A number of approximations can be made in Equation 2.18 for conditions en- countered in the solar atmosphere. Firstly we can again assume the plasma is in magnetohydrostatic equilibrium by replacing the left–hand side of the equa- tion with zero. Then, if we assume that the length scale of interest is much smaller than the density scale–height of the plasma, L0 << H, we can say that ρcg <<∇p, and remove it from the equation. Finally, if the plasma beta param-
∇p. This leaves us with, simply:
J×B= 0 (2.22)
which is known as the force–free approximation, and is widely used to model solar coronal fields.
Equation 2.22 tells us that, if the current density J is non–zero, then it must be parallel to the magnetic field B, or:
J=α(r)B (2.23)
where α(r) is a parameter which may vary over some space r. Three common forms of α are generally used in coronal field modelling. First, one can simply assume that α = 0, meaning there is no current, which is called the potential case. Second, α can be given a value which varies from one line of magnetic field to another, but is constant along each, called the linear force–free approximation (LFF). Finally, and most generally, α can be allowed any value at any point, called the non–linear force–free approximation (NLFF).
2.1.4.1 Potential Field Extrapolations
It is useful here to describe a common data analysis technique used to estimate the strength and topology of coronal magnetic fields, based on the force–free approximation of coronal plasma. Given a solar photospheric magnetogram such as those introduced in Section 3.2, magnetic fields can be extrapolated upwards into the corona, using either the potential, LFF or NLFF approximations. Here
(1982). An important property of potential magnetic fields is that, since the current density is zero, we have from Amp´ere’s Law:
∇ ×B= 0 (2.24)
which has the general solution
B =∇φ (2.25)
where ψ is a scalar potential. Now, from Gauss’ law for magnetism, we know that ∇ ·B= 0, and therefore φ satisfies the Laplace equation:
∇2φ = 0. (2.26)
With measured photospheric magnetic fields Bph from solar magnetograms, we
also have a boundary condition:
−nˆ· ∇φ(r) = Bph(r) (z = 0) (2.27)
where z and ˆn are the height above and unit vector normal to the photosphere, and r is the three–dimensional spatial coordinate of the volume above and in- cluding the photosphere, defined as wherez = 0. Finally, we apply the condition
φ(r)→0 as r→ ∞. (2.28)
If we define r0 as the location on the z = 0 surface such that r0 = (x0, y0,0), then given these conditions, the Green’s function G(r,r0) for this problem must satisfy:
∇2G(r,r0
) = 0 (z >0) (2.29)
G(r,r0)→0 as |r−r0| →0 (2.30)
−nˆ· ∇G(r,r0) = 0(r) (z = 0,r6=r0) (2.31)
Finally, we require that as we move towards the z = 0 surface, unit flux is maintained:
lim
z→0 Z
−nˆ· ∇G(r,r0)dS = 1 (2.32)
where dS is the surface element. These conditions require that −nˆ · ∇G(r,r0) is a delta function centred on r0:
−nˆ· ∇G(r,r0) =δ(r−r0) (2.33)
thereby establishing ourG(r,r0) as the Green’s function for our linear differential operator −nˆ· ∇. The explicit form of our Green’s function is then found to be
G(r,r0) = 1
2π|r−r0| (2.34)
which is mathematically equivalent to a representation of a magnetic monopole atr0.
B(r) =
i 2π|r−r
0
i|
3 (2.35)
whereφis the magnetic flux, and the subscript icorresponds to theith measured
magnetic charge. These charges can be purely simulated, or can be generated from a magnetogram by isolating regions of positive or negative flux based on a chosen threshold paramater. Each region is then simplified to a point with a location r0i and flux φi.
In order to determine topology based on these extrapolated field strengths, linkage between charges is found by tracing field lines rB using the equation
drB(s)
ds =
B[rB(s)]
|B[rB(s)]|
. (2.36)
An example of the resulting extrapolated field is shown in 2D in Figure 2.2 and in 3D in Figure 2.3. The former image represents a 2D slice of an mpole ex- trapolation of three colinear magnetic charges, while the latter shows a full 3D extrapolation in the case of a quadrupolar active region, produced by two parallel bipoles. Importantly, the topology produced by this code can be used to define separatrix surfaces, which are three–dimensional surfaces bordering regions of different magnetic connectivity. Separators occur at the intersection of these sur-
Figure 2.2: Potential magnetic field extrapolation from three colinear photo- spheric magnetic charges using the mpole algorithm. The solid horizontal line rep- resents the photosphere at z = 0, while curved solid lines represent extrapolated potential fields. Dotted lines represent the ‘mirror’ extrapolated fields underneath the photosphere, which can also be produced by the same method. However, the sub–surface field is generally treated as a cylindrical flux tube, as shown. Triangles represent the location of magnetic null points (Longcope, 1996).
faces, and so are lines in 3D space. Separators link magnetic null points, and play an important role in three–dimensional reconnecton theory, which we will cover in the following section.