Chapter 17ConvectionVersion 0417.1.K.pdf, 19 July 2004. The same as 0217.1.K.pdf except for using an updated,2004 header file.Please send comments, suggestions, and errata via email to [email protected], or onpaper to Kip Thorne, 130-33 Caltech, Pasadena CA 9112517.1 OverviewIn Chaps. 12 and 13, we demonstrated that viscosity can exert a major influence on fluidflow. When the viscosity is large, and the Reynolds’ number R = LV /ν is low, viscousstresses transport momentum directly and the behavior of the fluid can be characterized bysaying that the vorticity (ω = ∇ × v) diffuses through the fluid [cf. Eq. (13.3)]. As theReynolds’ number increases, the advection of the vorticity becomes more important. In thelimit of large Reynolds’ number, we think of the vortex lines as being frozen into the flow.However, as we learned in Chap. 14, this insight is only qualitatively helpful because highReynolds’ number flows are invariably turbulent. Large, irregular, turbulent eddies transportshear stress very efficiently. This is particularly in evidence at turbulent boundary layers.Thermal conduction is a similar transport process to viscosity when viewed microscop-ically and is responsible for analogous physical effects. If a fluid is an efficient thermalconductor, heat diffuses rapidly through it just like vorticity diffuses through a highly vis-cous fluid. In fluids with more modest conductivity, the accumulation of heat drives thefluid into convective motion and the heat is transported much more efficiently by this mo-tion than by thermal conduction alone. As the heat transport increases, the fluid motionbecomes more vigorous and, if the viscosity is sufficiently low, the flow can also become tur-bulent. Again these effects are very much in evidence near solid boundaries, where thermalboundary layers can be formed, analogous to viscous boundary layers.In addition to thermal effects that resemble the effects of viscosity, there are also uniquethermal effects—particularly, the novel and subtle combined effects of gravity and heat.Heat, unlike vorticity, causes a fluid to expand and thus, in the presence of gravity, tobecome buoyant; and this buoyancy can drive thermal circulation or free convection in anotherwise stationary fluid. (Free convection should be distinguished from forced convectionin which heat is carried passively by a flow driven in the usual manner by externally imposed12pressure gradients, for example when you blow on hot food to cool it.)The transport of heat is a fundamental charcteristic of many flows. It dictates the formof global weather patterns and ocean currents. It is also of great technological importanceand is studied in detail, for example, in the cooling of nuclear reactors and the design ofautomobile engines. From a more theoretical perspective, as we have already discussed, theanalysis of convection has led to major insights into the route to chaos (cf. Sec. 14.5).In this chapter we shall describe some flows where thermal effects are predominant. InSec. 17.2, we shall modify the conservation laws of fluid dynamics so as to incorporate heatconduction. Then in Sec. 17.3 we shall discuss the Boussinesq approximation, which is ap-propriate for modest scale flows where buoyancy is important. This allows us in Sec. 17.4to derive the conditions under which convection is initiated. Unfortunately, this Boussi-nesq approximation sometimes breaks down. In particular, as we discuss in Sec. 17.5, itis inappropriate for application to convection in stars and planets where circulation takesplace over several gravitational scale heights. Here, we shall have to use alternative, moreheuristic arguments to derive the relevant criterion for convective instability, known as theSchwarzschild criterion, and to quantify the associated heat flux. We shall apply this theoryto the solar convection zone.Finally, in Sec. 17.6 we return to simple buoyancy-driven convection in a stratified fluidto consider double diffusion, a quite general type of instability which can arise when thediffusion of two physical quantities (in our case heat and the concentration of salt) canrender a fluid unstable despite the fact that the fluid would be stably stratified if there wereonly concentration gradients of one of these quantities.17.2 Heat ConductionWe know experimentally that heat flows in response to a temperature gradient. When thetemperature differences are small on the scale of the mean free path of the heat-conductingparticles (as, in practice, almost always will be the case), then we can expand the heatflux as a Taylor series in the temperature gradient, Fheat= (constant) + (a term linear in∇T ) + (a term quadratic in ∇T ) + .... Now, the constant term must vanish; otherwisethere would be heat conduction in the absence of a temperature gradient and this wouldcontradict the second law of thermodynamics. The first contributing term is thus the linearterm, and we stop with it, just as we do for Hooke’s law of elasticity and Ohm’s law ofelectrical conductivity. Here, as in elasticity and electromagnetism, we must be on thelookout for special circumstances when the linear approximation becomes invalid and beprepared to modify our description accordingly. This rarely happens in fluid dynamics, soin this chapter we shall ignore higher-order terms and writeFheat= −κ∇T , (17.1)where the constant κ is known as the coefficient of thermal conductivity or just the thermalconductivity; cf. Sec. 2.8. In general κ will be a tensor, as it describes a linear relationbetween two vectors Fheatand ∇T . However, when the fluid is isotropic (as it is for thekinds of fluids we have treated thus far), κ is just a scalar. We shall confine ourselves to this3case in the present chapter; but in Chap. 18, when describing a plasma as a fluid, we shallfind that a magnetic field can make the plasma’s transport properties be decidely anisotropic,so the thermal conductivity is tensorial.In this section we shall incorporate heat conduction into the fundamental equationsof fluid dynamics. This can be accomplished most readily via the conservation laws formomentum and energy.On the molecular scale, the diffusing heat shows up as an anisotropic term N1in themomentum distributions N (p) = N0+N1of particles (molecules, atoms, electrons, photons,...); cf., e.g., Eqs. (2.112) and (2.116). This anisotropic term is tiny in magnitude comparedto the isotropic term N0, which has already been included
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