Ueckermann, M.P., 2014. *High Order Hybrid Discontinuous Galerkin Regional Ocean Modeling*. Ph.D. Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, February 2014.

Accurate modeling of physical and biogeochemical dynamics in coastal ocean regions
is required for multiple scientific and societal applications, covering a wide range
of time and space scales. However, in light of the strong nonlinearities observed in
coastal regions and in biological processes, such modeling is challenging. An important
subject that has been largely overlooked is the numerical requirements for
regional ocean simulation studies. Major objectives of this thesis are to address such
computational questions for non-hydrostatic multiscale flows and for biogeochemical
interactions, and to derive and develop numerical schemes that meet these requirements,
utilizing the latest advances in computational fluid dynamics.

We are interested in studying nonlinear, transient, and multiscale ocean dynamics
over complex geometries with steep bathymetry and intricate coastlines, from
sub-mesoscales to basin-scales. These dynamical interests, when combined with our
requirements for accurate, efficient and flexible ocean modeling, led us to develop
new variable resolution, higher-order and non-hydrostatic ocean modeling schemes.
Specifically, we derived, developed and applied new numerical schemes based on the
novel hybrid discontinuous Galerkin (HDG) method in combination with projection
methods.

The new numerical schemes are first derived for the Navier-Stokes equations. To
ensure mass conservation, we define numerical fluxes that are consistent with the discrete
divergence equation. To improve stability and accuracy, we derive a consistent
HDG stability parameter for the pressure-correction equation. We also apply a new
boundary condition for the pressure-corrector, and show the form and origin of the
projection method’s time-splitting error for a case with implicit diffusion and explicit
advection. Our scheme is implemented for arbitrary, mixed-element unstructured
grids using a novel quadrature-free integration method for a nodal basis, which is
consistent with the HDG method. To prevent numerical oscillations, we design a selective
high-order nodal limiter. We demonstrate the correctness of our new schemes
using a tracer advection benchmark, a manufactured solution for the steady diffusion
and stokes equations, and the 2D lock-exchange problem.

These numerical schemes are then extended for non-hydrostatic, free-surface,
variable-density regional ocean dynamics. The time-splitting procedure using projection
methods is derived for non-hydrostatic or hydrostatic, and nonlinear free-surface
or rigid-lid, versions of the model. We also derive consistent HDG stability parameters
for the free-surface and non-hydrostatic pressure-corrector equations to ensure
stability and accuracy. New boundary conditions for the free-surface-corrector and
pressure-corrector are also introduced. We prove that these conditions lead to consistent
boundary conditions for the free-surface and pressure proper. To ensure discrete
mass conservation with a moving free-surface, we use an arbitrary LagrangianEulerian
(ALE) moving mesh algorithm. These schemes are again verified, this time
using a tidal flow problem with analytical solutions and a 3D lock-exchange benchmark.

We apply our new numerical schemes to evaluate the numerical requirements of
the coupled biological-physical dynamics. We find that higher-order schemes are
more accurate at the same efficiency compared to lower-order (e.g. second-order)
accurate schemes when modeling a biological patch. Due to decreased numerical
dissipation, the higher-order schemes are capable of modeling biological patchiness
over a sustained duration, while the lower-order schemes can lose significant biomass
after a few non-dimensional times and can thus solve erroneous nonlinear dynamics.

Finally, inspired by Stellwagen Bank in Massachusetts Bay, we study the effect
of non-hydrostatic physics on biological productivity and phytoplankton fields for
tidally-driven flows over an idealized bank. We find that the non-hydrostatic pressure
and flows are important for biological dynamics, especially when flows are supercritical.
That is, when the slope of the topography is larger than the slope of internal
wave rays at the tidal frequency. The non-hydrostatic effects increase with increasing
nonlinearity, both when the internal Froude number and criticality parameter
increase. Even in cases where the instantaneous biological productivity is not largely
modified, we find that the total biomass, spatial variability and patchiness of phytoplankton
can be significantly altered by non-hydrostatic processes.

Our ultimate dynamics motivation is to allow quantitative simulation studies of
fundamental nonlinear biological-physical dynamics in coastal regions with complex
bathymetric features such as straits, sills, ridges and shelfbreaks. This thesis develops
the necessary numerical schemes that meet the stringent accuracy requirements for
these types of flows and dynamics.