(ocean-barotropic-gyre)=

# barotropic gyre

The barotropic gyre test case implements the Munk model for a simplified,
wind-driven gyre
([Munk 1950](https://doi.org/10.1175/1520-0469%281950%29007%3C0080:OTWDOC%3E2.0.CO;2)).
It is a variant of the Stommel model except it uses Laplacian horizontal
viscosity rather than drag to balance the wind stress. As such, this test case
may be used to verify the Laplacian viscosity term and the wind stress forcing
term.

The test case is barotropic.

Here, we linearize the momentum equations to permit comparison against an
analytical solution. The nonlinear solution would be a logical extension of
this case and should be qualitatively similar.

## description

A successful test shows the development of a western boundary current and
clockwise circulation. The comparison with the analytical solution is performed
via a streamfunction. The L2 norm is printed and the following figure is
produced:

```{image} images/barotropic_gyre_solution.png
:align: center
:width: 800 px
```

## mesh

The mesh is planar with dimensions given by the config options `lx` and `ly`
and the resolution given by the config option `resolution`. The defaults are
1200 km x 1200 km and a resolution of 20 km but different mesh dimensions and
resolutions may be tested. The resolution will be compared with the estimated
lateral boundary layer width to ensure that the western boundary current will
be adequately resolved.

The boundary conditions are non-periodic in both x- and y-dimensions.

## vertical grid

As this test case is barotropic, the default number of vertical levels is 1.
A different number of vertical levels may be specified by the user. The
bottom topography is flat so this is not a test case for which different
vertical coordinates would be informative.

```cfg
# Options related to the vertical grid
[vertical_grid]

# The type of vertical grid
grid_type = uniform

# Number of vertical levels
vert_levels = 1

# Depth of the bottom of the ocean
bottom_depth = 5000.0

# The type of vertical coordinate (e.g. z-level, z-star)
coord_type = z-star

# Whether to use "partial" or "full", or "None" to not alter the topography
partial_cell_type = None

# The minimum fraction of a layer for partial cells
min_pc_fraction = 0.1
```

## initial conditions

The initial condition is at rest. Since the test case is barotropic, only a
reference density is provided with the config option `rho_0`. A beta-plane
is indicated by the coriolis parameter and determined by the config options
`f_0` and `beta`.

## forcing

This test has a constant wind stress field following:

$$
\tau_x &= \tau_0 * \cos(\pi \frac{y}{l_y}) \\
\tau_y &= 0
$$

where `tau_0` is given by a config option.

```{image} images/barotropic_gyre_forcing.png
:align: center
:width: 500 px
```

### config options

```cfg
[barotropic_gyre]

# distance in kilometers between cell centers
resolution = 20

# Longituidinal domain length in kilometers
lx = 1200

# Latitudinal domain length in kilometers
ly = 1200

# Maximum amplitude of the zonal wind stress [N m-2]
tau_0 = 0.1

# Horizontal visocity [m2 s-1]
nu_2 = 4e2

# [s-1 m-1]
beta = 10e-11

# [s-1]
f_0 = 10e-4

# homogenous fluid density [kg m-3]
rho_0 = 1000
```

The config option `nu_2` specifies the del2 horizontal viscosity. This value
will be compared against the resolution to check for stability at set-up. All
other config options are explained further in previous sections.

### cores

The number of cores is determined by `goal_cells_per_core` and
`max_cells_per_core` in the `ocean` section of the config file.

(ocean-barotropic-gyre-default)=

## supported models

These tasks support only MPAS-Ocean.

## default

The default case is designed such that only a short forward run is performed.
However, when users set up the default case, a long forward run is also set
up and may be run along with the analysis step to compare the numerical
solution with an analytical solution. The user should change the
`barotropic_gyre: steps_to_run` config option to include desired steps.

### time step and run duration

The time step for forward integration is set as a function of the CFL
condition assuming a maximum velocity of 1 m/s and a function of the coriolis
parameter `f_0`. The stability parameter is set to 0.25 based on the stability
of MPAS-Ocean for this test case.

The run duration is 3 time steps for the `short_forward` step and 3 years for
the `long_forward` step.