1. Introduction
In numerical weather prediction and climate models, the calculation of precipitation processes is carried out by using bulk microphysics parameterizations for the sake of computational efficiency. In such models, the cloud microphysics are taken into account by describing their effects on a few bulk properties of the cloud, such as drop number density N and liquid water content L. These are moments of the hydrometeor size distribution function (“spectrum”), or are proportional to them. The moments obtained from the parameterized model are different from those calculated from the solution of the detailed microphysical (“spectral”) model (the numerically expensive forecast of the spectrum), as the same processes have to be described without information on the actual drop distribution. Consequently, the quality of a parameterization is judged by its ability to reproduce the results of the spectral model. Most parameterized models only forecast one or two moments by solving the respective budget equations. Any other moment of the spectrum has to be diagnosed from the prognostic moment(s). It turns out that the spatiotemporal structure of a moment depends on refinements of the basic parameterization method and also the selected prognostic moments, as demonstrated by Milbrandt and McTaggart-Cowan (2010, hereafter MM) and Wacker and Lüpkes (2009, hereafter WL). The diagnosis of moments is of special importance, because most parameterized cloud physics transformation rates are proportional to a diagnostic moment. The more accurate the model can represent the diagnostic moments, the better the agreement is between the transformation rates following the parameterized and the spectral model.
The quality of a parameterization can be studied best when considering a simplified model problem, where the results from the spectral model are easily available for comparison. That is the case when dealing with the sedimentation of drops. This problem can be treated analytically for the spectral case as well as the one-moment and two-moment parameterization (Wacker and Seifert 2001). It is an illustrative example demonstrating the strengths and weaknesses of a parameterization.
Because of the increase of the sedimentation velocity with drop size, a higher-order prognostic moment is shifted faster downward than a lower-order prognostic moment. Therefore on the forefront of the traveling signal, the mean particle size will become very large and exceed the value found in natural raindrop ensembles. In addition, the representation of the diagnostic moments shows severe deficiencies. In previous studies concerning two-moment parameterization, these problems are circumvented by artificially augmenting the drop number to limit the mean drop size (Cohard and Pinty 2000) and by manipulating either the moment-weighted sedimentation velocities (Mansell 2010) or the parameters of the spectrum (Seifert and Beheng 2006). Alternatively one can make the parameterization assumption of a narrow spectrum instead of a wide one. Sensitivity studies by MM and Milbrandt and Yau (2005a, hereafter MY) document the dependency of the propagation speeds (and consequently the mean particle mass) on the shape of the prescribed spectrum: for a narrower spectrum, the ratios of the moment-weighted sedimentation velocities decrease. Both MM and MY also propose a diagnostic relation between shape parameter and mean particle mass (i.e., the prognostic moments). It gives a high shape parameter (a narrow spectrum) when the mean particle mass is high, thus mitigating the increase of propagation speed with mean particle mass. In MY it is shown that the relative errors in the diagnostic moments are smaller for a diagnostic shape parameter than for a fixed one. Furthermore, Milbrandt and Yau (2005b) show that the results of this ansatz are close to that of a three-moment approach in a multiphase model, which also includes other processes (e.g., coagulation and condensation), though in general the relation between the shape parameter and the mean mass is not unique.
In this study, a different route is taken to control the moment-weighted sedimentation velocities. It is parameterization practice to apply a raindrop spectrum for diameters ranging from zero to infinity in order to define the moments. This assumption dates back to Kessler (1969) and was motivated by the easy mathematical treatment of the resulting integrals. In the calculation of moments, the contribution of the tail of the distribution increases with increasing order of moment. This tail reaches out to very large diameter ranges. The diameter of drops in nature, however, is limited. Therefore, from a physical point of view, it seems sensible to truncate the spectrum at a finite maximum drop diameter Dmax to eliminate any influence of unrealistically large drops. The Dmax should be chosen in a physically sensible way. Of course, this concept can also be generalized to other hydrometeor categories, where a realistic Dmax is either larger (e.g., for hail) or smaller (for certain types of ice crystals).
With this ansatz, the mean mass is bounded by construction. Furthermore, a priori calculations show that the ratios of the moment-weighted sedimentation velocities decrease for Dmax becoming smaller. It will turn out that the equations for the parameters of the size distribution function can be solved only iteratively, as also found by Lüpkes et al. (1989), who introduced finite lower and upper limits for the drop diameter in a parameterization of drop coagulation. Though the solution is straightforward in many situations, severe mathematical problems occur on the forefront of the traveling signal, where the mean particle size is large.
In this paper, we will study the influence of the chosen Dmax value on the spatiotemporal structure of the prognostic and diagnostic moments, the mean drop mass, and the rain rate. The performance of the new parameterization is compared with the methods of WL and MY. We present some statistics on the sensitivity of the results to the change in Dmax and furthermore on the proximity to the solution of the spectral model.
The paper is organized as follows. In the next section, we will outline the two-moment parameterization with a focus on the introduction of a finite maximum drop diameter (general remarks on the parameterization procedure can be found in appendix A). The results of the numerical simulations are presented in section 3, followed by the comparison with other models and the sensitivity studies in sections 4 and 5, respectively. Finally, section 6 discusses the results. All relevant formulas are compiled in appendix B.
2. Modeling fundamentals
Drop size distribution function according to (3) with (left) μ = 0 and (right) μ = 1. Bold lines signify the distribution calculated according to the prescribed Dmax, N = 3 × 10−3 cm−3 and L = 5 × 10−7 g cm−3. (left) Thin lines signify the distribution functions mirrored along D = Dmax/2 (i.e., negative slope parameter λ) with the same N, but increased L.
Citation: Monthly Weather Review 140, 5; 10.1175/MWR-D-11-00020.1
A crude measure for the propagation speed of a moment is the moment-weighted sedimentation velocity 
(left) Average moment-weighted sedimentation speeds for moment orders 0–6 calculated according to N = 3 × 10−3 cm−3 and L = 5 × 10−7 g cm−3 with varying Dmax. (right) As in (left), but showing the ratio of 
Citation: Monthly Weather Review 140, 5; 10.1175/MWR-D-11-00020.1
3. Results for finite Dmax
All model runs [spectral reference and parameterized; cf. (A2) and (2), respectively] are done for constant initial conditions for the distribution function and the moments between z = 8250 m and z = 9750 m. The values are N = 3 × 10−3 cm−3 and L = 5 × 10−7 g cm−3, representing typical conditions for widespread rain (Waldvogel 1974, see his Table 6), as in WL and MM (exceptions will be indicated). For numerical reasons, constant nonzero minimal values (N = 10−12 cm−3, L = 10−17 g cm−3) have to be prescribed elsewhere. Throughout the simulations, (2a) and (2b) are solved with a finite volume scheme of (Monotone Upstream-centered Scheme for Conservation Laws) MUSCL-Hancock type (Toro 1999).
All model runs are made for shape parameter μ = 0 and prognostic moments M0 = N and 
Concerning the diagnosis of moments, results are presented for M1 and M3.5. The first is a surrogate for a cloud physics transformation rate. These are proportional to moments of order approximately 0.5 ≤ l ≤ 2.5 (see MY; Milbrandt and Yau 2005b). The M3.5 is proportional to the sedimentation rate and together with M0.5 determines the evolution of the prognostic moments.
Figure 3 shows the evolution of the moments (both prognostic and reference) for four different Dmax (0.25, 0.5, 0.75, and 1.0 cm). The initial signal is a square wave for both the reference solution (thin lines) and the solution from the parameterized model (bold lines). For the reference solution, the discontinuous initial signal is shifted downward by the sedimentation of drops, and is smoothed out in the course of time due to the fact that big drops fall faster than small ones (also termed “gravitational sorting”). Note that the corresponding Dmax was used to truncate the spectrum. The results show that all solutions from the parameterized model differ from the spectral reference solutions.
Results of model runs for (left to right) four different Dmax (0.25, 0.5, 0.75, and 1.0 cm as indicated) and prognostic moments M0 and M3: Vertical profiles of (top) N and (bottom) L (bold lines) and solution from the spectral reference model (thin lines). The line style refers to the three time steps indicated.
Citation: Monthly Weather Review 140, 5; 10.1175/MWR-D-11-00020.1
The N-profiles (top row) from the parameterized model are generally in good agreement with the reference solution and the small differences are not visibly influenced by a change in Dmax. Matters are different for the higher-order moment L (bottom row): the smaller Dmax is, the closer the results are to the reference solution. The main discrepancies lie in the lower part of the signal, where the moment from the parameterized model has a considerable value (termed “lower front” because of the abrupt decline in value at the leading edge of the signal), which by far exceeds the nearly vanishing values of the reference solution. This makes the signal from the parameterized model more widespread than that of the reference solution. The amplitude of the lower front diminishes with growing maximum drop diameter. In the limiting case of Dmax → ∞, the lower front develops itself into a shock wave with a tiny amplitude, as discussed in WL (see also the results from WL in section 4).
The differences in the shapes of N and L have their origin in the different characteristic structures of the lower-order and higher-order prognostic moment. For Dmax = ∞ (i.e., the conventional parameterization), one can calculate the shape of the signal analytically (Wacker and Seifert 2001). This is not possible for finite Dmax, but our numerical results indicate that the structures of Mj and Mk also differ in this case.
When decreasing Dmax, the shift of L slows down, while N gets slightly faster. The dependence of the moment-weighted fall speed 
Concerning the influence of a change in Dmax on the results, it is remarkable that for N and Dmax ≥ 0.5 cm, the results do not differ visibly when Dmax is changed. In contrast, for the higher-order moment L, a change in Dmax from 0.75 to 1.0 cm still influences the results. In general, the solution from the parameterized model appears to be closer to the spectral reference solution when Dmax is small than when it was large. For details on the sensitivity of the results to the maximum drop diameter see section 5.
Figure 4 shows the first moment as a surrogate for a transformation rate (top row), the 3.5th moment proportional to the sedimentation rate (middle row), and the mean drop mass (bottom row), calculated from the same model runs as before.
As in Fig. 3, but for the diagnosed quantities (from top to bottom): M1 (surrogate for a cloud physics transformation rate), M3.5 (proportional to the sedimentation rate), and x = L/N (mean drop mass). (bottom) Note the different scales on the abscissas for x.
Citation: Monthly Weather Review 140, 5; 10.1175/MWR-D-11-00020.1
After a certain time, M1 as diagnosed from the prognostic moments M0 and M3 is damped (cf. the reference solution). This is in accordance with the classifications of WL, where a diagnostic moment of order between the prognostic ones would suffer from damping. The influence of a changing maximum drop diameter is only seen for fairly small Dmax. This is similar to the observations made with N, which also is a moment of low order.
In contrast, the results for M3.5 are visibly influenced by a change in Dmax. The influence of the maximum drop diameter is expressed in a variation of the lower front of M3.5, which has a very high amplitude and is more ahead of the reference solution, the higher Dmax is. The difference of M3.5 as diagnosed from M0 and M3 to the results from the spectral model gets smaller for decreasing Dmax. This explains why also the prognostic moments are closer to the reference solution in that case, as their evolution is in part determined by M3.5. According to WL, a moment of order 3.5, diagnosed from M0 and M3, should be subject to overshooting, which they define as “exceeding the initial values.” This is not the case here, so maybe the classification of WL is too strict. Sensitivity experiments regarding different orders of diagnostic moments (not shown here) yielded that the degree of overshooting depends on the order of the moment as well as on the model time. Overshooting appears first for higher orders (l ≥ 4 at t = 300 s) and for earlier times. For later model times the maxima of the prognostic moments are damped and it is possible that the diagnosed moments do not exceed their initial values. Overshooting in the sense of WL appears for, for example, the diagnostic moment M6 (radar reflectivity). Here, the overshooting decreases with decreasing Dmax, from a factor of about 22 for Dmax = 1.0 cm to a factor of about 2 for Dmax = 0.25 cm. This is a consequence of the diminishing ratio of 
The mean particle mass x by construction cannot exceed 
Rain rate as a function of time for prognostic moments M0, M3 (bold lines) for four different Dmax (0.25, 0.5, 0.75, and 1.0 cm as indicated). Spectral reference solution included (thin lines; the differences in the results for 0.5, 0.75, and 1.0 cm are below drawing accuracy). The rain rate is calculated 2500 m below the base of the initial concentration.
Citation: Monthly Weather Review 140, 5; 10.1175/MWR-D-11-00020.1
In the parameterized model, the rain event is shifted forward in time: the rain rate is overestimated in the early phase of precipitation, but underestimated in the phase after the maximum rain rate. These differences diminish for decreasing maximum drop diameter. The earlier onset of rain can be related to the earlier arrival on the ground of the lower front in L. Furthermore, the changes with Dmax in the results increase when Dmax gets smaller (see also section 5). The maximum rain rates are reproduced fairly well for all maximum drop diameters. Note that the reference solutions for Dmax = 0.5, 0.75, and 1.0 cm all appear as one line, because their differences are very small.
4. Comparison with other models
Now we turn to a comparison of the proposed method for finite Dmax with the existing parameterizations of WL (μ is constant) and MY (μ is a function of the prognostic moments). The latter methods both use Dmax = ∞ for the calculation of the moments and fluxes. The initial conditions are given by a raindrop spectrum following a Marshall–Palmer distribution [cf. (3)] for μ = 0, truncated at Dmax = 1 cm. The initial parameters n0 = 0.0798 cm−4 and λ = 26.6134 cm−1 lead to the number density and liquid water content used before (N = 3 × 10−3 cm−3 and L = 5 × 10−7 g cm−3). The geometrical setup is identical to that of section 3. The initial shape parameter μ = 0 has been chosen in order to facilitate the comparison of the present results with those of MY (their Fig. 3), although this somehow favors the methods likewise assuming μ = 0. A comparison of the methods’ results with the evolution of an initial spectrum with μ = 1 (Fig. 1) has also been done. The results are not shown here, but differences will be commented on in due course.
The outcome of the simulations with the different parameterization models is shown in Fig. 6. The methods from left to right are the parameterization of WL with μ = 0 [WL(μ = 0)]; the proposed method with Dmax = 1 cm and μ = 0 [Dmax(μ = 0)]; the parameterization of MY with a diagnostic relation for μ [MY(μdiag)]; and finally WL with μ = 5.8826, which is the diagnosed shape parameter of MY for the initial conditions [WL (μ ≫ 0)]. While the function for μ of MY is only valid for 0 and 3 as orders of the prognostic moments, MM have generalized the diagnosis of the shape parameter to arbitrary orders 
Comparison of (left to right) four different methods (see text for details) with prognostic moments M0 and M3: results for (top to bottom) L, M1, M3.5, and x (bold lines), and solution from the spectral reference model (thin lines). The line style refers to the three time steps indicated. (bottom) Note the different scales on the abscissas for x.
Citation: Monthly Weather Review 140, 5; 10.1175/MWR-D-11-00020.1
Results are shown (from top to bottom) for the higher-order prognostic moment L and the same diagnostic quantities as in section 3: M1, M3.5, and x.
Having a look at the results for the moments (L, M1, and M3.5), it appears that the qualitative description of the results is the same for the methods assuming μ = 0 and the methods assuming a higher shape parameter, respectively.
First, we compare the methods regarding their performance for L (top row). Both WL(μ = 0) and Dmax(μ = 0) show a significant damping of the maximum value (cf. the reference solution). Furthermore, the signal is broadened, so that the solution from the parameterized model has nonvanishing values of significant magnitude in such heights where the spectral reference solution still has nearly vanishing values. The behavior of WL(μ = 0) and Dmax(μ = 0) differs here: while for Dmax(μ = 0) the propagation speed of the moment is bounded due to the boundedness of the mean drop mass (leading to the abrupt decline in moment value on the lower front), the propagation speed for the moment in WL(μ = 0) is not bounded, resulting in a slow decline of the signal over several hundreds of meters. MY(μdiag) and WL(μ ≫ 0) do not yield a broad signal, hence they better reproduce the spatial position of the moment. However, they drastically overestimate the maximum value. This is in accordance with the results of MY (their Figs. 3d,f), where the two-moment method with a diagnostic μ also overestimates the reference value of the mass content.
Concerning the diagnostic moments, it turns out that with MY(μdiag) and WL(μ ≫ 0), the initial value is not identical to that of the reference solution, as those methods intrinsically assume a different (higher) shape parameter. Of course, this discrepancy also holds during the model runs.
Turning to M1 (second row), we see that WL(μ = 0) and Dmax(μ = 0) show a damping of the signal, whereas MY(μdiag) and WL(μ ≫ 0) overestimate the maximum value. The percentage of overestimation (approximately +110% at 600 s) is much higher than the percentage of underestimation (approximately −30% at 600 s). The spatial position is reproduced well by all methods, because the signals are narrow.
Regarding M3.5 (third row), the results are similar to the results for L. The methods assuming a constant μ = 0 yield values of significant magnitude on the lower front of the broad signal, where the reference solution vanishes. In contrast, the methods assuming a high shape parameter have a narrow signal, but exceed the peak value of the reference solution by about 30%.
So far, it can be stated that no model performs best of all. Both WL(μ = 0) and Dmax(μ = 0) have mostly broad signals with an underestimated peak value, whereas MY(μdiag) and WL(μ ≫ 0) yield narrow signals that considerably overestimate the reference solution. Note that the results for the latter two do not differ much. It seems that in regions of a high shape parameter, a variation in value has little influence on the moments’ profiles. Taking for reference an initial spectrum with μ = 1, also the initial values of the diagnostic moments for WL(μ = 0) and Dmax(μ = 0) are not reproduced correctly. The findings from above, however, do not change qualitatively. Yet the methods assuming a constant μ = 0 are now generally farther away from the reference solution while the methods assuming a high shape parameter get closer.
Last turning to a comparison of the results for the mean particle mass (again with the reference solution for a spectrum with μ = 0), we see that x is considerably overestimated by the methods using Dmax = ∞ and a constant μ. Because of the broad L signal extending to regions with a very low number density, the results for WL(μ = 0) are completely wrong in the context of cloud microphysics. Matters are better, but again not sensible in this physical context for WL(μ ≫ 0). For the mean mass, Dmax(μ = 0) and MY(μdiag) perform best, the latter differing considerably from WL(μ ≫ 0) for the first time. The proposed method exceeds the reference mean mass by about 50% at 600 s. The method of MY(μdiag) gives a much lower mean mass (approximately −75% at 600 s), which qualitatively also was a result of MY (their Figs. 3v,x). Comparing the results with the “μ = 1”-reference solution, the maximum value for the reference mean mass is reduced by nearly a factor of 3. Therefore, Dmax(μ = 0) now strongly overestimates the mean mass while the degree of underestimation with MY(μdiag) is reduced.
5. Sensitivity studies
We now investigate the sensitivity of the prognostic moments to the variation of Dmax. This will be done using three different mean masses of the initial concentration in order to diversify the standard test case. Of course, the initial mean mass cannot exceed 
Figure 7 shows the profiles of prognostic moments N (top) and L (bottom) at t = 600 s for eight different Dmax and three different initial conditions: 
Profiles of prognostic moments (top) N and (bottom) L at t = 600 s for eight different Dmax and three different initial conditions: L = 5 × 10−7 g cm−3 (all columns) and (left) N = 6 × 10−3 cm−3, (middle) N = 3 × 10−3 cm−3, and (right) N = 1.5 × 10−3 cm−3. Bold lines indicate profiles for Dmax = 0.25, 0.50, 0.75, and 1.00 cm; thin lines for Dmax = 0.125, 0.375, 0.625, and 0.875 cm. Mean mass 
Citation: Monthly Weather Review 140, 5; 10.1175/MWR-D-11-00020.1
The ηi is decreasing monotonically with increasing maximum drop diameter. It is lower for Mj=0 and the decline in value is more abrupt than for Mk=3. This indicates that the results are more sensitive to changes in Dmax in regions of a small maximum drop diameter than in regions of a large Dmax. Furthermore, these changes are the smaller, the smaller the order of the moment is. The sensitivity of the results to Dmax increases when augmenting the initial mean mass.
(left) Rate of change of the results [η, (8)]. (right) Difference to solution of spectral model [ζ, (9)]. Each for (top) N and (bottom) L and for three different initial conditions (indicated in line styles). See text for details.
Citation: Monthly Weather Review 140, 5; 10.1175/MWR-D-11-00020.1
Summing up, there seems to be a trade-off between the increased proximity of the solutions from the spectral model and bulk model (for a smaller Dmax) and the decreased sensitivity toward the exact value of the maximum diameter (for a bigger Dmax).
For both error measures ηi and ζi, a constant state is reached at a certain maximum drop diameter, which is smaller for the smaller-order moment. This behavior is due to the fact that for increasing order i, the maximum point of Dif(D) moves to regions of bigger diameters. As the function gets narrower for greater i, most of the contribution to the value of the integral is concentrated around that maximum point. When now truncating the spectrum at some fixed Dmax, a bigger portion of the moment’s value will be cut off for higher orders of moments. Likewise, for fixed order i, a smaller portion of the moment’s value will be truncated if Dmax is large. Therefore, changes in high values of Dmax do not influence the results for N = M0 as strongly as they influence the results for L ~ M3.
6. Discussion and conclusions
In this paper, a two-moment parameterization scheme for drop sedimentation was proposed, where the average sedimentation velocities were controlled by introducing an upper limit to the diameter range when defining the diagnostic moments. This is more physically realistic than the range used in conventional parameterizations, where an infinite Dmax is assumed for the sake of computational simplicity. The results have shown that the selection of the maximum drop diameter has a decisive influence on the structure of the moments’ profiles. When reducing Dmax, the ratio 
Cutting off the distribution function at Dmax < ∞ introduced technical difficulties, which however could be circumvented by a mirroring technique. Furthermore, the slope parameter λ of the drop size distribution had to be calculated from an implicit equation, which added further computational overhead to the numerical simulations. A possible alternative to artificially truncating the gamma distribution function would be the use of a distribution function that intrinsically has a finite support. One such distribution function is the beta function as used in spray theory (Watkins 2005). It has four free parameters, one of which is the maximum allowable drop diameter Dmax. When using a two-moment scheme, one of the remaining three free parameters has to be diagnosed from the other ones. It turns out that the results heavily depend on the choice of the diagnostic function for this third parameter. This makes the beta function unsuitable for use in two-moment schemes, but it may well be worthwhile investigating its performance in three-moment parameterization models.
We compared the proposed parameterization with the existing parameterizations of WL and MY. It could not be found that it outperformed the other methods in all aspects; hence a more physical setup of the method does not automatically lead to better results. Unlike the common two-moment parameterizations assuming a constant μ, however, the proposed parameterization and likewise the scheme of MY gave values for the mean mass in realistic ranges. A drawback of the new parameterization is its increased computing time, which is mainly due to the special treatment of the negative slope parameter values and the use of the incomplete gamma function. All methods were compared in their performance of simulating the evolution of a drop spectrum with μ = 0 and also μ = 1 (not shown). Based on those results, it seems sensible to select a μ for the bulk method, which is close to the approximate shape parameter of the simulation’s rain regime.
In applications, Dmax should be consistent with the actual maximum drop (or hydrometeor) diameter. Of course, this is not obligatory and in principle, Dmax can be varied freely. In literature (e.g., Rogers and Yau 1989; Pruppacher and Klett 1997), one can find a variety of statements about an upper bound for the drop diameter, depending on the ambient conditions. Values given for Dmax range from 0.45 to 1 cm. Our simulations have been done for the prognostic moments most commonly used in NWP models and an initial mean mass typical for widespread rain. The sensitivity studies of section 5 have shown that our results are not visibly (N) or only slightly (L) sensitive to changes in the maximum drop diameter around 0.75 cm. We therefore come to the conclusion that the exact maximum drop diameter is only of minor importance when it is selected close below 1 cm, but note that this sensitivity changes when greatly varying the initial mean mass from the value used here.
Acknowledgments
The authors thank the two anonymous reviewers for their helpful comments. This research was sponsored by the Deutsche Forschungsgemeinschaft Grant WA 1334/8-1.
APPENDIX A
Principles of Parameterization
The moment as a function of time and space can either be calculated from the budget equation in integral form in (A3) or from the integration of the analytic solution of the spectral budget equation in (A2). The latter will be called the reference solution.
For the numerical evolution of the moments according to (A3), it must be possible to write the fluxes Fi as a function of the prognostic moments. To that end, a self-preserving drop size distribution function has to be assumed, which in this case is the gamma distribution f(D) = n0Dμe−λD also given in (3). For constant n0 and μ = 0, this is the well-known Marshall–Palmer distribution (Marshall and Palmer 1948). When the parameters of the distribution function are known, the fluxes can be written in terms of the moments (see sections c and e in appendix B). The number of prognostic moments equals the number of parameters that can be varied freely.
As the parameters of f can be calculated from the prognostic moments, the drop size distribution function can be reconstructed for each (t, z). Consequently, it is possible to calculate any other moments that have not been forecasted (the so-called diagnostic moments, see section d in appendix B). The different moment-weighted sedimentation velocities of the prognostic moments Mj and Mk lead to the so-called diagnostic problems of overshooting (j, k < l or l < j, k) and damping (j < l < k) of the moment Ml—see WL for a thorough discussion.
The process of passing from a detailed model describing certain processes to a bulk model, which only describes the effects of these processes on some bulk quantities, is called parameterization. Here, we obtained a closed set of n equations in (A3) for n moments from (A2) by means of assumptions on f and υT. With these assumptions, (A3) is also called the parameterized model. Because the parameterized model has been obtained by using n moments of the distribution function, one also speaks of the n-moment parameterization or the method of moments.
APPENDIX B
Formulas for Parameters, Moments, and Fluxes
The following abbreviations are used: 
a. Critical and maximum mean mass
b. Explanation of the mirroring technique
c. Parameters
d. Diagnosed moments
e. Flux and moment-weighted sedimentation velocity
REFERENCES
Cohard, J.-M., and J.-P. Pinty, 2000: A comprehensive two-moment warm microphysical bulk scheme. II: 2D experiments with a non-hydrostatic model. Quart. J. Roy. Meteor. Soc., 126, 1843–1859.
Kessler, E., 1969: On the Distribution and Continuity of Water Substance in Atmospheric Circulations. Meteor. Monogr., No. 32, Amer. Meteor. Soc., 84 pp.
Lüpkes, C., K. Beheng, and G. Doms, 1989: A parameterization scheme for simulating collision/coalescence of water drops. Beitr. Phys. Atmos., 62 (4), 289–306.
Mansell, E. R., 2010: On sedimentation and advection in multimoment bulk microphysics. J. Atmos. Sci., 67, 3084–3094.
Marshall, J., and W. Palmer, 1948: The distribution of raindrops with size. J. Meteor., 5, 165–166.
Milbrandt, J. A., and M. Yau, 2005a: A multimoment bulk microphysics parameterization. Part I: Analysis of the role of the spectral shape parameter. J. Atmos. Sci., 62, 3051–3064.
Milbrandt, J. A., and M. Yau, 2005b: A multimoment bulk microphysics parameterization. Part II: A proposed three-moment closure and scheme description. J. Atmos. Sci., 62, 3065–3081.
Milbrandt, J. A., and R. McTaggart-Cowan, 2010: Sedimentation-induced errors in bulk microphysics schemes. J. Atmos. Sci., 67, 3931–3948.
Pruppacher, H., and J. Klett, 1997: Microphysics of Clouds and Precipitation. Kluwer Academic Publishers, 954 pp.
Rogers, R., and M. Yau, 1989: A Short Course in Cloud Physics. Pergamon Press, 290 pp.
Seifert, A., and K. Beheng, 2006: A two-moment cloud microphysics parameterization for mixed phase clouds. Part 1: Model description. Meteor. Atmos. Phys., 92, 45–66.
Toro, E., 1999: Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, 624 pp.
Wacker, U., and A. Seifert, 2001: Evolution of rain water profiles resulting from pure sedimentation: Spectral vs. parameterized description. Atmos. Res., 58, 19–39.
Wacker, U., and C. Lüpkes, 2009: On the selection of prognostic moments in parameterization schemes for drop sedimentation. Tellus, 61A, 498–511.
Waldvogel, A., 1974: The n0 jump of raindrop spectra. J. Atmos. Sci., 31, 1067–1078.
Watkins, A., 2005: The application of gamma and beta number size distribution to the modelling of sprays. Proc. 20th Annual Conf. of ILASS-Europe, Orléans, France, 103–108.
