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

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