Pederson Resistivity in the Chromosphere

Transcription

1 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

2 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.

3 Single fluid equations Single fluid + v.r v = rp + j B r.( n (1 n )ww) Ohm s Law E + v B = rp e ne + j B ne + ej n 2 e 2 en w ne n w B Ion-neutral slip w = n n j B n rp + 1 rp n n n

4 Single fluid equations Single fluid + v.r v = rp + j B r.( n (1 n )ww) Ohm s Law E + v B = rp e ne + j B ne + ej n 2 e 2 en w ne n w B Ion-neutral slip w = n n j B n rp + 1 rp n n n

5 Ohm s Law - fully ionised plasma For a fully ionised plasma a good approximation is E + v ^ B = k j k +? j? + 1 n e e j ^ B 1 n e e rp e Dropping the battery term Ohm s Law often written E + v ^ B = k j k +? j? + H j ^ ˆb Here H = B n e e is not a resistivity!? = m e n e e 2 e and k =0.51?

6 Ohm s Law - partially ionised plasma For a partially ionised plasma E + v ^ B = k j k + P j? + H j ^ ˆb Now the perpendicular resistivity is the Pederson resistivity P =? + 2 nb 2 (1 n ) 1 in n = n / - neutral fraction in - ion-neutral collision time Usually take? = k = for simplicity Cowling (1957), Braginski (1965)

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

8 Ohm s Law - Goodman E + v ^ B = j k + P j? + H j ^ ˆb Where P = (1 + ) H = M e = 2 nm e M i M e =! e ei + en = eb/m e ei + en M i =! i in = eb/m i in Goodman (2010) and others

9 Ionospheric example y j = k E k + P E CM,? + H E CM ^ ˆb N z B j y = P E y H E z j z = P E z H E y Steady state current system must have j z =0 j y = P 1+ 2 H 2P E y P 1+ 2 H2P sometimes called the Cowling conductivity

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

11 Resistivities - FALC Parallel, Pederson, Hall

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

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

14 Flux Sheet emergence 2D simulations of horizontal flux sheet in model atmosphere

15 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

16 Pederson resistivity in simulations Modified Saha Bifrost fit (Leenharts) - Significant differences in Pederson resistivity between models

17 Late-time Uniform Resistivity Density and B field

18 Pederson with Saha

19 Pederson with Bifrost fit

20 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 off.

21 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 = 10 4 m. Solid line is based on VAL-C. The dashed line on the updated C7 model. Arber, Botha & Brady (ApJ 2009)

22 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...

23 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