Exploring topographic and scaling properties of landscapes by dimensionally analyzing stream-power incision and linear diffusion landscape evolution models

Abstract

In this PhD project, I dimensionally analyzed a frequently used landscape evolution model (LEM) and I derived geomorphological interpretations of the characteristic scales and dimensionless numbers that were revealed by the analysis. I studied an LEM with terms of stream-power incision, linear diffusion, and uniform uplift, parameterized by an incision coefficient K, a diffusion coefficient D, and an uplift rate U, respectively. I also examined a version of the LEM that included an incision threshold θ. This model is mostly applicable to soil-mantled landscapes with gentle slopes, and it can reproduce salient features of fluvial landscapes such as ridge-and-valley topography and dendritic valley networks. The dimensional analysis of the LEM used characteristic scales that were defined in a novel way, based on the combination of two main premises. First, lengths and heights were assumed to be dimensionally distinct; thus, the LEM was assumed to have three fundamental dimensions, namely, length L, height H, and time T. Second, I defined the corresponding characteristic scales of length, height, and time (lc, hc, and tc) solely in terms of model parameters, not in terms of extrinsic properties of landscapes, such as the domain size or the total relief. Non-dimensionalizing the LEM with the three parameter-dependent characteristic scales removes three parameter-related degrees of freedom. Thus, in the version of the LEM without incision threshold, which includes three parameters, I obtained a dimensionless governing equation without any parameters. In the LEM version with an incision threshold, I eliminated three out of four parameter-related degrees of freedom and, thus, obtained a dimensionless equation with one parameter. This dimensionless parameter quantifies the importance of the incision threshold θ relative to the incision coefficient K and the uplift rate U, and I termed it the incision-threshold number Nθ. The dimensionless governing equation without incision threshold has only one solution for any given combination of boundary and initial conditions, because it has no parameters to be adjusted. For the dimensional space, this implies that if we rescale simulation domains, initial conditions, and time steps by the characteristic scales of lc, hc, and tc, respectively, then we can obtain solutions that evolve geometrically and temporally similarly regardless of the values of their parameters. Thus, we can explore the entire parameter space of the LEM with a single simulation per set of boundary and initial conditions, incurring significant savings of computational resources. The similarity of landscapes implies that, for two landscapes with different parameters, all corresponding distances, elevation differences, and durations will scale with lc, hc, and tc, respectively. Likewise, any landscape metric whose dimensions combine L, H, and T will scale with a corresponding combination of lc, hc, and tc. Based on this fact, we can use the definitions of characteristic scales, which depend only on model parameters, to straightforwardly deduce scaling relations between landscape metrics and the LEM parameters K, D, and U. These relations can be used to verify theoretical results and to test whether numerical results are influenced by boundary or resolution effects. The dimensionless form of the LEM with incision threshold includes one parameter, the incision-threshold number Nθ. Therefore, the four-dimensional space of the parameters K, D, U, and θ collapses to the one-dimensional parameter space of Nθ, and simulations with equal Nθ values can evolve similarly if they are rescaled by lc, hc, and tc. This also leads to computational savings. To interpret the characteristic height hc, I defined diffusion- and incision-specific height scales that quantify how the strength of these processes varies within a landscape. Specifically, these height scales give the total elevation change due to each process during one unit of characteristic time tc. I found that these height scales are equal to hc at special landscape points, specifically, at drainage divides and at hillslope–valley transitions. Furthermore, using these height scales, I derived a steady-state relationship between curvature and the steepness index. This relationship plots as a straight line, whose slope and intercepts depend on the characteristic length and height lc and hc, and can be viewed as a counterpart of the slope–area relationship for the case of landscapes that include diffusion. The curvature–steepness-index relationship is a quantitative prediction that one could use to test whether a landscape follows the incision¬–diffusion LEM and to estimate the values of lc and hc. Furthermore, the response of topography to changes in the values of LEM parameters can be visually expressed as simple shifts and rotations of the curvature–steepness-index line. To interpret the characteristic scales of length and time lc and tc, I examined the competition between advective and diffusive propagation of elevation perturbation. I found that lc and tc characterize points where these two modes of propagation are equally strong. To quantify the relative strength of advection versus diffusion, I used a Péclet number with a modified definition that can account for the scaling of basin length with basin area and, thus, for the convergence or divergence of topography. These interpretations of lc, hc, and tc remain valid when an incision threshold θ is included in the LEM. However, the curvature–steepness-index relationship and the Péclet number definition, which were used to derive these interpretations, need to be modified and to include the incision-threshold number Nθ. Consequently, an interpretation of Nθ is that it quantifies the effect of the incision threshold on the competition between processes. The incision-threshold number Nθ characterizes a landscape as a whole and can be used to compare the relative importance of the incision threshold θ in different landscapes. Within a given landscape, however, the relative influence of θ varies. I found that this influence can be quantified by the dimensionless ratio of θ to the steepness index, which I termed the fractional reduction in the rate of incision. I found that this dimensionless quantity can also quantify how the influence of the incision threshold on the Péclet number varies across points in a given landscape. Furthermore, plots of this quantity can be used together with curvature–steepness-index lines to visualize the effects of θ on landscapes. The results of this work show that the characteristic scales lc, hc, and tc, along with the incision-threshold number Nθ for the case that includes an incision threshold, are fundamental properties of landscapes that follow the LEM and shed light on topographic and scaling properties of these landscapes. I expect that additional properties of such landscapes can be examined using Nθ and the characteristic scales lc, hc, and tc. Furthermore, studies of landscapes that follow other LEMs might benefit from the results of this work by adopting the way in which I defined and interpreted characteristic scales and dimensionless numbers.