RUSSIAN ACADEMY OF SCIENCES KELDYSH INSTITUTE FOR APPLIED MATHEMATICS

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RUSSIAN ACADEMY OF SCIENCES KELDYSH INSTITUTE
FOR APPLIED MATHEMATICS

• NUMERICAL SIMULATION OF THE MAGNETIC FIELD
• CORONAL LOOP EVOLUTION
• Tatiana Yelenina
• 2006

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• Purpose
• Large time scales ideal MHD numerical simulation
  of the magnetic field loop evolution in the
  ''star disk'' system

• Topics under consideration
• Construction and verification of the high
  resolution scheme for the ideal MHD equations
• Investigation of the main features of the
  star-disk system evolution scenario

'Star disk'' system
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Scheme for 2D MHD equations

• Finite volume method
• Godunov type scheme, approximate Riemann solver
• To guarantee divergence-free magnetic field
  through the vector potential
• Explicit approximations of the forcing term

Tóth G.
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''Star corona disk'' system
Star mass ?, angular velocity ?, magnetic
moment ?
Corona perfectly conducting plasma
Disk Keplerian infinitely thin (h lt cs/?kltlt r)
conducting plane consisted of the dense cold
matter (?s ltlt Vk) Disk surface electrical
conductivity ? c2 / (2? ?t cs), Disk
surface magnetic diffusivity ? c2 / (2? ?)
?tcs ?t(cs / Vk)Vk
Shakura-Sunyaev ? - disk model
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Governing equations
Continuity equation
Momentum equation
Induction equation
Energy equation
div B 0
Tik ?vivk pdik BiBk /(4p) B2dik /(8p), g
-?Fg, Fg - GM / R, S p/ ??
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Boundary conditions

• In disk ? ? /2 ( v-Vk ) B? ( w ?
  ) Bf 0
• uB? ( w ? ) Br 0
• w ac vc2az2 / (c2 a2) 0
• Sd const
• At inner boundary RRin u V ef , B
  RRin 0, V O Rin sin ?
• At outer boundary RRout incoming wave
  strengths vanish
• On rotation axis flow symmetry conditions

Initial conditions

• Stellar magnetic field (dipole-like topology,
  with magnetic moment ? ) penetrates the corona
  and the disk.
• Corona and disk are in mechanical equilibrium
  with the force-free
• dipole magnetic field.

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Numerical simulation of the magnetic field
coronal loop evolution

• Purpose of the numerical experiments
  investigation of
• type of the system evolution steady state or
  periodic regime and its characteristics
• intensity of angular momentum transport from
  the disk
• disk surface magnetic diffusivity influence on
  the scenario of star-disk system evolution

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Initial distribution of the entropy S(r,z) and
angular velocity ?(r,z)
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Flow distribution at ? 0, t 50
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Flow distribution at ? 0.001, t 50
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Flow distribution at ? 0.005, t 50
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Magnetic field topology dependence on the disk
surface magnetic diffusivity ?
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Time dependence of the angular momentum fluxes on
the disk surface magnetic diffusivity ?
? 0
? 0.001
? 0.005
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Electric current density and magnetic field lines
in corona
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Conclusions

• Magnetic field coronal loop evolution consists in
  periodic reconnection of the magnetic field lines
  and plasmoid ejections which yields to
  oscillations of the flux angular momentum taken
  away from the disk.
• Intensity and periodicity of the process strongly
  depend on disk electrical conductivity.