ASIGNACIÓN SALARIAL $1’200

A magnetic field line is a curve tangent to the magnetic field at each point and is described by,

dr dℓ =

B(r(ℓ))

|B(r(ℓ))|, (2.68)

where r is a vector pointing from the origin to a point along the line andℓ is the arc length along the line. A given line can be determined by integrating the equation from a given starting point. While field lines are not an actual physical entity, they are useful

2.5 Magnetic Fields in the Corona

Figure 2.2: An EUV image of coronal loops in an active region (from Reale, 2010).

in describing the magnetic field topology of a given system. The density of field lines indicates the magnetic field magnitude in a given region.

A coronal loop is often modeled as a thin magnetic flux-tube, composed of a bundle of individual filaments. Since magnetic pressure should cause the loop to expand, the loop is thought to be twisted, causing magnetic tension to resist the expansion. Coro- nal loops are rooted in the photosphere at either end, forming a closed field geometry. An extreme ultraviolet (EUV) image of coronal loops within an active region is shown in Figure 2.2. In the photosphere, a flux-tube is in magnetohydrostatic equilibrium, where the internal magnetic and gas pressure balances the external gas pressure. In the photosphere, flux tubes are constantly swept into down-draft regions between su- pergranules (Simon & Leighton, 1964). Within flux tubes, flows are confined to the direction of the field and this leads to an evacuation of the flux-tube. These flows are

shown to be >250 m s−1 _{(Solanki, 1986). Since magnetic flux is frozen-in and total}

flux through a surface is conserved, any increase in external gas pressure leads to a compression of the flux-tube and results in an increase in the internal magnetic field. This process is called convective collapse (Roberts, 2000). A schematic of this situation is shown in Figure 2.3.

In active regions with simple configurations, coronal loops are often observed to be stable structures on the scale of τadv (Equation 2.37). Above the photosphere, flux-

tubes expand, as the ambient gas pressure drops off with height. This occurs rapidly within the chromosphere and more slowly in the corona as the pressure gradient levels off until the magnetic forces are in equilibrium. In the corona, β << 1 so plasma is

confined to flow parallel to the field lines. A given coronal loop exhibits hydrostatic equilibrium along its curvature. Coronal loops are bright because they are hot and heating may be due to small flares (DC) or waves (AC) along the loop (Reale, 2010).

The magnetic properties of coronal loops can be determined by extrapolating the magnetic field from magnetogram observations and comparing the result to loop obser- vations. To obtain a static solution, the sum of forces included in Equation 2.43 cancel out,

∑

F =−∇P +j×B+ρg= 0 (2.69)

Additionally, the “force-free” approximation is used, where each force is assumed to be negligible. Dimensional analysis shows the gravitational force is small compared to the pressure force. The pressure gradient is not important if scales are considered that are much less than the pressure scale height (H = kBT /µmHg), the distance over which

the pressure drops by 1/e. Since β <<1, the Lorentz force dominates over the other

forces,

0 = −∇P +j×B+ρg, (2.70)

2.5 Magnetic Fields in the Corona CHROMOSPHERE PHOTOSPHERE CONVECTION ZONE CORONA

F

_F

F

F

F

F

F +

F

ρ

ρ

B

B

B <

B

ρ <

ρ

Figure 2.3: A schematic of a coronal loop, with its foot points located at down-flow

regions between supergranules. The combination of gas and magnetic pressure within flux- tubes at the photosphere are balanced by exterior gas pressure. In the corona, the gas pressure drops off, leaving the magnetic pressure and tension forces to define structures. Hydrostatic equilibrium is achieved along the curve of a coronal loop.

Two solutions to this are j= 0 or that j∥B. In the former case, from Equation 2.29 this leads to,

∇ ×B= 0 , (2.72)

which is the current-free solution and results in “potential fields” that are curl and divergence free. For the latter case, the currents are field-aligned and Equation 2.29 can be written,

∇ ×B = µ0jBˆ =αBBˆ, (2.73)

Figure 2.4: An example of a spherical potential field source surface extrapolation using MDI line-of-sight magnetograms as input (courtesy of Marc DeRosa). Field lines are traced separately for closed (black), positive open (out of the Sun; green), negative open (into the Sun; magenta).

whereα is a scalar derived from Equation 2.73,

α= jµ0

B (2.75)

These fields can be “non-potential” and result in a twist in the field. For the potential case, α is set to 0,

2.5 Magnetic Fields in the Corona

soB can be written as the gradient of some scalar potential field,

B=∇Ψ (2.77)

It can be see that Ψ satisfies Laplace’s equation by taking the divergence of both sides of Equation 2.77,

∇ ·B = ∇ · ∇Ψ , (2.78)

∇ ·B = 0 , (2.79)

∴∇2_{Ψ = 0 (2.80)}

A solution to Equation 2.80 in spherical coordinates is the superposition of a series of spherical harmonics, Ψ(r, θ, ϕ) =∑ ℓ,m ( Am_ℓ rℓ+B_ℓmr−(ℓ+1) ) Y_ℓm(θ, ϕ) , (2.81)

whereris the radius from solar centre,Am_ℓ andB_ℓmare coefficients determining the im-

portance of each harmonic, andY_ℓm(θ, ϕ) are the pure harmonic modes. The subscripts

m andℓ define the number of sectors in the longitudinal (nϕ=m+ 1) and latitudinal

(nθ =ℓ+ 1) directions, respectively. The spherical harmonics are given by,

Y_ℓm(θ, ϕ) =Cℓ_mP_mℓ(cosθ)eimϕ, (2.82)

wherePℓ

m(cosθ) are the Legendre polynomials and,

Cℓ m= (−1)m [ 2ℓ+ 1 4π (ℓ−m)! ℓ+m)! ]1/2 (2.83)

by line-of-sight (LOS) magnetic field observations and an upper boundary is determined by an arbitrary “source surface” where the field becomes radial. Conventionally, this is usually 2.5R_⊙. These conditions stipulate a unique solution for the field. The coefficients can be solved by substituting Equation 2.81 into Equation 2.77 and applying the boundary conditions. An example spherical extrapolation1 using potential field assumptions is shown in Figure 2.4; this is called a potential field source surface (PFSS) extrapolation.

If α̸= 0 in Equation , then field aligned currents may exist. Taking the divergence of Equation ,

∇ ·(∇ ×B) =∇ ·(αB) = 0 , (2.84)

due to the solenoidal constraint. So, the gradient of α along B can not vary. Thus, a property ofαis that it must be constant along a given field line, but can vary between

field lines. Making the assumption thatα is constant over all space,

∇2_B=−α2_{B, (2.85)}

can be derived. This relation defines a “linear force-free” field. If α is not assumed

constant over all space,

∇2_B+α2_{B=B× ∇α, (2.86)}

and

B· ∇α= 0 , (2.87)