Thermal evolution of the Earth

As discussed above, there is evidence that a dynamo similar to that at the present day has existed through the bulk of Earth history. It is therefore natural to enquire whether the prolonged life of the dynamo places constraints on the thermal evolution of the Earth. Anticipating the results of the sections below, it has recently become clear that the constraints are quite strong: generating a dynamo requires relatively rapid cooling of the core, while producing an inner core of the correct present-day size requires relatively slow core cooling [Buffett (2002); Gubbins et al. (2003)]. The parameter space which allows these two opposing constraints to be satisfied is relatively restricted, and in particular appears to require both a young inner core (<« 1.5Gyr) and, less certainly, significant (O(100ppm)) potassium in the core.

As discussed above, powering a dynamo requires the core cooling rate to exceed a given value. The core cooling rate depends on the rate at which the mantle extracts heat from the core. The ability of the mantle to extract heat depends, in turn, on the rate at which the mantle is cooling, and thus the behaviour of the near-surface boundary layer. Plate tectonics on the Earth is an efficient way of cooling the mantle; other planets, in which lateral motion of the surface material does not occur, probably cool much more slowly. This link, between the top 100 km of the Earth's mantle, and the behaviour of the dynamo, is both suprising and of fundamental importance. It also means that the evolution of the Earth as a whole has to be investigated in order to investigate the evolution of the core.

Modelling the thermal evolution of the Earth is a challenging problem. Although 3D numerical mantle convection models can be run, doing so for 4.5 Gyr is not yet possible. Alternatively, parameterised evolution schemes [e.g. Butler and Peltier (2000)] can be adopted, which consider only globally-averaged properties and thus run very much faster, allowing a proper exploration of parameter space. The disadvantage of this approach is that complications, such as compositional layering or vertical viscosity variations, are less easy to include.

Figure 7 shows one such parameterised thermal evolution model, which generates a present-day thermal structure similar to Fig 1 (c) while permitting a dynamo throughout Earth history. Figure 7(a) shows the temperature evolution of the core and mantle, and demonstrates the slow cooling regulated by radioactive decay in the mantle. The kink in the central temperature at 3.5 Gyr is due to the inner core starting to solidify. Figure 7(b) shows the evolution of the heat fluxes with time. The model present-day surface heat flux matches the observed value, and the CMB heat flux is 9 TW, in agreement with the arguments presented above. The core heat flux is high early on because of the presence of 400 ppm potassium, the effect of which is discussed below. Figure 7(a) also shows the net entropy production rate as a function of time, which is always positive, indicating a dynamo could have operated over the whole of Earth history. Figure 7(b) shows the inner core

Fig. 7 Parameterized thermal evolution model, modified from Nimmo et al. [2004], with 400 ppm potassium in the core. (a) Temperature variation and entropy production rate as a function of time. Tm is the mantle temperature at the CMB, Tc is the core temperature at the CMB, Ti is the temperature at the centre of the planet, or at the ICB if an inner core exists. The kink in Ti at 3500 Myr is due to the onset of inner core solidification. (b) Heat output and inner core size as a function of time. Qm is the surface heat loss, HmMm the mantle contribution from radioactive decay and Qc the core heat loss. The inner core size is normalised by the core radius.

Fig. 7 Parameterized thermal evolution model, modified from Nimmo et al. [2004], with 400 ppm potassium in the core. (a) Temperature variation and entropy production rate as a function of time. Tm is the mantle temperature at the CMB, Tc is the core temperature at the CMB, Ti is the temperature at the centre of the planet, or at the ICB if an inner core exists. The kink in Ti at 3500 Myr is due to the onset of inner core solidification. (b) Heat output and inner core size as a function of time. Qm is the surface heat loss, HmMm the mantle contribution from radioactive decay and Qc the core heat loss. The inner core size is normalised by the core radius.

growth history, demonstrating that it is young (1.1 Gyr) and at the correct present-day size. The entropy production rate increases when the inner core solidifies, due to additional release of latent heat and compositional convection. Prior to inner core formation, the dynamo was maintained by the relatively rapid cooling rate of the core, plus radioactive decay.

The above model produces results compatible with our understanding of present-day Earth structure and geodynamo history. However, it does so mainly because of the presence of 400 ppm potassium in the core. Similar models run without potassium generally result in an inner core which is much too large. This is because the heat released by the potassium reduces the rate at which the core cools and the inner core grows. In the absence of potassium, the core cooling rate has to be significantly reduced to generate an inner core of the correct present-day size. However, a lower core cooling rate and an absence of potassium means a reduction in the rate of entropy production (Fig. 6). There is thus a tradeoff between getting the correct inner core size (which requires slow cooling) and generating enough entropy to drive the dynamo (which requires rapid cooling).

This tradeoff is shown explicitly in Fig. 8, which plots the mean entropy production rate against the present-day inner core size. Except at large inner core sizes, increasing the entropy production rate also results in a larger inner core. Adding potassium to the core shifts the curves to higher

I 700

g 600

1 500

5 400

6 300 S

S 200

u g 100

-100

decre cooling ■

increasing* potassium

A

a • K=0 i i i

i i i

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0.2 0.4 0.6 0.É Dimensionless IC radius

0.2 0.4 0.6 0.É Dimensionless IC radius

Fig. 8 Tradeoff between time-averaged entropy production and present-day inner core (IC) size, normalised by core radius, from Nimmo et al. [2004]. Open symbols have a minimum rate of entropy production < 0, indicating an at least temporary cessation of the dynamo. Different points are for different mantle viscosity structures (and hence core cooling rates). Increased cooling rates lead to higher entropy production and larger IC sizes. Adding potassium (K) allows smaller inner cores for the same entropy production rate. Vertical line denotes present day IC size.

rates of entropy production for a given inner core size, because potassium both delays core solidification and is an additional entropy source. The curves demonstrate that none of the models lacking potassium are able to match both the entropy and the inner core size requirements simultaneously. It also turns out that none of the models with a correct present-day inner core size resulted in a core older than 1.5 Gyr.

The results presented here depend on a large number of parameters, many of which are poorly known. Furthermore, as discussed above, the parameterised calculations are unlikely to capture the full complexity of convection in the Earth. Nonetheless, the two main results — that the inner core is young, and that potassium is likely present in the core — are relatively robust. For instance, appealing to initially hotter core temperatures (rather than potassium) to delay inner core formation fails because the mantle is more efficient at cooling the core at higher temperatures.

Other authors have derived similar results using different techniques. For instance, Buffett [2002] found that obtaining an ancient inner core required a present-day heat flux across the CMB much lower than that inferred (Fig. 1c). To solve this problem he posited a significant amount of radioactive heat production, either within the bottom mantle boundary layer or in the core. Both Roberts et al. [2003] and Labrosse et al. [2001] examined the evolution of the core for specified CMB heat fluxes, and concluded that an inner core age of 1 ± 0.5 Gyrs was most likely in the absence of any radiogenic heating.

In summary, whether or not a dynamo operates is ultimately controlled by the ability of the mantle to extract heat. There appears to be general agreement that both the present-day thermal structure of the Earth, and the maintenance of a dynamo, are compatible with a present-day CMB heat flux of about 9 ± 3 TW [Buffett (2003); Labrosse and Macouin (2003); Nimmo et al. (2004)]. This heat flux implies an inner core age of < 1.5 Gyr. Parameterised thermal evolution models suggest that maintaining a dynamo over Earth history while producing an inner core of the correct size is difficult (Fig. 8), because the dynamo requires rapid core cooling, while the small inner core requires slow core cooling. A possible resolution of this paradox, which is supported by experimental results, is the presence of 0(100) ppm potassium in the core, generating 1.5-3 TW of radioactive heating at the present day.

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