A characteristics-based method for solving the transport equation and its application to the process of mantle differentiation and continental root growth

Abstract

Summary Purely advective transport of composition is of major importance in the Geosciences, and efficient and accurate solution methods are needed. A characteristics‐based method is used to solve the transport equation. We employ a new hybrid interpolation scheme, which allows for the tuning of stability and accuracy through a threshold parameter εth. Stability is established by bilinear interpolations, and bicubic splines are used to maintain accuracy. With this scheme, numerical instabilities can be suppressed by allowing numerical diffusion to work in time and locally in space. The scheme can be applied efficiently for preliminary modelling purposes. This can be followed by detailed high‐resolution experiments. First, the principal effects of this hybrid interpolation method are illustrated and some tests are presented for numerical solutions of the transport equation. Second, we illustrate that this approach works successfully for a previously developed continental evolution model for the convecting upper mantle. In this model the transport equation contains a source term, which describes the melt production in pressure‐released partial melting. In this model, a characteristic phenomenon of small‐scale melting diapirs is observed (De Smet et al.1998; De Smet et al. 1999). High‐resolution experiments with grid cells down to 700 m horizontally and 515 m vertically result in highly detailed observations of the diapiric melting phenomenon.

Interaction between small‐scale mantle diapirs and a continental root

“…Furthermore, we investigated the effect of dehydration during partial melting. This extends earlier work by de Smet et al [1998, 1999, 2000a, 2000b] and de Smet [1999] based on purely Newtonian rheology.…”

“… a Here, η ref , P ref and T ref are reference viscosity, pressure and temperature, respectively (other symbols are explained in the caption of Table 2), and are used following de Smet et al [1998, 1999, 2000a, 2000b] and de Smet [1999], indicated by JdS. The components of composite rheology are based on data by Karato and Wu [1993], indicated by KW, and are chosen to be intermediate between their “wet” and “dry” parameters. …”

“…This results in the effective prefactors B k of , the values of which are specified in Tables 3 and 4. For models N1–N2 (Newtonian rheology), we specified a reference viscosity η ref at a reference pressure P ref and temperature T ref , and activation energy and volume, following de Smet et al [2000a], to allow direct comparison of the results.…”

“…From this one can predict a greater predominance of the dislocation creep component under higher temperature conditions, representative for the early Earth. This has motivated us to extend our previous work on growth mechanisms of continental roots, based on purely diffusion creep models [ de Smet et al , 1998, 1999, 2000a, 2000b; de Smet , 1999], with a detailed investigation of more complete viscous rheology including also the rheological effect of dislocation creep flow and sensitivity for the degree of dehydration induced by partial melting.…”

Transfer equations applying to layered media sounding problems

“…Wave equations, as well as the other methods of signal processing, permitting us to simplify signal decoding (Huang & Fehler 2000; Sato & Kennett 2000; de Smet et al 2000; Kennett 2000; Toomey & Bean 2000; Mendel & Goutsias 1986) are commonly used for layered media sounding problems. At present wave coupling and scattering are still the main problem of deep seismic sounding, and propagation through heterogeneous media is still are of the most important problems of theoretical investigations (Pavlenkova 1999).…”