In this book, we develop a goal-oriented adaptive method to improve the efficiency of tropical cyclone prediction. The term goal-oriented refers to a common desire in numerical modeling that a certain physical quantity of the solution, such as point-values, is of special interest and should be determined with high precision. In this case, discrete models can be tailored with respect to this specific request (e.g. by local grid-refinement) which allows for very efficient models.
We investigate an idealized scenario of two closely located, interacting tropical cyclones. During the first hours of development, their mutual influence on the storm structure and position is strong. Since small perturbations can have strong impact on the storm positions at later times, this scenario is an excellent benchmark problem for adaptive methods. We use a discretization based on a spacetime finite element method and present a new a posteriori error estimator for the error in user-defined functionals. We introduce several functionals defined as localized integrals of vorticity and kinetic energy that are strongly correlated to the storm positions. We discuss the perturbation sensitivity of these functionals, and estimate the corresponding error of approximate solutions. Using goal-oriented adaptation of the space and time discretization, a significant efficiency improvement in storm track prediction can be achieved for this scenario.