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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 a

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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:

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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 ≈ 0.2Ra

1/3

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 was

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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 RaQ

0.2 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 a consequence, plumes are a relatively unimportant part of the convecting system. (They may have an importance for volcanism that is out of proportion with their importance for cooling, however, and we will return to this.) Large Viscosity Variations and Stagnation. Viscous flows tend to distribute stress uniformly in regions away from buoyancy sources. This means, for example, that there is very little strain in regions of very high viscosity. If those regions of high viscosity are all connected together, e.g. in a planet-encircling layer, then a stagnant layer will form. This is what happens on one-plate planets (all solid planets other than Earth, to a greater or lesser extent). From the point of view of scaling

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laws for heat flow, it then makes sense to divide the upper "lid" of the planet into an immobile zone (conductive and non-circulating) and an active thermal boundary layer.

In this picture, the criterion for convection will be (approximately) Ra = gαΔT2 .δ2

3/ν(T-ΔT2/2)κ≈500 or 1000 (15.11) Notice that this makes use of the viscosity half way through the boundary layer. In practice, the formation of a stagnant region (meaning a region that exhibits very little flow) requires extreme viscosity differences (of about four orders of magnitude or more). Numerical simulations show that the appropriate choice of ΔT2 = 9/γ where γ≡-dnν/dT. One then finds (in the usual way) that F = 0.5k(gα/νi κ)1/3γ−4/3 (15.12) where νi must be interpreted as the viscosity immediately below the boundary layer. This is smaller than the previous estimate for constant viscosity convection by a factor of order (γΔT)4/3. This is a large factor! The implication of this is that convection with a stagnant layer is much less efficient than a system where the convection involves the surface layer.

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Relationship to Plate Tectonics Somehow, Earth manages to involve the surface layer despite the enormous viscosity contrast. This is not well understood. It is almost as if Earth had constant viscosity convection! Indeed, many numerical simulations have had partial success describing mantle convection using this approach. The formulae above suggest that (other factors being equal) planets with plate tectonics will “run colder” (i.e., cool more efficiently) than planets with a stagnant lid. As best we can tell, Earth is currently the only planet with plate tectonics. One way to think about plate tectonics is to ask where is the greatest impedance to motion? It might be at subduction zones. But the remarkable fact is that plates seem to sink at roughly the velocity you would predict if subduction zones were weak. In this sense, the fluid dynamical predictions above may have merit (i.e., some applicability). However, it is true that plates tend on average to be larger than you might have expected based on convection scaling, suggesting that there is indeed some inhibition to convection that “looks like” constant viscosity convection. Increased aspect ratio is disadvantageous. One way to think about this is to look at convection from the point of view of it’s energy budget: Gravitational energy release = viscous dissipation of the internal flow + work done deforming the plates (primarily at subduction zones) Ordinary thermal convection ignores the second term on the RHS. When this term becomes large, it is favorable to have fewer plates. When this term becomes prohibitively large, you have one plate and revert to “ordinary” thermal convection, but with a much smaller energy budget (because the temperature anomalies are smaller). It is possible that the efficiency of the plate tectonic style of heat expulsion will change with geologic time, so simple scaling laws are unlikely to be universal. The State of the Terrestrial Planets based on Simple Scaling Laws

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If we apply the Nu-Ra scaling to Earth (ignoring for now the profound question of whether this is legitimate for describing plate tectonics) we can ask what viscosity the earth must have to eliminate the mantle heat flow of around 70-80erg/cm2.sec. The nominal conductive heat flow is [4 x 105 erg/cm.sec.K] .[1500K]/[3000km], or about 2 erg/cm2.sec. Thus, Nu is about 35. The value of the Rayleigh number Ra ≈ [980].[2 x 10-5].[1500K].[3000km]3/[0.01][1 x 1022ν22] ≈ 8 x 106/ν22 (15.13) where ν22 is the viscosity in units of 1022 cm2/sec. Inserting in a Nu-Ra scaling law, we find that 35 = A(8000/ν22)

β (15.14) For example, A=2 and an exponent of 1/3 implies a viscosity of ν22 of around unity (actually about 1.5). This is plausible from the point of other independent measures of viscosity, primarily post-glacial rebound. So scaling laws seem to work! We will apply them below to simple thermal evolution models of the earth. Other terrestrial planets might more reasonably be described by the stagnant lid regime and the viscosity is smaller (interior is hotter), other things being equal. This might create a problem because it may predict excessive melting and the scaling laws will then break down. The scaling laws ignore the effect of melting. But we can still maybe compare stagnant lid planets of different size with each other. Plausibly, if x is the planet radius in units of Earth radius, say, then g∝x and F∝x, so νi ∝g/F3 ∝ x-2, so smaller planets are colder (higher viscosity) than larger planets. Of course, if x is small enough, this will begin to become more complicated because the boundary layer becomes a substantial fraction of the radius and/or lots of melting takes place and/or conduction can get the heat out. Thermal Evolution Models for Earth (or Earthlike Bodies)

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We have seen that despite all the complexities of real planets, scaling laws for heat flow seem to have some validity. The reason is that scaling laws focus on mean, global properties of the flow and are insensitive to the size of the convection cells and the form that the circulation takes (e.g. stable cells vs. chaotic motions, etc.) Lets proceed then to examine the consequences of the first law of thermodynamics for the thermal history of the earth:

�

ρCpdTdt

= Q0e− t /τ − 4πR2F /Vm

F ≈ k(T −Ts)R

RaRac

⎛

⎝ ⎜

⎞

⎠ ⎟ β

Ra ≈ gα(T −Ts)R3

ν (T)κ

ν(T) = ν 0 expATmT

⎡ ⎣ ⎢

⎤ ⎦ ⎥

(15.15)

In the first equation, the LHS represents the heat content of the mantle (T is some mean mantle temperature), the first term on the RHS is the heat input by radioactive decay with some average half life (of order 2 billion years) and the second term is the heat loss by convection, based on a scaling law that involves an Arrhenius relationship for the viscosity. Vm is the volume of the mantle and Tm is the melting point; A is of order 25. The rest should be self-evident. We can non-dimensionalize these equations into the following simple form:

�

dxdt

= 1τQe−t /τ − x

4 / 3

τKexp[−βA(1

x− 1x0)] (15.16)

where x=T/Tm, x0 is the current value of x (about 0.7), τQ is a characteristic time scale for heating the earth from absolute zero up to the melting point using the initial radiogenic heating rate (and is of order 1 to 2 billion years), and τK is a "Kelvin" time (of order 10 billion years), a measure of the time it would take for the earth to cool if the current heat flow were sustained by cooling alone. Two typical examples of the outcome of these calculations are shown on the next page.

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There are several general principles illustrated by these kinds of models: (i) Any initial high temperature is quickly eliminated by high heat flow. In the first example shown, the earth starts out at the melting point and rapidly cools because radiogenic heat production cannot sustain this state. In the second example shown, the earth starts out much cooler but rather soon merges onto the same temperature history as the hot case. The implication is that the earth tends to forget its initial condition. (ii) Because the radiogenic heating declines with time, so must the heat flow and the mantle temperature required to sustain that heat flow. Thus the total heat content of earth declines with time. This decay of the heat content contributes significantly to the heat flow. This was recognized long ago by Harold Urey, and accordingly we define the Urey Number to be the ratio of current total radiogenic heat production to current total heat output. In steady state, this would be unity for a planet that had little ability to store heat. In Earth, the calculations shown here predict that the Urey number is perhaps about 0.8. It is significantly non-unity despite the very small temperature decline per billion years because the earth has such a large heat content: enough to sustain the earth's current heat flow for ten billion years even if the earth had zero radiogenic heating! As we earlier discussed, geochemical arguments suggest an even lower Urey number (since we are unable to explain more than about one half the earth's heat flow by potassium depleted but otherwise “chondritic” heating rates.) Simple models do not explain the “observed” Urey number of 0.5. A layered model (in which the lower mantle layer is currently cooling rapidly) may succeed. Or maybe the current heat flow from Earth is higher than average (a fluctuation on a several hundred million year timescale). Or perhaps the correct description of plate tectonics is more complicated than the scaling laws used here, e.g. the models of Korenaga "Energetics of mantle convection and the fate of fossil heat," Geophys. Res. Lett., 30(8), 1437, doi:10.1029/2002GL016179, 2003 (iii) The models predict a cooling by less than 100K per billion years (perhaps only 40K/Ga at present). While small, this is enough to suggest much more melting and volcanism in the Precambrian. It is not well supported by geological evidence for continental regions, suggesting that most of the excess heat flow is taken up in the ocean basins. We have no geological record for early earth ocean basins, of course.

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(iv) The model predicts much more mixing of the mantle in the past than at present. To see this, note that the temperature drop driving convection does not change much through geologic time but the viscosity changes by a large factor. The Nu-Ra scaling laws indicate that F ∝ ν-1/3 and u ∝ ν-2/3 so it follows that u ∝ F2. This is a strong dependence; it means that convective velocities (and plate velocities?) may have been an order of magnitude larger at 3.5 Ga. This means that the preservation of isotopic anomalies is much harder than people often assume; since they tend to base those arguments of preservation on present day estimates of convective circulation. More happened in mantle dynamics in the first billion years than during all subsequent geological time. However ,there is some doubt about whether the scaling law used is correct (cf. Korenga paper referenced above). (v) Thermal evolution models can also be used to estimate the rate at which the core cools. It is commonly assumed (but not known for certain) that the cooling of the core drives growth of inner core freezing and is thus responsible (both directly and indirectly) for the energy source of the geodynamo. By these arguments, the heat flow through the core-mantle boundary is about 10% (or possibly even less) of the total heat flowing through the Earth's surface. In this sense, the mantle can truly be said to be a fluid heated from within rather than from below. Of course, we don't know for certain whether the core possesses any radiogenic heat sources. Current estimates of the energy budget for earth’s core (based on the need to maintain the magnetic field over geologic time) suggest a higher core heat flow and a possible need for additional energy sources in the core. Perhaps the core is cooling unusually fast “now” (meaning the last billion years, say) but that does not solve the problem for earlier times, since we know Earth has had a magnetic field for most of its history.

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Problems 15.1 Suppose that a planet has plate tectonics, but then the plates “freeze up.” Approximately how much time must elapse before the planet can re-equilibrate into steady state convection with stagnant lid regime? Solution: As explained in the text, plate tectonics is more efficient in heat transport by a factor of ~(γΔT)4/3 . But the planet has a fixed heat flux (roughly speaking) and that means it will adjust to a higher mantle temperature if it has stagnant lid convection. Let Tpt be the mantle temperature in the plate tectonic regime and Tsl is the mantle temperature in the stagnant lid regime for the same heat flow. Then (equating heat flow formulas for the two convective regimes), we get ν(Tsl)-1/3 ~ (γΔT)4/3 ν(Tpt)-1/3 where ν is the mantle viscosity. Taking the log and recalling that γ =-dlnν/dT; we get Tsl-Tpt ~4ln(γΔT)/γ. For γ-1 =40K and ΔT = 1200K this gives 540K. This is probably an overestimate but if it were correct then you would have to wait for radioactivity to heat up the interior by that much (or for the steady state heat flow to decrease by large factor). This would take several billion years or even not be possible (because of the decaying heat sources.) Realistically, the viscosity of the asthenosphere is lower by a factor of ten or so relative to the value appropriate to “plate tectonics” even for the same mantle temperature and as a result a more realistic estimate might be more nearly 100 or 200K and a billion year delay at the current epoch for a planet like Venus (where this very process has been suggested.) 15.2 One simple model for tidal heating of Europa’s ice shell assumes that the heating rate Q (in erg/cm3.sec) is uniform throughout the ice and that the tidal or radiogenic heating deeper down (in an ocean or a rocky core) is negligible. (a) Confirm that the steady state conductive solution for temperature T in this ice layer is T = Ts + Qz(D-z/2)/k where Ts is the surface temperature (100K), z is the depth, D is the ice thickness and k is the thermal conductivity (assumed constant). Since D<<Europa’s radius, you don’t need to worry about spherical geometry. (b)The ice is bounded below by an ocean, the temperature of which should be independent of Q. Why? Confirm that as a consequence, D ∝ Q-1/2. (c) Hence show that there is an upper bound to the value of Q for which convection will occur in the ice. (This is counterintuitive! One normally expects convection to be the mechanism of choice for high heat flows.) Assume that the parameter choices are such that one is always in the regime where the conductive solution predicts an ocean beneath the ice. (There will of course also be a lower bound to the Q for which convection can

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occur, but this is in the regime where there is no ocean and the ice is very thick but cold and highly viscous.) (d) Determine the thinnest ice layer for which convection will occur. For this purpose, assume that convection occurs in only the lowermost (warmest) half of the ice layer, where the mean viscosity is 1015 cm2/sec. [Use α =10-4, κ=10-2 cm2/sec, and work out g and ΔT. You can’t use the total ice thickness in the Rayleigh number because a lot of that ice is very cold and thus has high viscosity.] Comment: Convection in the ice shell of Europa is now widely accepted. However, the explanation of how Europa behaves requires a model for the tidal heating (chapter 17) and cannot be straightforwardly analyzed by treating the heat flow as uniform.