A Study of Numerical Errors in the Integration of Barotropic Flow on a Spherical Grid

The barotropic vorticity equation for horizontal nondivergent flow is integrated numerically over the northern hemisphere on a spherical grid and compared with the known analytic solution for harmonic initial conditions. At latitudes ≤45°N, points spaced each 5° lat. and 5° long. are used, the longitudinal mesh increment being increased to 10°, 15°, 20°, 30°, and 60° long. at higher latitudes. The pole is treated separately in surface spherical coordinates. In these solutions the truncation and relaxation errors have been isolated without the usual errors of initial data and boundary conditions. Variations of the over-relaxation coefficient α. from 0.28 to 0.44 and of the residue tolerance ∈ by a factor of 100 produced approximately 8 per cent variation in the magnitude of the maximum tendency error at t = 0. In a series of integrations to 48 hr, the value of the time increment Δt was varied from 1/4 hr to 3 hr, producing a variation in the magnitude of the maximum error of approximately 5 per cent. By far the largest part of the total numerical error is due to the space truncation of the finite-difference grid. This error in the 5° grid produces a phase lag of the predicted solution relative to the true solution of approximately 1° long. per day for the case of six symmetrical hemispheric waves moving eastward at 20° long. per day. A parallel integration with a 10° grid of similar geometry increased this truncation phase lag to approximately 4° long. per day.