STOCHASTIC COUPLING OF SOLAR PHOTOSPHERE AND CORONA ( Astrophysical J., 2012, 2013)
Jens Kleimann and Gunnar Hornig- Non-ideal MHD Properties of Magnetic Flux Tubes in the Solar...
Transcript of Jens Kleimann and Gunnar Hornig- Non-ideal MHD Properties of Magnetic Flux Tubes in the Solar...
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Introduction
Question and Strategy
Set of MHD Equations
Resistive Inflow
Quantifying
Summary
Appendix
Title Page
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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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Introduction
Question and Strategy
Set of MHD Equations
Resistive Inflow
Quantifying
Summary
Appendix
Title Page
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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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Introduction
Question and Strategy
Set of MHD Equations
Resistive Inflow
Quantifying
Summary
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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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Introduction
Question and Strategy
Set of MHD Equations
Resistive Inflow
Quantifying
Summary
Appendix
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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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Introduction
Question and Strategy
Set of MHD Equations
Resistive Inflow
Quantifying
Summary
Appendix
Title Page
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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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Introduction
Question and Strategy
Set of MHD Equations
Resistive Inflow
Quantifying
Summary
Appendix
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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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Introduction
Question and Strategy
Set of MHD Equations
Resistive Inflow
Quantifying
Summary
Appendix
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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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Introduction
Question and Strategy
Set of MHD Equations
Resistive Inflow
Quantifying
Summary
Appendix
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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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Introduction
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Set of MHD Equations
Resistive Inflow
Quantifying
Summary
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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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Introduction
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Set of MHD Equations
Resistive Inflow
Quantifying
Summary
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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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IntroductionQuestion and Strategy
Set of MHD Equations
Resistive Inflow
Quantifying
Summary
Appendix
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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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