A leaky rigid lid

Various models of the atmosphere use the rigid lid approximation. In reality the atmosphere does not have a definite top. Furthermore, mathematically the two problems are drastically different. In the study of linear problem in hydrostatic balance the rigid lid approximation leads to a problem with discrete spectrum, while a semi-infinite atmosphere problem has a continuous one. The energy that is projected onto the continuous part of the spectrum is disregarded in the rigid lid approach. We propose to fix this ``leak'' problem via changing the boundary conditions at the top of the rigid lid.