Sedimeniology ( 1987) 34 ,3 19-33 1

The importance of double diffusion to the settling of suspended material

T H E O D O R E G R E E N

Department of Civil and Environmental Engineering, University of’ Wisconsin, Madison. Wisconsin 53706, U.S.A.

ABSTRACT

The vertical fingering motions which can occur when warm, salty water overlies colder, fresher water are important to the vertical flux of salt in the ocean. An analogous and not uncommon situation is that where turbid water overlies clearer and denser water. For the case of an originally sharp horizontal interface between the two water layers (associated, as an example with an interflow due to a warm, muddy river entering a stratified lake), the resulting downward sediment flux is compared with that occurring by way of gravitational settling. Rules are proposed to ascertain when the fingering effect could be important to sedimentation. For example, it seems that fingering could be of consequence in the situation where the sediment has a diameter of 2 ym and the upper layer is about 1°C warmer than the (clear) lower layer, when the upper-layer concentration is 1 ppm. The effect of fingering on the residence time of a gravitationally stable turbid layer is calculated. Such layers may be more short-lived than commonly thought.

INTRODUCTION

The vertical motion of particles suspended in a fluid is fundamental to sedimentation, and of great interest in associated studies such as those of the vertical fluxes of chemicals in large water bodies. Gravitational settling is commonly used to account for observed particle distributions; many theories also invoke some turbulent transfer mechanism (e.g. Ichiye & Carnes, 1977). Here our attention will be directed towards a laminar-flow mechanism : that associated with double diffusion. This mechanism is often important to fluxes of dissolved substances in large water bodies, and may play a corresponding role in the case of suspended sediment.

’Salt fingers’ associated with vertical temperature and salinity gradients in water have become critical to the understanding of small-scale oceanic motions (Huppert & Turner, 1981). Whereas ten years ago it was unclear that such motions even existed outside of the laboratory, one can now argue strongly that the vertical flux of sea salt is dominated by fingering (Schmitt & Evans, 1978).

Houk & Green (1973) pointed out that a similar phenomenon occurs when water is made denser by the presence of suspended material, and conducted

simple laboratory experiments which gave estimates of the resulting vertical sediment flux. These ‘sediment fingers’ may well occur where vertical sediment- concentration gradients are associated with vertical temperature (or salt) gradients. Such cases are not at all uncommon, and may be associated with salt-wedge estuaries, or with the trapping at pycnoclines of the outer, more dilute portions of descending coastal turbidity currents (Drake, 1971). These midwater nepheloid layers play a role in several models of coastal sedimentation patterns (Pierce, 1976), in offshore sewage-disposal systems (Kelling & Stanley, 1976), the dumping of dredge spoils (Proni et al., 19761, and interflows caused by sediment-laden rivers debouching into lakes (Pharo & Carmak, 1979; Fukuoka & Fukushima, 1980). They may play a major role in the dispersal of finely ground ore tailings, when such tailings are disposed of in large lakes. In fact, the original reason for this study was to assess the effects of fingering on mid-water turbidity maxima associated with taconite processing by the Reserve Mining Company, which was then disposing of tailings in Lake Superior.

Below, more data describing sediment fingering

319

320 Theodore Green

shall be presented and the implications of a strict analogy between sediment and salt fingering consid- ered. The effect of such fingers on gravitationally stable layers of high turbidity will then be estimated; such layers may be far more short-lived than commonly thought.

THE DOUBLE-DIFFUSIVE FINGERING MECHANISM

For completeness, the fingering mechanism will be described, although similar descriptions are available elsewhere (e.g. Turner, 1973). Consider two homoge- neous fluid layers, separated by a horizontal interface. The upper layer is warmer, and has more suspended sediment, than the lower layer. Moreover, the upper layer is less dense than the lower layer. For simplicity, this model will be used throughout the present paper. It is suitable for the experiments described below.

Now suppose that some local disturbance causes a small dip in the interface. The fluid within this small, nascent ‘finger’ is thus, by the increase in interface area, brought into closer contact with the colder, less turbid water below. The conduction of heat is very much faster than the Brownian diffusion of sediment (hence the term double diffusion). Thus the fluid within the nascent finger loses its heat while retaining its sediment, becomes heavier than its surroundings, and therefore countinues to move downward. More of the upper-layer fluid follows, and a long, thin finger is formed. The entire process is one of laminar flow, although the associated fluxescan then drive turbulent, convective motions in the fluid above and below the fingering region.

Such initiating disturbances are always present, and can be decomposed in a Fourier sense. Each Fourier component has a certain growth rate; that growing the fastest usually dominates. Note that, by an obvious change in the above argument, upward- moving fingers are also to be expected. Laboratory investigations of salt and sediment fingers, and what field observations of salt fingers have been carried out, usually indicate a very regular pattern of alternately upward- and downward-moving fingers which are square in plan form (Turner, 1973).

A sharp interface is not necessary for fingers to form. In fact, much of the theory describing such motions assumes a base state in which the vertical gradientsof both temperature and salinity are constant in space and time. However, some of the analogies between salt and sediment fingers are simpler with an

originally sharp interface, and this case is considered below. Similar analogies could be drawn in the constant-gradient case. Since sharp interfaces are often readily formed during larger-scale flows (e.g. a starting interflow), this does not seem to be a serious limitation.

The details of the relation of fingering to turbulence are still emerging. Local turbulence and velocity shear have been found to be much less effective in destroying fingers than once thought. This is apparentIy due to the rapidity with which fingers form (e.g. Schmitt, 1979a), and the intermittent nature of turbulence in regions of high stability such as the thermocline. In fact, small-scale fluctuations are found to be most common in situations conducive to double-diffusive fingering (Gargett, 1976), and it is becoming clear that fingering is an important ingredient in intrusive horizontal mixing near oceanic fronts (Williams, 1981; Joyce, 1977; Voorhis et a/., 1976). Williams (1981) has proposed a qualitative model in which double diffusion and turbulence associated with the Kelvin-Helmholtz instabilities due to shear flow combine to enhance mixing between two water masses. Schmitt & Georgi (1982) find vertical mixing to be better correlated with double diffusion than with shear instabilities, in intrusions (which should be analogous to turbid interflows).

Laboratory work has also given some insight into the effect of a velocity field on fingering. Linden (1971) has shown that grid-generated turbulence above and below a fingering layer can act to increase the fingering fluxes, by increasing the vertical gradients of temper- ature and salinity. However, further increase in the turbulence intensity eventually destroys the fingers. A velocity shear across the interface has little effect on fingering fluxes, at least until the local Richardson number decreases to the point where Kelvin-Helm- holtz instabilities occur (Linden, 1974). The details of the interaction between such instabilities and fingering remain to be fully investigated. However, it is reasonable to suppose that fingering could retard the instability, by decreasing the velocity shear and thus increasing the local Richardson number.

The experiments of Houk & Green (1973) served to demonstrate sediment fingering, and are in accord with the above model. The top and bottom halves of an insulated beaker were separated by a thin, horizontal asbestos sheet. The warm, sediment-laden water was initially above this separator, and the colder, sediment-free water below. The separator had a large hole in it, initially covered by another thin asbestos plate. This plate was gently withdrawn to

Double diflusion of suspended material 321

begin the experiment, and the ensuing fingers (made visible by the sediment) photographed. Similar exper- iments had been carried out by Green & Kirk (1971) using dissolved salt instead of suspended sediment. Below I will show that these sediment and salt experiments exhibit a similar behaviour, and use this fact to support an analogy between the two cases, which is then employed to infer characteristics of vertical sediment fluxes associated with fingering.

This paper would be incomplete without a discus- sion of the stimulating observations of Bradley (1969, who argued that the rapid sedimentation encountered in some lakes was due to near-surface particles increasing the water density, leading to rapid vertical convection currents. Several qualitative but convinc- ing laboratory results were presented in support of this conjecture. No estimates of flux were attempted, but it is hard to discount the idea. In the limiting case of an originally sharp, horizontal interface between upper, particle-laden water and lower, particle-free water, this is an example of Rayleigh-Taylor instabil- ity (see, e.g. Daly, 1967).

However, the main reason for the present paper is to point out that the upper water need not be more dense than the lower water for the situation to be dynamically unstable, and to have a substantial vertical flux of suspended material. Moreover, the analogies with molecular double diffusion pointed out below will allow estimates of the flux.

SEDIMENT FINGERING I N A DENSITY-FLOW EXPERIMENT

The experiments mentioned above are probably the simplest possible. They are also somewhat artificial. More recently, we conducted a study of the effects of fingering on a starting surface density current (Schet- tle, 19781, which is more representative of occurrences in nature. A description of the fingers was obtained as a byproduct, and can be used for comparison with the results of Houk & Green (1973).

A small reservoir was built into the upper part of one end of an aquarium, as shown in Fig. 1. This reservoir was filled with warm, sediment-laden water, while the rest of the aquarium was filled to a slightly lower level with colder, sediment-free water. The colder water was heavier than that in the reservoir. When the water came to rest, the vertical plate separating the two water masses was withdrawn,

Fig. 1. The apparatus used to study the effects of sediment fingering on a surface density current. Before the start of an experiment, the light, warm, sediment-laden water is in reservoir A, while the rest of the tank C contains heavier, colder, clear water. To start the experiment, the gate B is pulled up, and the suspension in A spreads to the right, over the water in C. The entire tank is 122 cm long, 38 cm wide, and 38 cm high. Reservoir A is 37 cm long and 12 cm high.

resulting in a horizontal density current as the light water spread over the heavy water.

The propagation speed of this density current depended heavily on the original difference between the two water levels. However, in almost every case, fingering' first began in the rearmost third of the current, and usually before the front had reached the other end of the aquarium. The fingers then quickly spread over the entire current, while at the same time the velocity of the front diminished. Representative finger descent rates were determined from photo- graphs, and found to be quite constant over time, at least to a finger length of about 10cm. A typical sequence of events is shown in Fig. 2. When seen from above, the fingers appeared to be sheets aligned with the flow, rather than square. However, this was due to the bending of individual fingers in response to the velocity gradient near the bottom of the density current. The fingers would first bend with this velocity gradient, then become slightly larger in cross-section, but then descend vertically. Any further bending was associated with return flows due to the density current.

The observed behaviour of sediment fingers strongly implies that they are a continuum phenomenon (and not due to flocculation and subsequent settling, as seen by Sakamoto, 1972). On the other hand, although sediment-finger descent rates vary with concentration in the same way as those of salt fingers (see below), sediment fingers are unlike salt fingers in some respects. The principal visible difference is the rapid formation of pronounced bulbous tips on the fingers, and their ensuing transformation into very distinct vortex rings. These were also pointed out by Bradley in the corresponding Rayleigh-Taylor case. These rings then grow, become unstable, and begin to divide into smaller rings as they fall, in a fashion similar to

322 Theodore Green

Fig. 2. A typical experiment showing the effect of double-diffusive fingering on a surface density current. The light-tone water originally in reservoir A (see Fig. 1 ) had temperature40.1nC, and a sediment concentration of 8.4 x lo-'. The receiving, clear, fresh water in the main tank C had temperature 16.0C. The sediment was taconite, with a specific gravity of 2.97, and with diameters all less than 10 p. Thus, the density of the water originally in A was 0.9928 g ~ m - ~ , and that of the water in C was 0.9990 g cm 3 .

Each picture is 62 cm wide. The pictures were taken 1 minute apart. The top picture shows a typical density current moving to the right. However, sediment fingers form soon thereafter, and the associated downward sediment flux greatly decreases the horizontal spread of the turbid water.

that of the vortex rings induced by drops falling througha slightlylighter fluid (Hsu, 1957; Lugt, 1983). The rings also lose contact with the parent finger at an early stage. Unfortunately, the rings slow down as they develop, so that their breakup is usually obscured by other rings which catch up to them. An example of the development of an individual finger is shown in Fig. 3. Note that this is an 'early' finger; its behaviour may differ from those which form later, and are influenced by other fingers (and vortex rings) around them. Also, only side views of the fingers are available, which limits the ability to describe them.

The development of vortex rings was clearly associated with sediment concentration in the upper layer. For example, the rings developed quickly and were very pronounced for taconite tailings (d< 10 p) when the upper-layer sediment concentration was C=8.1 x but developed rather slowly when C=5.1 x Here and later, the concentration Cis defined as mass of sediment per mass of the suspension. Unless otherwise stated, both masses are measured in the same units. A similar difference occurred with kaolinite (d<2 p). A more complete discussion of the vortex rings is given by Green & Schettle (1986).

Double diflusion of suspended material 323

Fig. 3. The development and evolution of one sediment finger into a vortex ring, which then becomes unstable. The experiment is the same as shown in Fig. 2. Now the nine pictures are 15 s apart, and each picture is about 13 cm wide. Note that there is no indication of sediment settling out of the vortex ring.

A COMPARISON OF SALT- AND SEDIMENT-FINGER DESCENT RATES

It seems reasonably clear that there is at least a partial analogy between salt fingers and sediment fingers. Nonetheless, it would be comforting to see some comparable quantitative results in similar situations. This may be especially true in view of some apparent misconceptions of the double-diffusive mechanism among sedimentologists (see, e.g. Pierce, 1976).

A direct comparison of vertical fluxes would be most satisfying. However, such results are not yet available for sediment fingers. We can, however, consider a closely related and more easily measured quantity (which is also associated with the model considered in this paper): the descent rate V of individual fingers from an initially quiescent upper fluid into a quiescent lower fluid. Such descent rates have not received as much attention as fluxes, for obvious reasons. However, the above-mentioned results of Green & Kirk (1971), Houk & Green (1973), and Schettle (1978) are available. Also, Piascek & Toomre (1980) have computed descent rates of salt fingers associated with an originally sharp interface, and Linden (1973) reports a few measurements of such descent rates.

It is not clear how to scale the measured descent rates. Houk & Green, and Green & Kirk have treated the finger as a solid cylinder. The gravitational driving force was calculated by assuming only salt or sediment- associated buoyancy forces to be important, and assuming the salt or sediment difference between upward and downward moving fingers to be that between the two layers at the start of the experiment. This driving force was balanced against viscosity, where fluid velocities were taken proportional to finger descent rates. This argument is somewhat crude, but seems reasonably successful. However, it does rely on an independent estimate of the finger width.

Piascek & Toomre (1980) related V to R=cc AT/B AS, which is in accord with scalings used by several other authors for flux measurements. Here, AT is the temperature difference between the two layers, and A S the salinity difference. Both are positive. The quantitycc= - I/p?p/dTandB= l/pFp/aS, wherepis density. However, the relation is not dimensionally complete. The main difference in the two scalings seems to lie in whether the density difference associ- ated with the original temperature difference between the two layers has a major impact on the ensuing fingers, or serves mainly to allow the initial, statically stable configuration.

324 Theodore Green

The measured finger descent rates are plotted against both PC (or AS) and R in Figs 4 and 5. The calculated values of Piascek & Toomre are also shown. The results suggest that finger descent rates scale better with PC or A S than with R . Also, and more significantly for what follows, the variation of V with PC is much the same as that of V with p AS. For both salt and sediment, V is proportional to P A S or PC. Sediment-finger velocities are roughly twice salt-finger velocities for similar PC and p A S values. This could well be due to the much lower diffusivity of suspended sediment, and to the increased downward velocity induced by the vorticity associated with the rings. Also, the larger sediment-finger bulbous tips (and the vortex rings) suggest that the fluid velocity/finger velocity ratio is larger in sediment fingers than in salt fingers. Thus, it appears that the vertical flux of sediment associated with fingering is at least as large as that of salt, for comparable density differences between the two layers. Despite the experimental scatter, the relation between V and pC is, for the purposes of this paper, reasonably well approximated empirically by

V = 3 x 102pCcmsec- ' . (1)

Fig. 4. The finger descent rate Vversus the fractional density change due to suspended sediment, PC. For the salt-finger data, an equivalent value of P A S was used on the abscissa. Filled-in circles represent sediment fingers, open circles salt fingers. The open triangles represent the calculations of Piascek & Toomre (1980). Vertical bars denote the spread of Vover one experiment. Where such bars are not shown, the data point refers to an average over the experiment. The slanted line is one for which VccPC.

8

I J: A

I I

l o 2 R Fig. 5. The finger descent rate Yversus the ratio of fractional density changes due to temperature and sediment (or salt): R = a AT/PC. The symbols are the same as in Fig. 4.

The finger descent rate seems independent of the method of creating the two-layer system: the results of Schettle are in accord with those of Houk and Green. However, Linden (1973) found V - cm sec-', which is much lower than the other values. This may well be due to a higher initial level of turbulence in the fluid, associated with the grid- stirring technique used to create a sharp interface. Moreover, V is independent of P A S in Linden's experiments. This behaviour is clearly at odds with the quiescent-fluid results shown here, and suggests that further examination'of the effects of various starting conditions is in order.

THE IMPORTANCE OF SEDIMENT FINGERING

Encouraged by the similar behaviour of salt and sediment fingers seen above, I will apply to sediment fingers some of the mainly empirical formulas found in recent experimental studies of salt fingers. Although this may seem somewhat premature, sediment finger- ing is a more complex phenomenon (as is discussed below), and it may be some time before the more subtle effects of shape, size, and density of the suspended particles are elucidated. The results will be presented in a semi-quantitative, order-of-magnitude fashion, which should atone for some fallacies associ- ated with a direct analogy between the two situations.

Double di@sion of suspended material 325

The formulas are easily adapted: sediment concen- tration is substituted for salt concentration, and a Brownian diffusivity for the molecular diffusivity of salt. The Brownian diffusion of a dilute suspension can be represented by a gradient law with diffusivity ~,=k~T,(3np,vd) - ’ . Here, ko is the Boltzmann con- stant, and T, is absolute temperature (“K). The suspended particles are assumed to be spheres of uniform diameter d. The other symbols are defined below.

For most particle sizes of interest, IC, is much less than the diffusivity of salt, so that double diffusive processes may be more efficient for suspended ma- terial. On the other hand, we now must also consider whether or not the suspension can be considered a continuum, and whether or not the particles will follow the rather sharp curvatures found near the finger tips. Also, the effects of coagulation should be considered. None of these matters is straightforward, and most consideration here will be devoted to determining where such complications are not import- ant. I will first ask when fingers can occur, and then when the resulting vertical flux is greater than that due to settling. Then some ancillary questions associ- ated with the differences between suspensions and solutions will be considered.

For simplicity, and because of the experimental support seen above, I will consider only the case of an originally sharp horizontal interface separating upper warm, sediment-laden water from lower, colder, and more dense but sediment-free water. The changes needed to describe the case where salt replaces heat as the stabilizing substance are straightforward. The resulting expressions are easily generalized to the case where the lower water also contains sediment, with a different concentration, density and diameter. Results based on assuming a uniform-gradient initial state could also be obtained. Finally, for the dilute suspen- sion considered, the heat-transfer properties of the suspension are assumed to be not significantly different from those of pure water.

The quantities characterizing the initial state are ps = particle density

pw = water density p = density of the suspension

N=number of particles in a unit volume C= particle concentration (the mass of particles

in a unit mass of the suspension) AT= upper-layer temperature minus lower-layer

temperature

6P= Ps - P w

I C ~ = diffusivity of heat in water v = kinematic viscosity of water

Note that AT and 6p are always positive below. Also, the relation between p and C for small concentrations is now

P = P w (1 + BC),

where

The relation between C and the concentration when defined as particle mass in a unit volume of the suspension (0 is

The occurrence of fingering

Huppert & Manins (1973) show experimentally that double-diffusive fingers will grow at an originally sharp interface when

This criterion is also supported by an ad hoc argument based on the instantaneous instability of the time- dependent temperature and salinity profiles near the interface, calculated using the two-gradient time- independent stability theory of Baines & Gill (1969). An examination of this argument for the present situation suggests that it is not seriously affected by the slow settling of particles across the original interface. This is so because I C ~ > > I C ~ , so that the slowly descending (and diffusing) interface is moving through a uniform temperature gradient equal to that at the original position of the interface at the time instability occurs. Thus, it seems that there is a good chance that equation (2) also applies to suspensions. When more fundamental quantities are inserted, this criterion becomes

The associated sediment flux

The most convenient relation for the vertical sediment flux F, is that given by Schmitt (1979b):

BFs = ~ A ~ J I C T ) ” ~ (Bc>4’3. (4)

326 Theodore Green

Here, Fs is the mass of sediment crossing a unit horizontal area in a unit time, and A is a dimensionless constant - 1/20 (and could be much higher for R - 1). Other expressions for Fs have been proposed (e.g. Linden, 1973). However, that of Schmitt covers the widest range of R, and so is most likely to be accurate for the wide ranges of C and ATto be considered later. The flux due to settling is ps = pCu,, where the particle terminal velocity v, = gd26p( 18p,v)- ’ . Then the re- quirement that the fingering flux is much more important than the settling flux is

The finger velocity Vshould also be much greater than the particle settling velocity:

v >> t‘,

Continuum requirements

Now consider the characteristics needed to ensure that the suspension acts as a continuum, and that phenomena peculiar to suspensions do not dominate. First, the distance between particles should be much less than the smallest space scale of the motion. The obvious space scale is the finger width, D . Here and below, ‘much less than’ is interpreted as ‘<l/lO’, which seems fairly conservative. Then a reasonable criterion is ND3 > lo3. Since for dilute concentrations, pwC=psNnd3/6, this criterion can be written

(7)

Most estimates of D are couched in terms of the local vertical temperature gradient, and are difficult to apply to the model considered here. Although an extension of the argument of Huppert and Manins may be appropriate, it is much simpler (and in keeping with the spirit of this paper) to note that there is no difference between laboratory and field conditions for a perfectly sharp interface, and that all laboratory measurements show D to be on the order of 1 or 2 mm. Field observations of salt fingers indicate that D is sometimes an order of magnitude larger. This increase would loosen the restriction of Cgiven in equation (7).

Second, the particles must follow the flow in which they are embedded. Following Friedlander (1977), this will be so if the characteristic particle time psdz( I8 p,v) ~ ’ is much less than the smallest important time scale of the fluid motion. The experimental evidence points to this time scale being on the order

of D/w, where w is a typical vertical velocity. The estimate of w analogous to that of Stern & Turner (1969) is

vw - - g s c D2

Then the flow-following criterion becomes

Other requirements

Finally, it is necessary to state explicitly some restrictions inherent in the entire model. First, the warm suspension must be less dense than the under- lying water: R = a AT/(sC) > 1, or

(9) 6P C - < M A T . Ps

Second, the requirement that the fluid be Newtonian leads to

c<< 1. (10)

Third, it seems reasonable to require that the settling (which is still occurring) is laminar in nature. That is, the Reynolds number Re = u,d/v<< 1, or

Coagulation

The effect of coagulation is more difficult to ascertain. Again, I will focus on situations where such effects are not important. Also, even the simple model considered in this paper cannot be treated in a completely satisfactory manner, so that it seems premature to consider more realistic situations at this point. The main coagulation mechanisms are Brownian motion, shear flow, and differential settling. Because the particles are all the same size in the model, the last mechanism can be dismissed here. Of course, one could avoid coagulation effects altogether by postulat- ing a stable suspension. However, this seems over- restrictive, as will be seen below.

First consider Brownian motion. The most appro- priate question seems to be whether coagulation changes the size distribution of the initially uniform particles before fingering starts. To answer this, the time scale T for the onset of fingering at an initially sharp interface must be known. No one seems to have calculated this quantity, for the general case. In the

Double difusion of suspended material 327

experiments of Huppert & Manins (1973), fingering was observed to occur within minutes of the formation of the interface. The same was true in the work of Houk & Green (1973). Piascek & Toomre (1980) found a time scale of less than one minute in their numerical simulations. However, no systematic rela- tion of 7 to other physical quantities was given by any of these authors. In particular, it is unclear how T changes as PC or b A S becomes very small. The smallest PC of Houk & Green was about lo-'; the smallest A S of Huppert & Manins about Naturally occurring values of PC are closer to save near the mouths of some rivers. On the other hand, the fact that the T values observed by Houk & Green and by Huppert & Manins are not too different does suggest that T may be fairly insensitive to PC.

Foster (1965) analyzed the stability of a fluid layer of uniform temperature, the top of which is subjected to a sudden temperature decrease 6T at t = O . For a deep layer, the time scale for the onset of the ensuing convective motions was found to be

Foster's calculations could probably be extended with some effort to a doubly-diffusive interface. However, it is far simpler (and again in keeping with the spirit of this paper) to try to adapt the above result, bearing in mind that salt is the quantity causing instability, and that (by symmetry) AS12 is the initial salinity drop at a sharp interface. Also, R will be held constant when drawing the analogy, so that changes in b A S are always compensated a similar amount by those in c1 AT. Then it is not unreasonable to expect that

where Iz, is the diffusivity of salt, and where we expect A > 20, due to the presence of the opposing tempera- ture gradient. The above-mentioned experiments then suggest that A - lo3.

For the sediment fingers in our model, the corre- sponding estimate is

Note that this estimate can only be a.ccurate (and then, in an order-of-magnitude sense) for R > 1, since the experimental values giving A are in this range.

However, this case should represent several natural phenomena, such as interflows in lakes.

Now consider the time scale associated with Brownian coagulation. Following Smoluchowski (see e.g. Lerman, 1979), the time for an initially uniform- size suspension to change significantly is

Q = p,V(Nk,Td-', (15)

wherefis the fraction of collisions that result in larger particles.

Coagulation should be unimportant to the onset of fingering when z<<f. Whenfis given the typical large value 0.1 (O'Melia, 1985), this criterion gives

The inequality (16) hold strongly in almost any situation of interest. Thus it seems that for the model considered, coagulation does not affect the onset of fingering.

Because of the uncertainty of the estimate of 7 for a sharp interface (and because coagulation is so import- ant in many sedimentation models), it is also worth considering the case where Tand C vary linearly with depth. Here, firmer estimates of 7 are available. For example, when adT/dz - PdCldz) (which seems appro- priate for an interflow), Schmitt & Evans (1978) give

where z is the vertical coordinate. Then the criterion T K T becomes, when p,/p,=2.65, f = O . l , and the ambient temperature is 20°C,

d T dz - - 2 3 x lo-' ' C2/db,

where Cis in mg kg-', d i n k , and dTjdz in "C cm-'. Although this criterion is usually satisfied, one can imagine temperature gradients where it will not be, for very small d and large C. Thus, the question of the relative importance of fingering and coagulation is not settled. On the other hand, a small amount of coagulation is unlikely to inhibit fingering in any direct way.

The effects of coagulation on fully developed, descending fingers are now discussed. The appropriate question in this situation is how long will these fingers become before the coagulation changes the character of the suspension significantly. First consider again the effects of Brownian motion. The time for a finger to descend a distance h is 7 '=h /V . Then, using

328 Theodore Green

equation (l), the requirement z'<< .2. gives, for p,/ pw = 2.65 and f = 0.1,

h << 103d3 ( h in cm, d in p). (18)

Second, consider the coagulation due to shear. The change of N with time in a uniform shear flow is (Lerman, 1979)

N(t) = No e-",

where k=4fpwGC/nps, No is N at t = O , and G is the shear rate.

Then the time for a significant change in the suspension is

Again following Stern & Turner (1969),

G - - - - . w SBCD D v

Using equation (l), the requirement that z'<<z* then gives, with f = 0.1, D = 2 mm, and p,/p, = 2.65,

h << 0.3 C- ' (h in cm). (21)

Neither condition (18) nor condition (21) is very restrictive for reasonable values of C and d. It is far more likely that the observed bulbous tips and vortex rings will alter the fingers, or that an internal-wave instability (Stern, 1969) will come into play. Thus, at least for the model considered in this paper, the effects of coagulation do not seem to be important.

The situation where fingering may be important

In order for double-diffusive fingering to be important to the vertical flux of suspended sediment, equations (3), and ( 5 ) through (1 1) must be satisfied. Here, only parameter ranges common in large water bodies will be considered. Thus, AT will be at most 10°C (and usually much smaller), d will be between 0.1 and 50 IJ- (above which settling is not laminar), and sediment concentrations will be less than lo3 mg kg- ' (i.e. C <

although some rivers have a higher sediment load (see e.g. Biggs, 1978). The sediment density ps is taken to be 2.65 gm cm-3, and the various properties of water are evaluated at both 10°C and 20°C.

The resulting criteria are shown in Fig. 6. The condition (3) for the formation of figures does not appear: it is strongly satisfied for any reasonable AT. However, the condition for gravitational stability must be considered; its effect is shown for AT=O.l"C. In this case, a concentration greater than 10 or

n

0.1 1 10 d (p)

Fig. 6. The particle diameter/sediment concentration (i.e. d/C) region where the sediment-finger/salt-finger analogy suggests that fingering should be important to the vertical flux of suspended material. Lines (a) and (b) shoy where the ratio of the fingering flux F to the settling flux F is ten and one, respectively. The left side of the shaded stripe in each case denotes calculations for an average temperature T of 20°C, and the right side calculations for 10°C. The region where the sediment-water mixture can be considered a continuum lies to the left of line (c). The finger velocity equals the settling velocity along line (d). To the left of this line, V > v,. Again, the left and right bounds of the stripe are for temperatures of 20°C and 10°C.

The upper, horizontal dashed line bounds the gravitation- ally stable situations from above, for T=20"C and AT=O.l"C. The lower dashed line does the same, but for T= 10°C. Similar bounds for AT= 1°C are ten times higher. Thus, for example, point A denotes a gravitationally stable case (i.e. the upper fluid layer is lighter than the lower layer), in which the fingering flux is about five times greater than the settling flux.

The dots show some cases where fingers have been observed experimentally, using kaolinite suspensions. The stars show cases where fingers have been observed using taconite suspensions (Houk & Green, 1973). In all cases, the situation was gravitationally stable, and the particle diameter shown is the largest of a continuous distribution.

20 mg kg-' leads to gravitational (i.e. Rayleigh- Taylor) instability, and the flux laws discussed above no longer apply. The corresponding concentration for AT= 1"Cisabout 100mg kg- I . Alltheothergoverning conditions are independent of AT. The condition (8) that the particles follow the flow does not appear in Fig. 6, being easily satisfied everywhere in the (C, d) range of interest.

Double diffusion of suspended material 329

Conditions (5) , (6), and (7) are most important, and all lead to similar restrictions on the domain where fingering is important to the vertical flux. Of course, conditions (5 ) and (6) are related, and treating them separately testifies to our lack of understanding of double-diffusive processes in the case of a suspension. To some extent, V and v, (and F, and @J should be additive, but the manner in which settling and fingering interact is not at all clear.

For a sediment concentration of 100 mg kg-’, it seems that the particle diameters should be less than about 5 p for fingering to be as important as settling to the vertical flux, and less than about 2 p for fingering to dominate the vertical flux. For a concentration of 1 mg kg-’, these diameters decrease to about 2 p and 1 p. While these limitations on particle size are fairly severe, they do not argue for dismissing the fingering phenomenon in natural water bodies from further consideration. This seems especially true near the mouths of rivers. Fig. 6 suggests that fingering can play a role in vertical flux even for small sediment concentrations and temperature differences.

Some of the experimental points of Houk & Green are also shown in Fig. 6 . The other experiments had Cvalues larger than those shown. All points fall in the region where fingering should be important, which is comforting. It is especially interesting that fingers were observed to occur in the one situation lying in the band where F, = Fs.

RESIDENCE TIMES

As an example of the effects of fingering, the above relations are now applied to estimate the residence time of a gravitationally stable layer of turbid water. This turbid layer has concentration C, thickness H , and lies above clear water. It could be associated with an interflow, due to the horizontal motion of an originally downward moving turbidity current, and caused by the presence of a density interface (i.e. a thermocline) in the receiving water. An example in the case of a turbid river flowing into a lake is shown in Pharo & Carmak (1979; fig. 11). Here, C-10 mg kg-’.

The residence time is (Lerman, 1979) T = p H C / F , where F is the vertical sediment flux. When settling dominates, F = f i S and T= Ts = H/v,. However, when fingering dominates, and Schmitt’s relation (4) is used to estimate F, the residence time is given by

Now the parameter CGp/p, is a key to the process, whereas the settling estimate Ts is independent of C. Of course, this estimate is only expected to hold in the appropriate region of Fig. 6. However, then TF can be much smaller than T,.

The lateral extent of an interflow having an original horizontal velocity U is then on the order UTF, which can be much less than UT,. It may also be less than that estimated by models incorporating turbulent vertical transfers. Estimates of enhanced longitudinal disperion could also change when fingering is consid- ered (see e.g. Fischer et al., 1979). However, detailed consideration of such effects seems premature at this point.

CONCLUSION

The vertical flux of sediment from the turbid, gravitationally stable layers often observed in large water bodies can be dominated by double-diffusive effects, and may be significantly greater than com- monly thought. This is especially true for relatively high sediment concentrations and small particles, when the density of the turbid layer is only slightly less than that of the underlying water, and when coagulation does not dominate. Thus, for example, the residence time of sediment in turbid layers associated with the discharge of muddy rivers into stratified lakes may be much shorter than usually supposed, and the lateral coherence of such layers correspondingly smaller. These conclusions may also have significant consequences for typical models of coastal and estuarine sedimentation, such as that of Farmer (1971).

The conceptual model of double diffusion (i.e. of the conditions leading to sediment fingering) is quite idealized, and more work is needed to fully assess the implications of double-diffusive effects, especially for the low sediment concentrations usually found in nature. In particular, the effects of coagulation, the interactions between Stokes settling and fingering, and the overall effects of the vortex rings encountered in all our sediment-finger experiments need further study. Additional consideration of coagulation effects would seem to be especially important, in view of its importance to settling in estuaries (see e.g. Kranck, 1973 and 1980), and because the stabilizing effect of salt can replace or augment that of temperature in the double-diffusion mechanism in many coastal situa- tions, such as in estuaries where a ‘salt wedge’ exists (see e.g. Farmer & Morgan, 1953).

330 Theodore Green

ACKNOWLEDGMENT

I a m pleased to acknowledge enlightening conversa- tions with Dr. Kate Kranck and Professor J . R. Moore, who d id their best to correct my misconcep- tions about estuarine particles and nepheloid layers.

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(Manuscript received 10 October 1985 ; revision received 23 April 1986)