Forme et rugosité des surfaces karstiques. Conséquences pour une théorie spatiale et fractale de l'interface terrestre
Authors: Martin Philippe 
Published: November 2018

MORPHOLOGY AND ROUGHNESS OF KARSTIC SURFACES. CONTRIBUTION TO A SPATIAL AND FRACTAL THEORY OF KARST LANDSCAPES.
This text proposes an theoretical, hypothetical and speculative approach of the transformation of earth's surfaces. This reflection is based on the notion of otherness. Our approach uses two oppositions : levelled / roughness and karstic / non karstic .
The dimension of the roughness surfaces is understood between 2 and 3. The dimension of the surfaces of levelling is close to 2. Quantifications showed that massifs are limited by surfaces more or less irregular. In certain cases, the erosion transforms so a surface of levelling into rough surface. In that case initial shape is not preserved. The levellings on the karstic massifs (outliers often) seem better preserved (karstic immunity) than on the other rocks. This conservation would explain a weak value of the fractal dimension of air surfaces of karsts tested always with the same protocol (relation S∝P∆).
They were compared with the sutfaces of reliefs of basal complex. Three ideas summarise obtained results :
[I] The average of fractal dimensions of karsts are smaller than those of the relief of basal complex. [2] The dispersal of the mean values of surface of karst is lower to the dispersal of the mean values of basal complex.
[3] Distance between minimal and maximal values for karsts is much bigger than distance between minimal and maximal values for basal complex. To explain the weak roughness of karsts we made three hypotheses :
[a] These fragments would correspond to zones still not affected by the erosion (time problem).
[b] In such a system some changes on a plan would prevent changes on the another plan (spatial problem ).
[c] Initial shape is replaced by a similar shape (Platon 's Parménide).
The endokarst is described empirically and by analogy with the fractal model of SierpinskVs sponge as an unique surface infinitely folded up in a limited volume. So the growth of the karstic spaces in the endokarst , increases almost until the infinity, the size of the internal surface of the karst.
To find a theoretical base at the roughness and at the extreme size of these surfaces, we studied the report between the growth of a volume and the growth of the surface which limits this volume. Three theoretical models show that if sutfaces do not change, volume to be affected by unity of surface grows strongly.
Eroded volume depends on the size of the exposed surface.
If the eroded volume depends on the size of the exposed surface then time to erase a mountain could be, in theory, infinite. This is not acceptable because a massif can be erased in about 200 Ma. According to analogies with different morphogenesis (physical, biologic), we make the hypothesis that fractal character, of surfaces of the massifs corresponds to the necessity of increasing, as much as possible, the size of the surface subjected to the erosion so as to decrease the time of destruction of the relief.
This is coherent with the idea of a system far from the balance which tends to join the state of balance as quickly as possible by developing specific morphologies.
Distance between the relief and the lower limit of the potential of erosion is then introduced as a factor being able to explain the small roughness of high continental surfaces.
The reduction of the volume by erosion is cause (and not consequence ) of the decrease of the roughness. The surface can become less rough because volume decreases. The surface of levelling constitutes the final morphology which is transformed only very slowly. In this perspective the dynamics allows only the fulfillment of spatial rules.
In the case of the karst, because of the existence of the subterranean part of the karstic surface and its roughness, it is not useful that air part becomes very rough. Levellings would be preserved by geometrical uselessness to destroy them. They would not correspond to forms in respite as implies him the temporal analysis (hypothesis [a]), but to forms corresponding to a particular balance (hypothesis [b]) who would even be locally transformed (karstic levelling) into the same shape (hypothesis [c]).
This theoretical plan supplies with more an explanation on the visible contradiction between the speed of the karstic erosion and the durability of levellings.