Background

The defining property of fluids is their ability to deform. In the case of nearly incompressible fluids such as water, the deformation is associated with relatively small changes in density. For gases, on the other hand, changes in density are generally quite important. A fluid is treated as a continuum if the length scales of changes in fluid properties such as velocity are large compared with the atomic or molecular scale. In order to solve problems in fluid mechanics, it is necessary to solve the applicable continuum partial differential equations. For details of the derivations of the basic equations the reader is referred to Batchelor (1967).

Many solutions of these equations have been obtained and have been compared with observations. These solutions generally describe flows that are smooth or laminar. In fact, many observed flows are highly oscillatory in a random or statistical way. These flows are generally described as being turbulent. A fundamental understanding of turbulence does not exist; it is one of the major unsolved problems in physics. Under many circumstances laminar flows are unstable; small disturbances grow exponentially. These instabilities lead to turbulence.

Some sets of nonlinear differential equations give chaotic solutions, i.e., the solutions evolve in a nondeterministic manner. Infinitesimal disturbances can lead to first-order differences in flows. These solutions are associated with turbulence. However, true turbulence is so complex that not even the largest computers can resolve its structure. Three-dimensional disturbances exist on all scales from the molecular to those of the boundary conditions. In studies of mantle convection and other fluid problems in the Earth Sciences, the role of turbulent flow must always be kept in mind.