Sediment dispersion is a fundamental process embedded in the bedload of rivers but the rate at which it occurs is poorly known. This study quantifies the streamwise and vertical dispersion rates of gravels over a long flood sequence by using magnetically tagged clasts. Both grain virtual velocity and grain burial rate decline rapidly as a function of expended flow energy and attain limiting values that are a small proportion of initial rates. Differences between grain sizes lessen over time as equilibrium rates of dispersion are achieved. Relative rates demonstrate that streamwise dispersion dominates over vertical dispersion, although the latter persists as an essential part of the dispersion process.
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 Sediment dispersion is a fundamental process embedded in the bedload of rivers. This process accomplished through grain displacements dominates sediment texture in gravel-bed rivers. Accordingly, models that route bed material through channel networks [e.g.,Benda and Dunne, 1997; Davy and Lague, 2009; Gasparini et al., 1999] must characterize the grain-size dependent magnitude and variance of streamwise displacements to fully account for downstream fining [Pizzuto, 1995; Sternberg, 1875]. Local adjustment to bed sediment also entails vertical mixing that distributes gravels within the bed and causes vertical segregation of their sizes [Parker et al., 1982; Schick et al., 1987]. To date, nearly all empirical insights into natural gravel dispersion come from short-term studies that extend over several floods or years. Yet modeling dispersion over time scales that are meaningfully related to river adjustment requires a full understanding of the process and its equilibrium rate. In this paper the streamwise and vertical rates of gravel dispersion are quantified over a 19 year period to establish equilibrium rates in Carnation Creek, Canada, where most bedload fluxes occur under partial sediment transport conditions [Haschenburger and Wilcock, 2003; Wilcock and McArdell, 1997].
 Higher rates of streamwise dispersion can be expected under flow conditions that promote grain mobility [Habersack, 2001; Wilcock and McArdell, 1997] with finer gravels traveling longer distances than coarser gravels on average [Church and Hassan, 1992; Ferguson and Wathen, 1998; Haschenburger, 1996]. For gravels initially exposed on the bed surface, the rate of streamwise dispersion has been shown to decline over time, with the slowdown attributed primarily to the vertical mixing of bed sediment [Ferguson and Hoey, 2002] and its effect on grain mobility [Hassan et al., 1992; Wong et al., 2007]. While the depth of sediment mixing is known to increase with flow magnitude on an event basis [e.g., Carling, 1987; Haschenburger, 1999], the characteristic burial depth of gravels over time should reflect the magnitude of the largest event(s) as long as sediment supply maintains a relatively stable bed. The influence of vertical mixing on gravel mobility suggests that a characteristic rate of streamwise dispersion will not be achieved until initially exposed gravels are adequately mixed. In some rivers, the rate of vertical mixing can be substantial, with most surface gravels exchanged into the subsurface relatively quickly [Haschenburger, 2011b], and limited evidence suggests that finer gravels are initially mixed more rapidly and deeply than coarser gravels [Ferguson and Wathen, 1998; Haschenburger, 2011a]. Preferential storage in less mobile areas of the channel retards gravel dispersion [Dietrich et al., 1989; Ferguson et al., 2002; Pyrce and Ashmore, 2005] because the longer residence times reduce the dispersion rate for the entire bed.
 Streamwise and vertical dispersion rates of gravels were determined by tracking about 2500 magnetically tagged clasts deployed on the bed surface as four generations in Carnation Creek, which is located on the west coast of Vancouver Island (48°54′ 56″ N, 124°59′ 52″ W; Table 1 and Figure 1). Observations from both mobile and stationary tracers from 13 recoveries, completed between 1990 and 2008 (Figure 1), were used to compute rates. Streamwise dispersion was quantified as a grain virtual velocity, which is expressed as
where Vtr = virtual velocity up to recovery r, Lr = mean travel distance at recovery r, and Ttr = elapsed time up to recovery r. Vertical dispersion was quantified by a grain burial rate, which is given as
where Btr = burial rate up to recovery r and br = mean depth of tracer burial at recovery r. To consider dispersion based on the duration of sediment transport, effective rates of virtual velocity (Ver) and burial (Ber) were computed by substituting the duration of flow theoretically competent to entrain bed sediment (Ter) for Ttr. Hydrologic forcing was characterized by the total expenditure of flow energy. For Vtr and Btr, this flow energy was calculated as
where Ωtr = total flow energy expended up to recovery r, g = acceleration due to gravity (equal to 9.81 m s−2), ρ = water density (taken as 1000 kg m−3), Qj = discharge at hourly increment j, and S = energy slope estimated by the bed gradient. For Ver and Ber, total excess flow energy expended (Ωer) was computed by limiting Qj to values that exceed the 6.7 m3s−1 threshold for sediment entrainment (see auxiliary material for additional details on methods).
Table 1. Channel Characteristics of the 2.7 km Long Study Reacha
 Both virtual velocity and burial rate decline rapidly as a function of expended flow energy and attain limiting values that are a small proportion of initial rates (Figure 2). As Ωtr increases to 3000 GJ, Vtr and Btr decrease from 82 mm hr−1 to 10 mm hr−1 and 0.15 mm hr−1 to 0.02 mm hr−1, respectively, with further flooding causing a decline to asymptotes of 4 mm hr−1 and 0.003 mm hr−1, respectively, after about 8500 GJ. Similarly, Ver and Ber decline from maximum values of 1525 mm hr−1 and 3.7 mm hr−1 to about 400 mm hr−1 and 0.5 mm hr−1, respectively, after about 1000 GJ. Asymptotes of 215 mm hr−1 and 0.2 mm hr−1 are evident for Ver and Ber after 3000 GJ and 1800 GJ, respectively. Once limiting rates are achieved, streamwise rates are three orders of magnitude larger than vertical rates.
 As limiting rates are approached, differences between the dispersion rates of individual grain sizes decrease (Figure 3). Burial rates converge to a greater degree than virtual velocities such that only the latter exhibits statistically significant differences between grain sizes over the entire flood sequence (Figures 3a and 3b). At the limiting rates, dispersed gravel originates mostly from the subsurface as only about 10% of the recovered tracers were found on the bed surface. Of these buried tracers, about 80% moved between successive recoveries.
 Relations between rates and expended flow energy are well described by power functions (Figure 2). Rates of change for Vtr and Btr are larger than those for Ver and Ber, respectively, although they are not significantly different, and more variance is described by the Vtr and Btr functions (Figure 2, insets). Most scatter is concentrated in the early portion of the record, which generally reflects the particular flood histories of the tracer generations and the vertical mixing that was accomplished.
4. Discussion and Conclusion
 Downstream displacement dominates gravel dispersion. While streamwise dispersion progresses as flooding continues to transport sediment downstream, vertical dispersion is constrained by the depth of sediment exchange, which typically averages less than two times the thickness of the surface layer [Wilcock and McArdell, 1997]. The spatial extent of exchange is also restricted because, over a long flood sequence, most bedload is concentrated within only a portion of the available bed area [Haschenburger and Wilcock, 2003; Lisle et al., 2000]. Together these expectations about gravel exchange establish a general limit for the extent of vertical dispersion. Nonetheless, vertical dispersion impacts streamwise dispersion because the depth to which the bed is disturbed as flow increases affects virtual velocity through vertical exchange that releases and reburies grains [Hassan et al., 1992; Wong et al., 2007]. Therefore, in channels where gravels are mobilized from deeper depths for a given flow [Lisle, 1995], slower virtual velocities would be expected, influencing the relative magnitudes of streamwise and vertical dispersion. Additionally, flood regimes that exhibit a relative small range in peak discharge (e.g., snowmelt generated) may produce even more differentiation between rates of streamwise and vertical dispersion if only because the limited vertical mixing that controls vertical dispersion rates can be more easily exceeded by streamwise displacements.
 Exposed gravels dispersed from a point source and mixed with other bed sediment exhibit a pronounced initial reduction in dispersion rates [Ferguson et al., 2002; Haschenburger, 2011b; Hassan et al., 1992]. The rapid change in virtual velocity corresponds to tracer burial, which reached between 50% and 95% in Carnation Creek [Haschenburger, 2011a] due to the vertical mixing driven by the larger floods early in the sequence. Local changes in bed elevation that promote passive gravel exchange may augment vertical dispersion. However, in Carnation Creek, the frequency and magnitude of bed adjustments derived from local imbalances between scour and fill increase with flood magnitude [Haschenburger, 2006], suggesting that the potential to affect vertical dispersion is limited during smaller floods. Thus, for a population of initially exposed gravels, the timing of larger floods capable of more extensive vertical mixing helps determine how quickly dispersion rates evolve toward limiting rates.
 Limiting rates are substantially smaller than the initially observed rates. For Ver, the slowdown to 28% of the initial value is larger but broadly comparable to the 50% reduction reported for the Allt Dubhaig, Scotland [Ferguson et al., 2002], while the rates for the last recoveries are similar (215 mm hr−1 versus 240 ± 39 mm hr−1 for the six reaches in Allt Dubhaig (R. Ferguson, personal communication, 2010)). Results from a vertical exchange model for Allt Dubhaig [Ferguson and Hoey, 2002] suggest that tracers were probably sufficiently mixed by the end of the study to consider the velocity a good approximation of a limiting rate. Models focused on longer-term outcomes of bedload fluxes need to distinguish between initial rates, derived from surface gravels, and limiting rates, which represent the entire sediment bed.
 The limiting rates establish characteristic dispersion rates in a natural channel that reflect longer-term patterns of bedload fluxes. Given the temporal trend in dispersion rates and known patterns of vertical exchange in Carnation Creek [Haschenburger, 2011a], it can be argued that these rates are equilibrium rates, at least to the extent which is possible in natural rivers. Under equilibrium rates, size dependent dispersion persists only in the streamwise component, suggesting that the claim of size similarity that underpins many sediment transport formulae [Gomez and Church, 1989; Parker and Klingeman, 1982] is justified based on grain burial rate but not virtual velocity. Further, at these rates, the vertical dispersion rate is maintained largely through the release of subsurface gravels and their subsequent reburial. Thus, vertical exchange is an essential component of gravel dispersion, persisting under equilibrium rates, even though the streamwise component dominates the dispersion process.
 At the equilibrium rates, effective rates are two orders of magnitude larger than rates based on total elapsed time. Gravel dispersion is accomplished during about 2% of the total elapsed time. Carnation Creek is a relatively active gravel-bed river given an average of 15 ± 5 floods per year capable of transporting bedload. Dispersion driven by flood regimes of drier climates would occur over an even smaller proportion of time. Over long time scales, the process of dispersion is dominated by periods of non-events that control how quickly bed material is routed through channel networks.
 This research was supported by the Natural Sciences and Engineering Research Council of Canada (grants awarded to M. Church), Geological Society of America, University of Auckland Research Committee (grant 3603028), and National Geographic Society (grant 8241–07). The British Columbia Ministry of Forests operates the Carnation Creek field station. M. Church and M. Hassan deployed the yellow and orange generations and directed their initial recoveries. Over 40 students and colleagues have assisted with tracer recovery. Comments from M. Church and reviews by T. Lisle and an anonymous reviewer strengthened this paper.
 The Editor thanks Tom Lisle and an anonymous reviewer for their assistance in evaluating this paper.