## The equations of fluid motion

6.1. Differentiation following the motion

6.2. Equation of motion for a nonrotating fluid

6.2.1. Forces on a fluid parcel

6.2.2. The equations of motion

6.2.3. Hydrostatic balance

6.3. Conservation of mass

6.3.1. Incompressible flow

6.3.2. Compressible flow

6.4. Thermodynamic equation

6.5. Integration, boundary conditions, and restrictions in application

6.6. Equations of motion for a rotating fluid

6.6.1. GFD Lab III: Radial inflow

6.6.2. Transformation into rotating coordinates

6.6.3. The rotating equations of motion

6.6.4. GFD Labs IV and V: Experiments with Coriolis forces on a parabolic rotating table

6.6.5. Putting things on the sphere

6.6.6. GFD Lab VI: An experiment on the Earth's rotation

6.7. Further reading

### 6.8. Problems

To proceed further with our discussion of the circulation of the atmosphere, and later the ocean, we must develop some of the underlying theory governing the motion of a fluid on the spinning Earth. A differentially heated, stratified fluid on a rotating planet cannot move in arbitrary paths. Indeed, there are strong constraints on its motion imparted by the angular momentum of the spinning Earth. These constraints are profoundly important in shaping the pattern of atmosphere and ocean circulation and their ability to transport properties around the globe. The laws governing the evolution of both fluids are the same and so our theoretical discussion will not be specific to either atmosphere or ocean, but can and will be applied to both. Because the properties of rotating fluids are often counterintuitive and sometimes difficult to grasp, alongside our theoretical development we will describe and carry out laboratory experiments with a tank of water on a rotating table (Fig. 6.1). Many of the laboratory

FIGURE 6.1. Throughout our text, running in parallel with a theoretical development of the subject, we study the constraints on a differentially heated, stratified fluid on a rotating planet (left), by using laboratory analogues to illustrate the fundamental processes at work (right). A complete list of the laboratory experiments can be found in Appendix A.4.

FIGURE 6.1. Throughout our text, running in parallel with a theoretical development of the subject, we study the constraints on a differentially heated, stratified fluid on a rotating planet (left), by using laboratory analogues to illustrate the fundamental processes at work (right). A complete list of the laboratory experiments can be found in Appendix A.4.

experiments we use are simplified versions of ''classics'' of geophysical fluid dynamics. They are listed in Appendix A.4. Furthermore we have chosen relatively simple experiments that, in the main, do not require sophisticated apparatus. We encourage you to ''have a go'' or view the attendant movie loops that record the experiments carried out in preparation of our text.

We now begin a more formal development of the equations that govern the evolution of a fluid. A brief summary of the associated mathematical concepts, definitions, and notation we employ can be found in Appendix A.2.

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