# **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:

1. **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.

2. **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:

1. **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.

2. **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 :  
>
>$$ 
>\mathbf{D}= \left[ \begin{matrix}
>r & r' \\
>r'' & r'''
>\end{matrix} \right]
>$$
>
> Now the friction include other contribution (internal tides wave drag, sea ice, etc. ). See the [physical parameter](./physical.md) section for further information.

## Main Features

1. **Tidal Potential Forcing**

    The model includes **astronomical tidal potential** terms derived from the **Love numbers** and **equilibrium tide theory**.

2. **Load Self-Attraction (LSA)** and **ocean tide loading** effects

3. **[Dissipation Terms](./physical.md)**:

   - **Bottom friction**: Quadratic or linear parameterizations,
   - **Horizontal viscosity**: Smagorinsky or constant diffusion,
   - **Internal wave drag**
   - **Sea Ice**

4. **Nonlinear advection** and **Coriolis effects** 

5. **Open boundary conditions** (Flather, clamped, or radiation) for nested or global applications.

6. **Wetting/Drying**

   Handles **flooding/emerging** coastal areas via depth thresholds for time-stepping runs.

7. **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).

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