Static and dynamic solar coronal loops with cross-sectional area variations

The Enthalpy Based Thermal Evolution of Loops (EBTEL) approximate model for static and dynamic coronal loops is developed to include the effect of a loop cross-sectional area which increases from the base of the transition region (TR) to the corona. The TR is defined as the part of a loop between the top of the chromosphere and the location where thermal conduction changes from an energy loss to an energy gain. There are significant differences from constant area loops due to the manner in which the reduced volume of the TR responds to conductive and enthalpy fluxes from the corona. For static loops with modest area variation the standard picture of loop energy balance is retained, with the corona and TR being primarily a balance between heating and conductive losses in the corona, and downward conduction and radiation to space in the TR. As the area at the loop apex increases, the TR becomes thicker and the density in TR and corona larger. For large apex areas, the coronal energy balance changes to one primarily between heating and radiation, with conduction playing an increasingly unimportant role, and the TR thickness becoming a significant fraction of the loop length. Approximate scaling laws are derived that give agreement with full numerical solutions for the density, but not the temperature. For non-uniform areas, dynamic loops have a higher peak temperature and are denser in the radiative cooling phase by of order 50% than the constant area case for the examples considered. They also show a final rapid cooling and draining once the temperature approaches 1 MK. Although the magnitude of the emission measure will be enhanced in the radiative phase, there is little change in the important observational diagnostic of its temperature dependence.