ANALYSIS OF THE U AND V FIELDS IN THE VICINITY OF THE POLES1

The fields of the eastward (V) and northward (V) components of the wind are investigated mathematically. Rules are given to Maps whose point values are the products of the point valnes of two maps are also investigated. aid in the analysis of these types of maps. 1. THE PROBLEM The use of polar coordinate charts means that the niaps of U and V, the eastward and northward components of wind, have mathematical singularities at the poles. The existence of these singularities has been eithcr completely ignored or circumvented b2- some device such as, for example, not analyzing near the poles. A strict niat,hematical analysis of the li and V fields in the vicinity of one of these poles reveals information which has proved to be of great assistance to the analyst. Application of our rules considerably improved the analysis of a series of Northern Hemisphere niaps (Barnes [I]) in the region north of 75" N. Even though the following matheiiiaticnl development is around the North Pole, it is applicable to the pole of any polar coordinate systeni such as, for example, those frequently employed in special studies of hurricanes and extratropical storms where the radial, V, and tangential, U, velocity components are used. One of the ways of representing the horizontal wind field is to use the south to north component, V, and the west to east component, U. This representation is unique except at the north and south poles. A closer inspection reveals that, under the U, Vdecornposition, one is mapping a cylinder onto a spherical surface (e.g., 100-mb. surface). This mapping of a cylinder onto a coaxial spherical surface is unique (one-to-one) except at the poles. The top and bottom lines of the cylinder are mapped onto the poles so the mapping is not one-to-one at these points. Thus, special consideration must be given to quantities containing U or V in the neighborhoods of the north pole and the south pole.