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# ELEMENTARY THEORY OF ASSOCIATIONS BETWEEN ATMOSPHERIC MOTIONS AND DISTRIBUTIONS OF WATER CONTENT

EDWIN KESSLER III

## Abstract

Continuity equations are used to clarify relationships between air motions and distributions of accompanying precipitation. The equations embody simple modeling of condensation and evaporation with the following assumptions: (1) water vapor shares the motion of the air in all respects; (2) condensate shares horizontal air motion, but falls relative to air at a speed that is the same for all the particles comprising precipitation at a particular time and height; (3) the cloud phase is omitted.

After a review of one-dimensional models, the distributions of condensate in two-dimensional model wind fields are discussed with regard to instantaneous evaporation of condensate in unsaturated air and to no evaporation. The most nearly natural cases must lie between these extremes. The methods for obtaining solutions are instructive of basic interactions between air motion and water transport. The steady-state precipitation rate from a saturated horizontally uniform updraft column is shown to equal the sum of the vertically integrated condensation rate and a term that contains the horizontal divergence of wind. The latter term becomes relatively small as the ratio of precipitation fall speeds to updrafts becomes large. A basis for some studies of precipitation mechanisms, the equation N(V + w) = const., where N is the number of particles comprising precipitation at a particular point in space and time, V is their fall velocity, and w is the updraft, is shown to imply violation of continuity principles unless variations in w are quite small. Continuity equations are applied to radar-observed convective cells (generators) and their precipitation trails, and to radar-observed precipitation pendants (stalactites), and provide bases for estimating the strength, duration, and vertical extent of the associated vertical air currents. The stalactite study also discloses how horizontal variations of precipitation intensity arise during precipitation descent through a saturated turbulent atmosphere.

The continuity equations are powerful tools for illuminating fundamental properties of wind-water relationships. The conclusion discusses attractive paths along which this work should be extended.

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