Magnetic structure of the disk corona
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Magnetic structure of the disk corona. Slava Titov, Zoran Mikic, Alexei Pankin, Dalton Schnack SAIC , San Diego Jeremy Goodman, Dmitri Uzdensky Princeton University CMSO General Meeting , October 5-7, 200 5 Princeton. 2D case : field line connectivity and topology. BP separtrix

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Magnetic structure of the disk corona

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Magnetic structureof the disk corona

Slava Titov, Zoran Mikic,Alexei Pankin, Dalton Schnack

SAIC, San Diego

Jeremy Goodman,Dmitri Uzdensky

Princeton University

CMSO General Meeting, October 5-7, 2005

Princeton


2D case: field line connectivity and topology

BP separtrix

field line

NP separtrix

field line

normal

field line

disk

  • Flux tubes enclosing separatrices split at null pointsor "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!


Extra opportunity in 3D: squashing instead of splitting

  • 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 notapplicable;

    • other approach is required.

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


Squashing factor Q

  • 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.

  • (Titov, Hornig & Démoulin, 2002)

    • 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.


    Expansion-contraction factor K

    • 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.

    • 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.


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

    • Identify the regions subject to boundary effects.

    • Understand the effect of resistivity.

    • Identify the reconnecting magnetic flux tubes.


    Example (t=238)

    Numerical MHD

    log Q

    From the initial B(r)

    and vdsk(rdsk,t) only!

    From the

    computed B(r,t).

    1 2

    Exact ideal MHD

    -10 0 10


    Example (t=238)

    Numerical MHD

    K

    -1 0 1

    Exact ideal MHD

    -10 0 10


    Example (t=238)

    Numerical MHD

    log Q

    K

    1 2

    -1 0 1

    Exact ideal MHD

    -10 0 10


    Helical QSL (t=238)

    Magnetic field lines

    Launch footpoints


    Conclusions

    Evolving Q and K distributions make possible:

    • to identify the regions subject to boundary effects,

    • to understand the effect of resistivity,

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


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