## Brief Introduction To Dynamics

Mathematics is the easiest bit in physics.

â€”Pierre-Gilles de Gennes, Les Objets Fragile

WE NOW BEGIN OUR QUEST OF PROVIDING AN EXPLANAtion for how and why the ocean circulates the way it does and how and why it affects the climate. In this chapter, we'll explain some of the basic dynamical principles that determine the circulation; in the next chapter, we'll apply these principles to the circulation itself. The current chapter is a little more mathematical than the others in this book, but it requires no more sophistication on the part of the readerâ€”perhaps less in fact. Nevertheless, some readers may prefer to read the later chapters first, referring back to the material here as needed.

The atmosphere and ocean are both fluids and, although one is a gas and the other a liquid, their motion is determined by similar physical principles and described by similar equations: the Navier-Stokes equations of fluid dynamics. These are complex, nonlinear, partial differential equations that require the largest supercomputers to solve, but embedded within them are two important principles that reflect the dominant force balances in the atmosphere and ocean, namely hydrostatic balance and geostrophic balance. Hydrostatic balance represents the balance in the vertical direction between the pressure force and the gravitational force, and geostrophic balance represents the horizontal balance between the pressure force and the so-called Coriolis force, a force that arises because of the rotation of Earth. If we understand these forces, we will be able to understand a great deal about the motion of the atmosphere and ocean, so let's figure out what they are. We'll begin with the forces that arise (or appear to arise) as a consequence of Earth's rotation, namely centrifugal force and the coriolis force, and then consider the pressure force. It turns out that for most geophysical applications, the coriolis force is much more important than centrifugal force, but we need to understand the latter to understand the former, so that is where we begin.