A&A 371, 328-332 (2001)
DOI: 10.1051/0004-6361:20010315
I. McKaig
Department of Mathematics, Tidewater Community College, Virginia Beach, VA 23456, USA
Received 19 January 2001 / Accepted 28 February 2001
Abstract
The solar convection zone may be a mechanism for generating the magnetic fields in the corona
that create and thermally insulate quiescent prominences. This connection is examined here by numerically
solving a diffusion equation with convection (below the photosphere) matched to Laplace's equation
(modeling the current free corona above the photosphere). The types of fields formed resemble both
Kippenhahn-Schlüter and Kuperus-Raadu configurations with feet that drop into supergranule boundaries.
Key words: supergranulation - convection - prominences
The first mathematical models were developed by Menzel (1951) and Dungey (1953). These early models proved unrealistic. For example, Menzel's model predicted prominences that were as wide as they were tall. Better models were developed by Kippenhahn & Shlüter (KS) (1957) and Kuperus & Raduu (KR) (1974). These models have better withstood the test of time and are the basis for much work on quiescent prominences. For detailed accounts of the history and physics of prominences see the books "Solar Prominences'' by Tandberg-Hanssen (1974) and "Dynamics and Structure of Quiescent Solar Prominences'' edited by Priest (1989). Marvelous photographs of these magnificent structures can also be found on the internet - see for example http://mesola.obspm.fr (a site run by INSU/CNRS France) and http://sohowww.estec.esa.nl (the web site of the Solar and Heliospheric Observatory).
The field lines in the models by KS and KR are line tied to the photosphere (see McKaig 2001). In a previous paper (McKaig 2001) photospheric motions were simulated by a one-dimensional boundary condition and KS/KR type fields were obtained in the corona. In this paper we will numerically derive magnetic field lines in the corona from two-dimensional convection below the photosphere.
The basic equation of kinematical magnetoconvection, where a velocity field
is imposed on a
plasma and its effect on the magnetic field
computed, is
(1) |
In much of magnetoconvection the fluid fills the whole of the domain and the boundaries are assumed to be
perfectly conducting. This ties the magnetic field lines to the boundary and confines the field to the
region of fluid flow. Even though the equation is time dependent, an initially uniform field is expelled
by the convection to the boundaries and reaches a steady-state. In an attempt to have convection expel
magnetism from below the photosphere into the corona we will solve (1) with
(steady state has been achieved) on the interval -L<x<L and -d<y<d. The convection zone
will be y<0 where
will be prescribed, while y>0 will be a current free corona where
.
Figure 1: The stream function with magnetic field lines computed using the boundary conditions: g1(y)=g2(y)=0, h1(x)=0, h2(x)=100 | |
Open with DEXTER |
Figure 2: The stream function with magnetic field lines computed using the boundary conditions: g1(y)= | |
Open with DEXTER |
Figure 3: The stream function with magnetic field lines computed using the boundary conditions: g1(y)=g2(y)=0, h1(x)=100, h2(x)=0 | |
Open with DEXTER |
Figure 4: The stream function with magnetic field lines computed using the boundary conditions: g1(y)=g2(y)=0, h1(x)=0, h2(x)=100 | |
Open with DEXTER |
Figure 5: The stream function with magnetic field lines computed using the boundary conditions: g1(y)= | |
Open with DEXTER |
Figure 6: The stream function with magnetic field lines computed using the boundary conditions: g1(y)=g2(y)=0, h1(x)=100, h2(x)=0 | |
Open with DEXTER |
If a vector potential A(x,y) is introduced such that
,
then the contours given by
A=constant will give the
magnetic field lines, A being essentially a stream function for .
In the corona (y>0),
becomes Laplace's equation,
.
In the convection zone
(y<0), we cast (1) with
in 2-D which in terms of A gives a
diffusion equation with convection:
(2) |
(3) |
(4) |
All runs were made with and d=4. The first stream function used was . This models a deep convection zone with a downflow at the center. This stream function was used with different boundary conditions in Figs. 1, 2 and 3. The stream function has been overlayed with the field lines.
The next stream function, , has an upflow at the center and the same boundary conditions were applied in Figs. 4, 5 and 6. An exact solution to (1) with using this stream function was obtained by Parker (1963) for . Figure 6 resembles his solution.
Acknowledgements
This paper was written under the auspices of the Department of Mathematics and Statistics, Old Dominion University, Norfolk VA. I am grateful to Dr. John Adam, my Ph.D. advisor and the referee for their suggestions on improving this article.