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3. The total heat flux at the top of the convective region of the core q CMB is the sum of the conductive heat flux along the adiabat q ad and the convective heat flux q conv q CMB = q ad + q conv . q ad can be assumed to be laterally uniform. In contrast q conv is variable; here we assume it has a high and a low value q conv high and q conv low respectively. We suppose q conv high is found over some fractional area ƒ of the CMB. Then the mean heat flow at the CMB can be expressed as q̅ CMB = q ad + ƒ q conv high + (1 − ƒ) q conv low . We define the lateral variation of total heat flux as A = (q ad + q conv high )/(q ad + q conv low ) and the mean Nusselt number by Nu = q̅ CMB /q ad . Then the lateral variation of convective heat flux is given by B = q conv high /q conv low = [ ANu − 1 − ƒ ( A − 1)]/[ Nu − 1 − ƒ ( A − 1)]. Estimates for Earth are A ≤ 10 (from mantle convection calculations) and Nu = 1.1 [q̅ CMB = 20 mW/m 2 and q ad = 18 mW/m 2 (2)]. Assuming ƒ = 2.3% (the same as the experiments) for A = 1.5 we obtain B ≃ 7 and for A = 4 we obtain B ≃ 80. Hence a small lateral variation in q CMB implies a large lateral variation in q conv in the core.
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