The proposed research is directed toward some models of turbulence in fluids and magnetofluids. Deterministic and statistical equilibrium theories of coherent structures in turbulence will be explored using both analytical and numerical methods. These theories, which complement the cascade-oriented concept of a driven and dissipative system, provide models for the long-time, large-scale behavior of flows and fields governed by conservative evolution equations. Their equilibrium solutions are characterized by variational principles, such as a maximum entropy principle with constraints dictated by the conservation laws for the governing dynamics. Theoretical tools and computational algorithms will be developed to solve the constrained optimization problems that arise from these general principles. A variety of fundamental, prototype problems derived from specific physical applications will be treated with these methods; they include, for instance, the organized vortex structures in a shear layer, and the most probable state of a magnetically-confined plasma. Most of the tractable problems of this kind are essentially two-dimensional; however, some aspects of the analogous three-dimensional problems will be also tackled using extensions of the same methods. The principal aim of this research is to develop a basic understanding of the physics of turbulence. The turbulent medium can be an ordinary fluid, such as water flowing in a pipe or air streaming past a wing; or, it can be a magnetized plasma, such as ionized gas in a thermonuclear fusion reactor. Simplified mathematical models of these complex physical problems are formulated in order to investigate some fundamental properties of the phenomena they exhibit. Particular attention is focussed on the coherent or organized structures that emerge and persist in the presence of turbulent disorder. These structures are studied quantitatively by deriving and then solving equations that captu re the coarse-grained behavior of the system while partially ignoring its fine-grained fluctuations. In this way, useful predictions can be made of physical systems having a very large number of degrees of freedom, which otherwise would be beyond the grasp of current methods of mathematical analysis or computer simulation. An interdisciplinary approach will be adopted in which abstract analytical tools, modern computing techniques, and solid physical reasoning are employed in a unified manner.
