Most physically based erosion models compute on a high spatial and temporal resolution.
EROSION 2D uses time step of 10min, and an internal spatial resolution of 1m. Adapting
this model to a coarser resolution, i.e. to 1km2 as used in this study, means an information
loss concerning flow paths and runoff accumulation. Fig. 4.11 illustrates this problem. The
modelled surface runoff, which is passed from the soil component ofPROMETto the erosion
model, merely represents an areal mean value of the whole proxel (Fig. 4.11(a)). This could
be e.g. 1mm and quantifies, if multiplied with the unit area, the total surface runoff leaving
this unit (in an arbitrary time step). So calculating the erosion routines on this unit area means converting the given input surface runoff value to the internal computation unit of the erosion module. This leads theoretically (if other subscale factors, such as slope, are neglected) to the same result, as calculating the same routines repeatedly on a higher spatial resolution (see Fig. 4.11(b)), because the input parameters (based on another resolution)
1
(a) Representation of a single spa- tial computation unit, i.e. a whole proxel, with a surface runoff of 1 unit per area, e.g. 1 mm, distributed equally over the proxel.
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
(b) subscale representation of the same proxel as illustrated in (a). Each sub-unit has a surface runoff of 1 mm per sub-unit area. Since no subscale information is avail- able, the surface runoff is distributed evenly. 1 16 2 1 1 11 7 1 2 3 1 1 5 1 3 1
(c) subscale representation of the proxel in (a), but with subscale infor- mation on the flow paths and direc- tions. 1 2 1 2 3 64 3 56 4 3 4 3 2 1 2 1 1 2 1 2 3 48 3 40 4 3 36 3 2 1 2 1 1 2 1 10 1 1 12 13 32 3 27 3 2 1 2 1 1 6 1 3 1 1 1 1 10 3 5 3 2 1 2 1
(d) Illustration of (c) with doubled spatial resolution. (The arrows indi- cating the flow direction have been omitted for the purpose of clarity).
0 2 2 0 0 2 2 0 2 2 0 0 2 0 2 0
(e) Same representation of the proxel in (c), but with no subscale information on the flow directions.
2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 (f) Abstract representation of Fig. 4.12
Fig. 4.11: Subscale effects on flow concentration and particle detachment. The arrows indicate the flow paths of surface runoff. Gray colour denotes the areas where runoff concentrates.
are likewise converted to the same internal resolution. But calculating the erosion routines either based on a whole proxel, or on a multiple of sub-proxels, means neglecting any flow concentration. Fig. 4.11(c) shows the flow paths of the proxel, if subscale information was available. The difference to the previous illustrations is, that the sub-proxels do no longer contain the mean values, but runoff concentrates along the relief. The total discharge leaving the proxel accumulates in a single sub-proxel, which represents the outlet. It is obvious, that along the flow path the amount of collected runoff is much higher and therefore has much more energy to detach (and transport) particles, but the total amount leaving the proxel is the same. It is just the sum of the runoff discharging on a small outlet. This leads to the crucial question, which spatial resolution is required to adequately represent runoff behaviour and resultant particle detachment.
the development and evaluation of the three dimensional version of EROSION 2D, which is able to simulate flow paths. He tested the model EROSION 3D with identical input data sets,
varying only the spatial resolution from 2.5m to 100 m. The test site had an area of 0.78
km2 and the net erosion for the simulated event varied from 0.14 (at 100m) to 0.73 (at 2.5
m) tons per hectare, with a measured net erosion of approx. 0.3 tons per hectare. He con-
cludes, that the optimum resolution for his test site constitutes between 10 and 20m, as very
high resolution overestimates (inadequate process representation by the model) and very low resolution underestimates (inadequate representation of input parameters) net erosion. The main reason for the above mentioned overestimation of erosion, is the accumulation of surface runoff in flow paths. On a coarser resolution, runoff would be distributed over a larger area, whereas at fine resolution, the concentration leads to high discharge amounts and de- tachment rates. Such concentrated flow paths would have to be treated as drainage channel by the model EROSION 3D, in order to avoid further accumulation and particle detachment. This can be demonstrated with some theoretical considerations based on Fig. 4.11. As already noted, the the total runoff amount from Fig. 4.11(b) equals Fig. 4.11(c), but the forces acting on the subscale units increase. So if a grid with yet a higher resolution is assumed (see Fig. 4.11(d)), the flow concentration would once more increase, resulting in higher de- tachment rates. The important question then is, if this grid represents processes better, or if there are processes on this scale, which cannot be comprised (as e.g. sediment transport in channels), thus leading to simulation errors.
Another problem, irrespective of spatial resolution, is the availability of subscale information on measures, like pipeworks, drainage channels or vegetation strips, disturbing the drainage network. Imagining a vertical drainage channel in Fig. 4.11(c) separating the upper half from the lower half of the proxel, the channel would discharge all of the runoff from the lower part out of the proxel. It would be lost for the upper half, and the three affected, gray coloured proxels would only gain 2, 4 and 8 (instead of 7, 11 and 16) units of surface runoff along the flow path. von Werner (1995) actually encountered such problems on a catchment size of
0.78km2, which is less than the area of a single proxel of 1km2 , as used in this study. It is
virtually impossible to gain subscale information about drainage networks (or structures like vegetation strips with similar effects) on such an areal extent as the Upper Danube Basin.
Therefore an approximation of subscale flow concentration is introduced here, which is illustrated in Fig. 4.11(e). The basic assumption is, that the whole runoff volume concentrates on a certain definable area of the proxel. But as no subscale information on flow paths is available and the outlet of the proxel is unknown, a redistribution of the whole runoff volume
of the proxel is carried out, where thesum of the subscale unit runoff must equal the total
proxel volume. Thus the runoff is accumulated along its flow path, whereas the other sub- proxels do not receive any runoff in order to conserve the total outflow volume. Note, that there are no arrows drawn in Fig. 4.11(e) indicating the flow direction (like in Fig. 4.11(b)), as the flow direction essentially is irrelevant. The important aspect is, that usually a certain threshold of force acting on the soil, has to be exceeded to initiate particle detachment (as discussed below). Assuming, this threshold is overcome at a runoff volume above 1 unit, it can be seen, that the situation in Fig. 4.11(b), does not produce an detachment of soil. A comparison between Fig. 4.11(c) and Fig. 4.11(e), shows that detachment occurs on the same area, but to a different extent. The sum of total detached sediment in Fig. 4.11(c)
Fig. 4.12: Geometrical structure of the EROSION 2D test plot, whichSchmidt(1996) used for deriva- tion of the sediment mass flux equation.
would be much greater, even if detachment was a linear process, which it is definitely not. It is arguable, which of the two approaches delivers the “more correct” result, as this is again the scaling question mentioned above. Therefore this concept is further abstracted to a smaller scale.
Fig. 4.12 shows the set up Schmidt (1996) used for determining the sediment mass flux equation (Eq. (4.15) in Section 4.5.1.1). He performed experiments with surface runoff only, and experiments with both runoff and rainfall simulation. For all experiments he found a
threshold of approx. 3 ·10−5 m3/m·s, which has to be exceeded for initiation of sediment
mass flux. The sowing furrows can clearly be recognised, which implies, that runoff con- centrates there. EROSION 2D does not explicitly model rill erosion (Section 4.5.1.5) but the experimental set up and thus the derivation of the empirical equation implicitly include a kind of rill erosion due to the flow concentration. This means, that the flow concentration as pre- sented above, theoretically should be considered on the scale of this plot, too. Since the course of the furrows or rills on a larger scale can only be estimated, the abstracted concept of flow concentration without directional information (Fig. 4.11(e)) is used to describe this be- haviour on the micro scale. Fig. 4.11(f) shows the abstracted representation of the plot from Fig. 4.12.
4.5.2.2. Implementation in the Erosion Module
In order to implement the presented abstract, scale-invariant concept of flow concentration in
the erosion module, a flow concentration factor (fcf) is introduced. It is defined as the width
of flow concentrationwf [m] to the total width of the considered unit area wa[m]:
fcf =
wf
Thefcf is dimensionless and ranges theoretically from values above 0 to 1, where 1 can be visualised e.g. as sheet flow or a totally plane proxel surface, and very low values indicate e.g.
an area trenched by many small rills. So thefcf defines two fractions of a proxel, which are
treated differently by the erosion module. For the sake of simplicity the fractions will be termed rill and interrill area in the following. The technical implementation in the erosion module uses the factor to concentrate the flow on the given fraction, and computes the detachment and transport equations based on rill and interrill area. The calculation of the momentum flux of the runoff is limited to rill areas, but the momentum flux of the raindrops can act on the whole area. It is assumed, that particles detached by raindrops on the interrill areas (indicated by zero runoff units in Fig. 4.11(e) and Fig. 4.11(f)) are transported by splash into the rills, where they can be transported further on by concentrated flow.
So the processes of detachment and transport can be computed as described in the fol- lowing list:
1. The available amount of runoff is scaled with the givenfcf, i.e. is concentrated on a
virtual area.
2. With the concentrated runoff volume, velocity and momentum flux of the flow are cal- culated.
3. Momentum flux of rainfall is calculated and the rill and interrill areas are taken into account (as described below).
4. The total momentum flux is used to calculate the particle detachment. The resulting
detachment is reduced with the fcf to the actual rill area for the case of rill erosion,
whereas detachment by raindrops may act on the whole area.
5. The transport capacity of the concentrated flow is determined. For calculation of the maximum particle concentration only the momentum flux of the drops on the rill areas
is considered (instead of ϕr,α in Eq. (4.21)). Based on the total available amount of
runoff (qin Eq. (4.22)) on this proxel, the amount of sediment actually leaving the proxel
is computed.
For computation of the momentum flux of the raindrops, rill and interrill area are considered, but additionally also the partitions of direct throughfall and leaf drip as described in Sec- tion 4.5.4.1. Momentum flux for each partition is calculated separately after the following scheme:
1. Momentum flux depending on drop size and velocity is calculated.
2. Only in case of rill areas: raindrop diameter is compared to flow depth, and if necessary, momentum flux is reduced according to Eq. (4.25).
3. Rill and interrill areas are considered by multiplying the corresponding calculated mo-
mentum fluxes with their fractional area (given byfcf).
The reduction of the momentum flux of the raindrops, depending on flow depth in the rill areas is expressed by (Wicks & Bathurst, 1996):
ff d=e
1− h
Dd (4.25)
whereff d is the water depth correction factor,his the flow depth [m] andDd is the median
drop diameter [m]. The median drop diameter is either set to the leaf drip diameter, if drip
off occurs (see Section 4.5.4.1), or approximated for direct throughfall after Laws & Parsons (1943) with:
Dd= 0.00124·I0.182 (4.26)
whereI is the rainfall intensity [mm/h].