BAAS: POTENTIAL ERRORS IN DUST EMISSION MODELS 1. Dust emission models and the propagation of a typographical error

Transcription

1 BAAS: POTENTIAL ERRORS IN DUST EMISSION MODELS 1 Dust emission models and the propagation of a typographical error Andreas C.W. Baas Department of Geography King s College London

2 BAAS: POTENTIAL ERRORS IN DUST EMISSION MODELS 2 Abstract This letter reviews the propagation of typographical errors in a frequently cited aeolian sand transport equation of White that is used in many recent dust emission models. The erroneous equations may have led to an underestimation of dust emissions by 25-50% as compared with the intended algorithm. The correct White model is, in fact, only a reformulation of an original transport model by Kawamura. The replication of the error in literature is, ironically, not critical since the incorrect formulation performs no better or worse than other traditional transport models that correspond poorly with empirical measurements. Actual field transport rates could be nearly an order of magnitude lower than predicted and more accurate and realistic transport equations need to be developed that can provide a firm basis for dust emission models. Index terms Aerosols and particles Planetary atmospheres Constituent sources and sinks Modeling Keywords - dust aerosols - dust emissions - aerosol climatology

3 BAAS: POTENTIAL ERRORS IN DUST EMISSION MODELS 3 Introduction The purpose of this letter is to alert the atmospheric dust research community to a potential error in some of the numerical modelling of surface dust emissions conducted over the past 10 years. It has long been established that the entrainment of dust into the atmosphere is primarily due to the impacts of saltating sand grains during aeolian sediment transport, dislodging and ejecting dust particles into the airflow, rather than a direct mobilisation of dust by the wind itself [Gillette 1977]. Most recent dust models therefore relate a vertical flux of surface dust emission to a standard horizontal flux of sand transport, driven by the shear velocity of the wind. In the majority of models this horizontal sand flux is quantified using a transport equation published by White [1979]. In the aeolian research community, however, it has been known for some time that this published equation contains an unfortunate typographical error, and that this error has subsequently propagated through the literature, as highlighted by Namikas and Sherman [1997]. The propagation and potential implications of this error are discussed here for three reasons. First, because emission estimates and circulation models of atmospheric dust on a local and global scale are highly relevant to informing mitigation strategies and to evaluating possible effects of climate change. Second, because the propagation of this error and further subsequent typographical faults may be responsible for discrepancies between various dust emission models. Third, because it should be recognised that the correct formulation of the White equation, as intended by White, is a rearrangement of a transport model introduced by Kawamura [1951].

4 BAAS: POTENTIAL ERRORS IN DUST EMISSION MODELS 4 The Kawamura model Ryûma Kawamura introduced his model in a study of sand movement by wind in Reports of the Institute of Science and Technology of the University of Tokyo, in Japanese [1951], later translated by the Hydraulic Engineering Laboratory at the University of California, Berkeley [1964]. The model is a force balance analysis on a control volume in the saltation layer, assuming a number of relationships between shear stresses and the velocities and geometry of the mean saltation trajectory. Transport rate under saturated conditions, q, is quantified as the product of the mass of sand grains falling onto a unit area per unit time, G 0, and the mean horizontal saltation path length, L: q = G0 L [1] Empirical measurements show that there is a linear relationship between G 0 and excess shear velocity above the entrainment threshold, (U - U t ), such that: ( U ) G 0 U t ρ [2] where: ρ is fluid density, U is shear velocity, and U t is threshold shear velocity. Analysis of the ejection and impact velocities of saltating grains suggests that: L ( U U ) 2 + t [3] g where: g is gravitational acceleration.

5 BAAS: POTENTIAL ERRORS IN DUST EMISSION MODELS 5 Substitution of equations [2] and [3] into [1] leads to the original Kawamura transport equation: ρ q = K + g ( U U )( U U ) 2 t t [4] where: K is an empirical constant, estimated by Kawamura at Kawamura considered [4] a more elegant transport equation than Bagnold s [1941], because, unlike the latter, the transport rate reduces to zero at the threshold shear velocity. White [1979] proposed comparing transport rates on Mars and Earth by considering equivalent U t /U ratios. This was achieved by extracting U from the bracketed ( ) terms in [4], leading to: ρ U U q = K U g U 3 1 t 1 + t U 2 [5] Based on his experimental data White estimated the constant K at The form of [5] was, however, not the formulation that appeared in print as White s equation 22: q ρ U 1 U U + U 2 = 3 t t K U 1 2 g [6, incorrect]

6 BAAS: POTENTIAL ERRORS IN DUST EMISSION MODELS 6 The incorrect formulation in [6] was clearly due to typographical error as White did cite the correct Kawamura equation in the text (White s equation 19), in the abstract, and in his figure 9. The later-cited White model was therefore never intended to be different from the Kawamura equation, as indeed White did not claim to have developed a fundamentally novel transport model. He rather presented the reformulation using the U t /U ratio as the central contribution of his paper, allowing a comparison of transport rates in different planetary atmospheres. Propagation of the erroneous White equation The subsequent dispersal in aeolian literature of the incorrect formulation [6] has been documented by Namikas and Sherman [1997]. The incorrect form (with further typographical errors) appeared in two influential textbooks on aeolian geomorphology [Pye and Tsoar 1990, equation 4.48; Lancaster 1995, equation 2.12]. Although Blumberg and Greeley [1993, equation 8] published and flagged the correct form [5], the case was further confounded when two different typographical errors appeared in separate places in Greeley et al. [1996, table 2, equation 4]. Namikas and Sherman reflected on several reasons for the propagation of the erroneous formulation. The incorrect equation is dimensionally balanced and is visually similar to the proper form of the Kawamura equation [5]. The incorrect equation predicts significantly smaller transport rates than the original Kawamura model. Because field measurements routinely yield much lower transport rates, the incorrect equation conforms more closely to empirical data and therefore seems appropriate. Namikas and Sherman aptly remarked that it is both frustrating and humbling that a mistaken expression appears to be as successful at predicting field transport rates as any of the painstakingly theoretical constructs currently available.

7 BAAS: POTENTIAL ERRORS IN DUST EMISSION MODELS 7 Dust models and the White equation Complications surrounding the White model have re-surfaced recently in global dust emission simulations. The incorrect formulation [6] seems to have been included in a Martian global dust model by Pankine and Ingersoll [2002, equation 19]. An incorrect version of the intended White equation [5] appears in the Dust Entrainment and Deposition (DEAD) model of Zender et al. [2003, equation 10], where the quadratic power of the second bracketed term in [5] has been omitted: q ρ U 1 U U + U = 3 t t K U 1 g [7, incorrect] It is not clear whether this is a typographical error or whether this wrong formulation is part of the numerical simulation model. At first glance, the formulation presented by Marticorena and Bergametti [1995, second equation 27] also appears to be mistaken, because of the switch of + and signs in the bracketed terms of [6]: q ρ U 1 + U U U 2 = 3 t t K U 1 2 g [8] This formulation is in fact algebraically equivalent to the original Kawamura equation [4] and indeed correct. The switch of signs in [8] may have been a fortuitous typographical error, however, since the authors restate the derivation of the Kawamura model [1-3] by White [1979, equations 11-17] in their first set of

8 BAAS: POTENTIAL ERRORS IN DUST EMISSION MODELS 8 equations and the formulation of [8] does not naturally follow from that sequence. Since the authors cite White as the source, the question remains whether the faulty model [6] has in fact been included in the numerical simulations, or whether the (coincidentally?) correct equation [8] is used. Note that there are duplicate equation numbers between 27 and 30 in this publication. Equation [8] reappears in the global dust production model of Lunt and Valdes [2002, equation 6] and also in the dust aerosol model by Tegen et al. [2002, equation 3]. Not all dust models have been affected by these potential complications. The original Kawamura equation [4] is correctly quoted in the sandblasting model of Alfaro et al. [1997, equation 5], although it is attributed to White. The correctly intended White equation [5] is cited in the Martian dust cycle model of Newman et al. [2002, equation 9] although they suggest that the equation was developed out of Bagnold s [1941] work and the same model was later used by Basu et al. [2004]. Implications The possible inclusion of erroneous sand transport equations in numerical dust emission models may have significant implications for predictions and calibrations, particularly since the work of Marticorena and Bergametti [1995] and Zender et al. [2003] has propagated through literature as an influential and popular template for other dust simulations. In these models the vertical dust emission flux is proportional to the horizontal sand flux and results and predictions may therefore be subject to errors of the same order as the erroneous formulations in [6] and [7]. Figure 1 shows the horizontal sand transport rate as a function of these two incorrect formulations, the original Kawamura equation [4=5=8], as well as Bagnold s [1941], and Lettau & Lettau s [1977] for comparison:

9 BAAS: POTENTIAL ERRORS IN DUST EMISSION MODELS 9 Bagnold: 0.5 ρ d 3 q = K B U if : U > U t g D [9] where: d is grain diameter, D is reference grain diameter (0.25 mm), and K B is a constant estimated by Bagnold at 1.8 for naturally graded dune sand. q = K ρ d g D 2 Lettau & Lettau: ( ) L 0.5 U U t U [10] where: K L is a constant estimated by Lettau & Lettau at 4.2. The threshold shear velocity is calculated according to Bagnold s [1936] standard model: ρ = s ρ t A g d ρ U [11] 0.5 where: ρ s is mineral density of sand, and A is a constant estimated by Bagnold at during saltation. Assuming a sand grain diameter d of 0.25 mm (medium fine sand), ρ = 1.23 kg m - 3, ρ s = 2650 kg m -3 and g = 9.81 m s -2, the threshold shear velocity used in evaluating [4, 6-7, 9-10] for figure 9 is 0.20 m s -1. Kawamura s estimated value of 2.78 is used for K in [4, 6-7]. The figure shows how the incorrect formulations [6, 7] underestimate aeolian sand transport considerably compared with Kawamura s [4],

10 BAAS: POTENTIAL ERRORS IN DUST EMISSION MODELS 10 depending on shear velocity for this grain size. The incorrect White equation [6] yields rates 50% lower than [4] at shear velocities just above threshold and rates lower by 38% at U = 0.6 m s -1. The incorrect [7] yields rates 49% lower than [4] just above threshold and lower by 25% at U = 0.6 m s -1. This implies that some dust models may be underestimating dust emissions by 25 to 50% as compared with the intended algorithm. Discussion The White model is frequently cited in the dust modelling literature and it is important that the convoluted history of the typographical error is recognised. White s [1979] paper has been cited 97 times, as reported in the ISI Science Citation Index, whereas the paper by Blumberg & Greeley [1993], flagging the correct equation [5], has been cited only 18 times and the alert by Namikas & Sherman [1997] just twice. Despite the potential underestimation in the dust emission models, the crux of the matter is that none of our current sand transport models correspond well with empirical field measurements [Sarre 1987, Arens 1996, Sherman et al. 1998, Baas & Sherman 2005]. Actual aeolian sand transport rates may be nearly an order of magnitude lower than predicted and even the established models [4, 9, 10] display a great variability amongst each other. There are several possible reasons for these discrepancies. Models are calibrated and verified in wind tunnels that can not properly mimic the great spatial and temporal variability of transport observed in the field [Baas 2003]. The models treat the saltation layer as a uniform blanket of moving grains responding to a uniform and steady shear velocity, whereas in reality most transport occurs during brief discreet events associated with turbulence in the boundary layer flow. Furthermore, empirical measurements in the field are hampered

11 BAAS: POTENTIAL ERRORS IN DUST EMISSION MODELS 11 by various instrumentation challenges, complicating calibration of models in natural environments [Sherman & Baas 2004]. Although the propagation of the typographical error in [6] is unfortunate, ironically it is not critical in this context, since the erroneous equation performs no better or worse than other traditional models. This episode tells us that we must develop better and more realistic sand transport predictors that can provide a meaningful basis for dust modelling efforts. Let s get to work.

12 BAAS: POTENTIAL ERRORS IN DUST EMISSION MODELS 12 Reference List Alfaro, S.C., Gaudichet, A., Gomes, L., and Maille, M. (1997) Modeling the size distribution of a soil aerosol produced by sandblasting, J. Geophys. Res., 102(D10), Arens, S.M. (1996) Rates of aeolian transport on a beach in a temperate humid climate, Geomorphology, 17, Baas, A.C.W. (2003) The Formation and Behavior of Aeolian Streamers, PhD Dissertation, 412 p., University of Southern California, Los Angeles. Baas, A.C.W. and Sherman, D.J. (2005) Spatio-temporal variability of aeolian sand transport in a coastal dune environment, J. Coast. Res., in press. Bagnold, R.A. (1936) The movement of desert sand, Proc. R. Soc. London Ser. A- Math. Phys. Eng. Sci., 157, Bagnold, R.A. (1941) The Physics of Blown Sand and Desert Dunes, Chapman and Hall, London. Basu, S. and Richardson, M.I. (2004) Simulation of the Martian dust cycle with the GFDL Mars GCM, J. Geophys. Res., 109, E11006, doi: /2004je Blumberg, D.G. and Greeley, R. (1993) Field studies of aerodynamic roughness length, J. Arid Environ., 25, Gillette, D.A. (1977) Fine particulate emissions due to wind erosion, Transactions of the American Society of Agricultural Engineers, 20, Greeley, R., Blumberg, D.G., and Williams, S.H. (1996) Field measurements of the flux and speed of wind-blown sand, Sedimentology, 43, Kawamura, R. (1951) Study of sand movement by wind. Reports of the Institute of Science and Technology, University of Tokyo, 5, (in Japanese); Translated

13 BAAS: POTENTIAL ERRORS IN DUST EMISSION MODELS 13 as Report HEL-2-8 (1964), 38 p., Hydraulic Engineering Laboratory, University of California, Berkeley. Lancaster, N. (1995) Geomorphology of Desert Dunes, Routledge, London. Lettau, K. and Lettau, H. (1977) Experimental and micrometeorological field studies of dune migration, in Exploring the World's Driest Climate, edited by K. Lettau and H. Lettau, University of Wisconsin, Madison. Lunt, D.J. and Valdes, P.J. (2002) The modern dust cycle: comparison of model results with observations and study of sensitivities, J. Geophys. Res., 107(D23), 4669, doi: /2002jd Marticorena, B. and Bergametti, G. (1995) Modeling the atmospheric dust cycle: 1. Design of a soil-derived dust emission scheme, J. Geophys. Res., 100(D8), Namikas, S.L. and Sherman, D.J. (1997) Predicting aeolian sand transport: revisiting the White model, Earth Surf. Process. Landf., 22, Newman, C.E., Lewis, S.R., and Read, P.L. (2002) Modeling the Martian dust cycle. 1. Representations of dust transport processes, J. Geophys. Res., 107(E12), 5123, doi: /2002je Pankine, A.A. and Ingersoll, A.P. (2002) Interannual variability of Martian global dust storms, Icarus, 155, Pye, K. and Tsoar, H. (1990) Aeolian Sand and Sand Dunes, Unwin Hyman, London. Sarre, R.D. (1987) Aeolian sand transport, Prog. Phys. Geogr., 11, Sherman, D.J. and Baas, A.C.W. (2004) Earth pulses in direct current, in: Geography and Technology, edited by S.D. Brunn, S.L. Cutter, and J.W. Harrington, Jr., pp , Kluwer, Dordrecht.

14 BAAS: POTENTIAL ERRORS IN DUST EMISSION MODELS 14 Sherman, D.J., Jackson, D.W.T., Namikas, S.L., and Wang, J. (1998) Wind-blown sand on beaches: an evaluation of models, Geomorphology, 22, Tegen, I., Harrison, S.P., Kohfeld, K., Prentice, I.C., Coe, M., and Heimann, M. (2002) Impact of vegetation and preferential source areas on global dust aerosols: Results from a model study, J. Geophys. Res., 107(D21), 4576, doi: /2001jd White, B.R. (1979) Soil transport by winds on Mars, J. Geophys. Res., 84(B9), Zender, C.S., Bian, H., and Newman, D. (2003) Mineral Dust Entrainment and Deposition (DEAD) model: description and 1990s dust climatology, J. Geophys. Res., 108(D14), 4416, doi: /2002jd

15 BAAS: POTENTIAL ERRORS IN DUST EMISSION MODELS 15 Figure captions Figure 1: Aeolian transport rates for medium fine sand as a function of shear velocity predicted by traditional models [4, 9-10] and two incorrectly formulated equations [6-7].

16 BAAS: POTENTIAL ERRORS IN DUST EMISSION MODELS 16 Figure 1: Aeolian transport rates for medium fine sand as a function of shear velocity predicted by traditional models [4, 9-10] and two incorrectly formulated equations [6-7].