INFLUENCES OF SLOPE GRADIENT ON SOIL EROSION*

Transcription

1 Applied Mamatics Mechanics (English Edition, Vol 22, No 5, May 2001) Published by Shanghai University, Shanghai, China Article ID: (2001) INFLUENCES OF SLOPE GRADIENT ON SOIL EROSION* LIU Qing-quan (~]~j~), CHEN Li (~ J3), LI Jia-chun (~]~) (Institute Mechanics, CAS, Beijing , P R China) (Paper from LI Jia-chun, Member Editorial Committee, AMM) Abstract: The main factors influencing soil erosion include net rain excess, water depth, velocity, shear stress overl flows, erosion-resisting capacity soil. The laws se factors varying with Abstract slope gradient were investigated by using kinematic wave ory. Furrmore, critical slope gradient erosion was driven. The The one-dimensional problem motion a rigid flying plate under explosive attack has analysis shows that critical slope gradient soil erosion is dependent on grain size, soil an analytic solution only when polytropic index detonation products equals to three. In general, bulk a numerical density, surface analysis roughness, is required. runf In length, this paper, net rain however, excess, by utilizing friction coefficient "weak" shock behavior soil, reflection etc. The critical shock in slope gradient explosive has products, been estimated applying oretically small with parameter its range purterbation between method, 41.5 an ~ ~ analytic, 50 ~. first-order approximate solution is obtained for problem flying plate driven by various high explosives with polytropic indices or than but nearly equal to three. Key words: soil erosion; critical slope gradient; flow scouting capability; soil stability Final velocities flying plate obtained agree very well with numerical results by computers. Thus an analytic CLC formula numbers: with P333.5; two parameters O347.7; TU43 high explosive Document (i.e. detonation code: A velocity polytropic index) for estimation velocity flying plate is established. Introduction Soil erosion can be classified into many types due to different eroding forces. Among m surface Explosive runf driven erosion flying-plate is commonest technique one. ffmds The its important slope gradient use in is one study most behavior important materials factors affecting under intense surface impulsive flow loading, erosion. shock Under synsis same diamonds, rainfall condition, explosive welding surface flow cladding metals. The method estimation flyor velocity way raising it are questions could be drastically different on different slopes, thus erosion quantity could also be common interest. enormously Under different. assumptions It is indispensable one-dimensional to study plane detonation relationship between rigid flying slope plate, gradient normal approach surface soil solving erosion problem for soil erosion motion prediction flyor is to solve soil-water following conservancy system planning. equations Many governing scholars have flow investigated field detonation this problem products for decades. behind However, flyor (Fig. I): results are greatly different since complexity problem difference in viewpoint or object. Mainly y can be summarized in following three respects: ap +u_~_xp + au 1) Erosion law: In 1940, Zingg [1] analyzed field data got an empirical relationship au au 1 between soil erosion quantity slope gradient: y = ax b (where a = 0.065, b = 1.48, x is slope in degree y is soil erosion quantity). It shows that soil erosion as as quantity is increasing with slope gradient." TANG Li-qun CHEN Guo-xiang(1997) [2] have established a soil erosion relationship in ir small watershed runf sediment generation model. They also present a proportional relationship between soil erosion quantity slope where p, p, S, u are pressure, density, specific entropy particle velocity detonation products respectively, with trajectory R reflected shock detonation wave D as a boundary trajectory * Received F flyor date: as anor ; boundary. Revised Both date: are unknown; position R state parameters on Foundation it are governed item: by National flow field Natural I Science central Foundation rarefaction wave China behind ( ) detonation ; Foundation wave D by State initial Key stage Laboratory motion Soil Erosion flyor also; Arid position Agriculture F state parameters products Biography: LIU Qing-quan ( ), Associate Pressor, Doctor 510

2 Influences Slope Gradient on Soil Erosion 511 gradient. However, most field data indoor artificial rain experiments show that this kind relationshi p is kept valid only in a certain parameter range. If slope gradient exceeds a threshold value, relationship takes inversely proportional form. For instance, Yaur Klein(1973) [3] analyzed data soil erosion quantity on ramps with slope 15 ~, 19 ~ 25 ~ respectively found that such inversely proportional relationship really appears. The fact infers existence critical slope gradient in soil erosion. 2) Critical slope gradient: Renner(1936) [4] analyzed field data Boise River watershed, Idaho in America, found that percentage eroded area is different with slope gradient. When slope gradient exceeded 40.5 ~, soil erosion quantity starts to decrease instead. CHEN Fa-yang (1985)Is] conducted a 9 groups experiment by artificial rainfall in a 6m 2 wooden box with variable slope degree soil was naked red clay formed in Abstract Quaternary period. Under condition fixed rainfall intensity, he got critical slope gradient The one-dimensional soil erosion 25 problem ~. But Horton(1945) motion [6] obtained a rigid flying critical plate under slope explosive gradient attack 57 ~ in has his an analysis. analytic solution only when polytropic index detonation products equals to three. In general, 3 ) a Quantitative numerical analysis relationships is required. between In soil this erosion paper, rate however, slope by utilizing gradient: Based "weak" on shock data behavior interdll erosion reflection for two shock kinds in soil explosive on products, slope from 3 % applying to 50%, small Singer parameter Blackard purterbation method, an analytic, first-order approximate solution is obtained for problem flying (1982) [7] assumed a polynomial relationship between soil erosion quantity slope gradient. It plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final was quadratic velocities form flying Dw plate = 0.22 obtained sinQ agree very sin well with 2 Q numerical for silty clay, results whereas by computers. it was Thus cubic an form analytic Dw = formula with sinQ two parameters sin high 2 Q - explosive sin (i.e. 3 Q detonation for clay, velocity where Dw is polytropic intenml index) soil loss for quantity, estimation Q is velocity slope gradient. flying plate ZHANG is established. Ke-li Hosoyamada(1996) Is] studied runf sediment generation on ground with different slope gradient by indoor artificial rainfall. They also got a cubic formula 1. S = Introduction sinQ + ll9.6sin 2 Q sin 3 Q, where S is soil loss quantity on hillslope Q is slope gradient. Explosive driven flying-plate technique ffmds its important use in study behavior The differences in many researches indicate that re are a great many factors influencing materials under intense impulsive loading, shock synsis diamonds, explosive welding cladding soil erosion metals. process. The method Therefore, estimation it is flyor great velocity significance to way analyze raising more it are carefully questions influence common interest. slope gradient for predicting soil erosion on a hill.~lope. Under assumptions one-dimensional plane detonation rigid flying plate, normal 1 Hillslope Runf Erosion Main Influencing approach solving problem motion flyor is to solve following Factors system equations governing Hillslope flow runf field erosion detonation is such products a process,that behind sheet flyor flow (Fig. generated I): during rainfall scours soil surface. Its motive power is acting force surface flow, while erosion-resisting capacity is dependent on stability soil body. That ap +u_~_xp + au is to say, soil loss on hillslope occurs when scouring capability surface runf exceeds erosion-resisting capacity. The au au 1 whole process can be divided into three stages. Firstly, y when =0, rainfall intensity is greater than soil inf'fltmtion rate surface as ponding as capability, rain excess flows down hillslope under action gravity thus forms surface sheet flow. Then, when scouring ability is greater than erosion-resisting capacity soil, scour soil particles are initiated. And finally, scoured soil is transported downstream by overl flow. Therefore, where soil p, p, erosion S, u are rate pressure, indicates density, ratio specific between entropy flow particle erosion velocity ability (including detonation products scouring respectively, with trajectory R reflected shock detonation wave D as a boundary ability transportation capability) erosion-resisting capacity soil. trajectory F flyor as anor boundary. Both are unknown; position R state parameters If on we it are use governed r0, by flow bottom flow field shear I stress central to rarefaction represent wave flow behind scour detonation ability, wave D incipient by initial motion stage shear stress motion soil flyor particle also; rc position to represent F erosion-resisting state parameters capacity, products surface soil erosion rate on hillslope can be expressed as [~-] :

3 512 LIU Qing-quan, CHEN Li LI Jia-chtm E, = A(r0 - ro)v, (1) where A is a coefficient related to compacted dry bulk density soil ~,,, turbid water bulk density )'m, soil characteristic, etc. r0, rc are flow shear stress sediment incipient shear stress on hillslope surface respectively, v velocity surface flow on hillslope. The foregoing expression shows that surface flow erosion-resisting capacity soil are main factors affecting soil erosion. The flow shear stress velocity on a slope with different gradient could be very different under same rainfall condition, erosion- resisting capacity soil is different as well. As a result, soil loss rate is different correspondingly. 2 The Variation Scouring Ability Abstract Flow With Slope Gradient The Because one-dimensional tiny depth problem overl motion flow a rigid complicated flying plate boundary under explosive conditions, attack it has is a an tough analytic task solution to describe only when movement polytropic this index kind detonation flow appropriately. products equals Usually to three. one- In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock dimensional shallow water equation (Saint Venant equation) is used in modeling (Emmett, W. behavior reflection shock in explosive products, applying small parameter purterbation W., 1978) method, [9] an analytic, first-order approximate solution is obtained for problem flying plate driven by various high Oh explosives Oh with 3v polytropic indices or than but nearly equal to three. 3--f + v-j-~x + hff-~ = q, (2) Final velocities flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with 3v two parameters 3v 3h high explosive v (i.e. detonation velocity polytropic index) for estimation Ot velocity + v-f~x + flying g-ff~x plate + q is established. + g(sf - So) = O, (3) in which t is time, x distance, h v represent runf depth velocity, q * rain excess, So slope, Sf energy 1. slope, Introduction g gravity acceleration. nl~an$ When solving above equations, kinematic wave approximation is ten adopted. Explosive driven flying-plate technique ffmds its important use in study behavior Wolhiser Liggett [1~ have ever analyzed one-dimensional unsteady overl flow. They materials under intense impulsive loading, shock synsis diamonds, explosive welding cladding find that when metals. The kinematic method wave estimation number K > flyor 20( K velocity = SoL/hoFo), way (in raising which it So are is questions slope, L common is runf interest. length, h 0 is depth runf in distance L, F 0 is Froude number) F0 Under > 0.5, assumptions kinematic one-dimensional wave model can plane describe detonation overl rigid flow flying quite plate, well. normal In most approach solving problem motion flyor is to solve following system equations situations, this condition may be satisfied. Therefore we adopt kinematic wave model, governing flow field detonation products behind flyor (Fig. I): implying that gravity component along slope is equal to resistance force, above equation (3) could be simplified as ap Sf +u_~_xp = So + = au sin0. (4) 4.0 l0 -s 3.0 x x 10 -s 9 9 observed calculated au au Applying 1 kinematic wave ory, we have as numerically simulated process overl flow formation as J. L. M. P. Lima's In] indoor artificial rainfall experiment in (This experiment was done 1.0x 10 -s in a0.5m ~" wooden box with length lm, width 0.5m % slope degree. The rainfall intensity is 135mm/hr where p, p, S, u are pressure, density, specific entropy particle velocity detonation products respectively, with trajectory R reflected shock st detonation is Limburg wave st. D ) as a The boundary comparison t/n~n trajectory F flyor as anor boundary. computation Both are unknown; experiment position in a fairly R good state agreement parameters Fig. on 1 it Simulation are governed result by overl flow field (Fig. I 1) central exhibits rarefaction that wave behind kinematic detonation wave ory wave is D by initial flow stage by using motion kinematic flyor also; suitable position for describing F overl state flow. parameters products wave ory

4 Influences Slope Gradient on Soil Erosion The variation net raln excess with slope gradient Let us consider excess rainfall condition at first. The runf yield is determined by difference rainfall intensity st infiltration rate. Suppose rain is vertical, for a horizontal plane with rainfall intensity [ soil infiltration rate f, rainfall excess turns out q* =l-f. Neverless, for a hillslope re is an incline angle 0 between surface horizontal plane 90 ~ - 0 between surface rainfall direction. The actual area receiving precipitation varies with slope gradient. It leads to different rainfall amount on same length hillslope at different slope Abstract gradient. 1 rainfall As showed in Fig.2, if area hillslope is AC for The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic surface solution with only slope when 0, polytropic actual area index receiving detonation Fig.2 The products relationship equals to three. actual In general, precipitation a numerical is reduced analysis to AB( is required. = ACcosO). In this It paper, means however, by utilizing "weak" shock receiving precipitation behavior that actual area reflection receiving shock precipitation in explosive decreases products, with applying small parameter purterbation method, an analytic, first-order approximate solution is obtained for slope problem 0 flying slope. plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final velocities When rain flying intensity plate obtained (rainfall agree amount very on well unit with area numerical in unit time) results corresponding by computers. Thus to an horizontal analytic plane formula is I with, two actual parameters rain intensity high Io explosive on surface (i.e. detonation with velocity slope 0 is polytropic index) for estimation velocity flying I o plate = I" is established. cos0. (6) Assuming infiltration rate does not vary with slope, actual net rain amount on slope becomes q. = I o - f = lcos0 - f (7) Explosive driven flying-plate technique ffmds its important use in study behavior materials meaning under that intense net rain impulsive amount loading, on unit shock area in synsis unit time decreases diamonds, with explosive slope welding gradient. cladding 2.2 The metals. variation The method runf estimation depth with flyor slope velocity gradient way raising it are questions common The variation interest. rate discharge per unit width on slope gradient is dq/dx, namely Under assumptions one-dimensional dq _ plane detonation rigid flying plate, normal approach solving problem motion dx q* flyor = lcos0 is to - solve f. following system equations (8) governing flow field detonation products behind flyor (Fig. I): Therefore discharge per unit width at site L away from top slope gradient under condition uniform rainfall is ap +u_~_xp + au q = dx = (Icos0-f) L. au J0 ctx au 1 Suppose water depth re is h. Using We yield as q =vh. vh = (lcoso - f) L. Although motion overl flows is so complicated that law resisting force is where different p, p, from S, u are usual pressure, open density, channel specific flow. entropy In practical particle application, velocity however, detonation concept products ~ respectively, with trajectory R reflected shock detonation wave D as a boundary trajectory expression F flyor resistance as anor in an boundary. open channel, Both are such unknown; as Darcy-Weisbach position R formulai state Chgzy para- ~s meters formuls on or it Manning' are governed s formula by can flow ten field be I borrowed central rarefaction as an approximution wave behind for simplicity. detonation Here wave we D use by Manning' initial stage s formula motion uniform flyor also; open position channel flow, F state average parameters velocity products overl flows is as (5) C (9) (10) (11)

5 514 LIU Qing-quan, CHEN Li LI Jia-chun v = lh2/3 sin ~ O, (12) /t where n is Manning' s roughness coefficient, 0 slope gradient surface. Then depth overl flow at length L looks like h = [ n(lcos0 - f) L] 3/5 sin0.3 0 (13) While cos0 is decreasing with slope gradient 0, sin0 is increasing, we can say from (13) that roughness, rainfall intensity, runf length infdtration rate soil all would influence overl flow depth. Generally, flow depth increases with surface roughness, rainfall intensity runf length, but decreases with infiltration rate. 2.3 The variation overl flow velocity with slope gradient We still use Manning's resistance Abstract formula for uniform open channel flow, average velocity overl flows at length L is The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic solution only when polytropic v = lh2/3 index sin ~ 0. detonation products equals to three. In rt general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock behavior Substituting Eq. reflection (13) into shock above in expression, explosive products, we derive applying small parameter purterbation method, an analytic, first-order v = 1[ n(lcoso approximate - f) solution L]2/Ssin~ is obtained for problem flying (14) plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final To simplify velocities analysis, flying plate Eq. obtained (7) can be agree rewritten very well as with numerical results by computers. Thus an analytic formula with q. two = parameters Icos0- f = high (Iexplosive f)cos0- (i.e. (1- detonation cos0)f. velocity polytropic (15) index) for estimation velocity flying plate is established. For ordinary slope gradient (less than 45~ 1 - cos0 is great less than cos0. In considered Horton' s form overl flow, rainfall intensity is much greater than soil infiltration rate, infiltration rate decreases with time. until saturated, i.e., f---~ 0. So second Explosive term at driven fight flying-plate h side technique in above ffmds formula its important could be use ignored. in study So behavior materials under intense impulsive q* loading, = /0 shock -f~ synsis (l-f)coso. diamonds, explosive welding (16) cladding In this way metals. Eq. (14) The can method be simplified estimation to flyor velocity way raising it are questions common interest. Under assumptions v one-dimensional = l[n(i - f)l]2/ssin~176 plane detonation rigid flying plate, normal (17) n approach solving problem motion flyor is to solve following system equations This formula shows that relation between overl flow velocity slope governing flow field detonation products behind flyor (Fig. I): gradient is more complex. And roughness surface, rainfall intensity runf length all may influence velocity. Generally, it will increase with rainfall intensity runf length, while decrease with ap surface +u_~_xp roughness + au soil infdtration rate. Differentiate above formula au with respect au to 1 0, we have dv 1 [n(i -f)l] 2/5 0"3c~ - 0"4sin20 (18) d0- n as as sin~176 Letdv/dO = 0, we can conclude that maximum value overl flow appears at slope is about 40.9 ~. The velocity increases with slope in scope 0 ~ ~ 40.9 ~ decreases with slope 0 greater than 40.9 ~. The relation between v ( [ n ( I - f) L ] 2/5/n )- 1 where 0 is showed p, p, S, in u are Fig. pressure, 3. density, specific entropy particle velocity detonation products respectively, with trajectory R reflected shock detonation wave D as a boundary 2.4 The variation scouring ability overl flow with slope gradient trajectory F flyor as anor boundary. Both are unknown; position R state parameters Based on it are on governed kinematic by wave flow approximation, field I central rarefaction shear stress wave behind overl flow detonation r0 is wave D by initial stage motion flyor also; ro = )'hsino. position F state parameters products (19) Considering (13), we have

6 Influences Slope Gradient on Soil Erosion 515 v 0 = 7[ n(icos0 - f)/_,]3/ssin~ 0. (20) In (20), cos0 increases with 0, whereas sin0 decreases with 0. Which shows relation between shear stress overl flow hillslope is not monotonic. The runf depth will decrease with slope gradient while energy slope will increase with slope. The comprehensive consideration preceding two factors will determine variation shear stress. Simplifying (20) by using (17), we have Differentiating (21) with respect to 0 leads to v0 = 7[ n(i - f)ll3/ssin~176 (21) d r0 L ]3/5 0.7cos sin'- 0 do - 7[ n(i - f) Abstract sin0.30cos0.40 (22) By virtue dv0/d0 = 0, we trmd maximum value shear stress appearing at slope The one-dimensional problem motion a rigid flying plate under explosive attack has an about analytic 47.2 ~ solution In or only words, when when polytropic slope index is 0 ~ detonation 47.2 ~ products scouring equals ability to three. flow In general, increases a with numerical slope, analysis while is required. scouring In this ability paper, decreases however, with by utilizing slope when "weak" shock slope behavior exceeds 47.2 ~ reflection The relation shock r0/[7[n(l in explosive - f)l] products, 3/5 ] 0 applying is plotted in small Fig.4. parameter purterbation method, an analytic, first-order approximate solution is obtained for problem flying plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final velocities 0.8 flying plate ~ obtained agree very well with numerical results by computers. Thus "~ 0.6.~ -"~'~ an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic index) for estimation velocity flying plate is established. 0.4 ~,~ 0.4 Explosive driven flying-plate technique ffmds its important use in study behavior materials under 0 intense impulsive loading, 75 shock 90 synsis 0 diamonds, explosive welding 90 cladding metals. The method 0/ ~ estimation flyor velocity way 0/ raising ~ it are questions common interest. Under Fig. 3 assumptions The relationship one-dimensional slope plane detonation Fig.4 The relationship rigid flying slope plate, normal approach solving velocity problem fo overl motion flow flyor is to solve shear stress following overl system flow equations governing flow field detonation products behind flyor (Fig. I): Furrmore, Eq. (21) shows that actual scouring ability overl flow relates also to factors such as surface roughness, rainfall intensity, runf length st ap +u_~_xp + au infiltration rate, etc. au au 1 3 Variation Erosion-Resisting Capacity St With Slope as as The erosion-resisting capacity soil is muinly dependent on factors such as characteristics soil, plant coverage surface slope gradient, etc. Actually, soil characteristics plant coverage are ten very complicated. Here we mainly focus on effect where p, slope p, S, gradient u are pressure, on density, erosion-resisting specific entropy capacity particle soil. To velocity simplify detonation problem, products we use respectively, natural binding with force trajectory soil, R rl reflected to represent shock erosion-resisting detonation wave effect D as a boundary soil, use trajectory F flyor as anor boundary. Both are unknown; position R state parameters adhesion on force it are r~ governed to represent by flow erosion-resisting field I central force rarefaction vegetation. wave behind Generally detonation we have: wave D by initial r stage 1 = f(soil motion types, flyor gradation, also; chemical position components F state in soil, parameters... ), products (23) r2 = f(plant coverage percentage, vegetation types,--. ). (24)

7 516 LIU Qing-quan, CHEN Li LI Jia-chun The gravity soil is also an important factor on soil stability. Suppose representative grain size is d, friction coefficient is No. When on horizontal ground, friction resistance caused by soil particle gravity (within range one particle diameter) is r 3 = N0(T s - ~')d, (25) where ~'6, ~" represent soil dry bulk density water bulk density respectively. On surface with slope 0, particle gravity could be decomposed to two forces with different effect on soil particle. The component perpendicular to slope has effect leading to stability, while or component parallel to slope has effect leading to instability. The friction resistance caused by gravity component perpendicular to surface is r 3 = N0(~", - ~')dcos0, (26) force parallel to surface is Abstract F = (r. - ~') dcoso. (27) The one-dimensional problem motion a rigid flying plate under explosive attack has an Combining analytic solution all factors only toger when gives polytropic rise to index scour resisting detonation force products as follows: equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock rc = rl + r2 + (Y~ - Y)d(NocosO - sin0), (28) behavior reflection shock in explosive products, applying small parameter purterbation which shows method, that an analytic, erosion-resisting first-order capacity approximate decreases solution with is obtained slope for gradient problem 0, that flying is, plate soil driven stability by decreases various high with explosives 0. Eq. (28) with also polytropic shows that indices stability or than soil but nearly on equal slope to is three. related Final velocities flying plate obtained agree very well with numerical results by computers. Thus to soil characteristics, plant coverage grain size, etc. an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic index) for estimation velocity flying plate is established. 4 The Critical Slope Gradient Soil Erosion Most artificial rainfall experiments field observations show that soil erosion is proportional to 0 within certain slope 0. While soil erosion quantity decreases with 0 as 0 exceeds Explosive a certain driven value. flying-plate Since technique complexity ffmds its influencing important factors use in study difference behavior in materials method adopted under intense by different impulsive scholars, loading, shock critical synsis slope gradients diamonds, were greatly explosive different. welding Renner cladding (1936) [4] metals. obtained The method critical value estimation 40.5 ~ flyor. CHEN velocity Fa-yang(1985) way [5] raising got it it are 25 questions ~. Horton common interest. (1945) [6] analyzed this problem oretically in Without considering soil infiltration, Under assumptions one-dimensional plane detonation rigid flying plate, normal approach he analyzed solving runf depth problem slope motion gradient flyor relation is to solve derived following out relation system between equations flow governing shear stress flow field slope detonation gradient 0 products based on behind kinematic flyor wave (Fig. approximation. I): He assumed that critical slope gradient was 57 ~. CAO Wen-hong(1993) [1~-3 furr considered effects surface roughness, soil particle size, rainfall, infiltration runf length on flow shear stress ap +u_~_xp + au found that critical slope gradient was not a constant due to influence se factors au au 1 close to 41 ~. Although se analyses described problem quite carefully, but y all ignored erosion-resisting force transport capacity flow. As a matter fact, we know from as as above analysis, that besides flow scouring ability varying with slope gradient, erosion- resisting capacity soil also would vary with slope gradient, velocity flow transport sediment would vary with slope gradient as well. Therefore three aspects where factors p, should p, S, u be are taken pressure, into density, account specific entirely entropy in analysis particle soil velocity erosion. detonation products respectively, According with to trajectory above analyses, R reflected overl shock flow detonation shear wave stress D r0, as a boundary erosion-resisting trajectory F flyor as anor boundary. Both are unknown; position R state parameters capacity on re, it are governed flow by velocity flow v field are represented I central as rarefaction wave behind detonation wave D by initial stage motion Vo = Y[ n( flyor I - also; f)l]3/ssin~ position 0cos~ F state parameters products rc = rl + r~ + (Is - l)d(nocoso - sin0),

8 Influences Slope Gradient on Soil Erosion 517 respectively. Analysis I : v = l[n(i - f)l]vssin~176 n Consider merely effect overl flow shear stress erosion resisting capacity soil. Then we have E, ~ r0- re r0 - re = 7[n(l - f)l]3/ssin~ Ocos~ - rl - r2 - ()'s - )')d(nocoso - sin0). (29) Differentiating above equation with 8, by d( r0 - r~)/d0 = 0, we may obtain following Abstract relation about critical slope gradient 0 m _" The one-dimensional 0.6sin problem 20m - 0.7cos motion 20m a rigid flying 7~ - plate 7 under d explosive attack has an analytic (sin~ solution (30) Om only -- COS0"4 when 0m ) ( NO polytropic sin0= + index COS0m ) detonation )" [n(i products - f) equals L] 3/5" to three. In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock Thus behavior reflection shock in explosive products, applying small parameter purterbation method, an analytic, first-order approximate solution is obtained for problem flying Om = F(--)'s" ), )',d,n,i - f,l,no). (31) plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final velocities It is obvious flying that plate critical obtained slope agree gradient very is well dependent with numerical on many results factors by computers. such as Thus grain an size, analytic soil formula bulk density, with two surface parameters roughness, high runf explosive length, (i.e. detonation net rain excess, velocity polytropic friction index) for estimation velocity flying plate is established. coefficient soil. If friction coefficient No takes value 0.047, we could get relation between critical slope gradient 0= synsis coefficient (Fig.5(a)). We could see on hillslope same Explosive soil characteristic, driven flying-plate larger technique surface ffmds roughness its important use in longer study runf behavior length materials greater under intense net rain impulsive excess, loading, smaller shock critical synsis slope diamonds, gradient. explosive welding cladding metals. The method estimation flyor velocity way raising it are questions common 80 interest. 80 Under assumptions one-dimensional plane detonation rigid flying plate, normal approach 70 solving problem motion flyor is to 70 solve following system equations governing flow field detonation products o / behind flyor (Fig. I): J \ 60 / f \60 50 / / / ap +u_~_xp + au au as as (a) I f 50 / t / , - 7 d Fig.5 Variation critical slope gradient (0=) with [n(l - f) L] 3/5 where p, Analysis p, S, u are II : pressure, density, specific entropy particle velocity detonation products respectively, Consider with three trajectory factors, R scour reflected ability shock flow detonation r0, erosion-resisting wave D as a boundary capacity soil trajectory F flyor as anor boundary. Both are unknown; position R state parameters re on it velocity are governed runf by transporting flow field I sediment central rarefaction v, simultaneously, wave behind ignore detonation binding wave D force between by initial grains stage rl motion concretion flyor also; force position plant coverage F r2 state 9 Substitute parameters r0, re, products v into (1). With some manepulation soil erosion rate on hillslope takes form au (b)

9 518 LIU Qing-quan, CHEN Li LI Jia-ehtm E+ = A[ 7(I - f) LsinOcosO - a_ E n( I _ f)l]z/s(7+ _ 7)( Nosin~ - sinl"3ocos~ (32) n The above formula shows that factors influencing soil erosion are very complicated. The slope gradient is just one main factors. This is essential reason responsible for great difference in critical slope gradient by scientists. Differentiate (32) with respect to 0. Following relation de,/do = 0, we obtain relation with which critical slope gradient 0,, satisfies sin ~ 0mCOS 0'6 0 m (cos 2 0 m -- Sill 2 0 m ) NocoSOm(O.3cos:~Om - 1.4sin2Om) - sinom(1.3cos20m - 0.4sin20~) 7~ - u d Abstract (33) Y [ncl-f) L]3/5" The one-dimensional problem motion a rigid flying plate under explosive attack has an We analytic alsohave solution only when polytropic index detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock behavior reflection shock 0.= in explosive products, LN0) applying small parameter purterbation method, an analytic, first-order approximate solution is obtained for problem flying That is, critical slope gradient soil erosion is a variable that varies with grain size, plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final soil velocities bulk density, flying plate surface obtained roughness, agree very runf well with length, numerical net results rain by excess, computers. Thus sou an friction analytic coefficient, formula with etc. two parameters high explosive (i.e. detonation velocity polytropic index) Taking for estimation friction coefficient velocity No flying = 0.047, plate is we established. obtain relation between critical slope gradient 0m synsis coefficient 7, 7-7 [n(i -f)l] d 3/5' as in Fig.5(b). It also shows critical slope gradient decreases with surface roughness, runf length, net Explosive driven flying-plate technique ffmds its important use in study behavior materials rainfall excess. under intense impulsive loading, shock synsis diamonds, explosive welding cladding metals. The method estimation flyor velocity way raising it are questions In synsis coefficient y+ - y d unit roughness n is s/m 1/3, common interest. 7 [n(i -f)l] 3Is' unit Under net rainfall assumptions excess is m/s. one-dimensional Actually, plane value detonation surface roughness rigid flying is about plate, 0.02 normal ~ 0.06, approach solving problem motion flyor is to solve following system equations net rainfall excess is about 20mm ~ 100mm/h, surface length is about 20m ~ 1 000m. governing flow field detonation products behind flyor (Fig. I): 2"+ - 7 d They determine value [ n( I - f) L ]3/5 which generally is quite small, about 0.03 ~ 0.3. Therefore, we conclude ap that +u_~_xp critical + au slope gradient soil erosion should be generally in range 41.5 ~ ~ 50 ~ au. au 1 5 Conclusions as as 1 ) The shear stress overl flow, erosion-resisting capacity soil overl flow velocity are three main respects determining soil erosion on slope surface. However, influencing factors should include slope gradient, rainfall intensity, infiltration rate, soil where p, p, S, u are pressure, density, specific entropy particle velocity detonation products characteristics, grain sizes, surface roughness, plant coverage runf length. respectively, with trajectory R reflected shock detonation wave D as a boundary trajectory 2) F The slope flyor as is anor important boundary. factor Both influencing are unknown; overl position flow R generation state para- soil meters erosion. on it It has are governed significant by effect flow on field net I rain central excess, rarefaction overl wave flow behind depth, detonation flow velocity wave D by shear initial stress. stage Generally motion speaking, flyor also; net position rainfall excess F decreases state with parameters slope, products flow velocity first increases with slope, reaches maximum value when slope increases to

10 Influences Slope Gradient on Soil Erosion ~, n y decrease with slope. The same situation occurs to flow shear stress only difference is it reaches maximum value at slope gradient 47.2 ~. 3) The erosion-resisting capacity soil generally decreases with slope gradient. 4) The soil erosion on slope surface does have a critical slope gradient dependent on grain sizes, soil bulk density, surface roughness, runf length, net rain excess soil friction coefficient. However, it is generally between range 41.5~ 50 ~. For surface with same soil characteristics, rougher surface or longer runf length or greater net rain excess, smaller critical slope gradient. References: [ 1 ] ZinggA W. Degree length l slope as it affects soil loss in runf[ J]. AgricEng,1940, 21 : Abstract [ 2 ] TANG Li-qun, CHEN Gno-xiang. The dynamic model water soil generation in small The one-dimensional problem motion a rigid flying plate under explosive attack has watershed[ J]. Journal Hydrodynamics, Serials A, 1997,12 (2) : (in Chinese) an analytic solution only when polytropic index detonation products equals to three. In general, [ 3 ] Yair a numerical A, Klein analysis M. The is influence required. In surface this paper, properties however, on flow by utilizing erosion processes "weak" on shock debris behavior covered slopes reflection in an shock arid area[j]. in explosive Catena,1973 products,,1(1) :1-8. applying small parameter purterbation [ 4 ] Renner method, F G. an Conditions analytic, first-order influencing approximate erosion solution boise river is obtained watershed[ for Z]. V problem S Dept Agric flying Tech plate driven Bull, by 1936,528. various high explosives with polytropic indices or than but nearly equal to three. Final [ 5 ] velocities CHEN Fa-yang. flying The plate experiments obtained agree effects very well with slops numerical on soil results erosion[ by J]. computers. Chinese Soil Thus an analytic Water formula Conservasion, with two 1985, parameters (2) : high (in explosive Chinese) (i.e. detonation velocity polytropic index) [ 6 ] for Horton estimation R E. Erosional velocity development flying streams plate is established. ir drainage basins, hydrophysical approach to quantitative morphology, Geol Soc Amer Ball, 1945,56(3) : [ 7 ] Singer M J, Blackard J. Slope angle-interrill soil loss relationships for slopes up to 50% [J] 9 Soil Sci Soc Am J,1982,46(6) : [ 8 ] Explosive ZHANG driven Ke-li, Hosoyamada flying-plate K. technique Influence ffmds slope its important gradient on use interdll in erosion study shirasu behavior soil plays materials [J]. under Cond intense Plant impulsive Growth Jpn, loading, 1996, (73) shock :37 synsis diamonds, explosive welding cladding metals. The method estimation flyor velocity way raising it are questions [ 9 ] Emmett W W. Overl flow [ A ]. In: Kirkby M J Ed. Hillslope Hydrology [ C ]. John: common interest. Wiely Sons, Under assumptions one-dimensional plane detonation rigid flying plate, normal [ 10 ] Wolhiser D A, Liggett J A. Unsteady one-dimensional flow over a plane- rising hydrograph[ J]. approach solving problem motion flyor is to solve following system equations governing Water flow Resoyr field Res, detonation 1967,3(3) :753 products behind flyor (Fig. I): [11] LimaJLMP. Model KINqNF for overl flow on pervious surface[a]. In: AJParsonsEd. Overl Flow[ C]. UCL Press,1992. [ 12 ] CAO Wen-hong. On study ap +u_~_xp critical + slope au soil erosion[ J]. Soil Water Conservasion Bulletin, 1993,13(4) : 1-5. (in au Chinese) au 1 as as where p, p, S, u are pressure, density, specific entropy particle velocity detonation products respectively, with trajectory R reflected shock detonation wave D as a boundary trajectory F flyor as anor boundary. Both are unknown; position R state parameters on it are governed by flow field I central rarefaction wave behind detonation wave D by initial stage motion flyor also; position F state parameters products