Gravity-driven size segregation is important in mountain streams where a wide range of grain sizes are transported as bedload. More particularly, vertical size segregation is a multi-scale process that originates in interactions at the scale of particles with important morphological consequences for the river bed. To address this issue, a volume-averaged multi-phase flow model for immersed bi-disperse granular flows was developed based on an interparticle segregation force (Guillard et al., J. Fluid Mech., vol. 807, 2016, R1) and a granular Stokesian drag force (Tripathi & Khakhar, J. Fluid Mech., vol. 717, 2013, pp. 643–669). An advection–diffusion model was derived from this model yielding parametrisations for the advection and diffusion coefficients based on the interparticle interactions. This approach makes it possible to bridge the gap between grain-scale physics and continuum modelling. Both models were successfully tested against existing discrete element model (DEM) simulations of size segregation in bedload transport (Chassagne et al., J. Fluid Mech., vol. 895, 2020, A30). Through a detailed investigation of the granular forces, it is demonstrated that the observed scaling of the advection and diffusion coefficients with the inertial number can be explained by the granular drag force dependency on the viscosity. The drag coefficient is shown to be linearly dependent on the small particle concentration. A new scaling relationship for the segregation force including the small particle concentration and the pressure is proposed. Lastly, adding a size-ratio dependency in the segregation force fairly reproduces the DEM results for a large range of small particle concentrations and size ratios.

Observations show that glaciers around the world are in retreat and losing mass. Internationally coordinated for over a century, glacier monitoring activities provide an unprecedented dataset of glacier observations from ground, air and space. Glacier studies generally select specific parts of these datasets to obtain optimal assessments of the mass-balance data relating to the impact that glaciers exercise on global sea-level fluctuations or on regional runoff. In this study we provide an overview and analysis of the main observational datasets compiled by the World Glacier Monitoring Service (WGMS). The dataset on glacier front variations (∼42 000 since 1600) delivers clear evidence that centennial glacier retreat is a global phenomenon. Intermittent readvance periods at regional and decadal scale are normally restricted to a subsample of glaciers and have not come close to achieving the maximum positions of the Little Ice Age (or Holocene). Glaciological and geodetic observations (∼5200 since 1850) show that the rates of early 21st-century mass loss are without precedent on a global scale, at least for the time period observed and probably also for recorded history, as indicated also in reconstructions from written and illustrated documents. This strong imbalance implies that glaciers in many regions will very likely suffer further ice loss, even if climate remains stable.