This report will provide an overview of climate modeling from a mathematical perspective, particularly with respect to the use of partial differential equations. A visit to the Swedish Meterological and Hydrological Institute's Rossby Center for climate research in Norrkoping, Sweden, is at the foundation of our investigations. An introduction and a brief history section will be followed by a description of the Navier-Stokes equations, which are at

the heart of climate-related mathematics, as well as a survey of many of the

popular approximations and modeling techniques in use by climate researchers

today. Subsequently, a boundary value problem based on the one dimensional

compressible Euler equations will be discussed from an analytical as well as a

numerical point of view, especially with concern to the well-posedness of the

same.