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CHAPTER 28: TRACER STONES MOVING AS BEDLOAD IN GRAVEL-BED STREAMS

CHAPTER 28: TRACER STONES MOVING AS BEDLOAD IN GRAVEL-BED STREAMS. This chapter was written by Miguel Wong and Gary Parker It is preliminary: code will be added later. Tracer stones (painted particles) in motion during a flume experiment at St. Anthony Falls Laboratory (SAFL).

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CHAPTER 28: TRACER STONES MOVING AS BEDLOAD IN GRAVEL-BED STREAMS

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  1. CHAPTER 28: TRACER STONES MOVING AS BEDLOAD IN GRAVEL-BED STREAMS This chapter was written by Miguel Wong and Gary Parker It is preliminary: code will be added later. Tracer stones (painted particles) in motion during a flume experimentat St. Anthony Falls Laboratory (SAFL)

  2. BEDLOAD TRANSPORT-DOMINATED STREAMS Mountain streams not only convey water, but also transport large amounts of bed sediment, including sand and gravel, and in some cases cobbles and boulders. The transport of gravel and coarser material is primarily in the form of bedload, with particles sliding, rolling or saltating within a thin layer near the stream bed. Another characteristic of these streams is that bedload transport events are sporadic and are associated with floods. Thus in a perennial stream, significant bedload transport may occur for e.g. only about 5% of the time that water is flowing. When bedload transport occurs, or for that matter when the combined processes of particle entrainment, transport and deposition take place, the morphology of the stream may evolve toward a new channel shape or bed profile. Reliable and accurate estimates of the bedload transport rate are essential, therefore, to the quantification of such morphodynamic evolution. The common approach is to obtain these estimates via empirically derived relations, which are based on characteristic driving parameters (i.e. those of the flowing water) and the corresponding resistance properties of the bed material. Examples of some of these bedload transport relations are presented in Chapter 7.

  3. CHANNEL-AVERAGED DETERMINISTIC APPROACH One good example of a relation still extensively used in basic research and engineering applications is that of Meyer-Peter and Müller (1948). All terms defined below are deterministic and represent channel-averaged values. q*b [1] is the dimensionless volume bedload transport rate per unit width of stream (or Einstein number), t* [1] is the dimensionless bed shear stress (or Shields number), and t*c [1] is the critical Shields number for particle incipient motion. These variables are defined in Chapter 7. (The notation inside the brackets denotes the dimensions of the parameter preceding it.) In a 1D model of channel morphodynamic evolution, restricted to tracking the time variation of the longitudinal profile (bed elevation) of a study reach, one can simply make use of the Exner equation derived in Chapter 4.

  4. BUT THE STORY IS NOT ALWAYS THAT SIMPLE There are two limitations in the use of the channel-averaged deterministic approach. First, it does not contain the mechanics necessary to describe the displacement patterns of individual particles, hence it lacks the option of explicitly linking changes in the composition and surface configuration of the bed deposit with the overall evolution of the channel morphometry of the river (Blom, 2003). Second, bedload transport is intrinsically a stochastic process (Einstein, 1950). One alternative is the use of passive tracer stones (e.g., painted or magnetically tagged particles). The working hypothesis is that their (vertical and streamwise) displacement history may serve as a good indicator of the bedload transport response of a stream to given water discharge and sediment supply conditions (DeVries, 2000). Use of tracer stones in Shafer Creek, WA. Image courtesy P. DeVries and T. Brown.

  5. SEDIMENT CONTINUITY The Exner equation derived in Chapter 4 relates the time evolution of the bed elevation h [L] at a given streamwise location x [L] with the volume bedload transport per unit stream width qb [L2/T]: where lp [1] denotes bed porosity, and t [T] represents time.

  6. SEDIMENT CONTINUITY The Exner equation derived in Chapter 4 relates the time evolution of the bed elevation h [L] at a given streamwise location x [L] with the volume bedload transport per unit stream width qb [L2/T]: where lp [1] denotes bed porosity, and t [T] represents time. A different way to present the continuity equation for the bed sediment is in terms of the rate of exchange of particles between the bedload and the bed deposit. This is the entrainment formulation: where Db(x) [L/T] is the volume rate of deposition from bedload per unit bed area at location x, and Eb(x) [L/T] is the volume rate of entrainment into bedload per unit bed area at location x.

  7. ENTRAINMENT FORMULATIONS The last equation of Slide 6 reduces in the limit as Dx → 0 to: where Db [L/T] and Eb [L/T] represent the spatial averages of the entrainment and deposition rates, respectively. This form of mass continuity for the sediment in the bed deposit is completely equivalent to the form of Slide 5, i.e.

  8. ENTRAINMENT FORMULATIONS The last equation of Slide 6 reduces in the limit as Dx → 0 to: where Db [L/T] and Eb [L/T] represent the spatial averages of the entrainment and deposition rates, respectively. This form of mass continuity for the sediment in the bed deposit is completely equivalent to the form of Slide 5, i.e. In a form analogous to Slide 12 of Chapter 4 for suspended sediment, it can be shown that the entrainment formulation of mass continuity for the sediment in the (moving) bedload layer is: where x [L] is the volume concentration of bedload per unit bed area.

  9. ENTRAINMENT FORMULATIONS The last equation of Slide 6 reduces in the limit as Dx → 0 to: where Db [L/T] and Eb [L/T] represent the spatial averages of the entrainment and deposition rates, respectively. This form of mass continuity for the sediment in the bed deposit is completely equivalent to the form of Slide 5, i.e. In a form analogous to Slide 12 of Chapter 4 for suspended sediment, it can be shown that the entrainment formulation of mass continuity for the sediment in the (moving) bedload layer is: where x [L] is the volume concentration of bedload per unit bed area. The term ∂x/∂t can be neglected for most cases of interest, as seen from dimensional analysis and the observation that bedload particles are typically at rest far longer than in motion.

  10. THE ACTIVE LAYER The conservation equations of mass continuity presented in the previous slides are intended for bed sediment of uniform size in a 1D bedload-dominated stream. Hirano (1971) extended their application to size mixtures by introducing the concept of an active layer of thickness La [L], so that the time evolution of the bed sediment composition could be tracked in response to changes in the sediment supply, overall bed aggradation / degradation or flood hydrographs. The basic equations are presented in Chapter 4, and some applications are given in Chapters 17 and 18. The use of this concept has allowed the successful modeling of various morphodynamic situations. However, the basis for its formulation has two drawbacks (Parker et al., 2000). First, the exchange of particles between the bed deposit and the bedload is limited to a surface layer of well-mixed sediment and finite thickness (Fi and La in the upper sketch, respectively). Second, entrainment of bed sediment is represented by a step function (blue dashed region in lower sketch), with the sediment below the active layer (i.e., the substrate) participating only as the bed degrades.

  11. THE ACTIVE LAYER The conservation equations of mass continuity presented in the previous slides are intended for bed sediment of uniform size in a 1D bedload-dominated stream. Hirano (1971) extended their application to size mixtures by introducing the concept of an active layer of thickness La [L], so that the time evolution of the bed sediment composition could be tracked in response to changes in the sediment supply, overall bed aggradation / degradation or flood hydrographs. The basic equations are presented in Chapter 4, and some applications are given in Chapters 17 and 18. The use of this concept has allowed the successful modeling of various morphodynamic situations. However, the basis for its formulation has two drawbacks (Parker et al., 2000). First, the exchange of particles between the bed deposit and the bedload is limited to a surface layer of well-mixed sediment and finite thickness (Fi and La in the upper sketch, respectively). Second, entrainment of bed sediment is represented by a step function (blue dashed region in lower sketch), with the sediment below the active layer (i.e., the substrate) participating only as the bed degrades. Are these realistic assumptions? Is there any coupling between the bed sediment composition and bedload transport rate that is missed with this formulation?

  12. THE ACTIVE LAYER, VIRTUAL VELOCITY AND TRACERS Even under steady, uniform transport conditions, bedload particles constantly interchange with the bed. A moving particle is eventually deposited on the bed surface or buried, where it may remain for a substantial amount of time. Fluctuations in bed elevation may cause the grain to be exhumed and re-entrained, however. Now let vb [L/T] denote the mean velocity of a particle while it is moving, and vv [L/T] denote its virtual velocity averaged over both periods of motion and periods of rest. For typical gravel-bed streams, vv << vb, implying that a particle spends most of its time at rest (even during equilibrium transport).

  13. THE ACTIVE LAYER, VIRTUAL VELOCITY AND TRACERS Even under steady, uniform transport conditions, bedload particles constantly interchange with the bed. A moving particle is eventually deposited on the bed surface or buried, where it may remain for a substantial amount of time. Fluctuations in bed elevation may cause the grain to be exhumed and re-entrained, however. Now let vb [L/T] denote the mean velocity of a particle while it is moving, and vv [L/T] denote its virtual velocity averaged over both periods of motion and periods of rest. For typical gravel-bed streams, vv << vb, implying that a particle spends most of its time at rest (even during equilibrium transport). Two alternative statements of equilibrium sediment mass conservation can be stated using these velocities. Let  denote the volume of bedload particles per unit bed area, and let the active layer thickness La specifically denote a characteristic thickness within which buried bedload particles reside. Then, Here La and vv can be quantified in the field from measurements of the depth of burial of tracers and distance moved by tracers, both over a flood of known hydrograph.

  14. ONE FIRST STEP TOWARD A MORE GENERAL MODEL … • One ambitious goal is to establish a relation between the statistics of vertical and streamwise displacement of a group of identifiable particles (tracer stones), the channel hydraulics and the bedload transport rate of a stream (see e.g., Hassan and Church, 2000; Ferguson and Hoey, 2002). This could allow the investigation of, for instance, how the vertical structure of the bed deposit (i.e. its stratigraphy) influences the overall morphodynamic evolution of a mountain stream. • The material presented here corresponds to one of the simplest morphodynamic scenarios. The theoretical framework is developed with the aid of results from flume experiments. Simplifications considered in the analysis include: • Straight channel of constant width. • 1D normal flow approximation valid at geomorphic time scales. • Lower-regime plane-bed equilibrium transport conditions. • Bedload transport predominating. • Bed sediment of uniform size and given density, with constant bed porosity. • A particle located at a given elevation in the bed deposit can be entrained into transport only if the instantaneous bed surface is at that elevation.

  15. THE BED ELEVATION FLUCTUATES!!! A first important fact to recognize is that even for the case of lower-regime plane-bed conditions [equilibrium bedload transport and normal (uniform and steady) flow], the bed elevation fluctuates in time t at any streamwise location x (see e.g., Wong and Parker, 2005). Double-click on the image to run the video. Experiments at SAFL: Tracer stones (painted particles) are gradually entrained from ever-deeper locations and replaced with non-painted particles, resulting in an approximately constant mean bed elevation. rte-bookbedload.mpg: to run without relinking, download to same folder as PowerPoint presentations.

  16. HOW TO HANDLE THESE VARIATIONS? The variation in time t of the instantaneous bed elevation z' [L] at a given streamwise location x, can be tracked in terms of the fluctuations around its local mean value h. Thus, a new vertical coordinate system (positive downward) can be defined in terms of the variable y [L], which is boundary-attached to h: Let’s trace now a line at relative level y, parallel to the mean bed elevation h. The fraction of sediment + pores in this line (depicted by the sum of the thick green strips in the sketch above) is represented by PS(y) [1]. Hence, for time scales shorter than those corresponding to overall net bed aggradation or degradation, it can be intuitively argued that (Parker et al., 2000): All water All sediment + pores and that PS(y) is a monotonically increasing function ranging from 0 to 1.

  17. PROBABILISTIC CHARACTERIZATION Finding the shape of PS(y) is not necessarily important at this stage, but the assumption that this function is monotonically increasing from 0 to 1 is key. Thus, PS(y) defines a cumulative distribution function, in this case of the amount of sediment + pores at a relative level y. In a more physical context, PS(y) can be interpreted as the probability that the instantaneous relative bed elevation is less than or equal than y, with its associated probability density function pe(y) [1/L] computed from: which by definition must satisfy:

  18. MEASURING FLUCTUATIONS Simultaneous measurements of bed elevation fluctuations at 6 different streamwise locations in a flume were conducted for 10 different equilibrium bedload transport conditions (Wong and Parker, 2005). A sonar-multiplexing system was used for this purpose. Let em [L] denote the measurement error, fm [L] denote the measuring footprint, and D50 [L] represent the median size of the bed material. The respective values of em/D50 and fm/D50 were 0.020 and 1.00 for the four 0.5 MHz type probes used, and 0.005 and 0.53 for the two 1.0 MHz type probes used. The algorithm developed was successful in discriminating between particles in bedload motion and the actual bed elevation. Experiments at SAFL: foreground – Pulser + computer used for data acquisition and processing; background – cables and metal frames for ultrasonic probes in flume.

  19. MEASURING FLUCTUATIONS Simultaneous measurements of bed elevation fluctuations at 6 different streamwise locations in a flume were conducted for 10 different equilibrium bedload transport conditions (Wong and Parker, 2005). A sonar-multiplexing system was used for this purpose. Let em [L] denote the measurement error, fm [L] denote the measuring footprint, and D50 [L] represent the median size of the bed material. The respective values of em/D50 and fm/D50 were 0.020 and 1.00 for the four 0.5 MHz type probes used, and 0.005 and 0.53 for the two 1.0 MHz type probes used. The algorithm developed was successful in discriminating between particles in bedload motion and the actual bed elevation. Close-up picture of ultrasonic transducer probe for measuring bed elevation fluctuations. Major concerns in setting the probes were to avoid the entrainment of air bubbles below them and to keep their bottom as far as possible from the gravel bed.

  20. TIME SERIES OF BED ELEVATION Time series of bed elevation z', and thus of the bed elevation fluctuations y around the corresponding mean value h, were obtained for the 10 equilibrium test conditions. From the analysis carried out, it was found that for any given equilibrium state, the time series were stationary in their first two statistical moments. Aggregated time series were then formed for each equilibrium state. Time series of bed elevation at various points in a flume. Case (a) is for a relatively low bedload transport rate, and case (b) is for a relatively high bedload transport rate. Note that fluctuations in bed elevation increase from case (a) to case (b).

  21. NORMAL PROBABILITY MODEL Working with the aggregated time series of bed elevation fluctuations y, an empirical cumulative distribution was constructed for each equilibrium state. Based on Chi-square tests, it was found that a normal distribution model gave a good fit of the probability density function of bed elevation fluctuations, pe(y): Empirical vs. theoretical normal cumulative distribution function for a sample equilibrium test. where sy [L] is the standard deviation of bed elevation fluctuations, which was found to correlate with D50 and excess Shields number (t* - 0.055) as follows:

  22. ELEVATION-SPECIFIC ENTRAINMENT AND DEPOSITION The following probabilistic terms can be defined: pEnt(y) [1/L] = probability density function that a particle entrained from the bed deposit into bedload transport comes from a depth y relative to the mean bed elevation h pDep(y) [1/L] = probability density function that a bedload particle is deposited onto the bed deposit at a depth y relative to the mean bed elevation h The following properties must be satisfied: such that = entrainment rate from level y to level y + Dy and, such that = deposition rate at level y to level y + Dy

  23. ENTRAINMENT OF TRACER STONES A total of 80 flume runs with tracer stones were conducted under conditions of plane-bed lower-regime equilibrium bedload transport. They corresponded to 10 different equilibrium cases, 8 tests each, and durations ranging from 1 min to 120 min. The experimental procedure consisted of running the system until equilibrium was reached; seeding tracer stones in 4 spots, 4 layers per spot, about 200 particles per layer, with the color of tracers used as a proxy for initial vertical position; re-running the system for a predetermined duration; and, counting the number of particles displaced per color. Layered placement of tracer stones The main results can be summarized as follows: the longer the duration of competent flow and/or the larger the driving force (excess Shields number), the larger is the fraction of tracer stones displaced, and the deeper is the layer accessed.

  24. THIS EXPERIMENT LASTED 1-min ONLY flow direction “Top” yellow tracers on LHS of channel are quickly displaced (lower spot), while the same does not happen with “top” orange tracers on RHS. Then the situation is reversed, likely because on the RHS there are more tracer stones “exposed” than on the LHS (they are buried or already gone!).

  25. ENTRAINMENT PER LAYER The uniqueness of the experimental runs conducted at SAFL is that they allow a direct measurement of elevation-specific particle entrainment. By looking at the “additional” fraction of tracers displaced when comparing two runs of different duration but both corresponding to the same equilibrium conditions, entrainment rates can be computed. The plots to the left show the percents of tracers moved from each layer (top, second, third and bottom) as a function of experiment duration. Case (a) is for a relatively low bedload transport rate and case (b) is for a relatively high bedload transport rate. Note that particle entrainment per layer increases with bedload transport rate, i.e. from (a) to (b), as well as with experiment duration.

  26. “VANILLA” MASS BALANCE FOR TRACER STONES In the SAFL experiments, tracers of a given color are not replaced with stones of the same color once they are displaced. This is because all tracers that moved out of the system were captured at a sediment trap and were not permitted to re-enter the flume. Thus, in an entrainment formulation of mass balance for tracer stones in a control volume corresponding to their seeding position, the deposition term vanishes. Let Ltr [L] denote the thickness of a layer of tracers. The conservation equation for the fraction of tracers fbts(t) [1] per layer at time t then takes the form: Solving this ODE for Eb results in: Time evolution of the fraction of non-displaced tracers as a function of test duration and excess Shields number (t* - 0.055). All 4 layers are aggregated in the plot.

  27. “VANILLA” MASS BALANCE FOR TRACER STONES In the SAFL experiments, tracers of a given color are not replaced with stones of the same color once they are displaced. This is because all tracers that moved out of the system were captured at a sediment trap and were not permitted to re-enter the flume. Thus, in an entrainment formulation of mass balance for tracer stones in a control volume corresponding to their seeding position, the deposition term vanishes. Let Ltr [L] denote the thickness of a layer of tracers. The conservation equation for the fraction of tracers fbts(t) [1] per layer at time t then takes the form: The values of Eb determined in this way were found to correlate with D50 and excess Shields number (t* - 0.055) as follows: Solving this ODE for Eb results in: where R is the submerged specific gravity of the bed sediment [1], and g is the acceleration of gravity [L/T2] 28 different combinations of fbts(t)-pairs have been used to estimate the value of Eb for each experimental equilibrium state.

  28. ENTRAINMENT AND DEPOSITION FUNCTIONS The probability density function pEnt(y) that a particle entrained into bedload transport is removed from depth y, and the corresponding probability density function pDep(y) that a particle deposited from the bedload is emplaced at depth y were introduced in Slide 22. Here the following general forms for pEnt and pDep are assumed: where pB(y) is an appropriately chosen probability density function, and ybe and ybd represent offset distances from the mean bed (at y = 0) for the erosion and deposition functions, respectively. The experiments reported here allow for quantification of only the offset ybe at equilibrium conditions. It is possible, however, to speculate about the general relation between the offset ybe and the offset ybd under conditions that may or not be at equilibrium. Here it is assumed that

  29. ENTRAINMENT AND DEPOSITION FUNCTIONS contd. The assumptions of the previous slide thus give the following relations: Here y0 is an offset common to both functions, which could be greater than or less than 0. The case y0 > 0 biases both functions downward below the mean bed elevation. The case y1 > 0 biases erosion upward in the bed, and deposition downward in the bed (see sketch to the right). The above forms are assumed to be valid for both equilibrium and disequilibrium cases. At equilibrium, however, erosion and deposition must balance within every layer, i.e. so that y1 0 as equilibrium is approached. This point is illustrated in more detail in subsequent slides.

  30. EXPONENTIAL MODEL The data allow estimation of the probability density function pEnt(y) at equilibrium conditions. Specifically, it was found that pEnt(y) could be fitted to an exponential function of the form: Note that the entrainment function pEnt(y) is continuous in y, thus overcoming the step function approximation of the active layer formulation (Slide 10; see also Chapter 4). Moreover, according to the above equation pEnt(y) depends on the standard deviation sy of bed elevation fluctuations, and hence correlates with excess Shields number (t* - 0.055) (Slide 21). Exponential fitting for a sample equilibrium test. Setting ybd = y0 + y1, the corresponding form for the deposition function pDep(y) is:

  31. THE STORY SO FAR

  32. THE STORY SO FAR Predictors to complete a modified version of the Parker et al. (2000) formulation have now been developed up to the specification of forms for the parameters y0 and y1.

  33. PROBABILISTIC FORMULATION OF MASS CONTINUITY FOR SEDIMENT IN THE BED DEPOSIT As indicated in Slide 16, the control volume (strip with fill) is boundary-attached to the mean bed elevation h, which is free to move up or down in time. The conservation equation for the sediment in the bed deposit within any layer from y to y + Dy can then be expressed as follows: Time rate of change of mass in control volume Flux of mass going into the control volume Flux of mass going out from the control volume – =

  34. PROBABILISTIC FORMULATION OF MASS CONTINUITY FOR SEDIMENT IN THE BED DEPOSIT As indicated in Slide 16, the control volume (strip with fill) is boundary-attached to the mean bed elevation h, which is free to move up or down in time. The conservation equation for the sediment in the bed deposit within any layer from y to y + Dy can then be expressed as follows: Note how the stone (solid circle) is moved out of the control volume as the control volume is advected upward Apparent “convective” transfer as a result of moving from the green to the blue strip.

  35. DOES PS(y) HAVE TO BE STATIONARY? The conservation equation for sediment in the bed deposit presented in the previous slide reduces to: Further reducing with the relation between PS(y) and pe(y) of Slide 17:

  36. DOES PS(y) HAVE TO BE STATIONARY? The conservation equation for sediment in the bed deposit presented in the previous slide reduces to: Further reducing with the relation between PS(y) and pe(y) of Slide 17: 0 ??? The expression above could be simplified more if PS(y) is assumed to be stationary (independent of time), even under disequilibrium conditions. By doing so, however, the term on the LHS of the relation to the right becomes independent of y. The only way that this can be true is if the following condition is satisfied:

  37. DOES PS(y) HAVE TO BE STATIONARY? The conservation equation for sediment in the bed deposit presented in the previous slide reduces to: Further reducing with the relation between PS(y) and pe(y) of Slide 17: 0 ??? This is not only a very restrictive assumption for cases of non-equilibrium transport, but it can be seen from Slides 21 and 30 that even at equilibrium pe(y) differs in form from pEnt(y)! Thus in general PS(y) should not be expected to be stationary.The term PS/t should be expected to vanish only for equilibrium conditions.

  38. THE ENTRAINMENT FORMULATION FOR SEDIMENT CONTINUITY OF SLIDE 9 CAN BE RECOVERED FROM THE PROBABILISTIC FORMULATION Let’s integrate the equation for conservation of bed sediment presented in Slide 35 over the whole range of possible relative levels y :

  39. THE ENTRAINMENT FORMULATION FOR SEDIMENT CONTINUITY OF SLIDE 9 CAN BE RECOVERED FROM THE PROBABILISTIC FORMULATION Let’s integrate the equation for conservation of bed sediment presented in Slide 35 over the whole range of possible relative levels y : 1 1 1 This yields: Applying integration by parts to the term on the LHS of the expression above, and using the relation between PS(y) and pe(y) of Slide 17, it is found that:

  40. THE ENTRAINMENT FORMULATION FOR SEDIMENT CONTINUITY OF SLIDE 9 CAN BE RECOVERED FROM THE PROBABILISTIC FORMULATION Let’s integrate the equation for conservation of bed sediment presented in Slide 35 over the whole range of possible relative levels y : 1 1 1 This yields: Applying integration by parts to the term on the LHS of the expression above, and using the relation between PS(y) and pe(y) of Slide 17, it is found that: 0, assuming “thin” tails for pe(y) 0, because mean of y is 0 by definition

  41. THE ENTRAINMENT FORMULATION FOR SEDIMENT CONTINUITY OF SLIDE 9 CAN BE RECOVERED FROM THE PROBABILISTIC FORMULATION Let’s integrate the equation for conservation of bed sediment presented in Slide 35 over the whole range of possible relative levels y : 1 1 1 This yields: 0 Applying integration by parts to the term on the LHS of the expression above, and using the relation between PS(y) and pe(y) of Slide 17, it is found that: then, 0, assuming “thin” tails for pe(y) 0, because mean of y is 0 by definition

  42. TIME EVOLUTION EQUATION FOR PS(y) Substituting the entrainment formulation of bed sediment conservation, into the relation of Slide 35, and reducing, an equation for the time evolution of PS(y) is obtained: The first term in brackets on the RHS of the equation above indicates that when net deposition occurs at relative level y, the amount of sediment + pores at that level increases, a physically reasonable result. The second term in brackets on the RHS is less intuitive in its interpretation. Recalling the relation between pe(y) and PS(y) in Slide 17, it represents the (vertical) advection of mass due to overall bed aggradation or degradation.

  43. CASE OF EQUILIBRIUM BEDLOAD TRANSPORT The following conditions hold for equilibrium bedload transport: in which case reduces to and reduces to This justifies the statement made at the bottom of Slide 29.

  44. PROBABILISTIC FORMULATION OF MASS CONTINUITY FOR TRACER STONES IN BED DEPOSIT Making use again of a boundary-attached control volume, let’s now derive the sediment continuity equation for the fraction of tracer stones in the bed deposit at vertical position y, fb≡ fb(x,y,t) [1] (depicted by the sum of the green  blue solid squares): where ftr ≡ ftr(x,t) [1] denotes the fraction of tracer stones in bedload transport (depicted by the sum of the red solid squares). Reducing,

  45. REDUCTION WITH THE HELP OF THE TIME EVOLUTION EQUATION FOR PS(y) Expanding the conservation equation for tracer stones in the bed deposit presented in the previous slide, and using the relation between PS(y) and pe(y) of Slide 17:

  46. REDUCTION WITH THE HELP OF THE TIME EVOLUTION EQUATION FOR PS(y) Expanding the conservation equation for tracer stones in the bed deposit presented in the previous slide, and using the relation between PS(y) and pe(y) of Slide 17: according to the last equation of Slide 35

  47. REDUCTION WITH THE HELP OF THE TIME EVOLUTION EQUATION FOR PS(y) Expanding the conservation equation for tracer stones in the bed deposit presented in the previous slide, and using the relation between PS(y) and pe(y) of Slide 17: according to the last equation of Slide 35 Making the substitution indicated and cancelling out terms, it is found that: The expression above captures an interesting physical process that is not possible to describe with the channel-averaged formulation. It is represented by the convective term on the LHS, which implies that changes in the composition of the bed deposit are not only due to the direct effect of overall bed aggradation or degradation, but also due to the interaction between bed level change and the vertical variation of the “background” stratigraphy of the deposit.

  48. PROBABILISTIC FORMULATION OF MASS CONTINUITY FOR TRACER STONES IN BEDLOAD TRANSPORT In an analogous form, let’s proceed with the formulation of the sediment continuity equation for the fraction of tracer stones in the bedload layer, ftr: Note that the term in brackets on the RHS accounts for the total rate of entrainment of bed tracers into bedload transport from any relative level y. Reducing, Expanding the conservation equation above, and assuming that the time variation of x can be neglected:

  49. WITH THE HELP OF THE EXNER AND ENTRAINMENT FORMULATIONS The equivalent formulations of sediment continuity for the bed deposit presented in Slides 5 and 7 are: Replacing this in the last equation of the previous slide and cancelling out terms on its RHS, it is found that: Note that in this case the convective term on the LHS accounts for the streamwise imbalance in the amount of tracer stones transported.

  50. WITH THE HELP OF THE EXNER AND ENTRAINMENT FORMULATIONS The equivalent formulations of sediment continuity for the bed deposit presented in Slides 5 and 7 are: Replacing this in the last equation of the previous slide and cancelling out terms on its RHS, it is found that: Note that in this case the convective term on the LHS accounts for the streamwise imbalance in the amount of tracer stones transported. Predictors for two additional variables are needed in order to compute the time evolution of the tracer stones displacement patterns; these variables are the volume bedload transport rate qb, and the volume concentration of bedload per unit bed area x.

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