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(Jm2)KE (J m^{-2}) is used to determine the detachment of soil particles by raindrop impact and is defined as:

Equation 53

KE=KELD+KEDTKE=KE_{LD}+KE_{DT}

Where KELDKE_{LD} is the kinetic energy of the leaf drainage (Jm2)(Jm^{-2}) and KEDTKE_{DT} is the kinetic energy of the direct throughfall (Jm2)(Jm^{-2}).

The kinetic energy of the leaf drainage KELDKE_{LD} is based on Brandt (1990):

Equation 54

KELD={0for PH<0.15LD(15.8PH5.87)for PH0.15KE_{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

KEDT=DT(8.95+8.44log10I)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 h1)(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 mm h110 \space mm \space h^{-1} for temperate climates, 25 mm h125 \space mm \space h^{-1} for tropical climates and 30 mm h130 \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 Pet(mm)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

Pet=Pet cos SPe_t=Pe_t\space cos \space S

Where PetPe_t is the effective precipitation (mm) and S is the slope (°).

Leaf drainage is determined as:

Equation 57

LD=Pet CCLD=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=PetLDDT=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 m2)(F, kg \space m^{-2}) and the detachment of soil particles by runoff (H,kg m2)(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

Fi=Ki %i100(1GC)KE103F_i=K_i \space \frac{\% i}{100}(1-GC)KE\cdot 10^{-3}

With K the detachability of the soil by raindrop impact (gJ1)(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 Kc=0.1K_c = 0.1, Kz=0.5K_z = 0.5, and Ks=0.3 gJ1K_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 m2)(H, kg \space m^{-2}) is calculated as:

Equation 60

Hi=DRi%i100Qrout,t1.5(1GC) sin0.3S103H_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 mm1)(g \space mm^{-1}) and Qrout,tQ_{rout,t} is the routed streamflow on day tt (mm). The detachability of the soil DR for each texture class is included as a model parameter for which Quansah (1982) proposed DRc=1.0,DRz=1.6DR_c =1.0, DR_z=1.6, and DRs=1.5g mm1DR_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=44.1Nfi0.29DEP=44.1N_{fi}^{0.29}

Where NfN_f is the particle fall number (-), defined as:

Equation 62

Nfi=lυsiυdN_{fi}=\frac{l \upsilon_{si}}{\upsilon d}

Where ll is the length of a grid cell (m), υs\upsilon_s the particle fall velocity (ms1)(m s^{-1}), υ\upsilon the flow velocity(ms1)(m s^{-1}), and d the depth of flow (m). Particle fall velocities υs\upsilon_s are estimated from:

Equation 63

υs=118δ2(ρs+ρ)gη\upsilon_s=\frac{\frac{1}{18}\delta^2(\rho_s+\rho)g}{\eta}

Where δ\delta is the diameter of the particle (m), ρs\rho_s the sediment density(=2650 kg m3)(=2650\space kg \space m^{-3}), ρ\rho the flow density (typically 1100 kg m31100 \space kg \space m^{-3} for runoff on hillslopes; Abrahams et al., 2001), g gravitational acceleration (taken as 9.81 m s29.81 \space m \space s^{-2}) and η\eta the fluid viscosity (nominally 0.001 kg m1 s10.001 \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 2106 m s12 \cdot 10^{-6}\space m \space s^{-1} for clay, 2103 m s12 \cdot 10^{-3}\space m \space s^{-1} for silt and 0.02 m s10.02 \space m \space s^{-1}for sand.

The flow velocity υ\upsilon from Equation 62 is obtained by the Manning formula:

Equation 64

υ=1nd23S12\upsilon = \frac{1}{n'}d^{\frac{2}{3}}S^{\frac{1}{2}}

Where n' is the modified Manning's roughness coefficient (s m1/3s \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=nsoil2+nvegetation2n'=\sqrt{n^2_{soil}+n^2_{vegetation}}

The Manning's roughness coefficient for bare soil nsoiln_{soil} is set to 0.015 s m1/30.015 \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

nsoil=exp(2.1132+0.0349RFR)n_{soil}=exp(-2.1132+0.0349RFR)

Where RFR is the surface roughness parameter (cm m1)(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) nvegetationn_{vegetation} is obtained from the following equation (Jin et al., 2000):

Equation 67

nvegetation=d232gDNVn_{vegetation}=\frac{d^{\frac{2}{3}}}{\sqrt{\frac{2g}{D NV}}}

Where D is the stem diameter (m) and NV the stem density (stems m2)(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 nvegetationn_{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, nvegetationn_{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.

Figure 4. Surface and vegetation roughness options: (a) bare soil, (b) tilled soil (Equation 66), (c) regular vegetation (Eq. 67), and (d) irregular vegetation.

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

Gi=(Fi+Hi)(1DEPi100)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.

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