Transport time scales in soil erosion modelling

Unlike sediment transport in rivers, erosion of agricultural soil must overcome its cohesive strength to move soil particles into suspension. Soil particle size variability also leads to fall velocities covering many orders of magnitude, and hence to different suspended travel distances in overland flow. Consequently, there is a large range of inherent time scales involved in transport of eroded soil. For conditions where there is a constant rainfall rate and detachment is the dominant erosion mechanism, we use the Hairsine-Rose (HR) model to analyze these timescales, to determine their magnitude (bounds) and to provide simple approximations for them. We show that each particle size produces both fast and slow timescales. The fast timescale controls the rapid adjustment away from experimental initial conditions – this happens so quickly that it cannot be measured in practice. The slow time scales control the subsequent transition to steady state and are so large that true steady state is rarely achieved in laboratory experiments. Both the fastest and slowest time scales are governed by the largest particle size class. Physically, these correspond to the rate of vertical movement between suspension and the soil bed, and the time to achieve steady state, respectively. For typical distributions of size classes, we also find that there is often a single dominant time scale that governs the growth in the total mass of sediment in the non-cohesive deposited layer. This finding allows a considerable simplification of the HR model leading to analytical expressions for the evolution of suspended and deposited layer concentrations

Drag coefficient, <i>C</i>_<i>d</i>, for <i>R</i>₀ = 100 for different viscosity (<i>X</i>) and density (<i>P</i>) ratios.

<p>Roman values (dashes: No values provided) from Juncu [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194907#pone.0194907.ref030" target="_blank">30</a>], results from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194907#pone.0194907.e011" target="_blank">Eq (10)</a> are in italics, results in the rightmost column are from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194907#pone.0194907.e001" target="_blank">Eq (1)</a>.</p

Comparison of various numerical data sets with predictions of drag formulas for the case of the gas bubble.

<p>Comparison of various numerical data sets with predictions of drag formulas for the case of the gas bubble.</p

Drag coefficient, <i>C</i>_<i>d</i>, estimated by Oliver and Chung [11] for various viscosity ratios (<i>X</i>) over a range of Reynolds numbers (<i>R</i>₀) compared with predictions of Eqs (1) and (10).

<p>Drag coefficient, <i>C</i>_<i>d</i>, estimated by Oliver and Chung [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194907#pone.0194907.ref011" target="_blank">11</a>] for various viscosity ratios (<i>X</i>) over a range of Reynolds numbers (<i>R</i>₀) compared with predictions of Eqs (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194907#pone.0194907.e001" target="_blank">1</a>) and (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194907#pone.0194907.e011" target="_blank">10</a>).</p

Comparison of various numerical data sets with predictions of drag formulas for the case of the solid sphere.

<p>Comparison of various numerical data sets with predictions of drag formulas for the case of the solid sphere.</p

Comment on 'Recent advances on solving the three-parameter infiltration equation' by Prabhata K. Swamee, Pushpa N. Rathie, Luan Carlos de S.M. Ozelim and André L.B. Cavalcante, Journal of Hydrology 509 (2014) 188–192

A recent approximation to the three-parameter infiltration was compared with an existing approximation. The new approximation has a minimum relative error that is two orders of magnitude greater than the maximum relative error of the existing approximation