4. Slow wave evolution
In the case of linear waves and a dissipation-less medium, the third and fourth terms in Eq. (28) are neglected, and the equation is easily integrated:
The solution is shown as the solid curve in Fig. 2. The amplitude grows until the wave reaches the loop apex and then decays. The amplitude is represented by a symmetric function of with respect to the apex .
Consequently, the growth rate of the amplitude is determined by the balance between the stratification and the dissipation. Waves of shorter wavelengths (larger wavenumbers) grow slower than long wavelength waves. Sufficiently short wavelength waves, with , do not grow but decay with height. Dependences of the linear wave amplitude on the distance along the loop for different values of the normalized dissipation coefficient are shown in Fig. 2. Obviously, the dissipative waves are not represented by symmetric curves along the loop. Waves, descending from the loop apex have smaller amplitude than ascending waves.
In general, when all terms of Eq. (28) are significant, it is difficult to find out an analytical solution to the equation. However, the equation can be easily solved numerically. Dependence of nonlinear dissipative wave amplitudes on the distance along the loop is shown in Fig. 3. Nonlinear generation of higher harmonics transfers the wave energy to smaller scales, which are, according to subject to stronger dissipation. The nonlinear dissipation increases with the growth of the amplitude. Waves of stronger amplitudes are more non-symmetric with respect to the loop apex.
According to the theory presented above, the slow wave evolution is controlled by the wave parameters: the period and the relative amplitude, as well as by parameters of the loop: the radius, the temperature (which prescribes the sound speed and local scale height) and the dissipation coefficient. For the propagating disturbances observed in the coronal loops, some of the parameters are determined: the wave periods are 300-900 s, the relative amplitudes are about 2%, the loop radii are about 140 Mm, the temperature is 1.6 MK. The most unknown parameter is the dissipation coefficient, because both viscosity and thermal conduction have not been determined observationally yet. According to Braginskii's theory, the first viscosity coefficient for the plasma with the concentration cm-3 and the temperature 1.6 MK is g(cm s)-1. This parameter coincides with the parameter used above, which, neglecting the thermal conduction (), gives . Consequently, this is the least possible value of . In the presence of finite thermal condition this value can be higher. Also, as it has been suggested by Nakariakov et al. (1999) for shear viscosity, the actual MHD wave dissipation can be dramatically enhanced (by, e.g. micro-turbulence).
In Fig. 3, we show dependences of the slow wave amplitudes upon the distance along the loop for three different wave periods and two different dissipation coefficients. It is seen that the theory developed easily explains the observational fact the descending wave is not registered: its amplitude is much weaker than the amplitude of the ascending wave. Also, the ascending wave amplitude growth can be efficiently depressed by dissipative processes.
© European Southern Observatory (ESO) 2000
Online publication: October 30, 2000