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Astron. Astrophys. 330, 726-738 (1998)

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3. Equilibrium configuration

We use the coronal arcade model described in Oliver et al. (1993). In this model it is assumed that the coronal arcade is in static equilibrium ([FORMULA]), so it follows from Eq. (2) that the magnetic Lorentz force must vanish everywhere, i. e.

[EQUATION]

A particular solution to this equation is a potential magnetic field with zero current,

[EQUATION]

In a y -invariant system, the equilibrium magnetic field may be written as follows

[EQUATION]

where [FORMULA] is the unit vector along the y -direction and [FORMULA] is the vector magnetic potential whose modulus satisfies Laplace equation, [FORMULA]. The solution for A can be obtained by separation of variables under the condition that it does not diverge as [FORMULA]. Therefore, the magnetic field components are given by

[EQUATION]

[EQUATION]

with [FORMULA] the magnetic field strength at the coronal base ([FORMULA]) and [FORMULA] the magnetic scale height. The equilibrium magnetic field configuration is shown in Fig. 1. Since [FORMULA] at [FORMULA], the magnetic scale height can be expressed as

[FIGURE] Fig. 1. Equilibrium magnetic field configuration of a coronal arcade.

[EQUATION]

where the width of the arcade is given by 2 L.

Note that, with these components, the magnetic field satisfies the divergence-free equation, [FORMULA]. Moreover, the amplitude of the magnetic field decreases exponentially with z as

[EQUATION]

Since the equilibrium is potential, and we are ignoring the effect of gravity, there are no mathematical constraints to set the background gas density [FORMULA]. Consequently, we are free to adopt it as decreasing exponentially with height,

[EQUATION]

The quantity [FORMULA] is assumed to be constant, which implies that there is no variation of the density in the horizontal (x) direction. The parameter [FORMULA] denotes the pressure scale height, i. e.

[EQUATION]

It will be useful to introduce the ratio of magnetic to pressure scale height,

[EQUATION]

For the magnetic field and the gas density chosen here the square of the Alfvén speed is

[EQUATION]

where [FORMULA] is the square of the Alfvén speed at the base of the corona. So, when [FORMULA], or equivalently when the characteristic magnetic field scale length [FORMULA] is twice the scale height of the coronal plasma [FORMULA], the Alfvén speed is constant. On the other hand, choosing [FORMULA] greater or smaller than 2 corresponds to the case of Alfvén speed exponentially increasing or decreasing with height.

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© European Southern Observatory (ESO) 1998

Online publication: January 16, 1998
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