**3. Nonpotentiality of coronal magnetic field**

#### **3.1 Force-free magnetic field of steady corona**

Since the magnetic reconnection associated with solar flare eruptions takes place in the corona, the coronal magnetic field distribution is crucial for understanding the physical process of flares [1, 3, 4]. In fact, the photospheric magnetic field acts

as the bottom boundary condition of corona system and confines the magnetic structure of the corona [4, 5].

In the corona, the temperature is very high (about 106 K), and the density of plasmas is very low [4]. Thus, the magnetic force (Lorentz force) dominates the coronal system, and other forces (such as gravity force, pressure, etc.) can be neglected. Then, in steady state of the corona, the plasmas are distributed along the field lines, and the Lorentz force is zero. This condition is called force-free condition, and the corresponding magnetic field is called force-free magnetic field [6]. Note that the essence of the force-free field concept is the dominant role of the magnetic field in a plasma system.

The force-free magnetic field *B* can be described by the following equation:

$$
\nabla \times \mathbf{B} = \mathbf{\alpha} \mathbf{B}.\tag{1}
$$

The left part of Eq. (1) represents the electric current density vector:

$$
\mathbf{j} = \frac{1}{4\pi} \nabla \times \mathbf{B} \tag{2}
$$

(in electromagnetic CGS units). Then Eq. (1) indicates that the current density vector *j* and the magnetic field vector *B* are parallel to each other, and hence the Lorentz force is zero.

The proportional coefficient α in Eq. (1) is called force-free factor. The value of α can be positive, zero, or negative. If α > 0, the electric current density *j* and the magnetic field *B* have the same directions, and if α < 0, they have opposite directions. α = 0 means that ∇× *B* = 0, i.e., there is no electric current in the magnetic field (see Eq. (2)), and this special case is called potential magnetic field.

By taking the divergence of Eq. (1) and considering the divergence-free property of magnetic field, we have

$$
\nabla \mathbf{a} \cdot \mathbf{B} = \mathbf{0}.\tag{3}
$$

**197**

**Figure 4.**

**Figure 3.**

*Variation of Coronal Magnetic Field and Solar Flare Eruption*

**3.3 Various manifestations of nonpotentiality**

measure to quantitatively describe the nonpotentiality of coronal magnetic field [10, 11] (see **Figure 3** for an example diagram of the electric current density spatial

For a solar AR with single polarity magnetic field in photosphere, the coronal magnetic field is usually simple and tends to be potential field. The nonpotentiality

*Spatial distribution of electric current density in the source AR of the flare event on 13 December 2006 [11]. The electric current density values were derived from the 3-D coronal magnetic field data by using Eq. (2), and the coronal magnetic field data were calculated based on the force-free field model (see Eqs. (1) and (3)).*

*Shear of the transverse component Bt of the photospheric magnetic field around the PIL in the source AR of the flare event on 13 December 2006 [11]. The small arrows indicate the directions of Bt. The contours show the vertical component (and the two polarities) of the photospheric magnetic field. White contours represent the* 

*positive polarity, and black contours represent the negative polarity.*

*DOI: http://dx.doi.org/10.5772/intechopen.86168*

distribution in a solar AR).

Eq. (3) means that along each field line, α is a constant. (Note that for different field lines, the values of α can be different.) This is an important character of the force-free magnetic field in corona.

Eqs. (1) and (3) together give the mathematical expression of the coronal force-free magnetic field model (two variable quantities α and *B* constrained by two equations) [6]. Given the observed photospheric vector magnetic field as the bottom boundary condition, the 3-D coronal magnetic field in steady corona can be calculated numerically based on Eqs. (1) and (3) [9, 10, 17–20].

#### **3.2 Electric current and nonpotentiality of coronal magnetic field**

The dominant force in the corona is magnetic force which acts on the electric current. In potential magnetic field, there is no electric current and hence no magnetic force. Thus, it is not possible for the potential magnetic field to produce eruptive phenomenon since there is no force to accelerate the plasmas.

To yield solar eruptions, the existence of electric current or nonpotential magnetic field in the corona is a necessary condition. In solar ARs with complex photospheric magnetic field, the coronal magnetic field generally contains electric current around the PILs (note that the magnetic field structure in the corona is confined by the photospheric magnetic field). This property (deviation from potential magnetic field) is called nonpotentiality of coronal magnetic field and is commonly employed to reflect the activity level (degree of possibility to produce eruptive events) of solar ARs [7, 8, 10, 11, 21]. The electric current density *j* (see Eq. (2)) is a physical

*Variation of Coronal Magnetic Field and Solar Flare Eruption DOI: http://dx.doi.org/10.5772/intechopen.86168*

measure to quantitatively describe the nonpotentiality of coronal magnetic field [10, 11] (see **Figure 3** for an example diagram of the electric current density spatial distribution in a solar AR).
