AUTHOR=Hendrick Martin , Renard Philippe 

TITLE=Fractal Dimension, Walk Dimension and Conductivity Exponent of Karst Networks around Tulum

JOURNAL=Frontiers in Physics

VOLUME=4

YEAR=2016

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2016.00027

DOI=10.3389/fphy.2016.00027

ISSN=2296-424X

ABSTRACT=<p>Understanding the complex structure of karst networks is a challenge. In this work, we characterize the fractal properties of some of the largest coastal karst network systems in the world. They are located near the town of Tulum (Quintana Roo, Mexico). Their fractal dimension <italic>d</italic><sub><italic>f</italic></sub>, conductivity exponent <inline-formula><mml:math id="M1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover accent="true"><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mo>~</mml:mo></mml:mover></mml:math></inline-formula> and walk dimension <italic>d</italic><sub><italic>w</italic></sub> are estimated using real space renormalization and numerical simulations. We obtain the following values for these exponents: <italic>d</italic><sub><italic>f</italic></sub> ≈ 1.5, <italic>d</italic><sub><italic>w</italic></sub> ≈ 2.4, <inline-formula><mml:math id="M2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover accent="true"><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mo>~</mml:mo></mml:mover><mml:mo>≈</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>9</mml:mn></mml:math></inline-formula>. We observe that the Einstein relation holds for these structures <inline-formula><mml:math id="M3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover accent="true"><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mo>~</mml:mo></mml:mover><mml:mo>≈</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:msub><mml:mtext> </mml:mtext><mml:mo>+</mml:mo><mml:mtext> </mml:mtext><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>w</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>. These results indicate that coastal karst networks can be considered as critical systems and this provides some foundations to model them within this framework.</p>