# Module Hydrodynamic

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## Overview

The hydrodynamic model solves the primitive continuity and momentum equations for the surface elevation and 3D velocity field for incompressible flows, in orthogonal horizontal coordinates and generic vertical coordinates, assuming hydrostatic equilibrium and Boussinesq approximations.

Density is computed depending on salt, temperature and pressure, by the UNESCO equation of state (UNESCO, 1981). The model uses an ADI (Alternate Direction Implicit) time discretization scheme which minimizes stability restrictions, and is defined in an Arakawa-C type grid. In the bottom, shear stress can be computed with the assumption of a logarithmic velocity gradient:

In the free surface, a momentum flux can also be imposed in the form of shear stress.

Momentum, mass and heat transport is computed using a generic 3D advection-diffusion library including various advection schemes namely: first, second and third order upwind, centred differences and TVD (Total Variation Diminishing). Advection is solved in the three directions as a one-dimensional case and various time discretizations can be combined: explicit, semi-implicit or fully implicit.

## Boundary conditions

### Bottom

In the bottom, advective fluxes are imposed as null and diffusive flux of momentum is estimated by means of a bottom stress that is calculated by a non-slip method with a quadratic law that depends on the near-bottom velocity. So, the diffusive term at the bottom is written as:

$C_{D}$ is the bottom drag coefficient that is calculated with the expression:

where $k$ is von Karman constant and $z^{b}_{0}$ is the bottom roughness length. This quadratic law is derived from the logarithmic law of the wall near boundaries characteristic of boundary layers, as the bottom velocities are located half a grid box above the bottom. This term is calculated semi-implicitly following Backhaus (1985) for numerical stability reasons.

### Surface

#### Momentum

Diffusive flux of momentum is imposed explicitly by means of a wind surface stress, $\tau_{ w}$:

#### Mass

A water flux can be imposed (e.g. precipitation or evaporation) or computed (e.g. evaporation) at the surface of the water column.

### Open boundaries

Blumberg-Kantha radiation scheme for the open boundary conditions (Blumberg and Kantha, 1985).

### Moving boundaries (Drying and flooding)

Moving boundaries are closed boundaries that change position in time. If there are intertidal zones in the domain, some points can be alternatively covered or uncovered depending on tidal elevation. A stable algorithm is required for modeling these zones and their effect on hydrodynamics of estuaries.

### Land boundaries

In these boundaries the domain is limited by land. For the resolution used, this lateral boundary layer is resolved, so a impermeable, free slip condition can be used:

$\frac{{\partial \overrightarrow{{v_H }} }} {{\partial \eta }} = 0$

$\overrightarrow{v} \cdot \overrightarrow{n} = 0$

In the finite volume formalism, these conditions are implemented straightforwardly by specifying zero normal water fluxes and zero momentum diffusive fluxes at the cell faces in contact with land.

A non-slipling condition can also be used in lateral land boundaries.

## Outputs

### Maps (HDF5 format)

To write 3D results use keyword OUTPUT_TIME and define keyword:

OUTPUT_HDF           : 1


in each property that you whish to write results.

To write results only in the surface use keyword SURFACE_OUTPUT_TIME and define keyword:

OUTPUT_SURFACE_HDF   : 1


in each property that you which to write results. Surface results can be written with a different frequency than the normal 3D maps.