Determining the Transport of Magnetic Helicity and Free Energy in the Sun's Atmosphere

The most important factors determining solar coronal activity are believed to be the availability of magnetic free energy and the constraint of magnetic helicity conservation. Direct measurements of the helicity and magnetic free energy in the coronal volume are difficult, but their values may be estimated from measurements of the helicity and free energy transport rates through the photosphere. We examine these transport rates for a topologically open system such as the corona, in which the magnetic fields have a nonzero normal component at the boundaries, and derive a new formula for the helicity transport rate at the boundaries. In addition, we derive new expressions for helicity transport due to flux emergence/submergence versus photospheric horizontal motions. The key feature o four formulas is that they are manifestly gauge invariant. Our results are somewhat counterintuitive in that only the lamellar electric field produced by the surface potential transports helicity across boundaries, and the solenoidal electric field produced by a surface stream function does not contribute to the helicity transport. We discuss the physical interpretation of this result. Furthermore, we derive an expression for the free energy transport rate and show that a necessary condition for free energy transport across a boundary is the presence of a closed magnetic field at the surface, indicating that there are current systems within the volume. We discuss the implications of these results for using photospheric vector magnetic and velocity field measurements to derive the solar coronal helicity and magnetic free energy, which can then be used to constrain and drive models for coronal activity.