Pederson Resistivity in the Chromosphere Tony Arber, John Adams - - PowerPoint PPT Presentation

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Pederson Resistivity in the Chromosphere

Tony Arber, John Adams & Gert Botha

University of Warwick, UK

James Leake

George Mason University, USA

Overview

• Summarise alternate forms of Ohm’s law.
• Where are neutrals and Hall term important.
• Example of flux emergence where Pederson important.
• Example of emergence when Pederson un-important.
• Collapse of equilibrium field structures.

Single fluid equations

ρ ✓ ∂ ∂t + v.r ◆ v = rP + j ⇥ B r.(ρξn(1 ξn)ww) E + v ⇥ B = rPe ne + j ⇥ B ne + αej n2e2 αenw ne ξnw ⇥ B w = ξn αn j ⇥ B ξn αn rP + 1 αn rPn Single fluid momentum equation Ohm’s Law Ion-neutral slip

Single fluid equations

ρ ✓ ∂ ∂t + v.r ◆ v = rP + j ⇥ B r.(ρξn(1 ξn)ww) E + v ⇥ B = rPe ne + j ⇥ B ne + αej n2e2 αenw ne ξnw ⇥ B w = ξn αn j ⇥ B ξn αn rP + 1 αn rPn Single fluid momentum equation Ohm’s Law Ion-neutral slip

Ohm’s Law - fully ionised plasma

For a fully ionised plasma a good approximation is η? =

me nee2τe and ηk = 0.51η?

Dropping the battery term Ohm’s Law often written Here ηH =

B nee is not a resistivity!

E + v ^ B = ηkjk + η?j? + 1 neej ^ B 1 neerPe E + v ∧ B = ηkjk + η?j? + ηHj ∧ ˆ b

Ohm’s Law - partially ionised plasma

For a partially ionised plasma ηP = η⊥ + ξ2

nB2

(1 − ξn) 1 ρτin Now the perpendicular resistivity is the Pederson resistivity ξn = ρn/ρ - neutral fraction τin - ion-neutral collision time

Cowling (1957), Braginski (1965)

Usually take η? = ηk = η for simplicity E + v ∧ B = ηkjk + ηP j? + ηHj ∧ ˆ b

Ohm’s Law - Ionospheric

Cowling (1957), Braginski (1965)

σk = 1 η σP = ηP η2

P + η2 H

σH = − ηH η2

P + η2 H

Where Conductivities are related to resistivities above by

If no Hall term then Pederson resistivity is inverse of Pederson

• conductivity. This is true for all

following results.

Ionospheric researchers usually use

j = σkEk + σP ECM,? + σHECM ∧ ˆ b ECM = E + v ∧ B

Ohm’s Law - Goodman

E + v ∧ B = ηjk + ηP j? + ηHj ∧ ˆ b ηP = (1 + Γ)η ηH = Meη Γ = ξ2

nMeMi

Me = ωe νei + νen = eB/me νei + νen Mi = ωi νin = eB/mi νin Where

Goodman (2010) and others

Ionospheric example

N z y B

j = σkEk + σP ECM,? + σHECM ∧ ˆ b jy = σP Ey − σHEz jz = σP Ez − σHEy Steady state current system must have jz = 0 jy = σP ⇣ 1 + σ2

H

σ2

P

⌘ Ey σP ⇣ 1 + σ2

H

σ2

P

⌘ sometimes called the Cowling conductivity

Magnetisation - FALC

Fontela et al. (1993, 2002)

Electron magnetisation, ion magnetisation, neutral fraction, Γ

Resistivities - FALC

Parallel, Pederson, Hall

Flux Emergence

Leake et al. (2006), Arber et al. (2007)

Field-lines and photospheric B

When Pederson ηP included

• Field closer to force free

Horizontal line shows location of transition region

Flux Emergence

Leake et al. (2006), Arber et al. (2007)

When Pederson ηP included - Chromosphere heated

Including neutrals Ideal MHD η fixed by VAL-C

Flux Sheet emergence

2D simulations of horizontal flux sheet in model atmosphere

Modelling neutral fraction

Model from Rad-hydro (Leenharts et al., 2007)

• Oslo group used BiFrost for time dependent ionisation
• Found correlation between mass density and ionisation fraction
• J. Leenarts (2012) private communication

Pederson resistivity in simulations

• Significant differences in Pederson resistivity between models

Modified Saha Bifrost fit (Leenharts)

Late-time Density and B field

Uniform Resistivity

Pederson with Saha

Pederson with Bifrost fit

Why so little difference?

Current sheets between loops are most important dynamically. Current sheets heated by Pederson and Saha gives higher ionisation - urns Pederson off. Plasma explelled from heated current sheet regions and Bifrost fit gives higher ionisation - Pederson turns

• ff.

Collapse Timescales

Variation of decay time τ for non-force-free components of the magnetic field as a function of height above the photosphere for Bp = 0.12 T and Lp = 104 m. Solid line is based on VAL-C. The dashed line on the updated C7 model.

Arber, Botha & Brady (ApJ 2009)

Summary

Neutrals needed to get atmospheric stratification correct Neutrals may be needed to get B field structure and heating

Problems

• no chromospheric or coronal heating
• no dynamic ionisation/recombination
• no radiative heating or losses
• no conduction, shock heating, wave heating ....

Summary

Neutrals needed to get atmospheric stratification correct Neutrals may be needed to get B field structure and heating

Problems

• no chromospheric or coronal heating
• no dynamic ionisation/recombination
• no radiative heating or losses
• no conduction, shock heating, wave heating ....

The End Thank you