Magnetospheric Modeling

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Magnetospheric Modeling

• M. Wiltberger and E. J. Rigler
• NCAR/HAO

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Outline

• What is the magnetosphere?
• How do we model it?
• Possible areas where statistics can help us

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Solar Origins

• Solar Flares - abrupt release of energy
• localized solar region
• mainly radiation (UV, X-rays,?-rays)
• occur near complex sunspot configurations

• Coronal Mass Ejections (CMEs)
• Releases of massive amounts of solar material
• Usually with higher speeds and greater magnetic
  fields than surrounding solar wind
• Usually cause shocks in solar wind

• Solar Wind
• Steady ionized gas outflow with average velocity
  400 km/s
• Magnetic field direction variable
• Exact properties depend upon solar origins

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Earth's Magnetosphere

• The magnetosphere is region near the Earth where
  it's magnetic field forms a protective bubble
  which impedes the transfer of energy and momentum
  from the solar wind plasma

• A variety of different phenomenon
• Substorms
• impulsive energy release over hours
• Storms
• globally enhanced activity over days
• Radiation belts
• trapped particles which are omnipresent

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Magnetospheric Currents

• Magnetopause current systems are created by the
  force balance between the Earths dipole and the
  incoming solar wind

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Ionospheric Currents
Region 1
Region 2

• FAC from the magnetosphere close though Pedersen
  and Hall Currents in the ionosphere

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LFM Magnetospheric Model

• Uses the ideal MHD equations to model
  the interaction between the solar wind,
  magnetosphere, and ionosphere
• Computational domain
• 30 RE lt x lt -300 RE 100RE for YZ
• Inner radius at 2 RE
• Calculates
• full MHD state vector everywhere within
  computational domain
• Requires
• Solar wind MHD state vector along outer boundary
• Empirical model for determining energy flux of
  precipitating electrons
• Cross polar cap potential pattern in high
  latitude region which is used to determine
  boundary condition on flow

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Numerics of the LFM

• LFM solves ideal MHD eqs in conservative form
  using the Partial Interface Method
• PIM is hybrid scheme that balances the competing
  diffusion and dispersion errors by selectively
  adding diffusion to high order scheme
• Limitor keeps solution monotonic
• Provides nonlinear numeric resistivity

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Computational Grid of the LFM

• Distorted spherical mesh
• Places optimal resolution in regions of a priori
  interest
• Logically rectangular nature allows for easy code
  development
• Yee type grid
• Magnetic fluxes on faces
• Electric fields on edges
• Guarantees ??B 0

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Ionosphere Model

• 2D Electrostatic Model
• ??(?P?H)??J
• ?0 at low latitude boundary of ionosphere
• Conductivity Models
• Solar EUV ionization
• Creates day/night and winter/summer asymmetries
• Auroral Precipitation
• Empirical determination of energetic electron
  precipitation

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Calculation of Particle Fluxes

• Empirical relationships are used to convert MHD
  parameters into an average energy and flux of the
  precipitating electrons
• Initial flux and energy (Fedder et al., 1995)
• Parallel Potential drops (Knight 1972, Chiu
  1981)
• Effects of geomagnetic field (Orens and Fedder
  1978)

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Determining Energy Flux

• According to Lummerzheim 1997, it is possible
  using the UVI instrument on POLAR to determine
  both the characteristic energy and energy flux
• Energy flux is proportional to emission rates in
  the LBH bands
• Characteristic energy determines altitude of
  emission so it is determined by monitoring
  brightness of features that decay away and those
  that persist

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Estimating Optimal Parameters

• LFM electron flux/energy estimates are projected
  onto irregular grid corresponding to Polar UVI
  observations
• 1-D state vector constructed from all available
  observations time is simply treated as a third
  coordinate, in addition to MLT and ALAT
• Levenberg-Marquardt (nonlinear least-squares)
  algorithm adjusts model parameters a, b, and R to
  minimize the sum of the squared prediction
  errors

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Can advanced statistics help us?

• Are there better parameter estimation algorithms
  then Levenberg-Marquart?
• What can statistical comparisons with
  observations tell us about the bias present in
  our models?
• What are the best measures to monitor model
  improvement over time?

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Some Research Problems

• 1. How can uncertain model parameters be
  optimized to provide the best agreement, on the
  average, with observations?
• 2. How can model variability about the average,
  including information about scale sizes of this
  variability, best be compared with variability in
  observations to determine agreement or
  disagreement?
• 3. How can we improve the interpolation/extrapolat
  ion of observations of model input parameters in
  space and time to get complete specification of
  the boundary conditions?
• 4. In developing parameterizations of sub-grid
  phenomena, such as the transport of momentum and
  the creation of turbulence by breaking gravity
  waves, what is a good measure of intermittency,
  and how can its effects be parameterized?
• 5. How can relatively rare and sparse
  observations of extreme events like large
  magnetic storms be used to characterize
  upper-atmospheric behavior and test simulations
  for such events?
• 6. What can statistical comparisons tell us about
  underlying biases in our models?
• 7. What are the best measures to monitor model
  improvement over time?

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MI Coupling Eqs

• As described in Kelley 1989 The fundamental
  equation for MI coupling is obtain by breaking
  the ionospheric current into parallel and
  perpendicular components and requiring continuity
  
• Assuming no current flows out the bottom of the
  ionosphere we get
• Further assuming the electric field is uniform
  with height we get
• And finally using and electrostatic approximation
  in the MI coupling region we obtain

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Conductances from Particle Flux

• Spiro et al. 1982 used Atmospheric Explorer
  observations to determine a set of empirical
  relationships between the average electron energy
  and the electron energy flux
• Robinson et al. 1987 revised the relationships
  using Hilat data and careful consideration of
  Maxwellian used to determine the average energy