Prediction of the Standard Atmosphere Profiles of Temperature, Pressure, and Density with Height for the Lower Atmosphere by Solution of the (S-S) Integral Equations of Transfer and Evaluation of the Potential for Profile Perturbation by Combustion Emiss

摘要

This analytical solution, believed to be original here, to the 1D formulation of the (1905-1906) integral (S—S) Equations of Transfer, governing radiation through the atmosphere, is developed for future evaluation of the potential impact of combustion emissions on climate change. The solution predicts, in agreement with the Standard Atmosphere experimental data, a linear decline of the fourth power of the temperature, T~4, with pressure, P, and, at a first approximation, a linear decline of T with altitude, h, up to the tropopause at about 10 km (the lower atmosphere). From these two results, with transformation using the Equation of State, the variations of pressure, P, and density, ρ, with altitude, h, are also then obtained, with the predictions again, separately, in substantial agreement with the Standard Atmosphere data up to 30 km altitude (1% density). The analytical procedure adopts the standard assumptions commonly used for numerical solutions of steady state, one dimensionality, constant flux directional parameter (μ), and a gray-body equivalent average for the effective radiation absorption coefficient, k, for the mixed thermal radiation-active gases at an effective (joint-mixture) concentration, p. Using these assumptions, analytical closure and validation of the equation solution is essentially complete. Numerical closure is not yet complete, with only one parameter at this time not independently calculated but not required numerically for validation of analytical closure. This is the value of the group-pair (kp)_o representing the ground-level value of (kp), the product of the effective absorption coefficient and concentration of the mixed gases, written as a single parameter but decomposable into constituent gases and/or gas bands. Reduction of the experimental value of (kp)_o to values of k for a comparison with relevant band data for water and CO_2 shows numerical magnitudes substantially matching the longest wavelength bands for each of the two gases. Allowing also for the maximum absorption percentages, α°, of these two bands for the two gases, respectively, 39% for water and 8.5% for CO_2, these values then support the dominance of water (as gas and not vapor) at about 80%, compared with CO_2 at about 20%, as the primary absorbing/ emitting ("greenhouse") gas in the atmosphere. These results provide a platform for future numerical determination of the influence on the T, P, and ρ profiles of perturbations in the gas concentrations of the two primary species, carbon dioxide and water, and it provides, specifically, the analytical basis needed for future analysis of the impact potential from increases in atmospheric carbon dioxide concentration, because of fossil-fuel combustion, in relation to climate change.