Spectral Representation of the Vertical Coordinate in Three-Dimensional Atmospheric Models on Tropical Beta - and F-Planes
Abstract
This research examines the application of the spectral method to the vertical coordinate of atmospheric models. Specifically, we use the vertical normal modes as basis functions in spectral expansion of the vertical structure of dependent variables in an equatorial beta-plane and a tropical f-plane model of the atmosphere. Previous attempts by Francis (1972) and Machenhauer and Daley (1972) found that use of vertical spectral methods was associated with an unrealistic acceleration of velocities near the top of the model. Machenhauer and Daley (1972) showed that this acceleration was associated with temperature not zero at the upper boundary where pressure is zero. They concluded that artificial constraints on temperature were necessary to control the upper-level accelerations. We show that artificial constraints are not required when the model uses prognostic equations for velocity and geopotential instead of velocity and temperature. We use the vertical normal modes of Staniforth et al. (1985), and our analysis shows that these vertical structure functions are bounded while all derivatives are unbounded at the upper boundary. Further analysis shows that the boundary conditions from the vertical structure problem ensure that temperature vanishes at the upper boundary. This analysis suggests that any basis set, for which the inverse of the first derivative approaches zero more slowly than pressure at the upper boundary, is suitable for the vertical spectral expansion. Using these ideas we coded tropical beta - and f-plane models using the basic state temperature and vertical normal modes of Staniforth et al. (1985). During model development we encountered the following problems. (1) The meridional grid is dependent upon the vertical mode. (2) The vertical quadrature is problematic due to presence of integrand derivative singularities. (3) The convergence of the vertical spectral expansion is slow due to the singularity at the upper boundary. (4) The projection of vertical derivative quantities is problematic because derivatives of the vertical basis function are not elements of the basis set. Each problem is addressed in this thesis, and the results from numerical simulations of these models show that the upper level velocities do not accelerate unrealistically with this formulation. However, our results show that the vertical spectral method makes the model more sensitive to mass and velocity field imbalances present in the initial conditions or introduced during the integration.
- Publication:
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Ph.D. Thesis
- Pub Date:
- 1994
- Bibcode:
- 1994PhDT.......147M
- Keywords:
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- Physics: Atmospheric Science; Mathematics