Rotation of Earth plays a dominant role in the dynamics of its atmosphere and oceans on time scales of a day or more. In a rotating (non-inertial) frame, the dynamic effect of rotation is represented a virtual force, called the Coriolis force, in the Navier–Stokes equation. A dominant effect of rotation is to produce waves and flows that are themselves rotating relative to a frame of reference fixed to Earth. This tendency is most readily observed in the atmosphere, which contains large-scale rotating air masses called “highs” and “lows” by meteorologists, and an array of smaller and often stronger rotating systems, including hurricanes (or cyclones), tornadoes, water spouts, and dust devils. As we shall see, rotation of Earth plays a direct role in large-scale systems, but only an indirect role in the smaller systems.

In this chapter we will modify the perturbation equations given in § 7.3.2 so that they apply more specifically to the atmosphere and oceans and in doing so will encounter a number of concepts and effects that are introduced by rotation. The Coriolis force complicates the analysis of rotating fluids in several ways; apart from introducing an extra term in the Navier–Stokes equation, it couples the horizontal components of this equation, so that motions are generally not unidirectional. It will take some care and effort to decipher the dynamic constraints that rotation imposes on the flow, so we shall proceed carefully, beginning in § 8.1 with the discussion of a number of rotational concepts. Then in § 8.2, we will present the equations governing flow in a rotating fluid. The rotation of the fluid itself, represented by the vorticity, is an important quantity of interest; the vorticity equation is presented and discussed in § 8.3. Often horizontal motions in a thin layer of fluid are independent of the vertical coordinate and can be quantified by vertically averaged equations; the vertically averaged continuity equation is developed in § 8.4. Large-scale motions in the atmosphere and oceans may be categorized as geostrophic or quasi-geostrophic. Geostrophic flows result from a balance between the Coriolis and pressure terms, as described in § 8.5, while quasi-geostrophic flows result from a balance between the inertial, Coriolis and pressure terms, as described in § 8.6.