Dynamical equivalence and the departure-point equation in semi-Lagrangian numerical models



The kinematics and Newtonian dynamics of a continuous fluid may be presented in a number of ways. Given the kinematic relation between velocity and position, the dynamics may be formulated in terms of momentum or in terms of angular momentum and the scalar product of velocity and position. It is also possible to carry out the formulation in terms of momentum, angular momentum and the scalar product of velocity and position, in which case the kinematic relation becomes a deducible property. The three formulations are dynamically equivalent in the sense that the flow evolution is demonstrably the same in each. It seems desirable to build this elementary symmetry into discrete models. Two-time-level, semi-Lagrangian models are considered here, with emphasis on the departure-point equation (from which trajectory origins at the earlier time-level are determined). Once a rule has been chosen for the approximation of trajectory time-averages of vector and scalar products, dynamical equivalence implies a particular form of the departure-point equation. One such rule delivers a doubly-implicit equation that is a simple time-centred discretization of the kinematic relation. A second rule leads to a form that differs only slightly from the singly-implicit equation of Gospodinov and co-workers. It is noted that dynamical equivalence relates to the consistency and non-ambiguity of results rather than to their accuracy. © Crown copyright, 2003. Royal Meteorological Society