Collaborative Research: Study of turbulence in physical systems through complex singularities and determining modes

This project applies two recently introduced evolution equations to fundamental issues in hydrodynamic turbulence. The first equation is for the determining modes of a differential equation of Navier-Stokes type. A set of low modes is determining, if for any solution on the global attractor, the high modes of the solution for all time, can be produced from only the low modes (for all time). Extending this relationship to the Banach space of bounded functions from the reals into the low modes leads to a determining form, an evolution equation that is globally Lipschitz, and dissipative. The global attractor of the original differential equation is contained in the long time behavior of the determining form. This provides an alternative to the theory of inertial forms which is not known to be applicable to the Navier-Stokes equations. The other evolution equation is for the radius of analyticity in the spatial variable. We will use this equation to study the domain of analyticity for solutions on the global attractor and its effect on aspects of fluid flow such as intermittency.

Fluid motions at large scales, for instance in ocean and atmosphere, display erratic, seemingly incomprehensible (turbulent) features when viewed locally and for short times. However, over time, consistent global statistical patterns emerge. The purpose of this research is to develop a rigorous understanding of the nature and origin of such patterns, and of deviations from it, with a view towards diverse applications such as ocean modeling and weather forecasting. For instance, weather forecasting is done by solving a large system of equations which model the state of the atmosphere over a period of time.

The current state of the atmosphere is needed as input in order to compute the state in the future. As there are always limited measurements of the state of the system, this input inevitably contains some error. The inherent sensitivity of the system to such an error means that as time goes on the computed solution diverges from reality. The technique of data assimilation uses the limited measurements of the system at not just a single moment of time, but rather over an extended time period in the past, to arrive at a more accurate input. This project applies recent mathematical discoveries to develop new variations on this technique. It will lead to new algorithms that will be implemented and tested by the research team. Graduate students will play an essential role in this collaborative effort, thus serving to train future generation of scientists in the STEM disciplines.