Luthi 80 Some New Aspects of Two-dimensional Turbidity Currents

8/12/2019 Luthi 80 Some New Aspects of Two-dimensional Turbidity Currents

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Sedimentology (1980) 28, 97-105

Som e new aspects of two-dimensional turbidity currents

STEFAN LUTHIGeological Irtstitute, Swiss Federal In stitute of Technology (E TH ),Zurich, Switzerland

A B S T R A C TA theoretical consideration of two dimen sional underflows and surge-type turbidity currents resultsin a general momentum equation. A number of formulae in current use are special cases of thisequation, a mong which are the modified Chezy equation and Bagnold's criterion for autosuspension.Five dimensionless paramete rs ar e included : he R ichardson n umber R i (defined as the inverse squareof the Froude number), the friction coefficient cf, he slope p the dimensionless settling velocity ofthe sediment v , /u an d the changes in flow height with distance d D / d x . The latter is mainly a measureof the dilution by entrainment of ambient water.For chalk powder experiments on surge type turbidity currents and on the initial front of con-tinuous underilows the momentum equation is shown to be correct. Values for Ri range from about1.5 at 0 slope to abo ut 0.75 at 5 an d are slightly to substantially lower tha n values from earlierauthors. The two types of turbidity currents investigated show close similarity. A surprising attributeis their st ron g dilution even at very low-angle slopes. Pelitic sedimentation is possible from the upper,dilute part of the currents, graded intervals found at the base of turbidites can be explained as bed-load deposits from the lowermost, concentrated layer of the current; hydraulic jumps are expectedto be rare in surge-type turbidity currents and fronts of incipient underflows.

T H E O R YThe f low of a s teady tu rb id l ayer a long a slopingb o t t o m c o r r e sp o n d s t o a s i tua t ion know n in hydrau l -ics as an incl ined plume, with an add i t iona l f ac to r :the set t l ing tendency of the par t ic les . The governingequa t ions are: (1) the continuity, (2) the dif fus iona n d (3) the m om e ntum equa t ion (E l li son & T urner ,1959; Johnson, 1962, 1964). The m o m e n t u m e q u a -t ion fo r such underf lows r eads as follows:

C h a n g e i n m o m e n t u m = Grav i ty fo rce m inussettling of particles mi nus friction of bed

The pressure force on the l ayer due to chang ingd e p t h has been, neglected. Using the notat ion ofFig. 1, t h e m o m e n t u m e q u a t i o n r e a d s f o r t w o -dimensional f low:

d uaD)p 2 7 S . A p g DContribution No. 156 of the Laboratory of Experi-mental Geology, ETH.

0037-0746/81/02OCr0097 $02.00@ 1981 International Association of Sedimentologists

in which u is th e av erage velocity, D the layer depth ,A p = p 2 - p 1 the density difference between the flowand the am bien t f lu id , g th e Earth's gra vitationalcons tan t , v the set t l ing velocity of the par t ic les a ndcr he f r ic t ion factor . S s a correct ion factor acco unt-i n g for non-un iform d ensity distribution within thef low; s ince such a correct ion can also b e m a d e w h e ncalculating t he averag e density difference of the flowA p , S is in the fol lowing assumed to be containedin A p .In o r d e r to s implify E qn (1) i t is as sum ed tha t thevelocity of the underf low do es not chan ge withdis tance (uniform f low) , for which there is a m p l eevidence on low s lopes (Chkzy equat io n, see below)a n d on high slopes (from dimensional analysis , seeAppendix). A further simplification is obtained byintroducing the Richardson number Ri , def ined as

* g D . c o s pu2 'i =

which co r responds to the inverse square of theF r o u d e n u m b e r Fr as used, for examp le, by Middle-

? - 2


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98 S. Liithi- densityp.

Fig. 1. Sketch diagram for underflows in an infinitely deepenvironment of ambient fluid. D flow height; H waterdepth; mean underflow velocity; u, return flow vel-ocity; u mean velocity of ambient water entering theunderflow; ,8 slope.ton 1 966b) or by Kersey & Hsii (1 976). Eqn (1) cannow be rewritten as

Thus the change in momentum is basically a changein flow depth, and thus an incorporation of ambientwater into the underflow, which can be thought of astaking place through the fluid interface at a meanvelocity u (Fig. 1) . The change in flow rate Q = u . Dof the underflow with distance is

(3)where the entrainment rate E =u /u has been intro-duced. Eqn (3) is the continuity equation as given,for example, by Ellison Turner (1959) or Turner(1973). Again assuming that u is constant withdistance, Eqn (3) reduces to E = d D / d x and helpsto simplify the momentum equation t o the form

E+cf = Ri(tanl-:), (4)which consists entirely of dimensionless numbers.In the following it will be shown that a number offormulae to describe the flow of turbidity currentsare special forms of the general momentum Eqn (4).

Modified Chkzy equationThis is a formula from channel hydraulics adaptedto stratified flow by various turbidity currenttheorists (Daly, 1936; Ippen & Harleman, 1952;Kuenen, 1952; Stoneley, 1957; Menard, 1964;

Middleton, 1966~) . t says that the acceleratinggravity force acting on the steady turbidity currentequals the opposing frictional resistance and thusresults in uniform flow. Using the notation of Fig. 1this readswhere c is a combined friction factor, taking intoaccount the bottom friction and the friction at theinterface of the two fluids. By inserting again theRichardson number, Ri, defined in Eqn (2) oneobtains R i . t an p = cwhich corresponds to Eqn (4) if one neglects thesettling of the particles. The mixing with the ambientfluid, which in Eqn (4) is expressed by the entrain-ment rate E , can be regarded as implicitly containedin c? as friction a t the interface.

With increase in slope angle, however, bottomfriction seems to become relatively unimportantcompared to the effects at the interface (Turner,1973, chapter 6.2). Middleton (1966~)noted anincrease of the interface friction expressed as afraction of the bottom friction from 0.25 on aslope of 0.14 to about 1.0 on a slope of 2-3 in salt-water experiments. It is presumed in this paper thatthese data resulted from the increasing importanceof turbulent entrainment with higher slopes and notfrom pure skin friction at the interface. The Chdzyequation, when used with a conventional frictionfactor from the Moody diagram (e.g. Massey, 1972,fig. 7.2), should only be applied to underflows onvery low slopes where fluid entrainment is negligible.

A p g D sin /3 = c: .pa. ua,

Flow dominated by turbulent entrainment (Turner, 1973)On steep uniform slopes, Turner (1973, eqn 6.2.5)proposes the flow of inclined plumes or underflowsto be described by

Ri . tanp = E .This can again be shown to be a form of Eqn (4), inwhich the bottom friction and the settling of par-ticles have been neglected. Ellison Turner (1959)and Turner (1973) show this relation to hold truein experiments conducted on slopes between 12 and90 .

Bagnold's (1962) criterion for autosuspensionThis concept was developed by Bagnold (1962) forturbidity currents. I t states that the power needed bythe turbidity current to maintain full suspension


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Some new aspects of two-dimensional turbidity currents 99must be greater than, or equal to, the power avail-able from gravity, minus the power lost by thesettling of the particles, minus the power lost throughbottom friction. In the terms of Fig. 1, Bagnoldsformula (p. 318, bottom) reads

V.Ri.tanP-- 2 cfU

which is Eqn (4) without the entrainment term.The concept of autosuspension has led to diver-

gent opinions among sedimentologists. On the onehand the term autosuspension entered the vocabu-lary of sedimentology; on the other hand it wasnoted that the criterion was never experimentallyverified. Middleton (1966a), as one of the critics,found that it is based on very dubious assump-tions, and he reproached Bagnold for ignoringthe resistance at the upper interface and the sortingand concentration effects of the sediment. Hepointed out that the criterion remains an inequalityeven if the settling velocity is negligibly small,whereas experiments have shown that the equationis actually an equality, namely the modified Chkzyequation. With the foregoing theory it has beenshown that the criterion for autosuspension is anincomplete momentum equation and that it is justlyformulated as an inequality.

Dimensional analysis of Middleton 1966a)By examining the model laws of steady uniformturbidity currents, Middleton (1966a, eqn 9) arrivedat a functional relationship which in terms of thepresent paper reads

in which a is a fraction of cf accounting for thefriction at the fluid interface, and (T is a sortingparameter.

A comparison shows that this requirement isbasically met by our momentum Eqn (4). TheFroude num0er Fr is given in the form of theRichardson number Ri; a which accounts for inter-face effects, is contained on very low slopes (wherethere is negligible fluid entrainment) in the skinfrictipn coefficient q nd on higher slopes (wherefluid entrainment may be dominant) in the entrain-ment rate E; the sorting parameter u s not includedin Eqn (4), because we assume that the sorting ofsuspended sediment is not a major factor controlling

the flow of turbidity currents, although it may haveconsiderable effects on the deposition.

SURGE-TYPE TURBIDITY CURRENTSBesides the continuous-feeding underflow type ofturbidity current another type may be at least asimportant for detrital deep-sea sedimentation : theso-called surge-type turbidity currents, of which theGrand Banks event is the best-known example.These are characterized by a slumping process dur-ing which sediment becomes suspended and movesdownslope as a turbid front. Since the slumpingtakes place for a relatively short time, the feedingwith sediment soon stops and the current wanes.Such turbidity currents are divided into three differ-ent regions of gradually decreasing velocity andcompetence: the head, the body and the tail (e.g.Blatt, Middleton & Murray, 1972). This situation issketched in Fig. 2B. For a simplified physicalmodel, however, it seems better to assume that thesediment becomes immediately suspended andmoves downslope in a turbid cloud (Fig. 2A). Such

1.ilhe ld ?

+ , mring with ambientl iv id (entr rmsnl l

Fig. 2. Sketch of different turbidity currents, defining theterms used in the present article. A and C after Hop-finger & Tochon-Danguy (1977).


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100 S. Luth iand the momentum equation, again substituted withthe Richardson number Ri, reads

Fig. 3. Sampling device for flow density measurements.Suspension samples were sucked off with small steeltubes heights of 1, 4, 8, 12 and 16 cm after the fronthad passed.a 'pure head' turbidity current has been comparedto the front of an incipient underflow (Fig. 2 C )(Middleton, 1966b). The main difference fromturbidity underflows (Fig. 2 D ) is that moving frontsand surge-type turbidity currents are non-steady.From dimensional analysis it can furthermore beshown that surge-type turbidity currents are non-uniform, at least on steep slopes (see Appendix).

For the derivation of the momentum equation theturbid cloud as a whole must be considered ratherthan just a control volume as in the underflow case.The main forces acting on the suspended volume arethe same as in Eqn (1). For the change in momentum

d ( u 2 D l )ne obtains

P a 7where 1 is the length, considered to be proportionalto the height of the turbid cloud. On the assumptionthat the velocity changes are small compared withthe changes in height (and length) of the turbidcloud, the change in momentum reduces to

d Ddx.2. u2 . I 2

5 )In contrast to underflows, an entrainment rate E

is not introduced into the momentum equation,because this term would not lead to further simplifi-cation.

A comparison of the momentum Eqn ( 5 ) forturbidity-current surges with the correspondingEqn (4) for underflows shows that they differ in thechange of momentum, which is mainly from theincorporation of ambient water. The surge-typeturbidity current entrains water not only through itsupper surface but also through its front , which resultsin the term 2 d D / d x instead of d D / d x . Tt must bekept in mind, however, that turbidity-current surgeschange their momentum not only by an increase involume, but also by a decrease in velocity, even onuniform slopes. This can be shown theoretically (seeAppendix) and has experimentally been provedcorrect (Tochon-Danguy & Hopfinger, 1975).

EXPERIMENTSA series of experiments was performed in order:(1) to investigate the mechanics of surge-type tur-bidity currents, mainly in view of the theory pre-viously developed, and ( 2 ) to compare surge-typeturbidity currents with the fronts of incipient under-flows. For this purpose it was necessary to measurethe height, the velocity and the density of the flow.The density of flow has been long neglected byturbidity-current experimenters.

A flume 50 cm deep, 25 cm wide and 570 cm longwas used; at its upper end a gate separated the sus-pension from the freshwater in the tank before therun was started. The flume could be tilted from

.5 to 5 . The sediment used was a chalk powderconsisting mainly of coccolith fragments with amean equivalent diameter of 344pm and a meaneffective settling velocity (see Bagnold, 1966) ofabout 4 x cm sec-'. The quotient v,/u thus wasnegligibly small and no measurable deposition wasto be expected even at very low angles. The initialsuspension density was varied from about 1.01 toabout 1.3 g ~ m - ~volumetric concentration 18yo),a value at which the viscosity of suspensions becomeshighly elevated (Dangeard, Larsonneur Migniot,1965).


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Some new aspects of two-dimensional turbidity currents 101Two types of experiments were conducted in order

to model: (a) pure surge-type currents, with a smallvolume of suspension entering the freshwater tank(resulting in the types A and more often B in Fig. 2)and (b) fronts of underflows, with a larger volume ofsuspension (resulting in type C followed by D inFig. 2).

The experiments were photographed with a time-synchronized, motorized Nikon camera a t intervalsof one second; the frontal propagation and the flowheight could be determined directly from the nega-tives. The density of the flow was determined fromsmall quantities of suspension extracted at severalheights within the flow with the sampling devicedepicted in Fig. 3.

Flow heightThe development of the flow height, measured as themaximum height of the frontal part of the flow, isshown in Fig. 4 A as a function of distance for threedifferent runs. The initial flow height is about 0.45of the initial water depth Ho .On horizontal floors thethickness decreases slightly with distance but in-creases rapidly with increase in slope. There is nodifference between the surge-type flows and theunderflow fronts on the same slope. The increase ofthe flow height with distance dD/dx is plotted as afunction of the slope angle in Fig. 4B and showsthat curve 1 (in Fig. 4B) fits closely to the surge-type curve 2 for higher angles obtained from salt-water experiments by Hopfinger & Tochon-Danguy(1977). The increase of flow height with distance forunderflows is, however, much smaller, as seen fromcurve 3 (from Ellison & Turner, 1959).

VelocityThe velocity, measured as the frontal propagation,did not reveal any systematic relationship with dis-tance, mainly due to the relatively short experi-mental section of 3 m. The velocity dependence onslope was surprisingly small, while the densityseemed to be of greater importance. The velocityranged from about 10 cm sec-l for the most di-lute currents to about 40 cm sec-* for the densestcurrents.

DensitySome density profiles are shown in Fig. 5 . They weremeasured at the end of the experimental section,

D / H ,

I100 200 x cm 360

.

0 10

om

0 06

owcI - o m

Fig. 4. (A) Developmentof flow height D (related to initialwater depth H o ) with distance x , shown for three experi-ments on slopes of O , 3 and 5 . (B) The increase in flowheight with distancedDj dx for underflows and surge-typeturbidity currents, over a wide range of bottom inclina-tionsp. 1 :Luthi (this paper) from chalk powder turbiditycurrents. 2: Hopfinger & Tochon-Danguy (1977) fromsaltwater experiments. 3 : Ellison & Turner (1959) fromsaltwater experimentsdirectly after the passing of the frontal part of thecurrents.

At 0 the sediment is more or less evenly distri-buted over the entire flow height, only at low con-centrations is a regular but slight decrease withheight visible (Fig. 5A). At 3 a marked gradientcan be seen, which is due t o mixing with the ambientfluid (Fig. 5B). This trend of stronger dilution athigher slope-angles is well illustrated by Fig. 5C,which is a dimensionless representation of the den-sities of a series of runs with slopes from .5 to 4 .


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102 S.LiithiAll results illustrated in Fig. 5 are for fronts (or

incipient underflows); however, the experimentswith surge-type turbidity currents showed the samedensity relationships.

O D 0 1.0 3 P l J P .

Fig. 5. Vertical density distributions for large-volume (orfront-type) turbidity currents a11 measured at 3 m dis-tance from gate. y is height, piis density. A) Horizontalfloor, water depth 20 cm. The number near curve indicatesinitial density. (B) 3 , water depth 57 rn. Note dashedline which is a dilute cloud detached from a very denseslurry which flows like adebris flow. Generally, the densitydecreases strongly towards the interface. The number nearthe curve is the initial density. (C) Dimensionless rep-resentation of vertical density distribution. Initial densitycontrast was Apo = 0.05 g ~ m - ~ ,easuring location was3 m from gate. The flow thickness in all runs was approx-imately 0.4 times the water depth H.

Richardson number(Defined in Eqn 2): computations of the Richardsonnumber use the height of the frontal part, the frontalvelocity and an average density calculated from thedensity distribution profiles. Fig. 6 shows Ri as afunction of slope for runs with an initial density con-trast of 0.05 g Fronts and surge-type turbiditycurrents both show a decreasing trend with increasingslope, from about Ri = 1.5 on horizontal floors toabout Ri = 0.75 for 5 . Fronts generally seem tohave slightly greater values of Ri than surges. Theruns of both types with higher concentrationsobviously have lower Richardson numbers, asindicated by the larger dots in Fig. 6.It is interesting to compare these values with thesaltwater 'lockexchange' (or front-type) experi-ments reported by Middleton (1966b) and KerseyHsii (1976), who obtained for angles below 3average values for Ri of 1.8 and 2.0 respectively.This difference may be attributed mainly to thelack of density measurements during the runs, b ecause these authors assumed the density to be con-stant.

Generally we conclude, in agreement with Turner(1973), that heads tend to stabilize Ri around unity.For underflows Ri may vary much more. Ellison &Turner (1959) report values as low as 0.03 andLofquist (1960) obtained values on low slopes ofalmost up to Ri = 10. Hand (1974) found Ri < 1on slopes greater than 0.06 .

+ R i

I I I I I I00 1 29 30 4- 5*--

Fig. 6. The Richardson number Ri as a function of slopep for surge type (circles) and front-type crosses) turbiditycurrents. Initial density contrast is 0.05 g except forlarge circles 0.1and 0.2 g ~ m - ~ )nd large crosses (meanof ten rum with various densities).


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Some new aspects of two-dimensionalturbidity currents 103

Fig.7. Test of momentum equation. There is considerablescatter, but the theoretical Ri . an ,8 = 2 d D / d x + cf isquite closely approached (dashed line), with cf = 0.0050.There is no systematic difference between the surge-typeruns (circles) and the front-type runs (crosses).Test of momentum equationA plot of Ri.tan ,8 versus 2 d D l d x reveals that themomentum Eqn (5) is fairly well satisfied for boththe front- and the surge-type experiments (Fig. 7).Using linear regression one obtains, although withconsiderable scatter, a line following

2 d D / d x = 0.95 Ri . tan ,8- 0.0050,which indicates a reasonable friction factor ofci = 0.0050 (Reynolds numbers were around lo4 o106 .

DISCUSSIONIt was theoretically shown that a number of equa-tions used to describe the flow of turbidity currentsare special forms of a momentum equation. In aseries of experiments which is far from being exhaus-tive, this equation apparently holds true. The changein momentum which enters the equation takes placemainly as change in flow height: by incorporatingwater, the turbidity current increases its mass. Thisprocess can, of course, proceed in the oppositedirection but this was not investigated here.

The dilution of the flows, which is a direct func-tion of this fluid entrainment, is greater with steeperslopes. Underflows are less diluted than surge-typeturbidity currents on the same slopes, because thelatter entrain water not only through the upperinterface but also through their frontal and rearboundary. Fronts of incipient underflows seem verysimilar to surge-type turbidity currents. They bothshow significant density decreases with height atangles as low as 0.5 . In the upper part of thecurrents we find, therefore, a dilute zone, in whichthe fine sediment grades may prevail. Pelitic sedi-mentation from turbidity currents (Bouma intervalE), therefore, must not necessarily be attributed tothe passage of a tail, but can also be explained by asorting of the original sediment by the entrainmentprocess. Close to the bottom of the currents there isa considerable increase in density, and there mayeven be a thin layer with bed-load flow. Such a zonecan account for the graded interval A, found at thebase in the Bouma sequence of turbidites. This inter-val, as Sanders (1965) points out, does not settle outfrom suspension as do the overlying intervalsB to E.

The experiments are believed to be correctFroudian models for turbidity currents. Because thedilution is much lower in underflows, they may reachmuch lower Richardson numbers (or higher Froudenumbers). Surge-type turbidity currents on the otherhand have Richardson (and Froude) numbersaround unity. These findings can be applied tohydraulic jumps in turbidity currents, i.e. the pas-sage from Fr > 1 to Fr < 1, the role of which hasbeen a subject of controversy (Hand, 1974, 1975;Komar, 1971, 1975). While it seems reasonable toexpect hydraulic jumps in underflows, e.g. in theregion of the upper submarine fan, they will prob-ably not be very pronounced, if at all, in surge-typecurrents.

ACKNOWLEDGMENTSI wish to thank all persons who helped me in thecompletion of this work: A. Barfuss, J. Biihler,Professor Th. Dracos, Professor K. Hsu,A. Lambert,W. Schneider, A. Suter, H. P. Weber and Miss LucieWerffeli. Miss Carol Simpson and K. Kelts madeEnglish corrections. E. K. Walton and G. V.Middleton read a first version of this manuscriptand made useful suggestions.


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104 S . LiithiREFERENCES

BAGNOLD, .A. 1962) Autosuspension of transportedsediment: turbidity currents. Proc. R. SOC. U S A ,BAGNOLD, .A. 1966) An approach to the sedimenttransport problem from general physics. Prof. Pap.US eol. Surv. 422-1, 37 pp.BLATT,H., MIDDLETON, . & MURRAY,R . 1972)Origin of Sedimentary Rocks. Prentice-Hall, NewJersey. 634 pp.DALY, .A. 1936) Origin of submarine canyons.Am.J . Sci. 31,401-420.DANGEARD,., LARSONNEUR,. & MIGNIOT, . 1965)Les courants de turbidite, les coul&s boueuses et lesglissements: resultats dexperiences. C.r . hebd. Sianc.Acad. Sci., Paris, 261, 2123-2125.ELLISON, .H. TURNE R, .S. 1959) Turbulent en-trainment in stratified flows. J . Fluid Mech. 6 423-448.HAND, .M. 1974) Supercritical flow indensity currents.J . sedim. Petrol. 44, 637-648.HA N D ,B.M. 1975) Supercritical flow in density cur-rents: reply. J. sedim. Petrol. 45, 750-753.

HOLT,M . 1961) Dimensional analysis. In: Handbookof Fluid Dynamics (Ed. by V.L. Streeter), chapter 15.McGraw-Hill, New York.HOPFINGER,.J. & TOCHON-DANGUY,.-C. 1977) Amodel study of powder-snow avalanches. J. Glaciol.IPPEN, A.T. & HARLEMAN,. R . F . 1952) Steady-statecharacteristics of sub-surface flow. US Nut. Bur.Stand. 521,79-93.JOHNSON, M.A. 1 962) Physical oceanography : turbiditycurrents. Phys. Oceanogr. Sci. Progr. SO, 257-273.JOHNSON, M.A. 1 964) Turbidity currents. Oceanogr.Mar. Biol. Ann , Rev. 2, 31-43.KER S EY , . G . & HsU, K.J. 1976) Energy relations ofdensity-current flows: an experimental investigation

Sedimentology, 23,761-789.KOMAR, .D . 1971) Hydraulic jumps in turbiditycurrents. Bull. geol. Soc. Am. 82, 1478-1488.KOMAR, .D. 1975) Supercritical flow in density cur-rents: a discussion. J. sedim. Petrol. 45, 747-749.KUENEN,PH.H. 1952) Estimated size of the GrandBanks turbidity current. Am. J . Sci. 250,874-884.LOFQUIST, . 1960) Flow and stress near an interfacebetween stratified liquids. Phys. Fluids, 3 158-175.MASSEY, .S. 1972) Mechanics of Fluids. Van NostrandReinhold, London. 508 pp.MENARD, .W. 1964) Marine Geology of the Pacific.International Series of Earth Science, McGraw-Hill,New York. 271 pp.MtDDLEToN, G.V. 1966a) Small-scale models of tur-bidity currents and the criterion for autosuspension.J. sedim. Petrol. 36 202-208.MIDDLETON,.V . (1966b) Experiments on density andturbidity currents. I. Motion of the head. Can. J .Earth Sci. 3,523-546.MIDDLETON, .V. 1966~)Experiments on density andturbidity currents, 11. Uniform flow of density currents.Can.J . Earth Sci. 3 627-637.SANDERS,.E. 1965) Primary sedimentary structuresformed by turbidity currents and related resedimenta-tion mechanisms. In : Primary Sedimentary Structures

315-3 19.

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and their Hydrodynamic Interpretation (Ed. by G.V.Middleton), pp. 192-219. Spec. Publ. SOC. econ.Paleont. Miner., Tulsa, 12.STONELEY,. 1957) On turbidity currents. Verh. Kned. geol.-mijnb. Geneol. 18, 279-285.TOCHON-DANGUY,.-C. HOPFINGER, .J. 1975)Simulation of powder-snow avalanches. Proc. Grindel-wald Symp., Snow Mech. Symp., IAHS-AIHS Publ. 14,TURNER,J.S. 1973) Buoyancy Effects in Fluids. Cam-369-380.bridge University Press. 367 pp.

APPENDIXDimensional analysisThis is a means to eva lua te the func t iona l r ela tion -sh ip be tween an unkno wn quan t i ty and a num ber o fvar iables (see, for example, Holt , 1961). I n t h efol lowing i t is used to evalu ate the character is tics oftu rb id i ty cu r ren t s on steep slopes.UnderflowsI t is assumed th at th e dis tance travelled for a con t ro lvolum e of the denser layer is a funct ion of t ime t a n dof the buoyancy f lux A = ( A p / p ) g. Q (in which Qis the discharge in two dimensions) or x = f ( t , A)thus neglect ing the f r ic t ion at th e bot tom , the set t l ingof th e par tic les and var iat ions in s lope. Accordingto the n- theorem the dimens ionless form o f x is afunc t ion o f n dimensionless combinat ions , wheren is the num ber of var iables on th e right-hand side,a n d k i s the num ber of var iables with independentdimensions. As b o t h n a n d k are 2, i t fo l lows that

X f 0 ) = cons tan t .A1/3tI f th ere is no depos i t ion an d if th e buoy ancy f lux iscons tan t w i th t im e and w i th the use of u = dx/dt , i tfo l lows that u = cons tan t , o r tha t the f low o f theturbidi ty curren t is uniform. Similar reasoning withthe funct ion D = f t , A ) yields D - or D x ,which sho ws that the f low height increases propor-t ionally with distance. The dens i ty of course isinversely propor t ional to the f low height , whenceA p - / x . A com plete funct ion al relat ion ( includingal l var iables neglected in the foregoing section) readsw i t h t h e same n o t a t i o n as in the t ex t

where A is omit ted s ince A = ( A p / p ) g .D . u , of


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Some new aspects of two-dimensional turbidity currents 105which all quantities are separately listed. Becausen = 6 and k = 2 it follows that a function of fourdimensionless numbers should equal a dimensionlessrepresentation of x . This requirement is met by themomentum Eqn (4).

Surge-type turbidity currentsIt can be assumed that a certain volume B of sedi-ment becomes at once suspended, with B = ( A p /p ) g V in which Y is a volume in two dimensions).The same procedure as before yields

x = t ,BXS-= f 0 ) = constantt 2 . B

and if there is no deposition (B constant)

which shows that the flow velocity is not constant,but decreases with distance. Evaluating the func-tional relations for the flow height, the volume andthe density yields D x , V 9 Ap x - ~ .Againbottom friction, settling of particles and variationsin slope have been neglected. If they are considered,a similar procedure as for underflows gives themomentum Eqn (5) as a solution of the functionalrequirement.

(Manuscript received 8 October 1979; revision received 21 February 1980)

Luthi 80 Some New Aspects of Two-dimensional Turbidity Currents

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