Entrainment Theory

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The need for entrainment in parcel theory was pointed out in 1947 by Stommel, who observed that
the differences between temperatures in clouds and ambient temperatures were much smaller than
one would expect based on a non-entraining parcel. These potentially large temperature differences
are easy to see with the parcel model for this class, as well as the ability of entrainment to effectively
reduce the difference between environmental and parcel temperatures.
The entrainment that we included in the parcel model was assumed to have three special characteristics
air is mixed in instantaneously across the parcel so that it remains homogeneous (¿mix ¼ 0)

air is mixed in laterally, that is, with the thermodynamic properties of ambient air at the same
altitude (z) as the parcel (Ta (z), wva (z), etc.)



1. though the Lagrangian parcel models could reproduce the cloud top heights, they could not
simultaneously predict the correct liquid water (condensate) contents; if they were tuned to
reproduce the liquid water contents, they underestimated the cloud top levels; this is commonly
known as the “Warner problem” (based on Warner, 1970 although in later aricles (1972, 1975)
Warner also helped bring out the other two points below)
2. the predicted droplet spectra were expected to become more narrow with altitude, while observations
showed that the spectra either remained constant or broadened slightly with altitude
3. The inverse-radius dependence of ¹ disagrees with observations (e.g., decreasing liquid water
content) that cloud parcel dilution actually increases with height