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

0

)(

Bj r

• 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

Motivation

• 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

perpr jBj 0

)(

Motivation

Model

From K.A.P. Singh

• 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 a NLFFF (aka Yokoyama-Shibata current sheet)

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

Harris Current Sheet (J||=0)• 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.

N

++

+

NN

N+

Pressure Gradient

Lorentz Force Lorentz Force

zB

P

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||≠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

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

20

20

)()(2hBhL

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

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

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

Bx/Bz=Black: 0.1Green: 0.2Blue: 0.3Magenta: 0.4Magenta-ish: 0.5Red: 0.7Purple: 1.0