2 Linear Force-Free Fields

Linear force-free fields are characterized by

∇ × B = αB, (15 ) ∇ ⋅ B = 0, (16 )
where the force-free parameter α is constant. Taking the curl of Equation (15View Equation) and using the solenoidal condition (16View Equation) we derive a vector Helmholtz equation:
2 ΔB + α B = 0 (17 )
which can be solved by a separation ansatz, a Green’s function method (Chiu and Hilton, 1977Jump To The Next Citation Point) or a Fourier method (Alissandrakis, 1981). These methods can also be used to compute a potential field by choosing α = 0.

For computing the solar magnetic field in the corona with the linear force-free model one needs only measurements of the LOS photospheric magnetic field. The force-free parameter α is a priori unknown and we will discuss later how α can be approximated from observations. Seehafer (1978Jump To The Next Citation Point) derived solutions of the linear force-free equations (local Cartesian geometry with (x, y) in the photosphere and z is the height from the Sun’s surface) in the form:

∑∞ C [ πn ( πmx ) (πny ) Bx = -mn-exp (− rmnz ) ⋅ α---sin ----- cos ---- − m,n=1 λmn Ly Lx Ly ( ) ( ) ] − rmn πm--cos πmx-- sin πny- , (18 ) Lx Lx Ly ∑∞ C [ πm ( πmx ) ( πny ) By = − --mn-exp(− rmnz ) ⋅ α----cos ----- sin ---- + m,n=1 λmn Lx Lx Ly ( ) ( )] +rmn πn-sin πmx-- cos πny- , (19 ) Ly Lx Ly ∑∞ ( ) ( ) Bz = Cmn exp (− rmnz ) ⋅ sin πmx-- sin πny- , (20 ) m,n=1 Lx Ly
with λmn = π2(m2 ∕L2x + n2∕L2y) and √ --------- rmn = λmn − α2.

As the boundary condition, the method uses the distribution of Bz (x,y) on the photosphere z = 0. The coefficients C mn can be obtained by comparing Equation (20View Equation) for z = 0 with the magnetogram data. In practice, Seehafer’s (1978Jump To The Next Citation Point) method is used for calculating the linear force-free field (or potential field for α = 0) for a given magnetogram (e.g., MDI on SOHO) and a given value of α as follows. The observed magnetogram which covers a rectangular region extending from 0 to Lx in x and 0 to Ly in y is artificially extended onto a rectangular region covering − Lx to Lx and − Ly to Ly by taking an antisymmetric mirror image of the original magnetogram in the extended region, i.e.,

B (− x,y) = − B (x, y), z z Bz(x,− y) = − Bz (x, y), B (− x,− y) = B (x,y) (0 < x < L ,0 < y < L ). z z x y
This makes the total magnetic flux in the whole extended region to be zero. (Alternatively one may pad the extended region with zeros, although in this case the total magnetic flux may be non-zero.) The coefficients C mn are derived from this enlarged magnetogram with the help of a Fast Fourier Transform. In order for rmn to be real and positive so that solutions (18View Equation) – (20View Equation) do not diverge at infinity, 2 α should not exceed the maximum value for given Lx and Ly,
( 1 1 ) α2max = π2 ---+ --- . L2x L2y

Usually, α is normalized by the harmonic mean L of Lx and Ly defined by

( ) -1- = 1- -1-+ 1-- . L2 2 L2x L2y

For Lx = Ly we have L = Lx = Ly. With this normalization the values of α fall into the range √ -- √ -- − 2π < α < 2π.

View Image

Figure 4: How to obtain the optimal linear force-free parameter α from coronal observations. Image reproduced by permission from Figures 3, 4, and 5 of Carcedo et al. (2003Jump To The Next Citation Point), copyright by Springer.
 2.1 How to obtain the force-free parameter α

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