Investigating the left tail of the
droplet size distribution using PDF methods
Dr.
Christopher Jeffrey,
Atmospheric
Science Faculty Candidate
Determination of the evolution
of a parcel of moist air and droplets undergoing turbulent convection,
molecular diffusion and condensation/evaporation is a challenging problem
involving a range of spatial and temporal scales. Moment formulations
suffer from the well-known closure problem that information about statistical
moments of every order is needed to have closed non-linear advection
terms. In addition, condensation/evaporation introduces a new closure
problem where the Eulerian evaluation of the average
droplet radius appears in the advection-diffusion equation for relative
humidity (RH) but is not easily evaluated.
Probability density function
(PDF) methods offer a distinct advantage over moment approaches since
non-linear reaction terms like evaporation are more easily evaluated. An
equation for the evolution of RH can be written in terms of either the
conditional Laplacian or the conditional dissipation if
the PDF is spatially homogeneous. However, evaluation of either statistic using
a Gaussian mixing assumption leads to unphysical behaviour
in the evolution of the scalar PDF unless the PDF is strictly Gaussian
itself.
In this study I use a technique
called "mapping closure" (Chen et al., PRL, 1989) to evaluate the
conditional Laplacian in the PDF-equation for RH that
does not suffer from the deficiencies of a purely Gaussian closure. The
turbulent mixing of moist and dry air is studied and a universal limit
identified where the RH-PDF evolution is only a function of the Damkohler number (Da)--- the ratio of turbulent and reactive time scales.
The spirited debate over the nature of turbulent mixing in clouds is then
revisited [see references below], and I demonstrate that the neither the limits
of "homogeneous mixing" (small Da) nor
"inhomogeneous mixing" (large Da) produce
maximal dispersion in the left tail of the droplet size distribution.
Rather a Da of order one is required.
Observational measurements required to validate this prediction are suggested.
References:
1) Baker et al., "The
influence of entrainment on the evolution of
cloud
droplet spectra:
106, 581, 1980
2) Telford et al., "Entrainment
at cloud tops and droplet spectra",
JAS, 41, 3170, 1984
3) Paluch
& Knight, "Mixing and evolution of cloud droplet size
spectra
in a vigorous continental cumulus", JAS, 41, 1801, 1984
4) Chai
et al., "Comments", JAS, 42, 753, 1985
5) Paluch
& Knight, "Reply", JAS, 42, 758, 1985
6) Paluch
& Knight, "Does mixing promote cloud droplet growth", JAS,
43, 1994, 1986
7)
8) Paluch
& Knight, "Reply", JAS, 44, 2355, 1987