### 3.1 Low and Lou’s (1990) equilibrium

As an example we refer to Low and Lou (1990), who solved the Grad–Shafranov equation in spherical
coordinates for axisymmetric (invariant in ) nonlinear force-free fields. In this case the
magnetic field is assumed to be written in the form
where is the flux function, and represents the -component of the magnetic field , which
depends only on . This ansatz automatically satisfies the solenoidal condition (6), and
the force-free equation (5) reduces to a Grad–Shafranov equation for the flux function
where . Low and Lou (1990) looked for solutions in the form
with a separation ansatz
Here and are constants and is not necessarily an integer; and corresponds to a
dipole field. Then Equation (23) reduces to an ordinary differential equation for , which can be
solved numerically. Either by specifying or , 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 5(a). The
symmetry becomes less obvious, however, when the symmetry axis is rotated with respect to the Cartesian
coordinate axis; see Figures 5(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. (2006). 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.