# 2D sediment transport formulation¶

## Governing equations¶

Suspended sediment transport is modelled in two dimensions using an advection-diffusion equation (6). If solved in non-conservative form, the prognostic variable is the passive tracer concentration, $$T$$. The corresponding field in Thetis is called 'sediment_2d'.

The sediment model is defined by specifying parameters in ModelOptions2d.sediment_model_options. See SedimentModelOptions for a list of all available parameters.

A conservative suspended sediment transport model is also available. In this case, the equation is solved for $$q=HT$$, where $$H$$ is the total water depth. The conservative tracer model is specified using the SedimentModelOptions.use_sediment_conservative_form option.

Bedload transport is modelled in two dimensions using the Exner equation (1). It is solved for the bedlevel, $$z_b$$, which is the bathymetry. The corresponding field in Thetis is called 'bathymetry_2d'.

To activate the 2D sediment model, set the SedimentModelOptions.solve_suspended_sediment and SedimentModelOptions.solve_exner options to True.

## Spatial discretization¶

Thetis currently only supports suspended sediment in P1DG space. Lax-Friedrichs stabilization is used by default and may be controlled using the ModelOptions2d.use_lax_friedrichs_tracer option. The scaling parameter used by this scheme may be controlled using the ModelOptions2d.lax_friedrichs_tracer_scaling_factor option.

The function space used for the bedlevel is determined by that used for the bathymetry. Typically, this is P1.

## Temporal discretization¶

Thetis supports different time integration methods, set by the SedimentModelOptions.sediment_timestepper_type and SedimentModelOptions.exner_timestepper_type options. Use the ModelOptions2d.set_timestepper_type method to set the same timestepper for all 2D model components.

Time integrator

Thetis class

Unconditionally stable

Description

'ForwardEuler'

ForwardEuler

No

Forward Euler method

'BackwardEuler'

BackwardEuler

Yes

Backward Euler method

'CrankNicolson'

CrankNicolson

Yes

Crank-Nicolson method

'DIRK22'

DIRK22

Yes

DIRK(2,3,2) method

'DIRK33'

DIRK33

Yes

DIRK(3,4,3) method

'SSPRK33'

SSPRK33

No

SSPRK(3,3) method

Table 1. Time integration methods for 2D sediment model.