Discussion paper: The eigen oscillations of the solar active regions

2. Oscillations of Magnetic Arcades

[3] Let us consider the bipolar solar active region with the potential magnetic field

(4)

0, -

, -p /2p /2. The plasma density we take in the form

(5)

In the coronal conditions the small plasma beta approximation may be used. The plane x=0 is the photosphere surface, x>0 is the height above the photosphere, the parameter l₀ gives the active region scale. The magnetic field lines have the equation e^-x/l₀ cos(y/l₀)= cos(y₀/l₀) with some constant y₀ and formate the arches which are contained in the planes z= const.

[4] In this field there are the Alfvén waves propagating from one footpoint to another along the rays which coincide with the magnetic arches. Being reflected from the photosphere they form the standing waves that give the eigen Alfvén modes of the active region. The standing fast waves are formed on the rays similar to magnetic arches, their ends lie on the photosphere. The boundary condition on the photosphere is taken as

(6)

The standing waves are described by the symmetric and antisymmetric functions

(7)

The spectrum is

(8)

where k_t=(n+1/2)p for the even mode and k_t=np for the odd mode, the integer number n

1. The expression for the eikonal t ( r) is defined below. The oscillations are localized near the rays whose footpoints are (0, -y₀, z₀) and (0, y₀, z₀). The rays form the surfaces, each of them is characterized by the discrete set of eigen frequencies. The frequencies vary continuously from one surface to another. For the Alfvén wave we obtain the eikonal

Here v_A0=B₀/(4pr ₀)^1/2 is the Alfvén velocity at the corona basis. For the velocity and magnetic field amplitudes we have

where G(x) is an arbitrary function. The plasma motions are polarized along the tunnel of the magnetic arcade. The plasma distribution reveals itself in the space distribution of the eigen oscillations, for d <4 they are localized close to the apexes of the magnetic arches, and for d >4 they are localized close to the footpoints.

[5] For the fast waves we obtain the eikonal

The amplitudes of the fast waves are

The plasma motions are polarized in the plane of the magnetic arches. The fast rays obey the equation

If d <2, the rays rise infinitely upward in the corona and the fast waves propagate freely. In this case the frequency is arbitrary, i.e. the fast modes spectrum is continuous. If d =2, the rays are transformed into a horizontal straight line. We can formulate no obvious boundary conditions in this case. If d >2, the rays have the arch form and the waves may be reflected from the photosphere. Moreover, if 2<d <4 the fast rays arches are more gently slopping than the magnetic arches. In this case only a part of the fast waves are localized close to photosphere, one part of the spectrum is discrete and the other part may be continuous. If d >4, the fast rays arches are steeper than the magnetic arches, here all fast waves are localized and the spectrum is discrete fully.