turbulence introduces effects beyond scope of laminar Dynamo Theory 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 February 28, 2001 32
The β-effect: eddies move flux J = σ E+V B+ αb β B ( ) 1+ µσβ J = σ E+V B+ αb ( ) ( ) 0 σ τ σt =, β 1+ µσβ 3 0 corr v 2 Will there be a turbulent modification to Rm-crit in experiments? Rm RmT = 1 + µσβ February 28, 2001 33 0
Measured turbulence levels already indicate resistivity enhancement may be important Turbulence levels are measured on small experiment σ T β = 1 + τ corr 3 σ µσβ 0 v 2, February 28, 2001 34
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 B = ϕ RmB z February 28, 2001 35
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) v v α στ corr 2 1+ B ρ February 28, 2001 36
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 February 28, 2001 37
Thanks to: The David and Lucille Packard Foundation The Sloan Foundation NSF The Research Corporation The DoE Roch Kendrick, Jim Truit Erik Spence, Mark Nornberg Rob O Connell, Cary Forest February 28, 2001 38