Jens Kleimann and Gunnar Hornig- Non-ideal MHD Properties of Magnetic Flux Tubes in the Solar Photosphere

8/3/2019 Jens Kleimann and Gunnar Hornig- Non-ideal MHD Properties of Magnetic Flux Tubes in the Solar Photosphere

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Introduction

Question and Strategy

Set of MHD Equations

Resistive Inflow

Quantifying

Summary

Appendix

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Non-ideal MHD Properties of Magnetic

Flux Tubes in the Solar Photosphere

Jens Kleimann and Gunnar Hornig

VW Group Presentation, February 22, 2002

Topological Fluid Dynamics

Theoretische Physik IV

Ruhr-Universitat Bochum, Germany

Talk basis:

diploma thesis (April 1999 April 2000)

publication: Solar Physics, 200: 47-62, 2001
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1. Introduction

One major test case for MHD theory:

The Sun

Fundamental building blocks for surface structures:arching flux tubes with Binside Boutside.


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Solar flux tubes are usually modelled usingideal MHD ( 0). This results in:

Frozenness of field lines (plasma flowcannot cross tube surface)

Iso-rotation: = 0 = t (v/r) B = 0

(i.e., all field lines rotate rigidly.)

B field may get wound up infinitelyby convective footpoint motion:

=

Settings of this type are particularly relevant in the framework oftopological dissipation (Parker 1972).


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but:

Photospheric temperature is too low for sufficient ionisation! A pronounced non-ideal layer is present!

For = 0, we have: For = 0, we expect:

Tube interior is isolated Infinite twist/no static solution

Iso-rotation

Mass exchange through surface Finite twist at steady state

???


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2. Question and Strategy

Question:

Which changes of the tube properties are induced by thenon-ideal layer as compared to the ideal case?

Method:

Use given temperature variation with

height to derive resistivity profile

Compute a flux tube model using re-sistive MHD, focusing on one footpointonly

Impose stationary footpoint vortexwhile summit is held fixed

B field gets wound up, but slippagewill keep the resulting twist finite.

...and of course: Try to keep it SIMPLE!


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3. Set of MHD Equations

t v + (v ) v = P +j B + gt +

( v) = 0

j = E + v B E = t B B = j B = 0

Simplifications made:

stationary solutions t = 0 E =:

v vA := B/ neglect inertia term against induction. restriction to one single footpoint implies axial symmetry = 0 in

cylindrical coordinates [r,,z].


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4. Resistive Inflow

Let

Bp := poloidal Part ofB andv := flow across tube surface.

Then Ohms law

+ v B =

B

yields:

v =(x)

BP ( BP)BP2

Flow properties:

proportional to (cf. ideal case) independent of Bs direction and strength (B B leaves v unchanged) v e < 0 ( inflow!) in the generic case where (Bp/Bp) is small. take R as tubes radius v 1/R Total mass inflow indep. of R !


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5. Quantifying

From now on: assume cylindrical tube geometry (i.e., no change of cross sectionwith height.) Justified by

(h)nonideal layer tube height and observation of coronal loops.

Since = (z) and = (z), B can be shown to be force-free: j B = 0 .

Simple solution (with x := r/R)

Bff = B0

0,

x

1 + x2,

1

1 + x2

Resistivity (z) fitted according toVAL reference atmosphere; Density (z) decays exp(z/H).


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Total inflow M :=

( v) da 2 108 kg/s

Timescale :=Mtot

M

70 hR

100 km2

Toroidal slippage (contours of v): Scaling: v vs. vA, R {1, 2,...20km}

= 0 z(v/r) = 0 (cf. iso-rotation)

Footpoint vortex (cut along z = 0): v(x, 0)130 m/s

R

100 km

2x

(1 + x2).


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6. Summary

Ohms law enforces an inflow of fluid towards loci of higher field strength,which is independent of

the presence of additional forces (e.g. gravity) the tubes cross section and

the strength and direction of its B field.

Since v , this inflow occurs wherever the tube penetrates the coolphotospheric layer, in particular near the tubes footpoints.

A static flux tube of cylindrical shape has to be force-free if the ambientplasma temperature is horizontally stratified.The introduction of a resistive layer allows for stationary MHD solutions withfinite field twist and yields a marked deviation from the iso-rotational ruleknown from ideal MHD.

Although the flow magnitudes scaling law makes these effects possibly eithertoo small or too slow to be detected by (present) solar observations, they

may play an important role for small-scale structures such as the magneticcarpet.


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7. Appendix

References: Vernazza, Avrett & Loeser: 1981, ApJ Supp. 45, 635

Kubat & Karlicky: 1986, Ast. Inst. Czech. Bull. 37, 155

Counterexample for OutflowSetting

Bsp. :=cosh z

1 + (r cosh z)2

r sinh z, 0, cosh z

tube profile R(z) = 1/ cosh(z)

field drops as Bsp.z=0 1/(1+r2)

but: (j

B)sp.

=

P


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The Magnetic Carpet

Potential-field predictions for the structure of the magnetic field using data forthe magnetic data from SOHO/MDI, compared to the heating. Black/white

indicating polarity, underlying image: brightness observed at the same timeby SOHO/EIT at 195 Angstroms, with bright green corresponding to hotand dense regions and dark green corresponding to cooler ones.

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Jens Kleimann and Gunnar Hornig- Non-ideal MHD Properties of Magnetic Flux Tubes in the Solar Photosphere