VERTICAL SPECTRAL REPRESENTATION IN PRIMITIVE EQUATION MODELS OF THE ATMOSPHERE

Attempts to represent the vertical structure in primitive equation mod els of the atmosphere with the spectral method have been unsuccessful to date. The linear stability analysis of Francis showed that small ti me steps were required for computational stability near the upper boun dary with a vertical spectral method using Laguerre polynomials. Mache nhauer and Daley used Legendre polynomials in their vertical spectral representation and found it necessary to use an artificial constraint to force temperature to zero when pressure was zero to control the upp er-level horizontal velocities. This ad hoc correction is undesirable, and an analysis that shows such a correction is unnecessary is presen ted. By formulating the model in terms of velocity and geopotential an d then using the hydrostatic equation to calculate temperature from ge opotential, temperature is necessarily zero when pressure is zero. Thi s strategy works provided the multiplicative inverse of the first vert ical derivative of the vertical basis functions approaches zero more s lowly than pressure. The authors applied this technique to the dry-adi abatic primitive equations on the equatorial beta and tropical f plane s. Vertical and horizontal normal modes were used as the spectral basi s functions. The vertical modes are based on the vertical normal modes of Staniforth et al., and the horizontal modes are normal modes for t he primitive equations on a beta or f plane. The results show that the upper-level velocities do not necessarily increase, total energy is c onserved, and kinetic energy is bounded. The authors found an upper-le vel temporal oscillation in the horizontal domain integral of the hori zontal velocity components that is related to mass and velocity field imbalances in the initial conditions or introduced during the integrat ion. Through nonlinear normal-mode initialization, the authors effecti vely removed the initial condition imbalance and reduced the amplitude of this oscillation. It is hypothesized that the vertical spectral re presentation makes the model more sensitive to initial condition imbal ances, or it introduces imbalance during the integration through verti cal spectral truncation. It is also found slow spectral convergence pr operties for our vertical basis functions. It is concluded that a desi rable vertical basis set should have the following properties: 1) near ly uniform distribution of zeros and rapid spectral convergence; 2) ve rtical structure functions that are bounded at the upper boundary and a multiplicative inverse of the first derivative that goes to zero mor e slowly than pressure; and 3) expansions for derivatives of the basis functions that need to converge quickly.