manufactured solution

The manufactured solution tasks implement configurations for surface wave propagation with the rotating, nonlinear shallow water equations on a doubly periodic domain. These tasks are intended to utilize tendency terms embedded in the forward ocean model in order to produce the manufactured solution. This solution can be then used to assess the numerical accuracy and convergence of the discretized nonlinear momentum equation.

Currently, the there is only one task, the convergence test from Bishnu et al.(2023)

suppported models

These tasks support both MPAS-Ocean and Omega.

convergence

description

There are 3 versions of the convergence test case, convergence_space, convergence_time, and convergence_both corresponding to space, time, and space and time convergence tests. Tests involving spatial convergence run the manufactured solution simulation for 4 different resolutions: 200, 100, 50, and 25 km. Tests involving temporal convergence use the parameter dt_per_km at the base_resolution multiplied by refinement_factors (see Convergence Tests for more details on how to change resolutions or time steps tested).

The forward step for each resolution runs the simulation for 10 hours. The model is configured without vertical advection and mixing. No tracers are enabled and the pressure gradient used is the gradient of the sea surface height. Horizontal mixing and bottom friction are also neglected.

The analysis step computes the L2-norm of the difference between the simulated SSH field and the exact solution at the end of the simulation. It also computes the convergence rate with resolution.

The visualization step produces two plots: the convergence of the L2-norm with resolution and a plan-view of the simulated, exact, and (simulated-exact) SSH fields.

mesh

For each resolution, the init step generates and planar hexagonal mesh that is periodic in both the x and y directions.

vertical grid

Since this test case is a shallow water case, the vertical grid is set to a single layer configuration.

[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 = 1000.0

# 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 conditions are set to the following:

\[\begin{split} \eta = \eta_0 \sin(k_x x + k_y y - \omega t)\\ u = \eta_0 \cos(k_x x + k_y y - \omega t)\\ v = u \end{split}\]

forcing

N/A

time step and run duration

The time step is determined by the config option dt_per_km according to the mesh resolution. The run duration is 10 hours.

config options

The following config options are availiable for this case:

[manufactured_solution]

# the size of the domain in km in the x and y directions
lx = 10000.0

# the coriolis parameter
coriolis_parameter = 1.0e-4

# the amplitude of the sea surface height perturbation
ssh_amplitude = 1.0

# Number of wavelengths in x direction
n_wavelengths_x = 2

# Number of wavelengths in y direction
n_wavelengths_y = 2

# Time step per resolution (s/km), since dt is proportional to resolution
dt_per_km = 3.0

# Convergence threshold below which the test fails
conv_thresh = 1.8

cores

The number of cores is determined according to the config options max_cells_per_core and goal_cells_per_core.