The behavior of number concentration tendencies for the continuous collection growth equation using one- and two-moment bulk parameterization schemes

This paper presents a mathematical explanation for the non-conservation 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. In a previous paper four commonly used parameterization schemes were evaluated to determine if they would conserve the total number concentration Nt of growing particles when Nt physically should be conserved for processes such as continuous collection growth and vapor diffusion growth. The schemes evaluated are: (i) Scheme A: one-moment scheme: where q is predicted, n_o is specified as a constant, Nt and Dn are diagnosed, (ii) Scheme B: one-moment scheme: where q is predicted, Dn is specified as a constant, Nt and n_o are diagnosed, (iii) Scheme E: two-moment scheme: where q and Dn are predicted, Nt and n_o are diagnosed, and (iv) Scheme F: two-moment scheme: where q and Nt are predicted, Dn and n_o are diagnosed. In these schemes, q is mixing ratio, n_o is the slope intercept, and Dn is the characteristic diameter (inverse of the slope of the distribution). It was found that only scheme-F conserved Nt for vapor diffusion and the continuous collection growth processes for which Nt should be conserved. In this current work, only the continuous collection growth process is considered, for which, it is shown mathematically why Nt is conserved for scheme-F and why Nt is not conserved for the other schemes. Vapor diffusion growth is not shown as the results are qualitatively similar. An equation is derived, which is a conservation condition for Nt based on the partial time derivative of the definition of the third-moment. This equation makes it possible to explain why only scheme-F is conservative for Nt when it should be for continuous collection growth. The equation also explains why the other schemes are non-conservative. The mathematical description shows that the assumption of a constant n_o in scheme-A is physically inconsistent with the continuous collection growth process, as is the assumption of a constant Dn for scheme-B. In order to compensate for these inappropriate assumptions, the other variables obtained with these schemes are also often inaccurate. Scheme-E is also non-conservative but it seems this result is not owing to a physically inconsistent specification, but rather the solutions scheme's equations simply do not satisfy Nt conservation.