人文研究大阪市立大学文学部 8 1990 simulation of...

人文研究大阪市立大学文学部第 42巻第8分冊 1990年

Simulation of Gully Development

on Three-Dimensional

Model Slope

MASASHIGE HIRANO*

Abstract

-1-

Establishment of a reasonable three-dimensional model of slope de-

velopment is now one of the most important aims in theoretical geo・

morphology. The process of gully development on three-dimensional

model slopes was investigated here by means of computer simulation.

Rainsplash and overland flow dominate hillslope erosion over large

areas of the world. The former is responsi ble for hill top con vexi ty and

the latter causes the development of drainage systems.

Mass flux by rainsplash is approximately proportional to the slope

gradient， s， and that by overland flow is probably proportional to the

boundary shear which is given by some power function of the slope

gradient and the equivalent slope length， l. Therefore， the equation

describing slope development has the form:

δzδ -1θ~ . a --一一{(a+blmsn-I

)一一}+一一{(a+blmsn-I)一一)δt ax' .. --. --- ， ax δy ，，- • -- - ， ay

The exponent values of m = 2 and n = 1 were used here based on previ-

ous studies. A simpler form of the above equation is avai1able for n = 1. The equation was integrated numerically by means of finite dif-

ference method， where the equivalent slope length was obtained at

every step， successively approximating the elevation， Z， in a small

* Depart_ment of Geography. Osaka City University.

Sugimoto. Sumiyoshi-ku. Osaka， 558 ]apan

(571)

•

‘

-2-

rectangular portion of the slope ¥vith function. z=a.xy十a2X十 aa)'十a"，

and tracing the stream line on it. Starting.frαn an :initial slope Ylith a

small depression， the change of slope :fo:rm ¥¥fith time '¥vas calculated.

Gulley dcvelopment began ¥vhere the drainage area ¥vas a maximum at

the point just below the initial depression. ¥'ar.ious sets of the ¥1a剖alueおso

αand b i泊nthe equation ¥vere chosen and the results ¥¥Tere c∞ompar陀r閃ed.The drainage area under consideration inc:reases at L:、-i江fthe over-

land flo¥v iぬseff匂ecti旬、ve，and the area approaches斗 maxXiimm{U:口nla Httle afれte

erosion bcgins. T hcn， thc area decreases through t:ime. I:f the erosio口b

ovcrland flow 1S wcak， thc drainage area decreases ，monotonically. In

both cascs. ho¥vever， the drainage area finally conve:rg'、':0a definite

val ue con trolled by re，la ti ve effecti veness of overland flo¥v to r訂nsplash.

An im portant problem for fu ture ¥¥'ork :is to cj:礼円、 preL:礼 ly the nature

of thc stablc drainage systern corresponding to the stcady 試atesolut:iou.

I . INTRODUCTION

A slopc dcvclop often under a nv

u

¥vhich rJIl

con trad ictory or opposing cf.fects. l~or .instancc， 011 SOlHe hiUslope只

rainsplash 111akcs t.hc slopc crest roundcd. ¥vhHe overland flo¥¥' 'produ

local steepncss along channcls ¥vhich is the contr.asting feature t:o th

crcst roundness. 1'hc t¥VQ proccsses act sinlultaneously or internlitt.entl

on fl tht・ccべlilncnsional slope. 1'herc.fore， if ¥¥'e ，¥'ish to kno¥" ，vhat cond i t 1011 ¥vould gO¥ l'rn thc d{'¥¥'clopnlcn t o{ channels on a slooe，. 'it i

ncccssary t:o in¥rcsLigHte this thrcc-dì :nH~nsi

J¥n、も.;}opcis thrce-d I1nenslonal in natllr、h

ash co¥'crcd slopc on Nlt.St. l'lclcns ¥vher

nal problen1.

For instan ---a--届帽• .1 .sh

ndition

n

111()oth inH..ial

i1n川1叶troduccd‘1ぶt:cphr千H、¥veωrcdcposi t:ed by ，(1 PY吋rochlst比.じuHiesthen

oc¥'cloped on these t:cphra. El・osionproceeded lHore at tht

¥¥' hcrc s¥lrf.¥cl" 00¥、conccntra tes， and t:h{~ itritial n haメ

、tllP hasizcd， pl'o<1 ucing H thr(-~c-d Îlllension叫'fonn ¥vith a channel n

¥¥' () r k. ，L:;日川l川t正必訓(Gω;汀rt h(.、gullics 、W、"crep均zaa訓占企~rt

t. :'日lingpl~口.ls h (Collins t，日，¥引叩1日川n刊K川d凪1i1り)u川lntlC.1 nSG). '1、ih11ui1ls.iHit a {lis 1n1{G:c{t:sslilt-v tG むきtnbliミh

57

d開伽

MV

Simulation of Gully Development on Model Slope -3 -

a three-dimensional slope model covering the sim ultaneous action of

both processes， if we want to define a real topographic problem of this

kind.

ー//

:泳三~:認 W:3jjZ:':'200ぷ・

/ M.N.

，00

、-_ーーー‘・ーノ

/

。 4m 2 3

Contour intervol 10 cm

Fig.1Topographic map of a small drainage basin developed on Mt.St. Helens ash-covered slope.Measurement in summer，1982，by means of the device by Ishii (1981). Shadowed portions are logs.

(573)

veral

CQurse， h

n of three-dirnensional ~n 。n prop n

1976; j¥ hnerl， 197

dimensional， becau

o reduce th

cornplicated fonn of n1

frofll the difficul

rn

1 U1 ler for

r'land n τ

u

nH

nal slopes. t.hough Sn

1alvsis on

n threc.dinl

refnar'ka bl

herc nUI11cri1caUy， in

lenglh

The erosinal pr

lent 0・1type p.

1・1type to 2・2t.Yl

nal proce.ss in

lopc 'Icngt.h and

la'ber. 1"hc initial 111

the 1l18Xinlun

ucinf! U

n

1'he 111

hich ln

J・th1

1 slolle t

--a -t

・圃唖・且圃咽

lSU

u the in 、 lcn

rcpresen tati

thc dnJ'in

Ul Ullaet

pe ptて}cessrel

1でo¥vas foll

111 N IRT IL

d'inl('巳l川 dis亡叶hnr厄(¥、hva tn白~\nsp似)(、刊U"‘"t. nr‘0でぞssis E臥tε引2汀n、柁e引1r‘¥1a.， 1_

.. • i.io開司

Hcnt nnd A

drぞ'('>:1)1でssedi n 1t'1

) ntrod uC"i 1l~~ thC' '(~on 1 in u H y t¥l1 uutitH¥， ¥¥'l' h，¥


δ~ 1 a --一一(+qs) (2)

θtρδχ =

for the change of surface elevation Z， where qs is in weight and ρthe

average density of sediment. Here we should be sensitive at the double

signs of the right side of Eq.(2)， and we take negative one if qs is brought

in the direction of increasing X， and posi ti ve for the opposi te case.

The exponent values， m and n， change depending on the kind of

erosional processes， as shown in table 1 after Kirkby(1971). If there is no

runoff， we have qs=q=O， which brings qm=qsm=o if not m=O. The

enl !l(，~! proportionali ty constan t denoted by k or k' also depends on the specific

process.

jw

，

h

i

品目L

V

O

E

L

l

-『

叫

:

作

師

側

川

崎

洲

町

d，d

M

Mlい

L山

川

M

M

w

b

m

h

M

W

j

川if

巾川

Mり

Jル

削

刷

、

J

柄引刷

1

1

J

J

U

F

L

叫川

UHH

，N山

hmv

Table 1; Variation of exponents m and n in Eq. (1 b) due to type of erosional processes (after Kirkby， 1971). List of sources has been ommi tted in the presen t reference list.

Process m n sources

一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一

Soil creep 0

Rainsplash 0 1-2

C. Davison， 1889; Culling， 1963.

Schumm， 1964; Kirkby and Kirkby

(unpublished data).

Soil wash l.3-l.7 l.3-2.0 Musgrave， 1947; U. S. Agric.Res.Surv.， 1961;

Zingg， 1940; Kirkby， 1969.

Rivers 2-3 3 Derived from Leopold and Maddock， 1953.

一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一HM同H

AHHV

are3.例ntO ..

)f cOll

(575)

6

flux iぬSpr訂rop閃)o:r凶r礼川tiona'lto Rrad i怜en川、札tj加n工-a 口nd'¥f-dire氏c乱凶tions.F，of'ω E口la't，el

kind Qf di:fficulty ncver appears in the case of t¥vo-dimensiona

the direction of the streanl line coincides '¥vith that of the co-orl

• IS・

r 'Lo overCOJne the difficuUy 'in t町田圃<umensl.

lling， ¥V，C enl p'loy here a con vlcnti，ona'l nl1ethod 'to 0

t any tilllC and for any geolnetry. The:n1

integrating ，of streanl Hnes in a 10臼Irccta日gular

31titudinal distribution is 8'pproxI'lllat.ed br a sinlp

function as sho'¥¥'n lat引

1111. RA:INSIPLASH 10 10τHIEIR 0・'1τ)'PElPiR，QCESSES

1f ¥¥ r ，a n ash c<OVCI

v.anna hiHsi

lo~ r

rJan nd rainsol r pJ

land

nel'decl

nllnanl. un 11 Ul'len -里

・畠咽圃圃圃唱 lSDI

u k I可百 15 n

nl

'loreovcr.

fornl '" hel

n if n時 11n t:en s，e，

pacity is suH

司spl

、tlncernI1H! th rO5110 11al

nel

h， th

l，h cnsむ塁、 U1

ld the soil

flu丸 111

(1) 1 ，and nlOlnentulll Jll 一、E-E曙.---EB

• ••••••

:f rain drop depcndin

both itnportonl

i ncl i nation

II Ikno'¥vn thAt

ntxolHn)l t.hぞ nl

¥tnnt -皿咽

m-aa-，

畠曹 •

pn1il'd to 'this cast'. if th~ nH¥

Eq.{ 1 b). T'his set ofむNpOll'en't

(1'981) h日srcported the ('^pcrh~ncn

民lopesa11d lplIinklesasmalletr valtl

f口¥11)でntU'nn日ptlrt'Ill t.his Icnse i

i ra11SJIJBort-qhIS ltX PE

1 tYI

uH that，町司 i

l' s( I{:-'P白rrsl111

¥. ，，' ，


where k1 is the constant of proportion. By application of the continuity

condition to Eq.(4) we have the model EL

迎a

(5a)

[~ for the two-dimensional case， where z is elevation，ρ1 bulk density of the

soil， and the slope gradient， s， has been replaced by tan β=δz/δχ. This

is simply the diffusion equation which was first introduced by Culling

(1960，1963) to describe slope evolution by soil creep. Solifluction could

be another example of a 0・1type process， but experimental values are

not yet available to confirm this. The three-dimensional model for a 0・1

type process is， likewise， gi ven by

:ea日

叫

ιrH

a

d

F

'

Ft

・'相比

・LU

R広u

w

明

刊

MM

(5b)

not民

;Fbf

Such a process su bd ues the form of hill tops. Con vexi ty of di vides

produced by this kind of process is one of the essential features of slope

profile evolution as discussed by Gilbert(1909) for the case of soil creep.

The constant a in Eq.(5a) or (5b) is called the subduing coefficent

(Hirano， 1968). Thus， even if a depression is formed on a three-

dimensional slope， it is eradicated as time elapses if a 0・1type process

works exclusively.

The steady-state， two-dimensional profile under a constant rate of

lowering of the slope is obtained by substitution of δz/δt=-k本 intoEq.

(5a)， and the resulting profile is a parabola given by

ion ((

d回|

i droP

i~ ve-

inard

) slo~ ¥、』ノχ

凸

Uχ

〆rt¥

*一G

h一2

一一z

(6)

where 払 isa constant giving the rate of uniform lowering and Xo is the

total slope length.

0.4巾W

MMVF

IV. SHEETWASH AS A 1-1 TO 2-2 TYPE PROCESS

Sediment transport by steady overland f10w has been discussed by a

number of investigators， although most of the experimental and the-

oretical analyses have concerned river-bed process. Raundkivi(1967)

(577)

•

-8-

has stated that various kinds of sediment transport equations can be

summarized in the form of Eq.(la)， generally， and Henderson (1966) has

showed that Einstein bedload formula can be '¥vritten a

q!tJ. = k 'i!J雪 t (7)

which mcans 2・2type p.rocess. :ivlizuta.n:i(1970) also revie¥ved the bed-

Ioad transport formurae and found the range of exponent in Eq.(la) to

bc

'lll=O.9----1.6， n= 1.05，，，_'1.I.

Almost a same result: has been given by Kashi¥vaya(1980) concernin

the experIn1en tal r.il1 developlne:nt.

S tead y overland flo¥v on a slope is o:ften a:nalysed on the basis of the

Darcy-Weisbach equation，

ゆ一f

の)

for in.finitelv ¥vide sheet: f日lovホ、v九， the COI日11ttUttU;jiIn11U1itYCOI日lditionfo:r ove町erla:ndflo、".1可4

under rain.

Q=t.(1)v= (，・-1:)/1. (9)

and thc :ivlood y diagr.an1， ¥vhere .Dunne and })同trich(1980) sho¥ved that

f k lfν =一一=-ー， (10

λ弘 fJ必L、

holds good .fOf' 11leaSUnncnt.s of lanlinar flo¥'¥1 at: e:xpcriilllcnt:al sites in

ハfrica.lt is notcd hore that:

A =an"!a of hills]opc drai:nin反 toa short strip of contour ，vith th lcngth t.札

Q;;，.;..dischHrge of ¥vat:cr，

'r and 1: ~ rai nfa ll int:ensity and infHtatio:n capac:it)・

1J --tnCBn flo¥v veloci tv

nle:-lll f10¥V dcpth，

ρI g.，ν同 .¥vaterdcnslt ¥. gravit:atonlll acc(~lcra t.i()nt .anci kltlc(.ic ¥vat.er

v iscosi t.y!

f~ .Darcy-¥Veisbnch frictiOll fnct:or，

\' ~ 了 l~{'yno l (rs nutnber equHl t.o fJl

1¥ = ::1 rough ness i nciぞX1fol・Lhesurfn •

，:) i ~ ¥


Combining these relations for laminar f1ow， we have

zsD3 ...... ( (r-i)Kν(A ¥11

/3

Q=(r-i)A= 二， and D=1 。一{片KνL乙I<S '-W / J

(11a，b)

Using these relations， the bottom friction (boundary shear) is given by

?_/~ ( (r-i)Kν( A ¥11/3 2/ τ=gDs=ピ吋 (一)十 sペ (12)

l 2 ¥WノJ

Sediment discharge is assumed to be a power function of the bottom

friction， and we have

nrerr Qs2= (て一τcy， (13)

where τ'c is critical boundary shear under which no entrainment of soi1

particles occurs. Especially for τc<<τorτc与 0，we have

( A ¥a/3 _20./3 ___! ~ t.. L _ _20./3 f(γ-i)Kνγ/3 q=K2-)s ， withk2=♂~ V ~/ ~J. V} '-Wノ¥.Z )

(14a，b)

o~

As shown here， the essential morphological quantities concerned are

A/ωand s. Introducing the equivalent slope length， l， by

l=A/uん (15)

where l has the dimension of length， we have

仏:2= k2lmsn. (7 a)

Values of m and n depend onαin Eq.( 13) in this case， and ifα=3，we

.，an川川温，

刊

ω

w hich suggests that

m=1， n=1. (17 a，b)

This means that soilwash is an example of a 1・1type process approxi-

mately.

lt appears conclusively that combinations of exponents between 1-1

and 2・2types can be related to sheet wash. This result is in harmony

with the exponent values given in Table 1 after Kirkby(1971) which

(579)

range frOln }..} to 2・:2or more for soil ¥¥'ash ana flver

hough sonlC uncertainty on th

till argucd

It is 引nphasizedhere that. a 1・1

for slope evolution. As the sedinlcnt discharge for

iv'cn b

-

hCI ne of

. foll

ubl

e---0

••••• ミ叫-E・E

・-E・E・----

EE

hu

e

d

私自

t・、

-EEE

-EE---

r n

ponolng 111

Espccially fOI Lrail! I

X

1・Hn

111 l 陀

正

his i8 t hc Penckian _n

(Scheid ---E • 3

1

n1 n

B，ASI UAT，I

'Il is I iblc n ‘，E喧

‘.圃圃・・-a-E・‘

日 hvthc、8tH11l)f l hOSl' bv t

fOfC'J!oin tion he total 1 t. dl~，(' 1

f cour主芯s虻む'

rlHIl忌p叶'10

point u'Ild

ho't、。i

1 nd Shl'l't

-E・周e

盆，.... 一

、aE一

一-

-aE瞳・

-E喧-

E

L

B

‘，

-也膚温圃巴

• 、.1

QJU 一、..

宅

tl

‘岨・-E・・晶司圃冊、望語、‘

•• l

a----丞....

-司

-EE--‘

.• f一一‘

↑

a

，.‘ h;¥

h

l t

Dl'llotilHI 1 a-‘

主

-z一一'I

一、..

‘E・E・

-L、.. z 4.‘、

邑圃

EE--'Tも毒‘

.a咋品、E

・=----、am

a--芝、電一

atz、•. 哩

-掴圃・圃喧-一

-z・u

-噌』

F -・・E

・、.、一‘...

••••• -皿、一、圃皿喧-E--z

lhぞ1111tlxi11111111則11diU1111111P111


下

fx= (k1 +k2lmsn-l)SCOS ()，

/y= (k1 +た2lmSn-l) ssin ()，

w hich gi ve mass fl ux in χ・andy-directions. It is noted here that

ーヌZscos ()= ~z and ssin ()=三一. (20a，b)

δχ ay

Therefore， by application of continuity condition， we have

(21)

~Jì II where a=k1/ρ1 and b=た2/ρ2as before. This is the basic equation we use

here for simulation of gully development on three-dimensional slopes.

Eq.(21) red uces in the two-dimensional case to

az θ ( az . _ _( az ¥n) 一一=一ーヤ一一+bxm(.一一.) ~ (22) δt a.χr-a.χ ¥δχノj

l白11 and analytical solutions for a definite combination of the exponents

have been discussed by Trofimov and Moskovkin(1984) and by Hirano

(1984).

VI. SLOPE LENGTH CALCUしATION

•

In order to integrate Eq.(21) .numerically， it is necessary to calculate s

(slope gradient) and l(equivalent slope length) at any point on a given

slope. Considering the angle () referred to before， we have

(23a) 4ιrs

anu

a屯・τ

角

d

，if the stream lines are everyw here perpendicular to al ti tudinal con tour

fê~t! ， lines. This assumption is not unusual in hillslope modelling. Further，

we can calculate the gradient s at any point on the slope from the

relationship，

αα z s=τ一cos()+士一sin()， (23b)

αx ay

using () gi ven by Eq.(23a).

Derivatives contained here are approximated by finite differences:

川

w

(581)

'.

-12-

(24a) az . Z}-l.k -z} l.k

δx2dx'

(24b) k-I z δ2 ._ Zj.. k+l

δy 2L1y

ituated at regular where Z川 givesthe elevation at the lattlce point (j，k)

+

ー

ー

ー

space of L1 X and L1 y in x-and y-directions

θ

y

(℃

ω工一ωLω↑03)

; ra p h ic rC' pt・(，~ (， llt H tio n of t h(.' dt・21i1131日('，，1 rea and corresp ¥Vldth of OVCtland flO¥¥' aroulld a latth.:l! point. SUI11 of 81'

shado¥¥ cd strips :'Hld dotted part gives t h(' dr日inagcぉrea，Sl1tll of '1.0.日ndU)a Ri ves t hl、¥vid th of t h(' fl，，)¥¥' under cons'idcr-H t lOl¥，. '(、hl':rn只t、1()gl¥('民 thcdirect.ion 0:\ ぉ tt~anl lin(、.11.tlh、Istti POlll t.

EPI

、圃岨-a・・

・・圃膚極

II

(wa↑ershed)

Flg. 2

\~)~:) ')


ln order to get the equivalent slope length， l， which is equal to A/ω，we

may consider a narrow strip(shadowed portions) of slope bounded by

stream lines (A， B， and C) shown in Fig.2. lt is possible to trace the

stream lines starting from the points Ao， Bo， and Co in Fig.2， if the land

surface elevation is locally approximated by a simple mathematical

function. This tracing is carried out here by approximation of the land

surface elevation locally by

z=F(χ， y) =a1xy+a2x+a3y+α4' (25a)

except for the narrow zone along the watersheds where a function

incl uding hig her powers of χand y is better for accuracy.

Especially for Eq.(25a)， stream lines are given by

G(χ， y) =atY2+2a2y-alx2-2a3x-2C=O， (25b)

where the constant of integration， C， is so specified that a stream line

passes a gi ven poin t on the slope. Drainage area， A， is gi ven by

integration of the stream lines traced in this way. ln the case of Fig.2，

the areaA is given approximately by the sum of two shadowed portions

and the dotted part， where the areas of two portions were obtained

separatedly for systematic calculation over the lattice points. The width

of a narrow strip bounded by stream lines is gi ven by

w=ω1+ω2=.L1ycos 8+ L1xsin 8 (26)

approximately， where 8 is defined by Eq.(23a). Graphic representaion of

this relationship for unit rectangle has also been given in Fig.2. Then， l

is calculated at any lattice point.

This method was applied to a model slope given by

z(x， y) =Zr+ω(笠 )x(1 +bncos竺.)，T - - - ¥ 2YI。ノ ¥-• -v---x。ノ' (27)

and the result has been shown in Fig.3， where some representative

stream lines and the equivalent slope length calculated at lattice points

distributed with a regular spacing of L1χ=xo/l0 and L1Y=Yo/10 are

shown. Eq.(27) is introduced as an example of the initial model slope in

the following discussion. This initial slope has a surface depression.

(583)

‘

2 899 。↑ y=1.0

-14 -

Equivalen↑

s lope leng↑h

1.738 1 158

833 947 .776 741 .655.665714

-+0.05

+ 0.25

十0.15

+0.2

+ 0.1

/ 寸~一十一/ /~I /¥プ¥J

/¥ト

-1--T

J

0・

x

¥

.-へ-'¥ / /ノく々

¥0 _，，/

b ya//

と/・ぺ

•

-11111'

，，，， •

.

///

illl-

•

•

J'I・

-Bf+III-

.

y

1.0

0.5

X 1.0 0.5 。

.

Modcl inital slope and reprcsentative streanl l:ines on .I1.obtained numerically.Eq ui valen t slope lengt 11.1，lats111G lowest:portion of the rnodd slope has bGG11given i11tippet-pari，of the fiRUI・v ・

(58'0

Flg.3


VII. GULL Y DEVELOPMENT ON THREE-DIMENSIONAL MODEL SLOPE

吾

We start now from the initial form given by the function equal to Eq.

(27)， which satisfies the boundary conditions，

手=O， atχ=0， (28a) ax

字=0， at y=O， (28b) ay

学=0， at x=χ0' (28c) αx

Z=Zr' at y=Yo・ (28d)

The ini tial form and the represen ta ti ve stream lines traced on i t by the

method employed here have been shown in Fig.3 already. The initial

slope has a major watershed at y=O and minor one at x=O. A major

valley is situated at y=Yo and a depression along the line X=Xo. All of

these four boundary conditions hold at any time.

We consider here especiallY the case where the elevation Zr on the

boundary y=Yo changes with time by down-cutting of the river. If the

rate of down-cutting is given. by a constant， k*， the fourth boundary

condi tion becomes to

，

•

，

~

•

zr=zo-k.t， at Y=YI。 (28d')

~~ .，附則ofEq.(制).This means伽 tste的・S凶 emppm仙 edfinally，

and denudation of the whole slope proceeds at the constant rate k*.

In order to integrate the basic equation numerically， it is possible to

employ an explicit form of finite difference approximation， as we have

now high-speed electronic computers with a large memory which make

us free from the stability problem. The approximations used are the

レノ11 forward difference of t， namely，

az IZ山 -Z.;'¥ {‘.~‘ J (29a)

8t ¥ L1t ノケ.k

and central difference on x and Y:

必ル'

EE

・圃''

(585)

、

-16 -

40

35 、同、

“・・句崎町.

30r 4・・6ー...'- 、.， -，~" ini↑ial value

『- ~ ~

、

¥ト 2899 、、、2 5L ¥くミ℃〈;X¥ 、、、、

..;:・

20 10

10 x 10 grids

15

10

(2-1↑ype)

IOL.ー.L..-..1.

t=00002

A由畳L田島J_1_' .1. -l 1 1 • ...' .......

'__ ーーーーム田・• ..._，_ J-J.聞...L

0.2 0001 0002 001 002 0:

Fig. 4 Changc of t hc lnaXlIllUTTI drainagc arca ¥¥'it.h tinlC on a three-dinlcnsional t110dcl slopc. ¥vhrre t he valur of l/v olS in thr ordi-natr and tIInc in thc absclssa. Nuolcrals b， cur¥ cs gl¥ r theνalu of b G for the in let 111i lieni CAse(i)andbi111111tancollb CAbr i sk rcspcctivcly. Hc()¥，y linc dt 10¥¥・C1' posi t ion s ho¥¥' s C¥. t i nct ion 0'1 thc initi<:11 dep"c~~lon b¥' a 0.1 typc proccss 310n仁.

o2z _ ~ fよu-1・k-23j，K+Zj-1・パδ¥

I

l (!l~\:)2 .L (~~) b )

t~9c)

i n a d d i ti 0 n t 0 E q ~. ( ~ .!.l a! b) f () t・ 17・ t: h ぷ tl~p. Each o( t h(' .four boundar .

、‘，、'ooλv

、.a'，z・・、，t、、


conditions are also approximated by their finite difference expressions.

First of all， the accuracy of the numerical method on tracing stream

lines was examined， using the case w here a 0・1type process works

exclusively. This is called the tfirst' experiment. The steady-state so・

lution of Eq. (21) satisfying the four boundary conditions concides with

that of Eq. (5b) and is given by Eq. (6) with y instead of χ， and a check

of accuracy is easy. The heavy line in Figs. 4 to 7 give the change of 1

just below the initial depression with time. Though some instability at

later stage of calculation has been shown by broken part of the lines， the

result is fairly good. The drainage area decreases gradually， and the

initial depression disappears finally， resulting in a two dimensional

slope wi th uniform length eq ual to yo.

The tsecond' experiment relates the type of erosion， namely， the

intermittent and simultaneous erosions， where the exponent values of m

~ 2 and n = 1 were employed as the effect of slope length is to be

emphasized this time. In the intermittent case， overland flow takes place

1jNtimes as frequently as rainsplash， thougha=b in Eq. (21). Therefore，

b--a/N eventual1y. The simultaneous case means that overland flow

and rainsplash act always together， but the intensity is different， and b

hAa ，where λis chosen equal.to l/N for comparison. The A/ωvalue

clearly fluctuates in the intermittent case as shown by zig-zag line for A

:_ 0.05 in Fig. 4. The lower en velopes for the in termi tten t case ha ve been

shown by brocken lines. The A/w value in the corresponding simulta-

neous case occupies the uppermost fringe of the intermittent one， as

shown by solid lines in Fig. 4.

This means first that wash-out and fil1-back of gully as shown by

Beaty(1959) occurs alternatively in the intermittent case. The inter-

mittent case seems more realistic from this point of view， but， it is also

true that the simultaneous case is reliable and more useful for further

discussion concerning the gully development. Secondly， relative posi-

tion of the curves corresponding to both cases has been decided by

antecedent N steps of a 0・1type process to a step of a 2・1type one in the

intermittent case.

(587)

‘

-18-

1.0 ーー守ー -ーーーーー- - ，----司T

6.0

~~~ザーさV. -唱

iniHol VG lue 一

-~‘ 戸、..... 5....... 、n 晴、‘ 4..348 J 『』

10

I 10 l( 10 I 可弘司、....，." 、h 司... -

..... grids

2.0

O.~ (2-1 type)

、、‘、、7・、二『-、ー邑』ー、~1.0 --'-- .l_

_ 1.

ーー

，-.00002 00005 .0001 .0002 0005 .001 .002 .005 01 .02 05 GI 。2

Fig 5 ιhange of thc mc.1XlInUIn drainage arρe. ¥¥fith tinle on 1.0 x 0.5 nlodel slopc in thc slIn u1tancous case :¥. .'tations are sanle as in Fig. 4..しocationo.f the .rnaXiIl1u:nl value has been distin:guished b¥ OpCtl， sol id. and dottcd circles・

9.0r ー I I ー I ~・・ '

8.0ト initiol"olue 7.822

1.0

6.0

5.0

110"，，0 1

• • • • • • ・

3，0 grid3

•

2.0

。噌S (2・Itype)

1.0 1・，00002 .OOOO~ ，0001 .0002

.L-..o

OOO~ 001 02 01 l忌

Fig. 6 じh;al1n1g{Gt of 1t'lh!h1{:1nH川l'lai日111日1りd園.lelslopc in thc sit川nulb日¥11礼t'、ο¥1S C川日StQ: ¥ tο、tations 羽羽訓re58則I口ncns 1日n1

F'汁l反.5.


•

ー 13.0

110 initiol volue 10.332

5.0

01

-一一』ー-_←4回目ιo4b ・司、も i '一ーー-争ーー-ーーー事ー-司

'・-・-・、、、.

ト・¥B ヘ.、、

k、...._'¥.-・-・、..‘・・-・ー・・・噌』・ー・ー、

JI--11111111lada--

oω

vht4

・，.

0

一9

9.0

-7.0

0.5

30

(2・Itype)

10 t・.00002 ，OOOO~ 0001 0002 .0005 001 002

、、‘、.句-

.005 01 02 05 0.1 02

Fig. 7 Change of the maximum drainage area with time for 2.0 x 0.5 modcl slope. Location of the maximum value of drainage area has been distinguished by cross mark additionally to circles. lt is especially noted here that different value of L1 t brings different resu1ts as shown by a 0.4a curve (L1 t=0.00004) and a 0.4b curve (L1 t= 0.00003).

'fhc・third'simulation experiment was carried out changing the slope length given by Yo， and the effect of slope length on gully development

was evaluated. Fig. 5 to 7 should be compared for this purpose， where

}'o is 1.0. 1.5， and 2.0 for comstan t value of Xo = 0.5. The result given in

Fig.4 is useful too， as this shows the case of Xo =Yo. The constant bo in

Eq. (27) was 50 determined that the maximum slope gradient in

，f-dir'cciion is maintained constantly for all case5 initially. It is clear from

Fig. 4 a:nd Fig. 5， which show rather short slope5， that there is some

cr，itical value of b under which no increase of the drainage area in early

tage is detect，ed. lf b is large， the drainage area increases and arrives at

'the :nlax:imum soon after the start of erosion， and it decreases again. For

:long slopes shown i:n 'Fig. 6 and in Fig. 7， however， fill-back of the

depression :in the earliest stage ooccurs first， and the drainage area

decreases. The:n. a week maximunl appears. This might be due to the

(589)

‘

-20 -

naturc of the initial condition selected here. Even in the case ¥¥rhere b or

YO is large enough， the drainage area decreases gradually as ti'me goes 0'0

aftcr the maximum value of it ¥vas approached once. Th.is occurs re-

gardlcss to thc initial slope length.

l'he final value of drainage area converges to some stable value

depending upon the ratio. b/a.， if the slope length， :.Vo， is constanl τhe

largcr thc value of b and/or l. the faster the stable va:lue is approached.

Ho¥vcvcr. further expcr.iJncnt are necessary to state this conclusively，

bccause some instability has appeared in the nu:merical expe:rimen'ts

sho¥vn hcre. The ins以叫tabilityconles I日10tfrom lattice o:r t:iI日Ille-stepiI日lterva

but frorn latcral shifting of the channel and from bra口chin.s!¥vh:ich make

1 largcst at the middlc of the sJope. lt :is yet hard at the pr邸 entto

concludc ¥vhcather or not the shifing or branching occu同 reasonablht

bascd on thc sinlulation.

VlII. DISCUSSION

.1 considcred a lhree-dhnension

Ollnt: of local chan.Qe o.f dra:in

1 Inodel of slope developnl '

p

・'a

司

宥

EE晶

、圃岨-

ML

，、圃咽，

••• •••••

¥vith t:.itne on n

¥¥'e starl: ，[roln a slope ¥vith a slnall depression on it， the :result oi

tllltllcr.ical int:cgration ()f the basic equation sho¥vs t:hat t:he drain

arC，l on a r日thcrshort slope increases in the ear'イ1y sta:ge of develo

!p〉O Ir. VVliki {仁dcdthat thc intβcnsi ty of ovcrlan刊忙叶df日10収¥Vヘ， ~交dven bv b ‘ is la剖ar~er t.han

SOtlH'心riUcalv山Fη;日1h.川.1 t is clcar froln thc natu rc of basic (~quation that th

1・ill<.'cll¥ ~1111 e depcnds 011 thc shape. of the initinl slopc， especiaHy on

slopc len京t.h01' initiell drHina史CHrea. On a long slopc， fiU.bnck

O¥l、1・landf10¥¥ ¥vith cnough scdirnent discharge takes place at fiJでstt.hen"

111Gre a ppeat-8111e sn111G tGIldG11cv aS011a short，siOPE-

'l'he dt・aillaEG area w ilich once i11C陀 n司'it.1on

H bove dccrc~れlSCS (日1E ;Iβ山ihIn1瓦却raduallvin n¥.1n¥

tο)11 fu川i日rt.her :-日l川ndi t seen¥s 1.0 con velγ'ge to羽 dなf口inilnvohn陀1H{e3〈C:OI日lt:l1

and ¥10. ln actunl topogrHphy， stutP side ¥¥'aUs are fornled if once n s.!uH

:-¥¥'st.ern is f()tllll、d.Hnd COl¥cent:nltion of surL¥c(.' \\lnt(~r thcrc a亡亡elt?rnt


~r.

thc gully development. 1t is difficult to follow this process of accelerated

抑制 decpening of gullies completely by numerical experiment， as the knick

tI!s[:I1 point is smoothed out by approximation of land surface by a smooth

Inathematical function. Flow in a gully is rather turbulent than laminar

flow， and， if once the gully developed， different equation than Eqs. (10)

and (21) would be applied there.

In spitc of these problems， it is probable to consider that a gully has

devclopcd and maintaincd， if concentration of surface water associated

with incrcase of drainage area has once occurred in numerical experi-

111enl. From this point of view， the model slope seems to bear some

drainage system specific to the value of b against a . Precise and detailed

disCllssIon on this point scems to be future problem， because some

instability of the numerical solution appeared in the later stage of

-hnuu

at

，、.

imulation.

Especially on this account， an interesting point can be emphasized.

onsidcr， for instancc， thc steady-state solution satisfying Eq. (22) for m

.2 and '11 = 1. As azlδl=k. over the whole slope， we have

-aa句何

MM

‘drL

・去{(a+bx2)去}=k (30)

:0 The solution of this equation iβgiven by

=Zo+主Liog a・ ? (31) V . '2b

¥¥'hich satisfies a11 o.f the four boundary cond.itions. As δ'zl ay=O over the

.¥¥'hole slope. Eq. (31) satisfies Eq.. (21) too. From uniqueness of the

lution of Eq. (21)， :it is said 什omEq. (31) that no gully is maintained on

this slope in spit，c that ，crosio:n by overland :flO¥Y is considered， so long as

n :starts from a t¥¥'o-d:imensional in:itial slope given by f(x). This

u

appare:ntly a contrad:ictory situation against the facts

ble to dcvclop by Eq. (21.) starting from an initial de-

look

'lhat

n:n

ion. and that '¥'le can observe gully systems dev，eloped ，ev，en on

，'lth alnlost constant length.

rconlC this ，c，ontradiction， i) any antecedent condit:ion Hke the

ndition ¥¥':ith a d，epr民 sion.which permits steady gully system to

00

initial

(.591)

22

be is needed， i:f a or b is unifo:rm and ，vhole slope is homogeneous; or i

some non-unifoTrn condition on a and b or do，vn・slopechange of附 and

n nlight be expected.: 'In order that

actual slopes・

1'he :foJ"lner :is historical and the latter :is hydro-rnechanical proble

T'he latter concerns dO"'J1-slope chang，e ，of the nature of ov'erland flo¥¥

¥¥'hich is not perfectly followred here. These seem very nluch im∞r

ad ull m IS口lalntalnea0

o u日derstandthe origin of prωenl dav .s!ul1

lopes・

日1 elopi on

\VLEI)GE~/IENTS: l"he author ¥"ishes to express his thanks '10

Prof. 1'hon13S :Ounne， 'University of ¥Vashington and to D:r

ietrich， Un'ivers，ity of 'California for critical 陀 adingof nlanuscript an

for d:iscussions. l"hanks 剖'eallso d ue to Prof.丁akaYllkiIshii!

Educational Unh'ersity， ¥¥'ho kindly supplied t.he device for n}easur

rncnt of 111Icrotopography， and to l1r. Bernard HaHet for generou、皿、.

istancc in the c.ourse of study at the University of 、1¥':，aSlllnglOIl!

，vherc the early part .of this shllulation ¥v，as carried out in 1981

14'und for stay in thc 'Un'I1.cd Statcs '''35 supported by Osakaしll¥"L nト

vcrsitv.γhc paper ，¥'as prcsen民dat the First lnternational IConfe陀 n

n1iorphology， Scptenlber" )'985 'in Nlanchester，

REFE.REN

¥hn巴rt，F. (1976) Brier

f lnndfonn

ミnn

nl J}1"C hens'i ¥'e t. h

'F.. SUJ)lll. '~

'tnensionnl sinHdoUan

-島町田-a

、，‘ •.

lUlle nl

;(J()71IO'PI九 N.F.， Snppl. :1

13eaIV.C.11(1959)Slope lreiIでntby g'ull )'ing、Bul

arson. ~vt. j¥. ond K什k'by，; i~11. J. (l '97:2)

111iv.Press-lP13.47、

。Iltns.n. n.川 ldl)nl¥1¥{¥ 'T. (1 986) l~t・nsil)n of t叫lhrnf1も111thl' ~

-eEEa哩

4

、・圃岨縄•

U1¥

‘ ..

az

・‘、，-圃・a噌一・・a方.. 、‘

-z

‘


"“u‘ψ

aHUHV

・l

Mount St. Helens. Geol. Soc. Amer. Bull.， 97， 896-905.

Culling， W. E. H. (1960) Theory of erosion. jour. Geol.， 68， 336・344.

Culling， W. E. H. (1963) Soil creep and the development of hill-side slopes. ]our.

Geol.， 71， 127・161.

Dunne， T. and Dietrich， W. E. (1980) Experimental investigation of Horton

overland flow on tropical hillslopes. Z. Geomorph.， N. F.， Suppl. Bd.， 35， 60・80.

Gilbert， G. K. (1909) The convexity of hill tops. jour. Geol.， 17， 344・350.

Henderson， F. M. (1966) Open channel flow. MacMillan， New York， pp. 522.

Hirano， M. (1968) A Mathematical model of slope development. jour. Geosci.，

Osaka City Univ.， 11， 13・52.

Hirano， M. (1976) Mathematical model and the concept of equilibrium in con-

nection with slope shear ratio. Z. Geomorph.， N. F.， Suppl. Bd.， 25， 50・71.

Hirano， M. (1984) Exact solution of landform equation based on the transpor-

lational laws and continuity condition (in Japanese with English abstract).

]inbun-kenkyu (Bull. Fac. Lit. Osaka City Univ.)， 36. 31・42.

lshii， T. (1981) Microforms and slope processes of the Ashio Mountains in Central

]apan (in Japanese with English abstract). Chikei (Tγans. japan.Geomor. Union)，

2. 279-290.

。[

ob~lt.

-が叫‘川柳川柳川

舟

aHw-ui引F

・"11

1

M

H

H

V

軌

MU

Kashiwaya， K. (1980) A study of rill development process based on field expri-

menls (in ]apanese with English abstract). Geogr. Rev. japan， 53， 419・434.

1ifWli kirkby，M.J.(1971)Hillslope process response models based on the continuity

hferentt 1 cond i tion. Inst. Brit. Geogr・， Spee. PU b. 3 (Slopes form and process)， 15-30.

Mizulani， T. (1970) Erosional process of strato・volcanoin young stage of erosion

(in ]apanese with English abstract). Geogr. Rev. japan， 43， 297・309.

Raundkivi， A. J. (1967) Loose boundary hydraulics. Pergamon， New York (1st edよ

pp. 331.

Savat， ]. (1981) Work done by splash; Laboratory experiments. Eaγth. Surf.

Process.， 6. 275・283.

Scheidegger. A. E. (1961) Mathematical models of slope development. Bull. Geol.

Soc. .Amer.. 72. 37・50.

AH"vz

・川w・

chutnm， S. A. (1956) Evolution of drainage systems and slopes on badlands at

Perth Amboy， New Jersey. Bull. Geol. Soc.Amer.， 67， 597・646.

Schumm， S. A. (1964) Seasonal variations of erosion rates and processes on

hillslopes in We.stern Colorado. Z. Geomoゅh.，Suppl. Bd. 5， 215・238.

mith. T. R. and Bretherton， F. P. (1972) Stability and the conservation of mass in

(593)

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