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Effect of solar chromospheric neutrals on equilibrium field structures - T. Arber, G. Botha & C. Brady ( ApJ 2009). 太陽雑誌 会ー 22/01/10. From T. Matsumoto. Motivation. Coronal Field believed to be a force free field, or more precisely a nonlinear force free field (NLFFF)

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Effect of solar chromospheric neutrals on equilibrium field structures -T. Arber, G. Botha & C. Brady(ApJ 2009)



  • Coronal Field believed to be a force free field, or more precisely a nonlinear force free field (NLFFF)

  • Extrapolation from the boundary requires the boundary to have a NLFFF

  • But photospheric fields, where observations of magnetic field are most accurate, are not NLFFF

  • Extrapolations of photospheric fields give a good approximation of the coronal field

    • Somewhere in the upper photosphere / chromosphere the field becomes a NLFFF


  • What mechanism allows this to happen

    • Chromospheric neutrals may be important (as well as gravitaional stratification or plasma becoming low β)

    • This is a study of how Cowling resistivity affects chromospheric equilibrium fields (As Cowling resisitivity (Ambipolar diffusion) is known to produce Nonlinear force free fields - NLFFF)

  • α is a measure of the parallel current, they studied the evolution of α under Cowling resistivity for a 1 2/2 D current sheet where the amount of shear is varied


  • MHD equations (including Spitzer, Cowling and viscous terms)

  • Define height in atmosphere through density and temperature

  • These values also determine the value of the cowling resistivity (greatest in upper chromosphere)

  • b gives the amount of shear of the magnetic field. 0 is a Harris current sheet and 1 is aNLFFF (aka Yokoyama-Shibata current sheet)

  • Looking at an area of the atmosphere where Cowling resistivity dominates Spitzer resistivity (so Spitzer resistivity can be ignored)

From K.A.P. Singh

Harris current sheet j 0
Harris Current Sheet (J||=0)

Lorentz Force

Lorentz Force

  • Lorentz force is balanced by pressure gradient in a fully ionized plasma

  • If there is a neutral component in the plasma, this will flow along the lines of hydrodynamic force

  • The force on the ions (still frozen to the magnetic field) from the gas will decrease, meaning the ions move in the direction of the Lorentz force

  • The current sheet will collapse into a singularity.









Pressure Gradient

Current sheet with shear j 0
Current Sheet with Shear (J||≠0)

  • Cowling resistivity cannot work on the component of J that acts parallel to the magnetic field

  • Therefore only perpendicular current is dissipated

  • This leaves a current sheet that is force free, as the Lorentz force now balances inside the current sheet

  • The parallel current has increased.

Current sheet with shear j 01
Current Sheet with Shear (J||≠0)

  • The smaller the initial shear, the larger and thinner profile of α created

  • Implies that accuracy of observational estimate of α is heavily dependent on the initial field structure

Time dependence
Time dependence

  • Characteristic time scale for force free field to be created was found to be:

  • Takes about 10~20 minutes for a field above 800km to relax to a force free state.

Conclusion summary
Conclusion & Summary

  • Maximum value and decrease in FWHM of α more pronounced for small b (small amount of shearing of field)

  • Any shear in the initial field and Cowling resistivity is able to create a force free field

    • Estimated to take about 10~20 mins

  • This work studies a highly simplified setting and ignores the complex chromospheric dynamics and so only provides a handle on how Cowling resistivity would really affect flux emergence

Application to my work
Application to my work

Study of how the Kippenhahn-Schlueter prominence model evolves under Cowling resistivity


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