van den Berg et al., 2005, Earth Planetary Science Letters.

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274 A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 149 (2005) 259–278 in (6). This quantity represents an integrated measure of the variation with depth of the parameters controlling conductive heat transport, conductivity and resistivity. Since the depth variation of the resistance is monotonic, its dependence on the parameterization of the conductivity model, like the f parameter value, is more straight forward than the conductivity or resistivity profiles. Like the electric resistance of a layer corresponds to the electric voltage required to drive a unit current through the layer, the thermal resistance corresponds to the necessary temperature contrast to drive a unit heat flow through the layer. The simple behavior of the thermal resistance in the thermal boundary layers may provide a basis for the development of parameterized convection models including effects of variable conductivity on thermal history. Fig. 15 shows 1D depth profiles of conductivity, resistivity and corresponding thermal resistance through the top and bottom thermal boundary layers for different mantle convection models. Two cases with contrasting contribution of the radiative conductivity are shown. The green curves correspond to a purely phonon conduction case (f = 0) and the blue curves are for strongly enhanced radiative conductivity (f = 10). The black curves indicate the constant conductivity reference case. The trend in the models for enhanced radiative conductivity is clearly reflected in the monotonic resistance profiles. In the top thermal boundary layer, top frames, the contribution from the radiative conductivity controls the resistance profile (c) where the constant conductivity case is intermediate between the contrasting variable conductivity models. This reflects the trend in the cooling curves for the mantle shown in Fig. 5, which explains the strong impact of k rad in speeding up secular cooling, through its influence on the low conductivity zone and the resulting thermal resistance of the lithosphere. In the bottom boundary layer, bottom frames, a similar relation exists except that the constant conductivity case has the highest resistance values, corresponding to the minimum value of the conductivity shown in frame (d). 4. Discussion and conclusions We have developed a core–mantle coupling convection model within the framework of a cartesian 2D geometry. This model has many realistic transport properties built in, such as variable thermal conductivity and variable viscosity. It does not have surface plates, phase transitions and chemical heterogeneities. But, nonetheless, this study will shed some light on the nature of the thermo-mechanical structure in the deep lower mantle. 4.1. Summary of important findings Variable thermal conductivity affects both conductive and convective cooling mechanisms in the mantle. Introducing pressure- and temperature-dependent thermal conductivity along with temperature- and pressuredependent viscosity into the mantle convection model results in several important changes in the cooling behavior and mantle flow patterns: 1. The secular cooling rate of the mantle is lower, using the Hofmeister conductivity model (Hofmeister, 1999) (f = 1) than with constant thermal conductivity. Heat flux at the surface is reduced. Increased f (greater radiative contribution to thermal conductivity) tends to increase the cooling rate relative to the Hofmeister model. This thermal conductivity mechanism, acting in concert with partial melting (Korenaga, 2003) can help to retain a lot of the primordial heat of the Earth. Therefore, variable thermal conductivity can keep the Urey ratio low, which is consistent with highly depleted heat-producing elements in the mantle (Jochum et al., 1983), favored by geochemists. 2. We have shown that the cooling delay of the variable conductivity models is closely linked to the formation of a low conductivity zone (LCZ) at shallow depth. This LCZ results from the negative temperature derivative of the dominant lattice conductivity. These results imply that purely pressuredependent conductivity models, characterized by a monotonic increase of the conductivity with depth are not suitable for long-term thermal history calculation (Anderson, 1987; Steinbach, 1991; Solheim and Peltier, 1994; Tackley, 1996; van Keken, 2001; Butler and Peltier, 2002). 3. Our model results show a high fluctuation level of the heat flow from the core into the mantle, with higher fluctuations for models with enhanced radiative conductivity. We speculate that such fluctuations could leave an imprint in the paleomagnetic
A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 149 (2005) 259–278 275 intensity record, through a possible influence on the geodynamo. 4. A higher value of the initial CMB temperature, T CMB (0), leads to a more stable boundary layer due to the increase in radiative conductivity with temperature. It is important to note that core–mantle boundary temperature is also important for boundary layer stability (Sevre et al., 2002). 5. A higher initial core–mantle boundary temperature leads to faster secular cooling of the mantle and faster core cooling. 6. The local maximum of viscosity (viscosity hill) in the lower mantle and the region of low viscosity below the hill also change with variable k. The viscosity hill is smaller for variable k compared with constant k. Increased T CMB (0) (or decreased viscosity) also leads to a decrease in the size of the viscosity hill in the lower mantle and at the same time maintains a shallower viscosity gradient in the lower mantle. 7. There is a greater predominance of smaller convective upwellings for variable k, especially in cases with a high initial T CMB (0). In this model, large plumes are less prevalent, and the lower mantle may have less capacity to create superplumes. 8. The geotherm in the deep mantle becomes superadiabatic (da Silva et al., 2000) with enhanced values of radiative thermal conductivity, which controls the magnitude of the thermal gradient in the lower mantle (Yamazaki and Karato, 2001). Recent suggestion by Badro et al. (2004) has added support to the importance of radiative heat transfer in the deep mantle. These results clearly demonstrate that temperatureand pressure-dependent thermal conductivity, though small in magnitude, can not be neglected in mantle dynamics and planetary thermal evolution, through the combined nonlinear feedback interactions among thermal conductivity, temperature and viscosity. 4.2. Applications to the core We have found that with core coupling the heat flux at the CMB depends on both conductivity in the lower boundary layer and the evolving temperature difference across the lower boundary layer, both of which are significantly impacted by variable thermal conductivity. The heat flux at the core–mantle boundary shows large fluctuations and is reduced for variable thermal conductivity with a normal amount of radiative conductivity (f = 1). These results would suggest that the heat flux out of the core may actually be less than that predicted by models with constant conductivity. If less heat is conducted from the core into the mantle, this would delay draining the core’s internal energy supply. This would decrease the minimum necessary concentration of radioactive elements such as 40 K in the core. However, the CMB heat flux with variable conductivity is strongly dependent on initial CMB temperature and on the relative strength of radiative conductivity in the lower mantle, which is represented in this model by the parameter f (e.g. Dubuffet et al., 2002). These parameters are not known with enough certainty to provide tight enough constraints on the CMB heat flux. 4.3. The role in 1D parameterized convection calculations Because of the nonlinearities in mantle rheology, mantle thermal conductivity and mantle convection, thermal evolution with convection is definitely influenced by its initial temperature condition for a period usually called the thermal adjustment time (Solomatov, 2001). After the thermal adjustment time, thermal histories calculated from different initial conditions should converge. Early thermal evolution models based on parameterized convection, found a thermal adjustment time of less than 1 Gyr (e.g. Schubert et al., 1979); however, when more complex rheology is taken into account, the thermal adjustment time for parameterized models becomes much longer (see discussion by Solomatov, 2001). Our 2D convection results for different values of initial CMB temperature indicate that the initial thermal state affects mantle thermal evolution for a period of time greater than the age of the Earth. After around 5 Gyr, there is still a significant difference in interior temperature, core temperature, and thermal profiles for cases with different initial core CMB temperature. This implies that for a fully convective system with strongly temperature- and pressure-dependent viscosity and variable thermal conductivity, the initial thermal state is an important parameter to be considered, in modeling thermal evolution and this cannot be neglected, as in early hypotheses of fast thermal adjust-
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