# Boundary Conditions

Boundary Conditions

## Horizontal Boundary Conditions

ROMS comes with a variety of boundary conditions, including open, closed, and periodic. See Marchesiello et al. (2001) for a more thorough exploration of the options. Some options require a value for the boundary points from either an included analytic expression (Functionals) or from an external NetCDF file. Here, $\phi ^{\rm {ext}}$ represents the exterior value of a quantity $\phi$ .

This boundary condition is extremely simple and consists of setting the gradient of a field to zero at the edge. The outside value is set equal to the closest interior value.

### Wall boundary condition

ROMS now assumes a wall condition if no other boundary condition is chosen. This is a zero gradient condition for tracers and the surface elevation and zero flow for the normal velocity. For tangential velocities, the wall is treated as either no-slip or free-slip, depending on the value of gamma2 chosen by the user.

### Clamped boundary condition

Almost as simple is setting the boundary value to a known exterior value.

 $\phi =\phi ^{\rm {ext}}$ (1)

### Flather boundary condition

For the normal component of the barotropic velocity, one option is to radiate out deviations from exterior values at the speed of the external gravity waves ( Flather (1976)):

 ${\overline {u}}={\overline {u}}^{\rm {ext}}-{\sqrt {\frac {g}{D}}}\,(\zeta -\zeta ^{\rm {ext}})$ (2)

The exterior values are often used to provide tidal boundary contitions to the barotropic mode. However, there are times when only the tidal elevation is known. A reduced physics option is available for estimating ${\overline {u}}^{\rm {ext}}$ in that case.

### Chapman boundary condition

The corresponding condition for surface elevation was investigated by Chapman (1985), assuming all outgoing signals leave at the shallow-water wave speed of ${\sqrt {gD}}$ . This can be useful when using the Flather condition on the 2-D momentum equations.

 ${\frac {\partial \zeta }{\partial t}}=\pm {\sqrt {gD}}\,{\frac {\partial \zeta }{\partial \xi }}$ (3)

The time derivative here can be handled either explicitly or implicitly. The model uses an implicit timestep, with the term ${\frac {\partial \zeta }{\partial \xi }}$ being evaluated at the new timestep.

In realistic domains, open boundary conditions can be extremely difficult to get right. There are situations in which incoming flow and outgoing flow happen along the same boundary or even at different depths at the same horizontal location. Orlanski (1976) proposed a radiation scheme in which a local normal phase velocity is computed and used to radiate things out (if it is indeed going out). This works well for a wave propagating normal to the boundary, but has problems when waves approach the boundary at an angle. Raymond and Kuo (1984) have modified the scheme to account for propagation in all three directions. In ROMS, only the two horizontal directions are accounted for (with the recommended RADIATION_2D option):

 ${\frac {\partial \phi }{\partial t}}=-\left(\phi _{\xi }{\frac {\partial \phi }{\partial \xi }}+\phi _{\eta }{\frac {\partial \phi }{\partial \eta }})\right)$ (4)

where

 {\begin{aligned}\phi _{\xi }&={\frac {F{\frac {\partial \phi }{\partial \xi }}}{\left({\frac {\partial \phi }{\partial \xi }}\right)^{2}+\left({\frac {\partial \phi }{\partial \eta }}\right)^{2}}}\\\phi _{\eta }&={\frac {F{\frac {\partial \phi }{\partial \eta }}}{\left({\frac {\partial \phi }{\partial \xi }}\right)^{2}+\left({\frac {\partial \phi }{\partial \eta }}\right)^{2}}}\\F&=-{\frac {\partial \phi }{\partial t}}\end{aligned}} (5)

These terms are evaluated at the closest interior point in a manner consistent with the time stepping scheme used. The phase velocities are limited so that the local CFL condition is satisfied. They are then applied to the boundary point using equation (4), again using a consistent time stepping scheme. Raymond and Kuo give the form used for centered differencing and a leapfrog time step while ROMS uses one-sided differences.

The radiation approach is appropriate for waves leaving the domain. A check is made to see which way the phase velocity is headed. If it is entering the domain, a zero gradient condition is applied unless the next option is also specified.