PRINCIPAL RESULT ON THE NUMERICAL SIMULATION OF THE

3. NUMERICAL SIMULATION OF THE COLLECTIVE SHAPE EVOLUTION OF PEBBLES

3.2. CASE STUDY: WILLIAMS RIVER

3.2.6. PRINCIPAL RESULT ON THE NUMERICAL SIMULATION OF THE

87 together, regardless of the sampling site), then we ordered the particles by size and averaged them in this order so that each mean corresponds to approximately 300 particles. Results are shown in the y1–y2 plane in Figure 3.8b, numbers beside the data points refer to the mean size. The similar U–turn on this diagram (at an average size of 54.78 at the turning point) suggests that our simulation is realistic, i.e. particle size influences the abrasion process in the real river as well.

3.2.5.4. Future plans

In our current numerical simulation, we fitted two scalar parameters which reproduce the complete downstream evolution of shape and size distributions in the river and we justified our parameters based on physical considerations. However, our future plan is to couple our model with existing transport models like the widely-used concept of Shields stress (e.g. Frings 2004). Also, pure size-selective transport could be directly implemented in the model (e.g. Cui et al. 1996; Hoey and Ferguson 1994). Our final goal is to be able to predict parameter values, based on measurable field variables and/or laboratory experiments. Thus, we hope that our numerical model based on the box equations, in combination with existing transport models, can provide a framework for future shape and size evolution studies in various sedimentary environments.

Our other future plan is to understand shape evolution during different abrasion processes considering the new, equilibrium-based classification system, presented in Chapter 2. As every shape can be encoded by natural numbers (the numbers of equilibrium points), the change of this code on an individual pebble can be followed up with the appropriate mathematical and technical tools in well-controlled laboratory experiments as well as in a suitable abrasion model. Our goal is to understand the changes in the morphology through the changes of this code, and to assign different code sequences to different abrasion processes.

3.2.6. PRINCIPAL RESULT ON THE NUMERICAL SIMULATION OF THE

PRINCIPAL RESULT 4

(Relevant publication: Szabó, Fityus and Domokos 2013, under review)

Based on the box equations (Domokos and Gibbons 2012), I developed a flexible numerical model for simulating the collective shape and size evolution of large particle populations in different sedimentary environments. Using this model with only two, physically-based parameters, I reproduced the downstream evolution of pebble shape and size distributions measured along the Williams River in the 12 Australian pebble samples. Based on this simulation, which was run on the BME Supercomputer, the following statements can be made:

4.1. I complemented the stochastic box equations (Domokos and Gibbons 2012) by considering fragment production during collisional abrasion. By adding the fragments to the particle population, the abrasive effect of sand-size particles, which turned out to be important in the Williams River, could be directly simulated.

4.2. While most researchers explain the well-known exponential downstream fining in grain size, observed in numerous gravel-bed rivers, by the effect of size-selective transport, through the example of the Williams River I demonstrated that abrasion is efficient enough to produce this size decrease for small diminution coefficients.

4.3. I verified that the phenomenon predicted analytically by Domokos and Gibbons (2012), in which a pebble that first moves away from the sphere, turns back and finally converges to the sphere, is demonstrable in a real sedimentary environment as well.

SUMMARY AND PRINCIPAL RESULTS

The aim of this dissertation was twofold. Firstly, I applied a new, mechanics-based classification system (Várkonyi and Domokos 2006a) called primary equilibrium classification to describe the morphology of pebbles. I demonstrated the practical applicability of the system, as well as its close connection to the widely-used Zingg classification relying on length measurements (Principal Result 1). Based on the measurements of 9 different experimenters, I also verified that the hand experiments applied to count equilibrium points are consistent and reliable (Principal Result 2). In addition, I refined the primary equilibrium classification, by dividing the primary classes into secondary equilibrium classes, based on the topology of the so called Morse-Smale complex (Principal Result 3). We proved that all secondary equilibrium classes are non-empty (Domokos, Lángi and Szabó 2013). Secondly, I presented a new numerical model describing the collective shape evolution of large pebble populations, based on the recent theoretical result called the box equations (Domokos and Gibbons 2012), and I applied the simulation to the Williams River located in the Hunter Valley, Australia (Principal Result 4).

Throughout my principal results, statistical data of 3 pebble sample collections will be referred. These are:

1. The 24 European pebble samples

24 pebble samples collected from 11 European locations, each sample consists of 50 pebbles, meaning 1200 pebbles altogether. Samples correspond to different geological and geomorphological settings, so they represent various sedimentary environments, abrasion and transport processes and pebble lithology. I measured the axis lengths a>b>c on the pebbles and counted the stable and unstable equilibrium points. Detailed data on these samples can be found in Table 2.7 and Appendix A.

4

90 2. The 300 scanned pebbles

300 pebbles from the 24 European samples were scanned with a 3D laser scanner: 40-40 pebbles from samples T1, A1, K1, K3 and K4, 50-50 pebbles from samples N3 and N4. Detailed data on them can be found in Appendix A and B.

3. The 12 Australian pebble samples

12 pebble samples were collected along the Williams River, Hunter Valley, Australia. Altogether 1626 basalt pebbles were collected, average sample size is around 140. A map on the sampling places and a description of the sampling method can be found in Subsections 3.2.1 and 3.2.2. I measured the axis lengths a>b>c on the pebbles. Detailed data on the samples can be found in Appendix C.

PRINCIPAL RESULT 1

(Relevant publications: Domokos et al. 2010; Szabó and Domokos 2010; Domokos, Sipos and Szabó 2012)

Based on the primary equilibrium classes (Várkonyi and Domokos 2006a), I proposed a simplified classification (E-classification) to categorize pebble shapes. Relying on the statistical results of the 24 European pebble samples, I compared E-classes to the classical Zingg classes so that I generalized the Zingg system by introducing a parameter p separating the classes from each other. Based on the statistical results, the following statements can be made:

1.1. The optimal choice of p, at which the agreement between the two classifications is the largest, is not universal: for well-rounded pebbles it is higher (between 0.75 and 0.89), for angular shapes it is smaller (between 0.55 and 0.75).

1.2. The best agreement between E-classes and the generalized Zingg classes is 80%

on average (standard deviation: 10%), i.e. most of the information contained in the Zingg classification can be extracted from the simplified equilibrium classification.

1.3. I established a strong (R²=0.85) logarithmic relationship between the mean number of stable equilibria and the flatness (mean value of axis ratio c/b) of pebbles. This relationship also verifies that the number of equilibrium points is closely connected to the overall shape of pebbles.

PRINCIPAL RESULT 2

(Relevant publications: Domokos, Sipos and Szabó 2012; Domokos, Lángi and Szabó 2012)

Based on the algorithm identifying micro- and macro-equilibria (Domokos, Sipos and Szabó 2012) and data of the 300 scanned pebbles, I simulated the hand experiments on the BME Supercomputer, based on the hypothesis that the scalar parameter μ₀ introduced in the algorithm is a measure of the inaccuracy of the experimenting person during counting equilibrium points. By comparing the results of computer experiments to hand experiments, the following statements can be made, which verify the reliability of hand experiments:

2.1. Based on hand experiments of 9 different people, μ₀ values are typically higher for unstable equilibria (mean value: 0.0082) than for stable equilibria (mean value: 0.0016).

2.2. Parameter μ₀ varies in a narrow range for the same experimenter and also, the results of hand experiments can be modeled well (R²=0.74–0.97) by μ₀ (the mean value of μ₀). Thus, hand experiments are consistent and μ₀ describes the inaccuracy of the experimenting person well.

2.3. Based on hand experiments of 9 different people, μ₀ changes only slightly between different experimenting persons (mean value: 0.0049, standard deviation: 0.0019). Therefore, the number of stable and unstable equilibria counted in hand experiments varies also only slightly among different people: the standard deviation is very small (0.02) for small number of equilibria (2.06) and is acceptable (0.46) for larger number of equilibria (3.49) as well.

PRINCIPAL RESULT 3

(Relevant publication: Domokos, Lángi and Szabó 2013, under review)

I refined the primary equilibrium classification (Várkonyi and Domokos 2006a) by defining secondary classes, based on the topology of the Morse-Smale complex. Every Morse-Smale complex associated with a homogeneous, convex, rigid body can be represented by a 2-colored plane quadrangulation (Dong et al. 2006), we denote this graph class by QQQQ. I showed that class QQQQ can be generated inductively by a combinatorial operation called vertex splitting from the stem-graph P2 (the path graph with 2 vertices). With this, I generalized the result of Bagatelj (1989) and Negami and Nakamoto (1993), who showed the same for simple plane quadrangulations without

92 coloring. Also, this combinatorial result is a first building block in the proof of a more complex theorem (Domokos, Lángi and Szabó 2013). The theorem states that all secondary classes are non-empty, i.e. for every graph in QQQQ, there exists a corresponding homogeneous, rigid, convex solid.

PRINCIPAL RESULT 4

(Relevant publication: Szabó, Fityus and Domokos 2013, under review)

ACKNOWLEDGEMENTS

First of all, I would like to express my sincere gratitude to my supervisor, Gábor Domokos. He provided me the opportunity to work in a lively and stimulating research group, while he also encouraged and helped me to determine the direction of my future career. His continuous support and advices, the countless hours spent on discussions with me and his generosity in sharing his new ideas were invaluable for me during the last few years.

I am also grateful to my closest colleagues, András Sipos and Zsolt Lángi. They not only shared their expertise in numerical analysis and geometry with me, but I could also learn the benefits and challenges of collaborative working during our joint projects.

I am indebted to Stephen Fityus for his tireless work during the three-week field study under his supervision in Australia. His thorough plans, precision and encouragement helped us to perform a considerable amount of work during my short visit. Also, his and his family's care and generosity made my Australian experience unforgettable.

I gratefully thank János Geiger, Gary W. Gibbons, Douglas J. Jerolmack, Sándor Józsa, Zoltán Unger, Ákos Török and Péter Várkonyi for their valuable help, comments and/or suggestions on parts of this work, Richárd Kápolnai for sharing his knowledge on graphs with me and John Gibson for his kind help in the Australian field work. I also thank the staff of the Department for their support and for replacing me while I was absent in Australia. I would like to thank György Károlyi and Ákos Török for their helpful reviews and constructive comments during the ‘in-house’ defense of this work.

Also, I appreciate the many valuable remarks during the in-house discussion, especially the suggestions of Gábor Etesi, János Haas, Zsolt Gáspár and Orsolya Sztanó.

Last but not least, I am beholden to my friend Dóra Debkowski for her careful reading of the manuscript and for her continuous encouragement, especially in the last few months.

Pebbles scans were performed and the scanned point cloud pieces were fitted together by Mátyás Farkas, Csaba Haraszkó, Zsolt Hodosán, Zoltán Iváncsics, Benedek Kiss, Gábor Kovács, Tibor Nagy and Zsófia Salát. Equilibrium points on the 300 scanned pebbles were counted in hand experiments by Sarolta Bodor, Zita Csörge, Györgyi Dalmadi, Zoltán Dobi, Eszter Holczer, Kata Kovács, Attila Szalai and Orsolya Zelenai.

Research was supported by OTKA grants 72146 and 104601, and by grant TÁMOP - 4.2.2.B-10/1–2010-0009.

PUBLICATIONS RELEVANT TO THE PRINCIPAL RESULTS

Szabó T., Fityus S., Domokos G. (2013) Abrasion model of downstream changes in grain shape and size along the Williams River, Australia. Under review

Domokos G., Lángi Z., Szabó T. (2013) The genealogy of convex solids. Under review.

Preprint: http://arxiv.org/abs/1204.5494, Apr 2012

Domokos G., Lángi Z., Szabó T. (2012) On the equilibria of finely discretized curves and surfaces. Monatshefte für Mathematik 168, 321–345

Domokos G., Sipos A.Á., Szabó T. (2012) The mechanics of rocking stones: equilibria on separated scales. Mathematical Geosciences 44, 71–89

Szabó T., Domokos G. (2010) A new classification system for pebble and crystal shapes based on static equilibrium points. Central European Geology, 53, 1–19

Domokos G., Sipos A.Á., Szabó T., Várkonyi P. (2010) Pebbles, shapes, and equilibria.

Mathematical Geosciences, 42, 29–47

OTHER PUBLICATIONS IN THE SUBJECT OF THE DISSERTATION

Kápolnai R., Domokos G., Szabó T. (2012) Generating spherical multiquadrangulations by restricted vertex splittings and the reducibility of equilibrium classes. Periodica Polytechnica Electrical Engineering, accepted for publication

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In document A MECHANICS-BASED PEBBLE SHAPE CLASSIFICATION SYSTEM AND THE NUMERICAL SIMULATION OF THE COLLECTIVE SHAPE EVOLUTION OF PEBBLES (Pldal 87-111)