Averaged dynamics of the advection-diffusion equation and applications to ocean flows

This dissertation presents some aspects of an advection-diffusion equation and its applications to physical oceanography. We propose a perturbative scheme of averaging the advection-diffusion equation in the limit of vanishing diffusivity. Under the restriction that the time-dependence of the advective field is completely separable we construct an exact solution of the purely advective part via action-angle coordinates and treat diffusion as a perturbation using Lie transform techniques. The developed method is applied to a regularized vortical flow field which is periodically modulated in time. Numerical simulations of the vortical flow advection in presence of small diffusion are discussed. We present numerical evidence that the spectrum of the averaged time-independent advection-diffusion operator converges to the spectrum of the operator with fully enabled time dynamics. A formal generalization of the method for three-dimensional time-periodic flows is discussed.;We also discuss the importance of advection and diffusion in problems of transport and mixing in complicated dynamical systems, such as hydrodynamical systems, in particular describing ocean currents. We propose a method to visualize and analyze the structure of complex flows using data from HYbrid Coordinate Ocean Model (HYCOM) as an example. We present results of simulations obtained with highly parallel Co-array Fortran code that can be run on modern computing systems that support partitioned global address space (PGAS) programming model.