Statistics of turbulence in a rapidly rotating system

Turbulence raises many issues such as fundamental questions in mathematics, continuum mechanics in physics and various industrial problems. Turbulence is characterized as a state of fluid flow that is influenced strongly by nonlinear processes compared to dissipation. Turbulence of fluids with strong rotation is of interest in turbo-machinery and geophysical flows that occur in the earth’s atmosphere and oceans. Strong rotation can bring a turbulent system into a quasi two-dimensional (2D) turbulence. Rotation causes anisotropic turbulent motions on large scales. However, on small scales the turbulence is believed to be homogeneous and isotropic and that fluid motions are independent of rotation and large-scale topography. Despite this general belief, in our experiments we find that the energy spectrum in a rotating turbulent flow strongly depends on large-scale topography and a rotation. A 2D fluid system with forcing and dissipation neglected has a Hamiltonian structure with conserved quantities. These conserved quantities constrain the dynamics of 2D fluid. For a long time, it has been quite mysterious why only quadratic conserved quantities (energy and the square of vorticity) should be important in a statistical mechanical description of turbulence, especially, in 2D turbulence, where there are an infinite number of conserved quantities (the so-called Casimir invariants). Previous models of statistical mechanics of 2D turbulence have not explicitly taken into account statistical independence of macroscopic subparts, and consequently all or most of the conserved quantities have been used. However, experimental results support the use of only quadratic conserved quantities. Because of statistical independence, we show that only quadratic conserved quantities are crucial in statistical mechanics. In addition, we propose a statistical mechanical theory based on new coordinates that define statistically independent subsystems, and we compare the theory with experiments. Hamiltonian and action principles elucidate the physics in various fields, from quantum to plasma physics. Such a formulation has been used in plasma physics for the Vlasov-Poisson system to obtain fluctuation spectra. For a fluid, a similar process is possible. In this thesis, we use Hamiltonian principles to formulate the analogous fluctuation spectrum in the fluid case and compare it with experiments.