2D tracer formulation¶
Governing equation¶
The two dimensional tracer model solves an advection-diffusion
equation
(17).
If solved in non-conservative form, the prognostic variable
is the passive tracer concentration,
\(T\). The corresponding field in Thetis is called
'tracer_2d'.
A conservative tracer model is also available, given by
(1).
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
ModelOptions2d.use_tracer_conservative_form
option.
To activate the 2D tracer model, set the
ModelOptions2d.solve_tracer
option to
True. The tracer model may also be run independently
by setting the
ModelOptions2d.tracer_only
option to
True. The hydrodynamics will be defined by any initial
conditions specified for the horizontal velocity, or any updates
imposed by the user.
Spatial discretization¶
Thetis supports two different tracer finite element discretizations and associated stabilization methods, summarised in the table below.
Element Family |
Name |
Space |
Stabilization |
|---|---|---|---|
DG |
|
P1DG |
Lax-Friedrichs |
CG |
|
P1 |
SUPG |
Table 1. Finite element families and stabilization methods.
The element family is set by the
ModelOptions2d.tracer_element_family
option. Polynomial degrees other than one are not currently supported.
Lax-Friedrichs stabilization is used by default and may be
controlled using the
ModelOptions2d.use_lax_friedrichs_tracer
option. Note that it is only a valid choice for the 'dg' element family.
The scaling parameter used by this scheme may be controlled using the
ModelOptions2d.lax_friedrichs_tracer_scaling_factor
option.
If the 'cg' element family is chosen, then SUPG stabilization is used by
default. It can be controlled using the
ModelOptions2d.use_supg_tracer
option. In that case, it is advisable to set characteristic velocities and
diffusivities for your problem using the
ModelOptions2d.horizontal_velocity_scale
and
ModelOptions2d.horizontal_diffusivity_scale
options.
Temporal discretization¶
Thetis supports different time integration methods, set by the
ModelOptions2d.timestepper_type option.
Note that the same time integration method will be used for both the shallow
water equations and the 2D tracer model.
Time integrator |
Thetis class |
Unconditionally stable |
Description |
|---|---|---|---|
|
No |
Forward Euler method |
|
|
Yes |
Backward Euler method |
|
|
Yes |
Crank-Nicolson method |
|
|
Yes |
DIRK(2,3,2) method |
|
|
Yes |
DIRK(3,4,3) method |
|
|
No |
SSPRK(3,3) method |
|
|
– |
Solves equations in steady state |
Table 2. Time integration methods for 2D tracer model.