A.A. Stepanov $^{a}$ and S.V. Starchenco $^{b}$
$^{a}$ Department of Physics, Rybinsk State Academy of Aviation Technology, Russia. $^{b}$ Geophysics Observatory of United Institution of the Earth RAS, Borok, Russia
Heat flow at core-mantle boundary (CMB) determines the power and type of the geomagnetic dynamo. This flow is very badly estimated yet - from 1 TW (as the lowest possible sub adiabatic flow) to 20 TW (as the highest super adiabatic flow).Here we present reliable estimations of this CMB flow and other unknown heat parameters using well-known thermal data for the Earth’s mantle and surface. We use the known average superficial contents of the radioactive elements, well-accepted reference points of the temperature in the upper mantle and 44 TW as long-time value for the surface heat flow. CMB temperature is our only fitting parameter that we took at 3000, 4000 and 5000$^o$C. The Earth was divided on a few cooling spherically symmetric shells with uniform physical properties. We have determined the effective heat conductivity and heatsources solving nonlinear inverse problem for the second order heat-transfer equation with both boundary conditions on the outer boundary of a shell. The time-function of temperature, and also time-function of radioactive sources power distribution, was expanded in Taylor series up to its first order. That has provided us with good enough accuracy $\le 10$\% on the time-scale averaged over a billion years since characteristic heat transport time $\sim$ 10 billion years. Resulting nonlinear system has $2n$ equations ($n$ is the number of the shells considered) for effective heat conductivity $\kappa_n$ and for the heat source profiles $(q_0-q_1 t)q_n (r)$ with constant scaling values of $q_0, q_1$ known from the surface radioactivity. Solving the system we use that all the heat flows are continuous. Besides, on the shell’s boundaries, we relate temperature (prescribed by data) to its derivative by using the exact solutions for cooling sphere and spherical shell. We investigated large variety of one-shell and two-shell thermal models of the Earth in details. The most realistic models with CMB temperature in 4000 C have the heat flow at the core in interval from 4 to 7 TW. The effective heat conductivity grows with depth in the lower mantle and its average value is $\sim$60 W/(m K) for all one-shell models and from 20 to 70 W/(m K) for two-shell models. Major part of the radioactivity heat could be concentrated near the surface of the Earth. Considering in our particular model the linear decrease of that heat from the surface value to zero we get an effective depth at $\sim$15 km that match well to the estimations of different authors, who used others methodic. However, our realistic and rather small value of heat source power in the lower mantle grows a little with depth. The other unusual effect of our reliable modeling is growing of the modern CMB heat flow with time, while the flow on the Earth’s surface decreases with time.