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

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268 A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 149 (2005) 259–278 Fig. 7. Time evolution of the rate of secular cooling, d〈T 〉/dt, for different initial CMB temperatures, T CMB (0) = 3273 K (a), 3773 K (b) and 4273 K (c). A comparison is made between constant conductivity (black curves) and variable conductivity (f = 1). The lower absolute values of the variable conductivity models is consistent with the delay in secular cooling of the corresponding curves in Fig. 6. Increasing the initial CMB temperature results in a higher cooling rate in line with Fig. 6 and a much stronger time dependence of the the cooling rate. lated to the increasing fluctuations in the surface heat flow, which contribute to the secular cooling, with increasing Rayleigh number (van den Berg et al., 1993; van den Berg and Yuen, 2002). The delay in secular cooling of the variable conductivity cases, apparent in Fig. 6, is reflected in Fig. 7 by the consistently lower cooling rates for the variable conductivity curves labeled f = 1. In Fig. 8 we show time series of the temperature of the core heat reservoir for the same models as used for Fig. 6, i.e. three different initial core temperatures and five different conductivity models. These core cooling results show a similar trend as in Fig. 6 for the mantle. However, there is a difference in the sensitivity to the contribution of the radiative conductivity expressed in the amplification factor f. The core temperature seems more sensitive for increased f values. This is due to the fact that models with f = 5 (red curves) show already faster core cooling than the constant conductivity runs, whereas Fig. 6 shows that the switch to faster mantle cooling for increased f occurs later, between f = 5 and 10. An explanation of this different behavior is that the accelerating effect of f on mantle cooling, resulting in an increasing effect on the temperature contrast between mantle and core, is compounded with the enhanced cooling by radiative heat transport across the CMB for increased values of f. This sensitivity is also increasing for higher initial core temperatures, in agreement with the temperature dependence of the radiative conductivity. Time evolution of the CMB heat flux is shown in Fig. 9, for several model cases with the same initial core temperatures as in Figs. 6 and 8. The results show that the variable conductivity with f = 0 produces the lowest core heat flux. For the constant conductivity case, represented by the dashed line, the core heat flux is relatively high and for the variable conductivity cases with enhanced conductivity, for 10 the highest core heat flux is obtained. The trend in these core heat flux results is consistent with the corresponding core temperature curves shown in Fig. 8, in agreement with the fact that core temperature is obtained in our model by integrating the cmb heat flux according to (9). Evolution of the heat flux from the core has been studied mainly in parameterized models characterized by smooth time variations (Buffett, 2003). Our model results show a remarkably high fluctuation level of the core heat flux and one could speculate that such fluctu-
A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 149 (2005) 259–278 269 Fig. 8. Thermal histories of the core for the same models as shown in Fig. 5. Three curve groups are labeled with the initial core temperature. Each group includes results for different conductivity models. The trends in these curves illustrate the key role of radiative conductivity in controlling core cooling. The slowest core cooling is obtained for the f = 0 model without any radiative conductivity, k rad = 0, and the cooling rate increases with increasing relative contribution of k rad controlled by the amplification factor f. Furthermore this effect is stronger in a hotter earth in line with the temperature dependence ∂k rad /∂T > 0. ations could impact the geodynamo process and leave their marks in the paleomagnetic intensity record. 3.2. Influence from enhanced thermal radiative conductivity As discussed above, the parameter f represents a measure of the radiative contribution to the thermal conductivity, with f = 1 having the same value as the model presented by Hofmeister (1999). Fig. 10 compares the two-dimensional temperature fields for values of constant thermal conductivity, and variable conductivity with f = 1 and 0 (purely lattice conductivity) and f = 5 (enhanced radiative conductivity). It is clear that with variable k, the entire convective region is hotter, in comparison to constant k (Dubuffet et al., 1999, 2002). As was observed in the previous work using constant viscosity (van den Berg et al., 2002), we find that the Hofmeister variable conductivity results in a reduced overall convective vigor. The constant k case has stronger downwellings and earlier, betterdeveloped plumes. For different values of f, there is Fig. 9. Heat flow through the core–mantle boundary against time, for the same model cases as in Fig. 7. The core heatflow increases with the relative contribution of the radiative conductivity k rad in agreement with the core cooling histories shown in Fig. 7. Core heat flux is highly time dependent in these models with peak to peak values of about 100%. This is a result of the strong time dependence of cold downwellings cooling the hot core in these models which are largely cooled from above. still present a noticeable difference in the temperature fields between the constant k and variable k models, although less so in the enhanced k rad case f = 5, which has the fastest cooling rate of the variable k models considered here (Fig. 6). The viscosity profiles for the entire mantle are shown in Fig. 11 for constant conductivity, f = 1 and 5 and three different values of initial CMB temperature T CMB (0). It is interesting to note that a sharper low viscosity valley is produced by a lower temperature at the CMB. For constant viscosity Dubuffet et al. (2002) have noted that there is a bifurcation in the behavior of the convective solution, as f is increased beyond a certain value, which depends on the T CMB and on the amount of internal heating. In Fig. 12 we show the one-dimensional profiles of the horizontally averaged temperature 〈T 〉, the viscosity 〈η〉 and the thermal conductivity 〈k〉 for f = 0, 1 and 5. The temperature profiles show that the effect of introducing variable conductivity is to make the thermal
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van den Berg et al., 2005, Earth Planetary Science Letters.

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