Vertical Normal Mode Transforms: Continuous vs. Discrete

In a hydrostatic atmosphere it is often convenient to separate the horizontal and vertical dependence of various fields. Solutions of the vertical structure equation (a Sturm-Liouville problem involving the basic state temperature profile) are referred to as vertical normal modes, and expansions in terms of these modes may be formalized as vertical normal mode transforms. With a rigid-lid boundary condition at a positive top pressure, the vertical structure problem is regular, so the spectrum of vertical normal modes is discrete and the associated transform is an infinite series. However, if the top pressure is zero, the vertical structure problem is singular; the resulting spectrum may be partly continuous, in which case the associated transform involves an integral. The situation is analogous to the Fourier case: on a finite interval the Fourier "transform" is an infinite (Fourier) series, whereas on an infinite interval it is an integral.

This paper addresses two questions. First, what distinguishes between a discrete spectrum and a continuous spectrum when the top pressure is zero? We show that the key is the limit of the basic state temperature as pressure goes to zero: if this limit is zero then the spectrum is discrete, and if it is positive, then the spectrum is continuous. Since the latter case is more realistic, this leads to the second question: how well can continuous transforms (with top pressure zero) be approximated by discrete transforms (with a lid at a positive pressure)? To address this, we compare the continuous and discrete transforms for the case of constant static stability in log-pressure coordinates, and examine the distribution of energy among the normal modes in the limit as the top pressure goes to zero.