Khadeeja Afzal

• This thesis investigates the use and behavior of balance relations for studying the nearly-geostrophic flow in a bounded domain. The goal is to derive simple, fully nonlinear models for the large scale flow in the vicinity of basin boundaries and investigate their asymptotic behavior. This will help improve our conceptual understanding and provide benchmarks for the calibration of larger numerical ocean models. The balance models are those in which the Coriolis force balances the pressure gradient force in the limit of small Rossby number. These models are derived using a Lagrangian-based variational approach. The idea was first proposed by Salmon (1983), in which the author derived the approximate model for nearly geostrophic flow for the rotating shallow water equations. He applied the approximations on the Lagrangian of the parent fluid model and then took variations to get the Euler-Lagrangian equations, which he named as L1 balance model. Oliver (2006) generalized this idea and started with the arbitrary change in coordinates to the canonical coordinates, and then consistently truncated the transformation and the Lagrangian to a desired order. This approach gives the one-parameter generalized family of large-scale models (GLSG), among which Salmon’s L1 model is observed to be numerically well-behaved, as noted by Dritschel et al. (2017). In the current study, we employ the approach detailed in Oliver (2006) and derived the variational L1 balance model for the shallow water equations with constant Coriolis force in the vicinity of the boundaries. At the boundary, zero-flux is assumed in the normal direction and the variational derivation of the model suggests the geostrophic balance up to O(ε) in the tangential direction. We numerically investigated how well the balance dynamics capture the shallow water equations under specified boundary conditions. For this, we initialized the full shallow water equations with the balanced state and allowed it to advect until time T. We then compared the fields using their root mean square (r.m.s.) differences and observed their asymptotic behaviour. Furthermore, Eulerian time scales are also determined at which both the models can be compared. Notably, we observed that the physical boundary interactions result in a slowdown of the time scales when compared to the time scales in the case of periodic boundaries.