Two different methods are used to solve the stochastic collection equation (SCE) numerically. They are called linear discrete method (LDM) and bin shift method (BSM), respectively. Conceptually, both of them are similar to the well-known discrete method (DM) of Kovetz and Olund. For LDM and BSM, their concept is extended to two prognostic moments. Therefore, the \"splitting factors\" (which are constant in time for DM) become time-dependent for LDM and BSM. Simulations are shown for the Golovin kernel (for which an analytical solution is available) and the hydrodynamic kernel after Hall. Different bin resolutions and time steps are investigated. As expected, the results become better with increasing bin resolution. LDM and BSM do not show the anomalous dispersion which is a weakness of DM. / Es werden zwei verschiedene Methoden zur numerischen Lösung der \"Gleichung für stochastisches Einsammeln\" (stochastic collection equation, SCE) vorgestellt. Sie werden als Lineare Diskrete Methode (LDM) bzw. Bin Shift Methode (BSM) bezeichnet. Konzeptuell sind beide der bekannten Diskreten Methode (DM) von Kovetz und Olund ähnlich. Für LDM und BSM wird deren Konzept auf zwei prognostische Momente erweitert. Für LDM und BSM werden die\" Aufteil-Faktoren\" (die für DM zeitlich konstant sind) dadurch zeitabhängig. Es werden Simulationsrechnungen für die Koaleszenzfunktion nach Golovin (für die eine analytische Lösung existiert) und die hydrodynamische Koaleszenzfunktion nach Hall gezeigt. Verschiedene Klassenauflösungen und Zeitschritte werden untersucht. Wie erwartet werden die Ergebnisse mit zunehmender Auflösung besser. LDM und BSM zeigen nicht die anomale Dispersion, die eine Schwäche der DM ist.

We propose a mathematical procedure to derive a stochastic parameterization for the bulk warm cloud micro-physical properties of collision and coalescence. Unlike previous bulk parameterizations, the stochastic parameterization does not assume any particular droplet size distribution, all parameters have physical meanings which are recoverable from data, all equations are independently derived making conservation of mass intrinsic, the auto conversion parameter is finely controllable, and the resultant parameterization has the flexibility to utilize a variety of collision kernels. This new approach to modelling the kinetic collection equation (KCE) decouples the choice of a droplet size distribution and a collision kernel from a cloud microphysical parameterization employed by the governing climate model. In essence, a climate model utilizing this new parameterization of cloud microphysics could have different distributions and different kernels in different climate model cells, yet employ a single parameterization scheme. This stochastic bulk model is validated theoretically and empirically against an existing bulk model that contains a simple enough (toy) collision kernel such that the KCE can be solved analytically. Theoretically, the stochastic model reproduces all the terms of each equation in the existing model and precisely reproduces the power law dependence for all of the evolving cloud properties. Empirically, values of stochastic parameters can be chosen graphically which will precisely reproduce the coefficients of the existing model, save for some ad-hoc non-dimensional time functions. Furthermore values of stochastic parameters are chosen graphically. The values selected for the stochastic parameters effect the conversion rate of mass cloud to rain. This conversion rate is compared against (i) an existing bulk model, and (ii) a detailed solution that is used as a benchmark. The utility of the stochastic bulk model is extended to include hydrodynamic and turbulent collision kernels for both clean and polluted clouds. The validation and extension compares the time required to convert 50\% of cloud mass to rain mass, compares the mean rain radius at that time, and used detailed simulations as benchmarks. Stochastic parameters can be chosen graphically to replicate the 50\% conversion time in all cases. The curves showing the evolution of mass conversion that are generated by the stochastic model with realistic kernels do not match corresponding benchmark curves at all times during the evolution for constant parameter values. The degree to which the benchmark curves represent ground truth, i.e. atmospheric observations, is unknown. Finally, among alternate methods of acquiring parameter values, getting a set of sequential values for a single parameter has a stronger physical foundation than getting one value per parameter, and a stochastic simulation is preferable to a higher order detailed method due to the presence of bias in the latter. / Graduate / 0725 0608 0405 / davidc@uvic.ca

Two different methods are used to solve the stochastic collection equation (SCE) numerically. They are called linear discrete method (LDM) and bin shift method (BSM), respectively. Conceptually, both of them are similar to the well-known discrete method (DM) of Kovetz and Olund. For LDM and BSM, their concept is extended to two prognostic moments. Therefore, the \'splitting factors\' (which are constant in time for DM) become time-dependent for LDM and BSM. Simulations are shown for the Golovin kernel (for which an analytical solution is available) and the hydrodynamic kernel after Hall. Different bin resolutions and time steps are investigated. As expected, the results become better with increasing bin resolution. LDM and BSM do not show the anomalous dispersion which is a weakness of DM. / Es werden zwei verschiedene Methoden zur numerischen Lösung der \'Gleichung für stochastisches Einsammeln\' (stochastic collection equation, SCE) vorgestellt. Sie werden als Lineare Diskrete Methode (LDM) bzw. Bin Shift Methode (BSM) bezeichnet. Konzeptuell sind beide der bekannten Diskreten Methode (DM) von Kovetz und Olund ähnlich. Für LDM und BSM wird deren Konzept auf zwei prognostische Momente erweitert. Für LDM und BSM werden die\' Aufteil-Faktoren\' (die für DM zeitlich konstant sind) dadurch zeitabhängig. Es werden Simulationsrechnungen für die Koaleszenzfunktion nach Golovin (für die eine analytische Lösung existiert) und die hydrodynamische Koaleszenzfunktion nach Hall gezeigt. Verschiedene Klassenauflösungen und Zeitschritte werden untersucht. Wie erwartet werden die Ergebnisse mit zunehmender Auflösung besser. LDM und BSM zeigen nicht die anomale Dispersion, die eine Schwäche der DM ist.