Radiative temperature dissipation in a finite atmosphere -I. The homogeneous case

have developed a new approach toward solving problems of linear radiative relaxation of LTE temperature perturbations in a plane-parallel atmosphere of finite extent. We show that the mathematical problem is one of solving an integral eigenvalue equation, for which non-trivial solutions exist only for discrete values of the radiative relaxation time. The solutions for the spatial part of the perturbation constitute a complete and orthogonal set of basis functions, making it possible to solve more general problems of temperature relaxation. In applying this method to radiative relaxation in the middle atmosphere of earth. we show how the additional influences of photochemical coupling, advection by winds, and eddy diffusion by small-scale turbulence may be easily included using matrix perturbation techniques. We have solved the homogeneous integral equation for a wide variety of vertical thicknesses in an idealized homogeneous slab medium. Adopting a number of different analytic line profiles (rectangular, Doppler, Voigt, and Lorentz)we have obtained numerical solutions using an exponential-kernel method for solving the integral equation, The discrete eigenvalue "spectrum" is presented for vertical optical depths (O-10')at line-center, and is used in solving several initial-value problems for a decaying temperature perturbation. We find that the eigenvalue spectrum is bounded from above by the lowest-order eigenvalue, and bounded from below by the familiar transparent approximation. The dependence of the lowest even eigenvalue on optical depth and the relative separation of the higher eigenvalues are found to depend sensitively on the line profile.