The Proceedings of the Eighth International Conference on Creationism (2018)

given time is effectively transported outward from the spreading centre at the same rate as the lithospheric material. In the secular literature the model data (temperature profiles, heat fluxes and shrinkage) are plotted against geological time; distance per se is ignored. This is consistent with the use of these models as global reference models, to be compared with globally averaged data, given that plate spreading speeds vary considerably between ocean basins. Thus, for example, near the East Pacific Rise current spreading rates are much higher (~10 cm/year) than in the North Atlantic (~2 cm/year; Müller et al. 2008), and mid-ocean ridges are correspondingly wider in the East Pacific Rise. For our short time scale calculations, we treat the spreading rate as a free parameter, to which we assign a predefined profile in time. This in turn enables us to define the resulting bathymetry in terms of calculated shrinkage against distance. The vertical shrinking of the cooling lithosphere is essentially thermal contraction and is therefore calculated from the total net heat loss. However the surface depression is greater than would be calculated simply from the heat lost by a column of lithosphere. This is because the water loading increases as the lithosphere shrinks and becomes denser, and the water depth increases. In turn this is isostatically compensated by further depression of the ocean floor (for a derivation see Turcotte and Schubert 2002, section 4.23). Thus for lithospheric density ρ m and water density ρ w , there is a shrinkage enhancement factor γ≡ρ m /(ρ m -ρ w ) , such that the total shrinkage for a column of height L when the average temperature has fallen from T 1 to T m is given by For the data values given in Table 1, the enhancement factor is γ = 1.429. METHODS OF SOLUTION 1. Analytical solution The earliest and simplest model used to analyse ocean floor cooling is known as the infinite half space model (Turcotte and Schubert 2002, section 4.15), which has an analytical solution in terms of error functions. For surface temperature T 0 and deep-mantle temperature T 1 , this gives a heat flux to the ocean at time t of Integrating this expression with respect to time then gives the total heat lost (expressed in joules per square metre) to the ocean as Given that the heat loss for a finite-thickness plate can be expressed in the form comparison of equation (4) with (2) implies that for the half space model the net shrinkage is Parsons and Sclater (1977), in comparing the predictions of the half space model with bathymetry data, conclude that the model gives a good approximation to reality up to about 70 Ma in the conventional time frame. Beyond this it overpredicts the surface depression due to shrinkage, which indicates that shrinkage is limited by heat transfer from the underlying mantle into the lithosphere. This behaviour (a t ½ dependence of the bathymetry up to a limited time) was also seen in previous plate models; Parsons and Sclater’s plate model parameters were chosen to optimize the data fit up to about 160 Ma. Stein and Stein (1992) improved the fit with a much larger dataset and formal procedures for optimizing their input parameters. In both of these papers the solution of the heat diffusion equation was obtained analytically in terms of infinite Fourier series in z , the terms decreasing exponentially in x (distance from the ridge). Because of computational difficulties in evaluating such series, here we use a simple finite difference timestepping scheme which can readily be evaluated on an Excel© spreadsheet. For reference purposes Stein and Stein give approximate fitting equations based on their GDH1 model for surface heat flux and ocean depth against uniformitarian time. Their heat flux equation for time t ˂ 55 Ma (million years) is and for t ˃ 55 Ma, where t is measured in Ma and q ( t ) in Wm -2 . Their depth equation for t ˂ 20 Ma is and for t ˃ 20 Ma, where d ( t ) is measured in km and t again in Ma. We use these later as comparisons for the results of our spreadsheet modeling exercise. 2. Finite difference solution The heat diffusion equation belongs to the class of second-order partial differential equations (PDEs) designated parabolic . The simplest widely-used scheme for solving the finite difference equations generated by discretization of parabolic PDEs is the classic explicit method (Lapidus and Pinder 1982). Applied to the one-dimensional time-dependent heat diffusion equation, the term explicit here means that the temperature at any given point in space is updated in a timestep directly from local temperatures at the start of the timestep. By contrast implicit methods involve multiple temperatures at both beginning and end of the timestep, and updating a temperature value may involve iteration or matrix inversion. Thus explicit methods are simpler to set up and computationally faster than implicit methods, but tend to become unstable – the solution becoming wildly ridiculous – more readily than with implicit methods. W e employ uniform discretization in both t and z , such that the temperature at each point in time and space has two suffixes, i.e. T i,j where i (=0, 1, 2 . . .) denotes the point in time and j (=0, 1, . . . n ) the point in space, where i =0 refers to initial conditions ( t =0), j =0 to the surface ( z =0) and j = n to the plate bottom ( z = L ). In the classic Worraker and Ward ◀ Ocean floor cooling ▶ 2018 ICC 675