Dynamics of Locked Dynamo Simulations

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Dynamics of Locked Dynamo Simulations
Chris Davies, David Gubbins and Peter Jimack
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Evidence for lateral heat flux variations

• Timescale of geomagnetic reversals
• Four lobes of concentrated magnetic flux
• Preferred VGP paths during reversals
• Low SV in Pacific

(Jackson et al, 2000)
(Lowrie, 1997)
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Core-Mantle Interactions

• Timescales of core/mantle processes are radically
  different
• so they view the CMB in very different ways

CMB
(Gubbins, 2000)
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Tomographic boundary condition
Dominated by m2 harmonic
Heat Flux at CMB (Masters et al, 1996)
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The Problem

• Electrically conducting fluid confined to a
  spherical shell of thickness d ro - ri
• Rapid rotation
• Buoyancy-driven convection
• Inhomogeneous thermal boundary condition
• Solve for
• Magnetic field intensity, B
• Fluid velocity, u
• Temperature, T

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Governing Equations

• Induction
• Navier-Stokes
• Temperature
• Constraints

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Solution Method

• Toroidal/Poloidal decomposition
• For any toroidal/poloidal scalar A
• Finite difference discretisation in radius
• Solve for coefficients X and Y

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Boundary Conditions

• Velocity No slip at ri and ro
• Magnetic Field
• Assume an insulating mantle (
  ) so and B matches to a potential field
• Inner core is electrically conducting solve
  induction equation here with u 0

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Boundary Conditions, cont

• Let
• T0 is the spherically symmetric temperature
  distribution, and

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Nondimensional Numbers

• Ekman number
• Prandtl number
• Roberts number
• Buoyancy parameter
• Rayleigh number variable

• Ra measured as ratio to critical Rayleigh number
  for onset of convection without magnetic field

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Boundary Resonance

• Three distinct dynamical regions

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Locked solution Ra1.5Rc(example of region 1)
Br at the CMB averaged over 10 diffusion times
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Failed dynamo Ra1.9Rc

• Magnetic energy decays after 1.5 magnetic
  diffusion times
• Kinetic energy becomes steady at the point of
  dynamo failure

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Failed dynamo for Ra1.9Rc Flow Pattern
Ra1.9Rc
Ra1.5Rc
Ur Equatorial Plane
Ur Meridional Plane
Uf Equatorial Plane
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What about other dynamos?
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Comparison of Ra1.8Rc and Ra1.9Rc Energies
Ra1.8Rc
Ra1.9Rc
Magnetic Energy
Kinetic Energy
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Comparison of Ra1.8Rc and Ra1.9Rc Flow patterns
Ra1.9Rc
Ra1.8Rc
Ur Equatorial Plane
Ur Meridional Plane
Uf Equatorial Plane
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Is the Ra1.8Rc field locked?
Br at the CMB averaged over 8 diffusion times
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Summary of failed dynamo, Ra1.9Rc

• Localised convection planform not good for dynamo
  action
• but a very similar flow sustains a dynamo at
  Ra1.8Rc
• Possible difference due to ability of flow to
  stretch and twist magnetic field lines, thus
  sustaining the dynamo

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Transition dynamos
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Ra1.77Rc
Br at CMB averaged over 7 diffusion times

• ME and KE time-series show transition to
  oscillatory state after 3 diffusion times

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Flows before and after transition
After
Ur Equatorial Plane
Ur Meridional Plane
Uf Equatorial Plane
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Transition region for Ra1.77Rc
Kinetic energy for Ra1.77Rc
m-spectrum before and after transition
Time-series of selected wavenumbers

• M1 mode grows at point of transition while m2
  mode decays
• Have any trends appeared as Ra is increased?

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Cause of m1 instability(?)
M-spectrum for various wavenumbers
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Cause of m1 instability

• Energy in m1 mode increases as Ra increases
• Appears as though boundary-driven flow and
  convection are competing against each other
• Boundary-driven flow tries to keep planform large
  scale and succeeds up to Ra1.75Rc
• At Ra1.76Rc convection is strong enough to
  substantially modulate overall flow pattern
• The complex interaction yields an m1-dominated
  flow

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Implications

• Effect of thermal boundary condition decreases as
  we increase the Rayleigh number
• Does the dynamo return?
• Now the flow pattern is dominated by the
  convection, with the boundary condition providing
  a modulation
• As Ra increases we expect to see a planform that
  is similar to the homogeneous case

Ra2Rc
Ur in equatorial plane
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Magnetic field comparison
Ra2Rc at CMB
Ra1.5Rc at CMB
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Summary

• 3 dynamical regimes near Rc
• 1.43Rc lt Ra lt 1.75Rc Locked dynamos
• 1.76Rc lt Ra lt 1.8Rc Oscillatory dynamos
• 1.81Rc lt Ra lt 1.9Rc Steady dynamos
• Transition to oscillatory dynamo occurs when
  convection is strong enough to overcome effect of
  inhomogeneous boundary
• m1 mode becomes dominant while m2 mode
  associated with thermal boundary condition drops
• Realised as a localised convection planform
• Failure of dynamo at Ra1.9Rc
• Probably due to stretching/twisting properties of
  the individual flows
• Dynamo action returns at Ra2Rc

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Future work

• Greater understanding of m1 instability
• What physical processes cause it?
• Kinematic dynamo simulations
• Exploration of regime after secondary onset
• Does the dynamo fail again?
• Use of simplified thermal boundary conditions
• Do we see the same effects? At the same values of
  Ra?
• Can we isolate the role of individual boundary
  harmonics in dynamo failure or bifurcations?
• Dependence on other parameters
• Varying e, E