Continuity equations as expressions for local balances of masses in cloudy air

Abstract

The mathematical representation of the mass continuity equation and a boundary condition for thevertical velocity at the earth’s surface is re-examined in terms of its dependence on the frame ofreference velocity. Three of the most prominent meteorological examples are treated here: (a) thebarycentric velocity of a full cloudy air system, (b) the barycentric velocity of a mixture consisting ofdry air andwater vapour and (c) the velocity of dry air. Although evidently the physical foundation holdsindependently of the choice of a particular frame, the resulting equations differ in their mathematicalstructure: In examples (b) and (c) the diffusion flux divergence that appears in the corresponding massequation of continuity should not be omitted a priori. As to the lower boundary condition for the normalcomponent of velocity, special emphasis is placed on the net mass transfer across the earth’s surfaceresulting from precipitation and evaporation. It is shown that for a flat surface, the reference verticalvelocity vanishes only in case (c). Regarding cases (a) and (b), the vertical reference velocities aredetermined as functions of the precipitation and evaporation rates. They are nonzero, and it is shownthat they cannot generally be neglected.