158 14. Convection Theory; Adiabats and Boundary Layers Thermal Convective Instability We saw in the last chapter that in the presence of an unstable vertical density profile and no diffusive decay of the entity (heat, composition) responsible for it, there is a “direct” instability, i.e. all disturbances grow exponentially in time. When limited by viscosity, the characteristic timescale for exponentiation is about  ν/g(δρ/ρ)L, where the density anomaly is density gradient times characteristic lengthscale (i.e.  δρ/ρ~β/k; L ~ k−1). The exact definition of an unstable density profile must of course be given in the following way: Does an element of fluid, upon vertical displacement, find itself in an environment such that its buoyancy causes it’s motion to be accelerated? Clearly, this must ignore adiabatic expansions and contractions that arise because of the displacement to a new pressure; buoyancy (positive or negative) arises from deviations from adiabaticity or non-uniform composition. In this context, “adiabatic” means isentropic. In the specific case of thermal effects, we need to take into account the fact that the thermal anomalies are not merely advected but also spread diffusively. So let’s now do this. The heat diffusion equation including advection and assuming incompressible flow can be written:  ∂T∂t=κ∇2T − u .∇ T +QCp (14.1) where Q is the heat generation per unit mass. Let’s look now at the behavior of perturbations θ to the temperature field, assuming that the background state takes the form of a uniform temperature gradient relative to the adiabat ; i.e. T = Tad -βz + θ. (Notice that β is now a temperature gradient whereas previously we used it to express a density gradient. If positive, it represents an unstable case, assuming of course that the coefficient of thermal expansion α>0. Notice that β is not the actual temperature gradient because Tad is a function of z. In some cases, β will be tiny compared to dTad/dz!) The equation for linear perturbations becomes:  ∂θ∂t=κ∇2θ+ uzβ (14.2)159 (where, as always, non-linear terms are neglected because all the perturbation variables are infinitesimal). The associated density anomalies are:  δρ= −ρ0αθ (14.3) where α is the coefficient of thermal expansion. Treating temperature anomalies, etc. as varying as exp(ik.r+σt) as before, we get that  (σ+κk2)θ= uzβ (14.4) Looking back now at our analysis of the vorticity equation 13.20, we have:  (σ+νk2)(σ+κk2) =gαβ(kx2+ ky2)k2 (14.5) This has a different behavior from eqn 13.25 (RT instabilities): Positive solutions for σ are no longer assured for all positive values of the other parameters. Clearly, the onset of the instability (which you can think of as the lowest value of β for which σ≥0) is found by setting σ=0, and this corresponds to:  gαβ(kx2+ ky2)νκk6= 1 (14.6) But we can still ask about the best choice of wavevector. In a real system you must satisfy boundary conditions, and for a fluid confined between z=0 and z=L (say), this enforces a choice of π/L for the vertical component of the wavevector. (Actually, this is not obvious, but it turns out to be exactly so for “free” boundaries, meaning those for which the component of flow parallel to the boundaries is unimpeded). But we are at liberty to choose the horizontal components (if our fluid is not in a box with sidewalls). So the eigensolutions of interest have the form: u(x,y,z) = A sin (kxx) cos (kyy) cos(πz/L) v(x,y,z) = B cos (kxx) sin (kyy) cos(πz/L) w((x,y,z) = C cos (kxx) cos (kyy) sin(πz/L) (14.7)160 for x, y and z components of the velocity field. A, B and C are arbitrary but small and must satisfy Akx+Bky+C(π/L) =0 to assure divergence-free flow. Notice that the chosen form satisfies the boundary conditions. The lowest value of temperature gradient will be achieved for the choice of wavevector that maximizes the quantity y/(1+y)3 where y is defined as  (kx2+ kx2) / kz2 . This value of y is 1/2 (which means that the characteristic wavelength of the horizontal variation in temperature, etc. is 23/2L) and the resulting value of y/(1+y)3 is 4/27. The relative values of kx and ky are arbitrary. Of course, this is the same as minimizing the value of (1+y)3/y, which is the graph shown. So putting this all together, we get  Rac=27π44= 657.5..Ra ≡gαβL4νκ (14.8) where Ra is called the Rayleigh number and the subscript “c” refers to its critical value (the value below which no motion will be amplified). The “proof” above is actually technically incorrect, because we have not (1+y)3/y ≡gαβ(L/π)4/ νκ 27/4161 explicitly established that our solution satisfies the boundary conditions, but it turns out to be correct nonetheless for free boundaries. The critical Rayleigh number is different for other boundary conditions but typically of order a thousand. The system we have analyzed is called Rayleigh-Benard convection, because the “starting state” of a uniform temperature gradient is appropriate to heating a fluid from below. Notice that this result applies for both low viscosities and high viscosities. But the quantitative difference is enormous for planetary scales (large L). Physical Interpretation As our discussion of the Rayleigh-Taylor problem revealed, there is a characteristic timescale for growth of the flow. It is of order  ν/g(δρ/ρ)L. But of course, this must be shorter than the characteristic diffusion time  L2/κ since otherwise the instability will be shut off. Clearly the ratio of timescales gives something with the ingredients of the Rayleigh number. So the Rayleigh number can be thought of as the ratio of timescales (thermal timescale divided by dynamic timescale). If it is too small then the thermal diffusion will win out over the buoyancy driven flow and the disturbance will not grow (i.e., convection will not occur). Application of the Rayleigh Number Example # 1: Low Viscosity Suppose we have L~1000km, ν~κ~ 0.01cm2 /sec, α~ 10-5 K-1 and g~1000 cgs, then Ra ~ 1000 requires a