some are 七erraced fans， some are dissec七ed fans， s七ill o七hers are faul七ed fans， and in occasional cases 七here are 七ilted

ATheoretical Study of the Forms of Alluvial Fans

Teizo  MURATA

In七roducもion

    Among numerous alluvial fans， −that are found in Japan，

some are 七erraced fans， some are dissec七ed fans， s七ill o七hers are faul七ed fans， and in occasional cases 七here are 七ilted

diversified forms of alluvial fans are presen七．

    One of 七he mos七 impor七an七 1）roblems in 七he s七udy of

alluvial fans is 七〇 de七ermine what form an alluvial fan would assume under an ideal condi七ion withou七 any deforma七ion．  This would be a key七〇 elucida七e disloca七ion and deforma七ion which a fan is 七hough七 七〇 have undergone af七er its forma七ion．

    Th・au七h・r ha・wri七t・n ar七i・les in Japanese（1）。n七h。 ba，i。

form of alluvial fans．  This paper is 七〇 discuss fur七her 七he morphology of alluvial fans， based upon recen七 s七udies．

1． The Basic Form of Alluvial Fans

    The mos七 aI）1）ropria七e approach 七〇 s七udy land forms，which

is もhree−dimensionεし1 in nature， is accomplished・wi七h the aid

°f・・nt・ur lin… 1七i・n・七七・m・nもi・n tha七七h・m・rph・1・gi・al

reason， 七he au七hor intends 七〇 make a 七heoretical s七udy on もhe forms of fans with もhe aid of cop七δur lines．

    A fan is of七en expressed on a map by concen七ric contour lines wi七h七he cen七er at 七he mouth of 七he valley． This means that a fan has radial slopes from i七s cen七er a七 七he mou七h of 七h・vall・y in all direc七i・n・・Wha七is call・d a七ypi・al fan

    C・A・COTTON describes in hi・b・・k（2）七he shap。。f a fan as follows；

      ，，The surface of a fan resembles a por七ion of a low cone

    with iも・ap・x in七h・m・u七h・f七h・vall・y・r gully fr・m     whi・h七h・fan building・七ream・merges，もhe sl・pes b・ing

一33一

       lig・ 2 『Contouτ 1ines of ＆fan    Fig． 1  Contour lines of a si且91e fI鵬      ▼ith傷central cone （C）

        on a horizon七al plain．       亀n己1●，tersl cones （L）．

    This describes 七he above−menもioned characteristics of a fan wi七h concentric con七〇urlines well． All the fans， of w．hich 七he schema七ic diagrams and photographs are inser七ed in his 「book， are「buil七at an almos七 linear foo七 1ine of a moun＿

もain． This is expressed by schematic con七〇urs in the manner shown in Figure l． The au七hor refers 七〇 i七 as a single fan．

1七 is no七ed tha七 the contours of the fans of 七his kind are intersec七ed perpendicular 七〇 七he foot line of 七he mountains・

In o七her words， the fan of this kind is a single circular cone severed層by a I）1ane 七hrough its apex in shal）e． Actually，

however， not all 七he fans are formed along 七he linear foot line as is shown in Fi gure 1。 Ra七her， only a few fans fall

    Many fans have 七heir 七〇ps lying a七 some distance in the

moun七ains。 The ground plan looks as if a por七ion of 七he fan

is inser七ed in the moun七ains like a wedge． This is sho・wn in

contours are nσも considered七〇 be I）ar七s of concen七ric arcs・

In o七her words， in 七he case of 七he fan shown in Figure 2， 0nly

    A ques七ion arises as 七〇 wha七 七he shape of the lower I）or七ion of a fan is like・

    Figure 2 illustrates schema七ically a mountain range wi七h a linear foot line and the V＿shaped wedge where the upエ）er

もoward七he lowland， the ex七en七ion lines expressed as dot．ted lines in Figure 2 divide 七he lower Por七ion of the fan in七〇 three par七s．  The part which lies between 七he two lines is

the central cone are referred to as la七eral cones．

    The con七〇ur lines of 七he cen七ral cone are nearly 七he same

q・th・upP・r p・rti・n・f七h・fan…ncen七ri・arcs wh・se cent・r

一34＿

is at i七s al）ex．  The cen七ral cone is， 七herefore， looked upon as a par七 〇f 七he same circular cone as 七he ul）per por七ion．

    On 七he other hand， the contour lines of 七he la beral cones

of the lateral cones．

    The slopes of 七he 江a七eral cones are 七he same as 七hose of

七he corresponding heigh七 〇f the central cone．  1七 is illus1）ra−

ted l）y七he same in七erval of 七he contour li耳es for 七he lateral cones as for 七he cen七ral cone．  This fac七 implies 七ha七 the

central cone is a parもof a circular cone with i七s ver七ex a七七he

．of differen七 circular cones・ And ye七， all the cones involved

蟹撫

lF g°3 ｯ竃錦・亨言，とき8）．  ・・g・…d・・g−F・n（・・7・・…）・

 make a con七inous smooth surface， because each cone aも 七he

 as a whole  a coml）ound landform which is made up of 七hree  cOneS， cen七ral and lateral．

     The ，，wedge，， par七 〇f a fan may be bσrdered by refrac七ed

 foo七 lines， as shown in Figure 4． As many la七eral cones will

 being delimi七ed by s七raigh七 lines drawn in  bhe manner illus七＿

 ra七ed in Figure 3．

     Thus， it is necessary in the morphological s七udy of an  alluvial fan to divide a fan into a number of cones of dif−

 feren七 sizes．

     If the foo七 line o、f 七he bordering moun一七ains is expressed

 by curved lines （see Fi gure 4），七he whole fan can be considered  七〇 be composed of mahy infinitesimal la・teral cones． Ma七hema七ic−．

 ally，七he con七〇ur lines will lbecome involutes of the foo七 ］．ine．

一35＿

  2． Fac七〇rs 七〇 De七ermine 七he Forms of Alluvial Fans

      We have seen in the I）revi ous section tha七 a fan can be

  「ega「d・d・i七h・r a・a・ingle c・n・・r a・a・・ntin・u・ly・・mbin。d

Fhape・f sev・ral・・ne・・Thi・implies七hat七he shape・f a fan

  ユS determined

       by

       the configuraで）ion of もhe foo七 1ine of the   moun七ains which is in 七〇uch wi七h 七he fan．

f。。・

      Only七he specific configuration of 土he foot lines were d・al七Vi七h in th・previ・us secti・n． Namely，七h・ab。v。 di。−

cussed f・・t lines are bent in th・mann・r that th・y bec。m。

wid・r a・もh・y descend・If the f・・七lin・i・lik・七hi。，七h。

  fan is 七〇 be add．ed by new la七eral cones in i七s  lower   part．

      When a fan covers 七he whole width of 七he lowland and

「eaches七h・m・untain・n th・・PP・・i七e sid・， h・w will七he shap・

  of 七he fan 七urn ou七 ？  An examI）le is given in Figure 5．

  The

      River

      Azusa makes an ex七ensive fan from the moun土ains   on 七he wes七ern side of 七he Ma七sumoで）o＿daira Basin． The fan

「eaches the m・un土ain・・n the ea・t・rn・id・・f七h・Ba・in，

wher・the fur七her d・v・1・pm・n七・f七h・fan i・imp・d・d． Th。

−contours show a七ypical landform of a fan as far as 七he fan

 signs of deforma七ion of the fan under 七he influence of 七he  east）ern moun七ains． There are some opinions tha七 the erosion

・f七h・Riv・r Azu・a affect・七he shape・f the c・n七・ur・， f・r七h。

Riv・r run・al・ng七h・f・・t・f the ea・七・rn m・un七ain・， bu七th。

 contours do not indica七e the nature of the river wall。

P・藷§i呈9譜。鑑 ：釜1鵬醜翻留ぎP艦1誌dd・

皿the way・f七h・d・vel・pm・n七・f a fan d。t。rmines七h。 ar。al

i鞭難：購灘響llil叢i難羅1諜鞭

 difference

       b・tween th・m・untain・whi・h affec七七h。 f。rm。f a fan and th・se whi・h d・n・七． The m・un七ain。 whi。h have。u。h a

無h と號謂器1器餓蹴．七・。￡n畿七蓋1量

躍 g濫藷1二：fi。ざh温s濃dfl呈mlhlf竪。・ia：？・bui・ding

忌1藷鞭盛i：藷ll雛諮n驚illl…1轟

the

     foo七

      line a土 an obli que angle．

＿36＿

    工t can be easily distinguished the foot lines which affect

the contour lines are in七ersec七ed perpendicular to もhe foot line or not．

3． The Marginal Line of a Fan on a Map

    The previous sections dealt with七he rela七ion be七ween 七he mountains and七he shal）e of a fan． In 七his sec七ion， the author・w・ill discuss  bhe relation bet・ween 七he fan and the

    The boundary between 七he fan and the lowland is defined as the marginal line of 七he fan． Wha七 is 七he shape of the curve

shape？

    The form of a fan is， as has been discussed in 七he first sec七ion， regarded as a single cone or an assemblage of several cones． First， もhe author 七akes up a fan composed of a single

      ＼        、ノー壕

      ・−ri       糸 轡        tW〕

臨

lk，：lhtWl

2，w

線〜ノ？．・

・kX

LN

趣 ^fv奄?Dx

曳㌦

｣！ψ

ぐ

幽揩牙A

ipag、

』

．，7・『瘁C

羅、

 才陰

錨

    藻一・響・、・醐＼蛭讐上，，、、s／

Fig。 5  Azusagawa Fan （1370，000）．

麟

x量 ignc

勤

潔

雛

・輩藩矯飛誌

Fig．6 Schema七ic con七〇ur lines of a  fan whose areal ex＿

七ension is deli＿

mi七ed by surround−

ing mounもains．

一37一

cone，which can be 七reated as a purely ma七hematical problem．

The curve of 七he marginal line is 七hen determined by solving solid geome七rical equation． Complica一むed fans ・wi11 「be dealt with la七er， based upon 七he resul七 for single cone．

    The plane 七able me七hod of map making re（luires an accurate survey of七he marginal line， because 七he line makes a boundary of different 七〇pographic surfaces along ・which 七he gradien七 changes abrup七1y． But usually 七he marginal line is no七 ex＿

pressed on a 七〇1）ographic map． Moreover， we must pay a七七ention

phic maps of Japan published by七he C−eogral phi cal Survey Institu七e， Minis七ry of Cons七ruc七ion， Japan， in so large a scale as l ： 50，000 are in general considered to be surveyed

fans are no七 always expressed I）recisely on 七hese maI）s．

In 七his sense， 七he s七udy of the marginal line is impor七an七．

fr。mT ﾕ誰 ：2t七欝a藍aΣ。蒲翻1「a撮ζ。灘多！e蹴￡（91c「ease

main七ains 七ha七 a fan has s lopes of the same gradien七．） But in order 七〇 simI）1ify the ma七hema七ical 七reatment， the au七hor assumes 七ha七 a fan is a part of a circular cone and also 七ha七 the surrounding plain is an inclined I）lane。

    Le七 us consider εm or七hogonal coordina七es of three dimen一 七ions． We 七ake 七he vertical line passing through 七he 七〇p of a fan for 七he z＿coordinate， もhe origin， 0， at the intersection of 七he z＿coordina七e with七he inclined plane， 七he in七ersec七ing line be七ween 七he horizon七al plane 七hrough the origin and 七he inclined plane for the y−coordinate， and −the perpendicular line 七〇 七he y＿coordinate on 七he horizon七al plane for 七he x＿coordina七e． Le七 七he in七ersection poin七 be七ween the y＿

coordinaもe and 七he cone 「be 】）． Le七 七he leng七h OD be ♂．

Denoting 七he gradien七 〇f 七he cone by p，and 七he gradient of

（See Figure 7）．  The e（lua七ion of the cone is expressed as

pt薪＝P・Cl＿・

The e（lua七ion of 七he inclined plane is expressed as

z＝−9x

    Hence， 七he equa七ion of 七he in七ersec−ting line on a horizon七al plane be七・w・een 七he cone and 七he inclined plane，

七ha七i・， th・marginal lin・・f a fan・n a map・i・d・riv・d by・limina七ing七h・七・rm・fr・m七h・ab・v・tw・equa七i・n・・

Then，

一38＿

！＿Cl

    92

1＿     P2

d    92

1一 P2

＝1

    As 七he gradien七 〇f 七he fan， p， is greater 七han 七ha七 〇f 七he

plain， g，the a］bove equa七ion deno七es an ellil）se． The elemen−bs of 七his ellipse are，

己

major half axis ^α ＝＝

92

1＿    jP 2

d

minor half axis ^b＝＝

2．a

cen七er

   92

1嗣

Z A

E o

l      l     幽

l       ll       l

置      伽8      10      1

1

＼一一。一一一一／フi1−7        1          〆  1           ・ l      bl d

o

l  l    l

←

」   ，     L し  1    し

＼  1     、」

、 1      、   、

̲IF D y

x

x

yig。7 Schema七ic marginal line

of a single fan．

Fig．8

E i／O

，

ノ 〆

The marginal line of 8 fan in general．

＿39一

   If g is suξfi・i・n七ly・mall・r七han

ア   lr 2   ろ

eCCρntri・i七y  ＿．乳

f・a加eSS @÷÷1十1；）≒÷舞

       Suppose 七he gradi ent of 七he fan is five 七imes as grea七 as 七ha七 〇f the I）lain， then

      9   1

      P   5

      1       e≒

       50

     This implies 七haも， if the gradien七 ratio is・of 七his order，

七he ellipse is・nealy a circle wi七h 七he cen七er a七 a dis七ance of apPr。xima七。・y卸fr。m七h。。rigin。n七h。 x−。。。rdina七。．、n other words，七he cuで．va．−gf七he marginal line of a fan can

pra・七i・ally b・assum・d a・a・ir・1・， wh・se cen七・r i・di・placed

       ♂

191i。：畿ec翻⊇。f3呈も蓋s毛hlh：器e三七e留ξe邑課，bX．lh2h。。urv。P

con七〇urs． The only excep七ion is in 七he case when 七he plain is horizon七al， or g．＝0． Then， the marginal line becomes to be

na七es． Therefore， もhe marginal line and 七he con七〇urs are expressed by the same concen七ric arcs．

    Now， a fan in general is an assemblage of several cones

the aboye−mentioned curve for each cone． A diagram as is illus七rated in Figure 8，enables us to㎞ow七he curve of七he marginal line．

    We deno七e 七he posi七ion of 七he apex of 七he cen七ral cone on

a map by O， the cen七er of 七he ellipse of 七he marginal line by E， and one end of the marginal line of 七he cen七ral cone by G．

If we draw a line O，E，parallel −bo OE， and if we denote もhe

in七ersecもion poinも of OtE「wi七h GE by E曾， then

       O，E曾 3 0E ＝ 0曾G 3 0G

一40一

    The right side is 七he ratio be七ween 七he sizes of 七he

cen七ral and 七he con七iguous la七eral cones．  OE represen七s the dis七ance bet・w・een七he posi七ion of七he al）ex and七he cen七er of

the la七eral cone．  Therefore， E曾will be the cen七er of 七he marginal line （an ellipse） for the la七eral cone with its

cone is 七hus obtained．  The marginal line for 七he cen七ral cone and・七ha七 for 七he la｝teral cone con七acも continuously in

    The curve of 七he marginal line of any fan can be ctete r−

mined in 七his way． This， too， is affec七ed by 七he shape of 七he bordering foo七 line．

    Before we apl）ly七he 七heoretically derived resul七 七〇

actual faロs，七he following three poin七s are もo be remembered．

Firs七，七he actual fan is no七 exactly a cone i七self， for 七he

もhe fan． The curve of 七he marginal line of a fεm is de七er−

mined only by 七he landforms near the in七ersec七ing line between

七he fan have nothing to do wi七h the curve of 七he marginal

the gradient of a cone， bu七 for practical purposes 七he

gradi ent may refer only七〇 七he gradien七 〇f the lower par七 〇f

    Second， 七he surface of a fan never makes a smoo七h curved surface．  Instead， i七 has alwarys minor undula七ions on its surface． Bu七 七hey are neglected in 七he theoretical 七reatment，

because 七hey ar6 consid．ered七〇be of secondary order． The primary surface of a fan should be expressed by concen七ric

con七〇urs．

    Third， 七he surface of 七he I）1ain has been assumed as an

inclined plane． If 七he I）lain has a rela七ively grea七 sloI）e，

the marginal line of a fan −turns out 七〇 be considerably

complica七ed， and 七he 七heory fails 七〇 fi七 七he ac 七ual 七〇pography．

Only those plains with gen七le slopes can be looked upon as a plane．

    Now， here is an example of application of 七he theory 七〇

七he ac七ual fan． The G6dohara fan （Fig． 9） is located in the

regarded as an inclined plane as a whole， bu七 the par七 near

The margin of 七he G6dohara fan has been sUbject）七〇 the la七eral erosion 七〇 form a low escarpmen七． Howev er， if the

＿41＿

t｝hen．the 七heory can be apPlied to i七． Since 七he ra七io of 七he

gradien七〇f七he Ma七sumo七〇−daira Basin七〇七ha七〇f七he G6dohara fan is㎞own，七he curve of七he marginal line can easily be

perfectly． The southern lower par七 〇f the fan seems 七〇

工）rojec七 七〇ward七he basin so abnormally 七ha七 some special cause migh七 be suspeC七ed on 七his par七 〇f th6 fan． This suspec七 is not correc七． but this is an expeCted form from 七he above七heory．

㌦P一・e・・P

   〜／ NI勿ト」

糧

藪；

翻

｝．』1：へ

       1

      Fig． 9  G6dohara Fan （1：75，000）。

    皿1e aわove examples are

values are easily obtained from もhe topographic

ratio of 七he gradi en七 〇f 七he

na七ion of 七he x＿and七he y＿coordina七es）

marginal line are all knOWn．

Un㎞Own， i七iS eS七ima七ed in aerial ver七ical pho七〇graphs

        g

value −。

    Th・R・li・f F・rm Atla・（4）・・ntain・an aerial ph・七・graph・f an arid area in which several alluvial fans are clearly seen．

Fi gure lO is a sketch map of 七he land form elemen七s seen on the pho七〇graph． The apex and 七he marginal line of the fan are easily dis一七inguished on the pho七〇graph． The direction of

difficulty is found in determining 七he x＿ and 七he y＿ coordi＿

na七es and cil． If 一むhe marginal line is regarded as an arc，

もhe cen七er can be loca七ed on the x＿coordinate． So 七he value

      q  l  9

dIS七〇he㎞own．

measuring d and d・

        1

abou七  7   ・  That

abOu七Seven timeS Steep aS

aerial pho七〇graph，

ob七ain the ra七io

      、       Wadi Valley， Sou七hern       MorOCCO。

七he cases in which 七he neccessary

       maps． The   plain 七〇 tha七 〇f 七he fan （de七ermi−

      and 七he curve of the    Even when any one of them i s

some o七her way． For ins七ance，

 can l）e used for de七ermining 七h・e

Since

   The ra七io −bhus o「b bained turns ou七 七〇 be   is 七〇 say，七he fan in （lues七ion has a s lope       七he gradien七 〇f 七he plain．  An

  even if its scale is un㎞o叫enables us七〇

〇f 七he gradien七s．

＿42＿

Conclusion

，  An alluvial fεしn may generally be considered as an assem−

blage of several cones． The shape of a fan is affec一むed by 七he

marginal line of a fan， where 七he fan con−bac七s wiもh七he under＿

lying I）lain， is approxima七ely． an arc， under 七he condi七ion 七ha七

七he plain makes as inclined plane． The cen−ber of 七he arc does no七，、however， coincide wi七h the apex of 七he fan on a topo＿

graphic map・ It is disl）laced from もhe posi七ion of 七he apex 七〇 be

       ・       9

1。ca七。d a七七h。 di。七ance。f apPr・xima七・lyア Cl・fr・m七h・ap・x

    Based upon七he㎞owledge concerning七he basic form of an

七y of deformation by もil七ing． The principle is also useful

ing or flexure． The author will make a fur七her discussion on 七hese problems in some near fu七ure・

    The au七hor is indeb七ed 七〇 his colleagues for 七heir kind advises． He also wishels 七〇 express his 七hanks 七〇 Mr． Kazuo  Nakamura and Mr． Takao Kikuchi for assis七ance in 七his  English publica七ion・

REFERENCES

（1）

））

（∠3

（（

（4）

T．MURATA，（1931）： Theore七ical Considera七ion on 七he Shape

         No。7， PP．569−586

      （1931）：Relation Be七ween a Fεm qnd I七s Surround−

         ing Mountains． Geogr。 Rev・Japan Vol・7， No・8，

      PP．649−663

C．A． COTTON （1948）：Landscape。 2nd Ed。 PP・253−254 W。B． BULL （1964）： Geomorphology、 of Segmented Alluvial

         Geol． Surv． Prof． Paper 352−E ． PP。94−100

1n。七i七u七・G6・graphi・qu・Nati・nal・（1956）・R・li・f F・rm

      A七las， P．147

＿43一