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Appendix A Freezing of a cylindrical water-filled hole in cold ice
To compute hole closure rates and cooling curves for a water-filled cylindrical hole in cold ice, diffusion equations of the form
must be solved in ice and water. At the ice–water interface the boundary conditions are
c is the radius of the water-filled cylinder and T
m the melting temperature of ice. The remaining boundary conditions are
0 is the ambient ice temperature, q
1 (t) the strength of a line heating source, and r
0 its radius. Boundary condition (A5) allows the possibility of evaluating the effect of ohmic dissipation in the power cable leading to the thermal probe. In the calculations discussed above q
1 was negligible and the water phase was essentially isothermal at temperature T
m. Solutions for large values of q
1 have also been computed to determine whether line heating can be used to inhibit hole closure during thermal drilling.
In passing to a finite-difference approximation of (A1) it is convenient to introduce a logarithmic grid by the transformation R= In r; thus (A1) becomes
with T = T(R, t), and (A2) gives
c = ln r
Following the Crank–Nicolson approach, the solutions of Equation (A6) for times t and t + τ are averaged to reduce the discretization error giving as the finite-difference equation in the ith medium
where h is the space increment of the logarithmic grid, τ the time increment, λ
i = κi
, and θ is an averaging parameter which is usually set to the value θ = 0.5. The variables R and t take discrete values mh and nτ respectively, where m and n are integers. In Equation (A8) the right-hand side terms arc known and the left-hand side terms are unknown. If similar equations are written at each grid point one obtains a tridiagonal set of linear equations which can be solved for the unknown temperatures T(R, t + τ).
Infinitely large grids are not feasible so that the boundary condition (A4) is replaced by T(R
max, t) = T
max is some suitably large value of R =In r. At R
= In r
0 we have the condition
and at R
= In r
c, the ice–water interface, T (R
c, t) = T
m. The finite-difference equations (A8) are solved subject to the above boundary conditions and the migration of the ice–water interface is evaluated at each time step by substituting finite-difference approximations of •T(R
c, t)/•t and •T
c, t)/•t into Equation (A2). When the condition R
0 is satisfied, the water phase is considered to vanish and a simple one-phase problem results.
Appendix B Peaceman–Rachford numerical method
For a finite-difference grid with space intervals Δx, Δy and time step τ, the standard implicit finite-difference approximation to Equation (2) yields
=kτ/(Δy)2, and the variables x, y, and t have the discrete values x = iΔx, y = jΔy, and t = nτ for integer values of i, j,and n. Equations (B1) are implicit in both x- and y-directions and have five unknowns per equation. Direct solution of this system of equations requires the inversion of a large five-band diagonal matrix and is computationally expensive.
In the Peaceman–Rachford implicit-alternating-dircction method, two systems of equations are used in turn over successive time steps of duration τ/2. The first equation is implicit in the x-direction only, the second in the y-direction. Using the notation T*(x,y) to represent the intermediate values of T half-way through the time step τ, for implicit x we have
and for implicit y
The systems of Equations (B2) and (B3) have only three unknowns per equation and the implicit solution of each system merely involves the inversion of tridiagonal matrices for which simple and efficient algorithms are readily available.
Holding y constant, one equation of the form (B2) is written for each value of x and the resultant tridiagonal system of equations is solved simultaneously. Equations (B2) are solved in this manner once for each value of y to generate the complete solution T*(x,y). Equations (B3) are now solved by substituting the solution T*(x,y) obtained from (B2) into (B3). Holding x constant, one equation of the form (B3) can be written for each value of y and the new system of equations solved simultaneously. Equations (B3) are solved once for each value of x to generate T(x, y, t + τ), the temperature distribution advanced one full time step. This procedure is unconditionally stable for any value of τ and the discretization error is 0[τ