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

centre for fusion, space and astrophysics 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 fl uid equations Single fl uid momentum equation ✓ ∂ ◆ ∂ t + v . r v = �r P + j ⇥ B � r . ( ρξ n (1 � ξ n ) ww ) ρ Ohm’s Law ne + j ⇥ B + α e j n 2 e 2 � α en w E + v ⇥ B = �r P e � ξ n w ⇥ B ne ne Ion-neutral slip w = ξ n j ⇥ B � ξ n r P + 1 r P n α n α n α n

Single fl uid equations Single fl uid momentum equation ✓ ∂ ◆ ∂ t + v . r v = �r P + j ⇥ B � r . ( ρξ n (1 � ξ n ) ww ) ρ Ohm’s Law ne + j ⇥ B + α e j n 2 e 2 � α en w E + v ⇥ B = �r P e � ξ n w ⇥ B ne ne Ion-neutral slip w = ξ n j ⇥ B � ξ n r P + 1 r P n α n α n α n

Ohm’s Law - fully ionised plasma For a fully ionised plasma a good approximation is 1 1 E + v ^ B = η k j k + η ? j ? + n e e j ^ B � n e e r P e Dropping the battery term Ohm’s Law often written E + v ∧ B = η k j k + η ? j ? + η H j ∧ ˆ b B Here η H = n e e is not a resistivity! m e η ? = n e e 2 τ e and η k = 0 . 51 η ?

Ohm’s Law - partially ionised plasma For a partially ionised plasma E + v ∧ B = η k j k + η P j ? + η H j ∧ ˆ b Now the perpendicular resistivity is the Pederson resistivity ξ 2 n B 2 1 η P = η ⊥ + ρτ in (1 − ξ n ) ξ n = ρ n / ρ - neutral fraction τ in - ion-neutral collision time Usually take η ? = η k = η for simplicity Cowling (1957), Braginski (1965)

Ohm’s Law - Ionospheric Ionospheric researchers usually use j = σ k E k + σ P E CM, ? + σ H E CM ∧ ˆ b Where E CM = E + v ∧ B Conductivities are related to resistivities above by 1 = σ k If no Hall term then Pederson η resistivity is inverse of Pederson η P conductivity. This is true for all = σ P η 2 P + η 2 following results. H η H = σ H − η 2 P + η 2 H Cowling (1957), Braginski (1965)

Ohm’s Law - Goodman E + v ∧ B = η j k + η P j ? + η H j ∧ ˆ b = (1 + Γ ) η η P = M e η η H Where ξ 2 n M e M i Γ = = eB/m e ω e M e = ν ei + ν en ν ei + ν en = eB/m i ω i M i = ν in ν in Goodman (2010) and others

Ionospheric example j = σ k E k + σ P E CM, ? + σ H E CM ∧ ˆ b y N z j y = σ P E y − σ H E z B j z = σ P E z − σ H E y Steady state current system must have j z = 0 ⇣ 1 + σ 2 ⌘ j y = σ P E y H σ 2 P ⇣ 1 + σ 2 ⌘ sometimes called the Cowling conductivity H σ P σ 2 P

Magnetisation - FALC Electron magnetisation, ion magnetisation, neutral fraction, Γ Fontela et al. (1993, 2002)

Resistivities - FALC Parallel, Pederson, Hall

Flux Emergence Field-lines and photospheric B When Pederson η P included • Field closer to force free Horizontal line shows location of transition region Leake et al. (2006), Arber et al. (2007)

Flux Emergence Including neutrals Ideal MHD η fixed by VAL-C When Pederson η P included - Chromosphere heated Leake et al. (2006), Arber et al. (2007)

Flux Sheet emergence 2D simulations of horizontal fl ux 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 Modified Saha Bifrost fit (Leenharts) - Significant differences in Pederson resistivity between models

Uniform Resistivity Late-time Density and B fi eld

Pederson with Saha

Pederson with Bifrost fit

Why so little di ff erence? 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 off.

Collapse Timescales Variation of decay time τ for non-force-free components of the magnetic fi eld as a function of height above the photosphere for B p = 0.12 T and L p = 10 4 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 strati fi cation correct Neutrals may be needed to get B fi eld 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 strati fi cation correct Neutrals may be needed to get B fi eld 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