In this chapter we address one of the most difficult problems in physical science: turbulence. Nearly all geophysical flows are turbulent.Much of our knowledge of turbulent flows is based on observation; a first-principle theory of turbulence does not exist, though there are a number of heuristic theories. We will consider turbulence in the context of channel flow of water (think of rivers and streams), but the concepts and models considered apply to many other geophysical flows, including motions of the atmosphere and oceans, as well as engineered flows, such as flows over airplanes and flows over and within turbines and rockets.

As water flows down a sloping channel, the downslope gravitational force imparts momentum and kinetic energy to the fluid. The force is balance by a drag force exerted by the bottom and the kinetic energy is dissipated as heat by viscosity. This requires the action of molecular viscosity. The trouble is, the viscosity of water is very small: ν ≈10−6 m2·s−1. We have seen in § 22.2.2 that the speed of flow predicted by laminar-flow theory is very large; such speeds do not occur in natural settings. Turbulence invariably develops if a given flow is large or rapid enough and acts to retard the flow. In § 23.1 we discuss when and how turbulence arises and investigate the transition from laminar to turbulent flow. Then in § 23.2 we present and discuss the engineering approach to the quantification of turbulence.

Models of turbulent flow invariably characterize it as the sum of a “macroscopic” mean and a set of “microscopic” perturbations, though in practice this division is not very clean. The equations governing the mean flow, developed in § 23.3, contain perturbation terms which must be determined. While man-made channels can have relatively smooth bottoms, we will see in § 23.4 that natural channels are invariably hydraulically rough. The approach which is the best combination of simplicity and accuracy is mixing-length theory, coupled with a rough bottom; this is introduced in § 23.5. The resulting turbulent velocity profile for flow of water in a channel is investigated in § 23.6. This chapter concludes in § 23.7 and § 23.8 with some comments on the drag coefficient and the turbulent diffusion of heat.