A wide variety of fluid flows occurs in the atmosphere. This chapter introduces the basic fluid-dynamical laws that govern these atmospheric flows. The length scales of interest range from metres to thousands of kilometres; these are many orders of magnitude greater than molecular scales such as the mean free path, at least in the lower and middle atmosphere. We may therefore average over many molecules, ignoring individual molecular motions and regarding the fluid as continuous. ‘Local’ values of quantities such as density, temperature and velocity may be defined at length scales that are much greater than the mean free path but much less than the scales on which the meteorological motion varies.

In Section 4.1 we derive the mass conservation law (often called the continuity equation) for a fluid. In Section 4.2 we introduce the concept of the material derivative and the Eulerian and Lagrangian views of fluid motion. An alternative form of the mass conservation law is given in Section 4.3 and the equation of state for the atmosphere (an ideal gas) is recalled in Section 4.4. Then in Section 4.5 Newton's Second Law is applied to a continuous fluid, giving the Navier–Stokes equation. The Earth's rotation cannot be ignored for large-scale atmospheric flows, so its incorporation into the Navier–Stokes equation is discussed in Section 4.6. The full equations of motion for a spherical Earth and for Cartesian tangent-plane geometry are given in Section 4.7.