Modelling Landscape Evolution - Environmental Modelling: Finding Simplicity in Complexity - page 305

2010), only the undercapacity model has as yet been
implemented in large-scale landscape-evolution models
(Table 19.1).

Moreover, all these models include an empirical constric-
tion term that simulates reduction of glacier velocities
by the friction of valley sidewalls. However, it appears
that incorporation of more realistic higher-order ice-
dynamics models is required for further development of
glacial erosion models.

19.3.5 Glacial erosion

Several 1-D models for the evolution of glacial valleys
have been developed (e.g. Oerlemans, 1984; Harbor, 1992;
MacGregor et al ., 2000) but incorporation of glacial ero-
sion processes in landscape-evolution models remains
rare. The only fully published model is currently that of
Braun et al . (1999), subsequently modified by Tomkin
and Braun (2002) and Herman and Braun (2008). Both
the 1-D and planform models are built around a surpris-
ingly simple glacial erosion law, in which erosion by both
glacial abrasion and quarrying is linearly proportional to
the basal sliding velocity u s (Hallet, 1979, 1996):
∂

19.4 Model testing and calibration

As for nearly all models in the earth sciences, validation
of landscape-evolution models is impossible in a strict
sense, for at least three reasons (Oreskes et al ., 1994):
the modelled Earth surface is not a closed system; model
predictions are nonunique; and we have incomplete
access to the natural phenomena driving landscape
evolution (see discussion in Chapter 2). Model testing
therefore often boils down to calibration of parameter
values, which are tuned so that the model output appears
'reasonable'. Because the model algorithms inherently
incompletely describe real-world processes and because
we are incapable of completely describing the resulting
landscape, it is extremely difficult to assess whether
such exercises attain 'essential realism' or merely lead to
'apparent realism', as discussed earlier.
Early attempts at calibrating landscape-evolution-
model parameters have compared statistical and fractal
descriptions of real and model landscapes (Chase, 1992;
Lifton and Chase, 1992; van der Beek and Braun, 1998).
This approach remains fairly inconclusive, however,
because (1) real-world landscapes do not show the
correlation between landscape descriptors and (tectonic
or climatic) controlling parameters predicted by the
models, suggesting that the models incompletely capture
landscape complexity; (2) statistical landscape descriptors
may be overly general and thus nondiscriminatory (e.g.
Kirchner, 1993); and (3) the present-day landscape is only
a snapshot of a continuously evolving system, leading
to the problem of equifinality (e.g. Beven, 1996). Van
der Beek and Braun (1998) therefore suggested the com-
parison of spatial and temporal patterns of denudation
predicted by the models with observational (for example,
thermochronologic or cosmogenic) data, but the latter
again only provide partial insight because of limits on
sampling density and because the observables are only
indirectly linked to denudation history (in particular,
thermochronology data strongly depend on the thermal
structure of the crust, which itself is partially dependent
on denudation rates; cf. Braun et al ., 2006 amongst
others, for an in-depth discussion of this problem).

h

t = K g u s

(19.15)

∂

The basal sliding velocity depends on temperature (it is
zero for cold-based ice, which is frozen to the bed, and
non-zero for temperate wet-based ice), ice thickness and
surface slope, and basal water pressure (Bindschadler,
1983; Paterson, 1994). Basal temperature is calculated
assuming a conductive geothermal gradient in the ice and
basal water pressure is usually considered proportional to
ice overburden. The ice thickness H g and its spatial vari-
ation are calculated from a mass-conservation equation:

∂ H g

∂

=∇ F + M =∇ ( H g W [ u d + u s ]) + M (19.16)

t

where F is the vertically integrated mass flux [L 3 T − 1 ]
(which equals ice thickness H g times valley width W
times the sum of the vertically averaged deformation
velocity u d and the sliding velocity u s )and M is the
mass-balance term [L T − 1 ]. Therefore, glacial erosion
models require both a mass-balance model and an ice-
dynamics model. The latter is most efficiently calculated
using the 'shallow-ice approximation', i.e. supposing that
in-plane deformation of the ice can be neglected. This
approximation, developed for modelling ice caps, is only
reasonable when lateral variations in ice thickness and
surface slope are small; its use for modelling mountain
glaciation is severely limited. Tomkin and Braun (2002)
therefore modelled glacial erosion on a coarse (2 km) spa-
tial grid, whereas Herman and Braun (2008) developed
a modelling approach that interpolates between a coarse
grid for calculating ice thickness and velocity and a finer
one for calculating the evolution of surface topography.