NUMERICAL INTEGRATION OF THE PRIMITIVE EQUATIONS WITH A FLOATING SET OF COMPUTATION POINTS: EXPERIMENTS WITH A BAROTROPIC GLOBAL MODEL1

Abstract

A Lagrangean-type numerical forecasting method is developed in which the computational (grid) points are advected by the wind and the necessary space derivatives (in the pressure gradient terms, for example) are computed using the values of the variables at all the computation points that at the particular moment are within a prescribed distance of the point for which the computation is done. In this way, the forecasting problem reduces to solving the ordinary differential equations of motion and thermodynamics for each computation point, instead of solving the partial differential equations in the Eulerian or classical Lagrangean way. The method has some advantages over the conventional Eulerian scheme: simplicity (there are no advection terms), lack of computational dispersion in the advection terms and therefore better simulation of atmospheric advection and deformation effects, very little inconvenience due to the spherical shape of the earth, and the possibility for a variable space resolution if desired. On the other hand, some artificial smoothing may be necessary, and it may be difficult (or impossible) to conserve the global integrals of certain quantities. A more detailed discussion of the differencing scheme used for the time integration is included in a separate section, This is the scheme obtained by linear extrapolation of computed time derivatives to a time value of t0 + aΔt where t0 is the value of time at the beginning of the considered time step Δt and where a is a parameter that can be used to control the properties of the scheme. When choosing a value of a between ½ and 1, a scheme is obtained that damps the high-frequency motions, in a similar way as the Matsuno scheme, but needs somewhat less computer time and, with the same damping intensity, has a higher accuracy for low-frequency meteorologically significant motions. Using the described method, a 4-day experimental forecast has been made, starting with a stationary Haurwitz-Neamtan solution, for a primitive equation, global, and homogeneous model. The final geopotential height map showed no visible phase errors and only a modest accumulation of truncation errors and effects of numerical smoothing mechanisms. Two shorter experiments have also been made to analyze the effects of space resolution and damping in the process of time differencing. It is felt that the experimental results strongly encourage further testing and investigation of the proposed method.