Primitive Equation Modelling ofPlankton Ecosystems
413
values in each cell ofthe Eulerian grid. Usually (but not necessarily) these cells are
the same as those used to describe the environmental fields. Such Eulerian integration ofthe biological equations has disadvantages, including the following:
1) The Eulerian simulation cannot describe intra-population variability, and is
therefore liable to demographic bias.
2) Plankton movement relative to the water is described by biomass advection
equations, which can lead to numerical instability.
3) It is not possible to simulate inheritance of biologic al attributes through plankton lineages.
But more fundamental than these is the fact that the equations used in Eulerian
models describe changes in populations and not individual organisms. They are
demographic equations, rather than biological primitive equations. So Eulerian
modelling lacks the intrinsic scientific credibility that BPEs provide. Although the
equations in Eulerian and Lagrangian models address the same biological processes, such as photosynthesis and respiration in phytoplankton or diel migration
and foraging in zooplankton, the values ofthe parameters in the Eulerian equations
cannot be established from laboratory experiments. In practice, the values are normally treated as open to tuning by comparison with observations of emergent properties ofthe Eulerian simulation (Hassell et al 1976), a very fraught process (Solow
1995 and 18.17).
18.15.1 Using BPE simulations to tune Eulerian models
Nevertheless, Eulerian integration is Hobson's choice for some practical applications. It is therefore worth asking whether the parameter values used in Eulerian
models could be tuned by comparison with the emergent properties of an equivalent Lagrangian model, which is securely based on Biological Primitive Equations.
In principle this is an attractive option. A Virtual Plankton Ecosystem is much
more complete than data sets from observations, and offers many more features for
comparison with equivalent Eulerian simulations. However, no attempt has yet
been made to tune Eulerian models to VPEs, mainly because the number of tuneable biological parameters is so large: twenty-seven in the WB model. Tuning
involves optimisation in 27-dimensional parameter space. Standard procedures
exist for multi-dimensional optimisation (based, for example, on the use of genetic
algorithms) and can easily be implemented on affordable computer clusters. But, to
be effective the user must have a good a priori knowledge of the reference data set,
its "landscape" in parameter space (Kauffman 1993). We do now have a sufficient
knowledge of the Azores VPE. Fig. 18.9 shows how we might tune an Eulerian
model based on the Azores VPE. It will reveal how close it is possible to make
Eulerian and Lagrangian simulations of a WB plankton ecosystem.
413
values in each cell ofthe Eulerian grid. Usually (but not necessarily) these cells are
the same as those used to describe the environmental fields. Such Eulerian integration ofthe biological equations has disadvantages, including the following:
1) The Eulerian simulation cannot describe intra-population variability, and is
therefore liable to demographic bias.
2) Plankton movement relative to the water is described by biomass advection
equations, which can lead to numerical instability.
3) It is not possible to simulate inheritance of biologic al attributes through plankton lineages.
But more fundamental than these is the fact that the equations used in Eulerian
models describe changes in populations and not individual organisms. They are
demographic equations, rather than biological primitive equations. So Eulerian
modelling lacks the intrinsic scientific credibility that BPEs provide. Although the
equations in Eulerian and Lagrangian models address the same biological processes, such as photosynthesis and respiration in phytoplankton or diel migration
and foraging in zooplankton, the values ofthe parameters in the Eulerian equations
cannot be established from laboratory experiments. In practice, the values are normally treated as open to tuning by comparison with observations of emergent properties ofthe Eulerian simulation (Hassell et al 1976), a very fraught process (Solow
1995 and 18.17).
18.15.1 Using BPE simulations to tune Eulerian models
Nevertheless, Eulerian integration is Hobson's choice for some practical applications. It is therefore worth asking whether the parameter values used in Eulerian
models could be tuned by comparison with the emergent properties of an equivalent Lagrangian model, which is securely based on Biological Primitive Equations.
In principle this is an attractive option. A Virtual Plankton Ecosystem is much
more complete than data sets from observations, and offers many more features for
comparison with equivalent Eulerian simulations. However, no attempt has yet
been made to tune Eulerian models to VPEs, mainly because the number of tuneable biological parameters is so large: twenty-seven in the WB model. Tuning
involves optimisation in 27-dimensional parameter space. Standard procedures
exist for multi-dimensional optimisation (based, for example, on the use of genetic
algorithms) and can easily be implemented on affordable computer clusters. But, to
be effective the user must have a good a priori knowledge of the reference data set,
its "landscape" in parameter space (Kauffman 1993). We do now have a sufficient
knowledge of the Azores VPE. Fig. 18.9 shows how we might tune an Eulerian
model based on the Azores VPE. It will reveal how close it is possible to make
Eulerian and Lagrangian simulations of a WB plankton ecosystem.