25 August 2017 [Publications]
Reactive motion and inertia
Makarieva A.M., Gorshkov V.G., Nefiodov A.V., Sheil D., Nobre A.D., Bunyard P., Nobre P., Li B.-L. (2017) The equations of motion for moist atmospheric air. Journal of Geophysical Research Atmospheres, 122, 7300-7307. doi:10.1002/2017JD026773
How phase transitions affect the motion of moist atmospheric air remains controversial. In the early 2000s two distinct differential equations of motion were proposed. Besides their contrasting formulations for the acceleration of condensate, the equations differ concerning the presence/absence of a term equal to the rate of phase transitions multiplied by the difference in velocity between condensate and air. This term was interpreted in the literature as the “reactive motion” associated with condensation. The reasoning behind this reactive motion was that when water vapor condenses and droplets begin to fall the remaining gas must move upward to conserve momentum. Here we show that the two contrasting formulations imply distinct assumptions about how gaseous air and condensate particles interact. We show that these assumptions cannot be simultaneously applicable to condensation and evaporation. Reactive motion leading to an upward acceleration of air during condensation does not exist. The reactive motion term can be justified for evaporation only; it describes the downward acceleration of air. We emphasize the difference between the equations of motion (i.e., equations constraining velocity) and those constraining momentum (i.e., equations of motion and continuity combined). We show that owing to the imprecise nature of the continuity equations, consideration of total momentum can be misleading and that this led to the reactive motion controversy. Finally, we provide a revised and generally applicable equation for the motion of moist air.

The equation of motion for moist air in the presence of phase changes remains controversial in the meteorological and multiphase flow literature [Young, 1995; Drew and Passman, 1998; Ooyama, 2001; Bannon, 2002; Brennen, 2005]. Young [1995] for example reviewed this subject and highlighted a number of inconsistencies among published treatments. In the atmospheric sciences this problem received attention in the works of Ooyama [2001] and Bannon [2002]. But rather than resolving the issues these authors offered contrasting equations derived from first principles.

The two equations of motion differ in that Bannon [2002] includes the following term, cW, where W is the difference between condensate velocity (i.e. velocity of droplets) and air velocity and c is the rate of phase transitions. Bannon [2002, p. 1972] interpreted this term as the "reactive motion" arising during condensation: as the droplets begin to fall and thus gain a downward velocity, the remaining air gains an upward velocity so that the combined momentum of air plus droplets is conserved. Cotton et al. [2011, see their Figure 2.2] endorsed this interpretation.

Reactive motion

However, the reactive motion explanation appears counter to our knowledge of atmospheric processes: indeed, unlike a rocket which accelerates by reactive motion, i.e., by internal forces between the rocket and the expelled fuel, droplets upon condensation of water vapor are accelerated downward by a recognized and external force-gravity. As another example, consider a block of ice melting on a table made of open mesh. The block does not accelerate upwards as the melt water streams down.

Bannon [2002] mentioned the disagreement with Ooyama [2001] but did not identify either its cause or implications. If, as suggested by Cotton et al. [2011], the formulation of Bannon [2002] obeys momentum conservation, does the contrasting formulation of Ooyama [2001], used in global atmospheric models [Satoh, 2014], violate it? While some authors have argued that the reactive motion term is usually small [Monteiro and Torlaschi, 2007], Cotton et al. [2011] concluded that this term warranted further study. Indeed, irrespective of its magnitude, resolving the discrepancy between the formulations of Ooyama [2001] and Bannon [2002] is necessary for correct employment of the fundamental equations of momentum conservation to a moist atmosphere. We undertake such an analysis in this paper.

Fig. 2.2c of Cotton et al. 2011
Fig. 2.2c (modified) of Cotton, Bryan and van den Heever (2011) "Fundamental equations governing cloud processes". International Geophysics, vol. 99, pp. 15-22. "M" stands for total momentum per unit volume of the system gas+condensate (circles denote droplets).

The fifteen years that passed since the two contrasting equations were published saw no attempts to resolve the controversy. Given such an inertia, one should not be surprised that the discussion of the biotic pump physics got stuck ten years ago with the "big role of latent heat" argument and has shown no sign of advancing ever since (the most recent exchange on the latent heat topic can be found here.)

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