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Jerry M. Straka
,
Katharine M. Kanak
, and
Matthew S. Gilmore

Abstract

This paper presents a mathematical explanation for the nonconservation of total number concentration Nt of hydrometeors for the continuous collection growth process, for which Nt physically should be conserved for selected one- and two-moment bulk parameterization schemes. Where possible, physical explanations are proposed. The assumption of a constant no in scheme A is physically inconsistent with the continuous collection growth process, as is the assumption of a constant Dn for scheme B. Scheme E also is nonconservative, but it seems this result is not because of a physically inconsistent specification; rather the solution scheme’s equations simply do not satisfy Nt conservation and Nt does not come into the derivation. Even scheme F, which perfectly conserves Nt , does not preserve the distribution shape in comparison with a bin model.

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Jerry M. Straka
,
Matthew S. Gilmore
,
Katharine M. Kanak
, and
Erik N. Rasmussen

Abstract

One- and two-moment parameterizations are integrated over hydrometeor diameters D(0, ∞) for vapor diffusion and the continuous collection growth processes. For the conditions specified, the total number concentration of collector particles should be conserved. To address the problem, the gamma distribution function is used for the spectral density function. Predicted variables can include total mixing ratio q, total number concentration Nt , and characteristic diameter Dn (inverse of the distribution slope λ). In all of the cases, the slope intercept no is diagnosed or specified. The popular one- and two-moment methods that are explored include the one-moment method in which q is predicted, no is specified, and Nt and Dn are diagnosed; the one-moment method in which q is predicted, Dn is specified, and Nt and no are diagnosed; the two-moment method in which q and Dn are predicted and Nt and no are diagnosed; and the two-moment method in which q and Nt are predicted and no and Dn are diagnosed. It is demonstrated for the processes examined that all of the schemes 1) fail to conserve Nt for the collector particles when Nt should be conserved and 2) have other unphysical attributes, except for the two-moment method in which q and Nt are predicted. In recent years there has been a dramatic increase in the use of more-sophisticated microphysical parameterizations in cloud, mesoscale, and climate models, and it is increasingly important for a model user to be cognizant of the strengths and weaknesses of the parameterizations in complex models.

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