A Check on the Validity of Magnetic Field Reconstructions

Abstract

We investigate a method to test whether a numerically computed model coronal magnetic field ${\boldsymbol {B}}$ departs from the divergence-free condition (also known as the solenoidality condition). The test requires a potential field ${\boldsymbol {B}}_{0}$ to be calculated, subject to Neumann boundary conditions, given by the normal components of the model field ${\boldsymbol {B}}$ at the boundaries. The free energy of the model field may be calculated using $\frac{1}{2\mu _{0}}\int ({\boldsymbol {B}}-{\boldsymbol {B}}_{0})^{2}\mathrm{d}V$ , where the integral is over the computational volume of the model field. A second estimate of the free energy is provided by calculating $\frac{1}{2\mu _{0}}\int {\boldsymbol {B}}^{2}\,\mathrm{d}V-\frac{1}{2\mu _{0}}\int {\boldsymbol {B}}_{0}^{2}\,\mathrm{d}V$ . If ${\boldsymbol {B}}$ is divergence free, the two estimates of the free energy should be the same. A difference between the two estimates indicates a departure from $\nabla \cdot {\boldsymbol {B}}=0$ in the volume. The test is an implementation of a procedure proposed by Moraitis et al. (Solar Phys. 289, 4453, 2014) and is a simpler version of the Helmholtz decomposition procedure presented by Valori et al. (Astron. Astrophys. 553, A38, 2013). We demonstrate the test in application to previously published nonlinear force-free model fields, and also investigate the influence on the results of the test of a departure from flux balance over the boundaries of the model field. Our results underline the fact that, to make meaningful statements about magnetic free energy in the corona, it is necessary to have model magnetic fields that satisfy the divergence-free condition to a good approximation.