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# There is an area of thin mantle at the bottom of the ocean which is...

There is an area of thin mantle at the bottom of the ocean which is substantially raising the temperature of the sea bed due to a breach of molten material. In this assignment, you will use the heat equation to analytically solve for the transient temperature profile within the sea given a fixed Dirichlet boundary condition at the bottom of the ocean. The initial 1D temperature versus depth profile, at the instant when the ocean bed breach occurs, can be represented by a piece-wise function: the temperature is a constant Tα = 25◦C from sea-level d0 = 0m to a depth of d1 = 500m; there is then a sharp transition to a constant temperature of Tβ = 5◦C from d1 to the ocean floor d2 = 2600m; finally, the exposed mantle causes the temperature at d2 to be fixed at Tγ = 460◦C. The thermal diffusivity of sea-water can be approximated to κt = 1.6 × 10−2m2/s.

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1. Using the information given in the Introduction, brieﬂy summarise the problem information. (5%). Be sure to include the following: I Governing equation 0 Order and type of the PDE 0 Initial condition 0 Boundary conditions 0 Type of solution you expect to obtain 2. Derive the steady state analytic solution for the temperature as a function of ocean depth, showing all working. Produce a plot comparing the initial condition, steady-state solution and the initial condition that will be used for the homogeneous problem. All three plots should be overlaid. (5%) 3. Using the method of Separation of Variables you've seen in the lectures and tutorials, derive a solution to the heat equation for this problem in terms of unknown Fourier coefficients (i.e. do not apply the boundary conditions yet). In your working, clearly indicate: the spatial, F(a:), and temporal, C(t), functions and the eigenvalues. (5%)

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