An analysis of the vertical structure equation for arbitrary thermal profiles

Abstract

The vertical structure equation is a singular Sturm-Liouville problem whose eigenfunctions describe the vertical dependence of the normal modes of the primitive equations linearized about a given thermal profile. The eigenvalues give the equivalent depths of the modes. We study, for arbitrary thermal profiles, the spectrum of the vertical structure equation and the appropriateness of various upper boundary conditions. Our results depend critically upon whether or not the thermal profile is such that the basic state atmosphere is bounded. This is not surprising since, as we point out, the vertical structure equation is not meaningful at large heights because of the traditional shallowness approximations which are used to derive the primitive equations. The nature of the spectrum of a singular Sturm-Liouville problem depends only on the behaviour of the coefficients of the differential equation near the singular boundary. Spectral results therefore have no physical significance for unbounded atmospheres. For all bounded atmospheres we show that the spectrum is totally discrete, regardless of details of the thermal profile. For the barotropic equivalent depth, which corresponds to the lowest eigenvalue, we obtain upper and lower bounds which depend only on the surface temperature and the atmosphere height. All eigenfunctions are bounded, but always have first derivatives which become unbounded near the top. We prove that the commonly invoked upper boundary condition that vertical velocity must vanish as pressure tends to zero, as well as a number of alternative conditions, are well posed. For unbounded atmospheres, on the other hand, we show that typically there is a continuous spectrum, that the boundary condition of vanishing vertical velocity is not well posed, and that the eigenfunctions. if any. are unbounded.