Model Equations and Features¶
Equations¶
T-UGOm equations are derived from the classical shallow water equations where the advection and horizontal viscosity terms are neglected. The governing equations for time-stepping runs are:
Continuity:
\[ \frac{\partial \zeta}{\partial t} + \nabla \cdot \left( H \mathbf{U} \right) = 0 \]where:
\(\zeta\) the sea surface elevation,
\(H\) the bathymetry,
\(\mathbf{U} = (u, v)\) the depth-averaged horizontal velocity.
Momentum (linearized for tides):
\[ \frac{\partial \mathbf{U}}{\partial t} + f \mathbf{k} \times \mathbf{U} + g \nabla \zeta + \frac{g}{H} \mathbf{U} \cdot \nabla H + \frac{C_d}{H} \|\mathbf{U}\| \mathbf{U} = \mathbf{F} \]where:
\(f\) the Coriolis parameter,
\(g\) the gravitational acceleration,
\(C_d\) the quadratic bottom friction coefficient,
\(\mathbf{F}\) the external forcing (e.g., tidal potential, atmospheric pressure).
T-UGOm also support frequency-domain computation where the previous equation are developped assuming periodic solution for each tidal constituents. The governing equations for frequency-domain runs are:
Continuity:
\[ j \omega \alpha + \nabla \cdot H \mathbf{U} \]where:
\(j = \sqrt{-1}\),
\(\omega\) the tidal contituent frequency,
\(\alpha\) the tidal sea surface elevation,
\(H\) the bathymetry,
\(\mathbf{U} = (u, v)\) the depth-averaged horizontal velocity.
Momentum (linearized for tides):
\[ j \omega \mathbf{U} + f\mathbf{U} = -g \nabla \cdot (\alpha+\delta) + g \nabla \Pi - \mathbf{D}\mathbf{U} \]where:
\(f\) the Coriolis parameter,
\(g\) the gravitational acceleration,
\(\delta\) the ocean bottom radial displacement,
\(\Pi\) the total tidal potential (i.e. astronomical plus loading/self-attraction potential.
\(\mathbf{D}\) the drag tensor,
In the early version of T-UGOm, the complex tensor \(\mathbf{D}\) was limited to the bottom drag contribution and its coefficients depend only on the dominant wave currents :
\[\begin{split} >\mathbf{D}= \left[ \begin{matrix} >r & r' \\ >r'' & r''' >\end{matrix} \right] >\end{split}\]Now the friction include other contribution (internal tides wave drag, sea ice, etc. ). See the physical parameter section for further information.
Main Features¶
Tidal Potential Forcing
The model includes astronomical tidal potential terms derived from the Love numbers and equilibrium tide theory.
Load Self-Attraction (LSA) and ocean tide loading effects
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Bottom friction: Quadratic or linear parameterizations,
Horizontal viscosity: Smagorinsky or constant diffusion,
Internal wave drag
Sea Ice
Nonlinear advection and Coriolis effects
Open boundary conditions (Flather, clamped, or radiation) for nested or global applications.
Wetting/Drying
Handles flooding/emerging coastal areas via depth thresholds for time-stepping runs.
Non-linear tides determination
The spectral approach is clearly less convenient for the simulation of non-linear tides. Contrary to the astronomical components, the non-linear tides are mutually inter-dependent. \(M_4\) is the product of the interaction of \(M_2\) with itself, but also with \(M_6\). Reversely \(M_6\) is mainly due to the interaction of \(M_2\) with \(M_4\). An iterative approach is used to solve for a coherent set of of non-linear tides, so that mutual interactions are mostly taken into acccount.
Current tidal prediction (from the intermediate state of resolved tidal spectrum) is used to reconstruct fully non-linear bottom friction. It might be useful to extend this approach to the 2 other non-liear tide producing terms (transport divergence in mass conservation and advection).
Additional tests (based on \(M_4\) simulation) have shown that a linear parameterisation of non-linear tides loading potential and deformation might be more adequate than to use a prior LSA field (\(i.e.\) computed from a prior tidal solution).