Journal of the Meteorological Society of Japan. Ser. II

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For solving one-dimensional linear partial differential equations numerically, several methods are studied on the accuracy, the stability and the nherited errors of solutions. First, the time extrapolation scheme is discussed, and the 3 time-level method (the usual centered difference) and one of the 4-level methods are concluded to give satisfactory results. In particular, the latter is numerically stable even for the equation of frictional type. Second, concerning the formulas of approximating the space derivative by finite difference, the 3 gridpoint and the 5 gridpoint methods are examined. The differential equations treated involves the advection term, where the advecting flow is spatially variable. Namely, the term is consisted of the product of two space-dependent quantities, which produces new waves by their interaction. It is shown that the usual computing method, i.e., the “3-3pt” method (3 levels for time and 3 points for space difference) gives arise the small scale errors, which turn out to be the ruggedness of the resulted pattern. On the contrary, the “3-5pt” method is stable and much accurate for the equation excluding the friction term and the “4-5pt” method for any kind of equations. The fictitious waves in small scale are extraordinarily small by these methods, The measures whereby one can expect the occurrence of the error are presented. They are derived from the difference between the differential and the difference equations, and they are related to the degree of the spatial undulation of the quantity concerned. Finally, using the techniques discussed, some sample calculations of 500 hour forecast are performed with respect to the patterns moving on a one-dimensional ringwise grid, and the numerical solutions are compared with the analytical solutions.

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In order to get fundamental and quantitative information on the primitive equation of motion, numerical analysis was performed for six ten-day periods by making use of six-hourly data of upper air observations at three stations in Japan. The average magnitude of each term in the equation is presented in Section 3 through 7 together with ten-day mean value of each term.
Inertia force due to horizontal motion, i.e. -V•∇V, is an important factor in the equation, and its magnitude in zonal direction is as much as in meridional direction at the levels lower than 400mb. Magnitude of -ω•∂V/∂p is somewhat small. Ten-day mean of -ω•∂u/∂p suggests that zonal momentum is transported downward on the average at northern Japan. Acceleration derived from metric force is generally small, but the treatment of this term in time-averaged equation of motion in meridional direction has to be done carefully. Coriolis force and pressure gradient are main factors in the equation. So far as this analysis is concerned, observed wind is subgeostrophic on the whole. Frictional acceleration in the free atmosphere was estimated as a residue of other terms in the equation. But no definite conclusion is obtained about it. Further study on frictional force is necessary.

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Assuming that the barotropic vorticity equation ∂ξ/∂t+V•∇(f+ξ)=0 is valid at the non-divergent level, the author discusses the displacement of a circular model perturbation superimposed on a basic motion and gives a generalized formula of Rossby's long wave theory.
This theory is applied to the displacement of typhoon in the subtropical zone and the author proposes a graphical forecasting method of typhoon movement.

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