Equivalences and differences between the hydrological dynamical systems
of water budget, travel time, response time, and tracer concentrations
and the legacy of models' topology
Abstract
We present, by using previous results on extended Petri Nets, the
relations of various hydrological dynamical systems
($\mathtt{HDSys}$) derived from the water budget
($\mathtt{DynWB}$). Once
$\mathtt{DynWB}$ has been implemented, there exist a
consistent way to get the equations for backward travel time
distributions ($\mathtt{DynTT}$), for the forward
response time distribution ($\mathtt{DynRTD}$) and
for the concentration for a solute or a tracer
($\mathtt{DynC}$). We show that the
$\mathtt{DynWB}$ has a correspondence one to many
with the $\mathtt{DynTT}$. In fact to any one of the
$\mathtt{DynWB}$ equation correspond as many equation
as the input precipitation events times. The
$\mathtt{DynTT}$ is related to
$\mathtt{DynRTD}$ by the Niemi’s relationship and, in
presence of multiple, $n$ outputs, by the specification of $n-1$
partition functions, which determine which fraction of water volume,
injected in the control volume at a specific time $t_{in}$, goes
asymptotically into a specific output. The
$\mathtt{DynC}$, given
$\mathtt{DynTT}$, depends further on the
solute/tracer concentration in inputs. The paper clarifies the
complicate set of relations above by using an example from literature.
Upon the introduction of the appropriate information, it is also shown
how these ($\mathtt{HDSys}$) can be solved
simultaneously without duplicating calculations. It is also shown that
these systems can be solved exactly, under the hypothesis of uniform
mixing of water ages inside each reservoir within the system.