[1] High-speed imaging of coarse sand particles transported as bed load over a planar bed reveals that the particle activity, the solid volume of particles in motion per unit streambed area, fluctuates as particles respond to near-bed fluid turbulence while simultaneously interacting with the bed. The relative magnitude of these fluctuations systematically varies with the size of the sampling area. The particle activity within a specified sampling area is distributed in a manner that is consistent with the existence of an ensemble of configurations of particle positions wherein certain configurations are preferentially selected or excluded by the turbulence structure, manifest as patchiness of active particles. The particle activity increases with increasing bed stress far faster than does the average particle velocity, so changes in the transport rate with changing stress are dominated by changes in the activity, not velocity. The probability density functions of the streamwise and cross-stream particle velocities are exponential-like and lack heavy tails. Plots of the mean squared particle displacement versus time may ostensibly indicate non-Fickian diffusive behavior while actually reflecting effects of correlated random walks associated with intrinsic periodicities in particle motions, not anomalous diffusion. The probability density functions of the particle hop distance (start-to-stop) and the associated travel time are gamma-like, which provides the empirical basis for showing that particle disentrainment rates vary with hop distance and travel time.

[1] We provide a probabilistic definition of the bed load sediment flux. In treating particle positions and motions as stochastic quantities, a flux form of the Master equation (a general expression of conservation) reveals that the volumetric flux involves an advective part equal to the product of an average particle velocity and the particle activity (the solid volume of particles in motion per unit streambed area), and a diffusive part involving the gradient of the product of the particle activity and a diffusivity that arises from the second moment of the probability density function of particle displacements. Gradients in the activity, instantaneous or time-averaged, therefore effect a particle flux. Time-averaged descriptions of the flux involve averaged products of the particle activity, the particle velocity and the diffusivity; the significance of these products depends on the scale of averaging. The flux form of the Exner equation looks like a Fokker-Planck equation (an advection-diffusion form of the Master equation). The entrainment form of the Exner equation similarly involves advective and diffusive terms, but because it is based on the joint probability density function of particle hop distances and associated travel times, this form involves a time derivative term that represents a lag effect associated with the exchange of particles between the static and active states. The formulation is consistent with experimental measurements and simulations of particle motions reported in companion papers.

[1] Particles transported as bed load within a specified streambed area possess at any instant a distribution of velocities. This distribution figures prominently in describing the rates of transport and dispersal of particles. High-speed imaging of sand particles transported as bed load over a planar bed reveals that the probability density functions of the streamwise and cross-stream particle velocities are exponential-like. For quasi-steady conditions the exponential-like density of streamwise velocities reflects a balance among three fluxes in momentum space: (1) an advection of streamwise momentum whose magnitude and sign vary with the momentum state; (2) a diffusion of momentum from higher to lower values of momentum density; and (3) a drift of momentum from regions in momentum space having high average rates of generation of kinetic energy toward regions having low rates of generation of kinetic energy. The probability density of cross-stream velocities similarly reflects a balance of fluxes of cross-stream momentum. Whereas the average net force acting on particles is zero under steady conditions, the mean, variance and asymmetry of the distribution of forces acting on particles vary with the momentum state of the particles. Numerical simulations of particle motions that are faithful to these statistical properties reproduce key empirical results, namely, the exponential-like velocity distribution and the nonlinear relation between hop distances and travel times. The simulations also illustrate how steady gradients in particle activity, the solid volume of particles in motion per unit streambed area, induce a diffusive flux as described in companion papers.Citation: Furbish, D. J., J. C. Roseberry, M. W. Schmeeckle (2012), A probabilistic description of the bed load sediment flux: 3. The particle velocity distribution and the diffusive flux,

Climate change alters species distributions, causing plants and animals to move north or to higher elevations with current warming. Bioclimatic models predict species distributions based on extant realized niches and assume niche conservation. Here, we evaluate if proxies for niches (i.e., range areas) are conserved at the family level through deep time, from the Eocene to the Pleistocene. We analyze the occurrence of all mammalian families in the continental USA, calculating range area, percent range area occupied, range area rank, and range polygon centroids during each epoch. Percent range area occupied significantly increases from the Oligocene to the Miocene and again from the Pliocene to the Pleistocene; however, mammalian families maintain statistical concordance between rank orders across time. Families with greater taxonomic diversity occupy a greater percent of available range area during each epoch and net changes in taxonomic diversity are significantly positively related to changes in percent range area occupied from the Eocene to the Pleistocene. Furthermore, gains and losses in generic and species diversity are remarkably consistent with ∼2.3 species gained per generic increase. Centroids demonstrate southeastern shifts from the Eocene through the Pleistocene that may correspond to major environmental events and/or climate changes during the Cenozoic. These results demonstrate range conservation at the family level and support the idea that niche conservation at higher taxonomic levels operates over deep time and may be controlled by life history traits. Furthermore, families containing megafauna and/or terminal Pleistocene extinction victims do not incur significantly greater declines in range area rank than families containing only smaller taxa and/or only survivors, from the Pliocene to Pleistocene. Collectively, these data evince the resilience of families to climate and/or environmental change in deep time, the absence of terminal Pleistocene “extinction prone” families, and provide valuable insights to understanding mammalian responses to current climate change.