| The ocean is a giant body of stratified fluid. It covers more than 71% of earth's surface, averages about 4 kilometers deep, and accounts for over 97% of the water on earth [79]. Five major mechanical forcings drive the ocean into motion: wind and tidal forcing, global heating, bioturbulence and geothermals. Energy is pumped into the ocean, which supports motions on various length and time scales, making the ocean rich in dynamics, yet mathematically complicated to describe. The work presented in this thesis is dedicated to studying the effect of ocean turbulence on the energy budget in ocean mixing and dissipation, driven by various types of external forcings.; We aim to bound the long-time-averaged buoyancy flux and energy dissipation, using a variational approach. On one hand, we look at the idealized problem of mixing in stratified shear flow, driven by two parallel plates moving in opposite directions. In chapter 3, using the background method originated from Doering & Constantin (1992), and numerically solving the Euler-Lagrange equations, we find that the maximal buoyancy flux is independent of fluid viscosity. In order to improve the bound by seeking the dependence on overall stratification, in chapter 4, we impose a flux Richardson number constraint, and find that the buoyancy flux increases with bulk stratification until a threshold, where it approaches the result in chapter 3 from below.; On the other hand, we look at energy dissipation in stress driven flow. For a simple non-rotating case, in chapter 5, using the background method again, we find that the lower bound on energy dissipation is also independent of flow viscosity. The result is then extended to a rotating frame in chapter 6, with an alternative approach analogous to the background method. Although the new approach doesn't solve the Euler-Lagrange equations exactly, it nevertheless captures the major scaling of the bound under the circumstances given. In the inviscid limit, the energy dissipation is independent of fluid viscosity and rotation rate, but for turbulence in the intermediate Grashof number range, rotation plays an important role.; These studies are cornerstones of further understandings of ocean turbulence when combined forcing conditions are considered and more complex flow structures are involved. The bounding calculation in this thesis also give us a rigorous framework with which to compare the results of numerical simulations. |
