(omega-user-horz-operators)=

# Horizontal Operators

The horizontal discretization in Omega is a staggered numerical scheme known as
TRiSK. It defines discrete versions of basic differential operators (divergence,
gradient, and curl) as well as other operators that are needed by the scheme,
such as reconstruction of tangential velocity from its normal components. These
operators act on quantities located at the cells, edges, and vertices of an
MPAS mesh, and rely on the connectivity, geometric measures, and ordering
conventions defined in the
{ref}`MPAS mesh specification <mpas-mesh-specification>`. Omega
provides reference implementations of these operators, each in a separate C++
class. The class name describes the operator and the mesh element associated
with its result.

The following operators are currently implemented:
- `DivergenceOnCell`
- `GradientOnEdge`
- `CurlOnVertex`
- `TangentialReconOnEdge`
- `InterpCellToEdge`
- `VectorReconOnCell`

There are no user-configurable options; the method for `InterpCellToEdge` is located in {ref}`Surface Stress Forcing <omega-user-forcing-sfc-stress>`.

(omega-user-horz-operators-interp)=
## Interpolate cells to edges

`InterpCellToEdge` contains two methods shown in {numref}`fig-interp`:
- `InterpolateAnisotropic`
- `InterpolateIsotropic`

```{figure} images/interpCellToEdge.png
:name: fig-interp
:align: center
:width: 60%

For a scalar quantity at the edge location shown in red, either average the two neighboring `cellsOnEdge` shown in green (left, anisotropic) or take the average of the four `cellsOnVertex` of each of the two `verticesOnEdge`, weighted by `kiteAreasOnVertex`.
```

The `cellsOnEdge`, `cellsOnVertex`, `verticesOnEdge`, and `kiteAreasOnVertex`
fields, along with their required ordering, are defined in the
{ref}`MPAS mesh specification <mpas-mesh-specification>`; see in particular
{ref}`connectivity-ordering-requirements`.

The isotropic interpolation has been shown to be less accurate than the anisotropic method, with higher error about pentagons, and should be used only when spatial smoothing is desired, e.g., in the context of wind stress coupling where both `SfcStressZonal` and `SfcStressMeridional` are interpolated from cells to edges, see {ref}`Surface Stress Forcing <omega-user-forcing-sfc-stress>`.