Abstract. The scaling and similarity of fluvial landscapes can reveal fundamental
aspects of the physics driving their evolution. Here, we perform a
dimensional analysis of the governing equation of a widely used landscape
evolution model (LEM) that combines stream-power incision and linear
diffusion laws. Our analysis assumes that length and height are conceptually
distinct dimensions and uses characteristic scales that depend only on the
model parameters (incision coefficient, diffusion coefficient, and uplift
rate) rather than on the size of the domain or of landscape features. We use
previously defined characteristic scales of length, height, and time, but, for the
first time, we combine all three in a single analysis. Using these
characteristic scales, we non-dimensionalize the LEM such that it includes
only dimensionless variables and no parameters. This significantly simplifies
the LEM by removing all parameter-related degrees of freedom. The only
remaining degrees of freedom are in the boundary and initial conditions.
Thus, for any given set of dimensionless boundary and initial conditions, all
simulations, regardless of parameters, are just rescaled copies of each
other, both in steady state and throughout their evolution. Therefore, the
entire model parameter space can be explored by temporally and spatially
rescaling a single simulation. This is orders of magnitude faster than
performing multiple simulations to span multidimensional parameter spaces. The characteristic scales of length, height and time are geomorphologically interpretable;
they define relationships between topography and the relative strengths of
landscape-forming processes. The characteristic height scale specifies how
drainage areas and slopes must be related to curvatures for a landscape to be
in steady state and leads to methods for defining valleys, estimating model
parameters, and testing whether real topography follows the LEM. The
characteristic length scale is roughly equal to the scale of the transition
from diffusion-dominated to advection-dominated propagation of topographic
perturbations (e.g., knickpoints). We introduce a modified definition of the
landscape Péclet number, which quantifies the relative influence of
advective versus diffusive propagation of perturbations. Our Péclet
number definition can account for the scaling of basin length with basin
area, which depends on topographic convergence versus divergence.
Scaling and similarity of a stream-power incision and linear diffusion landscape evolution model