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Earth
, Volume 3 (1) – Feb 9, 2022

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Article The Theoretical Approach to the Modelling of Gully Erosion in Cohesive Soil Aleksey Sidorchuk Geographical Faculty, Lomonosov Moscow State University, GSP-1, Leninskie Gory, 119991 Moscow, Russia; ﬂuvial05@gmail.com Abstract: The stochastic gully erosion model (STOGEM) is based on a combination of deterministic mechanics and a stochastic description of the erosion control factors. The main proposition in the model is that the depth of the active surface layer of eroded cohesive soil is equal to one particle diameter, and the deposition of eroded particles is negligible. The erosion rate at the gully bed is calculated directly from the equation of the balance between driving and resistance forces acting on soil particles in ﬂowing water using the probability density functions of stochastic variables: ﬂow velocity, soil aggregate size and cohesion. Probability density functions of cohesion in the model vary through time and space during the erosion event due to the changes in soil composition—armoring and loosening. This theory is still far from achieving practical application, but opens up a new way for better understanding the experimental results of gully erosion and shows the direction for future investigations. Keywords: stochastic model; balance of driving and resistance forces; functions of stochastic variables; probability density functions 1. Introduction Citation: Sidorchuk, A. The Gullying is a form of linear erosion of loose and cohesive soils (and of rocks) by Theoretical Approach to the a concentrated water ﬂow [1]. It is possible to classify the forms of linear erosion by Modelling of Gully Erosion in Cohesive Soil. Earth 2022, 3, 228–244. their size and stage of development. The sequence of such linear forms begins with the https://doi.org/10.3390/ smallest ones—the rills on the slopes. With increasing depth, the larger linear forms are earth3010015 ephemeral gullies [2], which are usually destroyed by plowing, followed by typical gullies. This quantitative difference in the depth causes the main qualitative difference between Academic Editor: Ioannis Gitas active typical gullies and smaller linear erosion forms—the instability of gully walls. The Received: 28 December 2021 processes of gully bed incision and bank erosion cause the increase in gully wall inclination Accepted: 1 February 2022 and soil slumping and falling of different types. These processes are typical for the ﬁrst Published: 9 February 2022 stage of gully development. Gully formation is very intensive and its geometry (length, depth, width, area, volume) is far from stable and changes rapidly [3]. The ﬁrst stage lasts Publisher’s Note: MDPI stays neutral from 4 to 10 years in loose sands and frozen loams, and up to 100–150 years in typical with regard to jurisdictional claims in agricultural landscapes [4]. The gullies cut slopes to their whole potential (possible) length published maps and institutional afﬁl- iations. during this period, and their walls eventually stabilize. After gully wall stabilization, usually due to vegetation, gully development moves into the second stage [3]. The gully width increases slowly due to slow soil movement (creep). Slumping can still occur at the gully head, and so gully length can also slowly increase Copyright: © 2022 by the author. through gravitation processes and suffusion, not by ﬂuvial erosion [5]. At this stage, the Licensee MDPI, Basel, Switzerland. gully is transformed into a dry valley. Often sediments washed from the catchment are This article is an open access article deposited at the dry valley’s bottom. The walls of a dry valley become more and more distributed under the terms and gentle, its depth decreases, and the dry valley is transformed to a shallow linear hollow conditions of the Creative Commons on the slope. The formation of hollows due to sediment deposition in dry valleys takes a Attribution (CC BY) license (https:// signiﬁcant amount of time and is discovered in paleo-geographical investigations [6]. creativecommons.org/licenses/by/ 4.0/). Earth 2022, 3, 228–244. https://doi.org/10.3390/earth3010015 https://www.mdpi.com/journal/earth Earth 2022, 3 229 Gullies can be also classiﬁed by their position: gullies on the hillslopes, bank gullies on the valley slopes, and bottom gullies at the valley bed. Gullies can be discontinuous and continuous, differing by soils and type of vegetation on gully slopes [7]. Mathematical models of different levels of simpliﬁcation [7–10] describe all of the abovementioned forms, stages and processes. All of these models represent signiﬁcant em- pirical components in process description. The goal of the current paper is to formulate the general theoretical principles of only one, but the main, process of gully development in co- hesive soil—erosion by water. The main approach is a stochastic description of the synergy of the main destructive and stabilizing forces, which causes soil particle detachment. 2. Materials and Methods 2.1. Gully Erosion by Water The rate of gully incision is controlled by water ﬂow velocity, depth, turbulence, temperature, as well as soil texture, soil mechanical pattern, and the level of protection by vegetation. These characteristics are described by equations of mass conservation: ¶Q = C q + EW + E d CU W (1) w w b f ¶X 3 1 3 1 where Q = Q C is the sediment transport rate (m s ), Q is the water discharge (m s ); X is the longitudinal co-ordinate (m); C is the mean volumetric sediment concentration; C is 2 1 the sediment concentration of the lateral input; q is the speciﬁc lateral discharge (m s ); E is the erosion rate or the mean particle detachment rate (m s ); E —bank erosion rate (m s ); W is the ﬂow width (m); d is the bank height (m); and U is the sediment particles’ falling velocity in the turbulent ﬂow (m s ). During the ﬁrst stage of gully development in cohesive soil, the accumulation on the gully bed of soil particles, eroded inside the gully or from the lateral input, is usually negligible. Flow velocity of erosion initiation in cohesive soils is higher than the critical ﬂow velocity when the settling of detached particles occurs. The rate of bank erosion E is related to the rate of erosion E of the gully bed [10]. Therefore, the main term in Equation (1) is the erosion rate E. The soil erosion rate E is the mean rate of the lowering of the soil surface Z or the increase in volume DV eroded from the area S during the time interval T. DZ DV E = = (2) T ST The erosion volume DV is the sum of the volumes of individual eroded soil particles (aggregates) V with correction on soil porosity ": ai DV = V (3) ai 1 # i=1 After multiplying and dividing each component of the sum (3) by the product of the soil particle bottom area s and the duration of particle detachment t , Formula (2) takes i i the form: 1 V s t ai i i E = (4) 1 # s t ST i i i=1 where the term V /s t represents the instant and local rate of particle detachment , and the ai i i term s t /ST represents the probability density of this particle detachment. Therefore [11]: i i E = a p da (5) 1 # where p is the spatial-temporal probability density function (PDF) for . Equations (4) and (5) show that the mean soil erosion rate is the result of spatial (on the area S) and temporal Earth 2022, 3 230 (during the period T) averaging of instant and local detachment rates . The detachment rate possess all positive values, including zero (“detachment” of stable particles). 2.2. Instant and Local Rate of Soil Particle Detachment Erosion of cohesive soil is a rather slow process, even in gullies, as water mainly only comes into contact with, and thus only affects, the surface layer. The main proposition in the further consideration in this paper is that the depth of the active surface layer equals one particle diameter. The instant and local detachment is a discrete process: the surface becomes lower when and where a soil fragment is detached. A soil fragment becomes unstable, and its movement begins when and where the driving forces F exceed the dr maximum of the resisting forces F and the resultant force Q is more than zero [12]: res Q = (jF jjF j) > 0 (6) dr res The acceleration of such an unstable fragment within the soil surface layer in the direction of the resultant force Q (along the axis Z) is described by Newton’s second law: d Z V r = Q (7) a s dt where r is soil density. A soil fragment becomes detached when all links between particles are broken, i.e., when the soil fragment is removed from its initial position by a distance greater than its height D. The duration of detachment equals. 2r V D s a t = (8) The local and instant (average for the period t) detachment rate is D DQ a = = (9) t 2r V s a Equations (8) and (9) are valid for the simplest case when Q is independent of Z. Equations (8) and (9) describe the behavior of a single soil particle. It is evident that is impossible in general to predict the exact force balance or to foresee the geometry and mass of a soil particle or aggregate at a given point at a given time. The main variables in these equations are of a stochastic nature. Therefore, it is possible to describe these variables with probability density functions and use these functions to calculate the mean rate of detachment (erosion rate) with Equations (5) and (9) [13]. 2.3. Probability Density Function for the Rate of Detachment The detachment rate calculated with Equation (9) is the function of multiple factors: soil particle geometry and density, as well as driving and resistance forces. Each of these factors (often compound) is a stochastic variable characterized by a probability density function (PDF). To obtain the PDF for the function of random variables (in our case, for the particle detachment rate), it is necessary to use the appropriate calculation techniques. The main technique is the formula for calculating the mean E(Y) of a function of random variables Y = g(x , x , . . . x ) [14]: 1 2 Z Z E(Y) = E[g(x , . . . x )] = . . . g(x , . . . x ) p(x , . . . x ,)dx , . . . dx (10) 1 n 1 n 1 n 1 n For independent variables, the multivariable PDF p(x , . . . x ) is equal to the product 1 n of the PDFs of each of the random variables. Earth 2022, 3 231 2.4. The Main Resistance and Driving Forces The process of a single particle’s detachment from the surface of cohesive soil caused by water was described in detail by Mirtskhulava [15]. We shall follow this work with additions. The main resistance forces are: submerged weight of soil fragments, with slope inclination g taken into account, projected in the direction of driving forces b: F = gV (r r) cos g sin b (11) w a s and the geo-mechanical force of cohesion: F = CS (12) C b The latter is a combination of electro-chemical, capillary and friction forces, which is parameterized by the product of cohesion C (in terms of Coulomb law) and contact area between and within soil aggregates S . Hydrodynamic driving forces are the gradients of static and dynamic pressure, acting on the soil surface, depending on geometry of the soil surface, the shape and size of soil aggregates, the ﬂow depth and the velocity distribution. The ﬁeld of these gradients can be calculated with 3D hydrodynamic models. More often [15], these pressure gradients are calculated with the use of ﬂow velocity and aggregates geometry as drag (F ) and lift (F ) d l forces, which are parameterizations of longitudinal and vertical pressure gradients: F = C rS (13) d R d F = C rS (14) y a where C is the coefﬁcient of drag resistance; C is the coefﬁcient of uplift; U is the instant R y near-bed ﬂow velocity; r and r are the soil aggregate density and water density, respec- tively; S is the area of aggregate cross-section exposed perpendicular to ﬂow; and S is d a the cross-section area of the soil aggregate parallel to the ﬂow (vertical projection). The direction of the sum of drag and lift forces (the angle with the mean soil surface) is: C S y a b = arctan (15) C S R d With this parametrization, the local detachment rate is [13] (with modiﬁcation): q h i DQ D U a = = C rS + C rS g(r r)V cos g sin b CS y a R d s a b 2r V 2r V 2 s a s a (16) a = k U k D k C U D C rD g(r r) s DS where k is S C + S C , k is cos g sin b and k = . If the expression U a y d R D C 4r V 2r 2r V s a s s a under the square root is equal to zero or negative, then = 0. For this case, Equation (16) is a generalized form (with cohesion) of a well-known expression for incipient motion criteria (see, for example, [16]). The rate of detachment is the stochastic variable, which is the function of six other independent stochastic variables: ﬂow velocity, particle (aggregate) size, soil cohesion and three geometric-kinematical coefﬁcients k , k and k . In the case of turbulent ﬂow, C , C U D C R and k are constants [15]. If we assume a simple shape and composition of soil particles (aggregates), then the coefﬁcient k is also constant. Then, Equation (16) contains three independent stochastic variables: ﬂow velocity U, soil particle size D and force of cohesion k C, and Equation (10) takes the form: Z Z Z q Cmax Umax p p Dmax E = p(k C)dk C p k U d k U p(k D)dk D k U k D k C (17) C C U U D D U D C 1 # Cmin Umin Dmin Earth 2022, 3 232 where p( k U), p(k D) and p(k C) are probability density functions for ﬂow velocity, soil U D C particles vertical size and cohesion, respectively. In numerical calculations, the continuous quantities are replaced by discrete one, inﬁnitesimal by ﬁnite, and integrals by sums: R nX Xmax XdX ! DX å X . iX Xmin iX=1 2.5. Probability Density Function for the Factors of Soil Erosion When geometric–kinematical coefﬁcients and PDFs for ﬂow velocity, soil aggregate size and cohesion are known from the theory or experiment, the PDF for instant and local erosion rate a and, therefore, the mean rate of erosion can be calculated with Equation (17). Various combinations of the input mean values of the factors, as well as different PDFs, lead to a great variability of the resulting relationships between erosion and controlling factors. Measurements in different environments show [7,10,17] that the ﬂows in the gullies are usually shallow, turbulent and often supercritical. A large number of measurements of turbulent characteristics using different techniques in such shallow ﬂows [18,19] showed Gaussian PDF of ﬂuctuations of instant longitudinal velocity U: " # 1 1 U U mean p(U) = exp (18) 2 s s 2p where U is the mean velocity and s is the standard deviation of velocity ﬂuctuations. mean For small mean velocities, the function p(U) can be asymmetrical. The ratio of the standard deviation of velocities and kinematic velocity U increases from the ﬂow surface to the bottom [19]; s z = 2.3 exp (19) U H where H is the ﬂow depth and z equals the roughness height. According to Mirtskhulava [15], in shallow turbulent ﬂow C = 0.1 and C = 0.42; then k = (0.1S + 0.42S ). y a R U d The PDF of the aggregate diameters can be expressed as a lognormal distribution [20,21]: " # 1 (ln D D ) p = p exp (20) Ds 2p 2s The parameters in Equation (20), calculated with arithmetical values, namely the mean of soil particle height D and standard deviation s , are 0 1 @ A D = ln q (21) D + s " # s = ln 1 + (22) The direction of sum of drag and lift forces is b = arctan 0.238 , therefore h i g(r r) s S k = cos sin arctan 0.238 . 2r S s d The details of the inﬂuence of soil cohesion on erosion rate are not properly investi- gated. Mean values, measured with existing techniques, such as tore-vane and penetrome- ters of different kinds, are not well correlated with aggregate stability tests and measured rates of soil erosion (see the results of experiments in [22]). Even less is known about PDFs of aggregates cohesion. Mirtskhulava [15] determined a normal distribution function for the variability of soil cohesion, measured by a round stamp. The same PDF ﬁt to the textural tensile strength of one hundred 2–3 mm rounded aggregates [23], but the initial Earth 2022, 3, FOR PEER REVIEW 6 Earth 2022, 3 233 of aggregates cohesion. Mirtskhulava [15] determined a normal distribution function for the variability of soil cohesion, measured by a round stamp. The same PDF fit to the tex- tural tensile strength of one hundred 2–3 mm rounded aggregates [23], but the initial data, published there, show a distinct asymmetrical distribution. Our experiments, with the data, published there, show a distinct asymmetrical distribution. Our experiments, with the cutting of the surface of soil sample with a blade [24], showed the lognormal distribution cutting of the surface of soil sample with a blade [24], showed the lognormal distribution of soil strength (Equation 20). We shall use lognormal distribution in further calculations, of soil strength (Equation (20)). We shall use lognormal distribution in further calculations, keeping in mind that this distribution changes over time during an erosion event. keeping in mind that this distribution changes over time during an erosion event. 2.6. The Algorithm of Erosion Rate Calculation 2.6. The Algorithm of Erosion Rate Calculation Two opposite processes transform the initial spatial distribution of soil cohesion: Two opposite processes transform the initial spatial distribution of soil cohesion: soil soil armoring and soil loosening (Figure 1). The armoring due to the removal of unsta- armoring and soil loosening (Figure 1). The armoring due to the removal of unstable soil ble soil particles and aggregates leads to the following transformation of cohesion PDF particles and aggregates leads to the following transformation of cohesion PDF through through time: time: Figure 1. Algorithm of the calculation of the gully erosion rate with the stochastic model. The ex- Figure 1. Algorithm of the calculation of the gully erosion rate with the stochastic model. The planation is given in the text. explanation is given in the text. 1. The probability density p(k C) in the part of the cohesion PDF where resistance forces are less than driving forces decreases due to the erosion E of soil with particular iC cohesion C iC Earth 2022, 3 234 Z Z q Umax p p Dmax E = p k U d k U p(k D)dk D k U k D k C (23) iC U U D D U D C 1 # Umin Dmin The initial PDF transforms into intermediate PDF (p*) iC p [(k C) , t ]dk C = p[(k C) , t ]dk C 1 (t t ) (24) C i+1 C C i+1 C i+1 i iC iC 2. Simultaneously, the intermediate PDF of cohesion is transformed due to the exposition of fresh initial soil in the “windows” of the eroded surface layer to PDF of armored soil (p ) Z Z C C max max p [(k C) , t ] = p [(k C) , t ] + p (k C) p (k C)dk C p [(k C) , t ]dk C (25) a C i+1 C i+1 0 C 0 C C C i+1 C iC iC iC C C min min where the index ‘iC’ indicates a particular soil cohesion and related erosion rate, the index ‘í ’ indicates a sequence in time, and p (k C) is the PDF of cohesion in the initial soil. 0 C This process leads to the increase in the proportion of the surface covered by stable soil aggregates, and an increase in the mean soil cohesion. The rate of erosion decreases through time when armoring is the prevailing process. Soil loosening takes place due to the destabilization of stable aggregates when links be- tween and/or within soil aggregates are weakening. This is complicated process, including decrease of electro-chemical, capillary and friction forces in saturated soil, the removal by ﬂow of unstable soil particles and aggregates, which previously stabilized other aggregates, the widening of pores and cracks due to changes in water pressure in soil. Destabilization also occurs due to the vibration of stable aggregates caused by turbulent ﬂow [15]. The failure of links decreases the contact surface area S between and within soil aggregates over time. The rate of failure of links in soils can be described with the common exponential failure function [25]. dS = lS (26) dt The parameter > 0 must be estimated from experiments. The remaining contact area at time T is the product of the contact area at time T i+ i and the failure function (Equation (26)) k (T ) = k (T )[1 ldT] (27) C i+1 C i Loosening leads to an increase in the proportion of the surface covered by unstable soil aggregates, and leads to a decrease in the mean soil cohesion. The rate of erosion increases through time when loosening occurs at a higher rate than soil armoring. Soil loosening is the main cause of erosion at the ﬁnal stage of the erosion event, when the soil surface is nearly completely stabilized by armoring. The ﬁnal PDF of cohesion after the armoring and loosening cycle, as the PDF of function (27) [14], is equal to: p[(k C) , T ] = p [(k C) , T ] (28) C i+1 a C i iC iC 1 ldT The rate of gully erosion at a given moment is calculated from the equation of the balance between the driving and resistance forces acting on a soil particle in ﬂow- ing water (Equation (17)) after soil armoring and after soil loosening (Figure 1). The armoring–loosening cycle exists only in the numerical representation of the process of erosion. In reality, soil armoring and loosening is a uniﬁed and simultaneous process. Earth 2022, 3, FOR PEER REVIEW 8 Earth 2022, 3 235 2.7. The Materials for Comparison of Calculations with Measurements 2.7. The Materials for Comparison of Calculations with Measurements The calculations with any model must be compared with the measurements for pos- The calculations with any model must be compared with the measurements for pos- sible validation and calibration. Field experiments were performed in the low-altitude sible validation and calibration. Field experiments were performed in the low-altitude hills of Ballantrae Hill Country Research Station near Woodville, New Zealand (Figure 2). hills of Ballantrae Hill Country Research Station near Woodville, New Zealand (Figure 2). The local soil belongs to Pallic Soils of the Wainui series formed on loess [26]. The entire The local soil belongs to Pallic Soils of the Wainui series formed on loess [26]. The entire depth of the A + AB-horizon is 40–50 cm. The organic carbon content in the topsoil is 2.4%, depth of the A + AB-horizon is 40–50 cm. The organic carbon content in the topsoil is 2.4%, and 0.4% in the parent loess. The topsoil is well-structured and highly water-stable: wet and 0.4% in the parent loess. The topsoil is well-structured and highly water-stable: wet sieving of 2–4 mm aggregates for 30 min showed that 88% was retained for this class. The sieving of 2–4 mm aggregates for 30 min showed that 88% was retained for this class. The aggregate stability in the parent loess is much lower—the proportion of aggregates re- aggregate stability in the parent loess is much lower—the proportion of aggregates retained tained on the sieve was 7.6%. The mean size of soil aggregates obtained by dry sieving 4S on the sieve was 7.6%. The mean size of soil aggregates obtained by dry sieving D Ds≈ was 1.83 mm, with a standard deviation σD = 0.88 mm. The aggregates were flat- was 1.83 mm, with a standard deviation s = 0.88 mm. The aggregates were ﬂattened, tened, to a plate or ellipsoidal shape, with ≈1/3. The mean soil strength, measured with to a plate or ellipsoidal shape, with 1/3. The mean soil strength, measured with a tor-vane, was nearly equal for parent loess and topsoil: 52 and 51 kPa, respectively. The a tor-vane, was nearly equal for parent loess and topsoil: 52 and 51 kPa, respectively. The soil density r was 1460 and 1230 kg/m , respectively. soil density ρs was 1460 and 1230 kg/m , respectively. Figure 2. Experimental site at the Ballantrae Hill Country Research Station near Woodville, New Figure 2. Experimental site at the Ballantrae Hill Country Research Station near Woodville, Zealand. New Zealand. Two erosion plots were organized on two steep slopes (at straight parts), which ac- Two erosion plots were organized on two steep slopes (at straight parts), which curately represent the gully beds. The abandoned road on the parent loess was used to accurately represent the gully beds. The abandoned road on the parent loess was used to organize plot 1, at 17.4 m long and 0.55 m wide with an inclination of 0.235 (13.2°). Plot 2 organize plot 1, at 17.4 m long and 0.55 m wide with an inclination of 0.235 (13.2 ). Plot 2 was cut into the topsoil by the removal of the upper 5–7 cm of soil with grass cover to was cut into the topsoil by the removal of the upper 5–7 cm of soil with grass cover to avoid avoid grass roots’ influence on the erosion rate. The residual content of thin roots in the grass roots’ inﬂuence on the erosion rate. The residual content of thin roots in the soil was soil was less than 0.1% by weight (the initial content in the upper 5–7 cm was about 4%). less than 0.1% by weight (the initial content in the upper 5–7 cm was about 4%). Plot 2 was Plot 2 was 11.5 m long, 0.32 cm wide and had an inclination of 0.483 (25.8°). 11.5 m long, 0.32 cm wide and had an inclination of 0.483 (25.8 ). Experiments were conducted during November 2001. Five runs, each lasting half an Experiments were conducted during November 2001. Five runs, each lasting half hour, were performed at each plot with discharges of 1.48–11.2 L/s controlled with a V- an hour, were performed at each plot with discharges of 1.48–11.2 L/s controlled with a notch weir (Table 1). Flow width was measured at 8–10 cross-sections and the mean flow V-notch weir (Table 1). Flow width was measured at 8–10 cross-sections and the mean ﬂow velocity velocity wa was s measur measured ed using using sa salt lt i injections. njections. The me The mean an flow ﬂow depth depth was c was calculated alculated from from discharge, flow width and velocity. All flows were turbulent and supercritical. Samples discharge, ﬂow width and velocity. All ﬂows were turbulent and supercritical. Samples of water of water w withith sediment sedimen particles, t particles with , with a a total totvolume al volum of e of ab about ou 4t L, 4 L, wer were t e take an ken at the at the end end of of the plo the plot and t an at d at the the end end of the of the h half-an-hour alf-an-ho run. ur rThe un. Th sediment e sedimen concentration t concentration was estimated was esti- by means of the standard procedure of ﬁltering, drying, weighting and dividing based on mated by means of the standard procedure of filtering, drying, weighting and dividing water volume. based on water volume. Earth 2022, 3, FOR PEER REVIEW 9 Earth 2022, 3 236 The coefficient kU for ellipsoidal aggregates is 0.1 + 0.42 and varied in the range 0.026–0.04 for the first plot and 0.031–0.047 for the second (depending on Sa/Sd ratio Table 1. The main characteristics of the experiments in Ballantrae Hill Country. in the range 2–4). The standard deviation of velocity fluctuations was 𝜎 ≅2.1𝑈 (accord- ing to [19]). Q, l/s U, m/s W, m d, m U s E, m/s Re Fr The coefficient kD = cosγ sin arctan 0.238 was in the range 0.44–0.64 for * U Plot–Soil–Run the first plot and 0.24–0.35 for the second. 1–b2–3 3.13 1.19 0.46 0.0057 0.12 0.26 6020 5.0 2.29 10 1–b2–4 4.01 1.30 0.46 0.0067 0.13 0.28 2.33 10 7670 5.1 −4 The coefficient kC = 𝐷 = ≈ was 2.04 × 10 for the first plot and 1.54 1–b2–5 5.12 1.40 0.47 0.0077 0.14 0.30 2.29 10 9510 5.1 −4 × 10 for the second. The variation coefficient for cohesion was esti 7 mated as 0.59 in the 1–b2–6 6.86 1.53 0.49 0.0092 0.15 0.33 5.49 10 12,340 5.1 experiments with the cutting of the surface of the soil sample with a moving blade [24]. 1–b2–7 11.13 1.79 0.51 0.0121 0.17 0.38 1.66 10 19,070 5.2 2–b4–3 1.48 1.20 0.28 0.0043 0.14 0.31 4590 5.8 3.13 10 2–b4–4 2.22 Table 1. 1.56 The m 0.29 ain characterist 0.0049 ics of the 0.15 experim 0.33 ents in Balla4.8 ntrae Hill 10 Country.6750 7.1 2–b4–5 3.04 1.66 0.29 0.0062 0.17 0.38 8.54 10 9100 6.7 2–b4–6 4.01 1.91 0.30 0.0070 0.18 0.40 11,810 7.3 1.4 10 Q, l/s U, m/s W, m d, m U* σU E, m/s Re Fr Plot–Soil–Run 7 2–b4–7 5.92 2.15 0.31 0.0089 0.21 0.46 3.49 10 16,830 7.3 −7 1–b2–3 3.13 1.19 0.46 0.0057 0.12 0.26 2.29 × 10 6020 5.0 Key: Q—discharge, U—mean velocity, W—ﬂow width, d—ﬂow depth, U —kinematic velocity, s —velocity * U −7 1–b2–4 4.01 1.30 0.46 0.0067 0.13 0.28 2.33 × 10 7670 5.1 ﬂuctuation standard deviation (from Equation (19)), E—erosion rate, Re—Reynolds number, Fr—Froude number. −7 1–b2–5 5.12 1.40 0.47 0.0077 0.14 0.30 2.29 × 10 9510 5.1 −7 1–b2–6 6.86 1.53 0.49 0.0092 0.15 3r 0.33 5.49 × S 10 12,340 5.1 The coefﬁcient k for ellipsoidal aggregates is 0.1 + 0.42 and varied in the 16r S s a −6 1–b2–7 11.13 1.79 0.51 0.0121 0.17 0.38 1.66 × 10 19,070 5.2 range 0.026–0.04 for the ﬁrst plot and 0.031–0.047 for the second (depending on S /S −8 a 2–b4–3 1.48 1.20 0.28 0.0043 0.14 0.31 3.13 × 10 4590 5.8 ratio in the range 2–4). The standard deviation of velocity ﬂuctuations was s 2.1U −8 U 2–b4–4 2.22 1.56 0.29 0.0049 0.15 0.33 4.8 × 10 6750 7.1 (according to [19]). −8 2–b4–5 3.04 1.66 0.29 0.0062 0.17 0.38 8.54 × 10 9100 6.7 h i g(r r) s S −7 The 2–b4–6 coefﬁcient 4.01 k = cos 1.91 0.30 sin ar0.0070 ctan 0.238 0.18 0.40 was in1.4 × the 10 range 11,810 0.44–0.64 7.3 for 2r S −7 2–b4–7 5.92 2.15 0.31 0.0089 0.21 0.46 3.49 × 10 16,830 7.3 the ﬁrst plot and 0.24–0.35 for the second. 2/3 Key: Q—discharge, U—mean velocity, W—flow width, 3 1# d—flow depth, U*—kinematic velocity, ( ) S S 3 4 b b The coefﬁcient k = D = was 2.04 10 for the ﬁrst plot and 2r V 8r S 8r s a s a s σU—velocity fluctuation standard deviation (from Equation (19)), E—erosion rate, Re—Reynolds 1.54 10 for the second. The variation coefﬁcient for cohesion was estimated as 0.59 in number, Fr—Froude number. the experiments with the cutting of the surface of the soil sample with a moving blade [24]. The relationship between the erosion rate and the ﬂow velocity shows a higher erosion The relationship between the erosion rate and the flow velocity shows a higher ero- rate at plot 1 than at plot 2 for the same velocity (Figure 3). This is explained by the higher sion rate at plot 1 than at plot 2 for the same velocity (Figure 3). This is explained by the soil aggregate stability and organic matter content in the soil at plot 2. The power–law higher soil aggregate stability and organic matter content in the soil at plot 2. The power– functions describe these relationships well. The exponents are different for these two plots law functions describe these relationships well. The exponents are different for these two (5.2 for the ﬁrst and 4.1 for the second) and differ from the function commonly used in the plots (5.2 for the first and 4.1 for the second) and differ from the function commonly used majority of shear–stress erosion models (see, for example, [27]). in the majority of shear–stress erosion models (see, for example, [27]). Figure 3. The relationship between ﬂow velocity U and erosion rate E for plot 1 (circles) and plot 2 (triangles) in the experiments in Ballantrae Hill Country. Earth 2022, 3, FOR PEER REVIEW 10 Earth 2022, 3 237 Figure 3. The relationship between flow velocity U and erosion rate E for plot 1 (circles) and plot 2 (triangles) in the experiments in Ballantrae Hill Country. 3. Results 3. Results 3.1. General Numerical Experiments 3.1. General Numerical Experiments The The firs ﬁrstt se sett o off ca calculations lculations with with the the a algorithm lgorithm de described scribed ab above ove (F (Figur igure e 1 1) show ) show the the erosion rate change through time (Figure 4). In the numerical experiments for different erosion rate change through time (Figure 4). In the numerical experiments for different flow velocities, soil was “eroded” for 1800 or 3600 s. During this period, the mean cohesion ﬂow velocities, soil was “eroded” for 1800 or 3600 s. During this period, the mean cohesion of of the the so soil il s surface urface l laye ayer r and and the the ins instant tant er erosion osion r rate ate ch changed anged thr throu ough gh t time ime d due ue to to the the ac action tion of of so soililarmoring armoring an andd so soilil loosening. looseningThe . Thtemporal e temporal trend trend chan changed ged its sign its sidepending gn depending on on the sign of the cumulative effect of armoring and loosening. The most common scenario was the sign of the cumulative effect of armoring and loosening. The most common scenario a was general a gener decr al decre ease in ase soil inmean soil m cohesion ean coheduring sion dur the ingﬁrst the 500 firsts, 50 then 0 s, then a slight a slincr ightease incre and ase stabilization after about 1000 s in the experiment. The erosion rate decreased during the and stabilization after about 1000 s in the experiment. The erosion rate decreased during ﬁrst 5–20 s, then signiﬁcantly increased and stabilized after about 1000 s in the experiment. the first 5–20 s, then significantly increased and stabilized after about 1000 s in the exper- The temporal changes in the percentage of the area of erodible soil P , where the driving iment. The temporal changes in the percentage of the area of erodible E soil PE, where the forces were greater than the resistance forces, corresponded with the trends in erosion rate. driving forces were greater than the resistance forces, corresponded with the trends in The inﬂuence of the soil particle size on the erosion rate was much smaller than that of erosion rate. The influence of the soil particle size on the erosion rate was much smaller cohesion, and is thus not discussed further. than that of cohesion, and is thus not discussed further. Figure 4. Change through time in calculated speciﬁc soil cohesion force k C (A), erosion rate E (B) and the percentage of the area occupied by erodible soils after the armoring–loosening cycles P ,% (C) for different ﬂow velocities. The initial k is 17, the parameter of loosening = 0.001, and other parameters were obtained from plot 2. Earth 2022, 3, FOR PEER REVIEW 11 Figure 4. Change through time in calculated specific soil cohesion force kCC (A), erosion rate E (B) and the percentage of the area occupied by erodible soils after the armoring–loosening cycles PE,% Earth 2022, 3 238 (C) for different flow velocities. The initial 𝑘 is 17, the parameter of loosening λ = 0.001, and other parameters were obtained from plot 2. The The tem temporal poral tren trends ds in in the the mean mean cohe cohesion sion of of the the surface layer surface layer an and d in the in the erosion erosion rate rate ar are con e contr tro olled lled by chang by changes es in in cohesion cohesion PPDFs DFs (F(Figur igure 5 e)5 . A ). very A very sma small ll par part t of th ofe P the DF of PDFthe of cohesion force at the segment, overlapping the PDF of the driving force (Figure 5A), con- the cohesion force at the segment, overlapping the PDF of the driving force (Figure 5A), contr trols the ols the rate of rate of erosion erosion anand d the the pe per rcentage centage of ero of erodible dible soil soilsur surface face P P E (F (Figur igure e 4C). 4C). The The values values of of pr probability obability de density nsity (PD) (PD) ar are e ve very ry small small h her ere e (F (Figur igure e 5 5B B). ). Therefor Therefore, e, n numerical umerical experiments focusing on this segment require precise calculations. experiments focusing on this segment require precise calculations. Figure 5. Temporal changes in the PDF of soil cohesion for a flow velocity of 2.15 m/s. (A) PDF of Figure 5. Temporal changes in the PDF of soil cohesion for a ﬂow velocity of 2.15 m/s. (A) PDF driving force (green line) and PDFs of cohesion force, initial (red line) and at the stage of rapid of driving force (green line) and PDFs of cohesion force, initial (red line) and at the stage of rapid changes (black lines). Two modes show two stage of stabilization. (B) Temporal changes in proba- changes (black lines). Two modes show two stage of stabilization. (B) Temporal changes in probability 2 2 bility density of cohesion force for kCC = 4 m /s at the segment, overlapping with the PDF of driving 2 2 density of cohesion force for k C = 4 m /s at the segment, overlapping with the PDF of driving 2 2 forces. (C,D) Changes in the probability density of cohesion force for kCC = 15–20 and 30 m /s . Nu- 2 2 forces. (C,D) Changes in the probability density of cohesion force for k C = 15–20 and 30 m /s . merical experiments for initial 𝑘 𝐶 = 30, parameter of loosening λ = 0.0015, and other parameters Numerical experiments for initial k C = 30, parameter of loosening = 0.0015, and other parameters obtained from plot 2. C obtained from plot 2. The PD at the segment around the mode of PDF increases through time and the mode The PD at the segment around the mode of PDF increases through time and the position shifts to lower values of cohesion due to the loosening of soil at the surface (Fig- mode position shifts to lower values of cohesion due to the loosening of soil at the surface ure 5C). The PD at the main part of PDF to the right of the mode mostly decreases until (Figure 5C). The PD at the main part of PDF to the right of the mode mostly decreases ~500 s, and then stabilizes (Figure 5D). The part of the PDF with a soil cohesion force until ~500 s, and then stabilizes (Figure 5D). The part of the PDF with a soil cohesion force greater than the driving force controls the temporal changes in the mean cohesion of the greater than the driving force controls the temporal changes in the mean cohesion of the soil surface layer. soil surface layer. 3.2. The Comparison of Calculated Erosion Rates with the Measured 3.2. The Comparison of Calculated Erosion Rates with the Measured In the situation where one of the main factors of erosion, i.e., the resisting forces of In the situation where one of the main factors of erosion, i.e., the resisting forces of soil soil cohesion, are indefinite, it is possible to find these indefinite characteristics by com- cohesion, are indeﬁnite, it is possible to ﬁnd these indeﬁnite characteristics by comparing paring the measured rates of erosion and calculating them with Equations (17)–(28). the measured rates of erosion and calculating them with Equations (17)–(28). Numerical experiments were performed with input characteristics from Table 1. The Numerical experiments were performed with input characteristics from Table 1. The fields of erosion rate E at given flow velocities were calculated for a variety of initial soil ﬁelds of erosion rate E at given ﬂow velocities were calculated for a variety of initial soil cohesions kCC and parameters λ in the failure function. Examples of such fields, repre- cohesions k C and parameters in the failure function. Examples of such ﬁelds, represented sented by isolines of erosion rate E, are shown in Figure 6 for two runs in plot 1. The soil by isolines of erosion rate E, are shown in Figure 6 for two runs in plot 1. The soil erosion rate is characterized by equiﬁnality. The calculation for each pair of k C and at a given ﬂow velocity leads to a unique value of the resulting erosion rate E. At the same time, calculations for different pairs of initial soil cohesion and parameter at a given ﬂow velocity lead to the same resulting erosion rate, forming the isoline in Figure 6. The same measured rate of erosion at a given ﬂow velocity, shown by bold dashed isolines, can also Earth 2022, 3, FOR PEER REVIEW 12 erosion rate is characterized by equifinality. The calculation for each pair of kCC and λ at a given flow velocity leads to a unique value of the resulting erosion rate E. At the same Earth 2022, 3 239 time, calculations for different pairs of initial soil cohesion and parameter λ at a given flow velocity lead to the same resulting erosion rate, forming the isoline in Figure 6. The same measured rate of erosion at a given flow velocity, shown by bold dashed isolines, appear with different combinations of soil properties. This set of numerical experiments can also appear with different combinations of soil properties. This set of numerical ex- shows that the result of a single run with measured characteristics of ﬂow and erosion rate periments shows that the result of a single run with measured characteristics of flow and cannot be used to ﬁnd indeﬁnite soil properties (initial soil cohesions k C and parameters erosion rate cannot be used to find indefinite soil properties (initial soil C cohesions kCC and ), which are used in the proposed model. parameters λ), which are used in the proposed model. Figure 6. The fields in isolines of erosion rate E (m/s), calculated for varying initial cohesion 𝑘 𝐶 Figure 6. The ﬁelds in isolines of erosion rate E (m/s), calculated for varying initial cohesion k C and the parameter of loosening λ for two flow velocities from plot 1: (A) 1.4 m/s and (B) 1.79 m/s. and the parameter of loosening for two ﬂow velocities from plot 1: (A) 1.4 m/s and (B) 1.79 m/s. The measured rate of erosion at a given flow velocity is shown by a bold dashed isoline. The measured rate of erosion at a given ﬂow velocity is shown by a bold dashed isoline. The shape of the erosion rate isolines is different for different flow velocities. There- The shape of the erosion rate isolines is different for different ﬂow velocities. Therefore, fore, a combination of such isolines can probably narrow the region of indefinite soil prop- a combination of such isolines can probably narrow the region of indeﬁnite soil properties. erties. Keeping the initial soil cohesion constant in the numerical experiments for a given ﬂow Keeping the initial soil cohesion constant in the numerical experiments for a given velocity, it is possible to minimize the difference between measured erosion rates and calcu- lated flow ve ones loci with ty, iEquations t is possible (24)–(28) to minim byivarying ze the dparameter ifference betwe . The en m points easured in Figur erosion rates es 7 and 8 shows and cal the cula optimal ted ones wit erosion h rates Equat for ions each (24) ﬂow –(28) velocity by varyi from ng para the measur meter λ ements . The point at Ballantrae s in Fig- Hill ures Country 7 and 8 sh for ows the op different ticombinations mal erosion rate ofs fo k C r eand ach flow velocity . The intersections from the measuremen of erosion rate ts isolines narrow the range of such combinations for the set of experiments on the same at Ballantrae Hill Country for different combinations of kCC and λ. The intersections of soil erosion withrate diffisolin erentes n ﬂow arro velocities w the ra(gr nge o een f such borders combinations at Figures for 7 and the se 8).t of The experiments on soil at plot 1 2 2 is characterized by a cohesion force k C in the range of 12–14 m /s (or C = 60–70 kPa), the same soil with different flow velocities (green borders at Figures 7 and 8). The soil at 2 2 a parameter in the range of 0.0025–0.004 and the percentage of the area, occupied by plot 1 is characterized by a cohesion force kCC in the range of 12–14 m /s (or C = 60–70 erodible soils after 3600 s of the armoring–loosening cycles, in the range of 0.55–0.7%. kPa), a parameter λ in the range of 0.0025–0.004 and the percentage of the area, occupied The same types of calculations were performed for the ﬂow velocities and erosion by erodible soils after 3600 s of the armoring–loosening cycles, in the range of 0.55–0.7%. rates, measured at plot 2 (Figure 8). They show that the soil is characterized (see green 2 2 boarders) by an initial cohesion force k C in the range of 27–31 m /s (or C = 180–200 kPa), a parameter in the range of 0.0015–0.002, and the percentage of the area, occupied by erodible soils after 3600 s of the armoring–loosening cycles, in the range of 0.35–0.55% per second. Earth Earth 2022 2022 , 3 , 3, FOR PEER REVIEW 240 13 2 2 Figure 7. The possible combinations of initial specific soil cohesion force (kCC, m 2 /s 2 ) with parameter Figure 7. The possible combinations of initial speciﬁc soil cohesion force (k C, m /s ) with parameter λ in the failure function, which lead to the same resulting erosion rate at a given flow velocity (A), in the failure function, which lead to the same resulting erosion rate at a given ﬂow velocity (A), and of the percentage of the area, occupied by erodible soils after 3600 s of the armoring–loosening and of the percentage of the area, occupied by erodible soils after 3600 s of the armoring–loosening cycles PE,% (B). The green box shows the possible range of soil properties, which fits the measured cycles P ,% (B). The green box shows the possible range of soil properties, which ﬁts the measured flow velocity and erosion rate in plot 1. ﬂow velocity and erosion rate in plot 1. The same types of calculations were performed for the flow velocities and erosion rates, measured at plot 2 (Figure 8). They show that the soil is characterized (see green 2 2 boarders) by an initial cohesion force kCC in the range of 27–31 m /s (or C = 180–200 kPa), Earth 2022, 3, FOR PEER REVIEW 14 a parameter λ in the range of 0.0015–0.002, and the percentage of the area, occupied by Earth 2022, 3 241 erodible soils after 3600 s of the armoring–loosening cycles, in the range of 0.35–0.55% per second. 2 2 2 2 Figure 8. The possible combinations of initial specific soil cohesion force (kCC, m /s ) with parameter Figure 8. The possible combinations of initial speciﬁc soil cohesion force (k C, m /s ) with parameter λ in the failure function, which lead to the same resulting erosion rate at a given flow velocity (A), in the failure function, which lead to the same resulting erosion rate at a given ﬂow velocity (A), and of the percentage of the area, occupied by erodible soils after 3600 s of the armoring–loosening and of the percentage of the area, occupied by erodible soils after 3600 s of the armoring–loosening cycles PE,% (B). The green box shows the possible range of soil properties, which fits the measured cycles P ,% (B). The green box shows the possible range of soil properties, which ﬁts the measured flow velocity and erosion rate in plot 2. ﬂow velocity and erosion rate in plot 2. These results are in general accordance with the measured characteristics at these These results are in general accordance with the measured characteristics at these two plots. The cohesion, measured with a tore—vane, was about 50 kPa at both plots, two plots. The cohesion, measured with a tore—vane, was about 50 kPa at both plots, which is rather close to the estimates provided by modelling for plot 1. The much higher which is rather close to the estimates provided by modelling for plot 1. The much higher estimate for the cohesion for plot 2 is explained by the anti—erosion effect of the organic estimate for the cohesion for plot 2 is explained by the anti—erosion effect of the organic matter in the soil (2.4%). The parameter λ in the failure function, with controls the rate of matter in the soil (2.4%). The parameter in the failure function, with controls the rate soil loosening during erosion and the rate of increase in the area of eroded soil, which of soil loosening during erosion and the rate of increase in the area of eroded soil, which controls the rate of erosion, are both two—fold larger at plot 2 than at plot 1. This is also controls the rate of erosion, are both two—fold larger at plot 2 than at plot 1. This is also due to the anti—erosion effect of organic matter. These observations show that erosion at due to the anti—erosion effect of organic matter. These observations show that erosion at plot 1 was due to the destruction of within—aggregate links, and so small soil particles plot 1 was due to the destruction of within—aggregate links, and so small soil particles were moved from the soil surface. The erosion at plot 2 was due to the destruction of were moved from the soil surface. The erosion at plot 2 was due to the destruction between—aggregate links, and so rather large particles were collected at the end of the of between—aggregate links, and so rather large particles were collected at the end of plot (Figure 9). This difference in the erodibility of the soil at plots 1 and 2 is also well— the plot (Figure 9). This difference in the erodibility of the soil at plots 1 and 2 is also well—illustrated by the aggregate stability tests: wet sieving aggregates for 30 min showed that 88% was retained for soil from plot 2 and only 7.6% was retained for soil from plot 1. Earth 2022, 3, FOR PEER REVIEW 15 Earth 2022, 3 242 illustrated by the aggregate stability tests: wet sieving aggregates for 30 min showed that 88% was retained for soil from plot 2 and only 7.6% was retained for soil from plot 1. Figure 9. Soil aggregates and particles, collected at the end of plot 2 during the run 2–b4–3. Figure 9. Soil aggregates and particles, collected at the end of plot 2 during the run 2–b4–3. The The numer numerical ical experim experiments ent(Figur s (Figures es 7 and 7 and 8) show 8) show the possibility the possibto ility to estimate estimate the main the soil characteristics used in the model through gully erosion simulation in ﬂume or direct main soil characteristics used in the model through gully erosion simulation in flume or measurements of the gully erosion rate in the ﬁeld with a variety of ﬂow velocities on the direct measurements of the gully erosion rate in the field with a variety of flow velocities same soil. The number of such experiments is limited now, and there is a large ﬁeld for on the same soil. The number of such experiments is limited now, and there is a large field future investigations. for future investigations. 4. Discussion 4. Discussion The combination of classical deterministic mechanics—Newton’s second law—and The combination of classical deterministic mechanics—Newton’s second law—and the stochastic description of the erosion control factors (driving and resistance forces) has the stochastic description of the erosion control factors (driving and resistance forces) has led to the development of the stochastic gully erosion model (STOGEM). led to the development of the stochastic gully erosion model (STOGEM). The mean erosion rate is calculated with the algorithm in Figure 1 using the equation The mean erosion rate is calculated with the algorithm in Figure 1 using the equation of the force balance acting on a soil particle in a water ﬂow with the PDFs of ﬂow velocity, of the force balance acting on a soil particle in a water flow with the PDFs of flow velocity, soil aggregate size and cohesion, including several model parameters, such as k , k and U D soil aggregate size and cohesion, including several model parameters, such as kU, kD and k . The mathematical methods for the calculation of the PDF of the function of stochastic kC. The mathematical methods for the calculation of the PDF of the function of stochastic variables, knowing the PDFs of these stochastic variables [14], are used. The relationships variables, knowing the PDFs of these stochastic variables [14], are used. The relationships between the main factors of erosion (for example, ﬂow velocity) and erosion rate are between the main factors of erosion (for example, flow velocity) and erosion rate are the- theoretically determined within the model for a given combination of input data. This oretically determined within the model for a given combination of input data. This theory theory opens up a new way for better understanding the experimental results of soil opens up a new way for better understanding the experimental results of soil erosion, and erosion, and demonstrates the direction for future investigations. It is possible to estimate demonstrates the direction for future investigations. It is possible to estimate the main soil the main soil characteristics used in the model by gully erosion simulation in ﬂume or direct characteristics used in the model by gully erosion simulation in flume or direct measure- measurements of the gully erosion rate in the ﬁeld with a variety of ﬂow velocities in the ments of the gully erosion rate in the field with a variety of flow velocities in the same soil. same soil. The results of numerical experiments are consistent with the measurements in the The results of numerical experiments are consistent with the measurements in the labor- laboratory ﬂumes [15] and show the same stages of erosion development in cohesive soils. atory flumes [15] and show the same stages of erosion development in cohesive soils. The relationships between the rate of erosion and control factors are not pre—installed in The relationships between the rate of erosion and control factors are not pre—in- this model. This is the main advantage and its main difference from empirical USLE—type [28] stalled in this model. This is the main advantage and its main difference from empirical and “process—based” shear stress—type models [27]. STOGEM also contains novel ele- USLE—type [28] and “process—based” shear stress—type models [27]. STOGEM also ments compared to previously published stochastic models of soil erosion [29–32]. contains novel elements compared to previously published stochastic models of soil ero- The expression for cohesive soil erosion rate calculation (Equation (17)) can be naturally sion [29–32]. included into the equation of deformation in models such as GULTEM or DYNGUL [10], or The expression for cohesive soil erosion rate calculation (Equation (17)) can be natu- any other process—based model of gully erosion instead of using empirical shear—stress rally included into the equation of deformation in models such as GULTEM or DYNGUL type formulas. [10], or any other process—based model of gully erosion instead of using empirical The theoretical approach to gully erosion calculations is far from achieving direct shear—stress type formulas. application in soil conservation practice. There are many unknown processes and param- eters, mostly related to changeable and complicated matters, such as the soil. The list of possible improvements and clariﬁcations of the model is long. It is obvious that most of the parameters taken as constants in the above—described model (soil particle shape, porosity, Earth 2022, 3 243 etc.) are stochastic variables. The exponential failure function is the simplest option to model the process of soil loosening, and the wet—sieving test of aggregate stability shows that the Weibull function better explains this process. The different resistance to erosion of the links within and between soil aggregates must be investigated. The effect of raindrops on erosion by water in the gullies must also be taken into account. The general limitation of this model is the requirement of erosion of the surface layer particle by particle. This requirement may not be respected at high rates of soil loosening and high velocities. The processes of transport and the deposition of eroded soil particles are not taken into account, and the inﬂuence of such processes can be important [33]. The further comparison of the measured rates of gully erosion performed in the frame of this model with calculated ones will reveal other details necessary for stochastic modelling. 5. Conclusions The stochastic gully erosion model (STOGEM) is based on a combination of deter- ministic mechanics—Newton’s second law—and the stochastic description of the erosion control factors. The main proposition in this model is that the depth of the active surface layer of eroded cohesive soil is equal to one particle diameter, and erosion develops particle by particle. The processes of transport and deposition of eroded soil particles are not taken into account. The erosion rate at the gully bed is calculated directly from the equation of the balance between the driving and resistance forces acting on a soil particle in ﬂowing water. All factors of erosion—ﬂow velocity, soil aggregate size and cohesion—are regarded as stochastic variables, described by probability density functions. The probability density function of cohesion in the model varies through time and space during an erosion event due to the changes in soil composition, namely armoring and loosening. This theory is still far from achieving practical application, but opens up a new way for better under- standing the experimental results of gully erosion and demonstrates the direction for future investigations. Funding: This research was funded by Russian State Task 0110—1.13, N 121051100166—4 “Hydrology, morphodynamics, and geoecology of erosion—channel systems”. Conﬂicts of Interest: The author declares no conﬂict of interest. References 1. Ireland, H.A.; Eargle, D.H.; Sharpe, C.F.S. Principles of Gully Erosion in the Piedmont of South Carolina; Technical Bulletins 167374; U.S. Department of Agriculture, Economic Research Service: Washington, DC, USA, 1939. 2. Watson, D.A.; Laﬂen, J.M.; Franti, T.G. Estimating Ephemeral Gully Erosion; Paper No. 86-2020; American Society of Agricultural Engineers: St. Joseph, MI, USA, 1986; pp. 1–16. 3. Kosov, B.F.; Nikolskaya, I.I.; Zorina, Y.F. Experimental research of gullies formation. Exp. Geomorphol. 1978, 3, 113–140. (In Russian) 4. 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Earth – Multidisciplinary Digital Publishing Institute

**Published: ** Feb 9, 2022

**Keywords: **stochastic model; balance of driving and resistance forces; functions of stochastic variables; probability density functions

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