Sedimentology (2005) 52, 291–311 doi: 10.1111/j.1365-3091.2005.00698.x

Grain-size sorting in grainflows at the lee side of deltas

MAARTEN G. KLEINHANS Department of Physical Geography, Faculty of Geosciences, Universiteit Utrecht, PO Box 80115, 3508 TC Utrecht, The Netherlands (E-mail: [email protected])

ABSTRACT The sorting of sediment mixtures at the lee slope of deltas (at the angle of repose) is studied with experiments in a narrow, deep flume with subaqueous Gilbert-type deltas using varied flow conditions and different sediment mixtures. Sediment deposition and sorting on the lee slope of the delta is the result of (i) grains falling from suspension that is initiated at the top of the delta, (ii) kinematic sieving on the lee slope, (iii) grainflows, in which protruding large grains are dragged downslope by subsequent grainflows. The result is a fining upward vertical sorting in the delta. Systematic variations in the trend depend on the delta height, the migration celerity of the delta front and the flow conditions above the delta top. The dependence on delta height and migration celerity is explained by the sorting processes in the grainflows, and the dependence on flow conditions above the delta top is explained by suspension of fine sediment and settling on the lee side and toe of the delta. Large differences in sorting trends were found between various sediment mixtures. The relevance of these results with respect to sorting in dunes and bars in rivers and laboratory flumes is discussed and the elements for a future vertical sorting model are suggested. Keywords Delta, dune, fining upward, grainflow, kinematic sieving, vertical sorting.

INTRODUCTION sediment settling on the entire lee slope, and the grain fall sediment may partly be preserved rather than be reworked in the following grainflow Scope and objective (Jopling, 1965; Hunter, 1985a,b) (Fig. 1B). Herein The downstream propagation of subaerial and the focus is on grainflow-dominated conditions, subaqueous bedforms with a downstream slip although the two conditions cannot entirely be face at the angle of repose such as dunes, bars and isolated. (small) Gilbert-type deltas proceeds by deposition In mixtures of sand and gravel, vertical grain- of transported sediment on the upper part of the size sorting occurs due to grainflows, usually lee slope (grain fall), until a threshold is exceeded yielding a fining upward trend within the dune, and the sediment mass flows downslope (grain- bar or delta deposit. The vertical sorting may be flow). Grainflows refer to discontinuous mass modified by the grain fall sediment, which pro- flows in which the individual moving grains motes coarsening upward sorting. This process is collide and interact. Grain fall on the other hand well-known and has often been observed in refers to the continuous settling of single grains experiments and in the field (e.g. Allen, 1965, from the decelerating flow just downstream of the 1984; Brush, 1965; Termes, 1986; Love et al., bedform top, in which grain–grain interaction is 1987; Ribberink, 1987; Pye & Tsoar, 1990; Makse, much less important. The repeated grainflows 1997; Blom et al., 2000). However, no quantita- (Allen, 1965, 1970) cause these types of bedforms tive model has been developed that successfully to migrate downstream (Mohrig & Smith, 1996) predicts the vertical sorting in various conditions (Fig. 1A). In the case of higher suspension rates, and for different sediment mixtures. The motiva- the bedforms migrate partly due to suspended tion for this research is that quantification of

Ó 2005 International Association of Sedimentologists 291 292 M. G. Kleinhans

A Brinkpoint Topset Temporary grain fall deposit

Foreset Toe deposit

Toeset Grain flow deposit

B Brinkpoint

Foreset Toe deposit Fig. 1. (A) Sorting and definitions in sand dune deposits (after Hunter, 1985a). (B) Sorting in sand–gravel Toeset dune deposits, where only grainflow deposits are preserved on the lee Grain flow deposit Grain fall deposit slope. vertical sorting is needed for morphodynamic downstream of the brinkpoint, while smaller models for river dunes with sediment mixtures of grains may saltate or even be suspended and gravel and sand (e.g. Ribberink, 1987; Parker therefore take more time to settle and are depos- et al., 2000). Commonly these rivers are bedload- ited lower on the foreset slope. Thus the concen- dominated systems as the gravel rarely is suspen- tration, settling velocity (grain size) and ded in natural rivers. Although the process of accumulation of sediment decreases in the down- vertical sorting in dune beds is well known, stream direction from the top of the dune, bar or systematic experiments in which this has been delta (Jopling, 1965). A wedge of sediment builds measured, are scarce. Hence, the principal vari- up on the top of the foreset, with a much faster ables which govern the sorting have never build-up rate and coarser sediment at the top of unequivocally been determined from systematic the foreset than further downstream of the brink- measurements in nature or in the laboratory. The point. This sediment wedge fails above a certain objectives of this paper are: threshold and then transforms into a grainflow. The threshold is usually specified as a critical 1. to quantify vertical sorting, slope: the static angle of repose. When the 2. to describe the sorting processes in detail, grainflow stops moving, the sediment is depos- and ited at the dynamic angle of repose, which 3. to quantify the effects of the variables that usually is a few degrees smaller than the static control sorting. angle of repose (Allen, 1970). During the move- This is done experimentally for a range of flow ment of the sediment mass, kinetic sieving takes conditions, sediment transport and for various place: the small grains work down into the mixtures. grainflow by occupying the small pores, with the result that the large grains are worked upwards (Brush, 1965; Makse, 1997). Conse- Review quently, the large grains on top of the lee slope For an extensive review on vertical sorting by may roll downslope more easily than the small grainflow and grain fall the reader is referred to grains below the large ones. In addition, the fine Kleinhans (2002, 2004). Sediment arriving at the sediment below the large grains may provide a brinkpoint is deposited on the upper foreset slope smoother slope for the latter, leading to an (Fig. 1A). In fact, grain fall occurs over a very effectively smaller angle of repose of large grains limited downstream distance (Allen, 1965). Large on small grains than vice versa (Julien et al., grains in traction are deposited immediately 1993; Makse, 1997; Koeppe et al., 1998).

Ó 2005 International Association of Sedimentologists, Sedimentology, 52, 291–311 Grain flow size sorting 293

The effects of these processes on the vertical Allen (1970) found that for continuous grainflows sorting have not been studied quantitatively. The and high suspension rates, the fining upward qualitative effects reviewed by Kleinhans (2004) disappears or grades into coarsening upward will be summarized below and form the basis of a sorting, but this was for rather well sorted sand. dimensional analysis in the next section. Factors 4. The sediment on the lee slope beneath the affecting vertical sorting are: grain flow may be deformed (Tischer et al., 2001) 1. The characteristics of the sediment mixture and even be mobilized (Kleinhans, 2004). Large that is delivered to the top of the lee slope. Most immobile grains may be pushed down by the importantly, the larger the mixture standard front of the grainflow. Due to the kinetic sieving, deviation, the more there is to sort. This may the lee slope is covered with relatively coarse seem self-evident but there are important secon- sediment lying on relatively fine sediment, so the dary effects of mixture characteristics. Kinetic coarse sediment is relatively easily dragged down sieving is only effective if the grain-size differ- by the grainflow that runs over it. Since prefer- ences are large enough, e.g. more than about two entially the coarse sediment on the lee slope is in a bimodal mixture (Makse, 1997). Moreover, worked down, this process might lead to better the exact shape (unimodal, skewed, bimodal) of sorting. The occurrence of this hypothetical the grain-size distribution affects the kinetic remobilization mechanism depends on the thick- sieving and percolation, and the effective angle ness of the grainflows and the sediment mixture of repose for larger grains on top of smaller. characteristics. Angular sediment has a larger angle of repose, and consequently angular particles may deposit in a higher position along the lee slope than THEORY rounded particles of the same average grain size (Makse, 1997; Koeppe et al., 1998). In the case of Non-dimensional description of sediment a mixture with rounded small grains and angular mixtures, coordinates and vertical sorting large grains, this would decrease the vertical sorting. Grain density differences will also have The sediment (factor 1 of the review) is described an effect (Julien et al., 1993) but this is neglected by a grain-size distribution, which is character- here because the sediments commonly found in ized by the geometric and arithmetic mean sedi- rivers and used in the present experiments all ment size (Dg,wm) and the geometric and have approximately the same density. arithmetic standard deviation (rg,ra) of the mix- 2. The velocity of the flow above the delta, bar ture: or dune top relative to the settling velocities of all XN grain-size fractions in the mixture (Jopling, 1965). D ¼ 2wm ; w ¼ w p This affects the trajectory lengths of suspended g m i i i¼1 grains, and therefore the length of the wedge and ð1Þ XN the importance of grain fall on the lower lee slope ra 2 2 rg ¼ 2 ; ra ¼ ðwi wmÞ pi and toe (Allen, 1970). Given a constant static i¼1 angle of repose for a certain mixture, a longer wedge will contain more sediment. Consequently in which wm ¼ arithmetic mean size [related to the volume involved in the grainflow will be the sedimentological grain-size scale (/)as larger (Hunter & Kocurek, 1986) and the layering w ¼ )/], wi ¼ midpoint of a size fraction, in the cross-stratification will be thicker. In turn, pi ¼ proportion in size fraction i, rg ¼ geometric the effectiveness of kinetic sieving decreases. standard deviation, and ra ¼ arithmetic standard 3. The frequency of the grainflows, determined deviation. The angularity and bimodality of the by the migration celerity of the lee slope, which sediment is presently only described qualita- is in turn determined by the sediment transport tively. rate and the height of the lee slope (Allen, 1965, In order to compare bedforms with different 1970; Hunter & Kocurek, 1986). When the heights, the height z at which a sample was taken grainflows are fast enough to overtake the is made dimensionless as z* ¼ z/D (D ¼ bedform previous, still active, grainflows, the thickness height). The arithmetic mean size (wm,z) and of the flowing sediment becomes larger and the arithmetic standard deviation (ra,z) of the samples effectiveness of kinetic sieving decreases. For at height z are made dimensionless as higher lee slopes, the grainflow will be relatively w* ¼ wm,z ) wm,T and r* ¼ ra,z/ra,T, in which thinner and kinetic sieving again more effective. the wm,T and ra,T describe only the sediment

Ó 2005 International Association of Sedimentologists, Sedimentology, 52, 291–311 294 M. G. Kleinhans passing the brinkpoint and participating in the angle of repose. The measurement error of the grain flowing process at the sampling location. rather small dilation angle is, however, rather The reason for this choice of the sediment will be large and will not be used. The suspended explained in the materials and methods section. sediment settling velocity is ws, and the flow Note that w* must be made dimensionless by conditions above the top of the slope are con- subtraction rather than division for the following veniently characterized by the depth-averaged w reason. Since Dg ¼ 2 , with Dg ¼ geometric mean flow velocity (u) (factor 2). In factor 3 are the size, it follows that: bedform height (D) and celerity (c), given by ) w c ¼ [(1 kp)qb]/D, in which qb is the volumetric 2 m;z Dg;z w wm;zwm;T transport rate and k the porosity. An additional D ¼ 2 ¼ 2 ) D ¼ w ) D ¼ ð2Þ p 2 m;T Dg;T variable is the submerged specific weight of the q ) q q q So, the dimensionless arithmetic mean size (w*) sediment: Rg, with R ¼ ( s )/ (with ¼ den- q is related to the dimensionless geometric mean sity of water and s ¼ density of sediment) and size (D*) by D* ¼ 2w*. g ¼ gravitational acceleration. The thickness hg of To assess the effect of certain variables on the each immobilized grainflow is given by h ¼ sin (h)L/n, with h the dynamic angle of vertical sorting, the fining upward trend along g repose, L the total progradation or migration the foreset is expressed as the sorting slope of the length of the bedform and n the number of relative grain size w* versus the depth (1 ) z*) in grainflows. The frequency (f) of grainflows was the bedform (not the height z*, because the sorting slope is conveniently defined as positive determined as 1/Tf, where Tf is the average time for fining upward sorting with 1 ) z*). The between each grain flow onset. These variables sorting slope: can be grouped into various alternative non- dimensional numbers, of which the explanatory 1 dw capacity for vertical sorting will be tested on the SS ¼ ð3Þ ra;T d1ðÞ z data. The transport rate is made dimensionless with is calculated for all sampled layers except the the height: bottom and the top, to exclude the effects of the toeset on the sorting. By parametrizing the sorting qbð1 kpÞ C Tr ¼ pffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffi ð5Þ with this slope, a linear relation between w* and RgDD RgD height is assumed. For the benefit of modelling the sorting for the The celerity is made dimensionless with the full grain-size distribution rather than the Dg, this geometric mean sediment size to describe the analysis is extended by considering the distribu- progradation in terms of the size of grains depos- tion of each grain-size fraction. Again, the distri- ited on the lee slope, because the individual grain bution in z of each fraction pi,z depends in the flow laminae thickness and the kinetic sieving first place on its abundance in the mixture going scale with this grain size: over the top; pi,T. Therefore pi,z is made dimen- C sionless as pi,z/pi,T. The grain size of the fraction C ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð6Þ is made relative to the average sediment going RgDg;T over the top as wi ¼ wm;i;z wm;T . The sorting slope for fractions SFi* can be now computed for As detailed information on the grain flow each fraction: thickness (h ) (after it stopped; not in dilated g mobile condition) and frequency (f) is available = d pi;z pi;T and are related to the celerity C as f ¼ C/h ,a SF ¼ ð4Þ g i dð1 zÞ dimensionless grainflow number A* can be con- structed that potentially relates the sorting directly to the thickness of the laminae (as an Dimensional analysis alternative to C*): Vertical sorting itself and the representation of Rg Rghg the grain-size distribution (factor 1 and possibly 4 A ¼ ¼ ð7Þ Cf C2 in the review) have been discussed in the previ- ous section. The sorting slope characterizes the The geometric mean grain size, the mixture fining upward trend along the foresets. The dila- standard deviation and the length of the lee slope tion angle is the static angle minus the dynamic (D/ sin h, in which h ¼ dynamic angle of repose)

Ó 2005 International Association of Sedimentologists, Sedimentology, 52, 291–311 Grain flow size sorting 295 can be combined in a dimensionless sorting all sediment sizes in the vertical direction. The number. Assuming that a longer lee slope and a following non-dimensional grain fall number can higher standard deviation both lead to more be constructed: efficient sorting, the sorting number is: 1 u tan hð h1 þ DÞ G ¼ 2 ð9Þ rg;T D ws D S ¼ 2 ð8Þ Dg;T sin h

The product of the standard deviation and the MATERIALS AND METHODS length of the slope are made dimensionless with the square of the geometric mean size. The vertical sorting as a result of grain fall is The flow conditions at the top of the slope opposite to that of grainflows. Herein it was govern the toeset deposition process and the attempted to isolate the grainflow process as shape of the wedge. The following reasoning is much as possible in order to obtain the largest based on the ideas of Jopling (1965). The settling possible sorting. The experiments were therefore velocity of the finer sediment (say, of grains with done with a very small (Froude-critical) water a diameter that is 1 standard deviation smaller depth above the bedform compared to the water ) than the mean: wm,T ra,T) acts in the vertical depth downstream of the bedform to maximize direction (downwards) while the flow velocity at the flow expansion and minimize the height and the top gives the grains an initial momentum in length of suspended grain trajectories. This was the horizontal direction (downstream). It can now done by generating deltas rather than dunes. be seen that the path of the grains from the top to Deltas were created with different heights, sedi- the toeset deposit is related to u/ws and to the ment feed rates and sediments. horizontal distance between top and toe (D/tan h). Two sets of experiments were done. Experi- As the water depth h1 is very small relative to the mental series 1 was with a wide sand–gravel bedform height in the experiments reported here, mixture (N, see Table 1 and Fig. 2) to test the the initial grain height above the lee slope top effect of delta height (water depth h0) and celerity (Jopling, 1965) is simply assumed to be 0Æ5h1. For (related sediment discharge and delta height as larger water depths, this analysis should include c ¼ [(1 ) kp)qb]/D). These two were systematic- the flow velocity and sediment concentration of ally varied in combinations of three target water

Table 1. Basic data of the experiments.

h0 q h1 i C D qb* Tf Tg hdilat hg DgT ) ) ) No. (m) (m2 sec 1) (m) (–) (cm min 1) (m) (m2 sec 1) (sec) (sec) (°) (mm) (mm) Experimental series 1 N1 0Æ200 3Æ1E-04 0Æ008 0Æ063 7Æ92 0Æ18 3Æ9E-04 9Æ41 3Æ77 4Æ86Æ50 1Æ56 N2 0Æ203 2Æ9E-04 0Æ010 0Æ032 2Æ40 0Æ19 1Æ3E-04 20Æ65 3Æ48 4Æ88Æ59 1Æ33 N3 0Æ200 4Æ2E-04 0Æ010 0Æ066 10Æ71 0Æ18 5Æ4E-04 7Æ71 3Æ49 6Æ37Æ11 1Æ45 N4 0Æ300 4Æ5E-04 0Æ010 0Æ067 7Æ50 0Æ28 6Æ8E-04 – – 3Æ3– 1Æ44 N5 0Æ301 4Æ2E-04 0Æ010 0Æ067 4Æ79 0Æ28 4Æ5E-04 14Æ67 5Æ16 5Æ36Æ17 1Æ41 N6 0Æ299 3Æ7E-04 0Æ010 0Æ037 2Æ48 0Æ28 2Æ3E-04 21Æ81 5Æ56 5Æ05Æ68 1Æ34 N7 0Æ080 3Æ5E-04 0Æ011 0Æ036 8Æ43 0Æ07 1Æ3E-04 4Æ97 1Æ94 4Æ03Æ56 1Æ33 N8 0Æ081 3Æ3E-04 0Æ012 0Æ026 3Æ69 0Æ07 5Æ6E-05 7Æ66 1Æ89 4Æ82Æ47 1Æ23 N9 0Æ082 3Æ9E-04 0Æ011 0Æ014 5Æ27 0Æ07 7Æ7E-05 7Æ21 1Æ83 4Æ83Æ32 1Æ22 N10 0Æ080 5Æ2E-04 0Æ013 0Æ039 34Æ50 0Æ06 4Æ3E-04 3Æ00 1Æ76 4Æ88Æ63 1Æ47 N11 0Æ198 3Æ9E-04 0Æ011 0Æ034 3Æ53 0Æ18 1Æ8E-04 14Æ67 5Æ02 4Æ86Æ11 1Æ33 Experimental series 2 M1 0Æ200 4Æ4E-04 0Æ010 0Æ069 9Æ60 0Æ19 5Æ7E-04 9Æ92 3Æ56 5Æ09Æ26 3Æ71 U1 0Æ201 4Æ2E-04 0Æ010 0Æ069 9Æ09 0Æ19 5Æ1E-04 9Æ49 3Æ08 5Æ07Æ95 2Æ49 S1 0Æ200 4Æ2E-04 0Æ012 0Æ027 3Æ62 0Æ18 1Æ9E-04 19Æ15 7Æ07 2Æ06Æ18 0Æ30 S2 0Æ080 4Æ1E-04 0Æ010 0Æ036 26Æ67 0Æ07 3Æ6E-04 3Æ05 1Æ52 2Æ07Æ15 0Æ77 A1 0Æ200 3Æ4E-04 0Æ010 0Æ060 6Æ83 0Æ19 4Æ3E-04 12Æ84 2Æ95 8Æ37Æ68 0Æ88 C1 0Æ205 3Æ9E-04 0Æ010 0Æ086 12Æ00 0Æ19 7Æ1E-04 8Æ53 3Æ05 5Æ08Æ96 1Æ01 C2 0Æ208 5Æ8E-04 0Æ012 0Æ072 13Æ47 0Æ19 7Æ8E-04 7Æ20 2Æ88 5Æ07Æ96 3Æ59

*The volumetric bedload transport rate (qb) includes the pore space.

Ó 2005 International Association of Sedimentologists, Sedimentology, 52, 291–311 296 M. G. Kleinhans

A and Utrecht mixtures are subsets of the coloured 0·8 sediment to obtain narrower mixtures. The sand Normal mixture is a subset of the normal mixture from which most of the pea-gravel had been removed. 0·6 Colored The angular mixture is crushed granite and therefore angular compared to the other mixtures Minnesota 0·4 which had rounded grains. This applies mostly

Fraction to the coarser grains in the mixture as the sand Utrecht grains in A are as angular in those of the S and N

0·2 Sand mixtures. The experiments of the second set were all Angular done with the intermediate delta height of 0Æ2m, 0 except one where the aim was again to obtain a 0·01 0·1 1 10 100 very high celerity. The latter experiment (S2) Grain size (mm) tested whether the condition of much suspension B 1 and continuous grainflows leads to absence of σ vertical sorting (see Allen, 1970). There are two mix Dg g %gravel N 1·18 2·85 37 experiments with the bimodal C mixture of which 0·8 U 2·86 1·34 92 the flow discharge in C2 is 1Æ5 times larger than in C 2·71 2·36 88 C1. This experiment was done to test the effect of M 3·71 1·66 85 0·6 A 2·11 2·41 64 higher flow velocity over the delta top on the S 0·69 2·26 7 suspension and trajectory of the fine sand tail of the C mixture. The S2 and C2 experiments can be Probability 0·4 seen as exploratory and preliminary. 0·2 The experiments were done in a narrowed flume at St Anthony Falls Laboratory (Fig. 3), which had a length of 7 m, a depth of 0.38 m and 0 a width of 0.075 m. Water and sediment were fed 0·1 1 10 Grain size (mm) at a constant and adjustable rate at the upstream end of the flume. The water depth in the basin Fig. 2. Grain-size distributions of the sediment mix- tures fed into the flume in non-cumulative (A) and (h0) was fixed with the downstream weir. The flow on the delta developed to its own equilib- cumulative (B) sieve curves. Dg ¼ geometric mean size (mm), rg ¼ geometric standard deviation. All sedi- rium slope (i) and water depth (h1) for the given )3 ments had a density of approximately 2650 kg m and discharge and sediment load. The water depth h1 a porosity varying between 0Æ30 and 0Æ36. The U and M was measured through the glass wall and in the mixtures are subsets (two colour-coded fractions) of the middle of the flume, while the bed slope was C mixture, the S is a subset of N after removal of most of measured at the end of the experiments from bed the pea gravel in N, and A is crushed granite of which the fractions >1 mm is highly angular and <1 mm is as level measurement at every 0.1 m along the delta angular as in the N and S mixtures. at both sidewalls. The flow discharge (Q) was determined twice per run by measuring the time it took to fill a bucket of known volume. The flow depths (0Æ08, 0Æ2and0Æ28 m) and three target velocity (u) over the delta was determined from )1 celerities (3, 6 and 9 cm min ). Due to the water depth (h1), flow discharge and flume width limitations of the sediment feeder, higher celeri- (W) from u ¼ Q/(Wh1). The slope (i) was deter- ties were only applied with smaller deltas mined from the bed elevation along the alluvial (experiment N3 and N10). topset of the deltas. From the slope and the water Experimental series 2 was done with a variety depth (h1) the total shear stress (s) was deter- of sediment mixtures that were available, to mined with s ¼ qgh1i. The dimensionless study the effect of sediment size, angularity and (Shields) shear stress is computed as s* ¼ s/qDg. mixture bimodality (or skewness depending on The width/depth ratio was about 8 so a sidewall how the fine tail is described) (Table 1 and roughness correction was unnecessary. The deter- Fig. 2). The coloured sediment was convenient mination of the water depth through the glass because the grain-size fractions were colour- wall, however, is inaccurate: ±1 mm which cor- coded, and consisted mostly of rounded gravel responds for most experiments to ±10%. The and of a tail of fine (white) sand. The Minnesota water temperature was 10° ±1°C.

Ó 2005 International Association of Sedimentologists, Sedimentology, 52, 291–311 Grain flow size sorting 297

Discharge and sediment feeder A Slope i Water depth h1 Water depth Grain flow thickness h0 Sediment bed hg

Fig. 3. Outline of the Delta Flume at St Anthony Falls Laboratory B (vertical scale exaggerated and 2 topset samples length cut off in the drawing near Foreset sample the weir). (A) Case with 0Æ3 m high deltas showing the hydraulic Final disturbed parameters. (B) Case with 0Æ08 m sediment volume high deltas showing the initial Initial sediment bed disturbed sediment and the 2 bed sampling locations sediment sampling locations. Toeset sample

Grain flow thickness, duration and frequency sediment in the topset is slightly downstream were determined from analysing the timing of fining for i < 0.05 and downstream coarsening for onset and stopping of grain flows from close-up i > 0.05. Thus, the feeder sediment is not exactly digital videos, excluding the initial 0.15 m of the same as the sediment going over the brink- accreted sediment. The duration Tg of grainflows point to participate in the grain flows because of ) was determined as Tg ¼ timmobilizing tonset from size-selective sediment transport and deposition the time base of the videos. During each run, the in the upstream topset. Therefore the samples at angle of the prograding deltafront was measured all heights z at each sampling location in the manually just before and after the individual deltas, was averaged over the depth to determine grain flow events, to determine the static and the grain size distribution of the sediment parti- dynamic angles of repose. cipating in the grain flows. This average compo- After the water was drained from the flume, the sition therefore excludes the topset and toeset bed was photographed from both sidewalls and sediment. from the top of the flume. The bed surface elevation was measured for the determination of the slope and the total volume of sediment. At RESULTS two horizontal positions in the delta (at the downstream end and just upstream, Fig. 3B), the Basic data bed was sampled vertically in horizontal layers of 1.5–2 cm thickness over a length of 15–20 cm for The basic data are presented in Table 1. The main grain size analysis by dry sieving (comparable to body of the delta is the fining upward deposit collecting a core and slicing the core in thin from the grainflows (example in Fig. 4A). On top layers). The topset, the toeset (over a length of of the delta, there is an alluvial topset at an angle 0.2 m immediately downstream of the delta slip- equal to that of the water surface slope of the flow face) and the foreset (1–2 grains thick, between which transported the sediment from the inlet to topset and toeset) were sampled separately. The the brinkpoint of the delta (Fig. 4B). This slope first 0.1–0.15 m of the delta was not sampled as it was straight in all cases except in the most represented the initial accumulation phase, while upstream few cm where water and sediment beyond 0.15 m a simple Gilbert-type delta pro- entered the flume. In most experiments a toeset gressively developed down the flume. formed by settling of sediment that was suspen- In order to maintain a constant bed slope, some ded at the brinkpoint. When this sediment was sediment is deposited in the topset while the overrun by the prograding delta, it became a brinkpoint of the delta migrates downstream. The bottomset.

Ó 2005 International Association of Sedimentologists, Sedimentology, 52, 291–311 298 M. G. Kleinhans

lload ttrransporrtt A water level ttopsett:: bed

M1 t se re fo on g in ch an al av

bottomset: settled from suspension 1·2m 1·1m

B 0·4

ent N4 n experim delta top i 0·3 slipface Fig. 4. (A) Fining upward sorting in x x delta M1. The dark grains are large, x x 0·2 x x the light grains are small. Position of bottomset in deltas with finer sediment is indicated for clarity. 0·1 Scale (in m) at the bottom of the photograph is the same as in (B). (B) Examples of surface profiles of some experiments, which were used Height above base flume (m) Height above 0 1·75 1·5 1·25 1 0·75 0·5 0·25 to determine the alluvial slope and Distance along flume (m) the total volume of the delta (for determination of sediment flume base N1 N2 N4 N10 x M1 transport).

dynamic angle of repose of the lower lee slope Process description of sorting in the grainflow (32°). The following qualitative description is based on Second, the sediment mass moves downslope visual observations done during the experiments over a small distance (less than half the lee slope and verified with the videos (Fig. 5). The sorting length). The accommodation space thus created is in the grainflows in the N experiments is a three- filled by the continuous grain fall. During the stage process. The first process is the deposition small downslope movement of the sediment, the of sediment from grain fall just downstream of the gravel is worked upwards in the moving layer by delta top. The flow carries the sediment down- kinetic sieving. As a consequence, the lee slope is stream from the delta top, and meanwhile the covered with the coarsest sediment from top to grains settle to the slip face. Thus a wedge-shaped toe for most of the time. mass of mixed sediment is built up at the upper Third, at some point roughly the whole upper few cm of the lee slope from the sediment half of the lee slope sediment fails and moves ‘raining’ from suspension. Beyond the zone just downslope as a large grain flow, usually about downstream of the delta top, however, there was 8 mm thick in the N experiments. The statistics of very little suspended sediment, except in the N10 these large grainflows will be presented later. The and S2 experiments. For the smaller grains, the gravel on the lower half of the lee slope is partly trajectory is longer and the smallest grains are dragged along and partly pushed ahead of the even deposited downstream of the delta as a thin flow, contributing to the upward fining. The toeset. The toesets were less than 1 mm thick at plane at which the sediment fails is in the middle 0Æ1 m downstream of the delta toes, except in N10 of the previous grainflow deposit between its and S2 where it was a few mm thick. The static gravelly surface and the underlying finer sedi- angle of repose of the wedge-shaped mass is ment. This underlying sediment is sandy near the larger (37° for the N experiments) than the delta top and gravelly near the base, and is finer

Ó 2005 International Association of Sedimentologists, Sedimentology, 52, 291–311 Grain flow size sorting 299

Water surface than the lee slope surface along most of the slope. So mostly the gravelly top of the previous grain- Grain flow is both pushed and dragged downslope by fall the next grainflow, which contributes to the Traction load observed upward fining in the delta. As no fine sediment drapes were found on the lee slopes and no clear cross-lamination was observed, the Dynamic Toeset 1. angle of repose deposited sediment from grain fall on top of the grainflows must have been incorporated into the grainflows by kinetic sieving immediately. Sometimes bedload sheets were observed on the Wedge topset (but never downstream of the brinkpoint), building but the volume of sediment in these bedload sheets was much smaller than the volume of the wedge, so the grainflows were solely driven by Static angle wedge collapse. 2. of repose When the grainflow reaches the bottom of the flume (top of the toeset), the gravelly head of the flowing sediment mass stops moving first and within a second or so all the sediment becomes Small grain flows, immobilized. The surface of the stabilized grain- with kinematic flow is again gravelly, while the top of the delta is sorting Kinematically sandy. As the slope of the immobilized grainflow sorted deposit (dynamic angle of repose) is smaller than the of previous large static angle of repose, there is a new accommo- grain flow 3. dation space for sediment depositing from grain fall, and the process starts anew.

Initiation of large Overall vertical sorting grain flow, which drags underlying Figures 6 and 7 show the non-dimensionalized Delta top gravel position just vertical sorting data for the two experimental before previous series. In all experiments, the sediment was large grain flow sorted fining upward, and the r* at most heights 4. is smaller than unity because the sediment is vertically sorted and therefore had a smaller standard deviation than the total mixture going Large grain flow over the delta top. The notable exception is the with kinematic base of most of the deltas, which has a larger r* sorting because the fine sediment of the toeset is mixed with the coarse sediment of the lowest part of the Fining foresets. The deltas without fine sediment (M1 upward and U1) do not have this toeset. sorting 5. Two examples (Fig. 8A and B) demonstrate that the fractions are distributed along the lee slope as Fig. 5. Conceptual model of the sediment deposition expected in such a way that the net sorting is and sorting on the lee slope of the experimental deltas. fining upward, but the ‘structure’ differs between The three stages of the process are given: 1,2: grain fall creating the sediment wedge and the toeset, 3: small the mixtures. The finer sediment fractions are grainflows with kinetic sieving, and 4,5: large grain- better represented in higher positions in the delta flows that drag the top of the previous grainflow de- and the coarser sediment fractions in the lower posit downslope, leading to the fining upward. The positions. The presence of the finest fraction in dynamic and static angle of repose, the sorted deposit the lowest sampled layer was explained above of the previous large grainflow and the previous posi- (Fig. 8A and B). There is a structural difference tion of the delta front are introduced in the first four between N6 and C1,2: in the N6 delta the coarsest panels, but are present in all time steps in reality. fraction has a relatively higher abundance in the

Ó 2005 International Association of Sedimentologists, Sedimentology, 52, 291–311 300 M. G. Kleinhans

1

∆ = 0·07 m base) 0·8 N7 0·8 = N7 top 0 0·6 N8 0·6 = N8 N9 0·4 0·4 N9 N10 0·2 Smallest deltas 0·2 N10

∆ = 0·07 m relative height z* (1=top 0=base) 0 0 finer coarser better sorted worse sorted 1 -1 -0·5 0 0·5 1 1·51 0·6 0·8 1 1·2 1·4 1·6 relative grain size psi* relative standard deviation sigma* ∆ = 0·18 m base) Relative height z* (1 0·8 N1 0·8 =

N1 top 0 N2 =

0·6 0·6 N2 N3 0·4 0·4 N3 N11 0·2 0·2 N11

∆ = 0·18 m relative height z* (1=top 0=base) 0 0 finer coarser better sorted worse sorted 1 -1 -0·5 0 0·5 1 1·51 0·6 0·8 1 1·2 1·4 1·6 relative grain size psi* relative standard deviation sigma* ∆ = 0·28 m base) Relative height z* (1

= 0·8 0·8 top 0

N4 = 0·6 0·6 N4 N5 N5 0·4 0·4 N6 N6 0·2 Largest deltas 0·2

Relative height z* (1 ∆ = 0·28 m relative height z* (1=top 0=base) 0 0 –1·5 –1 –0·5 0 0·5 1 1·5 0·6 0·8 1 1·2 1·4 1·6 1·8 Relative grain size y * Relative standard deviation s * Fig. 6. The sorting in the deltas of series 1 with the N sediment mixture. The vertical axis is relative height z* ¼ z/D. ) The horizontal axis of the left-side panels is the relative sediment size w* ¼ wm,z wm,T which is the logarithm of the ratio of the sediment diameter at a certain height (Dg,z) and the diameter of all the sediment in the delta (Dg,T): w* 2 ¼ Dg,z/Dg,T. The relative mixture standard deviation is r* ¼ ra,z/ra,T. The topset and toe deposit have not been included. The bottomset, however, is included in the samples taken at the base of the deltas. The sorting curves have been grouped for delta height. For most N experiments, two curves are shown as the delta was sampled twice. lowest layer compared to the trend in the layers fractions are more abundant in the delta tops and above (Fig. 8A), whereas in the C deltas the coarse grain-size fractions more abundant in the coarsest two fractions have moderate abundances lower parts of the delta. The structure of the N over the full depth while the finer fractions show mixture is slightly modified by the delta height a much more pronounced distribution pattern and delta celerity. The celerity has the strongest (Fig. 8B). This is likely to be a result of the effect on vertical sorting for the larger deltas. The mechanism of dragging coarse sediment down- U, M and S mixtures, which are relatively narrow, slope by the next grainflow, which is absent in have different shapes (Fig. 8D) with less pro- C1,2 but clearly present in the N mixtures. nounced vertical sorting. The N, C and A mix- This structure in the fractionwise sorting data is tures, which are all relatively wide, have visualized in a different manner in Fig. 8C and D. comparable structures. It is also clear that the The N, C and A mixtures all show a ‘lazy assumption of a linear function in the definition S-shaped’ structure (schematized by the curves of the sorting slope (Eq. 4) is violated for the in Fig. 8C and D), in which the fine grain-size grain-size fractions approach.

Ó 2005 International Association of Sedimentologists, Sedimentology, 52, 291–311 Grain flow size sorting 301

1 ∆ = 0·18 m base)

=

0·8 0·8 M1 M1 top 0

= 0·6 0·6 U1 U1

0·4 C10·4 C1

0·2 C20·2 C2

∆ relative height z* (1=top 0=base)

Relative height z* (1 = 0·18 m 0 0 finer coarser better sorted worse sorted 1-1·5 -1 -0·5 0 0·5 1 10·6 0·8 1 1·2 1·4 1·6 relative grain size psi* relative standard deviation sigma* ∆ = 0·18 m (S2: ∆ = 0·07 m) base) 0·8 0·8 =

top 0 S1 S1 =

0·6 0·6

S2 S2 0·4 0·4 A1 A1 0·2 0·2 ∆ = 0·18 m

∆ relative height z* (1=top 0=base) Relative height z* (1 (S2: = 0·07 m) 0 0 –1·5 –1 –0·5 0 0·5 1 1·5 0·6 0·8 1 1·2 1·4 1·6 1·8 Relative grain size y * Relative standard deviation s * Fig. 7. Same as Fig. 6, now for the alternative sediment mixtures of experimental series 2.

AB 1 1

0·18 0·18 base)

= 0·8 0·8 N60·35 C1,2 0·35 top 0

=

0·6 0·71 0·6 0·71

0·4 1·41 0·4 1·41

2·83 2·83 0·2 0·2

Relative height z* (1 5·66 5·66 0 0 0123012345 Abundance at height z* / Total abundance = pi,z* / pi,T Abundance at height z* / Total abundance = pi,z* / pi,T CD 4 4 Decrease of proportion * i with height in delta A1 2 2 A1 C2 N4 C1 A1 S2 S2 C1 S2 S1 S1 0 0 C2 S2 S2 U1 N8 S2 M1 S2 U1 S2 S1 N10 A1 M1 S1 C2 M1 S2 S1 C1 S1,2 C2 Celerity: A1 –2 low high –2 C1,2 S2 N6 C1 C1 Height: low A1 C2 C2 A1 high C1 –4 –4 U1 Increase of Finer Coarser M1,U1

Sorting slope for fractions SF proportion fractions fractions –6 –6 –6 –4 –2 02–6 –4 –2 02 y y Relative grain size of fraction i* Relative grain size of fraction i* Fig. 8. Vertical sorting of grain-size fractions. (A) Data of N6 (legend indicates grain size of the fraction). (B) Data of C1 and C2. (C) Sorting slope of each fraction SFi in N experiments as a function of relative grain size of the fraction wi . Lines through the data were fitted by eye for selected experiments to show the differences between high and low deltas and high and low celerities. (D) As (C), for various sediments of experimental series 2.

Ó 2005 International Association of Sedimentologists, Sedimentology, 52, 291–311 302 M. G. Kleinhans

Effect of grainflow kinematics on vertical Dimensionless parameters affecting vertical sorting sorting: results The probability distributions of the duration of The relation between the dimensionless numbers grainflows, and interval between grainflows are and the vertical sorting in the foreset deposits is shown in Fig. 9. The length and variation of the shown in Fig. 10. Within the given range of grainflow duration (Tg) depends on the length of transports and celerities, the sorting slope ranges the lee slope: Tg is much longer for N6 than for from 2 (the maximum) to almost zero (the mini- N8, whereas N8 and N10 have almost the same mum) with increasing C* and Tr* and decreasing Tg even though their C* is very different. The S*. A significant correlation between one of these length and variation of the grainflow interval parameters and the sorting slope SS* indicates a (Tf), on the other hand, depends mostly on C*as contribution to the description of the sorting; a the slow N6 has both larger interval and significant correlation between the parameters, variation in that interval than N10 which is however, indicates redundancy of parameters. almost continuously grainflowing. These find- The correlations in Table 2 indicate that the C* ings indicate that the grainflows are initiated by and S* are the most relevant for vertical sorting, wedge failure, which depends on a fixed volume whereas the Tr* and G* are redundant. The of the wedge which would cause much variation correlation of A* with SS* is higher than that in the Tg. between C* and SS*, but data for A* are difficult

1 1

0·8 0·8 N6 N8 0·6 Large height 0·6 Small height Small celerity Small celerity Probability Probability 0·4 0·4 Duration of grain flow

0·2 Interval between grain flows 0·2

0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Class (seconds) Class (seconds) 1 1

0·8 0·8 N10 C2 Small height Intermediate height 0·6 Large celerity 0·6 Large celerity

close resemblance indicates

Probability 0·4 Probability 0·4 continuous grain flowing

0·2 0·2

0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Class (seconds) Class (seconds) 1 1

0·8 0·8

0·6 0·6

S1 A1 Intermediate height Probability 0·4 Probability 0·4 Intermediate height Small celerity Intermediate celerity

0·2 0·2

0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Class (seconds) Class (seconds) Fig. 9. Probability distributions of the duration of, and interval between grainflows for representative experiments. Generalized information on delta height and celerity is given in the panels.

Ó 2005 International Association of Sedimentologists, Sedimentology, 52, 291–311 Grain flow size sorting 303

A B

2 S1 N2 2 N2 S1 N5 N2 N5 N1 N1 N2 N6 N6 N11 N6 M1 M1 N11 N4 N6 N4 N5 N9 N5 1·5 N8 N9 1·5 N9 N9 A1 N8 N11 A1 N11 N7 N10 N1 N3 N10 N7 N3 N3 N8 N3 N8 N1 C2 C2 1 C1 1 C1 N7 N7 N4 N4 N10 S2 N10 U1 U1 S2 Sorting SS* slope 0·5 0·5 S2 S2

0 0 0·0001 0·001 0·01 1E4 1E5 1E6 1E7 1E8 Dimensionless transport Tr* Dimensionless sorting S* C D

2 N2 S1 2 S1 N2 N5 N5 N2 N1 N1 N2 N11 N11 N6 N6 M1 M1 N6 N4 N6 T3 N5 N9 1·5 N8 N9 A1 N5 N9 A1 T4 1·5 T1 N9 N8 N11 N11 N7 N10 N10 N3 N3 N7 N8 N1 N3 N1 N8 BK7 B2 T5 N3 C2 T6 1 C1 1 C2 C1 N7 N7 N4 S2 S2 U1 N10 N10 U1 B1 Sorting SS* slope 0·5 0·5 S2 E1 S2 E2

0 0·001 0·01 0·1 1E3 1E4 1E5 1E6 Dimensionless celerity C* Dimensionless grain flow number A* Fig. 10. Relation between dimensionless numbers and fining upward sorting. Sorting slope as computed from the data presented in Figs 6 and 7, versus the dimensionless celerity, transport and sorting numbers. The line in C and D shows the power functions of sorting slope versus C*orA* (see text). The C* panel includes data from literature (see Discussion).

Table 2. Correlations between the natural logarithm of why the behaviour varies between the mixtures, the dimensionless parameters. The significance is but it discourages the construction of a simple determined with the F-test at a probability level of 0Æ05 vertical sorting model for grain-size fractions for 30 data points, which gave a critical correlation of (discussed later). 0Æ36. Significant correlations are given in bold script. The predictive capacity of the dimensionless

S* Tr* C* A* G* SS* rg,bottom celerity C* for vertical trends in average grain size should be reasonable. The relation with the S*1Æ00 Tr* )0Æ45 1Æ00 grainflow number A* is given as well because C* )0Æ27 0Æ92 1Æ00 its correlation was higher and for the sake of A* 0Æ52 )0Æ96 )0Æ94 1Æ00 future work on cross-bedded deposits. In double G*0Æ19 0Æ42 0Æ56 )0Æ42 1Æ00 logarithmic space, the sorting slope SS* versus C* SS* 0Æ42 )0Æ56 )0Æ59 0Æ69 )0Æ26 1Æ00 or A* is a straight line, so power functions were ) ) ) rg,bottom 0Æ33 0Æ56 0Æ58 0Æ61 0Æ34 0Æ54 1Æ00 fitted: ðaÞ SS ¼14C035 018 ð10Þ to obtain in active process environments so the ðbÞ SS ¼17A regressed function of SS* versus C* is given as well for practical purposes. So the vertical sorting can now be computed as:

The sorting slopes SFi* (Fig. 8C and D) dem- 035 w r onstrate that the grain-size fractions behave in a ðaÞ Z ¼ a;T ½ð1 z Þ0514C 018 ð11Þ complex way. In general there is a trend of ðbÞ wZ ¼ ra;T ½ð1 z Þ0517A increasing SFi* with increasing relative grain size wi*, but the trend is reversed for the finest fraction The intercept [(1 ) z*))0Æ5] of the vertical except for the U, M and S mixtures. It is not clear sorting w* is thus simply taken at z* ¼ 0Æ5

Ó 2005 International Association of Sedimentologists, Sedimentology, 52, 291–311 304 M. G. Kleinhans because this is the average intercept for the data graphs, which is expected because the fine sand presented in Figs 6 and 7, which means that the in the bimodal sand–gravel mixture C was much grain size at 0Æ5 times the delta height is equal to more susceptible to suspension than the coarse the grain size averaged over the delta. sand in the N, A and S experiments. There is a The relation between the dimensionless num- weak relation between the sorting slope and the bers and the toeset deposition is shown in Fig. 11 dimensionless grain fall parameter (G*), but G*is and Table 2. The C experiments are outliers in all strongly correlated to C* and Tr*, making G* largely redundant.

A 1·8 C2 DISCUSSION

1·6 C1 The processes of sorting 1·4 N10 A1 N7 S2 For the first experimental series (N sediments), S2 N10 S1 1·2 N9 N7 three trends (apart from fining upward) in the N8 N3 N8 sorting are manifest that confirm and extend the N3 1 N11 N4 N9 N1 U1 findings and hypotheses from the review sec- N11 N5 N4 N2 N2 N6 N5 tion: * of lowest sampled layer sampled lowest * of 0·8 M1 σ N1 N6 1. The given observations on the sorting process 0·6 in grainflows agree with most records in literature 1·4 1·6 1·8 2 2·2 2·4 (e.g. Bagnold, 1954; Allen, 1965; Jopling, 1965; Dimensionless grain fall parameter G* Hunter & Kocurek, 1986; Pye & Tsoar, 1990; Makse, 1997; Koeppe et al., 1998; Tischer et al., B 1·8 2001; Kleinhans, 2002). The observation that the C2 sorting is established by drag of a large grainflow 1·6 C1 on the gravelly top of the underlying grainflow deposit confirms the hypothesis in the review 1·4 N10 section. The shearing plane of that large grainflow A1 S2 N7 S2 S1 was at the transition between gravel and sand in N10 1·2 N9 N7 the underlying grainflow deposit, which can be N8 N8 N3 N9 U1 explained with the smaller angle of repose of that 1 N11 N3 N11 N4 N5 N1 N4 gravel on the sand being smaller than the angle of N2 N2 N5 * of lowest sampled layer sampled lowest * of 0·8 N6 repose of the overlying sand on the underlying σ M1 N6 N1 gravel (Makse, 1997). Sohn et al. (1997) mentions 0·6 a comparable mechanism on a Gilbert type delta 0·001 0·01 0·1 foreset in a Miocene fan delta: protruding large Dimensionless celerity C* clasts are apt to be remobilized by new debris- C flows (their fig. 8A) and large clasts were found to 1·8

C2 have a small resistance to the substrate and a larger momentum and therefore outrun further 1·6 C1 downslope (their fig. 8B). 1·4 2. The relations of r* versus z* are more curved N10 A1 N7 S2 in the middle of the delta for D >0Æ1 than for S1 S2 N10 1·2 N9 D <0Æ1 (Fig. 6). This means that the sediment is N7 N8 N8 actually better sorted (smaller r*) in the top and N3 N3 1 N11 N9 N11 U1 N1 lower parts of the larger deltas than in the middle N5 N4 N4 N2 N2 N5 of the larger deltas and the whole of the small * of lowest sampled layer sampled lowest * of N6

σ 0·8 N6 M1 N1 deltas. So, for larger deltas, the sorting process is more efficient at the top and lower parts of the 0·6 0·0001 0·001 0·01 foreset. This is also the case for the S1 (large delta) Dimensionless transport Tr* and S2 (small delta) experiments. This can be Fig. 11. The relative standard deviation of the toeset interpreted as that the small deltas were too small sediment (r*) plotted versus the dimensionless celerity to develop fully sorted lee slopes, whereas for C* (A), (B) transport Tr* and (C) grain fall G* numbers. D >0Æ1 the vertical patterns in standard deviation

Ó 2005 International Association of Sedimentologists, Sedimentology, 52, 291–311 Grain flow size sorting 305 are similar indicating that these deltas were large thickness. So, the flow conditions on the alluvial enough for full development of the sorting. slope have no obvious effect on the sorting on the 3. The toeset is more prominently present in the foreset, but do have an effect on the toeset lowest deltas, the measured thickness (not shown formation. This agrees with the visual observation here) also indicates this. So, the fine sediment that suspension at the brinkpoint was limited settles from suspension on the flume base for the except in the deltas with the highest celerity (and lowest deltas and settles on the lee slope of the transport rate, and flow velocity above the delta higher deltas. top). The second experimental series with various sediment mixtures (Fig. 7) reveals four additional The effect of grainflow size and kinematics on observations: vertical sorting 4. The differences in sorting between all sedi- ment mixtures are much larger than the differ- Grainflow kinematics decrease vertical sorting ences in sorting for one mixture in various (Fig. 9). In addition, the S1 experiment shows a conditions. The mixtures with a smaller geomet- much larger variation in Tg than the other ric standard deviation (M1 and U1) show less experiments. An explanation is related to the fact vertical sorting; the sorting trend in U1 is very that S1 had much less gravel than the other small compared to that of wider mixtures. This sediments. For a large slope failure and grainflow was expected, because the more uniform the in the N and C experiments to take place, the sediment, the less there is to sort. There are other gravelly layer on the lower half of the slope had to differences between the mixtures, e.g. the contrast fail. As a grainflow cannot become thinner than between M1 and U1 with other mixtures one grain thickness, this involves thick grain- (Fig. 8D), that cannot be explained from the flows. In the S1 experiment such a strong gravelly parameters. surface layer was not present, and the slope could 5. When A1 is compared to the N series, it can therefore fail sooner. Thus both frequent thin and be observed that the angularity of the coarser part infrequent thick grainflows became possible. of the A mixture has no significant effect on the This explanation suggests that the grainflow sorting. The A and N sediment have comparable dynamics are an effect rather than a cause of the grain sizes and standard deviations, and the vertical sorting. In the present case, Tg depends prominent difference is the angularity of the mostly on the wedge volume which is itself larger half of the grains. constrained by the necessary force to mobilize 6. The toeset is much more strongly developed the gravelly surface layer on the slope. This if the finest fraction is much finer than the rest of means that the sediment transport just upstream the sediment (fine tail or bimodal grain-size of the brinkpoint determines the Tg. Indeed, the distribution) as in experiments C1 and C2. So, grainflow frequency f ¼ 1/Tf strongly depends on the flow conditions on the alluvial slope were the celerity C* (Fig. 12A). Consequently, the such that the gravel in the mixture C could be grainflows tend to become continuous with transported as bedload, while the fine sand was increasing C*; that is, the ratio of grainflow suspended and deposited in a well-developed duration and interval between subsequent grain- toeset. flows Tg/Tf approaches unity for the faster repe- 7. The replica-experiments C1 and C2 were titions of grainflows (Fig. 12B). done in the same conditions, except that the flow The grainflow thickness hg was found to discharge in C2 was 1Æ5 times as high as in C1, depend on the delta height (Fig. 12C). This and the alluvial slope of C2 consequently was confirms the findings of Allen (1970) and Hun- smaller while the flow velocity in C2 was 10% ter & Kocurek (1986), and demonstrates that the higher. The difference in sorting between C1 and present dataset is consistent with their results C2, however is negligible, so the flow conditions even though the data from the literature was on the alluvial slope in this range have a marginal obtained from dunes and the present from the effect on the sorting at the lee slope of the delta. deltas. However, the laminae thickness of fore- This is not true, however, for the toeset formation set deposits cannot be used for reconstruction of in the N series; the flow discharge has an effect. the delta height, because the dataset (Fig. 12C) For example, for experiment N10 the flow dis- is for small deltas and dunes and shows much charge was 1Æ6 times as high as in N8, and the scatter, and it can reasonably be suspected that toeset is much more developed in N10 than in N8, the laminae thickness depends on grain size as although the toeset is still only a few mm in well.

Ó 2005 International Association of Sedimentologists, Sedimentology, 52, 291–311 306 M. G. Kleinhans

A 100 Towards fractionwise prediction of vertical sorting

N10S2 The sorting model developed above could be

N7 7- modified to compute the vertical distribution of 10 N9 C2 N8 N3 C1 M1U1N1 27- 22-25- 17- grain-size fractions by integrating (4): A1 N11N5 23- 10- 24-6^S1 N2N6 5^3~/- 4- 21- 18-8^ Pi;z 9~ 19^ 2~ 15- ¼½ð1 z Þ05SFi ð12Þ

equency (/min) 20^ 1 14~/-13~/- Pi;T 26~ Fr 12~ where SFi* is a function of wi and C* according to 11~ Fig. 8. However, in this approach it would be 0·1 assumed that the fractions are linearly distributed 0·0001 0·001 0·01 0·1 with increasing and decreasing abundance in z* C* for fine and coarse fractions, respectively. Fig- B 1 N1 N10 ure 8 demonstrates, however, that this assump- N3 tion is violated for the N mixture. Moreover, the C2 S2 M1 N7 relation between the SF * and w depends on the 0·8 i i C1 characteristics of the sediment mixture, which N5 U1 N11 N6 S1 means that a simple function may give errors. For 0·6 N10 N9 the modelling of vertical sorting a non-linear A1 N8 S2 function, depending on undefined mixture char- N2 N3 0·4 N1C2N7 acteristics would be needed, which in addition M1 S1 C1 N11 N5 U1 conserves mass for all grain-size fractions. Unfor- N6 N8 N9 A1 0·2 tunately there is not enough data available now to N2 Grainflow duration / interval undertake such a task.

0 0·001 0·01 0·1 Applicability of the results to natural river C* C dunes 0·02 The performance of Eq. 11a was tested against the data of the experiments. The results (Fig. 13A) N10 C1N2 4- C2 0·015 M1 agree with the N data (Fig. 6) although the scatter A1U1 13~/- N3 S2 is a factor of 2. The ratio of predicted and fitted 14~/- 15-12~ N1 S1N11 N5 SS* for all available data (Fig. 13B) demonstrates 10- 17- 3~/- N6 8^ 6^ 5^ 0·01 18- 24- 11~ that 95% of the slopes are predicted within a 25- 9~

27- factor of 2 for various datasets. For verification in 22- 23- N7 7- N9 the case of sorting in dunes and bars (discussed 0·005 N8 2~ later), the vertical sorting slope SS* as a function 20^21- 26~

19^ of C* is also computed for the data of Termes Laminae (adv per grain flow) (m) (1986), Blom & Kleinhans (1999), Blom et al. 0 (2000), and Kleinhans (2002). For these data the 0 0·05 0·1 0·15 0·2 0·25 0·3 scatter is larger, in specific for larger C* and r Slipface height (m) a,T (Fig. 13C), but in general Eq. 11a predicts the Fig. 12. (A) Frequency as a function of dimensionless sorting well. The computations with A* have the celerity C*. Numbers and symbols denote their experiment numbers and bed states on top of their same predictive capacity for the experiments but dunes (: ripples, Ù: secondary dunes, -: upper plane cannot be done for the data from the literature. bed). (B) Ratio of grainflow duration and interval be- It is concluded that the sorting process is tween subsequent grainflows Tg/Tf as a function of C* mostly dependent on C* and the sediment mix- [+: computed as Tg/Tf, x: computed as the more extreme ture characteristics, but that the natural variation case with (T + r )/(T ) r ], where r ¼ standard g Tg f Tf is large. The variation is due to the variables that deviation). When the ratio approaches unity, the were not included in the dimensional analysis; grainflowing becomes continuous. (C) Laminae thick- ness (grainflow thickness) versus slipface height (delta notably, the mixture characteristics are not fully height) of the present data and those of Hunter & described with the mean grain size and standard Kocurek (1986). deviation alone (Fig. 8D). In addition, the sorting

Ó 2005 International Association of Sedimentologists, Sedimentology, 52, 291–311 Grain flow size sorting 307

A is the result of two simultaneous processes with 1 different variables: the grain fall process mainly 2, 6, 8, 11 depends on the flow conditions and mode of base)

= 0·8 sediment transport (bedload or suspended load) 5, 9 of the grain sizes on top of the delta, whereas the top 0

= 0·6 grainflow process mainly depends on the sedi- 1, 3, 4, 7 ment transport rate, slip face height and the sediment characteristics that determine the grain- 0·4 10 flow thickness and rate of failure. To assess the applicability of the present results to natural lee 0·2 slopes, the question of whether additional para-

Relative height z* (1 meters such as dimensionless shear stress require 0 consideration is addressed in the section on –1·5 –1 –0·5 0 0·5 1 1·5 applicability to natural river dunes. Relative grain size y * The applicability of the present results to river B dunes is tested by comparison of the experimen- 10 tal ranges of flow conditions and celerity with those on experimental and field data from litera- ture [Table 3, see Kleinhans (2004) for a complete E2 T5 description]. The comparison of the (suspended) E1 BK7 N4 S2 grain fall conditions is based on the dimension- T3 B1 T6 N8 C1N7 T1 N11 less shear stress and the median or mean grain N6 N8 A1 T4 1 N2 N6 N5 N9 C2 N3 S2 N11 N1 B2 N10 size of the sediment in the bedforms (Fig. 14A N2 N9 U1 N7 N3 N5 N4 N1 M1 N10 and B). The comparison of the grainflow condi- S1 tions is based on the dimensionless celerity C* and on the ratio of grain size over delta or dune height (Fig. 14C and D). The dimensionless shear stresses and grain Predicted / measured sorting slope SS* slope sorting measured / Predicted 0·1 sizes of the experiments presented here are 0·001 0·01 0·1 Dimensionless celerity C * usually in the range of the field and flume data from the literature. The celerities of the experi- C 10 ments are somewhat lower than the range of field and flume data. The conditions of the Calamus sand bed river are just below the criterion for E2 T5 suspension (Fig. 14), which means that an in- E1 S2 T3 crease of shear stress will not lead to an increase B1 BK7 N4 T6 C1 T1 of dune celerity but to the suspension of sediment C2 1 T4 U1 S2 B2 that previously moved with the dunes as bedload. N1 N4 N5 M1 Therefore, in the sand bed rivers the dimension- N10 S1 less celerity cannot become much larger than

N1,5,10 (once) observed in the Calamus river. It is expected that N2,3,6,7,8,9,11 (twice) A1 sandy gravel-bed rivers have the largest potential for vertical sorting, as the standard deviation of Predicted / measured sorting slope SS* slope sorting measured / Predicted 0·1 the sediment is the largest while conditions are 012not too near suspension of the finer fractions. In Standard deviation of sediment over top s a,T general, the conditions in rivers are rather well Fig. 13. Sorting model performance. (A) Predicted represented by the sorting data presented herein. vertical sorting based on C* (Eq. 11a); compare to Fig. 6. Finally, the sorting in the lee of deltas is similar The N experiments are grouped for comparable celeri- to the sorting in the lee of dunes except for the ties (experiment numbers given in the legend). (B) Ratio toeset. The main difference between foreset sort- of predicted and measured SS* (Eq. 11a) including data from literature (see text), plotted as a function of C*. ing in deltas and dunes is that the relative For perfect prediction the ratio should be 1. (C) Same as standard deviation r* of the dune shows a much (B), now plotted as a function of ra,T. stronger increase with depth than in the N experiments (Fig. 15, compare to Figs 6 and 7). This may be because the sediment sorting in the

Ó 2005 International Association of Sedimentologists, Sedimentology, 52, 291–311 308 M. G. Kleinhans

Table 3. Description of the datasets used in the comparison between the experiments presented here, experiments in literature and three field datasets ranging between sand-bed and gravel-bed rivers.

Grain Flow Flow Bedform size depth velocity Bedform celerity )1 * )1 Author Code* Condition (D50, mm) (m) (m sec ) s (–) height (m) (cm min ) Gabel (1993) C Calamus river 0Æ25–0Æ41 0Æ34–0Æ61 0Æ47–0Æ77 0Æ46–0Æ88 0Æ097–0Æ195 1Æ2–2Æ4 Dinehart (1992) NT North Toutle r. 22Æ2–36 1Æ4–2Æ24 2Æ14–2Æ74 0Æ1–0Æ24 0Æ143–0Æ41 67Æ2–255 Kleinhans (2002) R Rhine river 1Æ58–1Æ58 8Æ6–10Æ69 1Æ42–1Æ86 0Æ33–0Æ74 0Æ119–0Æ53 4Æ2–14Æ4 Blom & Kleinhans BK Dunes in flume 0Æ59–0Æ73 0Æ19–0Æ35 0Æ59–0Æ79 0Æ14–0Æ37 0Æ011–0Æ057 5Æ4–7Æ8 (1999) Blom et al. (2000) B Dunes in flume 1Æ34–1Æ34 0Æ16–0Æ32 0Æ63–0Æ82 0Æ13–0Æ21 0Æ024–0Æ062 6–18 Termes (1986) T Bar in flume 0Æ64–0Æ96 0Æ17–0Æ21 0Æ69–1Æ20Æ15–0Æ43 0Æ08–0Æ11Æ2–9Æ6 Kleinhans (2002) K Dunes in flume 1Æ12–1Æ56 0Æ13–0Æ21 0Æ78–1Æ08 0Æ24–0Æ25 0Æ018–0Æ035 55Æ8–81

*The code of the dataset as used in Fig. 12. Average flow parameters, for Termes the flow parameters above the bar.

A 10 B 10

Suspension criterion Suspension criterion

1 CCC 1 CCCC R CCC C R CC R * * C R CC T t t R BK R T T K K NT BK T B NT T NT NTNT BKT B NTNT 0·1 NT 0·1

Shields criterion Shields criterion

0·01 0·01 0·1 1 10 100 0·1 1 10 100

D50 (mm) Dg (mm) CD 0·1 0·1 K K NT NT NT NTNTNT NTNT NTNTNTNT NT NT NTNTNT

NTNT NTNT NTNT B NTNT T R T BK TTBK BK 0·01 R BK 0·01 RR R B R CCC R R C CCC CCCC C CRC CCC CCCC C CCC T C C Dimensionless celerity Dimensionless celerity Dimensionless T

0·001 0·001 0·001 0·01 0·1 1 0·001 0·01 0·1 1 D D D50 / Dg / Fig. 14. Comparison between field and experimental datasets from literature with the experiments presented here. (A, B) Sediment mobility. The Shields criterion for incipient sediment motion and the Bagnold suspension criterion are indicated. (C, D) Dune parameters. (A, C) Datasets summarized in Table 3. (B, D) Experiments presented in this paper. grainflows at the lee side of the dunes is affected ments with dunes of Blom & Kleinhans (1999). by the stronger counterflow between dunes. Yet This is explained by the difference in flow the vertical sorting is reasonably well predicted structure over deltas and dunes. Over deltas, with Eq. 11a within its limitation of linearity, there is uniform flow over the top of the delta and indicating that additional parameters such as then a sudden and large increase of the water dimensionless shear stress are not necessary. In depth in which the flow velocity reduces to addition, no toesets were found in the experi- almost zero. Over dunes the flow expansion is

Ó 2005 International Association of Sedimentologists, Sedimentology, 52, 291–311 Grain flow size sorting 309

1 Termes (1986) D = 0·10m 0·8 s g = 3·0

0·6

0·4 Relative height z* 0·2

0 –1·5 –1 –0·5 0 0·5 1 1·5 2 Relative grain size y * 1 Blom & Kleinhans (1999) Blom & Kleinhans (1999) 0·8 D = 0·06m D = 0·06m s s g = 3·1 g = 3·1 0·6

0·4

0·2

Relative height z* 0

–0·2

–0·4 –1·5 –1 –0·5 0 0·5 1 1·5 0·6 0·8 1 1·2 1·4 1·6 1·8 Relative grain size y * Relative standard deviation s * Fig. 15. Examples of vertical sorting in experiments reported in literature, plotted in the same way as Figs 6 and 7 for comparison. The celerity is within a factor of 2 that of the N1–N9 experiments, and the height is comparable with that of the N7–N9 experiments. The geometric standard deviation of the sediment is slightly larger than in the N experiments. The Termes data is from a bar in a flume, and the Blom & Kleinhans (1999) data from dunes in a flume. The dashed line is the prediction by Eq. 11a. much less extreme because the water depth over on the lee side of the delta. (i) Sediment arriving dunes is much larger than over deltas. As a at the top of the delta is deposited from grain fall consequence, the turbulent flow structure at the on the upper lee slope. This mass of sediment lee side of the deltas and dunes differ. McLean & moves downslope in grainflows. (ii) In small Smith (1986) found that the counterflow at the lee grainflows, the gravel is worked to the top of the side of backward steps is much weaker and flow by kinetic sieving. (iii) Large grainflows have extends further downstream than that at the lee their shearing plane at the transition from the side of dunes, while the turbulence intensity at gravel to the underlying sand, and drags the the lee side of periodic dunes was found to be gravel downslope which leads to a fining upward much stronger than for single backward steps. trend. In addition, the finest sediment deposition Increased turbulence intensity decreases grain from grain fall may take place downstream of the fall from suspension. Consequently, the grain fall delta, leading to a toeset that is preserved as a and toeset deposition is less or absent in dunes bottomset below the migrating delta. and counterflow affecting the foreset is stronger Two factors determine the vertical sorting: the between dunes. mixture characteristics and the delta celerity or laminae thickness. The efficiency of the sorting process in the experimental deltas mainly depends CONCLUSIONS on the grain-size distribution: if the sediment mixture is very wide, then the sorting is better The fining upward sorting trend in the deposits of developed than when the mixture is almost uni- deltas was studied by systematic experimentation form. If the mixture is bimodal or fine-tailed, and dimensional analysis with various sediment however, the finest sediment is likely transported mixtures. The results were found to be applicable in suspension and settles out on the lee side and toe to dunes and bars. The fining upward trend is of the delta as a toeset. A larger celerity of the delta caused by a three-stage sediment sorting process front leads to a decreased efficiency of sorting.

Ó 2005 International Association of Sedimentologists, Sedimentology, 52, 291–311 310 M. G. Kleinhans

Dimensionless functions have been developed p Probability or abundance fraction for prediction of vertical sorting from the mix- Q Flow discharge ture standard deviation and the celerity or the q Specific flow discharge laminae thickness of deposited grainflows. qb Volumetric sediment transport rate R Submerged specific gravity of sediment These functions reproduce the sorting in experi- ) (qs q)/q mental bars and dunes reported in the litera- S* Dimensionless grain sorting number ture. The data demonstrate a non-linear SS* Dimensionless sorting slope response of the finest and coarsest size-fractions SF* Dimensionless sorting slope for grain-size and structural differences between mixtures that fractions could not be related to average grain size or t Time mixture standard deviation. This is explained Tf Time interval between onsets of subsequent grainflows by the interaction between the grainflow and Tg Duration of movement of a grainflow grain fall process. Tr* Dimensionless transport rate u Depth-averaged flow velocity ws Settling velocity of sediment ACKNOWLEDGEMENTS W Width of the channel z Height D I thank Gary Parker for inviting me to St Antony Delta or dune height h Angle of repose Falls Laboratory and making these experiments kp Porosity of sediment possible, for teaching me to think in non- q Density of water dimensional numbers and for many stimulating r Standard deviation discussions. I thank Jeffrey Marr for making the s Shear stress 2 experiments possible. Benjamin Erickson provi- w Sediment size ( log-scale), related to /-scale ) ded the digital videos. I thank all those who as / ¼ w helped with flume modifications and the sieve Subscripts a, m Arithmetic samples. The discussions with Suzanne Leclair, g Geometric Chris Paola, James Kakalios, Astrid Blom and i Index of grain-size fraction Janrik van den Berg were appreciated. The s Sediment comments by reviewers Stephen Rice, Marwan T Top: sediment going over the top of the lee Hassan and Paul Carling improved the manu- slope script. The investigations were supported by the * Denotes dimensionless parameter Netherlands Earth and Life sciences Foundation (ALW) with financial aid from the Netherlands Organization for Scientific Research (NWO), and by the St Anthony Falls Laboratory. Please REFERENCES contact the author to obtain the full dataset in digital format. Allen, J.R.L. (1965) Sedimentation to the lee of small under- water sand waves: an experimental study. J. Geol., 73, 95– 116. NOMENCLATURE Allen, J.R.L. (1970) The avalanching of granular solids on dune and similar slopes. J. Geol., 78, 326–351. Allen, J.R.L. (1984) Sedimentary Structures, Their Character A* Dimensionless grainflow number and Physical Basis. Developments in Sedimentology. Else- C Celerity of lee slope vier, Amsterdam, The Netherlands, 30 pp. C* Dimensionless celerity Bagnold, R.A. (1954) Experiments on a gravity-free dispersion df Degrees of freedom of large solid spheres in a Newtonian fluid under shear. Proc. Roy. Soc. London, A225, 49–63. Dg Geometric mean grain size f Frequency of grainflows Blom, A. and Kleinhans, M.G. (1999) Non-uniform Sediment g Gravitational acceleration in Morphological Equilibrium Situations. Data Report Sand G* Dimensionless grain fall number Flume Experiments 97/98. University of Twente, Ri- jkswaterstaat RIZA, WL|Delft Hydraulics, University of h Basin water depth 0 Twente, Civil Engineering and Management, The Nether- h Water depth on top of the delta 1 lands. hg Immobilized grainflow thickness (perpendicular Blom, A., Ribberink, J.S. and Van der Scheer, P. (2000) to lee slope) Sediment transport in flume experiments with a trimodal i Water surface slope sediment mixture. In: Proceedings of the Gravel Bed Rivers L Progradation length of delta Conference 2000 (Eds T. Nolan and C. Thorne), 28 August–3 n Number September, New Zealand, Special public. CD-ROM of the

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