May 2005 Campaign Event: Introducing Turbulence

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May 2005 Campaign Event Introducing Turbulence
Rona Oran Igor V. Sokolov Richard Frazin Ward
Manchester Tamas I. Gombosi
CSEM, University of Michigan
Ofer Cohen HSCA
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Summary of Results from SHINE 2008

• Out of equilibrium flux rope model superposed on
  a steady state MHD corona.
• The dynamical solution presented had good
  agreement with the shock arrival time to earth.

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2008 Results Revisited

• However
• Steady state solution highly criticized for use
  of variable polytropic index.
• The polytropic model, although it achieved good
  agreement with ambient solar wind observed at
  1AU, is not self - consistent and distort the
  physics, especially at shocks.
• Work done on thermodynamic MHD models, most
  prominently by the SAIC group (see Lionello,
  Linker and Mikic, 2009) inspire further attempts
  at a self - consistent model which will improve
  the physical basis for our solution.

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Introducing Alfven Waves Turbulence

• Turbulent MHD waves have been suggested in the
  past as a possible mechanism both to heat the
  corona and to accelerate the solar wind.
• Hinode observations suggest energy input is
  sufficient to drive the solar wind acceleration
  and heating (e.g. Pontiue et. al. 2007) .
• Heating Alfven wave dissipation at cyclotron
  frequency
• ( likely intensified by the energy cascade
  process).
• Wind acceleration work done by wave pressure
  gradient force.

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MHD - Wave Turbulence Model

• We employ a wave kinetic approach for describing
  the transport of MHD waves in a background MHD
  plasma.
• The wave transport equation for narrow band wave
  trains is given by
• here I is the wave energy spatial and spectral
  density and ? is a specific wave mode.
• Advantage describes the spectral evolution.

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Modified MHD Equations

• Background momentum equation
• Background energy equation
• Where P?? is the wave stress spectral density.

wave stress gradient force
Work done by wave stress
wave energy dissipation
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WT Equation - Low Frequency Alfven Waves

• In this limit the wave stress spectral density
  becomes
• Thus the pressure is scalar and the total wave
  pressure is
• The WT equation takes the form
• 2-way coupling of the WT equation to the MHD
  equations
• Wave pressure acts on the background flow, while
  background solution determines wave propagation
  and spectral evolution.

Advection in space
Advection in frequency
Dissipation
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Computational Model

• The WT equation is fully coupled to the BATSRUS
  code in the SWMF.
• The spatial grid is a 3D block adaptive
  Cartesian grid.
• In each spatial cell we construct a uniform
  frequency grid whose range / resolution is
  defined by the user.
• Both parallel and anti-parallel propagating
  waves are considered (currently share the same
  grid).
• Solution of WT equation is performed by Strang
  splitting of the spatial and frequency operators.
• The solution in 2nd order accurate in space,
  time and frequency.

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Model Inputs

• Magnetogram driven potential field
  extrapolation.
• Spectrum Can be defined by the user. This
  allows the testing of various theories by
  comparing results to observations.
• In the current work, we assume a Kolmogorov
  spectrum as the initial condition, i.e.
• I ? k-5/3
• (in accordance with observations of
  mean-free-path of protons in the heliosphere).
  The initial distribution in space can be rather
  arbitrary, since it quickly advects according to
  the MHD state.
• Level of imbalance
• Itot I I-
• I ?Itot I- (1 - ?)Itot 0lt? lt1

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Inner Boundary Conditions

• Radial magnetic field - high resolution MDI
  magnetogram data provided by Y. Liu, Stanford.
• Specify Alfven waves Poynting vector at 1Rs
• Solar wind expansion factors
• WSA model terminal velocity at 1AU
• Impose conservation of energy along flux tubes
  (Suzuki, 2006)
• In-going waves which reach the inner boundary are
  absorbed.

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Simulation Outline

• Magnetogram - driven steady-state solution of
  the solar corona (up to 24 Rs). Free parameters
  of the model and the uncertainties are
• Mass density at solar surface.
• Magnetogram scaling factor.
• Spectrum
• fmin 1x10-4Hz
• fmax 100 Hz
• (in accordance with Pontieu et. al. 2007)
• Roe solver scheme, adaptive mesh refinement.
• Initial grid resolution 0.001 Rs near the Sun
• Currently the advection in frequency space is
  not presented.

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Results - CR2029
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Results - CR2029
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Conclusions

• First calculations in the SWMF of a steady state
  solar corona/ solar wind which is entirely driven
  by Alfven wave pressure and the boundary
  conditions fully driven by the WSA model.
• Fast solar wind speeds are too high - conversion
  rate of wave energy into heating is too low. This
  might be solved when we take the wave dissipation
  into account.
• Compared to the previous polytropic model, we can
  now describe the corona self - consistently
  without distorting the physics, especially shock
  wave compression ratio which depend on ?.

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Future Work

• Repeat our dynamical simulation of the CME with
  the emphasis on more realistic shock wave
• Full implementation of spectral evolution
• Extend the model to the Inner Heliosphere (IH)
  model to enable comparison of results to
  observations at 1AU.