Exponential Approximation for Daily Average Solar Heating or Photolysis

Abstract

Formulations of instantaneous solar heating or photolytic rates, for various atmospheric absorbers, as functions of altitude and sun angle are available in the literature. Such work can be integrated over the solar day to obtain the net daily effect. When incorporating these processes in long-range atmospheric forecasting models with large time steps or when assuming steady state, it may be desirable to replace the time integrals by daily average rates that are simple functions of latitude and season (sun declination). To accomplish this the integral over the solar day, which is shown to have the form of a modified exponential-integral function, is approximated by a pure exponential. This gives a daily average rate as a multiplication factor times the instantaneous rate evaluated at an appropriate sun angle. The multiplication factor and sun angle are analytically found by matching certain properties of the exponential-integral and exponential functions. The result is several choices for the fitting parameters in close analogy to the well-known a and b parameters for the substitute-kernel approximation. Even though the approximation is general, it is nonrational in that it does not become asymptotically accurate in a physically meaningful limit. Therefore its accuracy is investigated by a sample calculation using an instantaneous ozone heating formulation available in the literature. The approximation, with the different fitting parameters, is compared with the “exact” solar-day integrals for an appropriate range of altitude, latitude and season. It is shown that use of the approximation can easily lead to daily average rates that are within a few percent of “exact” values.