A study of how indicators for 2-D turbulence depend on the driving force in the Navier-Stokes equation

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This project is focused on the discovery of driving forces that supportthe Kraichnan theory of 2-D fully developed turbulence. In recent workthe investigators and their collaborators have identified critical wavenumbers expressed as averages of different norms of the solution to theNavier-Stokes equations (NSE). These wave numbers provide necessary, andnearly sufficient conditions for the Kraichnan theory to hold. This teamhas also localized to some extent the global attractor of the NSE in aplane spanned by two of these norms (one being the energy), to helpunderstand which driving forces produce these conditions. The proposedwork will combine this analysis with computational optimization to zero inon such forces, and then study in detail the turbulent features theyproduce. The mathematical treatment of turbulence is largely driven bythe heuristic theories of Kolmogorov, Batchelor and Kraichnan. Theapproach taken in projecting the attractor however, seems to be entirelynew. The information provided by this analysis will guide thecomputational component which otherwise would be confronted with a vastlandscape of possible driving forces to consider.Turbulence is readily observed in three-dimensional physical spacedomains. Most people think of a bumpy plane rides (in this case thedomain is the volume around the airplane). Like the swirls generated byrocks in a stream, rapidly changing patterns form in the air around theplane. Turbulence theories do not attempt to predict the precisedevelopment of these patterns, but rather find (a) consistent laws whichdescribe how, on average, energy is transferred to smaller length scales,and (b) critical length scales at which this this phenomenon changes. True 2-D flows in nature are less prevalent. The most prominent example,the earth's atmosphere, is actually a thin 3-D domain, whose behaviorapproaches that of a 2-D flow. The fate of energy over different lengthscales is more complicated for 2-D flow, though that of 3-D flow is insome sense embedded into it. Though 2-D experiments are difficult tocarry out in the laboratory, they allow for much finer study on acomputer. Of all 2-D flows, the one studied in this project is arguablythe most amenable to analysis and efficient simulation. Yet it isfundamental, not only to 2-D and nearly 2-D flows such as the atmosphere,but also to 3-D turbulence due to universality.