Collective Phenomena in Rarefied Ensembles of Flux Tubes

Abstract

In the previous chapters we have considered the properties of individual flux tubes and their dynamics due to interaction with the surrounding medium. The solar atmosphere, however, consists of ensembles of flux tubes randomly distributed in space and over their parameters. In this chapter we shall consider the time-dependent response of a randomly magnetized medium to propagation of acoustic waves and unsteady wave packets. We will see that the collective phenomena resulted from these interactions lead to clear morphological effects that are observable. The character of these effects, and the associated energy input and distribution, crucially depend on the magnetic filling factor of the medium and several other factors that will be specified.

Appendix

To perform the integration in (6.129) we introduce the function

$$\begin{aligned} \kappa (\eta )= \frac{\delta \omega }{v_\mathrm{{ph}}} \end{aligned}$$

(6.142)

and change the variable $\zeta -\zeta ' = x$. Then the double integral in (6.129) becomes (we need only its real part):

$$\begin{aligned} \text{ Re }\,I(\zeta )=\text{ Re }\,\int \limits _0^{\infty } \int \limits _{-\infty }^{\infty }{ \mathrm{d} (\eta - \eta _0) e^{i \kappa (\eta ) x} F \left( \zeta - x,t- \frac{x}{v_\mathrm{{ph}}}\right) \mathrm{d} x}. \end{aligned}$$

(6.143)

Now, the shortest way to integrate (6.143) is to use the definition of the $\delta$-function, namely

$$\begin{aligned} \delta (x)= \frac{1}{2\pi }\int \limits _{-\infty }^{\infty }{e^{i k x} \mathrm{d}k}, \qquad \delta [f(x)] = \frac{1}{|\mathrm{d}f/\mathrm{d}x|_0} \delta (x). \end{aligned}$$

(6.144)

Therefore, (6.143) can be written as

$$\begin{aligned} \text{ Re }\,I(\zeta )=\text{ Re }\,\int \limits _0^{\infty } { 2 \pi \frac{1}{|\mathrm{d}f/\mathrm{d}x|_0} \delta (x) F \left( \zeta - x,t- \frac{x}{v_\mathrm{{ph}}}\right) \mathrm{d} x}. \end{aligned}$$

(6.145)

which gives immediately

$$\begin{aligned} \text{ Re }\,I(\zeta )= \frac{\pi }{|\mathrm{d}f/\mathrm{d}x|_0} F (\zeta ,t). \end{aligned}$$

(6.146)

Substituting (6.146) into (6.143) we come to expression (6.130)

$$\begin{aligned} \text{ Re }\,I_2 (\zeta )= (\rho _\mathrm{i} + \rho _\mathrm{e}) g(\eta )|_{\eta _0} \frac{\pi }{|(\mathrm{d}/\mathrm{d} \eta )(\delta \omega /v_\mathrm{{ph}})|_{\eta _0}} F (\zeta , t). \end{aligned}$$

(6.147)