This thesis is concerned with modelling geophysical flows. The problems considered in this work include dispersion in flows through heterogeneous porous rocks, turbulent mixing in the surface layer of the ocean, and mixing in turbulent starting plumes. In chapters 2, 3 and 4, we study the longitudinal dispersion of a passive tracer by a two-dimensional pressure-driven flow through a layer of heterogeneous porous rock which is bounded above and below by impermeable seal rock. In chapters 2 and 3, we assume that the heterogeneity of the rock is due to localised regions of different permeability located at randomly assigned vertical positions within the otherwise uniform permeability layer. It is well known that in a porous layer of large cross-flow extent, such heterogeneity leads to Fickian-type dispersion. However, many porous rocks consist of relatively thin, laterally extensive layers. As a result, streamlines in the centre of the channel can be diverted upwards or downwards into regions of higher permeability, while streamlines near the boundaries are more restricted. We demonstrate that this results in a net cross-layer shear in the mean flow. We develop a depth-averaged model for the dispersal of a pulse of tracer by the flow, which shows that although at early times the Fickian dispersion dominates, at large distances downstream the spreading of the pulse of tracer is controlled by the shear. In chapter 4, we demonstrate this shear in a cross-bedded formation, focusing on the flow across an interface between two neighbouring zones of the rock. We explore the strength of this shear as a function of the permeability ratio across the interface and the interface angle. Finally, in chapters 5 and 6, we focus our attention on mixing in turbulent flows, considering two classes of problems -- turbulent mixing of a passive tracer in the ocean mixed layer and mixing in turbulent starting plumes. In chapter 5, we present results from high resolution numerical simulations of the ocean mixed layer to estimate an exact functional relationship between the turbulent fluxes and the gradients of a passive tracer. This functional form of the eddy diffusivity does not use any closure assumptions, and it highlights both local and non-local effects of mixing of a passive tracer. For simplicity, we restrict our focus to convection-driven mixing in an idealised two-dimensional surface layer of the ocean. In chapter 6, we explore the dynamics of turbulent starting plumes by analysis of a series of new small-scale laboratory experiments to describe the mixing and interaction between the plume head, the following steady plume, and the ambient. We find that the head of the plume ascends with a speed which is approximately 0.6 times the characteristic speed of the fluid in the following steady plume, and so the fluid released from the source eventually catches the head of the flow. On reaching the top of the plume, it recirculates and mixes in the plume head. We present results from new experiments to visualise the dispersion of the source fluid in the plume head, and propose a theoretical model to describe the dynamics of the plume head. We present our conclusions and discuss directions for future work in chapter 7.