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 (), so it follows from Eq. (2) that the magnetic Lorentz force must vanish everywhere, i. e.
A particular solution to this equation is a potential magnetic field with zero current,
In a y -invariant system, the equilibrium magnetic field may be written as follows
where is the unit vector along the y -direction and is the vector magnetic potential whose modulus satisfies Laplace equation, . The solution for A can be obtained by separation of variables under the condition that it does not diverge as . Therefore, the magnetic field components are given by
with the magnetic field strength at the coronal base () and the magnetic scale height. The equilibrium magnetic field configuration is shown in Fig. 1. Since at , the magnetic scale height can be expressed as
where the width of the arcade is given by 2 L.
Note that, with these components, the magnetic field satisfies the divergence-free equation, . Moreover, the amplitude of the magnetic field decreases exponentially with z as
Since the equilibrium is potential, and we are ignoring the effect of gravity, there are no mathematical constraints to set the background gas density . Consequently, we are free to adopt it as decreasing exponentially with height,
The quantity is assumed to be constant, which implies that there is no variation of the density in the horizontal (x) direction. The parameter denotes the pressure scale height, i. e.
It will be useful to introduce the ratio of magnetic to pressure scale height,
For the magnetic field and the gas density chosen here the square of the Alfvén speed is
where is the square of the Alfvén speed at the base of the corona. So, when , or equivalently when the characteristic magnetic field scale length is twice the scale height of the coronal plasma , the Alfvén speed is constant. On the other hand, choosing greater or smaller than 2 corresponds to the case of Alfvén speed exponentially increasing or decreasing with height.
© European Southern Observatory (ESO) 1998
Online publication: January 16, 1998