Abstract
In an attempt to understand the implications of Long's “generalized dimensional analysis,” this method was applied to the problem of the wind distribution in the planetary boundary layer. The assumption was made that the equations of motion, together with an appropriate set of boundary conditions, define a unique relationship among the wind, the stress, and certain other variables and parameters. This relationship was found to be more precisely specified by the generalized analysis than by ordinary dimensional methods, although when the solution is required to reduce to the logarithmic wind profile near the ground both procedures give identical results and yield a universal relationship among the latitude, the surface roughness, the stress, the geostrophic wind, and the depth of the planetary layer, which is remarkably similar to one found by Rossby and Montgomery by a completely different argument. That such a result may be found by purely dimensional reasoning is taken as an indication of the power of the dimensional method.
The solution achieved was possible only through the artifice of treating all vectors as if they were vectors in two dimensions only. The dimensional method in its present form does not appear capable of treating the more complete vector problem, although there are indications that when the foundations of dimensional analysis, which lie in invariance theory, are better understood, such problems too will lend themselves to solution by dimensional reasoning.
* Paper presented at the 237th National Meeting of the American Meteorological Society, April 19–22, 1965, Washington, D.C.
