Horton 1945 Erosional Development of Streams and Their Drainage Basins Hydrophysical Approach To Quantitative Morphology PDF

 

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EROSIONAL DEVE DEVELO LOPM PMEN ENT T OF STREAMS AND THEIR DRAIN AGE BASINS; HY HYDR DROP OPHY HYS SIC ICAL AL APPROACH TO QU QUAN ANTI TITA TATI TIVE VE MORPHOLOGY BY R O B E R T E. H O R T O N CONTENTS   ge

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A ckno wled gm ents. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of sy mb o ls used P l a y f a i r s la l a w. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   Quantitative p h y s i o g ra p h i c f a c t o r s G e n e ra l c o n si d e r a ti o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  

277 279 279 280 281 281

Strea m orders , D r a i n a g e d e n s i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Length of o v e r la n d flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   S t r e a m f r eq u e n c y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   Co Compo mposi siti tion on of draina drainage ge net . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   Laws of d r a i na g e c omposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   en g gtt h of s ttrr ea ea ms ms of a given order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   Total l en Channel Cha nnel-st -storag oragee capacity ...........  Gene Genera rall equat equation ion of co comp mpos osit itio ion n of stream s ys te ms Re Rela lati tion on of size of drai draina nage ge area to stream o r d e r Law of stream slopes. . . . . . D e t e r m i n a t i o n of o f ph ph ys i o gr a p h ic fa f a c t or s ffo or d drr a i n a g e ba ba s in s Relation Relat ion of ge geol olog ogic ic structures to d r a i n a g e composition , I n f i l t r a t i o n t h e o r y of s urfa c e runoff. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   General statement .

281 283 284 285 285 286 291 292 292 293 295 295 300 306 306

Inf iltra tion-c a pac ity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   Ov e r l a n d or s h e e t flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Law of o ve r l a n d flow. . . . . . . . . . . • • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .  Index of t u r b u l e n c e o v e r l a n d fl f low. . . . . . . . . . . . . . . . . . . . . . . . .. .. . .. .. .. .. .. .. . . .. .   Types of ov Rain-wave trains . . . . . . . . . . . . . . . . . . . . . . . .  . Profile of o v e r l a n d flow. . . . . . . . . . . . . . . . . . . . Sur f a c e erosion by o v e r l a n d flow , , Soil-erosion processes R e s i s t a n c e to erosion , Eroding force . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... ... .. .. .. .. .. Critica Crit icall len length gth xe Belt of no er eros osio ion n : Erosion rate Total erosion an d erosion depth .erosi sion on to s ~ o p e l e n g t h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     e ~ t ~ o n of .ero

307 309 309 312 312 313 314 315 315 317 319 320 324 324 325

Abstract

Ra m intensity an d erosion Tr ans ans p por or ttatio ation n and s e d i m e n t a t i o n

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HORTON EROSIONAL D EV EV E EL LO OP PM ME EN NT T OF S T TR RE EA AM S

Or Orig igin in and deve develo lopm pmeent of stre stream am sy syst steems and and their their vall valleeys by aque aqueou ouss erosion ... Rill cch hannels and rilled su s ur f a ce .. .... .... . . . . . . . . . . . .. .. .. . . . . . . . . . Origin of of rriill cch hannels. . . . . . . . . . . . . . . . . . . .................. Cross grading an and mi m icropiracy. . . . . . . . . . . ............... Hydr ydrophy physi siccal basis of geometric sseeries laws of stream numbers and stream lengths.. General statement ................. . .. . . . . . . . . . . . . . . . . . . . .

331 331 332 333 339 339

F i r s t stage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Second s t ag e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. . . . . . . . Subsequent stages . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adventitious st s t reams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stream de deve velo lop pment ment with progressiv sively increasing land exposure comp compet etit itio ion. n... . . . . . . . o f ssttream development. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . En d p o in t of Stream en entrance aan ng l e s . . . . . . .. . . . . .. . . . . . . .. . . . .. . . . . . .. . . . . .. . . . . . . .. . Dr a i n a ge p a t t e r n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asymmetri Asym metrical cal drainage drainage patterns . .. .. .. .. . . .. .. .. .. .. .. .. .. .. .. .. .. .. . . .. Perched or sidehill str eams R ej uve na ted str eams; epicycles of eerrosion.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D r a i n ag e b a s i n t o p o g r ap h y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... Mar gi na l belt of of no erosion; gradation of divides Interfluve hills and pla tea us Concordant st r e am and valley junctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

339 339 341 341 342 346 349 350 352 352 352 355 355 360 362

St re a m v a l l e y g r a d a t i o n Typical ovoid forms of drainage basins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Development of large drainage basins Davis stream erosion cycle References cited. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  

363 365 366 366 369

ILLUSTRATIONS Page

Figure

1. 2. 3. 4. 5. 6.

Well drained basin Flat san sandy dy area area poorly poorly drained . Bi Bifu furc rcaati tion on or relation ofstr fstream order to number ofstrea fstreamsin msin different drainage basins Relation of str e a m lengths to stream order in different drainage basins . Di Diag agra ram m of factor   p   . . Law of stream slopes . .

7. Drainage net upper Hiwassee River. . . 8. Graphical d deetermination o off stream characteristics. . . 9. Drain inaage pa patt ttee rns rns ofLa fLaurel Fork and Glady Fork C h hee at at River drainage basin . 10. St S t re am numbers an and st stream le l engths . 11. Drainage p a t t e r n s. . . . 12 12.. Relation Relation ofsurfac ofsurfacee runo runoff ff intensity  q, to av aver erag agee depth ofsurf ofsurfac acee dete detentio ntion n  13. Ha H alf section of o f a ssm mall d drrainage basin . 14. Ha Half pr profile of a va valley sl slope. . . . . 15. 15. Hort Horton on sl slop opee fu func ncti tion on fo forr su surf rfac acee eros erosio ion n  16 16.. Gr Grad adie ient nt and de degr greee of eros erosio ion n  17 17.. Re Rela lati tion on of erosi rosion on to slop slopee le leng ngth th   18. Relation of in i nitial infiltration ccaapacity to erosion . 19. Erosio ion n of sod sodded area in init itia iate ted d by the br brea eaki king ng down of grass cover in inten tense rains   20. Successive st stages of ri r ill channel development. . . . . . . . . . . . . 21. Port Portio ion n of Moon Mou ount ntai ain n Ariz Arizo ona California qu quad adra rang ngle le U. S. G. S . 23 2.. S Duecvceelsospivmeesnttago of vfarlillel yob by blyitcr cerraotsisongrading . 2 efsaova 24. Deve Develo lopm pmen entt of a stre tream from a rill system by cross grading  

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. 282 . 283 . 288 289 294 . 295 297 299 301 303 304

311 314 316

321 322 326

327 328 333 334 336

337 338

 

ILLUSTRATIONS

 

25. 26. 26. 27. 28. 29. 30.

De Development of a drainage net in a strea m basin Begi Beginn nnin ing g of er eros osio ion n on newl newly y expo expose sed d land .. . . . . . . .   . .   . . . . . . . . . . . . . . . . . . .... . . .   Developmen t of of ffiirst pa pair of tr t ri b u tar i es on on n neew st st r e am ssy ystem Lines of flow a ft ft eerr cross-grading of first pair of t ri ri b bu u t aarr y areas. . . . . . . . . . . . . . . . . . . . . . . .   De Development of lower p air s of main tr trib utaries Fi n al development of two a d j a c e n t drainage basins on newly exposed land

340 342 343 344 345 346

31. 32. 33. 34. 34. 35. 36. 37. 38. 39. 40.

End p o in t of str eam development End p o i n t of a definite str eam channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   Drainage basin of Pen n yp ack Creek. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   Cen Centrip tripetal etal dr drain ainag agee pattern   . ........ Perched or hillside stre am. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   Belts of no erosion Be Belt lt of no erosion a t a cross divide , ..  .... To pogr aphy of an interior cross divide. . . . . . . . . . . . . . . .. ... .. .. .. .. .. .. ... .. .. .. . Origin of ungraded or p ar tial ly graded interfluve hills and p lateau s. . . . . . . . . . . . . . . . . . . Gradation of ssttream valley. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

347 347 348 351 353 356 358 360 362 363

ABSTRACT The comp compos osit itio ion n of the the stream stream syst system em of a drain drainag agee basi basin n ca can n be ex expr pres esse sed d qua quanti ntitati tatively vely in term termss of st stre am or drain de bi bifu furca ratio, and strea lengt ra rati o.iess are of the 1st order; Stre Stream ream amorde oder, rder,rsdr aainag re age soe cdens honsit seity, ny, . that trcati he tion fon ingrat ertiio, p oan r dunst breamra ranm-len ch ched ed gth trib trh ibu u tar ttio. arie stream streamss whic which h re rece ceiv ivee 1s 1stt orde orderr tri tribut butarie aries, s, bu t these only, are of the 2d order; third order stre stream amss re recceive 2d or 1st and 2d order tr trib ibut utar arie ies, s, and so on, until, finally, the main stream is of the high highes estt ord rdeer and ch char arac acte teri rize zess the the order rder of the drai draina nage ge basin asin.. Two Tw o fundam fundament ental al laws laws co conn nnec ectt the the numb number erss an and d leng length thss ofstream ofstreamss of diff differ eren entt orde orders rs in a dra draina inage ge basin: (1) The law of stream numbers. This expresses the rela relati tion on between the nu numb mber er of stre stream amss of a give given n orde orderr an and d the the st stre ream am orde orderr in term termss of an inve invers rsee geom geomet etri ricc se seri ries es,, of whic which h the the bifu bifurca rcati tion on ra rati tio o rb is th thee base base.. (2) The law of st stre ream am len length thss expresses the average len length of strea tream ms of a given order in terms of st stre ream am order, average len leng th th of strea tream m s of the 1st order, and the strea tream m-le -length ratio. This law tak es the form of a direct geometric series. These two laws extend Playf aaiir s law an d give it a quantitative mean meanin ing. g.   heinfilt infiltrat ration ion theo theory ry of surf surfac acee runo runoff ff is base based d on tw two o fundam fundament ental al co conc ncep epts ts:: rate (1) Th There ere is a maximum or li limi miti tin ng a t wh whic ich h th thee soil, oil, wh when en in a given iven co con nditio ition n, ca can n ab abssorb orb rain as i t falls.   hi sis th thee inf infilt iltrati rationon-cap capacit acity. y.   t is a volume per unit of time time.. (2 (2)) When ru run noff takes takes place lace fro from any so soil il su surf rfac ace, e, the there re is a def efin init itee func functi tion onal al rela relati tion on betw etwee een n th e depth of surf surfac acee detent detention ion li or the qu quan antit tity y of wate waterr accu accumu mula late ted d on the soil surface, and the rate q of su surf rfac acee ru run noff or ch chan anne nell in infl flo ow.

For a given te terr rrai ain n there is a minimum len leng tth h Xc of over overlan land d flow flow requir required ed to prod produc ucee suff suffic icie ient nt runoff volume to in init itia iate te erosion. The crit critic ical al leng length th x depen depends ds on surf surfac acee slop slope, e, runo runoff ff int intens ensity ity,, infiltra infi ltration tion-cap -capacit acity, y, and and resistiv resistivity ity of th e soil to erosion. Th This is is the most impo importa rtant nt single factor involved in erosion ph phen enom omen enaa and, in part partic icul ular ar,, in connection with the de deve velo lopm pmen entt of stream syste systems ms an and d their their draina drainage ge basi basins ns by aqueo aqueous us eros erosio ion. n. Th Thee er eros osiv ivee fo forc rcee an d the rate a t wh whic ich h eros erosio ion n can can take take plac placee a t a dist distan ance ce x from from the the wa water tersh shed ed line line is directl directly y pr prop oport ortion ional al to the the runo runoff ff intensit intensity, y, in inch inches es-p -per er hour hour,, the the distan distance ce x a func functi tion on of the the sl slop opee an angl gle, e, and and a propo proporti rtiona onalit lity y fact factor or K whi hich ch repr repres esen ents ts the the quantit quantity y of ma mate teria riall wh whic ich h ca can n be torn orn lo loo ose an and d ero eroded per unit unit of ti tim me an and d su surf rfac acee area, rea, wit with unit runo runoff ff intens intensity ity,, slop slope, e, an and d ter terrai rain. n. The rate of erosion is the qu quant antity ity of mate materi rial al actu actual ally ly removed from the soil surface per unit of time an d area, an d this may be governed by eith either er the tran transp spor orti ting ng power of ov over erla land nd flow or the actual rate of er eros osio ion, n, whic whiche heve verr is smal smalle ler. r.   the q uan uanti tity ty of mater ateria iall torn loose and carried in susp suspen ensi sion on in overl overlan and d flowexc flowexcee eeds ds the the quantity whic which h can can be tra transp nspor orted ted,, de depo posi siti tion on or se sedi dime ment nta a tion on the soil surface will take place. On new ewlly ex exp posed osed terr terrai ain, n, resu result ltin ing g, for for exam xampl ple, e, fro from the the rece recess ssio ion n of a co coas astt lin line, shee sheett er ero osion ion occu occurs rs fi firs rstt wher wheree the the di dista stanc ncee from from the the waters watershed hed li line ne to the the co coas astt line line firs firstt exce exceed edss the the crit critic ical al len length gth Xc an and d sheet sheet ero erosi sion on sp spre read adss latera laterall lly y as th thee wi width dth of the the ex exp posed osed terrai terrain n incr increa ease sess. Eros rosion ion of su such ch a ne newl wly y expo expose sed d plane surf surfac acee init initially ially de deve velo lops ps a seri series es of shal shallo low, w, clos closee-sp spac aced ed,, shoest shoestrin ring g gull gullie iess or rill channels. The rills flow parallel with or are consequent on the original slope. As a resu lt of var ario iou us ca cau uses ses, th thee div ivid ides es betw betwee een n adj adjace acent nt rill rill ch chan anne nels ls are are brok roken down loca locall lly y, an and d the the flo flow in the the sh shal allo low wer ri rilll ch chan ann nels els more more rem remote ote fro from th thee in init itia iall ril rill is dive divert rted ed into into deep deeper er rill rillss more clos closel ely y

 

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DEVE DE VELO LOPM PMEX EXT T OF STREAMS

adjacent thereto, an d a ne new w system system of rill rill chan channe nels ls is de deve velo lope ped d having having a di direc recti tion on of flo low w a t an an angl glee towa ward rd the in init itia iall rill. Thi Thiss is called cross to the in init itia iall rill ch chan anne nels ls and prod produc ucin ing g a resultant slope to

grading.

With pro progr gress essiv ivee expos exposure ure of new new terrain terrain,, streams dev develo elop p fir first st a t poi points nts whe herre th thee len length gth of over ver land flow flow firs firstt exce exceed edss the the critic critical al leng length th Xc an d streams starting a t th thes esee poin points ts gene genera rall lly y beco become me th thee primary or h i g h ~ s t o r d e r s t r ~ s of the u lt lt im im aatt e d rai na g gee b ~ s i n s ev el el o op pm men entt of a rilled Th e d ev

surf surfac acee on each side of the mam stre stream am,, followed by cro cross ss--gr grad adiing, crea create tess late latera rall slopes to towa ward rd the an d

tributary

aalt length mai ain ne st strea m, gth ofoov n erlan thes ese e dsfllow opw esin the stre stream ams s dsl eope veelodi p, rect us usua uall yfi o nteexce oceed n eds eith either iditic e,ical points wh wher ere the thream, e length len overl and lo the new resu resulta ltant nt slop dire ctio ion nlly firs rst ex s ter he scrit cr len gth x

Cross-grading an d recr recros osss-gr grad adiing of a given po port rtio ion n of the area will co cont ntin inue ue,, acco accomp mpan anie ied d in each case by the d ev ev el el op op men mentt of a new o rd rd er er of tributary st stre ream ams, s, until until fi fin nall ally the length length of over overla land nd flo flow wit withi hin n the the rema remain inin ing g are areas as is ev ever eryw ywhe here re less ess than the crit critic ical al leng length th xc· These proc proces esse sess full fully y accou acc ount nt for the ge geom omet etrric-s ic-ser eriies laws of stre stream am nu numb mber erss an and d st stre ream am leng length ths. s. A bel beltt of no erosion exists ar arou ound nd the margin of each drai draina nage ge basin and in inte teri rior or suba subare reaa while the deve develo lopm pmen entt of the stre stream am syst system em is in progress, an and d this bel beltt of no erosion finally covers the en enti tire re area wh when en the the str stream eam de deve velo lopm pmen entt beco ecome mess comp compllete. ete. Th e dev develo elopm pment ent of inter interio iorr divi divide dess be bettween een sub subord ordin inate ate str stream eamss takes akes place lace as th thee resu result lt of com com petitive eros erosio ion, n, an d such divides, as well as the ex exte teri rior or divide surr surrou ound ndin ing g the dr drai aina nage ge basin, are ge gene nera rall lly y si sin nuou ouss in plan plan an d profile as a re resu sult lt of co comp mpet etit itiv ivee erosion on the two sides of the divide, with th thee ge gen neral eral res result ult t h at isol isolated ated hill hillss co comm mmon only ly occu occurr alon along g di divi vide des, s, parti particular cularly ly on cros crosss di divi vide des, s, a t their junctions junctions with long longit itud udin inal al divi divid des. es. Th These ese in inte terf rflu luve ve hill llss are are no t un uner erod oded ed areas, as th thei eirr summits ha d been subj subjec ecte ted d to more or less ess re repe peat ated ed cro cross ss-g -gra rad din ing g pr prev eviious ous to the de deve velo lopm pmen entt of the div ide on which t he he y are l ocate d d.. With in incr crea ease sed d exp expos osu ure of ter terrai rain n weake eakerr str stream eamss may may be abso absorb rbed ed by the st stro rong nger er,, larg larger er st strea reams ms that by com compet petiti itive ve er eros osio ion, n, an d the drai draina nage ge basi basin n grows in wi widt dth h at the same time it in incr crea ease sess in le leng ngth th. . any Ther Th e oismajor , how vereams r,ams, alw,ay ays s a tria tr ular arthat area areath ofedi dire rect dr drai aina nage toaina tnage hege cobaasi stinlinise us in inte term rmed iate with thiang e ngul be betw tween een an yere tw maj orestre st result fi fin n alctform fo rm ofge a drai dr as usua uall lly yedia ovo voi ite d or p e a r - s h a p e d . . The drai draina nage ge basi basins ns of the firs firstt-or ord der trib tribut utar arie iess are the las lastt developed on a given area, an d such str stream eamss often often have ave stee steepp-si side ded, d, v-sh v-shape aped, d, incise cised d chan channe nels ls adjo adjoin ined ed by bel belts ts of no ero erosi sio on. Th e end point of stream stream dev devel elop opmen mentt occur ccurss whe hen n the tributar tributary y su subar bareas eas hav avee been een so co comp mple lete tely ly subd subdiv ivid ided ed by succ cceessive orde orders rs of stre stream am deve develo lopm pmen entt that t h e re re nowhere r emains a l en g t h of overlan over land d flow flow ex exce ceed edin ing g the the critic critical al length length xc Str Stream eam chan channe nels ls ma may y, howe wev ver, er, cont contin inue ue to de dev velop elop to some extent thro through ugh head headward ward eros erosio ion, n, bu t s tr tr ea ea m chan nels do not, in general, ex t en d to the watersh wat ershed ed li line ne.. Valley an d stre stream am de deve velo lopm pmen entt occur toge togeth ther er and are closely rela relate ted. d. At a given cross secti ction th e valley can can n no o t grade below the str str eam eam,, and the valley supplies the runoff an d sedi sedimen mentt whic which h t og og et et her her d et et eerr mi mi ne ne the valley and str str eam eam profiles. As a r esu esullt of cross-grading ant ant ece eced d en en t to the d ev ev el el op op m men entt of new tr ib ibu tar tarii es, es, the t rrii bu bu tar tarii es es and t h hei eirr valleys are con concor cor d dan antt wi witt h the parent stream an d valley a t the time the new stre stream amss are formed and rema remain in co conc ncor orda dant nt th ther erea eaft fter er.. Val allley cros crosss sect sectio ions ns,, when grad gradin ing g is comp comple lette, an d except for fi firs rstt-or orde derr tr trib ibut utar arie ies, s, are gen er eral allly ss-sh shap aped ed on each side of the str str eam eam,, with a p oi oi n ntt of con contrafl aflexu exure on the u pp pp er er p or or ttii o on n of the sl slo ope pe,, and and downs ownslo lope pe from rom thi this point point the the fin final form orm is determ determin ined ed by a co comb mbin inat atio ion n of fact facto ors rs,, in inclu clud d ing er eros osiion rate, rate, tra trans nspor portin ting g pow ower er,, and and the the rela relati tive ve freq frequ uenci encies es of oc occu curr rren ence ce of st stor orms ms an and d run unof offf Th e lo ng it udi nal profile of a valley along the s ttrr ea of diff di ffere erent invtens itie aven ndn tlo hecatio cion. ron. ss tnakgi se sect cti ion ofnt th theeinte alle alnsit ley yies. are ars.e clo closel sely relate related, d, an d both are rela relate ted d to the resul resultan tantt esalom pebaaan give locat M an an y ar e aass on which meager s tr tr e aam m dev el op men t has t a ke ke n place, and which are commonly clas asssified as yo yout uthf hful ul,, are really ma matu ture re,, because the end poin pointt of st stre ream am deve develo lopm pmen entt and erosion for ex exis isti ting ng co cond ndit itio ions ns has alre alread ady y been reac reache hed d. When the end p o oii nt nt of s tr tr e aam m and valley g r ad ad aatt i o n has ar r i v ed in a given dr ai nag e basin, the rema remain inin ing g surface is us usua uall lly y concave up upwa ward rd,, more or les esss reme rememb mbllin ing g a segm segmen entt of a pa para raba balo loid id,, ribbed ribbed by cro cross an and d long longit itud udin inal al divi ivides des and and cont contai aini ning ng in inte terf rflu luve ve hi hill llss an d p llat atea eau u s. s. Thi Thi s is called a  graded su surf rfac ace, e, and and it is su sugg gges este ted d that the the term term peneplai peneplain n iiss not not appropri appropriate, ate, si sinc ncee th this is su surf rfac acee is ne neit ithe herr a plan planee nor ne near arly ly a plan anee, nor does it ap appr proa oach ch a plane as an ul ulti tima mate te li limi miti ting ng form. Th e hyd hydrop rophys hysica icall con concep cepts ts app applie lied d   str str eam eam and valley d ev ev el el o op p men mentt acco accou u nt nt for o bse bserr ved ved phenomena phen omena from from th e time of exposure of the t e rr rr a iin n . De ta ta il il s of these p h en o men a of s ttrr ea ea m an d va vall lley ey devel develop opmen mentt on a given iven area ma may y be mod odiified fied by geol eologi gicc struct structure uress and and sub subseq sequen uentt geo eollogi gicc change cha nges, s, as we well ll as loca locall vari variatio ations ns of inf infilt iltrat ration ion-cap -capaci acity ty an and d res resist istanc ancee to eros erosio ion. n. thiss pape paperr stre stream am de deve velo lopm pmen entt an d dra draina inagege-bas basin in topograp topography hy are are co cons nsid ider ered ed wh whol olly ly from from th thee In thi vi view ewpo poin intt of the op oper erat atio ion n of hydr hydrop ophy hysi sica call processes. In co conn nnec ecti tio on wi with th the Da Davi viss erosion cy cyccle th e sa same me subjec subjectt is treated treated larg largel ely y wit with h refer eferen ence ce to the effe effect ctss of ante antecede cedent nt geol eolog ogiic cond condit itio ions ns and and subseq uent geolo geologic gic cha chang nges. es. Th e two views ews bear much much the same same rela relati tion on as tw two o pic pictu tures res of th e same obje object ct ta taken ken in d i f f e r e ~ t ligh lights ts,, and one supp supple leme ment ntss the ot othe her. r. The Da Davi viss erosion cycle is, in eff effect, partial system the aigin t leas usually usua lly ass umed toeam begi begin n after the theent dev develo le ast e hydr hydrophy ophysical sical co conc ncep eptt assume ca carr rrie iessdstr stream deve de velo lopm pmen t belopmen ackpment to thteofor orig inal alt naewly expstream osed surface. th

 

ACKNOWLEDGMENTS

  9

ACKNOWLEDGMENTS

The au auth thor or is ind ndeebte bted to Dr. Howard A. Meyerhoff for many many helpful sugges tions and criticisms. Gra ratteful acknowledgment is also given Dr. Alfred C. Lane, who, wh o, mor moree than 40 years ago, gave the auth author or bo both th the incentive and an op oppo port rtun unit ity y to begin the stu study dy of dr draaina inage basins with respect to possible in inte terr rrel elat atio ions ns of th thei eirr hydrau hyd raulic lic,, hy hydr drol olog ogic ic,, hy hydro droph phys ysica ical, l, an and d ge geol olog ogic ic fe feat atur ures es.. LIST OF SYMBOLS USED A

of dr draa ina inage ge ba bass iin n in square miles. slopee an angl gle. e. a = slop distancee from from stream tip to wa wate ters rshe hed d line. c   distanc o   depth of sheet flowin flowin inc inches hes a t th e stream mar margin gin or at th e foot of a slope length to 0 average depth of sur urfa facc e de dete tent ntio ion n or ov ovee rrla land nd flow, in inc inches hes , on a unit strip of len length gth to flow in in inche chess a t a dis distan tance ce x from t h e cr est est of the slope or wa water tersh shed ed line. depth of sheet flow Dd   drai draina nage ge den densi sity ty or av aver erag agee len length gth of str strea eams ms p er area. a. er u n it it of are energy gy expe expend nded ed by fric fricti tion onal al re resi sist stan ance ce on soil surf surfac ace, e, It lbs per sq. ft. per sec. e   ener averag agee eros rosion over a given strip of unit width an d length to p e time..   o   aver err u n it it of time mater terial ial,, pre prefe ferab rably ly expr expres esse sed d in ter terms ms of depth of solid olid mat materia erial, l, e;   erosion rate or quantity of ma remove rem oved d pe perr hour by sheet erosion. total erosion   total solid m a ate teri riaa l re rem mov ovee d from a given s ttri rip p of unit length per unit of time time.. E begi ginn nnin ing g of rain, inches per hour. f   infiltration-capacity at a given time t from the be inim mu um m infi infilt ltra rati tion on-c -caa pa pa ccit ity y for a given te terr rraa iin. n. fe   m ini infiltration ration-capa -capacity city a t be begin ginnin ning g of ra rain in.. fo   initial infilt F  eros erosiv ivee forc forcee of ove overla rland nd fl flow ow,, lb lbs, s, pe perr sq sq.. ft ft.. F 0   tractive forc forcee of over overlan land d fl flow ow,, lb lbs. s. pe perr sq sq.. ft ft.. of su surf rfac ace. e. stream stre am freq freque uenc ncy y or number ofstream ofstreamss per per unit area. F i   rain intensity usually in cchhes per h oouu r. r. turbul ulee nc nc e or pe perc rcee nta ntage ge of the area covered by s he he eett flow on wh whic ich h th e fl flow ow is turI   index of turb bulent. coef effi fici cien entt in the the runo runoff ff equa equation tion where here 0 is used i n nss ttee aad d of 0 as th e depth of sheet flow. K;   co k pr prop opor orti tion onaa lit lity y fa facc ttor or re requ quir iree d to c onv onvee rrtt th e rate of pe rrff or or ma ma n ncc e of work in sheet erosion into into equi equivale valent nt quantity of m aate teri riaa l re rem mov ovee d per unit of tim time. prop opor orti tion onaa lit lity y fa facc ttor or which de dete term rmin inee s the time f e re requ quir ired ed for infi infilt ltra rati tion on-c -cap apac acit ity y to be K j = a pr redu reduce ced d from rom its in init itia iall value fo to i t s co nst nst an an t value f corres cor respon pondin ding g co coef effi fici cien entt (t (to o K in the e q qu u aatt iio o n for l am am iin n ar ar o ve ve rrll an and flow. K K .   constant or pr prop opor orti tion onaa llit ity y fa facc ttor or in e qua quati tion on exp xpre resssing ing runoff inte intens nsit ity y in term termss of depth 0 of ov over erla land nd flo flow. l«   a ve ve ra ra ge ge leng length th of s ttre reaa m mss of or orde derr o to   ma maxi ximu mum m leng length th of over overla land nd flow on a given area. en g gtt h of o v vee rl rl an an d flow or l een ng gtt h of flow over the g rro o un un d surface before the runoff becomes to   l en conce con centr ntrate ated d in defin iniite st strea ream m ch chan anne nels ls.. age l en en g gtt h hss of s ttrr eeaa m mss of 1st an d 2d orde rders rs,, etc. t t2 etc.   av eerr ag  L = extended stream le leng ngth th m eeaa ssur uree d along s ttre reaa m from outl outlee t and e xte xtend ndee d to wa wate ters rshe hed d line. total leng le ngth th o f trib tr ibut uta a r rie ies s o f or orde der r o Lo   exponent ent in th e equation: q expr pres essi sing ng the runoff inten intensi sity ty in term termss of depth of sheet J1   expon o M   ex flow along the s tre treaa m ma rrgi gin. n. n   su surf rfac acee roug roughn hnes esss fa fact ctor or,, as in the Manning formula. Y   o   number of s t re re a m mss of a given or d er in a d r ai ai n a g e bas in . total number number of strea streams ms in a dra drain inag agee basin. N v   re aam m s of 1st, 2d orders, etc. N 2 etc.   total number of st re   area

 

 

 

280 o q

R. E . HORTOK EROSlONAL DE DEVE VELO LOPM PMEN ENT T OF STREAMS

order of a given stream. = surface-runoff intensity usually in inch ches es per per ho hour ur.. qi = runof unofff int intens ensity ity in cu cubi bicc feet feet per per seco second nd from from a unit st stri rip p 1 foot wide and and with a slope le leng ngth th lo p = stream length length ratio/bifu ratio/bifurcati rcation on rati ratio o = Tl Tb bifu furc rcat atio ion n ratio or ratio of the average nu numb mber er of br bran anch chin ings gs or bi bifu furc rcat atio ions ns of st stre ream amss of a Tb = bi gi give ven n or orde derr to that of st stre ream amss of the next ext lower order.   is usually consta constant nt for al alll orders of =

str streams eams in a given basin. str str eam eam leng length th ratio or ratio of average le leng ngth th of st strreams eams of a given order to that of st stre ream amss of th thee n nex extt lo lowe werr orde order. r. I   = stre stream am leng length th rati ratio o using exte extend nded ed stre stream am leng engths. T = ratio of cha channel slope to ground slope for a given str trea eam m or in a given drainage age basi sin n, = se/seR = ini initia tiall su surf rfac acee resi resist stan ance ce to sh shee eett eros erosio ion, n, lb lbs. s. per per sq sq.. ft. R; = su subs bsur urfa face ce resi resist stan ance ce to sh shee eett ero erosi sion on,, lb lbs. s. per per sq sq.. ft., t., or resis esista tan nce of a lo lowe werr su surf rfac acee or ho hori rizo zon n of th thee soil to erosion afte afterr th thee surface ace laye ayer of resista tan nce R is re remo moved ved.. f. 5/ 5/3, 3, the larger exponent can readily be explained if part of the ove verl rlan and d flo flow is la lami mina narr flow, and this is quite certa rtain to occur where the flow is alte tern rnaate tely ly in thin thin films on ridges in the form of laminar flow, and thro throug ugh h depressi sio ons wholly or partly partly as turb turbul ulen entt flow. An inde index x of turb turbul ulen ence ce app ppli liccable to such a case is expressed by the equ quaati tion on::  

t

 3 -

M

27)

the flow is fully turb turbul ulen entt and   = 5/3, 5/3, this gives  = 1.0.   the flow is fully lamina lam inarr and and   = 3.0, .0, this this gives ives   = O. By tr tran ansp spos osit itio ion, n, the the runo runoff ff expo expone nent nt   can be expressed in terms of the index of tu turb rbul ulen encce:  

  =

3-

 l

28

Turbulence in overland flow increases downslope from the watershed line. In labor lab orato atory ry and fie ield ld-p -plo lott expe peri rime men nts with plo plott len lengths gths lo us usua uall lly y less less than 25 feet feet,, the flow over surfaces with withou outt vegetal cover is usually pa part rtia iall lly y turb turbul ulen ent. t. On long naturall sl natura slop opes es,, wi with th   much greater frequently lO feet or more the flow flow is full fully y tu turb rbul ulen entt except for extremely slight de dept pths hs or close to the head of the slope. TYPES TYP ES OF OVE VER RLA LAND ND FLOW FLOW

The stu The study dy of ov oveerla rland flo flow in accorda rdance wit ith h the the infi infilt ltra rati tion on theo theory ry ha hass rev revealed led vario rious phe hen nom omeena of mic icro roh hydra drauli liccs not com omm mon only ly pres presen entt in ordi ordina nary ry cha chann nneel flow.

Parti Pa rtiall ally y turbu turbulen lentt flow low desc scri rib bed may be con onsi sid dered red mixe ixed flo flow.

In gene nera rall it

cons consis ists ts of turbu turbulent lent flow flow in inte ters rspe pers rsed ed wi with th lamin laminar ar flow. If the area on which flow occurs is covered with grass or othe otherr close-spaced vege ta tati tion on,, the flow may be subd subdiv ivid ided ed.. Part of the the ener energy gy avai availa labl blee for for ov over erco comi ming ng resistance is expended on the grass blades and stems, reducing the am amou ount nt of energy ava vail ilab able le for expenditure on the soi soil surface. For the limiting con ond dit itio ion n of complete subdivisi sio on of the flow, all the resist staance to flow would be due to the ve vege geta tati tion on,, and the law of overland flow would be:  29)

The velocity of overland flow would be sensibly cons consta tant nt regardless of the de dept pth h of surface de dete tent ntio ion. n. Some experiments show subs substa tant ntia iall lly y this condition. Because of the increased resistance, the de dept pth h of surface de dete tent ntio ion n required to carr carry y a given ra rate te of runoff is very gre greatl tly y increased, and the velocity of overland flow is corre-

 

I N

I L T R

T I O ~

THEORY OF S U R F A C E RUNOFF

 

sp spon ondi ding ngly ly decr decreeased ased wher wheree there here is dens densee co cove verr of gr graass ss,, gr grai ain, n, or sim similar ilar ve vege geta tati tion on.. This is an impo import rtan antt fact in relat elatio ion n both both to surface runoff and soil erosion.   t is oft often foun found d in run unof offf-plo -plott expe experi rime ment ntss that the hydr hydrog ogra raph ph does no t have a smooth smoot h su surf rfac acee bu t is broken into irregular waves or surges. This ty type pe of flow m ay be de desi sign gnat ated ed surge surge flow flow an and d ma y be due to several causes:  1) Un Unde derr certa certain in hydra hydrauli ulicc co cond ndit itio ions ns steady steady flowcann flowcannot ot oc occu curr eve ven n on a smoo smooth th,, unch unchan angi ging ng sur surfac face Je Jeff ffre reys ys,, 1925 1925). ).  2) Plant debris, especially of the san andd-b burr urr type, may be loosened an d carried along with the flow, forming debris dams, be behi hind nd which the water piles up, and these hold back the wa te r temporarily and t he he n release it in relatively large volumes, prod produc ucin ing g ir irre regu gula larr waves Beut Beutne ner, r, Gaebe, and Hort Horton on,, 1940).  3)  3) Activ ctivee sur surfa facce er ero osion sion ma y prod produc ucee a succession of irre irregu gula larr waves due ei eith ther er to mud or mud-and-debris dams dams of the type la last st described, to the breaking down of divides between natura naturall depressions, or to the la late tera rall incaving of the walls of gullies Horton, 1939). Erosion may produce traveling mud dams or mud flows similar to those sometimes produced on a larger scale in m o u n t a in canyons by cloudburst storms. Wherever a mud or debris dam is formed, wa ter accumulates behind it unti un till pres presen entl tly y the dam moves down the slope with the accu accumu mula late ted d wate waterr behind it. I n case of surge flow or traveling back-water due to debris dams, there is often no co cons nsis iste tent nt rela relati tion on be betw twee een n depth of over overla land nd flow flow and run unof offf in inte tensi nsity. ty. RAIN-W RAI N-WAV AVE E TRAINS

On slopes which are not too flat, shallow flow in the form of a uniform sheet ma y be hydraulically hydraulica lly imposs impossibl ible. e. Th e flow t he he n takes the form of wave train s or series of uniform ormly spaced waves in which near nearly ly all the runoff is conc oncen enttrat rated ed.. The au auth thor or has twice observed such rain-wave trai trains ns in inte intens nsee sto torm rmss. They They occur most com monly in rains of high inten sity , p a rt ic u l a r ly those of the cloudburst type, c har acterized by th eir abil ity to t e ar ar up sod on slopes and c a r ry ry fences and o t h eerr large debris into stre stream am channels. Rain-wave trai trains ns occur only unde underr su suit itab able le hydr hydrau auli licc conditions and have been described in an anot othe herr pape paperr Hort Horton on,, 1939). Obs Obser erva vattio ion n strong str ongly ly indic indicate atess that ra rain in-w -wav avee trains trains ma y be impo import rtan antt in ini niti tiaating ting erosion on slop slopin ing g la land nds. s. If, for example, the waves are 6 feet ap apar art, t, then each wave contains

as much wate waterr as would be containe ned d in a length of slope of 6 feet with uniform flow. Th e successive waves, with thei theirr concentration of runoff and energy, can stri trike y m ay ay in sl sled edge ge-h -ham amme merr blow blowss on obstr obstruct uctio ions. ns. T h eey init itia iate te erosion where it would never nev er occu occurr from from the same ame runo runoff ff intens intensity ity with with steady steady flo low w. Th e diff differenc erencee between between the two cases is like that of brea breaki king ng a rock with a few sl sled edg gee-h hamm ammer blows, when a million taps with a pencil tip would expend the same amou amount nt of energy bu t produce no effect. The following types of sheet or over overla land nd flow ta take ke place:  y p

o

ow

 

Pure l a m i n a r

. . . . . .

Mix ixed ed lamina laminarr and turbulent Turbulent

Subd ivi de d or s u p e r t u r b u l e n t Surge flow , R a i n - wa v e t r a i n s

 

o

3 3 to 5/3 5/3 5/3 to 1

o to

1

to

 

1

Indefinite Indefinite

 

314

R

E

H O R T O ~

E R O S I O N A L D E V E L O P 1 l f E ~ T PR OFIL E

O F STR EA \IS

OF OVE OVE R RL L AN AN D FLOW

I t can readily be shown from the infiltration theo theorry t h a t the profile of sheet or over verlan land flow, or the rel elat atiion of de dept pth h 0 of surface dete detent ntio ion n to the distan tance x down slope from the wate waters rshe hed d line, is expressed by a simple para parabo boli licc or power fu fun nctio ction n

  u ile t FIGURE

l3

H i l f se tion of a small dr in ge basin

Illustrat Illus trating ing runo runoff ff phen phenom omen ena. a. Vert Vertic ical al scal scalee gre greatl atly y ex exag agge gera rate ted. d.

 Horton, 1938 . This relation is il illu lusstr trat ated ed on Figure 13. A similar power function expresses th e re rela lati tion on of velocity of overl verlan and d flow in term termss of dist distan ance ce from the water shed line Hort Horton on,, 1937 . Fo r tur turbul bulent ent fl flow ow::

OX

 

U X)3/  K 0,)2/a

1. 1.48 486 6  liz

12

 

 30

_

Vs

31

and, and, in general, for 0z in inches, for any type of flow: 32

and for velo eloci citty in feet per second:

VS U-

 

0.2836 -

n

-x K to

~

33

3M

 

lliFILTRATIO

where

 J

THEORY OF S U UR R FA FA C CE E RU U:: W FF FF

315

= supply ra rate te,, in inches per hour; 1 = tot total al length of slope on which over x  

landis falocw cicie cuien rsn,t deri in riv feveed t; fro nce, ning in gfefo etrmu , duola wnfor slroturbul pe bulent from w,at ater ersh edpro loinxi oeff oeoffic de from dthe thisetaMannin Man form fo tur entthflow fleow, an and dshed ap appr xie;   mate ma tely ly applicable to othe otherr types of flow. It s va valu luee is is:: K

1020y S

1

(34)

and the exponent: (35)

in which S is the slope, I the the inde index x of tu turb rbul ulen ence ce,, 1 the length of overland flow, and n is a roughness factor, of the same t y p e as the roughness factor in the Ma n nin g formula. The equations for turb turbul uleent flow are derived directly from the Manni nning formula and the law of c ont i nu it y and are rational. The equations for o th er types of flow are closely approximate. The d e p th th 0 as given by these equations is the to tota tall de dept pth h of surface detention, including depression storage. The equations for 0 fail a t po poin ints ts close to the wate waters rshe hed d line if, as is often the case, depression st stor orag agee persis istts to the watershed line. In the equations both for turb turbul uleent and othe otherr types of flow, it is assumed that the velocity varies as VS as for t u r b ul e nt flow. This may not be enti entire rely ly correct al alth thou ough gh numerous expe xperim riments ents indi indica cate te that, that, in mixed flow, most of the resistance is that due to turbulence. Theoretically, for la lam minar flow the velocity should va vary ry dire direcctly tly as the slope, not as vS-. Thes Th esee eq equa uati tion onss apply apply prima primari rily ly to steady steady  low Exper Experiment imentss sh show ow that they ar are, e, how ho wev eveer, clo closel sely ap appr prox oxim imat atee duri during ng the the ear arly ly st staages of runo runoff ff,, whil hile sur surfa facce dete deten n tion is building up to its maximum value. While detailed discussion of the effect of the the va vari riou ouss fac facto tors rs on su surf rfaace ce-r -ru unoff noff phen phenom omen enaa ca cann nnot ot be under undertak taken en her eree, com om pari pariso son n of the the eq equa uati tion onss sh sho ows that the the velo veloci city ty of overl verlaand flo flow inc incre reas asees, whil whilee the the d e p t h at a given po i n t x decreases, as the slope increases. Inc Incre reaasi sin ng roughness of the the su surf rfaace dec ecre reas asees the veloc elocit ity y bu t in inccre reaase sess the the depth depth of surf surfaace dete detent ntio ion. n. The equa equati tion onss for de depth pth and velocity profiles, in conj conjun unct ctio ion n with that for K s   are of fu fund ndam amen enta tall impo import rtan ance ce in re rela lati tion on to er ero osi sion onal al condi onditi tio ons, ns, si sin nce they they ex expr preess the the

tw two o fa fact ctor ors, s,   and which contro trol the eroding and tra trans nsp por orti tin ng power of sheet fl flo ow, in term erms of the the inde indepe pend nden entt va vari riab able less whic hich gover overn n surf surfaace ce-r -ru unof noff phen phenom omen ena. a. Ther Th eree are six va vari riab able les: s: (1) rain in inte tens nsit ity, y, i (2 (2)) intil intiltra tratio tionn-ca capa paci city ty.j .j ; (3) (3) le leng ngth th of overland over land flow flow,, 1 (4 (4)) sl slop ope, e,   j (5) surfac surface-r e-roug oughne hness ss factor factor,, n ; (6) (6) ind index of turb turbul ulen ence ce or ty type pe of overland flow, I. To appl apply y these equations to erosion and gra gradat dationa ionall problems one must must also have laws governing the relation of velocity and dept depth h of overland   low to the eroding and tran transp spor orti ting ng power of over overla land nd flow. ;

SUR SU R FAC FACE E EROS EROSIION BY OVER VERLAN LAND FLOW SO IL-ERO IL-ERO SI SIO O N PRO CESSES CESSES

There are always two and sometimes three dis isti tin nct bu t cl clos osel ely y related related proc proces esse sess invo involv lveed in su surf rfac acee ero rosi sio on of th thee so soil il:: (1) te tear arin ing g loos loosee of soil soil ma mate teri rial al;; (2) (2) transpor transportt

 

316

R. E

HORTON EROSIONAL D EV EV EL EL OP OP ME ME NT NT OF STREAMS

or removal of the eroded m at e r i a l by sheet flow; 3 deposition of the m a t er iiaal in transport or se transport sedi dime ment ntat atio ion. n.   3 do does not occur, the eroded mate materi rial al will be car ried into a stream. Every Ev ery farmer has noti notice ced d that the spots most vuln vulneerabl rablee to erosion are the st stee eepe perr port po rtio ions ns of the hill or valley slopes, ne neit ithe herr a t the crest nor a t the b ot ot to to m of the hill

overl nd flow

  teri l  

c

 

~ U 5 p ~ n 5

t

n

 e lt of no  elt eroeiorz

Active  ros on   ow t

~ p o ~

 I G U R

1 4 . H a l f prof profil ilee of a va vall lley ey slop slopee

t

o

o ~ d i m e n t

  llustrating soil-erosion processes

bu t in inte term rmed edia iate te..

All soils possess a cert certai ain n resi resist stiv ivit ity y to erosion, and this resi resist stiv ivit ity y grea eatl tly y by a ve vege geta tall cover, especially a good grass sod. The unde under r ma y be increased gr ly lyin ing g soil soil may may ha have ve a much much small malleer re resi sist stiv ivit ity y to ero rosi sion on,, an and, d, if the the surfa urface ce co cond ndit itio ions ns ar aree ch chan ange ged d by cu cult ltiva ivati tion on or othe otherw rwis isee so as to dest destroy roy the the sur urfa face ce res resis ista tanc ncee, ero rosi sio on will begin on land which has not hith hither erto to been subj subjec ectt to erosion. Figure   shows a half profile of a t yp ical s t r e am valley slope, with the vertical scale grea greatl tly y exaggerated. The line o e rep repres resen ents ts the the so soil il-s -sur urfa face ce profile flat in th thee re regi gio on 0 near near the crest, steepest in the region ab abou aboutt mid-l id-len engt gth h of the slope, relatively ively fl flat at a t the foot of the slope, in the region be Th e line odef represents an d relat th e surface of s heet or overland flow in an intense rain, the d e pt pt h of overland flow increa inc reasin sing g downsl downslope ope from from 0 towardf. I n the the regi region on oa no eros erosio ion n oc occu curs rs throughout throughout

 

SURFACE  ROSION BY OVER OVERLA LAND ND  LO W

  7

a distan distance ce Xc from the crest of the slope, and this is called the belt of no erosion. Here the energy of the sheet or overland flow is not sufficient to overcome the initial re sistance of the soil surface to erosion, even in the most intense storm. I n the belt ab mi midd-le leng ngth th of the the sl slop opee an and d wher wheree th thee slo lope pe is ste teep epes est, t, acti active ve eros erosio ion n oc occcurs urs. Beginning a t a th thee amount amount of mate materi rial al carr carrie ied d in susp suspen enssio ion n by the the over overla land nd flowis flowis prop propo ort rtio iona nall to the ordi ordina nate te between the dott dotted ed line ab c and   bc At a it is zero; at b it is represented by the vert vertic ical al inte interc rcep eptt bb Beginning a t a a given volume of wate ter, r, for example, the water flowing over 1 square foot of soil surface, picks up a ce cert rtaain amou amount nt of eroded soil matter matter and carries it in suspension. Passing over the next ad adja jace cent nt square foot of area the same water picks up an anot othe herr increment of soil matt ma tter er and holds it in suspension, and so on, the amou amount nt of material in suspension incre increas asin ing g until a t some some point b the the over overla land nd flow is fully charged with mate materi rial al in suspension and can carry no more. Between the poin pointt b and the stre streaam channel, no mater ial is carried away because any mater ial picked up m u sstt be replaced by an equal quantity quantity of mate materi rial al deposited fro from that alr alread eady y in susp suspen ensi sion on..   the the slope ope decreases as shown on the diagram, then the abilit y of the overland flow to carry away mate materi rial al may may decrease, in which case de dep position ion of mate materi rial al or sedi sedime ment ntat atio ion n on the surfa facce will occur instea tead of erosion. RESI RESIST STAN ANCE CE TO EROS EROSION ION

The physical factors governing soil erosion are:  1 init initia iall re resi sist stiv ivit ity, y,   i rain intensity, i; infiltration-capacity, f; velo veloci city ty an and d en ener ergy gy of over overla land nd flow or erod erodin ing g force,   The breaking down of the soil st stru ruct ctur ure, e, tearing the soil apart an and d lift liftin ing g or rolling soil parti rticle less or aggregates, requires the expenditu iture of energy. Erosion ca can n occu occurr a t a give given n lo loca cati tion on only only whe where the the amount amount of en eneergy rgy expe expend nded ed as fric fricti tion onaal re resi sist stan ance ce on the the soil oil su surf rfaace exc xcee eeds ds the the amoun mountt of ener energy gy requ requir ired ed to over overco come me th thee res to feluros ro io cu rs rien ssuorm sinoiitliaisl cre hsuirsntaendceupofinth toe asosiel mif mi idsion mna.ss A byn ienxtceenpsteiornaioncb efor fo fac faececarsuensow ffhbeeregitnhse, producing high initial erosion rate. Sustained erosion can occur only where the co cond ndit itio ion n abov abovee des descr crib ibed ed is fulf fulfil ille led. d. The term soil as related to surface erosion includes not only the soil subs ubsta tanc ncee bu t also the vegetal cover and the struct structure ures-p s-phys hysica icall and biolog biologicic-in in the surface

layers of soil. Soils are of two general classes: 1) indigenous, or those formed by weathering of underlying pare parent nt rock, eithe ther igneous or sedimentary. Such soils generally pr prev evai aill outside of glaciate iated d and loe oesss-covered regions ons. For some types of rock the fo form rmat atio ion n of soil in situ is extremely slow. After a shallow surface layer of soil is for formed, the the forma rmation of addi additi tion onal al soil is rest restri rict cted ed by the the prev reviou iously forme rmed soil soil co cove ver, r, bu t even in full exposure the ra rate te of soil formation from ma many ny types of co cons nsol olid idat ated ed and ig igne neou ouss rock rockss is so slo low w that when the the soil cover has been remove ved d the land becomes worthless. 2) Preformed and tra trans nsp porte orted d soils. These consist of rock materia rial comminuted by glacial or aeolian action and trans ransp porte rted and de posite ited. Such soil is ofte ften a mixt mixtur uree of tran transp spor orte ted d and indigenous soil mate materi rial al and in incl clud udes es se sedi dime ment ntar ary y soils oils depo deposi site ted d on lake lake or oc oceean floo floors rs an and d af afte terw rwar ard d ex expo pose sed. d. Trans Transpo porte rted d so soil ils, s, part particu icula larly rly th thos osee of glac glacia iall orig origin in,, ar aree ofte often n high highly ly fert fertil ilee a t the time they are laid down, as is evidenced by the growth of thri thrift fty y forest vegetation

 

318

R. E

HORTON EROSIONAL DE DEVE VELO LOPM PMEN ENT T OF STREAMS

within a few years on soils recently exposed by glacial re tre at. Erosion of trans ported port ed soils oils,, whil whilee a ser erio ious us mena nace ce,, is no t in ge gene nera rall so co com mplet pleteely de dest stru ruct ctiv ivee as   case of indigenous soils unless the land is gullied and scarified to such an ex exte tent nt as to make cul ulttiv ivat atio ion n impracticable. With equal runoff inte intens nsit ity y the resistance of soil mate ma teri rial al to erosion generally increases with the fineness of the soil par artticle icless or soil textur tex ture, e, th thee re resi sist stan ance ce be bein ing g smal smalll fo forr fi fine ne un unce ceme ment nted ed sand sandss b ut sohig sohigh h for for ceme cement nted ed hard hardpa pan n and tough clay that erosion ra rare rely ly if ever occurs even on bare soil. Resi Re sist stan ance ce to er eros osio ion n is is,, ho howe weve ver, r, go gove vern rned ed more more larg largel ely y by vegeta vegetall cove cover, r, biol biolog ogic ic st stru ruct ctur ures es,, and physical st stru ruct ctur uree of the soil in the surface layers than than by soil stru struc c ture. A soil which forms a h a r d crust on drying may be highly resi stant to erosion alth althou ough gh the same soil when newly cu cult ltiv ivat ated ed erodes easily. Th e cohe cohere renc ncee of soil soil part articl cles es and consequently the resistance to erosion is generally increased by the pres presen ence ce of coll colloi oida dall matter, matter, particul particularly arly that of vegetal origin. Vegetal cover is the most impor importa tant nt factor in re rela lati tion on to initial resistance to soil erosion. It s ef effe fect ctss on th e resi esistivi vitty of the soil to erosion are complex b ut include:

 1 Vegetal cover br brea eaks ks the force of ra rain indr drop ops, s, ther thereb eby y reducing the effect of the ene nerg rgy y offal offallin ing g rai ain n in bre reak akin ing g do down wn th thee cr crum umb b struct structure ure of the soil oil an and d pa pack ckin ing g the soil surface. surface. Fo r som some soi oilswi lswitth li litt ttle le co coh her eren ence ce,, brea breaki king ng do dow wn of the crum rumb struc struc ture by rain ain impa impact ct re redu duce cess th thee soil soil to a fl flui uid d con ondi diti tion on,, read adiily susc suscep epti tibl blee to er eros osio ion, n, while for ot othe herr soils packing of the soil surface tends to increase the resistance to erosion.  2 A grass sod oper operat ates es somewhat like a ca carp rpet et covering the un unde derl rlyi ying ng soil and tends stro strong ngly ly to in inhi hibi bitt erosion.  3 Fine soil pa part rtic icle less adhere to root hairs and plant plant roots near the soil surface an d ac t stro strong ngly ly as a soil binder. I n a forest similar effects are produced largely by the gras grasss cove coverr b ut are ac acce cent ntua uate ted d by differences in soil stru struct ctur uree as between natu natura rall or undisturbed and cultivated soils and by the presence of an undisturbed humus layer near the soil surface. In addition there is often a dense ma t t i n g of roots of trees, herbaceous veg vegeta tattion, and li litt tter er with within in a forest. Some of the runoff may be subsurface runoff and pass th thrrough ough this ma t of l i t t e r and roots bu t at so greatly reduced veloci ocity as to in inhi hibi bitt erosion. Fact Factor orss have been devised which stress the resist res istivi ivity ty of so soil il materi material al to er eros osio ion n in te term rmss of the the chem chemic ical al and and phys physic ical al comp compos osit itio ion n of the soil. Such factors are, however, in inad adeq equa uate te to express the re resi sist stiv ivit ity y of a given

te terr rrai ain n to erosion because of the pr pred edom omin inan antt effect of ve vege geta tati tion on and soil stru struct ctur uree and cond condit itio ion, n, whic which h are no nott re reffle lect cted ed in in inde dexe xess of the er erod odib ibil ilit ity y of the the soi oill ma mate teri rial al itself. Thee resi Th resist stiv ivit ity y of a gi give ven n te terr rrai ain n to sur urfa face ce er eros osio ion n ca can n be expre xpress ssed ed quantit quantitati ativel vely y in terms of the force in pounds per square foot required to in stitute erosion. T h is quantity can re read adil ily y be de dete term rmin ined ed on a given soil surface from mea easu sure reme ment ntss of the di dist stan ance ce from the wate waters rshe hed d line downslope to a poin pointt where erosion begins. Nearly all the factors which control resistance of a soil to erosion also control Nea infiltration-capacity of the soil. At a given point on a given slope and with a given rain inte intens nsit ity y the erosion rate is governed by various factors, one of the most impor tant of whic which h is th thee infil infiltr trati ationon-ca capac pacity ity of th thee soil soil.. I n ma many ny inst instan ance cess fa fact ctor orss wh whic ich h tend to promote a high resistance to erosion also tend to restrict or reduce the in-

 

SURF SUR F ACE

 ROSION

BY OV OVER ERLA LAND ND

 LOW

319

filt filtra rati tion on-c -cap apac acit ity y an and d vice vice vers versa. a. Co Cons nseq eque uent ntly ly op open en-t -tex extu ture red, d, co coar arse se,, sand sandy y soil soils, s, such as so soiils of sa san nd dunes, with little vegetal cover, may never be su sub bject to erosion even in the most intense rains although the soil has little resistance to erosion, be cause ause th thee in infi filt ltra rati tion on-c -cap apac acit ity y is so high high that little or no su surrface runoff ever occurs. ERODING EROD ING FO FORC RCE E

Ero rosi sion on by aque aqueou ouss age genc ncie iess in invo volv lvees three hree proc proceesses sses:: 1) disl dislod odgm gmen entt or teari earing ng loose of soil material and setting it in motion. This is called entra ntrain inme ment nt.. 2) tr trans anspo port rt of ma mate teri rial al by flui fluid d mo moti tion on..  3) sedi sedime ment ntat atio ion n or de depo posi siti tion on of the the tr tran ans s port po rteed ma mate terria ial. l. Let x = distance from divide or watersh sheed line ne,, measu surred on and along the slope  n  not ot horiz horizontal ontally); ly); 0 = dept depth h ofov ofoverl erland and flow a t x, in inch inches es;; WI = we weig ight ht per per cubi cubicc fo foot ot of wat ateer in ru runo noff ff,, inclu ncludi ding ng soli solids ds in susp suspen ensi sion on;; a = slo slope pe ang angle; le;  D = velocity of ov over erla land nd flow a t x, in feet per second. The energy expended in frictional re sistance per foot of slope length on a strip 1 foot wide running down the slope, per uniit of ti un time me,, fo forr ste stead ady y flow will ill be equa equall to the ene energ rgy y of th thee vo volu lum me of wa wate terr pa pass ssin ing g over a unit of area per unit of time. This is the pr prod odu uct of the weight, fall, and veloc ity, ity, or or:: e =

 

8

12

 

•

S in

 36)

a.

The energy equals the force times the distance moved. Hence the force exerted parallel with the soil surface per unit of slope length and width is: e

= - =  

8 WI -

12

S in

a.

 37)

Equation  37 is known as the DuBoys formula and is a ra rati tion onal al expression which may be us useed as a ba basi siss for for de dete term rmin inin ing g th thee erod erodin ing g fo forc rcee pe perr squa square re fo foot ot of soi soil surf surfac ace. e. F 1 is the force available to dislodge or tear tear loose soil material. Sometimes not all this this forc forcee is ex expe pend nded ed in tear tearin ing g loos loosee soil soil ma mater terial ial;; on a gr gras asss-co cove vere red d surf surfac acee a la larg rgee portion of it may be expended in frictiona nall resista stance on grass blades and ste stems and have little effect in tearing loose soil material unless or until the grass is broken over by the impact of the overland flow How oweever, ver, th thes esee fa fact ctor orss are be best st inc nclu lude ded d

in the re resi sist stiv ivit ity y R of the soil to erosion. As show shown n in conn connec ecti tion on wi with th surf surfac acee runo runoff ff,, for for tu turb rbule ulent nt flow the depth pth of over land flow a t th thee di dist stan ance ce x from from th thee wate atershe rshed d line line can can be exp xpre ress sseed in te term rmss of slop slope, e, ru runo noff ff in inte tens nsit ity, y, and and surf surfac acee roug roughn hnes ess: s:

6

s =   0 nX 3   . ..

 38)

SU

1020

The slope S is ord rdiinarily expressed as the tang tangen entt of the slope angle a or as tan a. Also Al so,, for for stead steady y ov over erla land nd flow IT = q in inches per ho hou ur. Su Subs bsti titu tuti ting ng these values in 38): li

qnx)36

  1020

=

1

• tano.• a .

 

320

R. E

HORTON EROSIONAL D EV EV EL EL OP OP ME ME N NT T OF S T TR RE A AM MS

Subs Su bstit titut utin ing g this value of   in (37) gives as the tot total al eroding force a t x: F

I

sin 12  Q8nX 102031S . ta n  3a 

WI

=

(39)

CRITICAL LENGTH ...   B E L T OF NO EROSION

As ind ndiicat cated, erosion will not occur on a slope unless the available eroding force exce exceed edss th thee re resi sist stan ance ce R of the soil to erosion. The eroding force increases down sl slo ope from the wa wate ters rshe hed d line (Equ (Equat atio ion n 39). Th e di dist stan ance ce from the wa wate ters rshe hed d line staance R; is call called ed to the p o i n t a t which the eroding force becomes equal to the resist the cr crit itic ical al dist distan ance ce and is de desi sign gnat ated ed XC Between this po poin intt and the waters tershe hed d line no erosion occurs. This st r ip a d j a c e n t to the watershed line, and immune to erosion, is desig esigna nate ted d the be belt lt of no erosi sio on. An expression for the width of the be belt lt of no erosion can re reaadil ily y be ob obta tain ined ed from equa equati tion on (39) by su subs bsti titu tuti ting ng R; for equati tion on for xc The runoff is free from sedi sedime ment nt F   making x =   c and solving the equa I wher wh eree eros erosio ion n be begi gins ns,, an and d 7C 1 = 62.4 lbs. per cu. ft. 1020

  ~

S I 3

62.4

= 65.0.

Subs Su bsti titu tuti ting ng this con const staant in (39) gives: Xc

C ~

=

r

where 1( 1(5) 5) is a function expressing the effect of slope on the critical length given by the equation ion: j S

=

sin a

tanO.3 a

(40) Xc

and is

 41

Numeri Nume rica call values of the slope func functio tionf nf(5 (5)) are given on Figure 15. Fo r sl slop opes es less less th haa n 2 0 0°° ,1 ,1 (5 (5 ) increases nearly in pr op or ti o n to the slope. The critical length Xc va vari ries es in inve vers rseely as the runoff inte intens nsit ity y q in inches per hour, inversely as the rough nes esss fac facto torr n and direc rectl tly y as the 5/3 po pow wer of the the resi resist staanc ncee R (eq (equat uation ion 40 40). ). Tabl Ta blee 6 gives num umeeric rical values of Xc for R = 0.01, 0.05, 0.10, 0.20, and 0.50 lb.

pe r sq squa uare re foot, for slope angles of S°, 10°, and 20°, an d for four differen rent runoff in

tensities. These are com omp pute uted for the roughness factor n = 0.10 bu t can easily be prod oduc uctt q n appl applie ied d to ot othe herr roughness factors, since the value of Xc is the same if the pr is the same. Renner ner (1936) observed the percentages of areas having ing different slope angles which were su sub bjec ectt to erosion in the Boise River drainage basin, Idah Idaho; o; his results are shown on Figure 16. The e x t e n t to which erosion occurred on a given slope increased to a maximum on a 4O degree slope and ther therea eaft fter er decreased to zero ap proa proach chin ing g a 90 90-d -deg egre reee sl slo ope ang ngle le.. A comparison of the results of Renn nneer s observations with the value of the slope £unc £u ncti tion onf( f(5) 5) is also given on Figure 15, and the two curves are in close agreement. Equa Eq uati tion on (41) for the slope function is rati ration onal al in that it is based dire direct ctly ly on funda-

 

SURFACE

EROSION

BY O V VE E RL RL A AN ND

321

FLOW

mental men tal ph phys ysic ical al la laws ws an d prin princi cip ple less wi with th resp respeect to sm smaall an d modera moderate te slo slope pes. s.

Fo r

an d

s u1941 ch41) s)losh peow s bot both h Re Renn nner er s ob obse serv rvat atio ions ns bs bse rv tio Fl eru Beut utne 19 show boou ut t e2rva 0 adti eognrseebsythFlet e etch acher m mo o un nan t dofBe ero sner ionr th at for slopes of less than a bo generally increases abou aboutt in prop propor orti tion on to the slope. Fo r st stee eepe perr sl slo opes the slope

80.

. . . . . . .. . . . .

.

r

~

c

eJ U  

c

AO I---II---

----f----

------

t-\-j-----i

.Q

s ;

W

1

.

0

H

l

~

1 FIG U RE

 

Gradient  

d

ercent

;

\

i

eo

IS.-Horton slop slopefun efunct ctio ion n fo forr su surf rfac acee er erosi osion on sina f s = la 3 ,where a = slope slope angle. angle. a

function 1 5 must be cons consid ideered red as em empi piri riccal al,, bu t its va vali lidi dity ty is confirmed by com pari pariso son n with Re Renn nner er s obse serrvations. This function is pr pred edic icaated ted on uniform turb turbu u lentt flo len flow. Fo r ver very y steep slopes su succh flow cann cannot ot occur.

critical leng critical length th Xc is the most im impo port rtan antt fac factor tor in relation to the physi siog ogrraph phiic develop opm ment of dra rain inag agee basins by erosion processes and also in rela elati tio on to erosion control. The value of Xc   Table 6) is highly sensitive to changes in the variables by which it is contro rollled, in pa part rtic icul ular ar the re ressist staance R; and the runoff inten tensi sitty q• With Wit h a newly cult cultiv ivat ated ed bare soi soil, with R; sm smal all, l, 0.05 .05 lb. lb. per squ squar aree foot foot,, for for exam exampl plee, with a 10-degree slope and a runoff inte intens nsit ity y of 1 in. per hour, the width of the belt of Th e

no erosion would be 35.1 feet, whereas on the same ter terrain rain,, bu t with a good, well square re foot, the developed grass sad to pro protec tectt the soi soil, and R increased to 0.5 lb. per squa belt of no erosion would be 15i3 feet wide. The width of the b e l t of no erosion va vari riees wit ith h th thee rain rain inte intens nsit ity, y, and, and, cons conseq eque uent ntly ly,, regi region onss wi with th freq freque uent nt st sto orm rmss of high igh rain i n t e ns iitt y are much more sub ject to erosion, o t he r things equal, than regions with lower rain intensi sittie iess and less fre frequ quen entt he heav avy y storms. Fu Furt rthe herm rmor ore, e, in a given

 

 

R. E . HORTON-EROSIONAL D EV EV EL EL OP OP ME ME NT NT OF STREAMS TA B LE

6.-Criticalle 6.-Cr iticallength ngth xcfo xcforr va vari riou ouss valu valuesof esof R

Slope angle a and runoff runoffinten intensitie sitiess q. with roughness factor n q

Ri

a  degrees)

I

0.5

I

=

0.10: x.

=

US Ri

q.n 65

5/3

in h s p r ho hour ur

1.0

I

1.5

2.0

I

l q n 0

-

10.66 4.80 2.28

I

5.33 2.40 1.14

\

6.67

10

I

5.00

2.67 1.20 0.57

3.55 1.60 0.76

.01 .01 .01

20

.05 .05 .05

5 10 20

153.4 70.2 32.6

76.7 35.1 16.3

51.16 23.41 10.87

38.3 17.6 8.2

.10 .10 .10

5 10

20

487.6 218.4 102.8

243.8 109.2 51.4

162.61 72.84 34.28

121.9 54.6 25.7

.20 .20 .20

5 10 20

1535.2 689.0 325.0

767.6 344.5 162.5

512.0 229.8 108.4

383.8 172.2 81.3

.50 .50 .50

5 10 20

7046.0 3146.0 1478.0

3523.0 1573.0 739.0

2349.8 1049.2 4929.0

1762.0 787.0 369.0

5 10

..

,

i

----

;/ 55

I

r-

t

t

t

t

•

7/

SG

«>

-

/

53

-

 

21

 

5

6-15

1

25

5 -.45

13 olIlO S5

 

C ; ~

t erosion

16.-Gradient 16.-Gr adient an and d   gr

 

14

8

Gradl< nt (Percent) ~ l t

35

r --

r-r-

(j ,

2lii-35

ONo crosicn F I G U RE

r--

16

a

4

t

58

~

r

x ~ ~

7G;-B5

 

86+

  G u l l ~ erosion of eros erosio ion n

 F.  F . G. R e n n e r )

lo loca cali lity ty,, long interv intervals als ma y elapse duri during ng which no erosion occurs on a given slope, than a ra rain in of sufficiently high in inte tens nsit ity y ma y cause erosion on a part of the slope.

 

SURFACE EROSION BY OV E ER R LA LAN D FLOW

323

For exam For exampl ple, e, wi with th a mod modera eratel tely y go good od gr gras asss-co cove vere red d sl slop ope, e, wi with th R 0.20 and a slope angle of 5°, a slope 700 feet in length would not be subject to erosion with runoff inte intens nsit itie iess less tha than n 1 in. per hour, whereas with 2 in. per hour runoff int inten ensi sity ty ne near arly ly the enti entire re lower half of the slope would be subj subjec ectt to erosion. Most Mo st slopes do no t have a uniform gradi ent from the watershed line to a st r e am bu t are fl flat atte test st near the su summ mmit it,, stee steepe pest st in the middle po port rtio ion, n, and again flat adjoin ing the stre tream. For such a slope the belt of no erosion will usually comprise all the upper, upp er, fla flatte tterr por portio tion. n. If, for example, the slop lope le len ngt gth h is 2000 feet, q = 1.5 in per hour, and the mid-portion of the slope has a gradi ent of 10° and a resistivi ty of 0.5 lb. per square foot, erosion will begin 1049 feet from the watershed line. If the lower 250 feet of the slope is fla latt tteer its gr graadie dient being 5° , th then en the length of over land flow required to produce erosion with this slope would be 2350 feet. Conse quently, no erosion would occur on the lower or flatter port ion of the slope. This example shows why erosion is ge gen nerall lly y confined to the ste teeeper, middle po port rtio ion n of a given slope Fig. 14 . As an anot othe herr exa xam mple, on a slope whic ich h has been mo mode dera rate tely ly well pr prot otec ecte ted d by grass co cove ver, r, wi with th R = 0.20, slop opee ang nglle 5°, and wi with th the li limi miti ting ng maxim imu um va valu luee of runoff inte intens nsit ity y for the given te terrrain ain 2 in. per hour, the width of the belt of no erosion in the maxi ma ximu mum m st stor orm m would be 384 feet.   the slope length was 400 feet, erosio ion n would occu oc curr on only ly a t the foot of the slope and a t ra rare re int interv ervals als..   the resi resist stiv ivit ity y of the soil was redu ducced, for example, by overgrazing, to 0.1 lb. per square foo oott or half its former value, then then the width of the belt belt of no erosion in a maximum st stor orm m would be reduced to 122 feet, and some erosion would occur on the lower part of the slope, while for a runoff inte intens nsit ity y of 1 in. per hour the width of the belt belt of no erosion would be 243fe 3feeet. Under the changed conditions some erosion would occur in all storms with runoff intensities exceeding 0.5 in. per hour. An area having a low resistance to erosion and on which erosion occurs over nearly the entire area in the more intense storms beco becom mes par partia tially lly im imm mun unee to ero rosi sion on in ligh lighte terr sto torrms. Be Betw twee een n st stor orm ms of ma maxi xim mum i nt ensit y, resistivity to erosion may be bui l t up by the growth of grass or trees so that when the next succeeding maximum st o r m occurs the surface resistivity is increased, and the areal ex exte tent nt of erosion great.lydi ydim minished. In this ma mann nner er Natu Nature re te nds to correct the deleterious effects of surface erosion. Another result of im porrtanc po tancee is the fact that on an area subject to erosion only in maximum storms, the

totall am tota amou ount nt of erosion over a given time interval a ce cent ntur ury, y, for exa examp mple le---ma may y be rela lati tive vely ly small, while, if the area is sub ubje ject ct to erosion in storms with mo mode dera rate te as well as maximum runoff int intensit itiies, then because of the much gr grea eate terr frequency of stor storm ms of lower runoff int nteensit itiies, the to total tal erosion will be enormously increased. No t unc nco ommonly the en enti tire re surface of the the soil is removed in a ce cent ntur ury y or less. Anot An othe herr factor tor of im impo porrta tanc ncee in relation to erosion is that the soil surface, if pro te tect cted ed by ve vege geta tati tion on,, has commonly a resistance to erosion many times gr grea eate terr tha than n the the un unde derrlyin lying, g, unp unpro rotec tected ted so soil il..   the surface prot protec ecti tion on is removed and a maxi mum st stor orm m occurs, erosion will then take place a t a r a te governed by the lower re sistance of the underlying soil. The soil once exposed to direct erosion ma y th then en be rapidly dly removed. Soil removed in the belt of erosion may eith either er be carried away or deposited fa fart rthe herr downslope. The ma mann nner er in which th theese com omb bined effects develop and cont contrrol the the for forms of vall valley ey cr cros osss sect sectio ions ns is cons consid ider ered ed late later. r.

 

324

R. E

HORTON EROSIONAL D EV EV EL EL OP OP ME ME N NT T OF STREAMS EROSION RATE

a fact factor or k. be intr introd oduc uced ed in equa equati tion on 37 to reduce the erosive force to terms of pth th in inches of soil ma t eria l requantity of soil removed, as, for example, the d e p  

moved per hour, the the n the erosion r a te a t the po poin intt

xwou ould ld be, be, ma maki king ng F

 

=

er·t:  42

This equation is rational in form and in fact if the r a te a t which hich soil soil ma mater terial ial is to m loos lo osee is pro propor portion tional al to the the forc forcee avail availabl ablee fr from om fr fric icti tion onal al re resi sist stan ance ce on the the soil soil surf surfac ace. e. I t relates, however, only to the ra rate te a t which soil ma mate teri rial al can be to m loos loosee and do does es not take into ac acco coun untt the ab abil ilit ity y of ove verrland flow low to trans transpor portt ma mate teri rial al in suspension. Equat Eq uation ion 42 is limited in its ap appl plic icab abil ilit ity y to cases where the erosion ion rat ratee is less than the the tra trans nspor portin ting g po pow wer er.. TOTAL EROSION AND EROSION DEPTH

Beyond Beyo nd the the crit critic ical al dist distan ance ce Xc and where the the overland flow is not loaded to capac with th soli solid d matter in tran transp spor ort, t, the erosion ra rate te at a given po poin intt x may be assum ssumed ed it y wi to be proportional to the net eroding force. Introducing a proport ionali ty factor h in equat equation ion  .39 to reduce erosion force to equi equiva vale lent nt dept depth h of solid soil ma mate teri rial al removed from the surface per un unit it of time, and making F 1 equal the erosion ra rate te er and su subt btra ract ctin ing g the value of F1 at Xc gives: _ k,Wl   q,n  3 e,. - 12 1020

The tota totall erosion betw tweeen

Xc

B

and in inte tegr grat atin ing g equa equati tion on 43 :

j

S

x

i

•  

X ~ /

43

and x is: E,

or, le lett ttin ing: g:

5

=

= -5 k;  Wi 8 12

IX

Xc

 q

e; dx n

 3 5

1020

j 5

44

45

Subs Su bsti titu tuti ting ng the slope length to for

X

46 47

The qua quanti ntity ty or ordi din nar aril ily y measured or measurable in the field is the tot total al erosion per storm. Eq Equa uati tion on 4 47 7 ccaan be used to work back from measured tota totall erosion to the phys ph ysic ical al char charac acte teri rist stic icss of the the ter terra rain in whic hich go gove vern rn eros rosion ion rate. rate. The average de dept pth h of erosion between Xc and x is: E

- xc) •

l

 

325

SURFACE EROSION BY OV OVER ERLA LAND ND FLOW

The average d e p t h of erosion is commonly expressed in terms of d e pt h on the entire area area,, no t merely the de dept pth h on the part of the area where erosion occurs. so expre resssed the average dep depth th of erosion is:

When  48

Th e coefficient B in this e qua t i o n is 5/ 8 of the coefficient of the te term rm con onta tain inin ing g x

in e q qua uati tio on 43 . Consequently, the average erosion de dept pth h over a given area is, for turbulent flow, 5 / 8 of the erosion de dept pth h for the same time in inte terv rval al a t th thee po poin intt x the value of Xc is dete determ rmin ined ed fr fro om fiel ield obse observ rvat atio ion, n, to toge geth ther er wit ith h the av aver erag agee eros rosion ion dep depth, th, the the sl slo ope le len ngth an d runo runoff ff inten intensity sity bein being g know known, n, it beco become mess poss possib ible le to de dete term rmin inee for a given field or area the erosion force F  and the con onsstan tant k Th e latter is: k

Er Eros osio ion n depth Er Erod odin ing g fo forc rcee

Thes hese equa equati tion onss fo form rm a prac practic tical al work orking ing bas asis is for for dete determ rmin inin ing g th thee erosi rosio on con consta stants nts R; and k and for co comp mpaari ring ng the erosion cond condit itio ions ns on a qu quan antit titati ative ve basis for diff iffer en t areas. RELAT REL ATION ION OF EROSION TO SLOPE LENGTH

The average de dep pth of erosion on a given slope increases as the 3 /5 power of the slope length minus the 3/ 5 power of the quantity Xc which is constan tant for a given slo lop pe and st stor orm m equa equati tion on 48 . Fo r examp xample le,, the the rela relativ tivee erosi rosio on rat rates es for for di diff ffer eren entt sl slop opee leng length ths, s, wi with th Xc = 100 ft ft.. and B taken a t uni unity, ty, are as foll follo ows ws:: 100 15.85 0.00 Fo r a soil with ith R; and

200 24.00 8.15

500 40.3 24.45

 

2

63.10 47.25

95.0 79.15

5000 166.2 140.35

each zero, the re rela lati tive ve erosio ion n rate ratess would be as shown in the second line of the table i.e. prop propor ortio tiona nall to t ~ 5 The a cctu tuaal rel relation of erosion to slo slope leng length th is, is, how oweever, ver, no t quit quitee so si simp mple le.. I n equa equatio tion n 48 aan nd in comp comput utin ing g th e fig figure ures giv given abo above, ve, surf rfaace-r ce-run uno off int intens ensity ity q has been assumed c o ons nsta tant nt for Xc

all slope lengths. Ot Oth he r things equal, runoff inte intens nsit ity y in a given sto torm rm decreases so some mewh what at as th thee slope length   increases. Also, if erosion rate is dete term rmin ineed as an average for a ye aarr or for several storms, the width of the belt of no erosion will vary in different storms. There will in general be more storm rmss producing erosion on the middle and lower than on the up uppe perr po port rtio ions ns of the slope. Comparable det ermi nat i ons of soil loss by erosion over a period of 4 to 7 years have been made by the U. S. Soil Conservation Service at several st at i ons, using slope lengths of 145.2, 72.6, and 36.3 feet. Some of the report rteed res result ltss are shown on Figu Figure re 17 Be Benn nnet ett, t, 1939, P: 152 Diff ffeerences of soil type, slo lop pe an d resistivity, ra rain infa fall ll,, and and infi infiltr ltrat atio ionn-ca capa paci city ty acco accoun untt for for dif iffe fere ren nces in soi oill lo loss ss for for a gi giv ven slo lop pe length a t the the different st stat atio ions ns.. Va Vari riat atio ions ns of these co cond ndit itio ions ns in di diff ffeerent rent po port rtio ions ns of the same slope len length acco accoun untt for for small differences in the rela relati tion on of slo lop pe length to

 

326

R. E

HORTON EROSIONAL DE DEVE VEL L OPM OPME E NT NT OF STREAMS

erosion a t a giv given stat statio ion. n.

I n al alll thes thesee ex exp per erim imen ents ts the the soil soilss we were re under under cult cultiv ivati ation on,,

p cot opne, raim ndentthse cvan alnuoest boef Xc sma part icul arly ly inatsio umm mme Trhoedurceisnugltscoorfn tohresco e tetx ann dirw ecetrley sm coamllp, apa redrtic wular ith equ equ n (4e8r.) because Xc is un unkn kno own wn,, and and Xc was pr prob obab ably ly much much grea greate terr in wint winter er than in summe summer. r. 120

I

 

\ 5

I

  l

8 Xc an d wi with thin in this area sheet flow will pro rodu duce ce erosion and a series of rill

 

O R I G IK IK AND D E V E L O P M E N T OF

STRE

lII

343

SYSTEMS

ch chan ann nels els pa para rall llel el wit with the di dire rect ctio ion n of th thee in init itia iall slo slope surf surfac ace. e.   hefi firrst ri rilll chan channe nell will be at dd , where the len length of overl erlan and d flow first exceeds the cr criitic ical al le len ngth gth xc. As the coast line recedes the belt elt in which erosion can occur will increase lat ater eral allly

an d lo long ngit itud udin inal ally ly,, an and d th thee syst system em of ri rilll ch chan ann nels els will will be exte extend nded ed corr corres espo pond ndin ingl gly y

 

r  

IGUR

27.-Devel 27.-D evelopme opment nt of fi firs rstt pair pair of tr trib ibut utar arie iess

o

lte dI

str stream eam sys system tem

in both dire direct ctio ions ns..   he rill a t dd wa wass fi firs rstt fo form rmed ed an d has been longest sub subje ject ct to erosion and will become the mast master er rill. Cross-grading will ta take ke place, producing ing new co comp mpon onen ents ts of slop slopee to towar ward d the ri rill ll dd an and d oblite obliterat rating ing the the orig origin inal al rill rill ch chan anne nels ls.. New Ne w ri rilll ch chan anne nells will will dev evel elo op foll follow owin ing g the new di dire rect ctio ion n of sl slop ope, e, on ea each ch si sid de of the the orig origin inal al stream ddt  Fi  Fig g. 27 .   n general the lengths of these new rill channels will increase proceeding down the slope from the line ee . At some p oi nt 0  Fi  Fig. 27 a new rill chan chann nel will have a gr grea eate terr length oq and grea greate terr runoff tha than n rills between o and   wi will ll have have deve develo lope ped d earl earlie ierr than rill chann annels ente enteri ring ng the parent parent strea stream m dd between 0 an d d   tth ther eref efo ore has gr grea eater ter runoff and a longer dura durati tion on of runoff

 .

in which to cut its channel than rills formed fa fart rthe herr down the slope. Such rill chan nels will su surv rviv ivee as a tributary strea tream m. Such a rill channel occurs on each side of the parent strea stream m in th e vi vicin cinity ity of 0 an and d cro cross-g ss-gra rad din ing g towa toward rd thes thesee tributary tributary stream streamss will also occur. Cross-grading of the areas a d j a c e nt to these two t ri but a ri e s will produc prod ucee cr cro oss-g ss-gra rad ded slopes on eith either er side of each tributary  F  Fiig. 28 , u n t il il t he r e is ag agai ain n a lo loccati ation on each of these areas fav avo orable able for the dev evel elo opmen entt of tr trib ibut utar arie ies, s, an and d new tri tribu butar taries ies wil will dev evel elop op,, us usua uall lly y one on ea each ch of the the two two prec preced edin ing g tri tribu butar taries ies,, as a t m, n, an d p  Fig. 28 . This process will continue unti untill finally there is no land surface above the mouth thss of the original tri tributari taries es where the length of overland flow exceeds the cr crit itic ical al leng length th xc.

 

R. E . H O R T O N E R O S ro A L DE DEVE VELO LOPM PMEN ENT T OF STREAMS

  44

the coas astt line recedes f ar ar the therr Fig. 29 , the area upslope from 0 0 on the r ig h t ha hand nd side of the str str eam eam dd is t ri ri bu bu ttaa ry ry to the s ttrr ea ea m   q Th Thee or orig igin inal al ri rill ll chan channe nels ls parallel paral lel with with dd up upsl slop opee fr from om 0 0 have have been ob obli lite tera rate ted, d, and the runoff from the area  

   

I

g

f

 

F IG U R E

afterr cros crosss-gr grad adin ing g of fi firs rstt pair pair of tri tribut butary ary area areass 2 B . -L -L i ne ne s o f flow afte

now ente enters rs st stre ream am   q As the co coaast line recedes a new sys ystt eem m of rill channels nd d th he e area 00 d d wi parallel paral lel with dd de deve velo lops ps do down wnsl slop opee fr from om 0 0 a n will ll be beco come me cros cross s 00 0

grad graded ed towa toward rd dd . When the length of o verland flow within the a r e a oo d d be come comess suff suffic icie ient ntly ly great a t so som me poin pointt q , a new tributary q r will de deve velo lop p al alon ong g th e line of the re resu sult ltan antt slope, an d its ba basi sin n will in turn be de deve velo lope ped d by cr cros osss-gr grad adin ing. g. Ther Th eree mu must st be a ce cert rtai ain n mini nim mum space or in inte terc rcep eptt bet betwe ween en tr trib ibut utar arie iess of the main s tr tr e a m to provide a d eeq qu uaa te te length of overland flow to p e r m it a lower tributary to

 

O R I G I ~

AND  

V

L O P M

~ T

OF STREAM SYSTEMS

345

t he main s t r e a m to develop.

Fur thermore, the t r i bu bu t a r y q r, hav having ing deve develo lope ped d much muc h late laterr th n the tribu tributa tary ry oq will ill exte ten nd its its drai draina nage ge are reaa latera laterally lly more ore slo slowly wly th n the l a t t er er , with the result th t the drainage basin will t en en d to have an ovoid

o ut l i n e

Fig. 29 .

 IGU R

29.-Developme 29.-De velopment nt of 1

1 er

pa pair irss of main main tr trib ibut utar arie iess

Another Anoth er stream m y al also so de deve velo lop p  t zz in the same manner as the s tr tr ea eam  t dd ,  h e de deve velo lopm pmen entt of thi this stream stream m y have begun ei eith ther er a litt little le earlier or a litt little le la late terr th n the stre streaam dd and the final location of the l at e r al divide between the two dr ai nage basins will be determined by the conditions of competition. The older strea str eam m will absorb the gr grea eate terr p rt of the area between the two streams. A mar ginal area of di dire rect ct dra rain inaage d d z is left between the two majo majorr dra raiinage basins.   the the leng length th of over overla land nd flow flow here bec eco omes mes su suff ffic icie ien nt, an inte interm rmed edia iate te su subo bord rdin inat atee stream will will de deve velo lop. p.

The appearance of the final st ream systems in the two drainage basins will be so some mew wha hatt as shown by Figu Figurre 30. Two maj or factors control the development not only of the drainage basin of a giv iven en stream stream bu t the systems of drainage basins tribu tributa tary ry to a new coast line:  1 Streams Streams de deve velo lop p su succ cces essi sive vely ly  t po poin ints ts where the length of overland flow be come comess greater greater th n th thee cr crit itic ical al len engt gth h xc.  2 Competit titio ion n results in the su surv rviv ival al of those st stre ream amss which have the earliest st rt or had the gre greate test st length of overland  low or both, and which are therefore able to absorb th thei eirr competito titorrs by cross-grading.

 

346

R. E

HORTON EROSIONAL D EV EV E EL LO OP PM ME EN NT T OF S TR TR EA EA M MS S EN

POINT  

STRE

M   EVELOPM ENT

St Stre ream am de deve velo lopm pmen entt on a new ewly ly exp expose sed d sl slo ope cont contin inue uess unti untill th thee gr great eatest est re rema main in ing length of overland flow is less than the the cr crit itic ical al di dist stan ance ce Xc required to instit institute ute erosion.

FIGURE 30. FinaJ

deve develo lopm pmen entt of tw two o ad adja jace cent nt dr drai aina nage ge ba basi sins ns on newl newlyy ex expo pose sed d la land nd

At a cer certain tain stage of grad gradat atio ion n Fig. 31 the st stre ream am oa has developed with a dr drai ain n ag agee basi basin n o d Be Befo fore re cr cros osss-gr grad adin ing g of this this ar area ea th thee cr criti itical cal leng length th Xc is is,, fo forr exam exampl ple, e, equal to that shown b y the line mm on the insert, and this is less than oa After cros crosss-gr grad adin ing g of the the ar areea oed this cr crit itic ical al len length is so some mewh what at reduced by increased resu result ltan antt slope and is now mn The grea greate test st lengths of overlan land flow on the areas long the slope lines de and ee bu t these are bo both th less than mn oca and oad are now alon

Hence no ad addi diti tio ona nall st stre ream amss will develop in the area oed The upp e r ends of the strea ms in a drainage basin will extend at least to the dis tance Xc from thei theirr wat ater ersh shed ed line, measu asured in the dir ireecti ction of slope. Th They ey may be exten ext ended ded clos closer er to the the wat waters ershed hed line line by hea headwa dward rd er eros osio ion, n, under under sui suitab table le cond condit itio ions ns.. Fo r str stream eamss to be pere perenn nnia iall a t the their ir sou sourc rces es th ther eree must be gr grou ound nd-w -wat ater er flo low w a t the head of the str trea eam m channel. I n regions where th ther eree is a perm perman anen entt ground-wat -water er horizon un unde derr the drai ain nag agee basin the most common conditio tion is that the s tr e am am is in inte terr m mit itte ten n t for a distanc e downstream from the p oin intt where its channel begins. Fi Figu gure re 32 show owss the the prof rofil ilee a t the head of a stream. The watershed line is a t a and a definite channel begins ins a t b T he re is a wa ter table und e rn e a th the he a dwa ter

 

ORIGIN  

~

D EV EV EL EL OP OP ME ME K KT T OF

STREAM

  47

SYSTEMS

c  

o

 

 

-

 

m

 IGUR

a

  ximum

31. E nd poi point nt of stre stre m deve develo lopm pmen entt

surface

n

m

 irumurn

 I G U R

32

E n d poin pointt of

defi defini nite te st stre re m h n nnel nel

 

348

R. E

HORTON EROSIONAL D EV EV E EL LO OP PM ME EN NT T OF STREAMS

be belt lt of no erosion ab the max axiimum gr gro ound und wat water er tabl tablee is a t cc and the minimum a t dd . Between cc and dd   the s ttrr e eaa m is i n te te rm rm i tt tt e nt nt . At c part of th e infiltration on the uppe upperr drainag agee area ent nter erss the stre stream am..

At times of maxi aximum surface runoff If

th e gro ground und water water fl flow ow ma y re repr pres esen entt a considerable fr frac acti tion on of the total total flow.

FIGURE

33.-Drainage 33.-D rainage ba basi sin n of Pe Penn nnyp ypac ackk Cree Creekk

Above Abo ve Valley Valley Falls Falls Pa. showin showing g subareas fro from m whi which ch surfac surfacee runoff runoff is derived. derived.

for example the gro rou und wate waterr flow is one fo four urth th of the total flo flow at e then then if the the ch chan anne nell ex exte tend nded ed a li litt ttle le farth farther er upslope to e the ma maxi ximu mum m runoff wou would be re redu duce ced d br u up pt an nd d someone fou fourth rth by elimination of ground wat ateer. Ther Theree is therefore an a br time ti mess co cons nsid ider erab able le ch chan ang ge in th thee total runoff a t abou aboutt the poin pointt where the maxi aximum level of the wate waterr table intersec sects the st stre ream am channel. Surface runoff plus ground und-

wate wa terr flow can genera rallly extend the ch chan ann nel upst upstre ream am fart farthe herr by head headwa ward rd erosion than could surface runoff alone. Hence the chan nel u su ally ends n e a r the p o i n t where gr grou ound nd wate waterr flow is no longer effective. Grou Ground nd wate waterr flow a t e  is inte interrmittent b u t it usua usuall lly y co cont ntin inue uess much uch lo long nger er than surf surfac acee ru runo noff ff an and d by maint maintain aining ing th e soil a t th e head of the st stre ream am ch chan ann nel moist and soft it promot omotes es extension of the ch chan ann nel by he head adwa ward rd erosion and bank caving. Th e final re resu sult ltss of st stre ream am de deve velo lop pment ment unde underr natural co cond ndit itio ions ns are are illust illustrat rated ed

by Figur e 33. Some of the s tr e a ms in the lower part of the basi basin n are clea earrly ad adve vent ntiitious. Ther Theree are several drainage basins su such as   and B where trib tribut utar arie iess have have

 

ORIGIN

 N

349

DEVELOPMENT OF STREAM SYSTEMS

developed only on one side of the pare parent nt st stre ream am leaving in this case an isolated plaateau pl teau in the interfluve area although the drainage development of the basin is evide viden ntl tly y ma matu ture re.. STREAM-E STRE AM-ENTRA NTRANCE NCE ANGLES

Fr From om ge geom omet etri rica call co cons nsid ider erat atio ions ns th thee foll follow owin ing g eq equat uatio ion n ha hass be been en obta obtain ined ed fo forr the the entrance angle between a tr trib ibut utar ary y and the higher-order str stream which it enters  Horton 932 4; coss

=

 s

tan

tan

 

wher here s iiss th thee en entr tran ance ce ang ngle le betwe tween the two two st stre ream ams; s; So is the channel slope of the parent pare nt or rec receiving ving st stre ream am;; SI is the ground slope or re resu sult ltan antt slope w wh hich is here assumed to be the same as the slope of the tr trib ibut utar ary y stream. Value Val uess of the en entr tran ance ce ang ngle le compute puted d by this eq equa uati tion on fo forr diff ffeeren entt value lues of the ratio s /sI/ are as follows: s /So

=

0 9

z

=

25.5°

0.8 36.8°

0.7 45.5°

0.6 37.0°

0.5 60.0°

0 4

66.2°

0.3 72.3°

0.2 78.3°

0.1

84.2°

As shown by Table 4 st stre reaam slopes are always less than the adja adjace cent nt ground slope an and d tr trib ibut utar arie iess should ente enterr the confluent stre stream am a t acute angles when the slopes of the channels of the tr trib ibut utar ary y and confluent stre streaams are nearly the same. Thee eq Th equat uatio ion n take takess on th thee indet indeterm ermin inate ate form form   if the two two slo slopes s and SI ar aree eq equa ual. l. This Th is me mean anss that the two streams will be parallel and will not join. Three cases will be con onsi sid dered red for purpo urpose sess of il illu lust stra rati tion on.. CASE i FLAT STREAMS DEVELOPED ON A FLAT AREA: When the pare parent nt stre reaam has devel velop oped ed and cross ross-g -gra rad din ing g has proc proceeeded ded to a poin pointt whe herre a pair pair of tr trib ibut utar arie iess develop tth he pare parent nt st stre reaam will in general have cut into the initial surface to some dept depth h and its str strea eam m sl slop opee in th thee vi vici cini nity ty of th thee de deb bouch ouchur uree of the the tr trib ibut utar arie iess will be mate ma teria rially lly less less st stee eep p than the original slope while the slopes of the trib tribut utar arie iess as th they ey ap appr proa oach ch the paren parentt st stre ream am will be mat ater eria iall lly y steeper than the the or orig igin inal al slop slope. e. As a cons conseq eque uenc ncee in inste stead ad of th thee rati ratio o sciSo bein being g cl clos osee to unity unity this this ra rati tio o wi will ll se seld ldom om have a value greater t h a n 1/2 or 1 / 3 and the tributaries will not en t er the main stream a t acute angles as would be the case if   and So we were re ne near arly ly eq equa uall bu t will more generally ente enterr the parent parent st stre ream am a t angles of 60° to 80° On ex extr trem emel ely y fl flat at

sur surfac faces in hum umiid reg regio ions ns a sw swaampy condi ondittio ion n oft fteen pre rev vails ils an and d stre stream am-e -ent ntra ranc ncee angl angles es are are bu t littl ittlee sub subje jecct to control by erosion conditions. On semiarid plains wheere li wh litt ttle le erosi rosion on occ ccu urs ac acut utee en entr tran ance ce angle ngless of tr trib ibut utar arie iess to the parent parent stream stream may som someti tim mes be obse observ rved ed.. CASE 2-FLAT VALLEY SLOPE WITH MODERATE TO STEEP AD ADJA JACE CENT NT GROU GROUND ND SLOPE: Unde nder these ese con ondi diti tion onss the rat ratio s so is nearly always low and the stre stream am

entr entran ance ce angles to the main or pa pare rent nt st stre ream am are commonly 60° or greater.   Deri Deriva vati tion on of this this eq equa uati tion on is gi give ven n corr correc ectl tly y in the the refe refere renc ncee ci cite ted d is i nco ncom mp pll et et e and not wholly co corr rrec ectt .

As the

In Inte terp rpre reta tati tion on of the the equa equati tion on as ther theree give given n

 

350

R. E.

HORTON EROSIONAL DEVELOPMEKT OF STREAMS

stream sy stream syst steem dev evel elop ops, s, the the sl slo ope of the the main main stream stream st stee eepe pens ns proc proceeed edin ing g upstr upstrea eam, m, and the l at er al ground slopes also steepen proceeding upstr eam. The r ati o s m y rem remain ain se sens nsib ibly ly co cons nsta tant nt,, or it may ei eithe therr incr increease or dec ecre reas ase. e. Most co comm mmo only nly

it decreases to som some exte extent nt.. Quite generally the entr entran ance ce angles of trib tribut utar arie iess to the main ain or in init itia iall stream stream are quite quite uni nifo form rm and ra rang ngee fro from 60° upwa pward, rd, dec ecre reas asin ing g some some what wh at upst upstre reaam. CASE   TRIBUTARIES   A ST STEE EEP P SL SLOP OPE: E: Tri Tribu butar tarie iess de deve velo lope ped d on the the sa same me slop slopee gener eneraally lly ru run n ne near arly ly par araalle llel, an and d   the main valley is rela lattive ively flat they they will en ente terr the pa pare rent nt stre stream am   t an angle of 90°, re rep pre rese sent ntin ing g a limiting condition which is ap proached   ut not often atta ttaine ined. Tr Triibutari tariees developed on the same la latteral ral slope may of course join and are especially likely to join where drainage de dev velopm opment is inci incipi pieent, nt, as on steep, rocky slopes and in se sem miarid regions where tributa tributary ry develop m e n t has been arr est ed at the end of the rill stage. Parallel t r i bu ta r ie s which join on a steep slope unde underr these conditions commonly have an ac acut utee angle of junc junctu ture re.. In this case the ratio s is close to unit unity. y. DRAINAGE

P TT

RNS

Much has been writ Much writte ten n regarding the forms of drainage patter patterns. ns. They They are usua usuall lly y classified as de dend ndri riti ticc treelike , re rect ctaangul ngular ar or trellised, ra rad dia iall, and cen entr trip ipet etal al.. The terms ra rad dial and centri tripetal commonly refer to the arra rrangement of a group of draina dra inage ge patterns or orig igin inat atin ing g   t or converging to a common p o i n t and do not refer in general to the p ttern in an individual drainage basin. All drainage patt patter erns ns of indi indivi vidu dual al dr draain inaage ba basi sin ns ar aree tre treel elik ike, e,   ut diff diffeere ren nt patterns patterns re rese sem mble ble the the bran branch ch ings of different kinds of trees and range from those with branches entering the pa pare rent nt stre streaam nearly at righ rightt angles, to those with tri tributa utarie ries nearly parallel and ent nteering ring th thei eirr pa pare rent nt st stre ream amss   t small angles. The form of the drainage p ttern depends to a large e xt en t on the relation of the slope of the p a r e n t s t r e a m to the result res ultant ant gr gro ound und sl slo ope after after cro ross ss-g -gra radi ding ng..   this ratio incre reaase sess with suc success ssiive cros crosss-gr grad adin ings gs,, streamstream-ent entran rance ce angl angles es of succ succes essi sive ve trib tributa utarie riess ar aree somew somewha hatt more more acute acu te fo forr su succce cess ssiv ivel ely y lo low wer-or r-orde derr st stre ream ams, s, affor ffordi din ng the the most most usual sual type type of de dend ndri riti ticc dr drai aina nage ge pattern. pattern. On a relatively flat surface the directions of r e su l ta tan t overland flow a f te te r the firs firstt cro ross ss-g -gra rad ding ing are nea earl rly y   t ri ght angles to the initi al stream, and the second series of str treeams developed ente enterr the pare parent nt st stre reaam nearly rly at right angles. Cross grad gradin ing g of the areas tribut tributary ary to these st stre ream amss produces   u t a slight change in the slop slopee rati ratio o s g so t h t the n ex t order of streams also enters the p a re re n t str eams

more or less ne neaarly rly   t r ight angles. In this way a rectangular drainage p ttern is developed. If, on a steep, sloping, original surface, the head headwa wate terr divide forms roughly an arc of a circle, th then en the first two trib tribut utar arie iess developed will en ente terr the pa pare rent nt st stre ream am from oppo opposi site te side sidess   t nearly the same point oint Fig. 34 . These streams will develop long tri tribut butari aries es nearl nearly y pa para rall llel el with ith th thee init initia iall strea stream, m, giv giving ing rise rise to a ce cent ntrip ripet etal al dra rain inag agee  Fig. g. 34 . p ttern  Fi On flat flat slop slopes es each each su succ cces essi sive ve cr cros osss-gr grad adin ing g of a give given n subare subareaa chan change gess the the dire direct ctio ion n of the next stream to develop on the area through an angle approaching 90° as a

 

O RI RI G GII N AND DE V E EL L O PM PME N T OF STREAM SYSTEMS

s

 

c a

t ~

 5

IGUR

34 Centripetal dra draina inage ge patter pattern n

Payne Pay ne Cree Creek k Ga Ga.. lI lIIu Iulk lky y Gap Gap qua quad. d. U. U. S. G. S

T

V. A.

limit and changes the direction of overland flow through a corresponding angle. Thee di Th dire rect ctio ion n of resultant resultant cr cros osss gr grad aded ed sl slop opee a t the end of a give iven stage beco com mes the direction of the stre stream am of the next succeeding stage. The dir ireecti tio ons of streams and of resu esulta ltant slopes will change through nearly a right angle with each successive st stag agee of str stream eam de dev vel elop opm men entt an and d cr cro oss gra rad din ing g an and d the the dir irec ecti tion onss of str trea eams ms and and of resu result ltan antt slopes tend generally to be the same in any two stages of st stre ream am development which are either both even numbered or both odd numbered.

 

352

R. E . HORTON EROSIONAL D EV EV E EL LO OP PM ME E NT NT OF ST RE RE A AM MS ASYMME ASY MMETRI TRICAL CAL DRAINA DRAINAGE GE PATTERNS

Because newly developed trib tribu uta tari ries es en ente terr th thei eirr pare parent nt st stre reaams at ac acut utee angles, t h e y divide t hei r t ri ri bu bu ta ta r y areas into two p a rt rt s such that th thee rem remaini ainin ng upsl upslop opee tribu tributa tary ry areas are larger than those on the downslope side, using the terms  u pslope and downsl slo ope wit with reference to the two sides of the tr trib ibut utar ary. y. Because of in ineq equ ual alit ity y of area, width, and slope on the two sides of a tr trib ibut utar ary, y, the next lower order of trib tribut utar arie iess may develop with two or three tr trib ibut utar arie iess on the upslope side and fewer or none on the downslope side, a common phenomenon, pa part rtic icul ular arly ly in moun tain tain areas. Since the average elevation of the upslope area is greater than that of the downslope area, this phenomenon is sometimes attrib attribute uted d to increase of rain rainfa fall ll with with elevat elevation ion.. I t may occu cur, r, how howev ever er,, as th thee resu result lt of dif iffe fere ren nces of tributary tributary area rea an and d len engt gth h of over overla land nd flo flowon the the upsl upslop opee an and d dow downs nslo lope pe si sid des of the parent st stre ream ams, s, inde indepe pend nden entl tly y of va vari riat atio ion n of rain rainfa fall ll or runoff on the dr drai aina nage ge basin. Burc Burch h Creek and Reels Creek drai draina nage ge basi sin ns Utica, New New York, quad quad., ., U. S. Geological Survey) af affo ford rd exa xam mple ples of as asym ymme metr tric ical al drai draina nage ge-b -bas asin ins. s. PERCHED OR SIDEHILL STREAMS

I n general, st stre reaams follow the bott bottom omss of the valleys in which th they ey are located. Small usually 1st order streams ar aree occa casi sion onaall lly y perc erched pr prec ecar ario ious usly ly on th thee si side de

slopes of graded valleys of higher-order streams. The course of such a st r eam is of ofte ten n more more nearl arly pa para rall llel el with ith the the an ante tece cede dent nt sl slo ope than wit ith h the cro ross ss-g -gra rade ded d sl slop opee. At the foot of the slope the strea stream m often turn turnss abrup abruptly tly and debouches into the parent parent stream a t nearly a r i gh t angle Fig. 35). E v i de nt l y gradation of the valley of the parent stream cd rea reach cheed the the st stag agee sho shown in th thee fig figur uree be befo fore re the sl slo ope be beccame ame st steeep below w th thee maxi maximu mum m le leng ngth th   of overl overlan and d fl flow ow enoug nough h to re red duc ucee the the cr crit itic ical al le leng ngth th Xc belo on the righ ight-h -haand side, and   beca became me greate greaterr than Xc only when grad radatio ion n of the valley slope had reached the end point. A weak str eam, ab th then en developed by microp micr opira iracy cy an and d cros crosss-gr grad adin ing, g, bu t owing to some local cause, such as increased re resi sist stiv iviity of the soil to erosion a t increased dept depth h below the original surface, this st r eam was unable to develop a valley of its own by f u r t h er cross-grading and so remained high above the pare parent nt st stre ream am on the ant nteece cede dent nt rilled surface, un unti til, l, with increasing volume and slope, it turn turneed nearly a ri rig ght angle as it entered the pa pare rent nt stream. RE REJU JUVE VENA NATE TED D ST STR REA EAM MS; EP EPIC ICYC YCLE LES S OF EROSION

I n the preceding sections it has been assumed that:  1  1)) Uplift or exposure of new

terrain terrai n took took pla place co cont ntin inuo uous usly ly thou though gh no t necessarily a t a uniform rate rate,, the region fi fin nally lly bec eco omin ming stable stable;; 2) the init initia iall resi resist staance R of the soil surface to erosion re mained constant. The effect of subsequent fu furt rthe herr elevation or subseq sequent sub sidence of an area on which a st stre ream am system has al alre read ady y developed has been exten sively sive ly disc discus usse sed d in co conn nnec ecti tion on wi with th the the Davis Davis eros erosio ion n cy cycl clee Woo Woold ldri ridg dgee an and d Morgan, Morgan, 1937) 937) an and d will ill no t be consi sid dered furt furthe herr here. Before leaving the general subj subjec ectt of stre stream am development and valley grad gradaation tion consi sid deration will be given to the effect of 1) differences between surface and su sub bsu surf rfac acee resi resist stiv ivit ity y to erosion, 2) changes in the surface res resis isttivi ivity to erosion.

 

AND D E EV V EL EL OP OP ME ME N NT T OF STREAM SYSTEMS

353

The te term rm reju rejuve vena nate ted d stre stream am is applied to a stre stream am sy syst stem em in which a renewed cycle of erosi osion beg begins and which ma y extend th e drainage n et af afte terr it has reached matu ma turi rity ty..

Rej ejuv uven enaation ion ma y r es ul t from several causes, a lt ho u g h in the D a v i s

 IGUR

35 Perc Perched hed or hmsi

str

m

sense the te term rm is ap appl pliied chiefly where it resu result ltss from widespread geologic changes such as renewed upli uplift ft,, folding, an d tilting. Acce Ac cele lera rate ted d or decr decrea ease sed d eros erosio ion n ma y re resu sult lt witho without ut an y such such geol geolog ogic ic chan change gess if th e or oriigi gina nall terrain terrain var varies ies in eros erosio iona nall res resis isti tivit vity y or in infi filt ltra rati tion on-c -cap apac acit ity y pr proc ocee eed d ing dow downwar nward d from the surface. Then Then,, as erosional gr grad adat atio ion n ta take kess place, changes in the cri criti tica call le leng ngth th of ove overlan rland d flow Xc will occur, and if these changes are abrupt they ma y resu result lt in impor importa tant nt effects, eith either er 1) mark marked ed increase in dr drai aina nage ge de dens nsit ity y an d extension and numb number er of minor tribu buttarie ries,   and z, decrease downward

 

354

R. E . H O R T O N E R O S I O N A L DEVELOPMENT OF S T RE RE A AM MS

from the surface, or 2) a ba ba n d do on m mee n t an d fo foss ssil iliz izat atio ion n of pre pre-ex -exis isti ting ng streams streams an d tributa tributarie ries, s, if R i an d Xc inc ncre reas asee wi with th in incr crea ease sed d gradat gradation ion.. third condition ma y also b ri ng a bo an d stre u uttuvenat chanation ngion es in eeroto siostrictly n rctly aatt e ge stic ream am des. velo veA lopm pmen entt which is more common thanborej rejuve du due stri geol olog ogic cau causes. se T h is is occurs where, as the r es u l t chiefly of climatic or c u lltt u r aall changes, t h e rree is a change in the surface-erosional resis sistivit vity or infiltrat atiion on--ca cap paci citty of the te terr rrai ain n which brings abo about ut changes in the cri rittic ical al length Xc an d in the co cons nseq eque uent nt develop m en en t of drainage. Accelerated erosion due to the removal or replacement of an initially r es is ta n t surface by a less re resi sist stan antt surface has been ap appr prop opri riat atel ely y de desc scri ribe bed d by Bailey 193 1935) as an epicycle of erosion. This te term rm is appr approp opri riat atee since it implies a mark marked ed chan chang ged in er eros osio iona nall an d gra gradat dation ional al activi activity, ty, su supe perp rpos osed ed on the normal normal er eros osio iona nall cond condit itio ions ns.. Changes in erosional co cond ndit itiion onss bro brough ughtt abou aboutt by dust storms an d the fo form rmat atio ion n of loess veneer on soil surfaces, and changes in erosional ac acti tivi vity ty resulting from im prope properr cul culti tivat vation ion of th thee soi soil, de defo fore rest stat atio ion, n, fi fire ress, or ov over ergr graz azin ing g of ra rang ngee lands ands,, af affo ford rd

ex exce cell llen entt ex exam ampl ples es of ep epic icyc ycle less of er eros osio ion. n. Where a less permeable an d more re si st an t surface l ay e r of soil or sod overlies weaker or more pe perm rmea eabl blee subsoil, th ther eree will be in effect two different va valu luees of Xc one pe pert rtai aini ning ng to the surface layer, the othe otherr to the unde underl rlyi ying ng mate materi rial al.. This occurs wheere well wh well-e -est stab abli lish shed ed gr gras asss or othe otherr vegeta vegetall cov over er ov over erllie iess a no nonc ncoh oheesive sive sandy so soiil or where th ther eree is a la laye yerr of loess or similar fine-textured material, with mode moderrat atee or high cohesiveness, overlying more perm permea eabl blee an d less cohesive mat ater eriial al,, such as sand.   the overlying resist sistiive ma mate teri rial al is br brok oken en thr hrou ough gh,, the va valu luee of Xc pert pertain aining ing to the un unde derrly lyin ing g mate materi rial al governs subs subseq eque uent nt stre stream am deve evelo lopm pmen entt. I n such cases the de deve velo lopm pmen entt of a dr drai aina nage ge ne t is likely to be e r r a ti c and sporadic. On much of the area area th ther eree ma y be b ut few stre stream ams. s. This will be tr true ue where the larg larger er or surficial value of er eros osiv ivee re resi sist stanc ancee R an d criti critical cal distance Xc are effective. At ot othe herr lo loccat atio ions ns where the smal smalle lerr su subs bsur urfa face ce values of R; an d Xc hav havee be beco come me ef efffect ectiv ive, e, active active an d extensive stre stream am devel velopm pmeent may ta take ke place. Ext xteensi nsive plains, for the most most part undis und istu turbe rbed d by erosion, may be diss dissec ecte ted d by ra rapi pidl dly y growing an d irreg irregularl ularly y branch ing sy syst stem emss of gullylike channels. This co cond ndit itio ion n exists in the Po Ponto ntotoc toc Ridge region of the Li Litt ttle le Tal alla laha hatc tchi hie, e, Mississippi, dr drai aina nage ge basin, where deep in inco cohe here rent nt sa sand nd is overlain with a t h i n ve neer of fine uniform loessal silt. In this region Xc for the unde nderly lyiing sand is pr prac acti tica call lly y zero, and stre stream am de deve vellopm opment ent may exte extend nd far above the Xc limit for the surface mat mater eria iall as a resul sult of hea eadw dwaard erosion. The au auth thor or has observed gullies in the Pon onto toto tocc Ridge region which in some cases have ex exttende nded

no t only to b ut some somewh what at be beyo yond nd th thee to topo pogr grap aphi hicc bo boun unda dari ries es of their their dr drai aina nage ge ba basi sins ns

  Happ et al 1940). This has resulted from the slumping of masses of e ar ar tth h from the n ea rly ve r tic a l and sometimes un dermined scarp formed by the erosion of the deep, inc ncoh oher eren entt sand. Th e de dest stru ruct ctio ion n of ve vege geta tati tion on by sm smel elte terr fumes early in the pr pres esen entt cent centur ury y in the vic iciinity nity of Duck Duckto town wn an d Co Copp pper er Hill, Tennessee, br brou ought ght about a new erosion cycle. Glenn s ea r l y r e p o r t 1911) an d the a u utt h ho o r s l at at er er obs observ ervati ations ons show show that forest and hills sometimes pr prote otect cted ed the sod locally even where the trees were killed,

 

O RI RI G IIN N AND D EV E L LO OP M ME EN T OF STREAM SYSTEMS

an d where the sod was prot protec ecte ted d no erosion occurred.

 

As described by Glenn (1911,

p. 78):  T he erosion starts near the b o ott tto o m of a slope, and where the soil is porous ra rap pidly idly cuts a steep sided gully to a d eep p th th of 5 to 12 feet below the surface, where the underl rly ying ing schist is as a rule still meas me asur urab ably ly fi firrm. After a gully has re reaached its limit in de dept pth h it widens unti untill its walls coalesce with th thee wa wall llss of adjace adjacent nt gull gullie iess, by whic which h time time most ost of th thee soilha soilhass been been re remo move ved. d.

Over much of the denuded area erosion has no t been as complete as that above described. Narr Narrow ow flat lands still pe pers rsis istt between the para parall llel el gullies, an d uneroded, nearly rly flat sum summits of the hills are conspicuous. I n some cases ses the gullies affo fford exce excell llen entt examp xample less of cross ross-g -gra radi ding ng in prog progre ress ss,, with with remnan remnants ts of th thee an antec teced eden entt ri rill ll surf surfac acee stil stilll visib isible le.. The erosional topo topogr grap aphy hy of this region was ess ssen enti tial ally ly matur maturee before denu denuda dati tion on took place wherever ther theree was a well-est staablished so sod d cover. Th e resi resist stiv ivit ity y of the unde un derl rlyi ying ng soil to erosi sio on is, however, so small that la laccki king ng pr prot otec ecti tion on,, th thee crit critic ical al distance Xc is re redu ducced ne near arly ly b u t not quit quitee to zero. Cons Conseequent uently ly the walls between initial parall initial parallel el ridg ridges es on st stee eep p sl slop opes es have have so some meti time mess co coal ales esce ced, d, as desc descri ribe bed d by Glen Glenn. n. With Wi thin in a few years afte afterr destru truction of the vegetati tio on the drainage de dens nsit ity y was in creased locally from ten to one hund hundre red d fold, an d where this occurred the end poin pointt of the new erosion cycle was quickly at atta tain ined ed.. Pontot otoc oc Ridge region in Mississippi an d in the I n th e gully formation in the Pont vicini vic inity ty of Duckt Ducktow own, n, Tenn Tenneess sseee, surf surfaace an d subsurf subsurface ace resist resistanc ancee R; di diff ffer ered ed,, th thee su surrface resista stance, being init initia iall lly y grea greate terr and the terr terrai ain n in init itia iall lly y st stab able le agai agains nstt ero sion. Reduct ion of surface resistance resulted from improper cu lti vation in the Pontotoc Ridg Ridgee re regi gion on an d fro from partial partial de destr struc uctio tion n of ve vege geta tall co cove verr by sme smelt lteer fu fume mess in the Duck Duckto town wn region, an d an active epicycle followed in each case. The forma tion of ar arro royo yoss on overgrazed land affords anot anothe herr example of an epicycle of erosi sio on where the value of Xc is less for unde underl rlyi ying ng soil than than for undi undist stur urbe bed d surface cover. DRAINAGE-B DRAIN AGE-BASIN ASIN TOPOGRAPHY TOPOGRAPHY MARGINAL BELT OF NO EROSION; GR GRAD ADAT ATIO ION N OF DI DIVI VIDE DES S

I n ad addit dition ion to co cont ntro roll llin ing g the the drai draina nage ge de densi nsity ty an and d th thee compo omposi siti tion on of th thee dr drai aina nage ge

pointt of devel velopmen ment of a st stre ream am sy syst stem em on a given area, pattern an d fixing the end poin th thee cr criti itica call dista distanc ncee Xc and the belt belt of no erosion which it produces govern the degree of gra rada dati tion on which can occur on a given area and the ex exte tent nt of gr graada dati tion on along an d ad adja jace cent nt to both both exte teri rior or an d int interi erior or wate waters rshe hed d lin lines or div ivid idees. the an angl glee be betw twee een n the the wate waters rshe hed d lin line aa (F (Fig ig.. 36A) 6A) and th thee di dire rect ctio ion n of over overla land nd   the

fl flow ow is A th then en for a given criti riticcal length Xc th ere will be a b el t of no erosion on th e given side of th e wat ater ersh sheed line havi having ng a widt width h w

Xc

sin A

This marg rgiinal belt belt of no erosion aa cc is rela relati tive vely ly perman permanen ent. t.   t is widest, ot othe herr th thin ings gs equal, where the di dire rect ctio ion n of over overla land nd flow is most near nearly ly norm normal al to th e water shed line; this is usually ar o un d the head wat ers of an exterior divide. The width of the marg margin inal al belt of no erosion decrea reases for a given Xc as the di dire rect ctio ion n of over overla land nd

 

R. E

356

RORTOK EROSIONAL DE DEVE VELO LOP PM MEN ENT T OF STREAMS

 

--...;a.

.....

_

 

9_

FIGURE

  A) Width of belt of n no o eerrosion.

36. Belts of no  ros on B) L o ng i t u d in al be belt of n no o eerrosion.

flow becomes more ne near arly ly pa para rall llel el with ith the dire direct ctio ion n of the the divid ividee, a cond condit itio ion n which commonly occurs along late latera rall segments of the main divide surr surrou ound ndin ing g a drai draina nage ge basin.   aa Fig Fig. 36 36B) B) repr repres esen ents ts the the ex exte terio riorr divi divide de a t the the head of a newly exposed area area,, then, the n, with with suff suffic icie ient nt newly ewly ex expo possed su surf rfac ace, e, stream streamss wi will ll dev develop elop,, starting starting a t   and e Th e enti entire re slope fro from ee to the o u tl tl et et is sub j ect to sheet erosion. Cross-grading begins a d j ac ac e nt nt to the st r eams and spreads l at er a l l y u nt nt iill t he r e remains a narrow beltff gg n ot ot y e ett cross-graded. Dash Dashed ed arrow rows indi indica cate te dire direct ctio ions ns of ov over erla land nd fl flo ow

 

DRAINAGE DRAI NAGE BASIN BASIN TOPOGR TOPOGRAPH APHY Y

  7

an ante tece cede dent nt to and solid olid ar arro row ws the the cor orre resspond pondin ing g dire direccti tion onss wit ith h cros crosss gr graading ding.. This belt has however been prev reviously su subj bjec ectt to sheet erosion since it lies downslope from the headward belt of no erosion and the direction of overland flow is parallel with the slope. The profile of the belt   gg is conc oncave ave an and d it li liees exc excep eptt   t its ends considerably below the original slope. The narrow belt   gg is still su subj bjec ectt to cross gra grad ding. Slight vari variat atio ions ns in surface conditions will dive divert rt most of the the surfa urfacce ru runo noff ff   t a give given n loc locat atio ion n as   t h into one st ream or the other. The divi divid de be betw twee een n th thee st stre reaams wil illl move move away way fr fro om the the st stre ream am into into whic which h the the div divers ersion ion occurs. The direction of overland flow on the dive divert rted ed area will swing around unti untill it is more or less parallel with th t on the adja adjace cent nt cross graded slope an and a belt of no erosion will develop on the side of the divide on which diversion occurs. This belt will have a width Xc cos A where   is the runoff angle between the div diverte erted d surface runoff and the antecedent slope. This angle will vary vary from zero to mm n and the width of the belt of no erosion on the given side will vary accordingly. At some other location j the st stre ream am ee will gain the  dv nt gein competition with gg and and th thee wate waters rshe hed d li line ne will will be de defl flec ecte ted d towardff . As a result the watershed line will become sinuous as shown by the dashed line on Figure 36B. In Inte terrmediat med iatee betwee between n h   n d j the stre reaams will divide the runoff more or less equally. The wat ater ersshed hed li lin ne will will cro ross ss the the cen ente terr of the the beltff gg ut   t this location most of the runo runoff ff will will have have been been divert diverted ed   t h an and d ther theree wil illl be les less eros erosio ion n than   t either h or j As a consequence of the competitive development of divides the width of the belt of no er eros osio ion n will will vary fr from om point point to point point gove govern rned ed loca locall lly y by the the sl slop opee the the di dire rect ctio ion n of overland  low and the amou amount nt of previously undi undive vert rted ed surface ru run noff orig origin inat atin ing g within the belt of no erosion. The watershed line will be sinuous in plan and profile and the watershed ridge will be broken up into a series of irregularly spaced hills often with flat crestal pla plate teau auss and ad adja jace cent nt hill crests will be   t abou aboutt the same elevations. The hills will be separated by saddles and both will be rounded not only as a result of the mann manner er of their development by aqueous erosion   u t als lso o by sec econ onda dary ry proc proces esse sess such such as earth earth sli lips ps and ra rain in impa impact ct eros erosio ion. n. A fa favo vora rabl blee lo loca cati tion on fo forr fl flat at top top inte interf rflu luve ve hi hill llss is   t the the junc juncti tion on of a long longit itud udin inal al and a cross divide. Such junctions commonly occur where there is an angle or bend in the p ar ar en en t divide. Under these conditions the flat tto op hill usually has an arm extending out onto the inte interi rior or divide. Fl Flat at top top hills and plat platea eaus us may also occur   t inte interm rmed edia iate te lo loca cati tio ons whe where ther theree is a re rela lati tive vely ly wid wide belt belt of no ero rosi sion on.. On Figure 37 aa and bb are adja adjace cent nt tri tribut butary ary str treeams which developed more or less simultaneously on the same side of the pa pare rent nt stream and which flow nearly parallel and cro rossswise of the original slope. When these strea reams have developed on an antec antecede edent nt sl slop opee cr cros osss gr grad adin ing g will will oc occu curr spr sprea eadi ding ng later laterall ally y on both both si side dess of