Transient geotherms
Numerical solutions
• Since the arrival of computers, which allow for large numbers of operation to be performed in routine, new techniques based on numerical algorithms have been designed to solve differential equations. These numerical recipes are based on "discretization" of the differential equation...
•Consider the 1D heat conduction-advection equation in a slab zl thick. The upper surface of the slab has at temperature To. The transient temperature is then described by the conductive-advective equation of heat balance. Here we consider a situation where internal radiogenic heat is creation at a rate A, heat is lost or gain by advection at speed U (since z increases downward, if erosion U<0, if sedimentation U>0), and heat is lost through phase transition solid-liquid (L<0 for partial melting, L>0 for crystallization) X the melt fraction is a function of T and therefore z.
re-arranging:
Iinitial temperature profile: T(z, 0)=0
Boundary condition C1: T(0, t)=T0
Boundary condition C2: dT/dz(zl, t)=-Qm/K
finally we get that: (1)
•An approximation of this differential equation can be obtained by replacing ∂t and ∂z with finite differences Δt and h. The central tenet of this computational technique is that for a given time t=n, the temperature at a depth z=i (Tin) can be calculated from knowledge of the temperature at depth z=i-h and z=i+h. In a similar fashion, for a given depth z=i, temperature at time t=n can be caculated from knowledge or the temperature at time t=n-Δt and temperature at time t=n+ Δt. The level a accuracy depend of the size of the time and space finite differences. There are many ways to express T(z, t) as a function of T(z-h, t), T(z+h, t), T(z, t- Δ), T(z, t+ Δ). Here we present the Crank-Nicholson scheme where, equation (1) is re-writen as:
(2)
where:
•The finite difference equation (2) is known as the Crank-Nicholson scheme. It is based on central differences for the spatial derivatives averaged forward in time over time steps n and n+1.