The incompressible Navier-Stokes equations (the momentum, continuity and scalar transport equations) are the fundamental equations of fluid mechanics. Of great importance to weather and climate studies is the thermohaline circulation, which is affected by gravity currents, hence we chose it as our application.; First, we studied the Navier-Stokes equations (without scalar transport) using non-autonomous dynamical systems techniques, and showed the existence of recurrent or Poisson stable motions under recurrent or Poisson stable forcing, respectively. This was motivated by observed periodic and recurrent motions in nature.; Next, we investigated the coupled Navier-Stokes and scalar transport equations (we may take the scalar to be salinity, say), with spatially correlated white noise on the boundary. We employed random dynamical system ideas, and showed that this system is ergodic under suitable conditions for mean salinity input flux on the boundary, Prandtl number and covariance of the noise. This addition of a random term to the boundary conditions was motivated by observed seasonal variations in the salinity flux in gravity currents.; The final part of this thesis are numerical simulations studying the effects of different boundary conditions on the entrainment behavior, salinity distribution and salinity transport properties of gravity currents. The finding is that gravity currents developing under Neumann and Dirichlet boundary conditions differ most in the way they transport salinity from the middle salinity parts (roughly the middle of the current) towards the fresher part (roughly the top of the current).; This study contributes to understanding the behavior of the Navier-Stokes Equations under time-periodic forcings, uncertain boundary conditions, and how gravity currents are affected by different boundary conditions.
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