3D tidal channel demo

This example demonstrates a 3D barotropic model in a tidal channel with sloping bathymetry. We also add a constant, passive salinity tracer to demonstrate local tracer conservation. This simulation uses the ALE moving mesh.

We begin by defining the 2D mesh as before:

from thetis import *

lx = 100e3
ly = 6e3
nx = 33
ny = 2
mesh2d = RectangleMesh(nx, ny, lx, ly)

In this case we define a linearly sloping bathymetry in the x-direction. The bathymetry function is defined as an UFL expression making use of the coordinates of the 2D mesh. The expression is interpolated on the P1 bathymetry field:

P1_2d = FunctionSpace(mesh2d, 'CG', 1)
bathymetry_2d = Function(P1_2d, name='Bathymetry')
depth_oce = 20.0
depth_riv = 7.0
xy = SpatialCoordinate(mesh2d)
bath_ufl_expr = depth_oce - (depth_oce-depth_riv)*xy[0]/lx
bathymetry_2d.interpolate(bath_ufl_expr)

Next we create the 3D solver. The 2D mesh will be extruded in the vertical direction using a constant number of layers:

n_layers = 6
solver_obj = solver.FlowSolver(mesh2d, bathymetry_2d, n_layers)

We then set some options (see ModelOptions for more information):

options = solver_obj.options
options.element_family = 'dg-dg'
options.timestepper_type = 'SSPRK22'
options.use_implicit_vertical_diffusion = False
options.use_bottom_friction = False
options.use_ale_moving_mesh = True
options.use_limiter_for_tracers = True
options.simulation_export_time = 900.0
options.simulation_end_time = 24 * 3600
options.fields_to_export = ['uv_2d', 'elev_2d', 'elev_3d', 'uv_3d',
                            'w_3d', 'w_mesh_3d', 'salt_3d', 'baroc_head_3d',
                            'uv_dav_2d', 'uv_bottom_2d']

We set this simulation to be barotropic (i.e. salinity and temperature do not affect water density), but we still wish to simulate salinity as a passive tracer:

options.use_baroclinic_formulation = False
options.solve_salinity = True
options.solve_temperature = False

We also want to see how much the salinity value deviates from its initial value:

options.check_salinity_overshoot = True

In this simulation we do not set the time step manually but instead use the automatic time step estimation of Thetis. Time step is estimated based on the CFL number, used spatial discretization and time integration method. We only need to define maximal horizontal and vertical velocity scales:

u_max = 0.5
w_max = 2e-4
options.horizontal_velocity_scale = Constant(u_max)
options.vertical_velocity_scale = Constant(w_max)

Next we define the boundary conditions. Note that in a 3D model there are multiple coupled equations, and we need to set boundary conditions to all of them.

In this example we impose time dependent normal flux on both the deep (ocean) and shallow (river) boundaries. We begin by creating python functions that define the time dependent fluxes. Note that we use a linear ramp-up function on both boundaries:

ocean_bnd_id = 1
river_bnd_id = 2

un_amp = -0.5  # tidal normal velocity amplitude (m/s)
flux_amp = ly*depth_oce*un_amp
t_tide = 12 * 3600.  # tidal period (s)
un_river = -0.05  # constant river flow velocity (m/s)
flux_river = ly*depth_riv*un_river
t_ramp = 6*3600.0  # use linear ramp up for boundary forcings


def ocean_flux_func(t):
    return (flux_amp*sin(2 * pi * t / t_tide) - flux_river)*min(t/t_ramp, 1.0)


def river_flux_func(t):
    return flux_river*min(t/t_ramp, 1.0)

We then define Constant objects for the fluxes and use them as boundary conditions for the 2D shallow water model:

ocean_flux = Constant(ocean_flux_func(0))
river_flux = Constant(river_flux_func(0))

ocean_funcs = {'flux': ocean_flux}
river_funcs = {'flux': river_flux}

solver_obj.bnd_functions['shallow_water'] = {ocean_bnd_id: ocean_funcs,
                                             river_bnd_id: river_funcs}

The volume fluxes are now defined in the 2D mode, so there’s no need to impose anything in the 3D momentum equation. We therefore only use symmetry condition for 3D horizontal velocity:

ocean_funcs_3d = {'symm': None}
river_funcs_3d = {'symm': None}

solver_obj.bnd_functions['momentum'] = {ocean_bnd_id: ocean_funcs_3d,
                                        river_bnd_id: river_funcs_3d}

For the salinity, we define a constant value and apply as inflow conditions at the open boundaries:

salt_init3d = Constant(4.5)
ocean_salt_3d = {'value': salt_init3d}
river_salt_3d = {'value': salt_init3d}

solver_obj.bnd_functions['salt'] = {ocean_bnd_id: ocean_salt_3d,
                                    river_bnd_id: river_salt_3d}

As before, all boundaries where boundary conditions are not assigned are assumed to be impermeable land boundaries.

We now need to define the callback functions that update all time dependent forcing fields. As the 2D and 3D modes may be treated separately in the time integrator we create a different call back for the two modes:

def update_forcings_2d(t_new):
    """Callback function that updates all time dependent forcing fields
    for the 2D mode"""
    ocean_flux.assign(ocean_flux_func(t_new))
    river_flux.assign(river_flux_func(t_new))


def update_forcings_3d(t_new):
    """Callback function that updates all time dependent forcing fields
    for the 3D mode"""
    pass

Because the boundary conditions of the 3D equations do not depend on time, the 3d callback function does nothing (it could be omitted).

We then assign the constant salinity value as an initial condition:

solver_obj.assign_initial_conditions(salt=salt_init3d)

and run the simulation:

solver_obj.iterate(update_forcings=update_forcings_2d,
                   update_forcings3d=update_forcings_3d)

As you run the simulation, Thetis prints out the normal simulation statistics and also prints out the over/undershoots in the salinity field:

    0     0 T=      0.00 eta norm:     0.0000 u norm:     0.0000  0.00
salt_3d overshoot 0 0
    1     5 T=    900.00 eta norm:    15.1764 u norm:     0.0000  1.23
salt_3d overshoot -1.00586e-11 2.58318e-11
    2    10 T=   1800.00 eta norm:    83.4282 u norm:     0.0000  0.39
salt_3d overshoot -3.13083e-11 3.42579e-11
    3    15 T=   2700.00 eta norm:   229.6974 u norm:     0.0000  0.35
salt_3d overshoot -6.35199e-11 6.6346e-11

Note that here the u norm is the norm of \(\mathbf{u}'\), i.e. the prognostic 3D horizontal velocity field (3D velocity minus its vertical average).

This tutorial can be dowloaded as a Python script here.