sedimentology_v49_p397

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Flow structure of turbidity currents

M . FELI XSchool of Earth Sciences, University of Leeds, Leeds LS2 9JT, UK (E-mail: [email protected])

ABSTRACT

A two-dimensional numerical model is used to describe the flow structure ofturbidity currents in a vertical plane. To test the accuracy of the model, it isapplied to historical flows in Bute Inlet and the Grand Banks flow. The two-dimensional spatial and temporal distributions of velocity and sedimentconcentration and non-dimensionalized vertical profiles of velocity, turbulentkinetic energy and sediment concentration are discussed for several simplecomputational currents. The flows show a clear interaction between velocity,turbulence and sediment distribution. The results of the numerical tests showthat flows with fine-grained sediment have low vertical and high horizontalgradients of velocity and sediment concentration, show little increase in flowthickness and decelerate slowly. Steadiness and uniformity in these flows arecomparable for velocity and concentration. In contrast, flows with coarse-grained sediment have high vertical and low horizontal velocity gradients andhigh horizontal concentration gradients. These flows grow considerably inthickness and decelerate rapidly. Steadiness and uniformity in flows withcoarse-grained sediment are different for velocity and concentration. Theresults show the influence of spatial and temporal flow structure on flowduration and sediment transport.

Keywords Flow structure, mathematical model, multiple grain sizes, steadi-ness, turbidity currents, uniformity.

INTRODUCTION

Despite the widespread interest in turbidity cur-rents, their flow structure remains poorly under-stood. Interpretations of deposits based onhorizontal and/or vertical distribution of para-meters such as velocity, sediment concentrationand grain size (e.g. Kneller & Branney, 1995), andflow process models (e.g. Peakall et al., 2000), useconceptual ideas about flow structure that arepartly based on observations of natural currents

or laboratory currents or on the output fromnumerical models. However, all observed andmodelled flow structures comprise only a limitedrange of turbidity currents. Natural turbiditycurrents for which the flow structure has beenobserved are mainly muddy or silty very-lowdensity currents in reservoirs (Tesaker, 1975; Fan,1986; Chikita, 1989; Samolyubov, 1990; Umedaet al., 2000). These currents are generally theresult of a continuous input, with velocities of theorder of several dm s)1, and volumetric concen-

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Vertical profiles presented in these studies weredetermined from measurement of flows generatedfrom a (quasi) steady input. Measurements atdifferent vertical positions were made at differenttimes and then combined to yield one verticalprofile, assuming that the horizontal structure didnot change (Taylors frozen flow field hypothesis).

Therefore, these profiles are only valid for steadyuniform flow so that, for flows that are not steadyor uniform, the flow structure remains unclear.Other problems with this methodology are thatflumes are of limited depth, and so turbiditycurrents can only grow to a thickness determinedby the flume, with counterflow at the top of theflume sometimes influencing the flow structure.Additionally, as most experimental currents areof very low concentration (a few g l)1), it isunclear whether experimentally derived flowstructure is applicable to high-concentration

flows. Thus, as for natural scale flows, flowstructures derived from laboratory flows are onlyvalid for a limited range of concentrations, andlong-term flow development, such as steadinessand uniformity effects, cannot be described.

Descriptions of the entire flow structure ofturbidity currents are not available. Alahyari &Longmire (1996) described the two-dimensionalvelocity structure in the nose of a saline densitycurrent but did not include density information.Kneller et al. (1999) described the velocity andTKE of a saline density current. Velocity timeseries at different heights in the flow were

transformed by Kneller et al. (1999) to give atwo-dimensional image of flow structure in theentire flow, assuming that the current is steady.Best et al. (2001) presented the only instantane-ous two-dimensional vertical visualization ofvelocity and turbulence characteristics of a tur-bidity current for several different times. Unfor-tunately, density was not measured. Allnumerical models that describe flow structureare one-dimensional and describe only verticalprofiles (Hinze, 1960; Stacey & Bowen, 1988;Eidsvik & Brrs, 1989; Brrs & Eidsvik, 1992).

These models are necessarily only valid foruniform flow, and unsteadiness or non-uniform-ity effects cannot be described by these models.Although such effects can be partly described bysome analytical or depth-averaged models (e.g.Fukushima et al., 1985; Parker et al., 1986; Dade& Huppert, 1994; Bonnecaze et al., 1996), neithertype of model can describe vertical flow structure.Currently, no multidimensional mathematicalmodels have been applied to describe flow struc-ture.

Thus, work on flow structure has so far beenrestricted to slow, low-density flows, and onlypart of the temporal and spatial evolution hasbeen described before. Thus, the influence of theflow structure on flow duration and sedimenttransport remains unclear. In this paper, flowstructure will be described using the two-dimen-

sional numerical model of Felix (2001), whichsimulates unsteady flow in a vertical cross-sec-tion of a turbidity current. This makes it possibleto describe the interaction of velocity, density andturbulence, their influence on steadiness anduniformity and the temporal and spatial evolu-tion of turbidity currents, something that is notpossible using available observed flow structuresor previous numerical models. Several simpleflows will be described, ranging from low to highsediment concentration and from single-compo-nent to multigrain flows. Only internal controls

on flow structure will be considered, and externalinfluences such as topography are not taken intoaccount. The first part of this paper describes theresults of the simulation of historical flows, thiscomparison with observed flows being necessaryto ensure that the model results shown in thesecond part of the paper describe a realistic flowbehaviour. In the second part of the paper, thetwo-dimensional structures of several computedflows that are not related to historical flows areexamined. Although the model used here is two-dimensional, the governing principles describedwill be the same for three-dimensional flows.

MODEL CHARACTERISTICS

The model describes flow in a two-dimensionalvertical plane (see Appendix for the equationsused). The turbidity current forms from an ini-tially static column of suspended sediment (zerovelocity and zero turbulence) and flows overvariable topography. Several grain sizes for non-cohesive sediment are modelled for low to highvolumetric concentration (up to several tens of

per cent). The velocity of the bulk fluid (sedi-ment + water) is calculated from a momentumequation derived from the NavierStokes equa-tion and simplified using the hydrostatic and thinshear layer approximations [Eq. (A2) in Appen-dix]. Sediment concentrations are calculated withan advection-diffusion equation for each grainsize [Eq. (A3)], and turbulence is taken intoaccount using the MellorYamada level 25 sec-ond-order closure model [Mellor & Yamada, 1982;Eqs (A5) and (A6)], which solves for TKE

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2002 International Association of Sedimentologists, Sedimentology, 49, 397419


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q2 u0iu0i (where ui is the fluctuation velocity

obtained from Reynolds averaging) and a lengthscale l. Sediment entrainment is calculated usingthe empirical function given by van Rijn (1984).The equations are solved numerically using afinite volume method with a second-order impli-cit backward differentiation formula method. For

a more detailed description of the model, see theAppendix and Felix (2001).

SIMULATION OF HISTORICAL FLOWS

To test the ability of the model to describeturbidity currents, model results need to becompared with measured field values. Althoughdeposits are not considered in the second part ofthe paper for the computational flows, for naturalturbidity currents, they are the only measured or

observed indicators of sediment transport capa-city. Therefore, to test whether the model cantransport sediment over realistic distances,deposit information needs to be used. As there isno data set for a single flow that gives velocity andsediment distribution in the flow (let aloneturbulence characteristics) and also the resultingdeposit, observations of different flows will beused to test different model components. As aresult of this, test results can only indicatewhether the model leads to realistic flow beha-viour, as the exact starting conditions of historicalflows are unknown.

In the first test, numerical results will becompared with deposit grain-size data from ButeInlet (British Columbia, Canada) given by Zenget al. (1991), thus testing whether the model cangive realistic grain-size distributions that arecomparable with those observed for natural flows.The measured grain sizes are from grab samples

that give an average for several flows. These grainsizes cannot be linked with the velocity measure-ments of individual currents and, therefore, onlythe general grain-size distribution will be consid-ered. In the second test, the 1929 Grand Banks(Newfoundland, Canada) turbidity current issimulated. Times of underwater cable breaks

given by Heezen & Ewing (1952) are comparedwith numerical results, thus testing whether themodel produces realistic velocities. Grain-sizeand thickness distributions of the deposit haveonly been described in general terms (e.g. Piperet al., 1988) and, therefore, only a rough compar-ison is possible. The depositional and velocitypatterns of these historical flows might be repro-duced by models that are simpler than the presentone, but the aim of this paper is to look at the flowstructure of turbidity currents, for which thepresent model has to be used.

BUTE INLET

Bute Inlet is a fjord in British Columbia, Canada,with a maximum depth of 660 m. It is 78 kmlong, and the average width is 43 km (Prioret al., 1986). At the fjord head is a delta complexsupplied by the Homathko and Southgate rivers,and several smaller fans are present at the sidesof the fjord. The slope of the fjord bottomdecreases from its maximum (005) near the fjordhead to the basin floor about 55 km away

(


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are reported from the basin floor beyond thisdistance. Channel depth ranges from 34 m nearthe delta front to 6 m near the end of thechannel. The average sediment load from theHomathko river is 15% gravel, 65% sand, 15%silt and 5% clay (Prior et al., 1986). The fjord-side fans also deliver gravel, sands and silts to

the channel (Prior & Bornhold, 1989). Sedimentin the upper 15 km of the channel is moderatelyto well-sorted, fine-grained sand. In the lowerpart of the channel, grain size decreases fromfine-grained sand to mainly clay in the distal partof the depositional area.

Bornhold et al. (1994) examined the possibleinitiation mechanisms of turbidity currents inBute Inlet by comparing high-velocity events inthe channel with earthquake data, storm data,hyperpycnal flow possibilities, human distur-bances, extreme low tides and failure of rapidly

accumulated sediment deposits, all of which cantrigger turbidity currents. Bornhold et al. (1994)concluded that the failure of rapidly accumulatedsediments was the most likely initiation process,which is simulated by the model initiation pro-cess used here.

Numerical modelling of Bute Inlet flows

To simulate flows in Bute Inlet, three differentinput concentration profiles are used (Fig. 2). Thegrain sizes range from 002 mm (silt) to 1 mm(coarse sand), so that all non-cohesive grain sizes

measured by Zeng et al. (1991) are included.Smaller grain sizes are observed in Bute Inlet (seeFig. 3), but mud is not included in the presentmodel. As an initial condition for the numericalmodel, all grain sizes are distributed uniformly ina static suspension column 1000 m long and with

an initial height of 50 m for input 1 and 75 m forinputs 2 and 3.

A comparison of the average grain size calcu-lated by the model and that measured by Zenget al. (1991; Fig. 3) shows that the shape of thegrain-size profile is similar for all the differentinput concentrations, but the differences in theaverage grain size are caused mainly by thedifferent input volumes. Larger input volumeslead to faster flows, so that coarser material istransported further. The observed coarse parti-

cles at 30 km (Zeng et al., 1991) result frominput from the valley-side fans, whose input isnot modelled here, thus causing the grain size todecrease regularly downslope from the deltafront. Figure 3 demonstrates that the model isable to simulate sediment transport distances

Fig. 2. Input concentrations and grain sizes for ButeInlet runs. Input grain sizes are given by the symbols onthe lines; the connecting lines are shown for conveni-ence only. The flows have different input volumes, asdescribed in the text.

Fig. 3. Average grain size in ButeInlet as measured and computed.Lines are model results; + symbolsare measured values from Zenget al. (1991). The small observedgrain sizes around 50 km from thedelta front are mud, which themodel does not take into account.The smallest grain size used in thesimulation is 002 mm, which isfound in the distal simulateddeposit.

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and grain-size trends on a scale observed in ButeInlet.

GRAND BANKS 1929 TURBIDITYCURRENT

Observations from the Grand Banks (Newfound-land, Canada) current are used to examine thecurrent velocity, as well as the deposit thick-ness. The turbidity current was generated by amagnitude 72 earthquake west of Grand Banks,with its epicentre on the continental slopeabove the Laurentian Fan (Piper et al., 1985).Instantaneous breaks of telephone cables on thecontinental slope within a radius of 100 kmfrom the epicentre were interpreted by Heezen

& Ewing (1952) to result from failure of sedi-ment on the slope. Continuing retrogressivesliding resulted in debris flows that transformeddownslope into a turbidity current (Cochonat &Piper, 1995; Piper et al., 1999). This turbiditycurrent flowed southwards down the LaurentianFan onto the Sohm Abyssal Plain and causedsequential breaking of another 11 cables, up to13 h after the earthquake. Velocities deducedfrom the timing of the cable breaks indicatevalues from 41 to 73 km h)1 (Piper et al., 1985).The turbidity current travelled %1700 km from

the epicentre, of which about 1300 km was onthe nearly horizontal abyssal plain (Horn et al.,1971).

Sediment erosion generated during this eventwas up to 60 m in the valley floor channels andabout 515 m over most of the upper part of theflow path (Piper et al., 1988; Hughes Clarke et al.,1990). Deposits of the current decrease from up to6 m of coarse sand and gravel in valleys on theLaurentian Fan to deposits over 1 m of fine sandand silt on the northern Sohm Abyssal Plain,

finally thinning to


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The modelled distances compare well with theobserved distances at the times of the cablebreaks, albeit not exactly, which results at leastpartially from the error made in projecting thecable breaks onto the two-dimensional profile.The difference at the beginning of the travelledpath is probably also caused by the differentinitiation process of the flow, with the modelledflows starting as a static suspension, whereas thereal flow started as slumps and slides. Thesecohesive masses might have accelerated fasterthan the modelled flow initially, hence leading tomodelled distances that are too small. Later, at

the time of the last cable break observation, whenflow would move from the channel to the uncon-fined abyssal plain, lateral spreading wouldbecome more important, leading to a slower flow.It is not possible to simulate this lateral spreadingwith the two-dimensional model, so the modelled

flow will be too fast once lateral spreading startsin the real flow. Despite these inaccuracies, themodelled flows reach approximately the distanceof the cables at the times of the observed cablebreaks, so the velocities of the modelled flows areof a realistic magnitude.

Total deposit thicknesses are shown in Fig. 7.The general thickness distribution is comparablewith the thicknesses mentioned by Piper et al.(1988): a deposit thickness of 16 m at thebeginning of the Sohm Abyssal Plain and severaldecimetres at the end of the plain. The modelleddistal thickness is greater than the observed

value, again because of lack of lateral spreadingin the model. The computed proximal thicknessis larger than the observed thickness, which is aresult of the difference in initiation process.Sediment will not deposit quickly from a cohe-sive slide, whereas in the model presented here,

Fig. 6. Comparison of observed andmodelled distances of propagationfor the Grand Banks flow. The timeand distance of cable breaks arefrom Heezen & Ewing (1952). Themodelled flow distances are basedon the position of the flow front.

Fig. 7. Total deposit thicknessesfor the Grand Banks model runs forthe four different inputs given inFig. 5.

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coarse material will immediately start depositingbecause of the initial lack of turbulence needed tokeep sediment suspended. The slumping thatcauses the erosion in the uppermost part of theflow path is also not modelled, again because ofthe initial condition of a static suspension. Theoverall thickness decrease in the deposit after the

initial phase is thus well simulated, showing thatthe model can realistically reproduce thicknesspatterns and thus sediment transport processes.

APPLICATION OF THE MODEL

Now that the model has been shown to model thegross characteristics of turbidity current flowrealistically, the flow structure of several simpleflows will be considered. The two-dimensionalspatial and temporal structure will be examined

in terms of the horizontal and vertical distribu-tion of velocity and sediment concentration andin terms of flow uniformity and steadiness.Vertical non-dimensionalized profiles of velocity,concentration and TKE of the different flows willbe compared. First, three flows will be comparedto see how a single component (salt, fine sedi-ment, coarse sediment) determines the spatialflow structure. Secondly, two multiple grain-sizeflows with identical grain-size distributions butdifferent initial concentrations will be examinedto understand the influence of sediment concen-tration magnitude on the spatial flow structure.

Finally, multiple grain-size flows with the sameinitial concentration but with different grain-sizedistributions will be examined at four differenttimes to determine the effects of grain-size distri-bution on the temporal flow structure.

Single-component flows

Small-scale experimental studies of turbiditycurrents have been based on both particulateflows and saline density currents. The latter arenot actually turbidity currents (in which particles

provide the density difference with the ambientfluid), but they are assumed to represent currentswith very fine particles (clay). Here, a comparisonis made of the spatial flow structure of a salineflow and two particulate flows, one with sedi-ment with diameter d 005 mm (silt flow) andone with sediment with diameter d 05 mm(sand flow). The density of the sediment is2650 kg m)3. In the saline flow, density q wascalculated for a flow at 5 C after van Rijn (1993)from

q 1000 1455Cl 00065T 4 04Cl2 1

where Cl is chlorinity and T is temperature (inC). At time t 0, the dense fluid (water + saltor water + sediment) is distributed uniformlythroughout a column 50 m high and 500 m long.The initial concentrations were: saline flow, 15%;

silt flow, 25%; and sand flow, 7%. Theseconcentrations were chosen so that all flows haveapproximately the same maximum velocity at thetime shown (all flows are shown 2000 s afterinitiation). All currents flow down a slope ofconstant gradient.

Flow field

Contour plots of downstream velocity and con-centration distributions are given in Figs 8 and 9respectively. The lowest velocity contours are01 ms)1 for all flows. This allows easy compar-

ison, although the velocity, of course, goes to zeroabove this, so these contours do not indicate theactual thicknesses of the currents.

All three flows show a similar velocity distri-bution (Fig. 8) on a large scale, with a high-velocity core immediately behind the head, closeto the bed, and velocity decreasing both upstreamand vertically in the current. A similar image isgiven by Kneller et al. (1999) by transformingvelocity time series of a density current and,perhaps more convincingly, by the instantaneousvelocity measurements in a turbidity current of

Best et al. (2001). However, this is only a large-scale similarity, and both horizontal and verticalstructure are different for the three flows. Close tothe bed, the vertical velocity gradient is higher forthe flows with particles that are coarser andheavier, while the horizontal gradient is lower.Higher up in the flow, where the concentration isvery low, the velocity gradient is low. Owing toincreased settling velocity, heavier and coarserparticles will concentrate close to the bed, whichincreases density and velocity over a narrowervertical range and leads to a high vertical velocitygradient. Salt and silt undergo greater mixing andwill therefore lead to lower vertical velocitygradients (Fig. 8A and B). For larger and/orheavier particles that are more difficult to mixand are therefore concentrated closer to the bed,the velocity maximum is located closer to the bed.

The concentration distribution (Fig. 9) is differ-ent from the velocity distribution. For the salineand silt flows (Fig. 9A and B), both vertical andhorizontal density gradients are comparable as, inthese flows, suspended material is distributed

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throughout the entire flow, and the flows have lowhorizontal and higher vertical concentration gra-dients. The situation is somewhat different for thesand flow (Fig. 9C and D), where most sedimenthas dropped out at the time shown. In this flow,sediment is only present in the nose of the flow,even though the horizontal velocity structureshows a low gradient. The loss of sediment fromthe flow as a result of deposition is thus faster thanthe loss of momentum resulting from reduceddensity. As a result of this difference in timescales, flow uniformity is not the same for velocity

and concentration in the sand flow. Concentrationat the tail of the flows has not decreased to zerobecause of the initial conditions of the model. Asthe flow starts as a static sediment suspension, notall the material gets transported with the turbiditycurrent, but some slowly settles out. Because ofthis, there is still some sediment left, whichresults in non-zero velocities.

The different horizontal flow structures arecaused by different mechanisms of momentumdecrease. Flow structure at a point is built up by

the passing of the current over time, if topograph-ic influence is ignored. If the structure of flowchanges slowly after the front has passed, the flowwill be (quasi) uniform. On the other hand, if theflow structure changes rapidly, the flow will notbe uniform. Hence, the flow uniformity is deter-mined by steadiness at a point. The flows shownhere all lose horizontal momentum as a resultof decreasing density difference both with theambient fluid and within the flow itself as a resultof settling and mixing. Loss of momentum isindicated, for velocity u > 0, by

@qu

@t q

@u

@t u

@q

@t< 0 2

where t is time, and can be caused by eitherdecelerating flow (@u/@t < 0) or by decreasingdensity (@q/@t < 0). Only one of these two termsneeds to be negative to result in an overallmomentum loss. In fine-grained flows, @q/@t willbe small, as sediment will drop out of suspen-sion slowly, so most of the momentum loss is a

Fig. 8. Downstream velocity contour plots of the single-component flows, 2000 s after initiation of flow. (A) salineflow; (B) silt flow; (C) and (D) sand flow. The sand flow is shown twice, once (C) with the same vertical scale as thesaline and silt flows, and once (D) on a larger scale to show the entire flow. Contours are in ms )1, and flow is from leftto right. All flows have the same contour intervals. The lowest contours (0 1 ms)1 for all flows) were chosen forconvenience of comparison of the different flows and do not define a top of the current.

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result of decreasing velocity (@u/@t is negative).In coarse-grained flows, @q/@t is large as a resultof rapid sediment fallout, so momentum loss iscaused by decreasing density and is not associ-ated with a large decrease in velocity (@u/@t issmall). In the salt and silt flows, momentum lossresults in high horizontal velocity gradients, asconcentration does not change, but velocity does.In the sand flow, on the other hand, momentumloss results in low horizontal velocity gradients,as concentration changes while velocity doesnot. By dropping out (coarse) sediment, turbiditycurrents create their own velocity steadiness,while, as a result of sedimentation, the concen-tration is not steady. Thus, velocity and concen-tration steadiness are not the same for theseflows. As the sediment has dropped out, theseflows do not slow down as result of a decrease indensity difference with the ambient and thusdriving force, but because of viscous dissipationof turbulence, which is a slower process thancoarse sediment fallout.

The thicknesses of the saline and silt flows donot change rapidly, although they do changeseveral tens of per cent backwards from the noseof the flow. The sand flow becomes much thickerthan the other two: all flows started out with thesame thickness, but the sand flow has grown toabout 25 times the thickness of the other twoflows at the time shown. In the sand flow, there islittle sediment left, so there is negligible stratifi-cation-induced inhibition of mixing (see below).This causes the nose to thicken more than in theother flows.

Non-dimensionalized vertical profiles

To compare the results of the calculations withresults from laboratory experiments, vertical pro-files of velocity, TKE and density are non-dimen-sionalized (Fig. 10). All profiles shown in Figs 8and 9 are taken through the high-velocity corebehind the nose of the flows. The vertical axes arenon-dimensionalized with the height z1/2, whichis the height above the velocity maximum where

Fig. 9. Concentration contour plots of the single-component flows, 2000 s after initiation of the flows. (A) saline flow;concentration denotes chlorinity. (B) Silt flow; concentration is volumetric fraction. (C) Sand flow on the same scaleas saline and silt flows; concentration is volumetric fraction. (D) Sand flow on a larger scale to show the entire flow(the same as in Fig. 8D). Flow is from left to right. The lowest concentration contours are different for the flows

because of their different concentrations.




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the velocity equals half the maximum velocityUmax, i.e. u(z1/2) 1/2Umax. This is the same non-dimensionalizing method as that used for turbu-lent wall jets (Launder & Rodi, 1983) and by Brrs

& Eidsvik, 1992). Velocity and TKE are non-dimensionalized with the maximum velocityUmax and U

2max respectively. Concentration is

non-dimensionalized by taking the difference Dqwith the ambient density and dividing this by themaximum difference (Dq)max. This way of non-dimensionalizing shows the similarities in theprofile structures, but the disadvantage is thatdifferences between flows are lost. For example,the low-gradient velocity profile of the saline flowgets squeezed, whereas the high-gradient velocityprofile of the particle flows gets stretched, so all

profiles are similar (Fig. 10A). A profile collapsetherefore does not mean that flow structures arethe same.

The velocity profiles in Fig. 10A all show avelocity maximum close to the bed. Above thevelocity maximum, the velocity decreases rapidlyupwards. There is a gradient inflection in theprofiles, and the velocity decreases more slowlyupwards from that point on. A similar inflectionis present in the experimental density currentprofiles of Buckee et al. (2001). The present

velocity profiles compare reasonably well withprofiles from experimental turbidity and densitycurrents (Parker et al., 1987; Garcia, 1994; Altina-kar et al., 1996; Buckee et al., 2001). The saline

and silt flows are thinner than the sand flow,which leads to a higher velocity gradient in theupper part of the flow for the saline and silt flowsthan for the sand flow, as the decrease from themaximum velocity to zero takes place over ashorter distance. Overall, the sand flow is theleast stratified flow and is therefore the thickest.Sand mainly accumulates close to the bed, so thedensity gradient is high, and the sand flow ishighly stratified close to the bed. However, inmost of the flow, little sand is present, so thedensity gradient is low, and most of the flow is

unstratified. This is certainly the case near the topof the flow. The effect of stable stratification isthat shear production of turbulence is counterac-ted by buoyancy dissipation, which inhibitsmixing of momentum, sediment and salt. Thelack of stratification results in increased mixing ofmomentum, which results in a thicker flow and alower velocity gradient in the upper part of theflow, seen in the sand flow (Fig. 10A).

The effect of stratification is also clearlyreflected in the TKE profiles (Fig. 10B). All three

Fig. 10. Non-dimensionalizedvertical profiles for the single-com-ponent flows. (A) velocity, (B) TKE,(C) concentration. Values used tonon-dimensionalize are: for thesaline flow, Umax 301 ms

)1,z1/2 4041 m; for the silt flow,Umax 302 ms

)1, z1/2 4545 m;and for the sand flow, Umax 309 ms)1, z1/2 4697 m. The sandflow is thicker than the other twoflows but is shown on the same scale

for comparison. Profiles are takenthrough the high-velocity core

behind the nose. The velocity andconcentration gradient inflectionsare at the same heights as the bulgesin the TKE profiles for the differentflows.

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flows show high TKE near the bed resultingfrom high production, and a minimum aroundthe height of the velocity maximum. Above theheight of the velocity maximum, all flows showa local increase in TKE, located at the height ofthe gradient inflection in the velocity profilewhere there is higher shear. A similar bulge is

also present in the experimental TKE profilesgiven by Buckee et al. (2001). Above the bulge,TKE is fairly constant before decreasing to zerotowards the top of the flow, which results fromshear going to zero. The decrease in TKE at theheight of the velocity maximum is a result ofreduced shear and thus less production ofturbulence, with the magnitude of the decreasedepending on the amount of stratification at thatheight. The decrease in TKE is small for thesand flow, which is least stratified, whereas thedecrease is larger for the more stratified saline

and silt flows. Thus, although the low velocitygradient of the sand flow results in less shearproduction of turbulence, as stratification isnegligible, the actual TKE values are similarfor all flows (Umax is similar for all three flows,so non-dimensionalizing does not affect com-parison of actual values; see Fig. 10 for actualvalues).

The density profiles (Fig. 10C) are different forall three flows. For the silt flow, the maximumdensity is near the bed and decreases rapidly upto the height of the gradient inflection in thevelocity profile and more slowly above this

height. The saline current shows the standardstepped profile (Ellison & Turner, 1959; Fietz &Wood, 1967; Alavian, 1986). Above the heightof the velocity maximum, the concentrationdecrease is slower, and the saline and silt flowsshow a comparable gradient. There is a slightgradient inflection in the density profile some-what above the height of the velocity maximum,at the height of the velocity inflection. Theconcentration in the sand flow is low and, eventhough the sediment is the coarsest, the concen-tration gradient changes less from the bed to the

top of the flow than for the other two flows.Garcia & Parker (1993) and Peakall et al. (2000)

suggested that the TKE minimum around theheight of the velocity maximum leads to slowupward diffusion of sediment. Garcia & Parker(1993) and Peakall et al. (2000) argued that, inthe case of an eroding turbidity current, thiswould lead to sediment accumulation at theheight of the velocity maximum, which thenresults in a stepped density profile. The steppedprofile would thus be a result of developing flow

that has not yet adjusted to the change inconcentration caused by erosion. Such a steppedprofile, however, is always present in conserva-tive saline density currents (material is neithergained by erosion nor lost by deposition) whereit is not caused by the accumulation of sediment.The stepped profiles are stable, as shown in the

experiments by Alavian (1986), and not a resultof developing flow. Samolyubov (1990) observedstepped profiles in a muddy current for botheroding and depositing flow stages. The presenceof salt (or clay) will cause a pycnocline at theheight of the velocity maximum as a result of thelow TKE, and this further inhibits mixing.Suspended clay or salt will thus go neither up(no mixing through the velocity maximum) nordown (no settling), resulting in a constant den-sity from the bed to the height of the velocitymaximum. Settling of sediment coarser than clay

would limit the effect of stratification on theinhibition of mixing at the height of the velocitymaximum and would thus smooth out theconcentration profile. The erosive currents inthe experiments by Garcia & Parker (1993), onwhich Peakall et al. (2000) based their interpret-ation, started as saline density currents. Theseflows would therefore already have a steppedprofile before erosion took place and, thus, thestep is the result of the salt rather than of thesediment.

Above the zone of low TKE, there is an increasein TKE and, therefore, more sediment can be kept

in suspension, which explains the density inflec-tion (Fig. 10C). Similar inflections in the densityprofile can also be seen in the experimentalresults of Altinakar et al. (1996). Above the bulgein the TKE, density decreases at a slower rate, andthe different density values result in differentstratification effects, as highlighted above. Thedensity profiles for the saline and silt flows aresimilar above the height of the velocity maxi-mum, and this is reflected in the same growthpattern of the flows, which is determined bystratification. The density profiles shown here are

for higher concentrations than the profiles ofexperimental currents (Parker et al., 1987; Garcia,1994; Altinakar et al., 1996) and show a highconcentration near the bed that decreasesupwards at a faster rate than the densities meas-ured in the experimental currents. As a result, thedensity gradient is higher, and the velocity maxi-mum is closer to the bed. The concentrationprofile of the sand flow is characteristic of verylow-density flows and compares favourably withthe experimental currents. Most experimental




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profiles therefore only show a limited range ofdensity distributions.

Multiple grain-size flows: differentconcentrations

The flows described so far have been single-

component flows, but natural flows transportmultiple grain sizes, which will interact todetermine the flow structure and, therefore, thefollowing tests detail runs with multiple grainsizes. In order to compare spatial flow structureof a low- and a high-concentration multiplegrain-size flow, two tests were run for differentinitial concentrations and the same topography.At time t 0, five different grain classes weredistributed uniformly throughout a column 50 mhigh and 1000 m long. The grains all had adensity of 2650 kg m)3 and diameters of 002,

005, 0

1, 0

5, and 1 mm. In the low-concentra-tion flow, all five grain sizes had an initial

volumetric concentration of 1% (yielding a totalconcentration of 5%), whereas in the high-con-centration flow, the initial volumetric concentra-tion was 10% for all grain sizes, and total

concentration was 50%. The currents initiallyflowed down a slope and then onto a horizontalbed.

Flow field

Velocity and concentration contour plots are

shown in Figs 11 and 12 for the low- and high-concentration flows respectively. The low-con-centration flow is shown 1600 s after initiation,and the high-concentration flow is shown 500 safter initiation. For comparison of the spatial flowstructures, the lowest velocity contour shown forboth flows is 05 ms)1, and the lowest concentra-tion contour is a volumetric fraction of 5 10)4.As the general flow structure is the same as for thesingle-component flows, only specific differenceswill be described.

The low-concentration flow is slower (maxi-mum velocity %3 ms)1; Fig. 11A) than the

high-concentration flow (maximum velocity% 14ms)1; Fig. 12A) because there is less sedi-ment in the low-concentration flow and thus thedriving force is smaller. The vertical concentra-tion gradient is low in the head, where a thick

Fig. 11. Low-concentration flow (total initial concentration 5%) contour plots, 1600 s after initiation of the flow.(A) Velocity contours in ms)1. (B) Total concentration contours expressed as a volumetric fraction. Flow is fromleft to right.

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concentration maximum can be seen, andincreases upstream into the body, where highconcentrations are present close to the bed only(Figs 11B and 12B). Both flows show highhorizontal velocity gradients and low horizon-tal concentration gradients, which indicate thatthe flow structure for both these flows is

dictated by the finer grains in suspension: theflow structure is governed by the grain-sizedistribution rather than by the magnitude ofconcentration.

Both flows grow to a comparable thickness overthe same travelled distance, although the high-concentration flow grows to this thickness in ashorter time. Owing to its higher density, thehigh-concentration flow is faster and thereforehas a higher shear production of turbulence.Although this is offset by the higher concentra-tion, it does not result in a thinner flow, as a

result of the TKE, which is of comparable mag-nitude to the TKE in the low-concentration flow(see below). This shows that high concentrationdoes not necessarily lead to less turbulence. Inhigh-concentration flows, turbulence may still bean important mechanism for keeping sediment insuspension.

Non-dimensionalized vertical profiles

Non-dimensionalized vertical profiles of the low-and high-concentration flows (Fig. 13) were takenthrough the high-velocity core behind the head ofthe flows (Figs 11 and 12) and are non-dimensio-nalized in the same way as for the single-compo-

nent flows. The general shape of the profiles issimilar to that shown in Fig. 10. The non-dimen-sionalized TKE values (Fig. 13B) are lower thanfor the single-component flows, which indicatesthat, as the velocities of the multigrain flows arehigher, the actual TKE is of the same order ofmagnitude. The high-concentration flow has asimilar non-dimensionalized density gradient tothe low-concentration flow, except near the bed(Fig. 13C), where concentration is much higherleading to a higher density gradient. Despite thedifference in actual densities, the similarity of the

profiles again emphasizes the effects of non-dimensionalizing: the two flows have approxi-mately the same non-dimensionalized verticalprofiles, showing that the sediment distributiondetermines the spatial flow structure independ-ently of the concentration magnitude. Althoughthe spatial flow structure of the two flows is

Fig. 12. High-concentration flow (total initial concentration 50%) contour plots, 500 s after initiation of the flow.(A) Velocity contours in ms)1. (B) Total concentration contours expressed as a volumetric fraction. Flow is fromleft to right.




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similar, the temporal flow structures are different,as indicated by the different times at which thetwo flows are shown. The high-concentrationflow will travel for a longer time than the low-concentration flow, as its maximum concentra-

tion at the time shown is still more than threetimes higher than the initial concentration of thelow-concentration flow.

Multiple grain sizes: different grain-sizedistributions

The previous flows were all shown at only onetime, which allowed the spatial flow structure to

be examined. In order to describe the temporalflow structure, two flows with the same initialconcentration but different grain-size distribu-tions will be compared at four different times.Flows are shown from initiation until they have

slowed down to %0

15 ms

)1

, when it would notbe possible to distinguish them from the ambientflow. For both flows, the initial concentration is15%, and the initial static column is 15 m high by50 m long. These values are smaller than those forthe previous flows and result in slower currents.One flow (termed fine-grained flow or FGF)contains a grain-size range with a mean diameterof 01 mm (very fine sand; see Fig. 14), and the

Fig. 13. Non-dimensionalizedprofiles for the low- and high-con-centration flows. (A) velocity, (B)TKE, (C) concentration. The valuesused for non-dimensionalizing

are: z1/2 6672 m and Umax 382 ms)1 for the low-concentra-tion flow; and z1/2 4631 m andUmax 1486 ms

)1 for the high-concentration flow. Profiles arethrough the high-velocity core

behind the nose.

Fig. 14. Input concentration andgrain-size distributions for fine-grained and coarse-grained flows.Total concentration is 15% in bothcases.

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other flow (termed coarse-grained flow or CGF)contains a grain-size range with a mean diameterof 05 mm (coarse sand). Flow initially takes placeon a slope and then continues on a horizontal bed(Fig. 15). Velocity and concentration profiles areshown at four different times, which are not thesame for the two flows, as their behaviour is

different.

Flow field

Velocity and concentration plots for the FGF areshown in Figs 15 and 16, and for the CGF inFigs 17 and 18. Velocities for these runs are of theorder of several dm s)1, which is comparable withobserved velocities of turbidity currents in fjordsand lakes (Lambert et al., 1976; Shepard et al.,1979; Zeng et al., 1991). Despite this low velocity,the currents flow for several kilometres.

The CGF is slower than the FGF over its entire

flow history (Figs 15 and 17). This slower velocitydevelops immediately after flow initiation and,thus, the momentum development is different forboth flows, even though the initial concentrationsare the same. The CGF increases in thicknessmore than the FGF, and the increase is faster.

After % 35 km of travel (t 2500 s for the FGF,Fig 15A; t 5000 s for the CGF, Fig. 17B), theCGF is about twice as thick as the FGF. Even att 20 000 s (Fig. 15D), the FGF has not grown tothe same thickness as the CGF flow has in ashorter time (t 15 000 s; Fig. 17D).

As coarser sediment settles out faster, the CGF

has less driving force than the FGF and is sloweralong its entire flow path. Initially, there is noturbulence to keep sediment in suspension, sothis initial settling also leads to a rapid differencein stratification for the two flows (Figs 16 and 18),which leads to faster growth in the flow thicknessof the CGF. This is a similar growth pattern to thatshown for the single-component flows (Figs 810), where the sand flow became much thickerthan the saline and silt flows. Although thispattern resulted partly from lack of stratificationfor the single-component flow, this is not the case

for the FGF and CGF (Figs 16 and 18). Initially,enough sediment is present in the CGF to producean effect of flow stratification on thickness devel-opment (Fig 18A and B). However, coarse sedi-ment is only present near the bed, so it has noeffect at the top of the flow: the thickness of the

Fig. 15. Velocity contour plots for the fine-grained flow for four different times after flow initiation. Contours are inms)1. The lowest velocity contour is 0 01 ms)1 for all four times.




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CGF increases considerably as there is no strati-fication to limit mixing. Only during later flow

stages is the concentration very low (Fig. 18D)and has little influence on the flow structure, as ithas ceased to be the main driving force. Altinakaret al. (1990) found similar growth patterns intheir experiments investigating a saline, a fine-grained and a coarse-grained flow: the saline flowgrew in thickness least, whereas the coarse-grained flow grew most.

The CGF decelerates faster than the FGF andproduces a less steady velocity structure. This isshown even more clearly by the temporal changein concentration. At t 15 000 s, virtually all

the sediment has dropped out of suspensionfrom the CGF (Fig. 18D) even though the velo-city is not yet zero (Fig. 17D). This demonstratesthat concentration is less steady than velocity;sediment fallout takes place over a shorter timescale than turbulence dissipation. The overallmomentum loss of the CGF is thus caused bysediment settling and is not accompanied by acomparable reduction in velocity, as discussedpreviously. This is not the case for the FGF. Att 20 000 s (Fig. 16D), sediment still remains in

the flow (maximum concentration is about 10times higher than for the CGF), and both velo-

city and concentration are relatively more steadythan in the CGF. Altinakar et al. (1990) alsofound a similar pattern in their experiments, inwhich initially a fine-grained and a coarse-grained flow had the same velocity, but thecoarse-grained flow decelerated faster and wasthus less steady.

Uniformity of velocity and concentration arealso different in both flows. At any given time,the concentration of the FGF is more uniformthan the concentration of the CGF (Figs 16 and18). Concentration decreases rapidly from the

front to the back of the CGF, with sedimentonly present in the nose during the final flowstages, even though velocity is non-zero over alarger area. In the FGF, the sediment distribu-tion resembles the velocity distribution moreclosely, so that uniformity of velocity andconcentration are more similar than in theCGF. However, in general, velocity is moreuniform than concentration. Dissipation ofmomentum is clearly a slower process thanthe settling of sediment, so that the flow pattern

Fig. 16. Concentration contour plots for the fine-grained flow at four different times after flow initiation. Contours arein volumetric fraction. The lowest concentration contour is 10)6 for all times; the other contours are variable.

412 M. Felix



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caused by a certain sediment distribution is still

present after that particular sediment distribu-tion has changed as a result of sedimentation.Similar behaviour was seen for the single-com-ponent flows.

The FGF has a lower horizontal and a highervertical velocity gradient than the CGF, whichappears to be the opposite of the results fromthe single-component flows, in which thepresence of fine grains resulted in higherhorizontal and lower vertical gradients thanfor coarse grains (Figs 8 and 9). However, theFGF and CGF are the result of the interaction

of several grain sizes. Thus, although thecoarsest grains in both flows drop out ofsuspension, leading to increased steadiness, inthe FGF, the fine grains stay in suspension sothat the driving force does not diminish. How-ever, in the CGF, even the finest grains arerelatively coarse and settle out of suspensionrapidly so that, although settling leads toincreased steadiness, this is counteracted bythe loss of driving force, which leads to anoverall less steady flow.

DISCUSSION

The results of the simulations, together withobservations from laboratories and nature, showthat the flow behaviour of turbidity currents ismainly determined by the sediment in the flow,which is not surprising as this is the driving forceof turbidity currents. However, the results alsoshow a clear interaction of velocity, turbulenceand concentration (Fig. 19). The principles thatgovern the flow structure will be the same forthree-dimensional flows so, although flows mightspread out horizontally and thus decelerate at a

different rate, the same basic flow patterns andresults from the present two-dimensional modelwill be valid and can be used to describe flowstructure in three-dimensional flows. No clay waspresent in these simulations but, for small quan-tities, this may be expected to influence flowbehaviour in the same way as salt or very finenon-cohesive sediment. Concentration directlydetermines the velocity of the current, as higherconcentration results in a faster flow (see Fig. 19).The shape of the velocity profile is determined by

Fig. 17. Velocity contour plots for the coarse-grained flow at four different times after flow initiation. These times aredifferent from those for the fine-grained flow, as the flow shown here decelerates faster, but times in (A) and (B) arethe same as in Figs 15, and 16A and B. Contours are in ms )1. The lowest velocity contour is 0 01 ms)1.




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the grain-size distribution: coarser particles lowerthe height of the velocity maximum (see Fig. 8).

Velocity influences TKE as a result of shearproduction, which is determined by the shape ofthe velocity profile: high-velocity gradients lead tolarge shear production. Turbulence keeps sedi-ment in suspension and mixes it throughout theflow. The density distribution in turn leads tostable stratification, which dampens turbulenceand, if stable enough, will extinguish turbulence.

The level of turbulence influences velocitythrough mixing of momentum, which influences

the shape of the velocity profile (see Fig. 10).Although for small particles, velocity effects, suchas rotational lift force and added mass force, influ-ence the distribution of particles, these effects arenegligible for coarse sediment. Thus, for a simpledescription, the velocity can be assumed to haveno direct effect on particle distribution, but onlyto act on it through turbulence (Fig. 19).

Fig. 19. Simplified diagram ofinteractions between velocity,turbulence and sediment concen-tration and their effect on turbiditycurrent flow structure. Note thatvelocity only influences concentra-tion through turbulence. See text forfurther explanation.

Fig. 18. Concentration contour plots for the coarse-grained flow at four different times after flow initiation. Contoursare in volume fraction. The lowest concentration contour is 10)6.

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Stratification seems to become important whenthe concentration is higher than a volumetricfraction of%10)3, the limit given by Hinze (1972).For higher concentrations, particles will influ-ence flow behaviour. Most experimental currentsand those observed in reservoirs have a concen-tration around this limit of 10)3 and, hence, the

sediment will only have a limited influence onflow behaviour. Many of the flows described herehave a concentration higher than 10)3 and show astronger influence of sediment than those des-cribed from field observation and experiments.High stratification does not necessarily mean thatthe flow is laminar, as TKE is advected anddiffused from other parts of the flow, and shearproduction takes place over shorter time periodsthan viscous dissipation. The term stratifiedshould be used with care, as stratification willin general be different at the top and bottom of the

flow, leading to different behaviour in differentparts of the flow. For example, the transfer ofmass and momentum by mixing at the top of thecurrent determines the thickness of the current,this being independent of the amount of mixingnear the bed. This type of behaviour is importantfor a process such as flow stripping, as mentionedby Peakall et al. (2000).

Although the magnitudes of velocity and thick-ness of the flows in the computational runs(Figs 8, 11, 12, 15 and 17) are of the same orderas reported for natural flows (Heezen & Ewing,1952, 1955; Menard, 1964; Normark & Dickson,

1976; Shepard et al., 1977; Piper & Savoye, 1993),many of the characteristics of the vertical profilesare similar to small-scale laboratory flows (Buc-kee et al., 2001). Turbulence behaves differentlyat such different scales (Kuenen, 1937; Tennekes& Lumley, 1972), but the interaction of velocity,turbulence and density appears to lead to similarflow structures. Not all parts of the generalpattern are always clear in laboratory experi-ments, owing to the limited number of measure-ment points both at the top of the current andnear the bed. The flow structure derived from

such experiments must also be used with caution,as the magnitudes of the variables are obviouslydifferent between field and laboratory, so sedi-ment transport and deposition are difficult toscale realistically in laboratory experiments.

The temporal and spatial flow structures ofturbidity currents are determined by thegrain-size distribution and the fine-grained orcoarse-grained nature of the suspended sediment.Fine-grained and coarse-grained are somewhatrelative terms; compare, for example, laboratory-

scale and natural-scale flows and their respectiveability to transport certain grain sizes. Differentgrain sizes will lead to a coarse-grained beha-viour or fine-grained behaviour at each of thesetwo scales. A combination of different grain sizesleads to a combination of their respective influ-ences on different aspects of flow behaviour, such

as thickness growth and steadiness.The presence of fine-grained material leads to

large stratification at the top of the current, therebylimiting mixing, resulting in little increase in flowthickness (Figs 8A and B, and 15). Coarse-grainedmaterial does not result in great stratification at thetop of the flows, as most sediment is located nearthe bed, causing such flows to grow in thicknessmore than fine-grained flows (Figs 8C and D, and17). The degree of mixing in multiple grain-sizeflows depends on the relative concentrations offine and coarse material: predominantly coarse-

grained flows will be thicker than predominantlyfine-grained flows. Fine-grained flows are some-times thought to be thick, slow flows, whereascoarse- grained flows are thin, fast flows (e.g.Bowen et al., 1984), but it is not generallyexplained how these flows developed. Coarse-grained flows are actually thick flows, but mostsediment is present near the bed so, if thesecurrents flow in a canyon or deep channel, thiswould lead to little deposition high above thechannel bottom. This effect is not the result of flowthickness but of sediment distribution within theflow. Fine sediment will be present higher in the

flow and, thus, even for a thinner flow, sedimentwill be deposited higher above the channel bottomthan coarser material, leading to the impressionthat fine-grained flows are necessarily thicker.Fine material drops out of suspension slowly, soflows have a long time to grow in thickness.However, as shown by the results here, this stillleads to less growth than in coarse-grained flows.Slow flows will increase little in thickness, asminimal shear-generated mixing takes place. It isperhaps more likely that thick, slow, fine-grainedflows are a remnant of initially coarse-grained

flows that grew rapidly in thickness and subse-quently lost their coarse material. Towards theend of their life span, flows do not become thinner,but only the magnitude of velocity, turbulence andconcentration decreases (Figs 1518). Thickness-es of natural turbidity currents determined bymeasurement of velocities higher than a certaincritical velocity thus only measure graduallydecreasing heights of the flow where this criticalvelocity is exceeded, but this does not mean thatthe flow thickness is decreasing.




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Run-out distance and time are thus clearlyinfluenced by the temporal and spatial flowstructure of a turbidity current. The same is truefor sediment transport characteristics. Similar tothe vertical flow structure, the horizontal andtemporal flow structures depend on both thegrain-size distribution and the ability of turbidity

currents to keep certain grain sizes in suspension.Although the flows in the present model start as asurge, the resulting turbidity currents are long-lived, something more expected from a currentwith a sustained input (e.g. from a river). How-ever, by losing sediment by grain settling, turbid-ity currents create their own steadiness. In fine-grained turbidity currents, material stays in sus-pension for a long time, so the driving force doesnot diminish substantially. Momentum loss (@qu/@t < 0, for u > 0) for fine-grained flows is mainlycaused by velocity decrease (@u/@t < 0), which

will decrease competence and thus be accompan-ied by decreasing density. This density decreasewill mainly be taken up by the coarsest sedimentfraction, and most of the driving force of thecurrent will remain in suspension. In coarse-grained flows, a large proportion of the sedimentrapidly falls out of suspension, and momentumloss is thus mainly the result of a density decrease(@q/@t < 0), so that the velocity decreases little.By depositing sediment, the flow might evenaccelerate: an overall momentum decrease as theresult of sediment fallout can be accompanied byan increase in velocity, as long as the negative

term u(@q/@t) is of larger magnitude than thepositive term q(@u/@t). Such behaviour has, infact, been observed in the experiments of Parkeret al. (1987), where a number of flows showedacceleration while deposition took place. Sedi-ment deposition will lead to self-induced sus-tained flows and thus long flow durations,although flows will eventually slow down be-cause of loss of a driving force. In terms of thechange in sediment load, autosustaining currentsare the result of the opposite mechanism toigniting currents, in which long flow duration is

the result of uptake, rather than loss, of sediment(Parker, 1982). Sediment deposition will operatedifferently for different grain sizes because ofdifferences in flow competence. If velocity doesnot decrease (i.e. does not change or increases),this can lead to a sharp break between what theflow can and cannot carry in suspension, whichmight be reflected by a grain-size break in theresulting deposit. If velocity does not vary, flowcompetence will not change, so only materialbelow the competence limit will settle out. If

velocity increases, the competence of the flowincreases. As the material that cannot be trans-ported will have dropped out of suspension, avelocity increase will then lead to stronglyreduced deposition, as all suspended materialwill be above the competence limit. This might bereflected by a sharp grain-size break in the

resulting deposit as long as the transport capacityis not exceeded.

CONCLUSIONS

1 The spatial and temporal flow structures ofturbidity currents show a clear interactionbetween velocity, turbulence and sedimentdistribution. The spatial flow structure is mainlydetermined by the grain-size distribution,whereas concentration does not have a large

influence. The temporal flow structure is deter-mined by both grain-size distribution and con-centration, the latter influencing the overallsteadiness and flow duration, whereas bothinfluence sediment transport characteristics.

2 Fine grains influence flow behaviour in adifferent way from coarse grains, but the overallflow behaviour is determined by the grain-sizedistribution and the interaction between fine andcoarse grains. Fine-grained flows have low ver-tical velocity and concentration gradients (thevertical flow structure changes slowly) and highhorizontal velocity and concentration gradients

(the horizontal flow structure changes rapidly).These flows show little increase in flow thick-ness because of limited mixing at the top of theflow, which is caused by the presence of sedi-ment throughout the flow. Owing to slow sedi-ment fallout, the flows decelerate slowly, andvelocity and concentration have similar steadi-ness and uniformity. In contrast, coarse-grainedflows have high vertical velocity and concentra-tion gradients and low horizontal velocity gra-dients and high horizontal concentrationgradients. These flows grow in thickness con-

siderably, as sediment is present mainly near thebed and thus has little influence on mixing at thetop of the flow. Owing to rapid sediment falloutand the resulting loss of driving force, theseflows decelerate quickly. Steadiness and uni-formity are different for velocity than for con-centration. Interaction of fine and coarse grainswill lead to intermediate behaviour where, forexample, thickness increase is determined by thecoarse grains, whereas flow duration is deter-mined by the fine grains.

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3 Because concentration and velocity do notneed to have the same steadiness and uniformitybehaviour, they can influence flow behaviour in-dependently. Momentum loss can be caused byboth decreasing velocity and decreasing density,and these two mechanisms need not occur at thesame time because of the different time scales on

which they take place. Settling of coarse sedimentis a faster process than dissipation of turbulence,and momentum can thus decrease by only de-creasing density, which leads to autosustainedcurrents where velocity is steady even thoughconcentration is not. Because fine grains settleslowly, the overall steadiness of fine-grainedflows is larger than that of coarse-grained flows.

ACKNOWLEDGEMENTS

Jaco Baas, Henry Pantin, Jeff Peakall, ZoltanSylvester and reviewers Brian Dade and ananonymous reviewer are thanked for their com-ments on earlier versions of this paper, whichhelped greatly to clarify it.

APPENDIX: MODEL DESCRIPTION

The model calculates bulk fluid (sediment +water) velocity and is valid for low concentra-tions (much less than 1% volume fraction of

solids) to moderately high concentrations (%50%)of non-cohesive particles for multiple grain sizes.The model cannot take clay into account and isinvalid for higher concentrations, such as wouldbe found in debris flows (either cohesive or non-cohesive). In both these cases, the flow rheologywould change from Newtonian to somethingmore complicated, such as a Bingham rheology,which cannot be simulated by this model. Therun starts as an initial static column of uniformlydistributed suspended sediment. Initial velocityand turbulence are zero. The numerical methodused to solve the equations allows for a varyingtopography, albeit no vertical or very steepslopes. For a detailed derivation and descriptionof the model, see Felix (2001). The followingequations are solved.

Continuity equation

@ui@xi

0 A1

where u is bulk fluid velocity, and x is theco-ordinate direction.

Momentum conservation equation

@qui@t

@quiuj@xj

@P

@xi giq

@

@x3lr KM

@ui@x3

A2

where q is density, P is hydrostatic pressure, andlr is the viscosity of the waterparticle mixturecalculated using the empirical equation of Krieger& Dougherty (1959), g is gravitational accelera-tion, and KM is a turbulent mixing term calculatedfrom the turbulence model. The empirical equa-tion for the viscosity of the waterparticlemixture deals with the effect of particleparticleinteraction at high concentrations.

Concentration equation for grain size class k

@ck@t

@ckui wskdi3

@xi @

@x3 KH

@ck@x3

A3

where ck is the volumetric fraction of grain-sizeclass k, wsk is settling velocity of grain-size classk, and KH is a mixing term calculated from theturbulence model. The settling velocity wsk isdetermined using the empirical equation ofDietrich (1982). The relationship of Richardson& Zaki (1954) is used to account for the effectof concentration on settling velocity. Sedimententrainment is calculated using the empiricalequations of van Rijn (1984):

Es 0:00033qk

ffiffiffiffiffiffiffiffiffiffiffiffiffiDgd50

qd0:3 T

1:5 A4

where d* d50(Dgl2)1/3 and T* (sbscr)/scr, l

is the viscosity of water, sb is bottom shearstress, and scr is critical shear stress for erosion.Although this equation was not specificallyderived for turbidity currents, it produces realis-tic results in the present model.

The equations of the MellorYamada second-order turbulence closure model for turbulentkinetic energy q2 and turbulent length scale l are

@c0q2

@t

@c0uiq2

@xi

@

@x3c0Kq

@q2

@x3

2c0

@u

@x3

2

2gKH@q

@x3 2c0

q3

B1l

9

4q2le

Xnk1

ck

d2k

A5




22/23

@c0q2l

@t

@c0uiq2l

@xi

@

@x3c0Kq

@q2l

@x3

lE1

@u

@x3

2 lE1gKH

@q

@x3

c0q3

B11 E2

l

jL 2

" #

9

4q2lle X

n

k1

ck

d2k

A6

where c0 is the volume fraction of the water, Kq isa mixing term calculated from the turbulencemodel, B1 166, E1 18 and E2 133 areempirical constants, j 04 is the von Karmanconstant, and Lis the distance from the wall. Theturbulence model determines the advection anddiffusion of TKE and thus mixing of the currentwith the ambient fluid, at both the top and thefront of the current. TKE and length scale are bothinfluenced by stratification, and the mixing coef-

ficients for mass and momentum, which arecalculated from those variables, are therefore alsoinfluenced by it.

The variables that are changed for each specificrun are topography (for the historical flows, theobserved topography was used), initial sedimentdistribution (grain sizes and volume fractions)and initial size of the sediment column (see maintext and Figs 2, 5 and 14 for these data for thesimulations shown). For the historical flows,grain-size distributions were used that cover theentire range of observed non-cohesive grain

sizes.

NOMENCLATURE

TKE turbulent kinetic energy;FGF fine-grained flow;CGF coarse-grained flow;c concentration;Cl chlorinity;g constant of gravitational acceleration;K turbulent mixing coefficient;P pressure;q2 turbulent kinetic energy;t time;T temperature;u velocity;Umax maximum velocity;x co-ordinate direction;z1/2 height above velocity maximum where

u 05Umax;lr viscosity of sedimentwater mixture;q density.

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Manuscript received 22 March 2001;revision accepted 29 October 2001.


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