ALBERT

Description: The evolution of a hydrometeor ensemble („cloud“ ) can be described using a balance equation for its size spectrum. In numerical weather prediction or climate models, however, this approach is too time consuming. It is therefore necessary to capture, if only approximately, the on-going microphysical processes in a cloud using a parame- terised form of modelling. The parameterisation of sedimentation alone is already a demanding task. If its standard form is used in a two-moment scheme, the mean mass of the hydrometeors will be too large for a cloud physics context. Existing approaches try to avoid excessively large mean masses by altering dynamically the parameterisation assumptions in the running model. In this work, a new fundamental approach is presented: the assumption of a spectrum containing particles of all sizes is replaced by its truncation at a particle size realistic in cloud physics. The calculation of integrals over the truncated spectrum requires a new technique, which demands higher computational effort. In this work, comparisons of the new parameterisation with already existing parameterisations are made while changing the initial conditions and the advection scheme. With the new method, the parameterisation error is reduced by up to 50 %. The new paramerisation also constitutes a great improvement compared to the standard form when transferred to modelling an ensemble of solid hydrometeors. This is illustrated using a particle type commonly found in polar regions. Furthermore, for the first time a B-distribution is used as a basis for a cloud microphysics parameterisation. Its domain of definition is bounded by construction. This distribution, however, appears not suitable for use in two-moment schemes, because one free parameter has to be derived from model data. Extending the sedimentation model with drop collisions and starting with a cloud droplet spectrum, it is not possible to judge the quality of the sedimentation paramete- risations, because the coagulation rates are dominated by the choice of the sedimetation velocity for small droplets. If the initial spectrum already contains a sufficient number of raindrops, however, application of the new method again reduces the parameterisation error by up to 50 %.

Description: In common cloud microphysics parameterization models, the prognostic variables are one to three moments of the drop size distribution function. They are defined as integrals of the distribution function over a drop diameter ranging from zero to infinity. Recent works (by several authors) on a one-dimensional sedimentation problem have pointed out that there are problems with those parameterization models caused by the differing average propagation speeds of the prognostic moments. In this study, the authors propose to define the moments over a finite drop diameter range of [0, Dmax], corresponding to the limitation of drop size in nature. The ratios of the average propagation speeds are thereby also reduced. In the new model, mean particle masses above a certain threshold depending on Dmax lead to mathematical problems, which are solved by a mirroring technique. An identical, one-dimensional sedimentation problem for two moments is used to analyze the sensitivity of the results to the maximum drop diameter and to compare the proposed method with recent works. It turns out that Dmax has a systematic influence on the model's results. A small, finite maximum drop diameter leads to a better representation of the moments and the mean drop mass when compared to the detailed microphysical model.

Description: In numerical weather prediction models, parameterisations are used as an alternative to spectral modelling. One type of parameterisations are the so-called methods of moments. In the present study, two different methods of moments, a presumed-number-density-function method with finite upper integration limit and a quadrature method, are applied to a one-dimensional test case ('rainshaft') for drop sedimentation. The results are compared with those of a reference spectral model. An error norm is introduced, which is based on several characteristic properties of the drop ensemble relevant to the cloud microphysics context. This error norm makes it possible to carry out a quantitative comparison between the two methods. It turns out that the two moment methods presented constitute an improvement regarding two-moment presumed-number-density-function methods from literature for a variety of initial conditions. However, they are excelled by a traditional three-moment presumed-number-density-function method which requires less computational effort. Comparisons of error scores and moment profiles reveal that error scores alone should not be taken for a comparison of parameterisations, since moment profile characteristics can be lost in the integral value of the error norm.

Description: This paper examines different distribution functions used in a three-moment parameterization scheme with regard to their influence on the implementation and the results of the parameterization scheme. In parameterizations with moment methods, the prognostic variables are interpreted as statistical moments of a drop size distribution, for which a functional form has to be assumed. In cloud microphysics, parameterizations are frequently based on gamma- and log-normal distributions, while for particle-laden flows in engineering, the beta-distribution is sometimes used. In this study, the three-moment schemes with beta-, gamma- and log-normal distributions are tested in a 1D framework for drop sedimentation, and their results are compared with those of a spectral reference model. The gamma-distribution performs best. The results of the parameterization with the beta- and the log-normal distribution have less similarity to the reference solution, particularly with regard to number density and rain rate. Theoretical considerations reveal that (depending on the type of the distribution function) only selected combinations of moments can be predicted together. Among these is the important combination of “number density, liquid water content, radar reflectivity” for all three distributions. Advection or source/sink terms can only be calculated under certain conditions on the moment values (positivity of the Hankel–Hadamard determinant). These are derived from mathematical theory (“moment problem”) and are more restrictive for three-moment than for two-moment schemes.

Description: For the treament of clouds and precipitation in weatherforecast or climate models, a direct spectral representation of the microphysical processes is computationally infeasible. Therefore, their effects have to be considered in an approximative, parameterised way. It is then possible that the model results are not sensible in a cloud microphysics context. In order to avoid this, in this dissertation a new parameterisation of sedimentation (two moment method) is developed and tested in a 1D model. The parameterisation is based on a hydrometeor distribution with bounded domain and is implemented in two ways: 1) truncated Gamma-distribution: compared to already existing parameterisations, the model results are considerably improved (for drops as well as for ice crystals). Investigations include the computation time, the sensitivity of the results on the upper hydrometeor size limit and an optimisation concerning this limit. The development of a rain cloud is simulated by adding drop collisions to the test case. 2) Beta-distribution: It contains an additional free parameter. Its diagnostic relationship has a strong impact on the results.