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The relationship between α and the cross‐correlation of cloud fraction
Author(s) -
Astin I.,
Di Girolamo L.
Publication year - 2006
Publication title -
quarterly journal of the royal meteorological society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.744
H-Index - 143
eISSN - 1477-870X
pISSN - 0035-9009
DOI - 10.1256/qj.05.209
Subject(s) - cloud fraction , exponential function , weighting , altitude (triangle) , exponential distribution , function (biology) , mathematics , fraction (chemistry) , correlation , exponential decay , correlation function (quantum field theory) , meteorology , statistical physics , atmospheric sciences , cloud computing , statistics , physics , geometry , mathematical analysis , cloud cover , chemistry , organic chemistry , evolutionary biology , computer science , acoustics , biology , operating system , spectral density , nuclear physics
Abstract The cloud overlap weighting parameter, α, relates the averaged combined cloud fraction between two altitude levels to the averages derived under the random and maximum overlap assumptions. In several recent studies, it is found that α decreases approximately exponentially with separation distance between the two altitudes. This note provides a mathematical basis for these observed behaviours. An analytic expression for α is derived. It is found that α is solely a function of the cross‐correlation function between cloud fractions at the two altitudes, and the individual averages and variances in cloud fraction at the two altitudes. Only the first of these is a joint function of the two altitudes of interest and therefore determines the form of the dependence of α on the height separation between the two altitudes. Where the variances in cloud fraction are small compared to their respective averages, it is found that α is approximated by the cross‐correlation function. As the cross‐correlation function is exponential wherever and whenever the physical thickness of clouds follows an exponential distribution, it is suggested that, where observations show a strong exponential dependence of α on separation distance between two altitudes, this indicates times and regions where cloud thickness follows an exponential distribution. Copyright © 2006 Royal Meteorological Society