This proposal is for theoretical research on the higher statistics of Navier-Stokes turbulence. The principal tool to be used is called mapping approximation. It is a systematic mathematical procedure applied to the fluid equations of motion. Nonlinear transformations of fields with Gaussian statistics are exploited. In previous applications to several problems, notably to the advection of passive or chemically active scalar contaminant fields by turbulent flows and to Burgers' one-dimensional model of turbulence, it has yielded quantitatively accurate predictions of statistics that describe extreme intermittency of small scales. The proposed work involves further development of mapping approximation and its application to intermittency of vorticity in incompressible turbulent flows. Ultimate goals include the determination of scaling exponents for vorticity.
Turbulence, the disordered motion of fluids, is an ubiquitous phenomenon in geophysics and industry. It has been an outstanding challenge to mathematicians and physicists for over a century. Examples where turbulence plays a crucial role are weather systems, airplane flight, the dispersal of atmospheric pollutants, and industrial chemical reactors. The complexity of turbulent motion is so extreme that fully detailed computer solutions of real turbulent flows are not possible; the demands on memory and speed are too extreme. In order to reduce computation to maneagable size, theoretical models of the statistics of the smaller scales of turbulent motion are essential. The research to be carried out under this proposal brings recently developed new tools to bear on the statistics of small scales. These tools have proved to be powerful in related problems. Improved models of the small scales of turbulence that can be used in computer simulations are an objective of this research.
