168 15. Scaling Laws and Thermal Histories (Terrestrial Planets) Boundary Layer Analysis Although the heuristic argument presented at the end of last Chapter turns out to be basically sound, it is devoid of fluid dynamics (except for the critical Rayleigh number criterion). In particular, it says nothing about the large-scale circulation. We want to be able to think through the circulation for the whole system, including boundary layers. In the end, you must be guided by experiment; it is not possible to deduce this by pure thought! The following picture based on theory and experiment emerges for steady state Rayleigh-Benard convection: Now this 2D picture of a convecting layer has three regions that we must understand: the interior region away from the boundary layers and upwelling or downwelling motions, the boundary layers where the heat escapes or enters the entire fluid, and the buoyant sheets which are the localized regions of temperature anomaly and buoyancy anomaly in the interior of the layer. In the interior region, there is no source of vorticity (i.e. ∇ρ x g is zero). So we expect that the vorticity is roughly constant within each cell. If the upwellings and downwellings have a velocity of magnitude u, then the intervening flow will be a space-filling constant vorticity flow, which is nothing other than a rigid body rotation. However, the sense of this rotation clearly switches from one cell to the next. So the change in vorticity across a169 downwelling is about 2u/(L/2) or 4u/L, where L is a cell width (≈ cell height also). In the region of density anomalies, the vorticity ω obeys d2ω/dx2 = -(g/η)d(Δρ)/dx (15.1) from which it follows that the change in vorticity from the interior flow to the peak of the density anomaly is about gα(ΔT/2).δ/ν, which should be 2u/L. If the thickness of the thermal anomaly is 2δ in total, then; u ≈ 0.25gαΔTδL/ν (15.2) In the boundary layers we expect by continuity that roughly the same velocity applies. We thus expect the fluid to develop a cool surface layer that thickens by thermal diffusion. In the time it takes for the fluid to traverse top or bottom, about L/u, the thickness must become δ, whence δ ≈ 2(κL/u)1/2 (15.3) (where the factor of two comes from the z/2(κt)1/2 argument of the error function in the conductive solution).We also expect the mean heat flow to be about the total heat eliminated per unit area during this traverse, which is ~ 0.25ρCpΔT. δ, divided by the time in which it was eliminated, L/u. Using the fact that L/u≈ δ2/4κ , this can be rewritten F ≈ kΔT/δ (15.4) (This is of course equivalent to saying that the average boundary layer thickness is δ/2.) It is convenient to reference this relative to the conductive heat flow for the same temperature drop by defining a dimensionless number called the Nusselt Number. By definition, Nu = F/Fcond ≈ L/δ (15.5) because Fcond = kΔT/L. Recall also that we have a Rayleigh number for the entire system:170 Ra = gαΔTL3/νκ (15.6) Putting all these equations together (and doing a somewhat better job with the constants than I have done), one finds that: Nu ≈ 0.2Ra1/3 u ≈ 0.2(κ/L)Ra2/3 (15.7) These are asymptotic equations, i.e. they apply at very large Ra. The Nu-Ra relationship can be rewritten as F ≈0.2(gα/νκ)1/3ΔT4/3 (15.8) and notice (as expected) that this result is independent of the depth of the fluid. Recall that our previous heuristic argument argued for setting gα(ΔT/2)δ3/νκ ≈ 500 and F≈ k ΔT/δ. This predicts the same result except for a factor of two. The case analyzed above involves the particular assumption of a Rayleigh-Benard system with a free boundary condition, but our earlier heuristic argument suggests that the form of the results are more generally applicable. As a general principle, we can always expect that it is possible to write Nu in the form Nu = A(Ra/Rac)β (15.9) where A is some constant, typically a little larger than unity, and the exponent β is usually in the range 0.15 to 0.4. Obviously this exponent is very important at large Ra; it is often somewhat smaller than 1/3, especially if one attempts to allow for viscosity variation. The value of 1/3 is special: It is the only value for which the heat flow is independent of the depth of the fluid (cf. our heuristic model). Since the planets are primarily internally heated (not heated from below), it is important to ask how these results can then be applied. The answer is that they are still roughly correct if one uses a Rayleigh number defined in terms of the actual temperature drop in the system. Indeed our heuristic model was171 conceptually the same as an internally heated model (and different from the bottom heated case). Sometimes people like to use a Rayleigh number based on internal heating. If the heating rate per unit volume is Q then the steady state temperature rise predicted by the diffusion equation is about QL2/k, so the Rayleigh number is then by convention RaQ ≡ gαQL5/kνκ (15.10) Since it is clear that the heat flow should still be independent of L it follows that Nu varies as RaQ0.2 rather than RaQ1/3 . But this is a cosmetic difference, devoid of physical content. (Notice that the Rayleigh number defined in terms of internal heating can be much larger than that defined in terms of actual temperature difference, because ΔT may be much smaller than QL2/k. For example, QL2/k ~30,000K for Earth’s mantle.) The important difference between systems heated from within and heated from below lies in the nature of the upwellings and downwellings. When the fluid is heated from below, it has localized thermal anomalies in upwellings as well as downwellings. When the fluid is heated only from within, it has only the localized downwellings, since there is no heat flow and therefore no thermal boundary layer at the base. Real planets have mostly internal heating and (at least if one assumes whole mantle convection) with only the small bottom heating that arises from core cooling. As