Electrodynamic Coupling of Active Region Corona with the Photosphere

Abstract

In the previous chapters we studied response of the chromosphere and corona to magnetic activity in the underlying quiet sun regions and both types of plages, unipolar and mixed-polarity ones. In this chapter we turn to the question how the chromosphere and corona respond to magnetic activity in the active regions where filling factor is close to unity. Our consideration will be based on a specific case favorably caught in multiwavelength observations showing the entire process from the birth and evolution of a compact active region to formation and dynamics of coronal structures above it. We then discuss a general theory based on energetically open systems of currents that may be driven into various dynamic forms via nonlinear processes with continuous flow of matter and energy. Depending on the system parameters these may be long-living steady loops showing subtle oscillations, loops in the relaxation regime, and the periodically flaring and exploding loop systems. The theory also predicts that the EUV loops must have a filamentary structure and allows one to estimate the limiting currents and critical radii of elemental filaments associated with the stability criteria.

Appendix: Method of Slow Variables for van der Pol Equation

In a real time (16.48) has a form

$$\begin{aligned} \frac{\mathrm{d}^2 i}{\mathrm{d}t^2}-\mu (1-i^2)\frac{\mathrm{d}i}{\mathrm{d}t}+ \omega _0^2 i = 0 \end{aligned}$$

(16.70)

and belongs to the class of nonlinear equations of the type

$$\begin{aligned} \frac{\mathrm{d}^2 i}{\mathrm{d}t^2}+ \omega _0^2 i = f\left( i,\frac{\mathrm{d} i}{\mathrm{d} t}\right) \end{aligned}$$

(16.71)

with

$$\begin{aligned} f\left( i,\frac{\mathrm{d} i}{\mathrm{d} t}\right) = -\mu (i^2-1)\frac{\mathrm{d} i}{\mathrm{d} t} \end{aligned}$$

(16.72)

The method of slowly varying phase and amplitude is based on the usage of transformation (see e.g. Hagedorn 1988):

$$\begin{aligned} i=A(t) \text{ sin } (\omega _0t + \psi ), \nonumber \\ \frac{\mathrm{d} i}{\mathrm{d} t}=A(t)\omega _0 \text{ cos }(\omega _0t + \psi ) \end{aligned}$$

(16.73)

which replaces (16.71) by the system of a simple integro-differential equations:

$$\begin{aligned} \frac{\mathrm{d} A}{\mathrm{d} t}=\frac{1}{2\pi \omega _0}\int \limits _0^{2\pi }{f(A,\psi ) ~\text{ cos } (\phi +\psi ) ~\mathrm{d}\phi }, \nonumber \\ \frac{\mathrm{d} \psi }{\mathrm{d} t}=-\frac{1 }{2\pi \omega _0 A}\int \limits _0^{2\pi } {f(A,\psi ) ~\text{ sin }(\phi +\psi ) ~\mathrm{d}\phi } \end{aligned}$$

(16.74)

With (16.72) we have

$$\begin{aligned} \frac{\mathrm{d} A}{\mathrm{d} t}=\frac{\mu A}{2\pi }\int \limits _0^{2\pi }{(1- A^2\text{ sin }^2\phi ) ~\text{ cos }^2\phi ~\mathrm{d}\phi }, \nonumber \\ \frac{\mathrm{d} \psi }{\mathrm{d} t}=-\frac{\mu }{2\pi }\int \limits _0^{2\pi }{(1- A^2\text{ sin }^2\phi ) ~\text{ cos }\phi ~\text{ sin }\phi ~\mathrm{d}\phi } \end{aligned}$$

(16.75)

or, equivalently,

$$\begin{aligned} \frac{\mathrm{d} A}{\mathrm{d} t}=\mu \frac{A}{2}\left( 1-\frac{A^2}{4}\right) , \nonumber \\ \frac{\mathrm{d} \psi }{\mathrm{d} t}= 0 \end{aligned}$$

(16.76)

With the nondimensional time $\tau =t/\sqrt{LC}$ and $\epsilon =\mu /\sqrt{LC}$, (16.76) becomes (16.51).