104
5 2D Shallow-Water Modelling
5.5.9 Additional Exercise for the Reader
The reader is encouraged to create a different bathymetry and to explore the lake’s
circulation for a variety of wind directions.
5.6 Movement of Tracers
5.6.1 Lagrangian Versus Eulerian Tracers
Lagrangian tracers are non-buoyant flui parcels that move passively with the f ow.
In practice, the trajectory of Lagrangian float are predicted by means of displacement distances per time step derived from the velocity at floa locations. Eulerian
tracers, on the other hand, are concentration field being subject to advection and
mixing by currents.
5.6.2 A Difficul Task
The numerical simulation of advection of Eulerian concentration field is a challenging and difficul task and there are many potential sources of errors that can
occur. Some numerical advection schemes trigger unwanted numerical diffusion,
while other schemes lead to unwanted numerical oscillations.
5.6.3 Eulerian Advection Schemes
The advection equation for depth-averaged tracer concentration B in the presence
of depth-averaged horizontal fl w with components u and v is given by:
∂B
∂t
= −u
∂B
∂x
− v
∂B
∂y
(5.20)
Using the product rule of differentiation, this equation can be reformulated as:
∂B
∂t
= −
∂(uB )
∂x
−
∂(vB )
∂y
+ B
∂u
∂x
+
∂v
∂y
(5.21)
The temporal change of B can be discretised using a simple time-forward iteration. Also the last term can be formulated in a straight-forward explicit manner.
Several options are available for the treatment of the remaining terms. These options
are described in the following for the x-direction. Analog recipes apply for the
y-direction.