6.1 Upward integration method

This straightforward method was proposed by Nakagawa (1974) and it has been first computationally implemented by Wu et al. (1985, 1990). The basic idea of this method is to reformulate Equations (2View Equation) – (4View Equation) and extrapolate the magnetic field vector into the solar corona. The method is not iterative and extrapolates the magnetic field directly upward, starting from the bottom layer, where the field is measured. From B (x,y,0) 0 one computes the z-component of the electric current μ j 0z0 by Equation (13View Equation) and the corresponding α-distribution with Equation (14View Equation). Then the x- and y-components of the electric current are calculated by Equation (11View Equation):
μ0jx0 = α0Bx0, (49 ) μ0jy0 = α0By0. (50 )
Finally, we get the z-derivatives of the magnetic field vector with Equations (3View Equation) and (4View Equation) as
∂B ∂B ---x0 = μ0jy0 + ---z0, (51 ) ∂z ∂x ∂By0- ∂Bz0- ∂z = ∂y − μ0jx0, (52 ) ∂B ∂B ∂B ---z0 = − ---x0 − ---y0. (53 ) ∂z ∂x ∂y
A numerical integration provides the magnetic field vector at the level z + dz. These steps are repeated in order to integrate the equations upwards in z. Naively one would assume to derive finally the 3D magnetic fields in the corona, which is indeed the idea of this method. The main problem is that this simple straightforward approach does not work because the method is mathematically ill-posed and the algorithm is unstable (see, e.g., Cuperman et al., 1990 and Amari et al., 1997Jump To The Next Citation Point for details). As a result of this numerical instability one finds an exponential growth of the magnetic field with increasing height. The reason for this is that the method transports information only from the photosphere upwards. Other boundary conditions, e.g., at an upper boundary, either at a finite height or at infinity cannot be taken into account. Several attempts have been made to stabilize the algorithm, e.g., by smoothing and reformulating the problem with smooth analytic functions (e.g., Cuperman et al., 1991; Démoulin and Priest, 1992; Song et al., 2006). Smoothing does help somewhat to diminish the effect of growing modes, because the shortest spatial modes are the fastest growing ones. To our knowledge the upward integration method has not been compared in detail with other NLFFF codes and it is therefore hard to evaluate the performance of this method.
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