The growth of magnetic field energy in conducting fluids

1
The growth of magnetic field energy in conducting
fluids

• Philip Livermore
• Andrew Jackson

School of Earth Sciences University of Leeds, UK
2
Introduction

• The Earths magnetic field
• Generated in fluid outer core by buoyancy driven
  motion

• Field is complex, but hope
• to understand large scale features by simple
  models

3
What we know
Observations Under various assumptions Surface
measurements
Core-Mantle Boundary Experiments Laboratory
dynamos So far spherical dynamos
unsuccessful Numerical simulations E Viscous
force/Coriolis force 10-15 in Earth Numerical
problems, cannot resolve scales
4
Magnetic field generation

• A magnetic field B evolves according to

Stretching advection Diffusion
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Choice of flow

• Spherical geometry
• Flow defined in electrical
  insulator
• Non-slip BCs
• Simple cases axisymmetric / no inner core

Poloidal
Toroidal
Streamlines
Azimuthal flow intensity
Meridional section f0
6
Eigenmode analysis

• To investigate stability of induction equation
• Make ansatz
• Write

• For tgtgt1 the most unstable eigenmode dominates
• Stability depends on the existence of a growing
  eigenmode
• Issues
• sensitivity
• not necessarily interested in tgtgt1, only when
  Bgtgt1
• on shorter time scales, other effects may become
  important

7
Subcritical growth

• Consider
• combination of
• decaying modes
• Superposition of decaying non-orthogonal
  eigenmodes can produce transient growth
• Induction equation is non self-adjoint

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Stability

• Self-adjoint problems
• eigenvectors orthogonal - no transient growth
• stability completely determined by eigenvalues
• Non self-adjoint problems
• eigenvectors not orthogonal
• transient growth may occur even when all
  eigenmodes decay
• Energy methods look at onset of any growth

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The magnetic energy equation

• Magnetic energy
• Take

• LHS gives

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Energetic instability

• Define

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A Spectral method

• Spectral method - write
• where Bi is divergence-less and satisfies all
  BCs

• Bi has 3 components, but
• Write in poloidal/toroidal decomposition

• Expand scalars as

• Recombined Chebyshev polynomials
• Each Bi is either toroidal/poloidal with a

particular harmonic and radial form.
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A variational method

• Suppose is attained at some B
• Perturb it by
• No 1st order changes in

• Expand both B and dB as
• Generalised self-adjoint eigenvalue problem for
  B

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Examples at critical Rm
stretching
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Summary of energetic analysis

• Rmc O(10)
• Robust
• in that small changes in the flow do not affect
  the results
• Physically attributable to stretching by the flow

15
Transience

• Described the onset of instability
• How large can these get before decay?
• What time scales are relevant?

• Discretise induction equation with basis Bi

• Solution is

• Matrix exponential

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Energy envelopes

• Calculate for a given time

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Axisymmetric toroidal flow example

• Cannot sustain magnetic fields indefinitely

(1 Dipole diffusion time 20,000 years for the
Earth)
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Conclusions

• Energetic instability is robust and predicts
• critical Rm O(10).
• In non self-adjoint systems eigenmode analysis is
  not the full story, and transient instabilities
  may lead to transition to the non-linear regime
  where the Lorentz force is important even in
  eigenvalue-stable flows.
• Large planets or galaxies have such long
  diffusion timescales that linear analysis may
  tell us little.