GULTEM - The Model to Predict Gully Thermoerosion and Erosion (Theoretical Framework)

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1 Thi paper wa peer-reviewed for cientific content. Page In: D.E. Stott, R.H. Mohtar and G.C. Steinhardt (ed) taining the Global Farm. Selected paper from the 10th International Soil Conervation Organization Meeting held May 4-9, 1999 at Purdue Univerity and the USDA-ARS National Soil Eroion Reearch Laboratory. GULTEM - The Model to Predict Gully Thermoeroion and Eroion (Theoretical Framework) ABSTRACT The three-dimenional hydraulic model GULTEM wa developed to predict the rapid change of gully morphology in the early tage of gully development. It i baed on the digital elevation model analyi of flowline; calculation of runoff due either to nowmelt or to rainfall; and the olution of the equation of ma conervation and gully-bed deformation for different type of oil (including frozen oil). The model of the hallow landlide tability wa ued to predict the gully idewall inclination. The tochatic method of detachment rate etimation ued in GULTEM i baed on calculation of the probability of exce driving force above reitance force in the flow, which erode coheive oil. The method explain the ubtantial difference in the type of relationhip between detachment rate and flow velocity (or hear tre and tream power) for different oil. The model can be ued to chooe the appropriate ytem of land conervation meaure and to fit utainable land-ue condition to catchment with high gully-eroion potential. INTRODUCTION The oil conervation to heet and rill eroion approach i fundamentally different how we approach to gully eroion. In the firt cae, a oil conervation chedule include field morphological, hydrological and oil invetigation, laboratory analyi of oil eroion propertie, laboratory experiment, and engineering calculation before the olution for oil conervation meaure can be found. A large number of field and laboratory method can be found in handbook, and many mathematical model are available for oil eroion calculation. In the econd cae, oil conervation meaure are deigned with le information. Field invetigation conit mainly of morphological meaurement in the gullie. Engineering calculation include table lope etimation for gully ide and infilling. The number of mathematical model neceary to predict gully eroion i very much le than for heet and rill eroion calculation. A practical reaon for thee difference i not clear. The ignificance of gully eroion ha been well documented. The volume of the gullie on the Ruian Plain i about 4 x 10 9 m 3, i.e. about 4% of the whole volume of eroion ince 1700 AD (Sidorchuk, 1995). In Autralia, with mainly pature land, the volume of gully eroion amount to 16 x 10 9 t a -1 Alekey Sidorchuk* (Waon et al., 1996). In Wetern Europe, ephemeral gully eroion can meaure up to 30-40% and up to 80% of the total eroion volume (Poeen et al., 1996). Gullie detroy the fertile topoil layer, and the urrounding land are damaged with more evere heet and rill eroion. A gully i a linear deep eroion feature with an active head cut, untable ide wall, ubject to ma movement, and a non-graded longitudinal profile, with temporal water flow. A gully i often a tranient form of relief: It can be initiated a a rill on a lope, can be tranformed into an ephemeral gully, and, if not cultivated, enlarge into a typical gully. After long-term evolution a gully become table. ch a mature linear eroion feature with a graded profile and table idewall i called balka in Ruia. With a good ground water upply, a table gully can become a mall creek. During period of active bottom and head cut eroion, a balka or creek can be tranformed into a reactivated gully, and during an accumulation period, a gully can be completely filled with ediment, thu becoming a hallow elongated depreion at the lope (o called zero-order valley or lozhbina ). There are two main tage of gully development, controlled by different et of geomorphic procee. At the firt tage of gully initiation, hydraulic eroion (and thermoeroion in area with permafrot) i predominant at the gully bottom, and rapid ma movement occur along the gully ide. During thi period, when the morphological characteritic of the gully (length, depth, width, area, and volume) are far from table, gully channel formation i very intene. In the later tage, ediment tranport and edimentation are the main procee in the gully bottom. The gully width increae due to lateral eroion, and low ma movement tranform the gully ide. The experiment of B. Koov, I. Nikol kaya and Ye. Zorina (1978) on gully formation in and how that the firt tage i relatively hort and take about 5 % of the gully lifetime. More than 90 % of gully length, 60 % of gully area and 35 % of gully volume are formed at thi period. Gully morphology at the lat tage (the greatet part of a gully life) i nearly table. One of the main area of recent intenive anthropogenic gully eroion i the Yamal Peninula in Wetern Siberia, in the area of ga field with permafrot. The rate of gully growth i a much a 0-30 to 00 m year -1 (Sidorchuk, 1996; Sidorchuk and Grigor ev, 1998). Thee gullie are a real danger for contruction and ga tranportation facilitie, and their activity ha led to a regional ecological *Alekey Sidorchuk, Laboratory of Soil Eroion and Fluvial Procee, Geographical Faculty, Mocow State Univerity, Mocow, Ruia. idor@ya.geogr.mu.u

2 catatrophe. In recently exploited ga field of the Yamal Peninula the initial tage of gully development are typical: for example, a gully near the main exploitation camp (called PBB) did not exit in 1986, but a 40-m - long hallow, elongated depreion on the lope did. After the camp contruction in , eroion and thermoeroion were initiated due to the increae in urface runoff. In 1988, the gully length wa 450 m. In 1989 it wa 740 m, and in 1990 the length wa 940 m. The head of the gully reached the camp building. Filling of the gully head by heavy loam from the bank by bulldozer attempted to top the gully growth. Neverthele, in 1995, the gully wa 5 m longer, than in GULTEM model The 3D hydraulic gully thermoeroion and eroion model GULTEM wa developed for the initial tage of gully evolution. At thi tage, eroion (and thermoeroion at the area with permafrot) i predominant at the gully bottom, and rapid hallow landlide occur on the gully ide. Gully channel formation i very intenive, and the morphological characteritic of the gully (length, depth, width, area, and volume) are far from table. On the marine terrace of the Yamal Peninula, compoed from frozen loam and and with ubtantial ice content, thi tage lat 4-10 year and anthropogenic gullie cut into the terrain to cover their entire length. The GULTEM i baed on a net of flowline, evaluated from a topographical digital elevation model (DEM). The multi-layered oil texture (including topoil with the vegetation cover) i derived from DEM of the top urface of each layer with a imilar texture. The runoff due to nowmelt and rainfall i calculated with phyical-baed hydrological model. The input to the GULTEM model include topographic, hydrologic, hydraulic and oil mechanic data. Topography i decribed by elevation and ditance from the gully mouth along the longitudinal profile of each flowline on initial lope (including exiting gullie). The runoff along thee flowline i calculated with the hydrological model. The multi-layer oil propertie are ued in the model. The input for each layer include: the elevation of the bae of the layer along the ame flowline; oil bulk denity; oil coheion, angle of internal friction and fatigue to rupture; the ize of the water table aggregate; oil moiture and thin vegetation root content (for a topoil). During a nowmelt or raintorm event, flowing water i aumed to erode a rectangular channel in the topoil or at the gully bottom. The longitudinal profile tranformation in pace and time and gully bed widening are calculated in term of the ma-conervation equation with Lax-Wendroff predictor-corrector cheme. The tability criterion i determined for each calculation tep. The numerical cheme tability i obtained by change of a time tep: At each time Figure 1. Sytem of model for gully morphology prediction in condition of utainable land ue.

3 tep, width, depth, velocity and critical hear tre for the oil both in the gully bed and in the bank are calculated, along the flow. Thi allow calculation of the oil aggregate detachment rate, and the rate of the flow inciion and widening. Between water flow event, a gully cro - ection i quickly tranformed by hallow landlide. After each flood event, the model tranform the rectangular bottom trench to a trapezoidal hape with a hallow landlide tability model. Numerical experimentation how that the model decribe the real proce of gully longitudinal and cro-ection profile evolution in time and pace on a gully bain. A it i enitive to of oil erodibility variation, field invetigation and careful calibration of the model are neceary to predict gully eroion. The main parameter, controlling GULTEM calculation of eroion and thermoeroion (relief, water flow, oil mechanic, and vegetation cover), correpond to the main argument for oil conervation meaure. The numerical experiment provided by the model can be ued to chooe the correct land conervation meaure and to fit utainable land-ue condition to catchment with high gully-eroion potential (Fig. 1). The ytem of model ued for preparation of input data for GULTEM and for gully morphology calculation, wa decribed in the paper of A. Sidorchuk and A. Sidorchuk (1998). The baic principle of oil eroion ued in GULTEM are preented here. Theoretical framework of GULTEM The rate of gully eroion i controlled by water flow parameter (velocity, depth and turbulence) and oil texture (mechanical pattern and protection by vegetation). Thee characteritic are combined in equation of ma conervation (1) and deformation (), which can be written in the form: Q AC + = Cwqw + M0W + MbD CVf W (1) X t Z ε () t ( 1 ) = CV f M 0 Here Q = QC i ediment dicharge (m 3 /), Q = water dicharge (m 3 /); X = longitudinal co-ordinate (m); t = time (); C = mean volumetric ediment concentration; A = flow cro-ection area (m ); C w = ediment concentration of the lateral input; q w = pecific lateral dicharge; M 0 = upward ediment flux (m/); M b = ediment flux from the channel bank (m/); Z = gully bed elevation (m); W = flow width (m); D = flow depth (m); V f = oil aggregate fall velocity in the turbulent flow (m/); ε = poroity of the oil at the gully bed. The firt term in the left part of equation (1) define the ediment budget in the channel reach, the econd term i the ediment torage in the flow. The right part of (1) define the ediment flux: the firt term i lateral flux, the econd one i upward flux, the third i ediment flux from the bank, and the forth i downward flux. Equation () define the change of gully-bottom elevation according to the ediment budget. The ediment torage in the flow i uually very mall and can be neglected. The fall velocity in highly turbulent flow in the young gullie i often cloe to zero, and the rate of edimentation for thee environment i alo negligible. In thi cae, equation (1) i a firt-order ordinary differential equation, and equation () i a firt-order partial differential equation with variable coefficient. The olution of thee equation depend on the form of the term that decribe ediment fluxe. To undertand the eroion proce, the upward flux of oil aggregate and particle (detachment rate) i of major importance. Upward flux (detachment rate) The detachment rate (D r ) i the product of the concentration (C ) of active oil aggregate in the bed layer with the thickne and the mean vertical velocity of oil aggregate (U ): D r = C U (3) Sediment concentration i the ratio between the volume of active oil aggregate V a and the volume of the fluid V in the near bed layer (with the thickne and the unit area S): C ) =V a /( *S). The volume of active oil aggregate can be written a the product of the number of active oil aggregate N on their mean volume V m : V a =NV m. The unit area can be preented a the product of the number of oil aggregate M, expoed at the flow bed on the unit area, on their mean area S m : S=MS m. Therefore, near bed ediment concentration can be repreented a: NVm C = (4) MSm The ratio N/M i the probability (P d ) of oil aggregate detachment for a given unit time dt = / U, and the ratio V m /S m i ome meaure of the mean oil aggregate height D m. D C m = P d (5) For the near bed layer with thickne equal to aggregate height D m, the probability of detachment i equal to the ediment concentration in the near bed layer. After H. A. Eintein (194), it i a function of the meaure of the tranport rate for the cae of non-coheive ediment. Mirtchoulava (1988), Nearing (1991), Larionov (1993), Wilon (1993a, 1993b) and Lile et al. (1998) formulated probabilitic concept of detachment for coheive oil. The following theoretical tochatic decription of oil eroion i an amplification of the model lited above, with the ignificant addition of the tochatic variable that govern thi complicated proce. The probability of oil aggregate detachment i equal to the probability of the exce of driving force above reitance force in the flow. Driving force are drag force (F d ), lift force (F l ), negative turbulent dynamic preure (F dp ), and pore water preure (F pw ). Reitance force are ubmerged weight (F w ), friction force (F f ), tatic preure (F p ), poitive turbulent dynamic preure (F dp ) and coheion (F c ). After Mirtkhoulava (1988) and Borovkov (1989):

4 U Fd = CRρSd (6) U Fl = C y ρ (7) Um Fdp = 3. 5λρSb (8) Fpw = gρsbz p (9) w = gv u ( ρ ρ ) F (10) F (11) f = ft gvu Fp = gρsbd (1) = (13) Fc C0S b Here C R i the coefficient of drag reitance; C y i the coefficient of uplift; U i the actual near-bed flow velocity, and U m i it mean value; λ i the coefficient of hydraulic reitance; S d i the cro-ection area of oil aggregate, perpendicular to flow; ρ and ρ are the oil aggregate denity (containing pore) and water denity repectively; S u i the cro ection area of oil aggregate, parallel to the flow (vertical projection); S b i the area of oil aggregate that i olid with native oil and other aggregate; z p i capillary preure height; f t i the friction coefficient; d i water depth; C 0 i oil coheion. A probability of detachment i greater than zero, when the um of the driving force i more than the um of reitance force: Fd + Fl + Fpw m Fdp Fw Ff Fp Fm Fc >0 (14) or Sb Ψ = U + k pw z p k wf Dm ρ Sb m kdp λ U m Sb C0 k p d kc ρ Sb > 0 (15) The value of the coefficient can be obtained from Mirtkhoulava (1988) and Borovkov (1989): g ( + ft ) g kpw = 40;k dp = 7;kwf = 4; ( kcr + Cy ) ( kcr + Cy ) ( kcr + Cy ) (16) g 4 kp = 40;k m = 48;k c = 4. kc + C kc + C kc + C ( ) ( ) R y R Driving and reitance force are tochatic variable, and their um Ψ ha ome tochatic ditribution with the probability denity function p Ψ. Therefore, probability of detachment P d can be calculated with the formula: P = d p Ψ d Ψ (17) 0 The vertical velocity of oil aggregate i the econd component of the formula (3) for the detachment rate calculation. The moment of aggregate detachment acceleration can be derived from the expreion: y R y Dm ρ U = Ψ z (18) In the near bed layer of the flow with thickne, an aggregate accelerate from zero velocity to it maximum value, U. The integral of (18) give a imple expreion for the near bed vertical velocity of aggregate: U = Ψ D m ρ (19) In turbulent flow with random vertical velocity, it mean value in the um of the field of poitive force may be calculated with the formula: ρ pψ Ψ 0 Dm ρ U = pψdψ 0 ( ρ ) dψ (0) Theoretical analyi of the tochatic mechanic of oil aggregate eroion in water flow how that in the field of random driving and tabilizing force the detachment rate can be calculated a product of Eq. (17) and (0) a: = ρ Dr pψ Ψ d 0 Dm Ψ (1) It i important to note, that in the cae of the initial tage of gully evolution, the rate of edimentation i very mall and the detachment of the aggregate and particle occur from the urface of native oil. The role of edimentation can be important at the later tage of gully growth, and here the recommendation of Hairine and Roe (1991) mut be taken into account. The probability of a function of tochatic variable can be calculated if the probabilitie of the individual variable are known (Gnedenko, 1954). A probability of product Z of tochatic variable X and Y i derived with the integral: 0 1 Z ( ) = ( ) + 1 Z pz Z X px X py dx X px ( X ) py dx () X 0 X A probability of um Z of tochatic variable X and Y i derived from the function: pz ( Z) = px ( X) py ( Y) dx= px ( X) py ( Z X) dx (3) We hall work out a implified cae, where four characteritic are taken a tochatic variable: velocity U, coheion C h, aggregate ize D m and oil conolidation I =S b /S u, and all other are parameter. Therefore the probability denity function for tochatic variable mut be etimated theoretically or experimentally. A probability denity function p U for actual near bed velocity U with mean value U m and tandard deviation σ U i often decribed by the normal ditribution (Mirtkhoulava, 1988). Then frequency of z=u /σ will be defined by firt order non-central χ ditribution (Pugachev, 1979). Borovkov (1989) howed that σ U i related to dynamic velocity: σ U = 3.0 u. The ratio of the oil aggregate area S b, where aggregate

5 i olid with the native oil or other aggregate, to the aggregate vertical projection area S u : I =S b /S u i the meaure of the oil conolidation. The difference between thee two area i the area of micro-crack, which cut looe individual aggregate from native oil. ch micro-crack filled with the ice are often formed in the frozen oil. The relative volume of micro-crack i approximately the difference between bulk oil poroity and tructural within aggregate poroity. The oil conolidation i oppoite to oil fatigue, generated in the oil under a dynamic action of turbulent flow (Mirtkhoulava, 1988), and mainly due to flow velocity ocillation and dynamic preure rapid change. It ditribution depend on oil texture, coheion and the intenity of turbulent ocillation. Overview of laboratory and field experiment of Mirtkhoulava (1988) how that for a wide range of different oil, (I ) mean ha an aymmetrical ditribution. Beta-ditribution will therefore be ued in further calculation. It i evident that the oil conolidation or fatigue, a defined above, need for further invetigation. Analyi of the laboratory data of Mirtkhoulava (1988) how that a gamma-ditribution can be ued to decribe the ditribution of coheion within a ample of oil. Mirtkhoulava data alo howed that the coefficient of variation C v =σ C /C m for thi ditribution i contant and equal ~ 0. for wide range of oil characteritic. In thi cae, the ditribution curve for actual coheion i determined only by one parameter: mean coheion of oil. The ditribution denity of oil aggregate ize within a ample of oil generally fit a lognormal ditribution with the parameter, related to mean aggregate diameter D m and it tandard deviation σ D. The parameter of ditribution curve for flow velocity, oil coheion, conolidation and aggregate ize vary in pace at the flow bed and in the oil profile. Thee parameter can alo change in time during the proce of eroion. Mirtkhoulava (1988) define four tage in coheive oil eroion: 1) rapid eroion of the initially weakened oil urface; ) low eroion of the native oil body; 3) acceleration of eroion due to the increae of oil fatigue in the ocillating turbulent flow; 4) the eroion rate tabilization. It i evident that two lat tage can be repeated in a erie for each frehly expoed layer of eroded oil. The parameter of the flow velocity ditribution curve may change in thee erie due to oil urface roughne variability. Flow turbulence can increae at the beginning of accelerated eroion tage, when the oil urface become more irregular, and decreae at the low eroion tage of due to moothing of the oil urface. Soil conolidation will decreae with the increae of the flow turbulence due to higher lever of dynamic influence of the drag and lift force on oil aggregate tability. With the rapid removal of the weakened aggregate conolidation increae. Thi elforganiing interconnection of eroding flow and eroded oil need further invetigation, which can be more ueful by tochatic approach. RESULTS AND DISCUSSION The analytical form of (1) i rather complicated, and ha to be olved numerically with a certain et of input data. A FORTRAN program (available from the author) wa written for thee calculation. The input data conited of mean bed velocity U m, mean oil coheion C 0, mean oil conolidation I, mean aggregate diameter D m and it tandard deviation σ D. The hydraulic reitance coefficient λ, flow depth d, pore water preure height z, aggregate denity (with poroity) ρ alo mut be known. Numerical experiment were carried out to analyze the influence of thee four tochatic factor on the detachment rate. A firtorder non-central χ ditribution wa ued to decribe a probability of quare of velocity, a gamma ditribution wa ued for oil coheion, a lognormal ditribution wa ued for aggregate ize, and a beta-ditribution wa ued to decribe a probability of oil conolidation. The range of flow bed velocity wa m -1, the range of coheion wa 1-60 kpa, oil conolidation ranged from 0.1 to 0.9, aggregate mean ize from 1 to 10 mm. Other parameter were contant: flow depth wa 0.01 mm, pore preure height wa m, the hydraulic reitance coefficient wa 0.01, aggregate denity wa 1600 kg m -3, and aggregate ize tandard deviation wa 0.3D m. The detachment rate increaed with flow velocity (Fig. ). Thi increae in eroion rate cannot be decribed with an often-ued imple power function D r ~ U n m. Theoretical calculation howed that in the relatively low velocitie, the detachment rate increae more rapidly than in the relatively high velocitie. A imilar effect wa decribed by Larionov (1993) and by Nearing et al. (1997) on the bai of obervation of empirical oil eroion meaurement. The current theory explain thi phenomenon. The detachment rate increae i controlled by oil coheion (Fig. a), by aggregate ize (Fig. b) and, very ignificantly, by oil conolidation (Fig. c). Detachment rate increaed more rapidly with flow velocity for more conolidated oil with high coheion, large aggregate, and high oil conolidation. Decreae of oil conolidation and the aggregate ize led to a decreae of the exponent in power law of detachment rate veru flow velocity. Calculation alo how great difference in the type of oil eroion in the relatively high and relatively low flow velocitie. When flow velocitie are relatively high and driving force increae ignificantly over tabilizing force, oil propertie (coheion, aggregate ize, oil conolidation) are le important in determining the oil-eroion rate (Fig. ). Thi implie that the time and pace random variability of thee factor, which alway exit in natural condition, will not lead to major change in eroion rate. When flow velocitie are relatively low and driving force only lightly increae over tabilizing force, oil propertie are very important in determining oil eroion rate. Even the mall time and pace random variability of thee propertie may lead to ignificant change in eroion rate. To verify the theoretical reult, two et of data were

6 Figure 3. Comparion of calculated (line) and meaured (circle) rate of detachment of active oil aggregate in near bed layer for (A) laboratory experiment of Nearing et al. (1991) and (B) field experiment in Brook Creek gully (Sidorchuk, 1998). Different oil conolidation I caue different type of relationhip between the detachment rate D r (m/) and near bed flow velocity U (m/). The coheion of Paulding and Ruel oil in the laboratory varied from 1 kpa (thin line, white circle) to kpa (thick line, black circle), the aggregate ize varied from 0.47 mm (olid line, mall circle) to.07 mm (broken line, large circle). The coheion of oil in the Brook Creek gully varied from 0-30 kpa (white circle) to kpa (black circle). Figure. Influence of (a) oil coheion C 0, (b) aggregate ize and (c) oil conolidation I on the relationhip between detachment rate and flow velocity. ued: the laboratory meaurement of Nearing et al. (1991) of the detachment rate for the Ruell ilt loam (fine-ilty, mixed, meic Typic Hapludalf) and Paulding clay (very-fine, illitic, nonacid, meic Typic Haplaquept) in the USA, and the field meaurement of the detachment rate for pebbly loam in Brook Creek gully, Autralia (Sidorchuk, 1998). The detachment rate, hydraulic flow parameter, oil coheion and aggregate ize were publihed for thee experiment. Soil conolidation i unknown for both data et. Optimiation calculation were performed to etimate unknown oil conolidation value. The ame procedure of optimization wa ued by Wilon (1993b) for unknown parameter of imilar type. A bet fit of meaured detachment rate in Nearing et al. (1991) experiment with calculated one (Fig. 3a) wa obtained for (I ) mean =0.78. Although the oil coheion wa rather low (1- kpa), the conolidation of thee oil wa rather high. That led to very rapid change of detachment rate with velocity: D r ~ U The influence of aggregate ize wa not o obviou. A bet fit of meaured detachment rate in natural condition in Brook gully with calculated one (Fig. 3b) wa obtained for (I ) mean =0.1. Although the oil coheion wa rather high (30-50 kpa), a conolidation of natural oil in the gully wa low. That led to le exponent in the power law between detachment rate and velocity: D r ~ U.5 3.

7 CONCLUSION The tochatic method of detachment rate etimation wa ued in the gully eroion and thermoeroion model GULTEM. It i baed on calculation of the probability of exce of driving force above reitance force in the flow that erode coheive oil. The explicit relationhip of the hydraulic characteritic of the flow (actual flow velocity, water depth, dynamic preure) and the mechanical propertie of the oil (coheion and conolidation) with the oil aggregate detachment rate allow to give an explanation of a great difference in type of relationhip between detachment rate and flow velocity (hear tre, tream power) for different oil. In high flow velocitie, when driving force increae ignificantly above tabilizing force, the rate of eroion increae with flow velocity i relatively low. The influence of oil propertie (coheion, aggregate ize, oil conolidation) variability i alo le important in determining the oil eroion rate of high relative flow energy (Fig. ). Thi may be the main reaon for the greater predictive capability of exiting oil eroion model for high-energy event. With low flow velocitie and with driving force only lightly increaed above tabilizing force, the eroion rate peedily increae with flow velocity. Soil property variability caue ignificant change in oil eroion rate, and thi influence grow with the increae of oil coheion, conolidation and oil aggregate ize. Even minor patial and temporal random variability of thee propertie may lead to ignificant change in eroion rate. Thi may be the reaon why rather high error in oil eroion calculation are found even with detailed phyical baed model for low eroion rate. The explicit ue of the main parameter that control calculation of eroion and thermoeroion (relief, water flow, oil mechanic, and vegetation cover) in GULTEM model allow the model to be ued a a tool for etablihing of oil conervation meaure. Numerical experiment with the variable input can be ued to chooe the correct land conervation meaure for catchment with high gully eroion potential. The tochatic component in the model give the opportunity to take into account the patial and temporal variability of the main eroion and oil conervation factor. ACKNOWLEDGEMENTS The author i very grateful to Prof. T. Mitkhoulava for valuable talk during the 1995 GCTE Soil Eroion Network meeting. The paper wa dicued with Dr. Dino Torry and Mark Nearing, who made ignificant comment to the text. The language of the paper wa edited by Anne M. Autin. REFERENCES Borovkov, V.S Fluvial procee and flow dynamic at the urban territorie. (In Ruian). Gidrometeoizdat., Leningrad. Eintein, H.A Formula for the tranportation of bed load. Tran. Amer. Soc. Civil Eng. 107: Gnedenko, B.V The probability theory. (In Ruian). Gotekhizdat., Mocow. Hairine P. and C. Roe Rainfall detachment and depoition: ediment tranport in the abence of flowdriven procee. 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