Over the last several years, the strongly inhomogeneous nature of atmospheric and oceanic mixing in diverse situations has become increasingly apparent, as high resolution numerical simulations and observations begin to represent accurately the detailed spatial structure of air and water masses and their constituents. Mixing is now known to be confined to distinct latitudinal regions, often separated by sharp gradients that indicate dynamical transport barriers. Inhomogeneous mixing by waves and eddies in atmospheres and oceans is intrinsically linked to the presence of zonally aligned jets, which not only arise as a result of the eddy mixing, but also organize the mixing in distinct latitudinal regions. The combined effect is a dynamical feedback that is now known to operate under very general conditions. Inhomogeneous mixing is important for the transport of constituents such as water vapour, carbon dioxide, ozone, heat, and salinity; their inhomogeneous re-distribution impacts both global radiative balances and regional climate change. Despite recent advances, a complete understanding of the way zonal jets organize inhomogeneous mixing, in particular the vertical structure of such mixing, remains elusive. Progress in understanding the horizontal structure of jets and mixing has been made recently, in particular by focusing on the potential vorticity, a key dynamical quantity that contains information about both horizontal rotational motion and density stratification. The aim of this project is to build on that recent work to develop a complete theory for the vertical structure of jets and mixing. In doing so, it will contribute to our understanding both of the structure of the dominant jet structures in the atmosphere and oceans, as well as providing predictions of how they will reach dynamical equilibrium under different forcing conditions, conditions that may change in a changing climate. It is anticipated that the new theory will allow us to assess the robustness of predictions made by climate models, which are now beginning to accurately represent the complexities inherent in jet structures. As well as advancing our fundamental understanding of basic dynamical processes, we will study four specific issues of current importance in climate science: (i) systematic transport of trace chemicals within the stratosphere; (ii) the coupling of the stratospheric and tropospheric circulations; (iii) the consequences of a climatic shift in the tropospheric jet stream; and (iv) inhomogeneous transport and mixing associated with jets in the Southern Ocean. The project highlights how advances in fundamental science can be effectively combined with directed goals driven by specific applications.