Applying stable isotopes to examine food-web structure: an overview of analytical tools

Stable isotope analysis has emerged as one of the primary means for examining the structure and dynamics of food webs, and numerous analytical approaches are now commonly used in the field. Techniques range from simple, qualitative inferences based on the isotopic niche, to Bayesian mixing models that can be used to characterize food-web structure at multiple hierarchical levels. We provide a comprehensive review of these techniques, and thus a single reference source to help identify the most useful approaches to apply to a given data set. We structure the review around four general questions: (1) what is the trophic position of an organism in a food web?; (2) which resource pools support consumers?; (3) what additional information does relative position of consumers in isotopic space reveal about food-web structure?; and (4) what is the degree of trophic variability at the intrapopulation level? For each general question, we detail different approaches that have been applied, discussing the strengths and weaknesses of each. We conclude with a set of suggestions that transcend individual analytical approaches, and provide guidance for future applications in the field.

A preliminary analysis of the correlation of food-web characteristics with hydrology and nutrient gradients in the southern Everglades

We estimated trophic position and carbon source for three consumers (Florida gar, Lepisosteus platyrhincus; eastern mosquitofish, Gambusia holbrooki; and riverine grass shrimp, Palaemonetes paludosus) from 20 sites representing gradients of productivity and hydrological disturbance in the southern Florida Everglades, U.S.A. We characterized gross primary productivity at each site using light/dark bottle incubation and stem density of emergent vascular plants. We also documented nutrient availability as total phosphorus (TP) in floc and periphyton, and the density of small fishes. Hydrological disturbance was characterized as the time since a site was last dried and the average number of days per year the sites were inundated for the previous 10 years. Food-web attributes were estimated in both the wet and dry seasons by analysis of δ15N (trophic position) and δ13C (food-web carbon source) from 702 samples of aquatic consumers. An index of carbon source was derived from a two-member mixing model with Seminole ramshorn snails (Planorbella duryi) as a basal grazing consumer and scuds (amphipods Hyallela azteca) as a basal detritivore. Snails yielded carbon isotopic values similar to green algae and diatoms, while carbon values of scuds were similar to bulk periphyton and floc; carbon isotopic values of cyanobacteria were enriched in C13compared to all consumers examined. A carbon source similar to scuds dominated at all but one study site, and though the relative contribution of scud-like and snail-like carbon sources was variable, there was no evidence that these contributions were a function of abiotic factors or season. Gar consistently displayed the highest estimated trophic position of the consumers studied, with mosquitofish feeding at a slightly lower level, and grass shrimp feeding at the lowest level. Trophic position was not correlated with any nutrient or productivity parameter, but did increase for grass shrimp and mosquitofish as the time following droughts increased. Trophic position of Florida gar was positively correlated with emergent plant stem density.

Ecohydrology, evapotranspiration and hydrogeochemistry of carbonate mangrove wetlands

Coastal environments can be highly susceptible to environmental changes caused by anthropogenic pressures and natural events. Both anthropogenic and natural perturbations may directly affect the amount and the quality of water flowing through the ecosystem, both in the surface and subsurface and can subsequently, alter ecological communities and functions. The Florida Everglades and the Sian Ka'an Biosphere Reserve (Mexico) are two large ecosystems with an extensive coastal mangrove ecotone that represent a historically altered and pristine environment, respectively. Rising sea levels, climate change, increased water demand, and salt water intrusion are growing concerns in these regions and underlies the need for a better understanding of the present conditions. The goal of my research was to better understand various ecohydrological, environmental, and hydrogeochemical interactions and relationships in carbonate mangrove wetlands. A combination of aqueous geochemical analyses and visible and near-infrared reflectance data were employed to explore relationships between surface and subsurface water chemistry and spectral biophysical stress in mangroves. Optical satellite imagery and field collected meteorological data were used to estimate surface energy and evapotranspiration and measure variability associated with hurricanes and restoration efforts. Furthermore, major ionic and nutrient concentrations, and stable isotopes of hydrogen and oxygen were used to distinguish water sources and infer coastal groundwater discharge by applying the data to a combined principal component analysis-end member mixing model. Spectral reflectance measured at the field and satellite scales were successfully used to estimate surface and subsurface water chemistry and model chloride concentrations along the southern Everglades. Satellite imagery indicated that mangrove sites that have less tidal flushing and hydrogeomorphic heterogeneity tend to have more variable evapotranspiration and soil heat flux in response to storms and restoration. Lastly, water chemistry and multivariate analyses indicated two distinct fresh groundwater sources that discharge to the phosphorus-limited estuaries and bays of the Sian Ka'an Biopshere Reserve; and that coastal groundwater discharge was an important source for phosphorus. The results of the study give us a better understanding of the ecohydrological and hydrogeological processes in carbonate mangrove environments that can be then be extrapolated to similar coastal ecosystems in the Caribbean.

Organic carbon sources and their transfer in a gulf of Mexico coral reef ecosystem

Coral reefs face unprecedented threats throughout most of their range. Poorly planned coastal development has contributed increased nutrients and sewage contamination to coastal waters, smothering some corals and contributing to overgrowth by macroalgae. My approach to assessing the degree to which coral reef ecosystems have been influenced by terrestrial and anthropogenic organic carbon inputs is through the use of carbon (C) and nitrogen (N) stable isotopes and lipid biomarkers in a marine protected area, the Coral Reef System of Veracruz: Parque Nacional Sistema Arrecifal Veracruzano (PNSAV) in the southwest Gulf of Mexico. Firstly, I used a C and N stable isotope mixing model and a calculated fatty acid (FA) retention factor to reveal the primary producer sources that fuel the coral reef food web. Secondly, I used lipid classes, FA and sterol biomarkers to determine production of terrestrial and marine biogenic material of nutritional quality to pelagic and benthic organisms. Finally, I used coprostanol to determine pollutant loading from sewage in the suspended particulate matter. Results indicate that phytoplankton is the major source of essential metabolite FA for marine fish and that dietary energy from terrestrial sources such as mangroves are transferred to juvenile fish, while seagrass non-essential FA are transferred to the entire food web mainly in the rainy season. Sea urchins may be the main consumers of brown macroalgae, especially in the dry season, while surgeon fish prefer red algae in both dry and rainy seasons. C and N isotopic values and the ratio C:N suggest that fertilizer is the principal source of nitrogen to macroalgae. Thus nitrogen supply also favored phytoplankton and seagrass growth leading to a better nutritional condition and high retention of organic carbon in the food web members during the rainy season when river influence increases. However, the great star coral Montastrea cavernosa nutritional condition decreased significantly in the rainy season. The nearest river to the PNSAV was polluted in the dry season; however, a dilution effect was detected in the rainy season, when some coral reefs were contaminated. In 2013, a new treatment plant started working in the area. I would suggest monitoring δ¹⁵N and the C: N ratio in macroalgae as indicators of the nitrogen input and coprostanol as an indicator of human feces pollution in order to verify the efficiency of the new treatment plant as part of the management program of the PNSAV.

Geochemical and stable isotope composition of Middle Campanian marl-limestone rhythmites of the Lehrte West Syncline near Hannover (Lower Saxony, northern Germany)

A cyclic marl-limestone succession of Middle-Late Campanian age has been investigated with respect to a Milankovitch-controlled origin of geochemical data. In general, the major element geochemistry of the marl-limestone rhythmites can be explained by a simple two-component mixing model with the end-members calcium carbonate and 'average shale'-like material. Carbonate content varies from 55 to 90%. Non-carbonate components are clay minerals (illite, smectite) and biogenic silica from sponge spicules, as well as authigenically formed zeolites (strontian heulandite) and quartz. The redox potential suggests oxidizing conditions throughout the section. Trace element and stable isotopic data as well as SEM investigations show that the carbonate mud is mostly composed of low-magnesium calcitic tests of planktic coccolithophorids and calcareous dinoflagellate cysts (calcispheres). Diagenetic overprint results in a decrease of 2% d18O and an increase in Mn of up to 250 ppm. However, the sediment seems to preserve most of its high Sr content compared to the primary low-magnesium calcite of co-occurring belemnite rostra. The periodicity of geochemical cycles is dominated by 413 ka and weak signals between 51 and 22.5 ka, attributable to orbital forcing. Accumulation rates within these cycles vary between 40 and 50 m/Ma. The resulting cyclic sedimentary sequence is the product of (a) changes in primary production of low-magnesium calcitic biogenic material in surface waters within the long eccentricity and the precession, demonstrated by the CaCO3 content and the Mg/Al, Mn/Al and Sr/Al ratios, and (b) fluctuations in climate and continental weathering, which changed the quality of supplied clay minerals (the illite/smectite ratio), demonstrated by the K/Al ratio. High carbonate productivity correlates with smectite-favouring weathering (semi-arid conditions, conspicuously dry and moist seasonal changes in warmer climates). Ti as the proxy indicator for the detrital terrigenous influx, as well as Rb, Si, Zr and Na, shows only low frequency signals, indicating nearly constant rates of supply throughout the more or less pure pelagic carbonate deposition of the long-lasting third-order Middle-Upper Campanian sedimentary cycle.

Rare earth element concentrations and geochemistry of carbonate minerals from the Logatchev Hydrothermal Field and the Gakkel Ridge

The ultramafic-hosted Logatchev Hydrothermal Field (LHF) at 15°N on the Mid-Atlantic Ridge and the Arctic Gakkel Ridge (GR) feature carbonate precipitates (aragonite, calcite, and dolomite) in voids and fractures within different types of host rocks. We present chemical and Sr isotopic compositions of these different carbonates to examine the conditions that led to their formation. Our data reveal that different processes have led to the precipitation of carbonates in the various settings. Seawater-like 87Sr/86Sr ratios for aragonite in serpentinites (0.70909 to 0.70917) from the LHF are similar to those of aragonite from the GR (0.70912 to 0.70917) and indicate aragonite precipitation from seawater at ambient conditions at both sites. Aragonite veins in sulfide breccias from LHF also have seawater-like Sr isotope compositions (0.70909 to 0.70915), however, their rare earth element (REE) patterns show a clear positive europium (Eu) anomaly indicative of a small (< 1%) hydrothermal contribution. In contrast to aragonite, dolomite from the LHF has precipitated at much higher temperatures (~100 °C), and yet its 87Sr/86Sr ratios (0.70896 to 0.70907) are only slightly lower than those of aragonite. Even higher temperatures are calculated for the precipitation of deformed calcite veins in serpentine-talc fault schists form north of the LHF. These calcites show unradiogenic 87Sr/86Sr ratios (0.70460 to 0.70499) indicative of precipitation from evolved hydrothermal fluids. A simple mixing model based on Sr mass balance and enthalpy conservation indicates strongly variable conditions of fluid mixing and heat transfers involved in carbonate formation. Dolomite precipitated from a mixture of 97% seawater and 3% hydrothermal fluid that should have had a temperature of approximately 14 °C assuming that no heat was transferred. The much higher apparent precipitation temperatures based on oxygen isotopes (~ 100 °C) may be indicative of conductive heating, probably of seawater prior to mixing. The hydrothermal calcite in the fault schist has precipitated from a mixture of 67% hydrothermal fluid and 33% seawater, which should have had an isenthalpic mixing temperature of ~ 250 °C. The significantly lower temperatures calculated from oxygen isotopes are likely due to conductive cooling of hydrothermal fluid discharging along faults. Rare earth element patterns corroborate the results of the mixing model, since the hydrothermal calcite, which formed from waters with the greatest hydrothermal contribution, has REE patterns that closely resemble those of vent fluids from the LHF. Our results demonstrate, for the first time, that (1) precipitation from pure seawater, (2) conductive heating of seawater, and (3) conductive cooling of hydrothermal fluids in the sub-seafloor all can lead to carbonate precipitation within a single ultramafic-hosted hydrothermal system.

Helium and oxygen isotope ratios of ODP Sites 193-1188 and 193-1189

Fluid mixing processes and thermal regimes within the Snowcap and Roman Ruins vent sites of the PACMANUS hydrothermal system, Papua New Guinea, were investigated using 3He/4He ratios from fluid inclusions within pyrite and anhydrite and the d18O signature of anhydrite. Depressed 3He/4He ratios of 0.2-6.91RA appear to be caused by significant atmospheric diffusive exchange, whilst He-Ne diffusive fractionation precludes correction using 20Ne. 40Ar/36Ar ratios of 295-310 are elevated above seawater, indicating the majority of argon is seawater derived but with a magmatic component. d18O anhydrite ratios are 6.5 per mil to 11 per mil for Snowcap and 6.4 per mil to 11.9 per mil for Roman Ruins. Using oxygen isotope fractionation factors for the anhydrite-water system, the temperatures calculated assuming isotopic equilibrium at depth are up to 100 °C cooler than fluid inclusion trapping temperatures. It is likely that anhydrite is precipitated rapidly, preventing d18O equilibration. By comparing new d18O values for anhydrite with corresponding published 87Sr/86Sr ratios, seawater is inferred to penetrate deep into the Snowcap system with little conductive heating. A simple fluid mixing model has been constructed whereby the differing venting styles can be explained by a plumbing system at depth which favors delivery of end-member hydrothermal fluid to the high temperature sites.

Trace elements in benthic foraminifera of sediment core V30-49

Eastern tropical Atlantic benthic foraminiferal Ba/Ca and Cd/Ca data from core V30-949 (3093 m) reveal large inferred changes in nutrient concentrations of deep Atlantic waters during the last 250 kyr. Relative changes in North Atlantic Deep Water contribution to this site are estimated by scaling the V30-49 Cd/Ca record to values of modern end-member water masses; these estimates agree well with the relative structure and timing of circulation changes in the eastern tropical Atlantic reconstructed from a d13C record-based mixing model (Raymo et al., 1997, doi:10.1029/97PA01019). Temporal differences between V30-49 Cd/Ca and Ba/Ca records suggest that the Ba/Ca record reflects changes in circulation with an additional increase in the Ba composition of deep Atlantic water masses during glacial episodes, possibly resulting from increased productivity. Similarity between the d13C and Ba/Ca records suggests that carbon isotopes in the deep glacial Atlantic also reflect productivity increases.

Bulk and molecular proxies of sediment core GeoB6518-1

A 20 kyr long sediment sequence from the Congo deep sea fan (core GeoB 6518-1), one of the world's largest deep sea river fans, has been analysed for bulk and molecular proxies in order to reconstruct the marine, soil and plant organic carbon (OC) contributions to these sediments since the last glacial maximum. The bulk proxies applied, C/N ratio and d13Corg, ranged from 10 to 12.5 and from -24.5 to -21 per mill VPDB, respectively. As molecular proxies, concentrations of marine derived alkenones and terrestrial derived odd-numbered n-alkanes were used, which varied between 0.2 and 4 µg/g dry weight sediment. In addition, the branched vs. isoprenoid tetraether (BIT) index, a proxy for soil organic matter input, was used, which varied from 0.3 to 0.5 in this core. Application of binary mixing models, based on the different individual proxies, showed estimates for terrestrial OC input varying by up to 50% due to the heterogeneous nature of the OC. Application of a three end-member mixing model using the d13Corg content, the C/N ratio and the BIT index, enabled the distinction of soil and plant organic matter as separate contributors to the sedimentary OC pool. The results show that marine OC accounts for 20% to 40% of the total OC present in the deep sea fan sediments over the last 20 kyr and that soil OC accounts for about half (45% on average) of the OC present. This suggests that soil OC represents the majority of the terrestrial OC delivered to the fan sediments. Accumulation rates of the plant and soil OC fractions over the last 20 kyr varied by a factor of up to 5, and are strongly related to sediment accumulation rates. They showed an increase starting at ca. 17 kyr BP, a decline during the Younger Dryas, peak values during the early Holocene and lower values in the late Holocene. This pattern matches with reconstructions of past central African humidity and Congo River discharge from the same core and revealed that central African precipitation patterns exert a dominant control on terrestrial OC deposition in the Congo deep sea fan. Marine OC accumulation rates are only weakly related to sediment accumulation rates and vary only little over time compared to the terrigenous fractions. These variations are likely a result of enhanced preservation during times of higher sedimentation rates and of relative small fluctuations in primary production due to wind-driven upwelling.

Linear Hyperspectral Unmixing Using L0-norm Approximations and Nonnegative Matrix Factorization

Spectral unmixing (SU) is a technique to characterize mixed pixels of the hyperspectral images measured by remote sensors. Most of the existing spectral unmixing algorithms are developed using the linear mixing models. Since the number of endmembers/materials present at each mixed pixel is normally scanty compared with the number of total endmembers (the dimension of spectral library), the problem becomes sparse. This thesis introduces sparse hyperspectral unmixing methods for the linear mixing model through two different scenarios. In the first scenario, the library of spectral signatures is assumed to be known and the main problem is to find the minimum number of endmembers under a reasonable small approximation error. Mathematically, the corresponding problem is called the $\ell_0$-norm problem which is NP-hard problem. Our main study for the first part of thesis is to find more accurate and reliable approximations of $\ell_0$-norm term and propose sparse unmixing methods via such approximations. The resulting methods are shown considerable improvements to reconstruct the fractional abundances of endmembers in comparison with state-of-the-art methods such as having lower reconstruction errors. In the second part of the thesis, the first scenario (i.e., dictionary-aided semiblind unmixing scheme) will be generalized as the blind unmixing scenario that the library of spectral signatures is also estimated. We apply the nonnegative matrix factorization (NMF) method for proposing new unmixing methods due to its noticeable supports such as considering the nonnegativity constraints of two decomposed matrices. Furthermore, we introduce new cost functions through some statistical and physical features of spectral signatures of materials (SSoM) and hyperspectral pixels such as the collaborative property of hyperspectral pixels and the mathematical representation of the concentrated energy of SSoM for the first few subbands. Finally, we introduce sparse unmixing methods for the blind scenario and evaluate the efficiency of the proposed methods via simulations over synthetic and real hyperspectral data sets. The results illustrate considerable enhancements to estimate the spectral library of materials and their fractional abundances such as smaller values of spectral angle distance (SAD) and abundance angle distance (AAD) as well.

Relative contribution of natural productivity and compound feed to tissue growth in blue shrimp (Litopenaeus stylirostris) reared in biofloc: Assessment by C and N stable isotope ratios and effect on key digestive enzymes

The aim of this study was to assess the relative contribution of natural productivity and compound food to the growth of the juvenile blue shrimp Litopenaeus stylirostris reared in a biofloc system. Two experiments were carried out based on the same protocol with three treatments: clear water with experimental diet (CW), biofloc with experimental diet (BF) and biofloc unfed (BU). Shrimp survival was significantly higher in biofloc rearing than in CW rearing. The contribution of the biofloc to shrimp diet was estimated through measurement of carbon and nitrogen stable isotope ratios in shrimp and food sources. Different isotopic compositions between feeds were obtained by feeding natural productivity with a mixture rich in fish meal and the shrimps with a pellet containing a high level of soy protein concentrate. Using a two source one-isotope mixing model, we found that the natural productivity of the biofloc system contributed to shrimp growth at a level of 39.8% and 36.9%, for C and N, respectively. The natural food consumed by the shrimps reared in the biofloc system resulted in higher gene expression (mRNA transcript abundance) and activities of two digestive enzymes in their digestive gland: α-amylase and trypsin. The growth of shrimp biomass reared in biofloc was, on average, 4.4 times that of those grown in clear water. Our results confirmed the best survival and promoted growth of shrimps using biofloc technology and highlighted the key role of the biofloc in the nutrition of rearing shrimps. Statement of relevance In this study, we have applied an original protocol to determine the respective contribution of natural productivity and artificial feeds on the alimentation of the juvenile blue shrimp L. stylirostris reared in biofloc system by using C and N natural stable isotope analysis. Moreover, we have compared, in shrimp digestive gland, the α-amylase and trypsin enzyme activities at biochemical and molecular levels for two different shrimp rearing systems, biofloc and clear water. In our knowledge, the use of molecular tool to study the influence of biofloc consumption on digest process of shrimp was never carried out. We think that our research is new and important to increase knowledge on biofloc topic.

Simultaneous estimation of the nutritional contribution of fish meal, soy protein isolate and corn gluten to the growth of Pacific white shrimp (Litopenaeus vannamei) using dual stable isotope analysis

The nutritional contribution of the dietary nitrogen, carbon and total dry matter supplied by fish meal (FM), soy protein isolate (SP) and corn gluten (CG) to the growth of Pacific white shrimp Litopenaeus vannamei was assessed by means of isotopic analyses. As SP and CG are ingredients derived from plants having different photosynthetic pathways which imprint specific carbon isotope values to plant tissues, their isotopic values were contrasting. FM is isotopically different to these plant meals with regards to both, carbon and nitrogen. Such natural isotopic differences were used to design experimental diets having contrasting isotopic signatures. Seven isoproteic (36% crude protein), isoenergetic (4.7 kcal g−1) diets were formulated; three diets consisted in isotopic controls manufactured with only one main ingredient supplying dietary nitrogen and carbon: 100% FM (diet 100F), 100% SP (diet 100S) and 100% CG (diet 100G). Four more diets were formulated with varying mixtures of these three ingredients, one included 33% of each ingredient on a dietary nitrogen basis (diet 33FSG) and the other three included a proportion 50:25:25 for each of the three ingredients (diets 50FSG, 50SGF and 50GFS). At the end of the bioassay there were no significant differences in growth rate in shrimps fed on the four mixed diets and diet 100F (k=0.215–0.224). Growth rates were significantly lower (k=0.163–0.201) in shrimps grown on diets containing only plant meals. Carbon and nitrogen stable isotope values (δ13C and δ15N) were measured in experimental diets and shrimp muscle tissue and results were incorporated into a three-source, two-isotope mixing model. The relative contributions of dietary nitrogen, carbon and total dry matter from FM, SP and CG to growth were statistically similar to the proportions established in most of the diets after correcting for the apparent digestibility coefficients of the ingredients. Dietary nitrogen available in diet 33FSG was incorporated in muscle tissue at proportions representing 24, 35 and 41% of the respective ingredients. Diet 50GSF contributed significantly higher amounts of dietary nitrogen from CG than from FM. When the level of dietary nitrogen derived from FM was increased in diet 50FSG, nutrient contributions were more comparable to the available dietary proportions as there was an incorporation of 44, 29 and 27% from FM, SP and CG, respectively. Nutritional contributions from SP were very consistent to the dietary proportions established in the experimental diets.

Geochemical Assessment and Separation of Source Waters in the Upper Boulder River Watershed Near Boulder, MT

Environmental samples were collected at three surface water sites between 5/21/2011 and 11/21/2014 along the Upper Boulder River near Boulder Montana. The sites were located at Bernice (within the mountain block), near the High Ore drainage (near the mountain block/basin transition), and at the USGS Gauging Station near Boulder, Montana (within the basin). The parameters measured in the field were SC, temperature, and alkalinity with occasional pH measurements. We collected samples for anions, cations, and stable isotopes in the catchment. We identified endmembers by sampling snow and groundwater and determined from available data an approximate endmember for rain, snow, and groundwater. We used temporal and spatial variations of water chemistry and isotopes to generate an endmember mixing model. Groundwater was found to always be an important contributor to river flow and could increase by nearly an order of magnitude during large snowmelt events. This resulted in groundwater comprising ~20% of total river flow during snowmelt at all sites. At peak snowmelt we observed that near surface water contributions to the river were from a mixture of rain and snow. Soil water, though not sampled, was hypothesized to be an important part of the hydrologic story. If so, the endmember contributions determined in this study may be different. Groundwater may have the highest variation depending on water chemistry of shallow soil water.

Study of the equatorial Atlantic Ocean mixing layer using a one-dimensional turbulence model

The General Ocean Turbulence Model (GOTM) is applied to the diagnostic turbulence field of the mixing layer (ML) over the equatorial region of the Atlantic Ocean. Two situations were investigated: rainy and dry seasons, defined, respectively, by the presence of the intertropical convergence zone and by its northward displacement. Simulations were carried out using data from a PIRATA buoy located on the equator at 23º W to compute surface turbulent fluxes and from the NASA/GEWEX Surface Radiation Budget Project to close the surface radiation balance. A data assimilation scheme was used as a surrogate for the physical effects not present in the one-dimensional model. In the rainy season, results show that the ML is shallower due to the weaker surface stress and stronger stable stratification; the maximum ML depth reached during this season is around 15 m, with an averaged diurnal variation of 7 m depth. In the dry season, the stronger surface stress and the enhanced surface heat balance components enable higher mechanical production of turbulent kinetic energy and, at night, the buoyancy acts also enhancing turbulence in the first meters of depth, characterizing a deeper ML, reaching around 60 m and presenting an average diurnal variation of 30 m.

Axial mixing in rotating drums using magnetic resonance imaging using bran as a model for solid state fermentations

Magnetic resonance imaging (MRI) is an easily automated, reliable technique to investigate axial mixing within rotating drums. Moist bran can be clearly differentiated from dry bran using MRI allowing a non-segregating tracer for axial mixing. For a 20-cm diameter drum, the axial dispersion coefficient in the particle bed was 0.51 cm s(-2). Axial dispersion is scale-dependent.