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This chapter describes a linear stability analysis (that is, solving for the critical Rayleigh number Ra and mode) that allows readers to check their linear codes against the analytic solution. For this linear analysis, each Fourier mode n can be considered a separate and independent problem. The question that needs to be addressed now is under what conditions—that is, what values of Ra, Prandtl number Pr, and aspect ratio a—will the amplitude of the linear solution grow with time for a given mode n. This is a linear stability problem. The chapter first introduces the linear equations before discussing the linear code and explaining how to find the critical Rayleigh number; in other words, the value of Ra for a and Pr that gives a solution that neither grows nor decays with time. It also shows how the linear stability problem can be solved using an analytic approach.

This chapter explains how to write a postprocessing code, and more specifically how to study the nonlinear simulations using computer graphics and analysis. It first considers how to compute and store results in a file during the computer simulation, assuming the Fourier transforms to x-space are done within the main computational code during the simulation. It then describes the postprocessing code for reading these files and displaying the various fields, along with the use of graphics software packages that provide additional, more sophisticated visualizations of the scalar and vector data. It also discusses the computer analysis of several additional properties of the solution, focusing on measurements of nonlinear convection such as Rayleigh number, Nusselt number, Reynolds number, and kinetic energy spectrum.

Recall that we briefly examined the stability of thermally stratified fluids in Chapter 6. Our essential conclusion was that a condition in which hot, light fluid resides beneath cooler, heavier fluid is a potentially unstable configuration, inasmuch as gravitational forces are no longer balanced by forces associated with the vertical pressure gradient. In this regard we obtained a necessary (but insufficient) condition for instability, which stated that the magnitude of the vertical temperature gradient must exceed the adiabatic lapse rate. Further recall, however, that such a condition does not necessarily lead spontaneously to overturning (convection). It is possible for a fluid to possess a steady temperature gradient over its vertical extent, in excess of the adiabatic rate, such that the fluid remains static and merely acts like a thermally conducting solid, and we concluded that this static conduction state represents an unstable equilibrium. We then characterized the tendency for instability in terms of the Rayleigh number Ra. Let us now extend our treatment to a description of thermally driven convection motions. We will concentrate on Rayleigh–Bénard and Hele–Shaw configurations (Chapter 6), and the buoyancy-driven flows that occur within these configurations due to steady heating and cooling of the boundaries. In such configurations, convective motions typically occupy the full region between the boundaries, and vary from steady two-dimensional rolls at small Rayleigh numbers to complex turbulent motions at large Rayleigh numbers. In addition, we will briefly consider buoyancy driven flows that arise from localized temperature and compositional variations near a single solid boundary. Envision, for example, the roof of a magma chamber. For a sufficient contrast in temperature between the magma and country rock, melting of the magma roof occurs. If the density of the melt that is produced is less than that of the underlying magma, thermally driven convection may begin within the buoyant layer of melt, which in turn advects heat upward from the underlying hot magma to the roof. Alternatively, if the density of the melt is greater than that of the underlying magma, compositionally driven convection may occur, which in turn brings hot magma to the roof.

Thus far, we have implicitly assumed that chemical species move only by diffusion. In fact, a number of external forces can affect mass transport, with significant and interesting effects on chemical waves. In this chapter, we consider three types of fields: gravitational, electric, and magnetic. These always exist, though their magnitudes are usually very small. As we shall see, small fields can have surprisingly large effects. Gravity is a ubiquitous force that all living and chemical systems experience. People largely ignored the profound effect that living with gravity has upon us until humans spent significant time in space. Bone loss and changes to the vascular systems of astronauts (Nicogossian et al., 1994) are still not well understood. Eliminating the effects of gravity is not easy. Enormous cost and effort have been expended to simulate gravity-free conditions in drop towers, parabolic airplane flights, or in Earth orbit. A simple calculation seems to suggest that gravity should have negligible influence on chemical reactions. The mass of a molecule is on the order of 10-26 kg, which translates into a gravitational force of about 10-25 N. We can compare this with the force of attraction between the electron and the proton in a hydrogen atom, which is of the order 10-8 N. Even allowing for shielding effects, the electrostatic forces that cause chemical bonds to be made and broken will always be many orders of magnitude stronger than gravitational forces. So gravity does not affect the fundamental atomic and molecular interactions, but it can drastically alter the macroscopic transport of heat and matter through convection, or macroscopic fluid motion. Natural convection is the movement of fluid as the result of differences in density, so that denser fluid sinks and less dense fluid rises. This motion is resisted by the viscosity of the medium, which acts like friction does in slowing the motion of solids. The study of convection is an entire area of physics, and we will touch only on a few aspects. The reader is referred to some excellent texts on the subject (Tritton, 1988; Turner, 1979).

This system was first developed by Lorenz (1960b). It is an elegant system that provides an introduction to the concepts of spectral modeling, based on the use of double Fourier series representations of the basic equations in a doubly periodic domain. Here we examine the barotropic vorticity equation.