# Aeolian Ripple Migration and Associated Creep Transport Rates

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

_{*}/u

_{*t}(u

_{*}is shear velocity and u

_{*t}is fluid threshold shear velocity). Data from previous studies provided 34 usable cases from four wind tunnel experiments and 93 cases from two field experiments. Original data comprising 68 cases were obtained from sites in Ceará, Brazil (26) and California, USA (42), using combinations of sonic anemometry, sand traps, photogrammetry, and laser distance sensors and particle counters. The results supported earlier findings of distinctively different relationships between ${u}_{r}/\sqrt{gd}$ and u

_{*}/u

_{*t}for wind tunnel and field data. With our data, we could also estimate the contribution of creep transport associated with ripple migration to total transport rates. We calculated ripple-creep transport for 1 ≤ u

_{*}/u

_{*t}≤ 2.5 and found that this accounted for about 3.6% (standard deviation = 2.3%) of total transport.

## 1. Introduction

^{th}and early 20

^{th}Centuries [13,14,15,16,17,18]. King [19] appears to have been the first to perform (quasi) controlled experiments on aeolian ripple formation. Although there have been many attempts to understand the nature of ripple formation, there have been relatively few field or wind tunnel experiments designed to measure the geometries and kinematics of aeolian ripples, their response to changing wind conditions, and their contribution to total sand flux during transport events. The lack of data representing this fundamental boundary response motivated the research projects reported in this study, where the goals were: 1) to measure ripple lengths and heights, coincident shear velocities and total sand transport rates, 2) to incorporate our observations with those from other studies, and 3) to derive statistical models of key process-response relationships between wind speed, ripple migration, and sand transport. This work was focused on typical aeolian ripples with heights and lengths of the orders of 0.01 and 0.1 m, respectively, with mean grain sizes (d) between about 0.1 and 0.5 mm that we believe to be in equilibrium or near equilibrium with the wind. We did not consider megaripples [20,21,22] or transverse aeolian ridges [23,24,25], although both of these bed forms have received considerable and appropriate attention in the literature.

_{r}is the mass of sand transported by ripple movement, P is porosity, ρ

_{s}is sand density (2650 kgm

^{−3}), u

_{r}is the ripple migration rate, and H is ripple height. We assume P = 0.40 based on the data presented in Louge et al. ([53], Figure 3). Several have equated q

_{r}with the rate of transport via creep, q

_{c}[48]. More generally, we posit that q

_{r}≤ q

_{c}to allow for conditions where some creep may move through a system faster than the bedforms move. Previous studies have found the proportion of creep transport to total transport to vary across a range from 2% [54] to 57% [55] on Earth. Bridges et al. [50], based on image analysis, estimated that relationship to be about 25% for megaripple migration on Mars. In wind tunnel and field experiments, Bagnold [56,57] found the proportion to be 25%, and this value has been widely accepted as a reasonable approximation.

## 2. Previous Studies

^{−1}, but did not include a corresponding elevation. We averaged the migration rate measurements to obtain a single value, 0.18 ms

^{−1}, to use with the one wind speed and used the Law of the Wall to convert the wind speed measurement to an equivalent shear velocity, ${u}_{*}$:

^{−3}), and $d$ is sand grain size.

^{−1}, which we converted to shear velocities and threshold shear velocities using Equations (2)–(4).

^{−1}in winds of about 9 ms

^{−1}. They did not report the wind measurement height, so we again assumed that it was at 1.2 m above the sand surface and used this value for our assessments.

^{−1}measured at 0.10 m height) associated with ripples approaching equilibrium. For this study we used the averaged migration rate data (0.03 mms

^{−1}) from the longest duration (30 min) to represent conditions closest to equilibrium. Mean grain size in this study was 0.15 mm.

^{−1}in both environments. This is a well-constrained data set and the authors were the first, that we are aware of, to recognize a systematic bias in ripple response between wind tunnel and field experiments, although they did not develop arguments describing why this occurs. For a given shear velocity, they found that wind tunnel ripples typically migrate at rates 3 or more times faster than their natural counterparts. Their wind tunnel data were used by Durán et al. [29] in the development of a simulation model, discussed below.

## 3. Study Sites and Field Methods

^{−1}and blowing parallel with the long axis of the trough. The sands were dry and unconsolidated, with grain sizes averaging about 0.3 mm. The equilibrium ripple lengths were about 90–110 mm, and the heights were about 6–8 mm. More information about this field site can be found in [72] or [73].

_{50}= 0.40 mm). The equilibrium ripple wavelengths were about 0.1 m and the heights averaged about 8 mm. The field site and sediment characteristics are further described in [76].

## 4. Data Analysis

^{−1}s

^{−1}based on the trap size and sample duration. The sand samples were returned to the laboratory for ¼ phi dry sieving and the derivation of grain-size statistics using GRADISTAT [83].

## 5. Results

^{nd}and 24

^{th}caused those records to be discontinuous, with runs of 20 and 18 min on the 22

^{nd}and 21, 22, 24, and 9 min on the 24

^{th}. Photos were taken with an average temporal frequency of one minute or less: 98 images for the 21

^{st}, 71 on the 22

^{nd}, 167 for the 24

^{th}, and 114 on the 26

^{th}. From these records, we were able to extract 26 sample records of 8-min intervals each.

_{*}/u

_{*t}and u

_{r}/(gd)

^{1/2}and are segregated into wind tunnel and field measurements following precedent [68]. Linear regression analysis was used to test for statistically significant relationships. We found a wind tunnel relationship where ${u}_{r}/\sqrt{gd}=0.0088{u}_{*}/{u}_{*t}-0.0026$, with $n=41$, $R$

^{2}$=0.85$ and $P0.0001$. For the field data, the model was ${u}_{r}/\sqrt{gd}=0.004{u}_{*}/{u}_{*t}-0.0023$, with $n=17$4, $R$

^{2}$=0.52$ and $P0.0001$. We calculated 99.9% confidence limits for the two regression slopes, finding 0.0030 and 0.0049 for the field data, and 0.0069 and 0.011 for the wind tunnel data. Because the confidence bands do not overlap, these results support a conclusion that field conditions were not replicated at a one-to-one scale in the wind tunnels and the samples that we compared were not drawn from the same general population. We also performed a number of non-linear regression analyses to determine if there was an alternative distribution that improved the statistical strength of both models. The only improvement in the coefficients of determination was when logging u

_{*}/u

_{t}. For the wind tunnel data, R

^{2}= 0.85, and for the field data, R

^{2}= 0.57.

^{2}= 0.86 and P < 0.0001. These results, other than the offset, are essentially unchanged from the previous analysis. For the field data, the new model was ${u}_{r}/\sqrt{gd}=0.0040{u}_{*}/{u}_{*t}-0.0026$ with n = 161, R

^{2}= 0.63 and P < 0.0001, only modestly different. The 99.9% confidence limits for the regression slopes were 0.0064 and 0.0108 for the wind tunnel data and 0.0032 and 0.0048 for the field data. These uncertainty estimates are also similar to the estimates based on all data. We also performed regression analysis using just the new (Jericoacoara and Oceano) field data and obtained ${u}_{r}/\sqrt{gd}=0.0034{u}_{*}/{u}_{*t}-0.0030$ with n = 68, R

^{2}= 0.44 and P < 0.0001, suggesting that our data conformed to the relationship indicated by Sharp’s observations; i.e., the slope of the relationship was almost the same with all data and with just the Jericoacoara and Oceano data. As mentioned above, we performed non-linear regression analyses to determine if there was an alternative distribution that improved the statistical strength of both models for all quality-controlled data. Logging u

_{*}/u

_{*t}produced the only improvement for both data sets, increasing the coefficients of determination to 0.87 for the wind tunnel data and 0.68 for the field data.

_{*}/u

_{t}data were drawn from the same population and the probability in both cases was found to be less than 0.0001. It is not known if the differences can be ascribed to the different methods used at the two field sites, differences in grain sizes, some combination of the above, or other unknown factors. We plotted % transport against dimensionless shear velocity (Figure 5). Regression analysis suggests a weak positive relationship (${q}_{r}/q$ = 0.025 u

_{*}/u

_{*t}- 0.006; R

^{2}= 0.10, p < 0.01), although the basic linear regression requirement of homoscedasticity was violated by our data set, bringing the finding into question. That there is a trend is supported, however, by the differences in means of the two data sets.

## 6. Discussion

^{2}= 0.94); for field data, the relationship was substantially different, ${u}_{r}=0.039\left({u}_{*}-{u}_{t}\right){\left(\rho /{\rho}_{s}\right)}^{0.5}+0.061$ (R

^{2}= 0.70). We also used regression forced through the origin (i.e., forcing ${u}_{r}=0$ at ${u}_{*}={u}_{t}$) to represent the constraint that ripple migration should cease when the threshold of motion is not exceeded, and found that the relationship for wind tunnel observations was ${u}_{r}=0.10\left({u}_{*}-{u}_{t}\right){\left(\rho /{\rho}_{s}\right)}^{0.5}$ (R

^{2}= 0.90); for field data ${u}_{r}=0.046\left({u}_{*}-{u}_{t}\right){\left(\rho /{\rho}_{s}\right)}^{0.5}$ (R

^{2}= 0.66). In each instance, the laboratory and field relationships were statistically distinct. Our findings here, coupled with those of Andreotti et al. [68] demonstrate once again the scaling dichotomy and reemphasize the importance of using field data for the derivation of models. We found that wind tunnel data indicated ripple migration rates that were about two to three times faster than field equivalents. Because of the relatively constant scaling of ripple-migration creep transport with total transport, this also indicates that wind tunnel data predict sand transport rates that are two to three times greater than those produced by comparable conditions in nature, at least over the range of u

_{*}/u

_{t}from about 1 to 2.5. Scaling issues such as these may be especially important when using wind tunnel data to predict aeolian transport on extra-terrestrial surfaces, an approach that we recognize as substantially flawed.

_{r}≤ q

_{c}. The literature reports substantial variability in ${q}_{c}/q$. Bagnold [56,57], in wind tunnel and field experiments, and Willetts and Rice [7], in wind tunnel experiments, among others, found that creep comprised about 25% of total transport. From field data, Chepil [103] reported ${q}_{c}/q$ as 15.7% for fine dune sand. In wind tunnel experiments, Horikawa and Shen [104] found 20%, and they cited the results of Ishihara and Iwagaki [105] who found a range from 6.5 to 16.6% in field experiments. From their wind tunnel data, Dong et al. [106] estimated the creep transport fraction to range from about 4% to 29%, depending on grain size, and averaging 9%. In field studies, Nickling [107] found ${q}_{c}/q$ to be about 1.3%–3.6%, averaging 2.3%, and Kang et al. [108] indicated that the proportion of creep transport was about 4%–11%. Wang and Zheng [54] modeled creep flux and their results indicated creep fractions usually in the range of 2% to 14%. The wind tunnel research of Cheng et al. [55] indicated that ${q}_{c}/q$ decreased as shear velocity increased, falling from 57% at u

_{*}= 0.26 ms

^{−1}to 19% at u

_{*}= 0.56 ms

^{−1}. In studies of ripple migration and creep (reptation) transport on Mars, Bridges et al. [50] estimated ${q}_{c}/q$ at 25%. Yizhaq et al. [109] used the transport rate model of Kok and Renno [110] to estimate ${q}_{c}/q$ at Eagle Crater on Mars, with a result that the proportion should rise from about 3% to 7% over approximately the range of u

_{*}/u

_{*t}covered in this study. It has been indicated, however, that this scaling relationship has been found to be highly variable on Mars, ranging from 5% to 91% [111]. Our results compare favorably with those of other field studies on Earth, especially if we consider our ${q}_{r}/q$ to represent minimum estimates for ${q}_{c}/q$. Our regression results, supported by our difference of means tests, indicate a weak positive relationship with u

_{*}/u

_{*t}explaining, statistically, 14% of the variability in ${q}_{r}/q$ when we combine data from our two field experiments. The slope of the regression line, if it is reliable, matches closely with the Yizhaq et al. [109] model results. More field observations are needed to validate or refute the relationship.

## 7. Conclusions

- There is a linear relationship between shear velocity and ripple migration rate. This is evidenced in the regression analyses performed with the dimensionless variable pairs u
_{*}/u_{t}and u_{r}/(gd)^{1/2}, and for the dimensional variable pairs ${u}_{r}$ and $\left({u}_{*}-{u}_{t}\right){\left(\rho /{\rho}_{s}\right)}^{1/2}$. - The dimensionless and dimensional migration rates from wind tunnel studies are statistically distinct from those found in field experiments. For a given shear velocity, ripple migration rates in wind tunnels are about two to three times faster than those found in the field.
- The proportion of total sand transport that can be attributed to ripple migration (creep) averages 3.6% in our field studies, comparable to findings in other field studies. We found evidence that ${q}_{r}/q$ increases weakly with shear velocity.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Assessing Ripple Measurement Accuracy and Optimal Averaging Intervals for Linking Ripple Migration Rates with Shear Velocity Estimates

_{*}estimates. Averaging ripple migration rates over such a short period, however, had the effect of making small absolute measurement errors (one or two millimeters) into larger relative errors (10 – 20%, or more) of migration rates. Averaging over excessively long periods, on the other hand, had the effect of masking the dynamic response of the ripples to changes in wind conditions. Therefore, we tested the influence of shear velocity on ripple migration using averaging intervals of 5, 6, 7, 8, 9, 10, 11, and 12 minutes using linear regression analysis.

_{r}, or, following the parameterizations of [59],${u}_{*}/{u}_{*t}$ with u

_{r}/(gd)

^{0.5}. In both cases we found the optimal averaging interval to be eight minutes based on the coefficients of determination found with the eight intervals tested (Figure A1). When comparing just ${u}_{*}/{u}_{*t}$ with u

_{r}/(gd)

^{0.5}a ten minute interval produced results that were almost as strong as those from eight minutes, but the longer interval was not as strong with ${u}_{*}$ and u

_{r}, and it also reduced the number of samples available for analysis.

**Figure A1.**Comparison of coefficients of determination (R

^{2}) found using ${u}_{*}$ or ${u}_{*}/{u}_{*t}$ as independent variables with u

_{r}or u

_{r}/(gd)

^{0.5}as the respective dependent variables, for averaging intervals of 5 to 12 min. The 8 min interval produced the strongest results in both cases.

## Appendix B. Processing the Laser Distance Sensor (Sick Dt35) Time Series to Resolve Ripple Heights and Migration Rates

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**Figure 1.**Basic configuration for ripple migration study at Jericoacoara, Brazil, October 2008. Wind direction was right to left. The 0.80 × 0.80 m ripple reference grid is in the center of the photograph and at center right are the 0.40-m spaced pair of reference pins.

**Figure 2.**Configuration for ripple migration study at Oceano, USA, May–June 2015. Wind direction was right to left. The distance sensors are in the center of the photograph with orange cables.

**Figure 3.**Dimensionless migration rates for equilibrium ripples as a function of dimensionless shear velocity, segregated as wind tunnel (black symbols: n = 41; R

^{2}= 0.85) or field data (red symbols: n = 174; R

^{2}= 0.52).

**Figure 4.**These analyses included only data that met the quality control criteria. The dimensionless migration rates for equilibrium ripples as a function of dimensionless shear velocity, segregated as wind tunnel (black symbols: n = 34; R

^{2}= 0.86) or field data (red symbols: n = 161; R

^{2}= 0.63).

**Figure 5.**The percent of total transport carried by ripples compared to dimensionless shear velocity. The proportion was approximately independent of u

_{*}/u

_{*t}for the independent data sets but may indicate a weak dependence for the combined data based on the means of the two populations.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Sherman, D.J.; Zhang, P.; Martin, R.L.; Ellis, J.T.; Kok, J.F.; Farrell, E.J.; Li, B. Aeolian Ripple Migration and Associated Creep Transport Rates. *Geosciences* **2019**, *9*, 389.
https://doi.org/10.3390/geosciences9090389

**AMA Style**

Sherman DJ, Zhang P, Martin RL, Ellis JT, Kok JF, Farrell EJ, Li B. Aeolian Ripple Migration and Associated Creep Transport Rates. *Geosciences*. 2019; 9(9):389.
https://doi.org/10.3390/geosciences9090389

**Chicago/Turabian Style**

Sherman, Douglas J., Pei Zhang, Raleigh L. Martin, Jean T. Ellis, Jasper F. Kok, Eugene J. Farrell, and Bailiang Li. 2019. "Aeolian Ripple Migration and Associated Creep Transport Rates" *Geosciences* 9, no. 9: 389.
https://doi.org/10.3390/geosciences9090389