3D solar magnetohydrostatic structures

Pergamon

Phys. Chem. Earth, Vol. 22, No. 5, pp. 405-409, 1997 © 1997 Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 0079-1946/97 $17.00 + 0.00 PII: S0079-1946(97)00167-5

3D Solar Magnetohydrostatic structures T. N e u k i r c h School of Mathematical and C o m p u t a t i o n a l Sciences, University of St. Andrews, St. A n d r e w s KY16 9SS, U.K.

Received 10 May 1996; accepted 7 March 1997

Abstract. A discussion of an analytical method used to model coronal structures self-consistentlyin three dimensions is given. This methods does not only allow the calculation of the magnetic field from given photospheric flux distributions, but also gives the plasma pressure, density and temperature. This allows a detailed comparison of the models with different kinds of observations. The test of theoretical models against observations is a crucial step in establishing a quantitative understanding of the corona. For a genuine three-dimensional modeling, input from stereoscopic observations will be very important. © 1997 Published by Elsevier Science Ltd

1

Introduction

The coronal magnetic field is the major structuring field in the coronal plasma. Since the plasma may carry currents which again serve as a source for the magnetic field, a selfconsistent treatment is necessary to model coronal structures. Simple three-dimensionalmodels are based on potential and force-free magnetic fields; however, these models do not contain any information about the plasma parameters like plasma density, temperature or pressure. This information is necessary to compare the models with observations. This is different for the three-dimensional analytical equilibria calculated by Low (1985, 1991, 1992, 1993a,b), Bogdan and Low (1986) and Neukirch (1995). These equilibria extend the class of linear force free fields to non-force-free cases and provide at the same time information on the magnetic field and the plasma parameters. Solutions calculated by Bogdan and Low (1986) have been used by Bagenal and Gibson (1991), Zhao and Hoeksema (1993, 1994), Gibson and Bagenal (1995) and Zhao et al. (1997) to model the large-scale structure of the corona. This work provides an extension to the well-known 'hairy ball'

Correspondence to: T. Neukirch 405

models based on potential field extrapolation of photospheric magnetic field measurements. In the present contribution we investigate the possibility of modelling local structures like arcades and loops with the same method. Whereas in the cases mentioned above the natural coordinate system is spherical, in our case a Cartesian coordinate system is appropriate. 2

Method

We briefly outline the method as presented in Neukirch (1995). Starting from the magnetohydrostatic (MHS) equations in the form j x B - V p - a v e = 0,

(l)

v × fi

(2)

=

v . B = o.

(3)

Here, p denotes the pressure, p the plasma density, and ~b the gravitational potential, which we assume to be Gz. We now assume that the current density is written as (Low, 1991) = a/~ + V F x V~p.

(4)

In Eq=(4), a is constant and F is a linear function of V¢ • B = GB, with an additional arbitrary dependence on F(V¢. B,¢) = ~(¢)V¢. B

(5)

Following the treatment in Neukirch (1995) we may reduce the problem to solving one partial differential equation for Bz :

AB~ + ~(z)L2Bz + c~2Bz = 0.

(6)

In Eq. (6) ~(z) = G2t~(~) and L 2 = -(02/Ox 2 + 02/0y2). It is now possible to proceed in two ways: a) Bz is expanded as a series of eigenfunctionof L 2 or b) a Green's function for Eq. (6) is used. In the present paper we will use the possiblity

406

T. N e u k i r c h

m

Q~ - - 4

Fig. 1. Three-dimensional plot of magne~c field lines for three different values of the parameter ~o for the same boundary conditions at the photosphere (z = 0). The top row shows the force-free field (~0 = O) under two different viewing angles. The middie row shows the field for ~o = - 0 . 0 0 4 and the bottom row shows the field for ~o = -0.008.

3D Solar Magnetohydrostatic Structures

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Fig. 2. The plasma density in the plane x = L/2. The plot on top shows the plane parallel background model (~o = 0). Even for the relatively small changes of ~o considerable deviations from the plane parallel case may he seen in the middle and bottom plots.

0.0

Fig. 3. The plasma density in the plane y = 0. The plot on top shows the plane parallel background model (~0 = 0). Again the deviations from the plane parallel case may bee seen in the middle and bottom plots.

408

T. Neukirch

a). The remaining equation for the z-dependent coefficients u:m (z) of the expansion is a Schrtdinger-type equation with ~(z) as the "potential":

d2Ulm dz---T - + [)~tm (~(z) - 1) + a 2] Utm = 0.

1.5

(7)

Here, At,,~is an eigenvalue of L 2 for a specific eigenfunction Etm(x, y). Once Bz is known, all other quantities can be derived from that. The magnetic field is given by

=

--qt~(z)VEt,~ }2 t,m ,~tm

2.0

× ~+

1.0

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0.0

1 dqt'~VElm + qt~(z)Et,~(x,y)¢~ )~tm dz

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(8) 2.0

with qtm defined as

qt,~(z) = .atmu:,,~ --0) O) tz) . . . . -t- ,at,nut, (2) ( ~~ )tz) ,,

(9)

1.5

where A t : are constants and ul,~ are the two linearly independent solutions of Eq. (7). The pressure and the plasma density are given by

1.0

p(x,y,z)

=

(10)

0.5

(ll)

p(x, y, z)

--

0.0

(1 2)

(1 2)

p o ( z ) - 2~n~(z)B2(x,y,z)

I (dpo d~c B~ G

~'z

dz 2#0

+~/~.VB~ /~o ]

The plasma temperature can be defined as well if a suitable equation of state is prescribed.

Fig. 4. The plasma temperature in the plane x = ~0 # 0 are shown.

L/2. Only the cases with

3 A Simple Example The above solutions still contain two arbitrary functions, one determining the non-force-free part of the current density (~(z)), whereas the other one (po(z)) is related to a "background" plasma. Low (1991, 1992) has calculated solutions for ~ = 1 + b e x p ( - z / L ) . The solutions of Eq. (7) for this ~ are Bessel functions of real order with an exponential function as argument. A simpler class of solutions is obtained if we take ~(z) = ~0 = constant. For a = 0 solutions of this type have been discussed by Low (1985). These solutions can also easily be compared with the linear force-free fields for illustrative purposes because they are formally similar to those solutions, however, with different coordinate dependences for the same value of a. For simplicity, we assume in the following that Bz is given by B= =/30 sin(kx) [exp(-loz) + bx CoS(ky)e x p ( - / l z ) ] (12) with lo2 l~

= =

k~(1-~o)-Ce2 (k~ + k~)(1 - ~o) - a 2

kx

=

7r/L;

k u = 2rr/L

( L " 30000km)

(13) (14)

A similar solution for the force-free case has been used by Dtmoulin et al. (1989) as a simple model for quiescent prominences. Here, we want to investigate how the additional current changes the configuration if its strength is increased. As background model, we assume an isothermal atmosphere with a temperature of 2 • 106 K and a baseline particle density of 5 • 1014 m -3. In Figure 1, we show a set of magnetic field lines for three slightly different values of ~0 starting at ~0 = 0 and two viewing angles. However, since the value of ~0 decreases only relatively the field line configuration stays more or less the same. In Figure 2 we show the density in the plane x = L/2. For ~0 = 0 the plane parallel background density is recovered. Though the change of ~0 is relatively small, the plasma density shows considerable deviations from the plane parallel case for ~0 ~ 0. A second cut through the three dimensional density structure is shown in Figure 3 (y = 0). Again, the deviations from the plane parallel force free case can be clearly seen. The variation in temperature is shown for the same planes as for the density in Figures 4 and 5. Here only the cases for ~0 ~ 0 are shown, because the temperature is constant in the case ~0 = 0. Due to the exponential decay of the model magnetic field chosen for this purely illustrative example, the deviations from the force free case are confined

3D Solar Magnetohydrostatic Structures

409

e.g. the plasma density. Another advantage of this method is that it can be used to calculate solutions having a nonlinear dependence of the current density on the magnetic field (Neukirch, 1997). The price one has to pay for being able to calculate analytical solutions is that it is not possible to impose a plasma equation of state a priori. Therefore, it has to be checked a posteriori whether a solution class is physically sensible or not. However, in the previous applications of the method (Bagenal and Gibson, 1991; Zhao and Hoeksema, 1993, 1994; Gibson and Bagenal, 1995; Zhao et al., 1997) this does not seem to have played an important role. -0.4

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Acknowledgement. The author thanks PPARC for financialsuppotx. 2.0

References

1.5

Bagenal, F. and Gibson, S., Modeling the large-scale structure of the solar corona, J. Geophys. Res., 96, 17663-17674, 1991.

1.0

Bogdan, T. J. and Low, B. C., Three-dimensional structures of magnetostatic atmospheres. II. Modeling the large-scale corona, Astrophys. J., 306, 271-283, 1986. D~moulin, P., Priest, E. R. and Anzer, U., A three-dimensionalmodel for solar prominences,Astron. Astrophys., 221,326-337, 1989.

0.5

Gibson, S. and Bagenal, E,, J. Geophys. Res., 100, 19865-19880, 1995. 0.0 -0.4

-0.2

0.0

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Low, B. C., Three-dimensional structures of magnetostatic atmospheres. 1. Theory,Astrophys. J., 293, 31--43, 1985. Low, B. C., Three-dimensional structures of magnetostatic atmospheres. III. A general formulation,Astrophys. J., 370, 427-434, 1991.

Fig. 5. The plasma temperature in the plane ~/= O.

Low, B. C., Throe-dimensional structures of magnetostatic atmospheres. IV. Magnetic structures over a solar active region, Astrophys. J., 399, 300-312, 1992.

to a relatively low height. However, this would be changed if different boundary conditions were taken into account as for example a set of discrete dipoles below the photosphere. These cases require the application of the Greens function method mentioned above.

Low, B. C., Three-dimensinnal structures of magnetostatie atmospheres. V. Coupled electric currents systems. Astrophys. J., 408, 689692, 1993a. Low, B. C., Three-dimensional structures of magnetostatic atmospheres. VI. Examples of coupled electric current systems,Astrophys. J., 408, 693-706, 1993b. Neukireh, T., On self-consistentthree-dimensionalanalytic solutions of the magnetohydrostaticequations,Astrott Astrophys., 301,628--639, 1995.

4

Conclusions

It has been shown, how analytical three-dimensional selfconsistent M H D equilibria can be calculated in Cartesian coordinates. We have discussed a special illustrative example which shows how the inclusion of gravitation and pressure leads to three-dimensional deviations from the plane parallel background atmosphere. Three-dimensional solutions like these should be very useful to model three-dimensional structures in the solar corona, especially if data input from different viewing angles like from a stereoscopic mission is available. It is planned to extend this method to be able to include more realistic boundary conditions like flux concentrations on the photosphere. The advantage of this method is the prompt avallibility of a magnetic field model together with the plasma parameters. This knowledge could be used to compare models derived from one set of observations e.g. the magnetic field with another set of observations giving

Neuldrch, T., Nonlinear self-consistentthree-dimensionsl arcade-like solutions of the magnetohydrostatieequations,Astron. Astrophys., in press. Zhao, X. and Hoeksema, J. T., Uniquedeterminationof model coronal magnetic fields using photospheric observations, Solar Phys., 143, 41-48, 1993. Zhac, X. and Hoeksema, J. T., A coronal magnetic field model with horizontal volume and sheet currents, Solar Phys., 151, 91-105, 1994. Zhao, X., Hoeksema, J. T. and Scherver, P. H., Improved inputs to coronal magnetic field models, Solar Phys., submitted.