4.6 Consistency criteria for force-free boundary conditions

After Stokes inversion (see Section 4.1) and azimuth ambiguity removal, we derive the photospheric magnetic field vector. Unfortunately there might be a problem, when we want to use these data as the boundary condition for NLFFF extrapolations. Due to Metcalf et al. (1995), the solar magnetic field is not force-free in the photosphere (finite β plasma), but becomes force-free only at about 400 km above the photosphere. This is also visible in Figure 2View Image from Gary (2001), which shows the distribution of the plasma β value with height. Consequently, the assumption of a force-free magnetic field is not necessarily justified in the photosphere. Unless we have information on the magnetic flux through the lateral and top boundaries, we have to assume that the photospheric magnetic flux is balanced
∫ Bz(x,y,0 )dxdy = 0, (30 ) S
which is usually the case when taking an entire active region as the field-of-view.

In the following, we review some necessary conditions the magnetic field vector has to fulfill in order to be suitable as boundary conditions for NLFFF extrapolations. Molodenskii (1969); Molodensky (1974Jump To The Next Citation Point) and Aly (1989Jump To The Next Citation Point) defined several integral relations, which are related to two moments of the magnetic stress tensor.

  1. The first moment corresponds to the net magnetic force, which has to vanish on the boundary:
    ∫ ∫ BxBzdxdy = ByBzdxdy = 0, (31) S∫ S ∫ 2 2 2 S(Bx + B y)dxdy = S B zdxdy. (32)
  2. The second moment corresponds to a vanishing torque on the boundary:
    ∫ ∫ 2 2 2 x(B x + B y)dxdy = xB zdxdy, (33) ∫S ∫S y(B2x + B2y)dxdy = yB2zdxdy, (34) S ∫ ∫S yBxBzdxdy = xByBzdxdy. (35) S S

The total energy of a force-free configuration can be estimated directly from boundary conditions with the help of the virial theorem (see, e.g., Aly, 1989Jump To The Next Citation Point, for a derivation of this formula)

∫ 1-- Etot = μ (xBx + yBy)Bzdxdy. (36 ) 0 S
For Equation (36View Equation) to be applicable, the boundary conditions must be compatible with the force-free assumption. If the integral relations (31View Equation) – (35View Equation) are not fulfilled then the data are not consistent with the assumption of a force-free field. A principal way to avoid this problem would be to measure the magnetic field vector in the low-β chromosphere, but unfortunately such measurements are not routinely available. We have therefore to rely on photospheric measurements and apply some procedure, dubbed ‘preprocessing’, in order to derive suitable boundary conditions for NLFFF extrapolations. As pointed out by Aly (1989) the condition that α is constant on magnetic field lines (12View Equation) leads to the integral relation
∫ ∫ f (α )Bn ⋅ dA = f(α )Bn ⋅ dA, (37 ) S+ S−
where S+ and S− correspond to areas with positive and negative Bz in the photosphere, respectively, and f is an arbitrary function. Condition (37View Equation) is referred to as differential flux-balance condition as it generalizes the usual flux-balance condition (30View Equation). As the connectivity of magnetic field lines (magnetic positive and negative regions on the boundary connected by field lines) is a priori unknown, relation (37View Equation) is usually only evaluated after a 3D force-free model has been computed.
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