Magnetic structureMagnetic structureof the disk coronaof the disk corona

Slava Titov, Zoran Mikic, Alexei Pankin, Dalton SchnackSAIC, San Diego

Jeremy Goodman, Dmitri UzdenskyPrinceton University

CMSO General Meeting, October 5-7, 2005Princeton

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2D case: field line connectivity and topology

normal field line

NP separtrix field line

BP separtrix field line

Flux tubes enclosing separatrices split at null points or "bald-patch" points. They are topological features, because splitting cannot be removed by a continous deformation of the configuration. Current sheets are formed at the separatrices due to footpoint displacements or instabilities.

All these 2D issues can be generalized to 3D!

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Differences compared to nulls and BPs: •squashing may be removed by a continuous deformation, •=> QSL is not topological but geometrical object, •metric is needed to describe QSL quantitatively, •=> topological arguments for the current sheet formation at QSLs are not applicable;

other approach is required.

Extra opportunity in 3D: squashing instead of splitting

Nevertheless, thin QSLs are as important as genuine separatrices for this process.

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Definition of Q in coordinates:                                                           

where a, b, c and d are the elements of the Jacobian matrix                                                                          

D and then Q can be determined by integrating field line equations.

  Geometrical definition:

Infinitezimal flux tube such that a cross-section at one foot is curcular,

then circle  ==>   ellipse:

      Q = aspect ratio of the ellipse;

Q is invariant to direction of mapping.

Squashing factor Q

(Titov, Hornig & Démoulin, 2002)

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  Geometrical definition:

Infinitezimal flux tube such that a cross-section at one foot is curcular,

then circle  ==>   ellipse:      K = lg(ellipse area / circle area);

K is invariant (up to the sign) to the direction of mapping.

Expansion-contraction factor K

Definition of K in coordinates:                                                           

where a, b, c and d are the elements of the Jacobian matrix                                                                          

D and then Q can be determined by integrating field line equations.

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What can we obtain with the help of Q and K?

1. Identify the regions subject to boundary effects.

2. Understand the effect of resistivity.

3. Identify the reconnecting magnetic flux tubes.

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Example (t=238)Exact ideal MHD Numerical MHD

log Q

1 2

-10 0 10

From the initial B(r)and vdsk(rdsk,t) only!

From thecomputed B(r,t).

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Example (t=238)Exact ideal MHD Numerical MHD

K

-1 0 1

-10 0 10

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Example (t=238)Exact ideal MHD Numerical MHD

log Q

1 2

-10 0 10

K

-1 0 1

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Helical QSL (t=238)

Magnetic field lines Launch footpoints

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Conclusions

Evolving Q and K distributions make possible:

1. to identify the regions subject to boundary effects,

2. to understand the effect of resistivity,

3. to identify the reconnecting magnetic flux tubes (helical QSL).