Discussion paper: The kink oscillations of the thin nonuniform coronal loops

[2] It is well known that the coronal magnetic flux tubes are twisted. In addition to the longitudinal magnetic field they also contain an azimutal component. The magnetic tubes undergoe expansion in the rarefied solar atmosphere. If the tube expands, the azimutal field B_j sim r^-1 on the periphery of the tube is formed. The longitudinal magnetic field persists only in the central part of the tube [Parker, 1979]. The mathematical difficulties that arise in describing of such coronal tube force us to use its crude model in which the magnetic field has only a longitudinal component in the central part and only an azimutal component on the periphery.

[3] We consider a cylindrical tube of radius a with the plasma density r_0m=r₀/(ar)² in which a central part of radius b (br_0i, the other part of the tube is called a shell. The plasma density in the surrounding corona is r_0e<r_0i. The equilibrium magnetic field has the follow distribution

The Alfvén speeds in the cord and in the corona are V_Ai, V_Ae. V_Am² =B₀²/4pr₀ is the Alfvén speed scale in the shell. The equilibrium conditions for the tube are

[4] Let us seek the solutions of linear ideal MHD equations for the cool plasma in the form of cylindrical modes f(r, t)=f(r) exp(imj +i k_zz -iwt), where k_z is the longitudinal wave number and w is the frequency. They are expressed through the magnetic pressure perturbation P(r)=p(r)+ B(r) B₀(r)/4p. The solution in the cord and in the corona can be expressed in terms of the Bessel equation solutions (for the kink-mode m =1):

where

In the thin tube ( k_za

1 ) approximation we take the principal terms of the Bessel and McDonald functions at r=0. In the shell the radial equation for P have the form [Appert et al., 1974]

Two linearly independent solutions of this equation have poles of the first and the third orders at z=0.

The solution in the shell is

where F and G are arbitrary constants and the coefficients have the following approximate expressions

Using the principals terms in the solutions and the boundary conditions

we obtain the dispersion equation in the zeroth order approximation [Mikhalyaev, 2005]

It should be borne in mind that the tube parameters in this equation are constrained by equilibrium conditions. This equation have two real solutions that describe undamped oscillations. One of them describes a fast magnetosonic wave whose phase speed exceeds the Alfvén speed in the corona. Therefore, it propagates radially into the surrounding corona, i.e., is radiated by the tube. The damping manifests itself as the effect of the next order with respect to the small k_za. In the first approximation, the dispersion equation has a complex solution with a relatively small imaginary part. We write the complex frequency as w=w₀(1+ epsilon

), where w₀ is the solution of the dispersion equation in the zeroth order approximation, and the dimensionless quantity epsilon

gives the first correction. Its imaginary part determines the damping coefficient -w₀ Im epsilon

, while the ratio Q=-1/2 Im epsilon

is the Q-factor of the oscillations. For epsilon

the following expression holds

[5] We applied the results obtained to the oscillations of solar coronal loops. As the corona is characterized by Alfvén speeds much larger than the sound speed, we have chosen V_Ae=700 km s^-1. The density in the cord r_0i=5r_0e, and the characteristic density r₀=5r_0e was chosen for the shell. The scale parameter a =0.25 cm^-1. The Q -factor increases with decreasing wave number, i.e., with increasing tube length L. For example, at the tube radius a =12 Mm and the cord radius b =2 Mm, the Q-factor increases from 19.7 to 84.9 as the tube length changes from 11 Mm to 230 Mm. The oscillation period takes on values within the range 239 to 497 s. The Alfvén speed in the shell is the same, 939 km s^-1. The Q-factor and the period increase with cord radius. If b changes from 1 to 4 Mm (at a=12 Mm and L=130 Mm), then the period increases from 270 to 328, while the Q-factor increases from 18.1 to 190. Our calculations show that Q-factors close to their observed values can be obtained [Nakariakov et al., 1999; Ofman and Aschwanden, 2002]. Thus, a double magnetic flux tube with a strongly twisted magnetic field in the shell can serve as an acceptable model for coronal loop, and the observed fast damping of transverse loop oscillations can be explained in terms of the effective radiation of fast magnetosonic waves into the surrounding corona by the loop.