3.1 Low and Lou’s (1990) equilibrium

As an example we refer to Low and Lou (1990Jump To The Next Citation Point), who solved the Grad–Shafranov equation in spherical coordinates (r,𝜃,φ ) for axisymmetric (invariant in φ) nonlinear force-free fields. In this case the magnetic field is assumed to be written in the form
( ) --1--- 1-∂A- ∂A- B = rsin 𝜃 r ∂𝜃 er − ∂r e𝜃 + Qe φ , (22 )
where A is the flux function, and Q represents the φ-component of the magnetic field B, which depends only on A. This ansatz automatically satisfies the solenoidal condition (6View Equation), and the force-free equation (5View Equation) reduces to a Grad–Shafranov equation for the flux function A
2 2 2 ∂-A-+ 1 −-μ-∂--A + Q dQ- = 0, (23 ) ∂r2 r2 ∂μ2 dA
where μ = cos𝜃. Low and Lou (1990) looked for solutions in the form
1+1∕n dQ 1∕n Q (A ) = λA (α = --- ∼ A ) (24 ) dA
with a separation ansatz
P (μ) A (r,𝜃) = -----. (25 ) rn
Here n and λ are constants and n is not necessarily an integer; n = 1 and λ = 0 corresponds to a dipole field. Then Equation (23View Equation) reduces to an ordinary differential equation for P (μ), which can be solved numerically. Either by specifying n or λ, the other is determined as an eigenvalue problem (Wolfson, 1995).The solution in 3D space is axisymmetric and has a point source at the origin. This symmetry is also visible after a transformation to Cartesian geometry as shown in Figure 5View Image(a). The symmetry becomes less obvious, however, when the symmetry axis is rotated with respect to the Cartesian coordinate axis; see Figures 5View Image(b)–(d). The resulting configurations are very popular for testing numerical algorithms for a 3D NLFFF modeling. For such tests the magnetic field vector on the bottom boundary of a computational box is extracted from the semi-analytic Low-Lou solution and used as the boundary condition for numerical force-free extrapolations. The quality of the reconstructed field is evaluated by quantitative comparison with the exact solution; see, e.g., Schrijver et al. (2006Jump To The Next Citation Point). Similarly one can shift the origin of the point source with respect to the Sun center and the solution is not symmetric to the Sun’s surface and can be used to test spherical codes.
  Go to previous page Go up Go to next page