The local mass balance equations of cloudy air are formulated for a model system composed of dry air, water vapor, and four categories of water condensate particles, as typically adopted for numerical weather prediction and climate models. The choice of the barycentric velocity as reference motion provides the most convenient form of the total mass continuity equation. Mass transfer across the earth’s surface due to precipitation and evaporation causes a nonvanishing barycentric vertical velocity ws and is proportional to the local difference between evaporation rate and rain plus snow rate. Hence ws vanishes only in the special situation that evaporation and precipitation balance exactly. Alternative concepts related to different reference motions are reviewed. However, the choice of the barycentric velocity turns out to be advantageous for several reasons. The implication of the nonvanishing total mass transport across the earth’s surface is estimated from model simulations for two extreme weather situations: a polar cold air outbreak and a tropical cyclone. While the effect is small in the first case, it is important in the latter. The large precipitation rates in the tropical cyclone case cause a loss of atmospheric mass, which corresponds to a vertical velocity at the surface larger than −20 mm s−1, and an instantaneous drop in pressure, which if sustained for 6 h, would correspond to about 49 hPa; this demonstrates the necessity of using the correct formulation of the mass balance in simulation models for moist air.

Abstract:In this paper, similarity hypotheses for the atmospheric surface layer (ASL) are reviewed using nondimensional characteristic invariants, referred to as -numbers. The basic idea of this dimensional -invariants analysis (sometimes also called Buckingham's -theorem) is described in a mathematically generalized formalism. To illustrate the task of this powerful method and how it can be applied to deduce a variety of reasonable solutions by the formalized procedure of non-dimensionalization, various instances are represented that are relevant to the turbulence transfer across the ASL and prevailing structure of ASL turbulence. Within the framework of our review we consider both (a) MoninObukhov scaling for forced-convective conditions, and (b) Prandtl-Obukhov-Priestley scaling for free-convective conditions. It is shown that in the various instances of Monin-Obukhov scaling generally two -numbers occur that result in corresponding similarity functions. In contrast to that, Prandtl-Obukhov-Priestley scaling will lead to only one number in each case usually considered as a non-dimensional universal constant.Since an explicit mathematical relationship for the similarity functions cannot be obtained from a dimensionalinvariants analysis, elementary laws of -invariants have to be pointed out using empirical or/and theoretical findings. To evaluate empirical similarity functions usually considered within the framework flux-profile relationships, so-called integral similarity functions for momentum and sensible heat are presented and assessed on the basis of the friction velocity and the vertical component of the eddy flux densities of sensible and latent heat directly measured during the GREIV I 1974 field campaign.

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 and water 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.

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