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(that will change with wetting and drying). Since velocities require a length scale in order to simulate the
timing of boundary fluxes, the scale and geometry of the control volume are naturally taken into account
(even if there is necessarily some uncertainty about the subsurface geometry and velocity distributions).
9.5 Inferring Scale-Dependent Hysteresis from Simplified Hydrological
Theory
An analytical treatment of the scale-dependent hysteresis of hillslope scale runoff responses has been
produced by Norbiato and Borga (2008). Their analysis was limited to horizontal kinematic subsurface
fluxes on slopes of a certain range of shapes (assuming a parabolic width function and an exponential
surface slope function) under conditions of homogeneous conductivity and uniform recharge (which also
means that delays in the unsaturated zone can be neglected). For nine different shapes of slope their
analysis represents the hysteresis in the storage-discharge relationship as a non-dimensional functional
form (Figure 9.2). For each slope shape, the length of the slope and the uniform recharge rate control the
scaled discharge from the slope.
Norbiato and Borga did not consider cases where the water table rises to the surface and allows a
saturated contributing area to develop. They speculate about the hysteresis in the fast runoff response that
would result in this case. This has been demonstrated by the work of Martina
et al.
(2011) who develop
relationships between storage and saturated area for use in the lumped version of TOPKAPI (Ciarapica
and Todini, 2002; Liu and Todini, 2002). These relationships are derived from the fully distributed version
(see Section 6.5) in which the length scale is considered explicitly. The parameters of the lumped version
Figure 9.2
Normalised storage (
/Q
max
) relationships for four hill-
slope forms: 1. divergent concave; 3. divergent convex; 7. convergent concave; and 9. convergent convex
(after Norbiato and Borga, 2008, with kind permission of Elsevier).
r
(
t
)
=
V
(
t
)
/V
max
)
) versus flux (
u
(
t
)
=
Q
(
L, t
)
