Question

Consider the situation of an area of land being heated periodically at the surface. A model for the temperature Θ(x, t) of the soil at a distance x metres below the surface at time t is ∂Θ ∂t = D ∂ 2Θ ∂x2 , where Θ(0, t) has an annual sinusoidal variation about a mean temperature of −5 ◦ C with amplitude 10◦ C, and ∂Θ ∂x → 0 as x → ∞. (a) Verify by substitution that the following function satisfies both the heat equation and the boundary conditions of the model for an appropriate value of k: Θ(x, t) = −5 + 10e −kx cos(ωt − kx), where ω ' 2 × 10−7 rad s−1 is a constant that corresponds to the annual temperature variation. Calculate explicitly the value of k needed for this to be a solution of the model if the soil has diffusivity constant D = 4 × 10−7 m2 s −1 . [6] (b) Find the depth at which there is permafrost (that is, where the ground is permanently below 0◦ C). [3]

Answer 1

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