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physical quantity) is usually expressed in terms of a continuous variable referred to a
space point, the cell must correspond to a portion of space small enough so that a
single value may be attributed to full space of the cell. The continuity of the variable
is not a problem; in practical case the utilised variables have a finite number of
significant digits and a finite range of permitted values, then the set of utilised values
could be extremely large, but always finite.
As the state of the cell can be decomposed in substates, the transition function may
be split in many local interactions: the “elementary” processes. Such local interactions
could be inhomogeneous in space and/or time: the opportune dimension of a cell can
vary for different local interactions; very fast local interactions need a step
corresponding to short times on the same cell size; the appropriate neighbourhoods
for different local interactions could be different. An obvious solution to these
problems is the following: the smallest dimension of a cell must be chosen among the
permitted dimensions of all the local interactions. Then it is possible to define for
each local interaction an appropriate range of time values in correspondence of a
CA
step; the shortest time necessary to the local interactions must correspond to a step.
It is possible, when the cell dimension and the
CA
step are fixed, to assign an
appropriate neighbourhood to each local interaction; the union of the neighbourhoods
of all the local interactions must be adopted as the
CA
neighbourhood. A lookup table
could be unpractical to describe the local interactions. Each local interaction may be
espoused by means of procedures involving the proper substates and neighbourhood
of the local interaction.
Considering these premises, it is expedient to consider the transition function step
divided in as many phases as the elementary processes; the substates involved in each
process will be updated each time at the phase end.
2.2 A Practical Approach for Modelling Surface Flows
A delicate point concerns the modelling of flows in a
CA
context. Solutions to this
problem were proposed especially for microscopic simulation (e.g., Boltzmann lattice
[7]). A practical approach for modelling surface flows is here proposed.
Macroscopic phenomena involving surface flows may be often modelled by two-
dimensional
CA
, when the third dimension, the height, may be included as a property
of the state of the cell (a substate). This condition permits to adopt a simple, but
effective strategy, based on the hydrostatic equilibrium principle in order to compute
the cell outflows [4].
Let us focus for simplicity on a single cell (individuated as the “central” cell). It
can be considered limited to the universe of its neighbourhood, consisting of
m
cells:
the central cell and of the remaining cells (individuated as the “adjacent” cells). Index
0 individuates the central cell, indexes 1, 2 … m-1 individuate the adjacent cells.
On the basis of this assumption, the outflows from the central cell to the adjacent
cells depend on the hydrostatic pressure gradients across the cells, due to differences
in heights (for trivial instance, altitude plus debris thickness).
Two quantities must be identified in the central cell: the fixed part (
q
[
0
]) and the
mobile part (
p
) of the height. The mobile part represents a quantity that could be
distributed to the adjacent cells (the debris thickness, for instance), whilst the fixed
