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

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 ε(th). 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.