  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 of our 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.
