Effect of Topography on Thermoelastic Deformations of the Earth’s Core: Temperature Field Corrections

The article presents a preparatory stage for solving the thermoelasticity problem for a halfspace with relief. The effect of the relief on diurnal and season variations of the temperature in the upper layer of the crust induced by temperature variations the in atmosphere is studied. The case of weak two-dimensional relief is discussed. In this paper, we say that relief is weak if: (1) the angle of inclination of a relief element to the horizon is small; (2) the thickness d of the heated layer is small compared to the radius of curvature of the line of the relief. For the diurnal mode, we have d ≈ 15 cm, for the seasonal mode, d ≈ 3 m. The heat equation with a boundary condition of the first kind is considered, and an approximate analytical solution is obtained. The results are compared with the numerical solution, which can be considered. The approximate formula gives a satisfactory result if: (1) the angle of inclination of a relief element to the horizon does not exceed 0.2 rad; (2) the curvature of the line of relief does not exceed 2.4 × 10^–2 m^–1 for the diurnal mode and 1.2 × 10^–3 m^–1 for the seasonal mode. In this case the relative discrepancy between the numerical and approximate analytical solutions is less then 4% if the depth ≤20d. Temperature variations at a depth of 20d are already almost completely absent: the amplitude does not exceed ~10^–9–10^–8°C. Therefore, the underlying layers do not significantly effect on displacements and tilts of relief elements located near the surface.