EARTH SURFACE PROCESSES AND LANDFORMSEarth Surf. Process. Landforms 34, 1320–1321 (2009)Copyright © 2009 John Wiley & Sons, Ltd.Published online 2 June 2009 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/esp.1816

John Wiley & Sons, Ltd.Chichester, UKESPEarth Surface Processes and LandformsEARTH SURFACE PROCESSES AND LANDFORMSEarth Surface Processes and LandformsThe Journal of the British Geomorphological Research GroupEarth Surf. Process. Landforms0197-93371096-9837Copyright © 2006 John Wiley & Sons, Ltd.John Wiley & Sons, Ltd.2006Earth ScienceEarth Science99999999ESP1816Research ArticleResearch ArticlesCopyright © 2006 John Wiley & Sons, Ltd.John Wiley & Sons, Ltd.2006Exchange

Response to Kinnell’s ‘Comment on “A transport-distance approach to scaling erosion rates: III. Evaluating scaling characteristics of MAHLERAN” ’ Response to Kinnell’s comment

John Wainwright,1* Anthony J. Parsons,1 Eva N. Müller,2 Richard E. Brazier3 and D. Mark Powell4

1 Sheffield Centre for International Drylands Research, Department of Geography, University of Sheffield, Sheffield, UK2 Institut für Geoökologie, Universität Potsdam, 14 415 Potsdam, Germany3 Department of Geography, University of Exeter, Exeter, UK4 Department of Geography, University of Leicester, Leicester, UK

Received 6 November 2008; Revised 19 February 2009; Accepted 3 March 2009

* Correspondence to: John Wainwright, Sheffield Centre for International Drylands Research, Department of Geography, University of Sheffield, Winter Street, SheffieldS10 2TN, UK. E-mail: j.wainwright@sheffield.ac.uk

Kinnell comments that we misinterpret his analysis in thedevelopment of his empirical relationship relating flowdepths and drop sizes (Kinnell, 1993). Beyond any potentialmisunderstanding of his original paper, Kinnell’s comment issomewhat tangential to the statement he cites. Our issue issimply that the theory is not well supported by the datapresented, and furthermore given that his approach is basedon assumptions about transport mechanisms that we believeare unwarranted, we felt that it was unwarranted to implementhis functional form as a further test of the scaling charac-teristics of MAHLERAN.

Our original statement is indeed inaccurate. It would havebeen better to say that the analysis and results of the originalpaper suggested that the peak occurs at a flow depth ofapproximately twice the raindrop diameter. This value isobtained by the value of the relationship relating the inflexionpoint or critical flow depth in the analysis to the depth of flow(Kinnell, 1993. equation 10). The value of two is a successivelypoorer approximation as drop sizes increase beyond reasonablelimits for natural rainfall. Comparable analyses in Kinnell (1991)and Kinnell and Wood (1992) suggest that the theoreticalvalue of the ratio is a function of 6/π albeit over a limitedrange of raindrop diameters. Kinnell (1993) does indeed focushis analysis and produce specific comments about transportrate in flow, but his subsequent analyses imply that this canbe used to back-estimate detachment rates with the samefunctional form (Kinnell, 1993, equation 17, and discussionon pp. 1101–1102), at least when flow depths are less thanabout three drop diameters. In an earlier paper, he also citesthe results of Moss and Green (1983) which ‘indicate thattransport rates peak when drops impact flows that are betweentwo and three drop diameters deep’ (Kinnell, 1991, p. 161).

There is evidence for a decline in transport when the flowdepth is between one and 1·5 times the raindrop diameter(Kinnell, 1991), although only the data presented for thecoarsest particle size (0·9 mm diameter) show a convincingcurve with decreasing flow depth. The lack of any further datato support similar statements in Kinnell (1993) and subsequentexpositions of this work led to our belief that there isinsufficient empirical support to warrant their testing withinthe MAHLERAN framework. Thus, we believe that the point ofour original statement still holds.

More fundamentally, there is the issue of the use ofsediment concentration as an analytical method for theinvestigation of any sediment-transport mechanism, as forexample used by Kinnell in equation 2 of his comment. Whileuseful for simplicity (and often a consequence of the ways inwhich sediment transport is measured as a concentration in atransporting medium, regardless of the transport mechanism),in the case of raindrop detachment and unconcentrated flowtransport, it does not reflect the full interaction of particleswith the flow. Not only does it not account for the sum ofvectors of the combination of the ballistic trajectory ofparticles from their ejection within the flow-velocity field, butsignificantly it assumes that all particles are lifted into theflow. There is no justification for this assumption, and it islikely that particle creep is common when sediment particlesare large relative to the flow depth and/or when raindropimpact is directed into the slope (e.g. through wind action),as are mechanisms of rolling or other forms of bedload transport(as a function of flow velocity and the relative vector of theraindrop impact and subsequent ejecta). Indeed Kinnell (1991,p. 161) himself notes the possibility of movement as ‘bed load’,without incorporating it in further analyses. There is some

Copyright © 2009 John Wiley & Sons, Ltd. Earth Surf. Process. Landforms 34, 1320–1321 (2009)DOI: 10.1002/esp

RESPONSE TO KINNELL’S COMMENT 1321

justification of the assumption of suspension in Kinnell’s (2005)review based on the grounds that the ‘ratio of the [effectiveaverage particle travel distance] values [observed for twoparticle sizes] (1:2·96) was close to the ratio of the measuredsettling velocities of the two materials (1:2·75)’ (p. 2822).Given the uncertainties in measuring both sets of values aswell as the structural uncertainty in Kinnell’s model, it isunclear how this difference of 7·6% should be evaluatedeither in support of or against the model.

In terms of being an artefact [sic], the issue is simplywhether Kinnell’s assertion noted earlier about the use ofconcentration data to back-calculate erodibility as a functionof raindrop size and flow depth is tenable. We would notdisagree with the reasons cited by Kinnell in this comment forthe ‘perceived’ form of this function. However, they dependfundamentally on considering the detachment process differentlywhen transport by splash occurs from when transport is byunconcentrated overland flow, and the subsequent interactionbetween mass detachment and flow transport. Therefore theassertion about using the same functional form for back-calculating erodibility cannot hold; this is the artefact towhich we refer.

We do not disagree with Kinnell’s concluding paragraph.Indeed, it re-emphasizes points made in our discussion thatmuch better data are required for developing erosion models,and that these data must be collected in combination withthe development of more robust theories. We did not intendour statement as a critique of the original data, but merely thefact that (as Kinnell says elsewhere) better data are requiredfor evaluating this or any other model of raindrop erosion. A

significant limitation here is the methodologies available formeasuring transport in very shallow flows. Kinnell’s modelmay provide useful approximations for models that employsediment concentration, which was explicitly specified asthe reason for developing the equations (Kinnell, 1993,p. 1099). However, as we noted in Wainwright et al. (2008),we have reason to doubt the general validity of such modelsand therefore it would be inappropriate to take our analysisof potential alternative detachment models further, includingthis one.

References

Kinnell PIA. 1991. The effect of flow depth on sediment transportinduced by raindrops impacting shallow flows. Transactions of theAmerican Society of Agricultural Engineers 34: 161–168.

Kinnell PIA. 1993. Sediment concentrations resulting from flowdepth/drop size interactions in shallow overland flow. Transactionsof the American Society of Agricultural Engineers 36: 1099–1013.

Kinnell PIA. 2005. Raindrop-impact-induced erosion processes andprediction: a review. Hydrological Processes 19: 2815–2844.

Kinnell PIA, Wood JT. 1992. Isolating erosivity and erodibilitycomponents in erosion by rain-impacted flow. Transactions of theAmerican Society of Agricultural Engineers 35: 201–205.

Moss AJ, Green P. 1983. Movement of solids in air and water byraindrop impact–effects of drop-size and water-depth variations.Australian Journal of Soil Research 21: 257–269.

Wainwright J, Parsons AJ, Müller EN, Brazier RE, Powell DM, Fenti B.2008. A transport-distance approach to scaling erosion rates:III. Evaluating scaling characteristics of MAHLERAN. Earth SurfaceProcesses and Landforms 33: 1113–1128.