Construction of Economical Meshes for Tropical Cyclone Forecasting

The development and motion of tropical cyclones (TCs) is influenced by processes on a wide range of spatial and temporal scales. The motion of TCs is mainly determined by the flow on large scales while the storm's intensity is regulated primarily by processes on the convective scale. Due to restrictions of available computer resources only a limited range of these scales can be resolved on a uniform grid.

Mesh adaptation techniques are a promising way to tackle such problems. We are investigating techniques for automatic local mesh adaptation controlled by goal functionals. These techniques are based on the idea that in many cases one is primarily interested in certain physical quantities that can be computed from the solution, for instance the cyclone position in TC forecasts. In these cases, methods for efficient numerical calculation of these quantities are better suited than methods that lead to good approximations with respect to global norms. Often such quantities of interest can be described in terms of a so-called goal functional, which typically represents local features such as point values or integrals over small sub-domains. The sensitivity of the goal functional with respect to perturbations of the solution can be used to construct optimal meshes on which the quantity of interest can be approximated very efficiently.

The investigated techniques are based on the Dual Weighted Residual method, which represents a generic framework for goal-oriented error control and mesh adaptation. It includes an a posteriori error estimator to quantify the error of approximate solutions with respect to the user-defined goal functional. Furthermore, cell-wise error indicators can be defined that represent each cell's error contribution related to the space or time discretization. This allows for the construction of economical discrete methods by adaptation of the space-time mesh.

We apply these goal-oriented adaptive techniques to idealized scenarios of TC dynamics described by a nondivergent barotropic model. For these scenarios, we construct economical discretizations for the efficient prediction of the storm tracks. To this end, we define adequate goal functionals that are correlated with the storm position and can be used for error estimation. Approximate solutions are determined based on a space-time finite element method using stable finite elements in space and a continuous Galerkin-Petrov method in time. Some details related to the discretization and the corresponding goal-oriented error estimator are given. Our numerical results are discussed with respect to the relevant error norms. In particular, we demonstrate that using goal-oriented mesh adaptation in space and time, considerably fewer grid cells are required to achieve a given accuracy in the prediction of the cyclone position than on uniform grids.