**3.3 Various manifestations of nonpotentiality**

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

#### **Figure 3.**

*Nanofluid Flow in Porous Media*

structure of the corona [4, 5].

Lorentz force is zero.

of magnetic field, we have

force-free magnetic field in corona.

as the bottom boundary condition of corona system and confines the magnetic

mas 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:

∇× *B* = α*B*. (1)

(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

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

By taking the divergence of Eq. (1) and considering the divergence-free property

∇α ∙ *B* = 0. (3)

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

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

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

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

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

(see Eq. (2)), and this special case is called potential magnetic field.

calculated numerically based on Eqs. (1) and (3) [9, 10, 17–20].

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

eruptive phenomenon since there is no force to accelerate the plasmas.

K), and the density of plas-

<sup>4</sup>*<sup>π</sup>* <sup>∇</sup><sup>×</sup> *<sup>B</sup>* (2)

In the corona, the temperature is very high (about 106

*j* = \_\_\_1

**196**

*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)).*

#### **Figure 4.**

*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.*

**Figure 5.**

*Twisted field lines of the coronal magnetic field in the source AR of the flare event on 13 December 2006 [11]. Closed field lines are in white and open field lines are in black. The white and black background contours show the two polarities of the photospheric magnetic field. The coronal magnetic field was calculated based on the force-free field model.*

is more prominent around the PILs of ARs with complex bipolar photospheric magnetic field. The typical characteristic of the photospheric vector magnetic field associated with the nonpotential magnetic field in the corona is the shear of the transverse component (denoted by *B*t) of the photospheric magnetic field. For potential magnetic field, the direction of *B*t is generally vertical to the PILs, whereas for nonpotential magnetic field, the direction of *B*t tends to be parallel to the PILs. The shear property reflects the direction deviation of the real *B*t from the potential *B*t [7] (see **Figure 4** for a diagram illustration).

Owning to the shear of *B*t in photosphere, the field lines in the corona also show shear behavior around the PILs, i.e., the field lines tend to be parallel to the PILs. For a closed field line which starts at positive polarity, shears around the PIL, and ends at negative polarity, the three segments (start, shear, and end segments) of the field line compose a twist (S-shaped or inverse S-shaped) morphology (see **Figure 5** for an example diagram of the twisted field lines in corona). The twisted field lines indicate the existence of electric current in the corona and are another manifestation of the nonpotential magnetic field [8].

## **4. Variation of coronal magnetic field and solar flare eruption**

#### **4.1 Quasi-steady evolution of the corona**

Because the photospheric magnetic field confines the magnetic structure in corona, along with the evolution of the photospheric magnetic field, the coronal magnetic field also evolves accordingly as the response to the variations in photosphere. The evolution in the photosphere is relatively slow owing to its relatively dense plasma, and the evolution in the corona is fast for its very tenuous condition. Thus, the corona can catch up the variations in photosphere promptly. If there are no eruptions on the Sun (quiet state), the coronal evolution (along with the evolution of the photosphere) can be quasi-steady and approximated by the force-free condition [4].

Provided a time series of observed photospheric vector magnetograms of a solar AR, by numerical modeling of the coronal magnetic fields based on the force-free field model for each magnetogram, we can obtain a time series of 3-D coronal magnetic field data associated with the photospheric magnetograms. These time

**199**

**5. Conclusion**

*Variation of Coronal Magnetic Field and Solar Flare Eruption*

series coronal magnetic field data quantitatively describe the quasi-steady evolution of the corona [9, 10]. By deriving the electric current density distributions from these coronal data, the time evolution of the nonpotentiality in the corona can also

During the period of quasi-steady evolution of the corona, the plasmas in the corona are in quasi-equilibrium state, and the topological structure of coronal magnetic field evolves continuously. In some situations, the variations of photospheric magnetic field may cause sudden changes of topological structure of coronal magnetic field at certain sites in the corona [1, 3, 11]. The plasmas at these sites lost equilibrium and are ejected from their original positions. The magnetic reconnections occur beneath the erupted plasmoid, and then the flares are initiated. After the flare eruptions, the corona returns to equilibrium state and continues its

During the period of flare eruptions, the plasmas are in dynamic state, and the force-free condition is not satisfied. Out of the flare eruption period, the force-free

Because the plasmoid ejection associated with the flare initiation needs magnetic force and the magnetic force (Lorentz force) only acts on the electric current, the sites of flare initiation are always located in the areas with strong electric current which in turn concentrates around the PILs of ARs. That is why the flare phenomenon is connected with the nonpotentiality (indicated by the electric current) of

Not all magnetic energy in the solar atmosphere can be depleted by flare eruptions. For example, the magnetic energy of a potential magnetic field cannot be consumed since no flare eruptions can occur in the potential magnetic field. Only the magnetic energy associated with the electric current in a nonpotential magnetic field can be accessed by flare eruptions. This part of available magnetic energy (energy bundled with the electric current) is called the free magnetic energy [4, 22]. During a flare eruption, a fraction of the total electric current around the PIL is ejected out together with the plasmoid eruption, and the corresponding part of the free magnetic energy is released. After the flare eruption, the total free magnetic energy decreases, and the twisted field lines around the PIL relax to a certain extent owing to the loss of electric current and the depletion of free magnetic energy [11]. A proportion of the released magnetic energy is converted to the electromagnetic emission which manifests as sudden brightening across a broad range of electromagnetic wave spectrum, and that is why the flare phenomenon is named [2]. Other parts of the released magnetic energy are converted to the mechanical energy of the erupted plasmoid and are also carried off by the high-energy particle radiation [2, 4]. The erupted plasmoid might lead to the coronal mass ejections (CMEs) accompanied with solar flares, and the high-energy particles might lead to the solar energetic particle (SEP) events associated with solar flares [12–14].

As a prominent eruptive phenomenon happening in solar atmosphere, solar flares usually come from solar ARs which possess strong and concentrated bipolar

field model is a well approximation to the coronal magnetic field [11].

**4.2 Sudden change of coronal magnetic field structure and flare initiation**

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

quasi-equilibrium evolution [3, 11].

solar ARs, and flares always occur above the PILs.

**4.3 Release of magnetic energy by flare eruption**

be revealed [10, 11].

*Nanofluid Flow in Porous Media*

**Figure 5.**

*force-free field model.*

is more prominent around the PILs of ARs with complex bipolar photospheric magnetic field. The typical characteristic of the photospheric vector magnetic field associated with the nonpotential magnetic field in the corona is the shear of the transverse component (denoted by *B*t) of the photospheric magnetic field. For potential magnetic field, the direction of *B*t is generally vertical to the PILs, whereas for nonpotential magnetic field, the direction of *B*t tends to be parallel to the PILs. The shear property reflects the direction deviation of the real *B*t from the

*Twisted field lines of the coronal magnetic field in the source AR of the flare event on 13 December 2006 [11]. Closed field lines are in white and open field lines are in black. The white and black background contours show the two polarities of the photospheric magnetic field. The coronal magnetic field was calculated based on the* 

Owning to the shear of *B*t in photosphere, the field lines in the corona also show shear behavior around the PILs, i.e., the field lines tend to be parallel to the PILs. For a closed field line which starts at positive polarity, shears around the PIL, and ends at negative polarity, the three segments (start, shear, and end segments) of the field line compose a twist (S-shaped or inverse S-shaped) morphology (see **Figure 5** for an example diagram of the twisted field lines in corona). The twisted field lines indicate the existence of electric current in the corona and are another manifesta-

potential *B*t [7] (see **Figure 4** for a diagram illustration).

**4. Variation of coronal magnetic field and solar flare eruption**

Because the photospheric magnetic field confines the magnetic structure in corona, along with the evolution of the photospheric magnetic field, the coronal magnetic field also evolves accordingly as the response to the variations in photosphere. The evolution in the photosphere is relatively slow owing to its relatively dense plasma, and the evolution in the corona is fast for its very tenuous condition. Thus, the corona can catch up the variations in photosphere promptly. If there are no eruptions on the Sun (quiet state), the coronal evolution (along with the evolution of the photosphere) can be quasi-steady and approximated by the force-free condition [4]. Provided a time series of observed photospheric vector magnetograms of a solar AR, by numerical modeling of the coronal magnetic fields based on the force-free field model for each magnetogram, we can obtain a time series of 3-D coronal magnetic field data associated with the photospheric magnetograms. These time

tion of the nonpotential magnetic field [8].

**4.1 Quasi-steady evolution of the corona**

**198**

series coronal magnetic field data quantitatively describe the quasi-steady evolution of the corona [9, 10]. By deriving the electric current density distributions from these coronal data, the time evolution of the nonpotentiality in the corona can also be revealed [10, 11].
