CHAPTER 1

A Model of Rayleigh-Bénard Convection

There are two basic types of fluid flows within planets and stars that are drivenby thermally produced buoyancy forces: thermal convection and internal gravitywaves. The type depends on the thermal stratification within the fluid region. TheEarth's atmosphere and ocean, for example, are in most places convectively stable,which means that they support internal gravity waves, not (usually) convection (butsee Chapter 7). On warm afternoons, however, the sun can heat the ground surface,which changes the vertical temperature gradient in the troposphere and makes theatmosphere convectively unstable; the appearance of cumulus clouds is an indicationof the resulting convective heat (and moisture) flux. Thermal convection likelyalso occurs in the Earth's liquid outer core, which generates the geomagnetic field,and, on a much longer time scale, in the Earth's mantle, which drives plate tectonicsand, on a much shorter time scale, initiates earthquakes and volcanic eruptions.Thermal convection is seen on the surface of the sun and likely occurs in the outer30% of the solar radius, where solar magnetic field is generated. Below this depthbuoyancy likely drives internal gravity waves. Rotation strongly influences the styleof the convection and waves in all of these examples except the mantle, which isdominated by viscous forces.

Computer simulation studies, over the past few decades, have significantly improvedour understanding of these phenomena. Some studies, like those for theatmospheres of the Earth and sun, have provided physical explanations and predictionsof the observations. Others, like those for the deep interiors of the Earth andsun, have provided detailed theories and predictions of the dynamics that cannot bedirectly observed. As computers continue to improve in speed and memory, computerprograms are able to run at greater spatial and temporal resolutions, whichimproves the quality of and confidence in the simulations. Numerical and programmingmethods have also improved and need to continue to improve to take fulladvantage of the improvements in computer hardware.

1.1 BASIC THEORY

We begin with a simple description of the fundamental dynamics expected in afluid that is convectively stable and in one that is convectively unstable. Then wereview the equations that govern fluid dynamics based on conservation of mass,momentum, and energy.

1.1.1 Thermal Convection and Internal Gravity Waves

The thermal stability of a fluid within a gravitational field is determined by itshorizontal-mean (i.e., ambient) vertical temperature gradient. The classic way ofdescribing this is to consider a fluid in hydrostatic equilibrium, i.e., the weightof the fluid above a given height (per cross-sectional area) is supported by thepressure at that height. Therefore, the vertical pressure gradient is negative. (Asusual, "vertical" here and throughout this book refers to the direction of increasingheight or radius, opposite to that of the gravitational acceleration.) In the interiorsof planets and stars the horizontal-mean density and temperature also decreasewith height. The question is how does the vertical temperature gradient of this fluid(atmosphere) compare with what an adiabatic temperature gradient would be.

Consider a small (test) parcel of fluid (Fig. 1.1) that, at its initial position (1), hasthe same pressure, density, and temperature as the surrounding atmosphere at thatposition. Imagine raising the parcel to a new height (2), fast enough so there is noheat transfer between it and the surrounding atmosphere but slowly enough that itremains in pressure equilibrium with its surroundings; that is, its upward velocity ismuch less than the local sound speed. Assuming this process is reversible and alsoadiabatic since there is no heat transfer, the parcel's entropy remains constant whilerising; that is, this is an isentropic process. However, since it remains in pressureequilibrium with the surroundings, its density and temperature both decrease as itrises because the decrease in pressure causes it to expand. If, when reaching its newhigher position (2), its temperature has decreased more than the temperature of thesurrounding atmosphere has decreased over that change in height, its density therewill be greater than the density of the surrounding atmosphere there (assuming atypical coefficient of thermal expansion). Therefore, the parcel will be antibuoyantand, when no longer externally supported, will fall. As the parcel falls its temperatureincreases faster than the surrounding temperature and when it passes the initialposition its temperature exceeds the temperature of surrounding atmosphere, causingthe now buoyant parcel to eventually stop falling and then to start rising. Thisprocess of...