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# Soil erosion processes
The soil erosion model is based on the Modified MMF model (Morgan and Duzant, 2008), which determines total soil erosion from detachment by raindrop impact and detachment by runoff and considers cell deposition. The next paragraphs provide a detailed description of all these processes.
#### Estimation of rainfall energy
The total kinetic energy of the effective precipitation $$KE (J m^{-2})$$ is used to determine the detachment of soil particles by raindrop impact and is defined as:
*Equation 53*
$$
KE=KE\_{LD}+KE\_{DT}
$$
Where $$KE\_{LD}$$ is the kinetic energy of the leaf drainage $$(Jm^{-2})$$ and $$KE\_{DT}$$ is the kinetic energy of the direct throughfall $$(Jm^{-2})$$.
The kinetic energy of the leaf drainage $$KE\_{LD}$$ is based on Brandt (1990):
*Equation 54*
$$
KE\_{LD}=\begin{cases}
0 &\text{for } PH < 0.15 \\
LD(15.8\sqrt{PH}-5.87) &\text{for } PH \ge 0.15
\end{cases}
$$
Where *LD* is the leaf drainage (mm) and *PH* is the plant height (m), specified for each landuse class.
The kinetic energy of the direct throughfall is based on a relationship described by Marshall & Palmer (1948), which is representative of a wide range of environments (Morgan, 2005):
*Equation 55*
$$
KE\_{DT}=DT(8.95+8.44 log\_{10}I)
$$
Where *DT* is the direct throughfall (mm) and *I* is the intensity of the erosive precipitation $$(mm \space h^{-1})$$. The intensity of the erosive precipitation is a model parameter and varies according to geographical location. Morgan & Duzant (2008) proposes $$10 \space mm \space h^{-1}$$ for temperate climates, $$25 \space mm \space h^{-1}$$ for tropical climates and $$30 \space mm \space h^{-1}$$ for strongly seasonal climates (e.g. Mediterranean, tropical monsoon).
The leaf drainage *LD*, i.e. precipitation that reaches the soil surface as flow or drips from the leaves and stems of the vegetation, and direct throughfall *DT* i.e. precipitation that reaches the soil surface directly through gaps in the vegetation cover, from Equation 54 and 55, are obtained from the effective precipitation $$Pe\_t (mm)$$, also known as throughfall. The effective precipitation from the hydrological model is first corrected for the slope angle, following Choi et al. (2017):
*Equation 56*
$$
Pe\_t=Pe\_t\space cos \space S
$$
Where $$Pe\_t$$ is the effective precipitation (mm) and *S* is the slope (°).
Leaf drainage is determined as:
*Equation 57*
$$
LD=Pe\_t\space CC
$$
Where *CC* is the canopy cover (proportion between zero and unity). The canopy cover is either introduced by a landuse-class specific tabular value or determined by the vegetation module. When the vegetation module is used, the canopy cover is obtained from the *LAI* (Equation 6), maximized by 1.
Direct throughfall becomes:
*Equation 58*
$$
DT=Pe\_t-LD
$$
#### Detachment of soil particles
Detachment of soil particles is determined separately for raindrop impact and accumulated runoff and is subsequently summed. The detachment of soil particles by raindrop impact $$(F, kg \space m^{-2})$$ and the detachment of soil particles by runoff $$(H, kg \space m^{-2})$$ are determined for each of the soil texture classes separately and subsequently summed. The detachment of soil particles by raindrop impact is calculated as:
*Equation 59*
$$
F\_i=K\_i \space \frac{% i}{100}(1-GC)KE\cdot 10^{-3}
$$
With *K* the detachability of the soil by raindrop impact $$(g J^{-1})$$, *i* the textural class, with *c* for clay, *z* for silt and *s* for sand, and *GC* the ground cover (-). The detachability of the soil for each texture class is included as a model parameter, for which Quansah (1982) proposed $$K\_c = 0.1$$, $$K\_z = 0.5$$, and $$K\_s = 0.3 \space g J^{-1}$$. The ground cover *GC*, expressed as a proportion between zero and unity, protects the soil from detachment and is determined by the proportion of vegetation and rocks covering the surface. The ground cover is set to 1 in case the surface is covered with snow, which is determined from the snow module.
The detachment of soil particles by runoff $$(H, kg \space m^{-2})$$ is calculated as:
*Equation 60*
$$
H\_i=DR\_i \frac{%i}{100}Q\_{rout,t}^{1.5}(1-GC)\space sin^{0.3}S \cdot10^{-3}
$$
Where *DR* the detachability of the soil by runoff$$(g \space mm^{-1})$$ and $$Q\_{rout,t}$$ is the routed streamflow on day $$t$$ (mm). The detachability of the soil *DR* for each texture class is included as a model parameter for which Quansah (1982) proposed $$DR\_c =1.0, DR\_z=1.6$$, and $$DR\_s = 1.5 g \space mm^{-1}$$.
#### Immediate deposition of detached particles
A proportion of the detached soil is deposited in the cell of its origin as a function of the abundance of vegetation and the surface roughness. The percentage of the detached sediment that is deposited (*DEP*) is estimated from the relationship obtained by Tollner et al. (1976) and calculated separately for each texture class:
*Equation 61*
$$
DEP=44.1N\_{fi}^{0.29}
$$
Where $$N\_f$$ is the particle fall number (-), defined as:
*Equation 62*
$$
N\_{fi}=\frac{l \upsilon\_{si}}{\upsilon d}
$$
Where $$l$$ is the length of a grid cell (m), $$\upsilon\_s$$ the particle fall velocity $$(m s^{-1})$$, $$\upsilon$$ the flow velocity$$(m s^{-1})$$, and *d* the depth of flow (m). Particle fall velocities $$\upsilon\_s$$ are estimated from:
*Equation 63*
$$
\upsilon\_s=\frac{\frac{1}{18}\delta^2(\rho\_s+\rho)g}{\eta}
$$
Where $$\delta$$ is the diameter of the particle (m), $$\rho\_s$$ the sediment density$$(=2650\space kg \space m^{-3})$$, $$\rho$$ the flow density (typically $$1100 \space kg \space m^{-3}$$ for runoff on hillslopes; Abrahams et al., 2001), *g* gravitational acceleration (taken as $$9.81 \space m \space s^{-2}$$) and $$\eta$$ the fluid viscosity (nominally $$0.001 \space kg \space m^{-1} \space s^{-1}$$, but taken as 0.0015 to allow for the effects of the sediment in the flow; Morgan & Duzant, 2008). When Equation 63 is applied to the three texture sizes of 2 μm for clay, 60 μm for silt and 200 μm for sand, this gives respective values of $$2 \cdot 10^{-6}\space m \space s^{-1}$$ for clay, $$2 \cdot 10^{-3}\space m \space s^{-1}$$ for silt and $$0.02 \space m \space s^{-1}$$for sand.
The flow velocity $$\upsilon$$ from Equation 62 is obtained by the Manning formula:
*Equation 64*
$$
\upsilon = \frac{1}{n'}d^{\frac{2}{3}}S^{\frac{1}{2}}
$$
Where *n'* is the modified Manning's roughness coefficient ($$s \space m^{-1/3}$$), which is a combination of the Manning's roughness coefficient for the soil surface and vegetation, defined as (Petryk and Bosmajian, 1975):
*Equation 65*
$$
n'=\sqrt{n^2\_{soil}+n^2\_{vegetation}}
$$
The Manning's roughness coefficient for bare soil $$n\_{soil}$$ is set to $$0.015 \space s \space m^{-1/3}$$, as suggested by Morgan and Duzant (2008) (Figure 4a). For tilled conditions (Figure 4b) the following equation is applied:
*Equation 66*
$$
n\_{soil}=exp(-2.1132+0.0349RFR)
$$
Where *RFR* is the surface roughness parameter $$(cm \space m^{-1})$$. Values for *RFR* are tillage implementation specific and can be obtained from Morgan & Duzant (2008) Table IV.
The Manning's roughness coefficient for regular spaced vegetation (Figure 4c) $$n\_{vegetation}$$ is obtained from the following equation (Jin et al., 2000):
*Equation 67*
$$
n\_{vegetation}=\frac{d^{\frac{2}{3}}}{\sqrt{\frac{2g}{D NV}}}
$$
Where *D* is the stem diameter (m) and *NV* the stem density $$(stems \space m^{-2})$$. Stem diameter and stem density may be difficult to obtain for certain landuse classes with irregular spaced vegetation (e.g. forest, shrubland), therefore, users may opt to use tabular values for $$n\_{vegetation}$$, e.g. from Chow (1959) (Figure 4d). Preliminary model runs showed that Equation 67 results in unrealistically high flow velocity values for landuse classes where the stem density is very low, such as in orchards. Therefore, in these conditions where the influence of vegetation on flow velocity is negligible, $$n\_{vegetation}$$ can be set to 0.
Equations 62, 64 and 67 require a flow depth *d*, a model parameter that can be used in the model calibration. The value *d* for should be taken such that it corresponds to a water depth from runoff generated within the cell margins, i.e. without accumulation of flow from upstream located cells.
#### Sediment taken into transport
The amount of sediment that is routed to downstream cells is determined from the sum of the detached sediment from raindrop impact (Equation 59) and accumulated runoff (Equation 60), subtracting the proportion of the sediment that is deposited within the cell of its origin (Equation 61):
*Equation 68*
$$
G\_i=(F\_i+H\_i)\Big(1-\frac{DEP\_i}{100}\Big)
$$
The amount of sediment that is routed to downstream cells is the summation of the individual amounts for clay, silt and sand.