Fractal Measures of Sea, Lake, Strait, and Dam-Reserve Shores: Calculation, Differentiation, and Interpretation
aa r X i v : . [ phy s i c s . g e o - ph ] S e p Fractal Measures of Sea, Lake, Strait, and Dam-Reserve Shores:Calculation, Differentiation, and Interpretation
Dilara Yılmazer, A. Nihat Berker,
1, 2, 3 and Y¨ucel Yılmaz
4, 2 TEBIP High Performers Program, Board of Higher Education of Turkey,Istanbul University, Fatih, Istanbul 34452, Turkey Faculty of Engineering and Natural Sciences, Kadir Has University, Cibali, Istanbul 34083, Turkey Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Department of Geological Engineering, Istanbul University, Maslak, Istanbul 34469, Turkey
The fractal dimensions d f of the shore lines of the Mediterranean, the Aegean, the Black Sea, theBosphorus Straits (on both the Asian and European sides), the Van Lake, and the lake formed bythe Atat¨urk Dam have been calculated. Important distinctions have been found and explained. I. INTRODUCTION
The fractal dimension d f gives the amount of materialin an object as function of its linear extent: If the linearextent is changed by a factor of b , the amount of materialchanges by a factor of b d f . Thus, physically, the fractaldimension d f subsums important structural and histori-cal information on the object. We thus expect the fractaldimension of a shoreline (or any line boundary) to be be-tween 1 (a straigh line) and 2 (a curve compactly coveringa surface), and to reflect important information. We havethus calculated the shores of Turkey: The shores of theMediterranean, the Aegean, the Black Sea, the Bospho-rus Straits (on both the Asian and European sides), theVan Lake, and the lake formed by the Atat¨urk Dam. Aswe shall see below, we have indeed found distinctive re-sults, leading to cogent explanations and associations. II. METHOD
On a given shore line between two specific points, wewould expect L = lim G → N ( G ) · G, (1)where G is the length of the unit ruler used in the mea-surement, N ( G ) is the number of unit rulers spanningthe shore between the two points, and L is the actualshore distance between the two points. However, in hisclassic work on the border between Portugal and Spain,Richardson [1, 2] found that N ( G ) · G did not convergein the limit G →
0, but that M = lim G → N ( G ) · G d f (2)did converge. Subsequently, Mandelbrot [3] interpreted d f , generally a non-integer number, as the fractal dimen-sion of the shore line. Simply set, this is the consequenceof the shore line not being comprised, at any length scale,of consecutive small linear units.Substituting Eq.(1) into Eg.(2),log L = log M + (1 − d f ) log G. (3) Shores N , N , N d f Line Goodness RMediterranean 416,244,180 1.21 0.9489Aegean 391,234,170 1.20 0.9787Black Sea 224,144,106 1.08 0.9963Bosphorus Asia 96,61,43 1.16 0.9854Bosphorus Europe 85,56,41 1.05 0.9576Bosphorus Eur + Asia 181,117,84 1.11 0.9807Thrace 87,55,41 1.09 0.9453Van Lake 136,85,55 1.31 0.9492Atat¨urk Dam Lake 485,252,170 1.51 0.9914TABLE I. Calculated fractal dimensions d f of the shore linesof the Mediterranean, the Aegean, the Black Sea, the Bospho-rus Straits (on both the Asian and European sides), the VanLake, and the lake formed by the Atat¨urk Dam, using unitlengths of G , G , G = 1 , . , R shows the goodness of the linear fits, also seenin Fig. 2 The fractal dimension d f is found by fitting the slope ofthis function for varying G . III. APPLICATION: DISTINCTIVE FRACTALDIMENSIONS
We have calculated by this method to obtain the frac-tal dimensions of the different outer and inner shores ofTurkey. On maps of different sizes appropriate to theshore object, we have measured the shore lines of theMediterranean, the Aegean, the Black Sea, the Bospho-rus Straits (on both the Asian and European sides), theVan Lake, and the lake formed by the Atat¨urk Dam, us-ing unit lengths of G = 1 , . , and 2 cm.The Mediterranean was calculated from Data to theSyrian border. The Aegean was calculated from Dat¸cato C¸ anakkale. The Black Sea was calculated from theBulgarian border to the Georgian border. The totalityof the Van Lake and Atat¨urk Dam Lake shore lines werecalculated. In addition, the Thracian shore line was cal-culated the Greek border to Sedd¨ulbahir. The results aregiven in Fig. 2 and Table I. FIG. 1. Tectonic map of Anatolia and the surrounding regions showing the major faults (black lines) and the earthquakeepicenters (yellow circles). The white arrows indicate relative motions of the different regions of the Arabian, Anatolian, andAegean Plates. The abbreviations are LV for Lake Van, AD for the lake of the Atat¨urk High Dam, and B for the Bosphorus.(Modified after [6]).
IV. DISCUSSIONS: ORIGINS AND GEOLOGY
The goodness of the linear fits, as seen from Fig. 2 andthe last column of Table I, clearly shows the validity ofthe concept of fractal dimension, which indeed turns outto be more than 1 (a line) and less than 2 (a surface).Furthermore, our specific results lead to cogent explana-tions. The fractal dimension of the Atat¨urk Dam Lakeclearly stands out with the maximal value of d f = 1 . R = 0 . d f = 1 .
3. This is supported bythe difference in the morphotectonic patterns of theseregions as outlined in the following paragraphs.Anatolia is one of the most strongly deformed conti-nental regions of the World. This is manifested by twogeological features: 1-Morphology 2-Earthquakes (Fig.1).Therefore, the landforms are young, formed primarily af-ter the Late Miocene. The two mountain ranges, thePontide and the Taurus in the North and the South re-spectively lying along with the shores, rise steeply likea Wall and separate the Central Anatolian Plateau fromthe sea realm. The coastal regions are tectonically veryactive and display zigzagging patterns formed as a re-sult of the conjugated pairs of faults of medium (1-10 km) and big (10-100 km) scale [4–7]. The Lake Van onthe other hand represents an erosional flatland on theelevated Eastern Anatolian High Plateau, which is laterfilled with water when the broad valley floor was dammedby edifices of the young volcanoes, i.e., the Nemrut andKirkor volcanoes [4–7].All other fractal dimensions of the shore lines are about d f = 1 .
2. This consistency in itself is an important fact.Finally, one would wonder that the Bosphorus was alsothe result of the flooding of a meandering river [4–7],some 8,000 years ago, but does not show the high fractaldimension. The explanation could be that the Bosphorusis singularly lacking in important branches. Therefore,the Bosphorus represents an ancient meandering rivervalley which was flooded by the sea from the Black Seaabout 8000 years ago [4–7].
V. CONCLUSION
It is seen that fractal dimensions can easily yield im-portant classifications for shore lines.
ACKNOWLEDGMENTS
Support by the TEBIP High Performers Program ofthe Board of Higher Education of Turkey and by theAcademy of Sciences of Turkey (T ¨UBA) is gratefully ac-
FIG. 2. Logarithmic plot of the shore length L versus theunit ruler G . The goodness R (see Table I) of the linear fit,shown here, gives the validity of the fractal dimension d f .The value of the slope gives 1 − d f , also given in Table I. Theline fits are, from top to bottom on the left, for the Atat¨urkDam Lake, Mediterranean, Agean, Black Sea, Bosphorus Eu-rope and Asia, Van Lake, Bosphorus Asia, Bosphorus Europe,Thracian. Notice how the shores of the Atat¨urk Dam Lakeand (to a lesser extent) the Van Lake stand out by their slopeand, therefore, by their high fractal dimension d f , which hasa geological explanation. knowledged. [1] L. F. Richardson, The problem of contiguity: An appendixto statistics of deadly quarrels, General System Yearbook, , 139-187 (1961).[2] D. G. Tarboton, R. L. Bras, I. Rodriguez-Iturbe, The frac-tal nature of river networks, Water Resources Research, (8), 1317-1322 (1988).[3] B. Mandelbrot, How long is the coast of Britain? Statisti-cal self-similarity and fractional dimension, Science, NewSeries (3775), 636-638 (1967).[4] Y. Yılmaz, Y. G¨uner, F. S¸aro˘glu, Geology of the quater-nary volcanic centres of the east Anatolia, J. Volcanol.Geotherm. Res. , 173-210 (1998).[5] Y. Yılmaz, E. G¨oka¸san, A. Y. Erbay, Morphotectonic de-velopment of the Marmara region, Tectonophysics ,5170 (2010).[6] ˙I. C¸ emen and Y. Yılmaz, eds., Active global seismology-neotectonics and earthquake potential of the East-ern Mediterranean region, Geophysical Monograph , American Geophysical Union, Wiley Press, 306 p., ISBN:978-1-118-94498 (2017).[7] Y. Yılmaz, Morphotectonic development of Anatolia andthe surrounding regions, in Active global seismology-neotectonics and earthquake potential of the EasternMediterranean region, ˙I. C¸ emen and Y. Yılmaz, eds., Geo-physical Monograph225