Global Modeling of Atmospheric Flow by Spectral Methods

Computational Aspects of Numerical Models for Weather Prediction and Climate Simulation

A comparison is made of integrations of the primitive equations on the sphere using a second-order finite-difference model, and spectral models with both triangular and rhomboidal truncation. The initial conditions comprise a baroclinically unstable mid-latitude jet, and a perturbation of small amplitude. Increasing the resolution in each model gives a good estimate of the exact solution, and thus of the errors involved in each integration.

No one method has a superiority in all respects. Spectral integrations with truncations at zonal wave-number 16 give a more accurate description of amplitudes and phases in the growing wave than does a finite-difference integration using a 5° × 3° grid, but a much poorer description of the fronts that form as the disturbance approaches maturity. Large-scale changes to the zonal-mean state are predicted to greater accuracy using the spectral models, but smaller-scale changes are resolved better by the finite-difference model. The spectral models require less computing time, and less storage. For these experiments, rhomboidal truncation is favoured.

A comparison of spectral and finite-difference simulations of a growing baroclinic wave

The f-plane semi-geostrophic equations have provided much insight into atmospheric motions and we wish to see if more can be gained by removing the f-plane restriction. As a first step we begin by considering the shallow-water semi-geostrophic equations on a sphere and describe a numerical solution procedure for them based on a semi-Lagrangian predictor-corrector method. It is demonstrated, numerically, that the method appears to converge to a true solution of the semi-geostrophic equations for fixed resolution, and this suggests that there is a unique solution to the continuous equations. The results from several idealized and real-data test problems are presented and show that the scheme is stable and accurate. The results are compared with those obtained from a shallow-water semi-Lagrangian primitive-equation model. The simulations are broadly similar but the semi-geostrophic solution maintains stronger anticyclones than the primitive-equation model. This is in agreement with results expected after considering f-plane baroclinic wave simulations with the effects of spherical geometry taken into account.

A shallow‐water semi‐geostrophic model on a sphere

Smooth-Particle Hydrodynamics as Applied to 2-D Plume Convection

Integrations of the shallow water equations on the sphere using a second order finite difference method and a spectral method are compared. By increasing the resolution, a good estimate of the exact solution may be made, thus allowing an estimate of the accuracy of each integration. The particular initial conditions used are a Rossby wave of zonal wavenumber 4 which moves with little change in shape or amplitude and a Rossby wave of zonal wavenumber 8 which undergoes large changes within 5 days. All the models perform reasonably for the wavenumber 4 integration. A 5° times; 3° grid for the finite difference simulation is insufficient to resolve the breakdown in wavenumber 8 despite there being 9 points per zonal wavelength. A spectral model with a truncation at wavenumber 16 uses less storage and takes the same computing time as the grid-point model. However, it is able to predict quite accurately the breakdown.

A comparison of grid‐point and spectral methods in a meteorological problem

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A comparison of grid‐point and spectral methods in a meteorological problem