Goal oriented adaptivity for tropical cyclones

The development and motion of tropical cyclones (TCs) is influenced by processes on a large range of scales. TC motion is determined by the environmental flow on scales of several thousand kilometers whereas the genesis is particularly sensitive to inner-core convection on scales below several kilometers. The structure and intensification are influenced by the large-scale and the convective scale. Thus a multi-scale approach is essential to represent a tropical cyclone adequately.

Adaptive techniques are a promising way to tackle such challenging multi-scale problems. We are investigating techniques for automatic local mesh adaptation controlled by goal functionals. The sensitivity information obtained with goal oriented methods is used to refine the mesh automatically in a way that minimizes the error of the numerical solution with respect to a arbitrary physical quantity of interest. The sensitivity information presents not only a way to optimize the mesh, but can also provide insights into the processes that affect the development and motion of TCs (similar to singular vectors).

Here we present adaptive calculations of time-varying problems related to TCs. Highly idealized TC scenarios are investigated, as the TC problem in its entirety is too complex and not sufficiently well understood to serve as a test problem. For this purpose we define barotropic TC problems that could benefit considerably from goal-oriented adaptivity. We perform non-adaptive high-resolution reference model runs for these scenarios and compare them to adaptive calculations using different goal functionals.

In one of the scenarios we consider the interaction of a cyclone with a jet stream. The circulation of the TC excites Rossby waves on the strong potential vorticity gradient in the upper troposphere that is associated with the jetstream and which can be interpreted as a front. This has consequences for the motion and intensity of the cyclone as well as for the downstream development. Model runs for different vortex types and jet speeds indicate that resonant frontal waves -- those whose phase speed matches the zonal translation speed of the cyclone -- are decisive for the interaction. The frontal wave spectrum excited by a cyclone on an initially unperturbed front is dominated by waves that are in resonance in the initial phase and these waves have also the largest impact on the cyclone motion.

A pre-existing resonant wave on the jet modifies the interaction significantly. In this case the model evolution depends sensitively on the initial position of the cyclone relative to the troughs and ridges. We identify a bifurcation point located on the trough axis. Arbitrarily small displacements from this position determine whether a cyclone is advected towards the front and accelerated in the zonal direction, or is repelled from the front and decelerated. The difference in the cyclone position for these two cases increases by up to 2000km per day. We discuss the properties of frontal waves that can lead to such bifurcations.