From turbulence to landscapes: Logarithmic mean profiles in bounded complex systems

We show that similarly to the logarithmic mean-velocity profile in wall-bounded turbulence, the landscape topography presents an intermediate region with a logarithmic mean-elevation profile. Such profiles are present in complex topographies with channel branching and fractal river networks resulting from model simulation, controlled laboratory experiments, and natural landscapes. Dimensional and self-similarity arguments are used to corroborate this finding. We also tested the presence of logarithmic profiles in discrete, minimalist models of networks obtained from optimality principles (optimal channel networks) and directed percolation. The emergence of self-similar scaling appears as a robust outcome in dynamically different, but spatially bounded, complex systems, as a dimensional consequence of length-scale independence.

Figure 1

An example of the steady-state surface obtained from the numerical solution of Eqs. (1) and (2). The simulation domain is $l_x=700$ m by $l_y=100$ m and has 1-m grid spacing. The boundary condition at the edges is constant elevation $z=0$ m. On the $y\text{−}z$ plane, the mean-elevation profile (black line) and the ensemble of profiles (red lines) are shown. To minimize the effect of domain size along the $x$ axis for the calculation of mean-elevation profile we used the surface within $100\text{m}\le x\le 600\text{m}$ as highlighted by the dashed box.

Figure 2

The landscape surface and the mean-elevation profile for different ${\mathcal{C}}_{\mathcal{I}}$ and $m$. (a–d) The normalized elevation $z$ for ${\mathcal{C}}_{\mathcal{I}}=10$, ${10}^2$, ${10}^3$, and ${10}^4$ and $m=0.5$. The shaded segments at the boundaries are discarded for further extraction of mean-elevation profiles. (e–g) The average profile for a wide range of ${\mathcal{C}}_{\mathcal{I}}$ and $m=0.5$, $m=0.7$, and $m=0.9$. As ${\mathcal{C}}_{\mathcal{I}}$ increases (higher relative proportion of fluvial erosion to diffusive transport), the surface becomes more dissected with branching channels. This results in the evolution of the mean-elevation profile toward a flatter shape.

Figure 3

The logarithmic profile in numerical simulations. (a) The dimensionless profiles corresponding to $0\le y\le l_y/2$ in a semilog space for $m=0.7$. The dimensionless quantities are defined as $\eta =\frac{K{y}^{m+1}}{D}$ and $\phi =\frac{z}{z_{*}}$, with $z_{*}={\overline{z}}_{\max }$. Increasing ${\mathcal{C}}_{\mathcal{I}}$ leads to the emergence and further expansion of the logarithmic segment in the profiles. The dashed line is the fitted logarithmic line to the profile with ${\mathcal{C}}_{\mathcal{I}}={10}^5$ and the slope is reported as $\kappa$. (b) The relationship between $\kappa$ and $m$. The slope $\kappa$ (average of simulations with ${\mathcal{C}}_{\mathcal{I}}\ge {10}^4$) decreases monotonically with $m$, suggesting that higher values of $m$ result in flatter logarithmic profiles. The data points from a physical experiment are also shown and lie close to the results from the numerical simulation.

Figure 4

The logarithmic profile in the experimental and natural landscapes. (a) The specific catchment area $a$ for an experimental landscape, and (b) its mean-elevation profile represented by the nondimensional quantities $\eta$ and $\phi$. The fitted line to the logarithmic segment of the profile is shown by the red line with the slope reported as $\kappa \approx 0.12$. (c) The digital elevation model of a basin at the Calhoun Critical Zone in South Carolina, USA, and (d) its normalized mean-elevation profile which exhibits a logarithmic scaling with $\kappa \approx 0.11$.

Figure 5

The logarithmic profile in optimal channel networks (left) and directed percolation (right). Panels (a) and (b) show an optimal channel network (OCN) with $\gamma =0.5$ and a directed percolation (DP) network generated numerically in a 500 m by 100 m domain with 1-m grid spacing. The lines represent the connection between nodes (links) and their width is proportional to the drainage area $A$ at their end. Panels (b) and (e) show the constructed surface elevations from the OCN and DP configurations. Panels (c) and (d) show the mean-elevation profiles computed for $100\text{m}<x<400\text{m}$ in linear (inset) and logarithmic horizontal axis for $0\le y\le l_y/2$. The existence of a logarithmic scaling is evident as marked by the fitted lines in green.

Figure 6

The dependence of the slope of logarithmic scaling on the parameter $n$. (a) The logarithmic scaling persists for a range of $n$ and the slope $\kappa$ increases with higher $n$. (b, c) The nondimensionalized mean-elevation profiles for two cases with the same $m=0.9$ and ${\mathcal{C}}_{\mathcal{I}}={10}^5$, but different $n$. In both cases the logarithmic scaling is easily detectable.