Contents

104 5 2D Shallow-Water Modelling5.5.9 Additional Exercise for the ReaderThe reader is encouraged to create a different bathymetry and to explore the lake’scirculation for a variety of wind directions.5.6 Movement of Tracers5.6.1 Lagrangian Versus Eulerian TracersLagrangian 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 displacementdistances per time step derived from the velocity at floa locations. Euleriantracers, on the other hand, are concentration field being subject to advection andmixing by currents.5.6.2 A Difficul TaskThe numerical simulation of advection of Eulerian concentration field is a challengingand difficul task and there are many potential sources of errors that canoccur. Some numerical advection schemes trigger unwanted numerical diffusion,while other schemes lead to unwanted numerical oscillations.5.6.3 Eulerian Advection SchemesThe advection equation for depth-averaged tracer concentration B in the presenceof 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( ∂u+ B∂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 optionsare described in the following for the x-direction. Analog recipes apply for they-direction.