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Abstract. We present a method for deriving the ensemble-averaged reactive solute transport equation for unsteady, non-divergence-free flow field conditions. Our approach uses a cumulant expansion, Lie group theory, and time-ordered exponentials to develop the ensemble-averaged transport equation. The cumulant expansion is in powers of a arc, where a measures the magnitude of the perturbations of the transport and reaction operators and rc is the correlation time of these perturbations. Because the cumulant expansion avoids secular terms (terms in powers of time), the problem can be closed by rationally truncating the expansion. The truncated terms can be shown to be of lower order than those terms that are kept, provided that a particular constraint (in terms of the Kubo number) is met. The use of Lie group theory allows one to automatically combine the Eulerian and Lagrangian approaches. A particular time-ordered exponential that arises in the analysis can be interpreted as a translation operator that possesses a welldefined algebra. These translation operators appear in the second-order (covariance) terms of the cumulant expansion, and their effect is to shift one of the terms of the covariance functions relative to the other along the trajectory formed by the ensembleaveraged velocity field. This approach does not require neglecting the local dispersion tensor and has the advantage that no integral transformations are conducted; therefore all results are expressed in terms of.real space variables.

Global climate change can be reproduced in detail by using three-dimensional general circulation models (GCMs). However, such complex models require super-computers and extensive hours of computational time for a single attempt at reproducing long term climate change. An alternative approach is to make simplifying assumptions that retain the essential physics for the desired simulation. Energy balance and radiativeconvective models are examples of such models. The model in this study follows the simplified approach using physics-based climate processes as well as interactions between atmospheric and hydrological processes. The vertically and latitudinally averaged mean temperature and mean water vapour content between 30°N-50°N latitudes are considered as atmospheric state variables while soil and sea temperatures and water storage amount are considered for describing the behaviour of the hydrological system. Temperatures in both the atmosphere and ground are calculated by a thermal energy equation that considers the physically-based processes of shortwave radiation, longwave radiation, sensible heat flux, and latent heat flux. Precipitation and evaporation processes transport moisture between the atmosphere and ground. In this study, the radiation parameterization of the simplified climate model is tested in the investigation of the various effects of global warming due to doubling and quadrupling of C0 2 . Changes of temperature, soil water content, evaporation rate and precipitation rate are investigated by numerical experiments. The simplified climate model provides acceptable simulation of climate change and holds promise for practical investigations such as the interactions of physical processes in the evolution of drought phenomena.Un modèle simplifié du climat combinant les processus atmosphériques et hydrologiques Résumé Le changement du climat planétaire peut être reproduit en détail en utilisant des modèles tridimensionnels de la circulation générale. Mais des modèles aussi complexes exigent des ordinateurs puissants ainsi que de nombreuses heures de calcul pour ne simuler qu'un seul scénario d'évolu-tion à long terme du climat. Une alternative serait de faire des hypothèses simplificatrices ne retenant que les processus physiques essentiels pour la simulation désirée. Les modèles "bilan d'énergie" et "radiatif-convectif" sont des exemples de tels modèles. Le modèle présenté dans cette étude suit l'approche simplifiée en prenant en compte des processus climatiques fondés sur des phénomènes physiques comprenant des interactions entre des processus atmosphériques et hydrologiques. Dans cette étude, nous mettons à l'épreuve la paramétrisation des phénomènes radiatifs du modèle climatique simplifié dans notre étude des différents effets du réchauffement global résultant du doublement ou du quadruplement du C0 2 . Les modifications de la température, de l'humidité du sol, de l'évaporation et des précipitations sont testées grâce à des expériences numériques. Le modèle simplifié du climat fournit u...