Vertical Spectral Representation in Primitive Equation Models of the Atmosphere

Abstract

Attempts to represent the vertical structure in primitive equation models of the atmosphere with the spectral method have been unsuccessful to date. The linear stability analysis of Francis showed that small time steps were required for computational stability near the upper boundary with a vertical spectral method using Laguerre polynomials. Machenhauer 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 upper-level horizontal velocities. This ad hoc correction is undesirable, and an analysis that shows such a correction is unnecessary is presented. By formulating the model in terms of velocity and geopotential and then using the hydrostatic equation to calculate temperature from geopotential, temperature is necessarily zero when pressure is zero. This strategy works provided the multiplicative inverse of the first vertical derivative of the vertical basis functions approaches zero more slowly than pressure. The authors applied this technique to the dry-adiabatic primitive equations on the equatorial β and tropical f planes. Vertical and horizontal normal modes were used as the spectral basis functions. The vertical modes are based on the vertical normal modes of Staniforth et al., and the horizontal modes are normal modes for the primitive equations on a β or f plane. The results show that the upper-level velocities do not necessarily increase, total energy is conserved, and kinetic energy is bounded. The authors found an upper-level temporal oscillation in the horizontal domain integral of the horizontal velocity components that is related to mass and velocity field imbalances in the initial conditions or introduced during the integration. Through nonlinear normal-mode initialization, the authors effectively removed the initial condition imbalance and reduced the amplitude of this oscillation. It is hypothesized that the vertical spectral representation makes the model more sensitive to initial condition imbalances, or it introduces imbalance during the integration through vertical spectral truncation. It is also found slow spectral convergence properties for our vertical basis functions. It is concluded that a desirable vertical basis set should have the following properties: 1) nearly uniform distribution of zeros and rapid spectral convergence; 2) vertical structure functions that are bounded at the upper boundary and a multiplicative inverse of the first derivative that goes to zero more slowly than pressure, and 3) expansions for derivatives of the basis functions that need to converge quickly.