A Successive Substitution Method for the Evaluation of Trajectories Approximating the Parcel Path by a Linear Function of Space and Time

Abstract

Parcel trajectories can be determined by the time integration of the velocity using the Picard successive substitution procedure. However, this scheme can rarely be applied in numerical problems because it implies (in general) an increasing difficulty in solving the analytical solution in each iteration. This strong restriction to its practical use can be overcome by approximating the velocity to a simple function where the analytical integration can be performed as the iteration advances. The usual procedure adopted in meteorological research is to compute the trajectories using the Petterssen scheme. In this work it is shown that approximating the velocity by a linear time function along the trajectory in the Picard method yields a numerical algorithm formally identical to the Petterssen scheme. An alternative method is proposed following the philosophy of approximating the velocity in the Picard method in order to simplify the integration. In this approach, the parcel path is assumed to be a linear function of space and time, and the velocity along this path at each iteration is computed using an interpolating polynomial. Only one analytical integration is necessary, and the successive integration depends upon the coefficients of the interpolating function adopted for the velocity. For cubic spline, the alternative algorithm can be represented in a suitable vectorial form convenient in designing the numerical code. The method is tested on idealized situations with high time and space velocity variations using either Lagrangian or Eulerian coordinates. The results for the alternative and Petterssen schemes are similar in the Eulerian coordinate, but the superiority of the alternative scheme is evident in the Lagrangian coordinate.