Geoscience Reference
In-Depth Information
There is an analogy here with the top-down sub-grid parameterisations of atmosphere and ocean models
mentioned above; but rather than the representation of the momentum dissipation due to turbulence, the
hydrologist is confronted with energy and momentum dissipation due to friction and viscosity at all the
internal boundaries for surface and subsurface flows. It is difficult to theorise about the latter because
of the sheer complexity of all the internal boundaries across a wide range of pore sizes in the soil and
roughness conditions on the surface. Thus specifying the “physics” at this scale is inherently difficult and
inherently uncertain. Indeed, the hydrological closure problem might therefore actually be considered
harder than the representation of atmospheric turbulence (even if the problem of representing the effects
of inhomogeneous turbulence is accepted as one of the most difficult problems in geophysics). That
does not mean that this is not a valuable strategy to take. While inital solutions to the closure problem
might be lacking in both sophistication and accuracy, hydrological science can work to gradually develop
improvements.
9.2.1 The REW Balance Equations
Examples of the mass, energy and momentum balance equations have already appeared earlier in this
topic (see for example Equations (3.2), (5.4) and (5.5)) but, before the formulation of the REW framework,
no-one had developed these equations in a consistent way for multiple processes. In fact, all the balance
equations can be reduced to the simple form (Reggiani and Rientjes, 2005):
∂t
=
i
∂ψ
Q
ψ
+
R
+
G
(9.1)
ψ
t
Q
ψ
ψ
i
where
is the mass, energy or momentum,
is time,
is a flux of
for the
th internal or external
boundary,
is an internal production term. Examples of such balance
equations are given by Reggiani and Rientjes (2005) for multiple processes within a REW. The expansion
of the terms of these equations can be done in a variety of ways but the essence of the closure problem
immediately becomes apparent. Even this simplest representation of the catchment requires the flux terms
R
is a source or sink term, and
G
i
Q
ψ
to be defined for all
i
to close the system of equations. In some cases, it will be possible to make
a plausible argument that some of these terms are negligible but, in most cases, representations of these
boundary fluxes are required, hence the closure problem is rather important.
9.2.2 The REW Closure Problem
It is evident that closure of all the mass and energy balance equations will not be easy in most applications.
(What, for example, would be the effective kinetic energy added to a control volume by the input of
rainfall? What is the effective dissipation of energy in macropore flows over a control volume?) This
does not affect the physical principles involved, only their practical implementation and application
(Beven, 2006b). We should, in principle, be able to close the mass, energy and momentum balances. We
might, in practice, find that difficult to do when there is uncertainty about how to represent many of the
flux terms, even those such as rainfalls and stream discharges that it might be feasible to measure. It
might, indeed, at the present state of knowledge, only be possible to estimate partitioning between the
required boundary fluxes and closure to a certain degree of approximation and uncertainty.
The functional requirement here is that the representation of each boundary flux for each process
component should properly reflect the relationship between that flux and the changing state of the system
(the
constitutive relationships
for a process). This immediately raises new possibilities about how to
think about the process representations. We would expect the closure fluxes to be a hysteretic function
of storage, for example, i.e. the fluxes would be different depending on the past sequence of wetting or
drying of a control volume. In general, we would expect the discharge for a given storage to be lower
