As an example of a two-dimensional heat conduction problem, we solve for the temperatures beneath the surface in a region where there are lateral variations in surface temperature. Such surface temperature variations can arise as a result of topographic undulations and the altitude dependence of temperature in the Earth’s atmosphere. Horizontal surface temperature variations also occur at the boundaries between land and bodies of water, such as lakes or seas. A knowledge of how surface temperature variations affect subsurface temperature is important for the correct interpretation of borehole temperature measurements in terms of surface heat flow. Consider again a semi-infinite half-space in the region y ≥ 0. The surface is defined by the plane y = 0. For simplicity, we assume that the surface temperature Ts is a periodic function of x (horizontal distance) given by Ts = T0 + ∆T cos 2πx λ , (4.53) where λ is the wavelength of the spatial temperature variation. The situation is sketched in Figure 4–16. We also assume that there are no radiogenic heat sources in the half-space, since our interest here is in the nature of the subsurface temperature variations caused by the periodic surface temperature. To determine the temperature distribution, we must solve Equation (4–51) with the boundary condition given by Equation (4–53).
We assume that the method of separation of variables is appropriate; that 4.12 Subsurface Temperature 267. An undulating surface topography results in surface tempera- ture variations that extend downward.
T (x, y) = T0 +X(x)Y (y).
In order to satisfy the surface boundary condition, we must have X(x) = cos 2πx λ ; (4.55) that is, the horizontal variations in temperature are the same at all depths.
When Equations (4–54) and (4–55) are substituted into Equation (4–51), we obtain
0 = −4π2 λ2 Y + d2Y dy2 , (4.56) which is an ordinary differential equation for Y . The general solution of this equation is Y (y) = c1e −2πy/λ + c2e 2πy/λ, (4.57) where c1 and c2 are the constants of integration. Since the temperature must be finite as y → ∞, we must require that c2 = 0. To satisfy the boundary condition given in Equation (4–53), it is necessary that c1 = ∆T .
The solution for the temperature distribution in the half-space is T (x, y) = T0 + ∆T cos 2πx λ e−2πy/λ. (4.58) The temperature disturbance introduced by the surface temperature variation decays exponentially with depth in a distance proportional to the horizontal wavelength of the surface temperature variation.
The energy equation is linear in the temperature. Therefore, solutions to the equation can be added, and the result is still a solution of the energy equation. This is known as the principle of superposition. If the temperature in the continental crust is given by Equation (4–30) but the surface temperature has a periodic variation given by Equation (4–53), the temperature 268 Heat Transfer distribution in the crust is obtained by adding Equations (4–30) and (4–58): T = T0 + qmy k + ρH0h 2 r k (1 − e−y/hr) + ∆T cos 2πx λ e−2πy/λ. (4.59)
This result satisfies the applicable energy equation (4–49) and the required surface boundary condition (4–53).
The analysis in this section can also be used to determine the effect of small amplitude, periodic topography on the near-surface temperature distribution. This problem is illustrated in Figure 4–17. The topography is given by the relation h = h0 cos 2πx λ (4.60)
We assume that the atmosphere has a vertical temperature gradient β so that the surface temperature Ts is given by Ts = T0 + βy y = h.