[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} 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}=r_{0}/(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 [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_{z}z -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) B_{0}(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): [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 r_{0}=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. Powered by TeXWeb (Win32, v.2.0). |