Scale Consideration of Planetary Motions of the Atmosphere¹

1 This work was carried out while attached to the South African Weather Bureau, Pretoria. It was also presented as a lecture at the conference on Numerical Weather Prediction in Stockholm, in June, 1957.

Abstract

The application of scale considerations to atmospheric motions indicates that the equations hitherto used in numerical prediction are unfit for detailed prognosis on the planetary (or very large) scale.

It appears that the dominant terms of the vorticity equation in this case are the divergence term and the β-term, which means that the vorticity equation here displays the same quasistationary character as the undifferentiated equations of motion. An attempt at expressing more than the quasistationary character of the planetary flow leads to undue complication of the equation, even requiring inclusion of the twisting terms.

As for the application of the geostrophic approximation, it is confirmed that the β-term in the divergence equation is important in the representation of the planetary scale. Further, though metrical terms (containing the reciprocal of the earth's radius) are important on this scale in spherical coordinates, it is shown that, in using the geostrophic approximation and a square grid on a conformal plane projection of the earth's surface, no metrical terms appear.

Remarks are also included on a relation of Prandtl, which loses its validity on the planetary scale, and also on the use of the Richardson number to characterize large-scale atmospheric motion.

Notes

1 This work was carried out while attached to the South African Weather Bureau, Pretoria. It was also presented as a lecture at the conference on Numerical Weather Prediction in Stockholm, in June, 1957.