Topographic roughness is a popular yet ambiguous metric used in surface process research for many applications that indicates something about the variation of topography over specified measurement intervals. In soil- and sediment-mantled settings topographic roughness may be framed as a competition between roughening and smoothing processes. In many cases, roughening processes may be specific eco-geo-hydromorphic events like shrub deaths, tree uprooting, river avulsions, or impact craters. The smoothing processes are all geomorphic processes that operate at smaller scales and tend to drive a diffusive evolution of the surface. In this article, we present a generalized theory that explains topographic roughness as an emergent property of geomorphic systems (semi-arid plains, forests, alluvial fans, heavily bombarded surfaces) that are periodically shocked by an addition of roughness which subsequently decays due to the action of all small scale, creep-like processes. We demonstrate theory for the examples listed above, but also illustrate that there is a continuum of topographic forms that the roughening process may take on so that the theory is broadly applicable. Furthermore, we demonstrate how our theory applies to any geomorphic feature that can be described as a pit or mound, pit-mound couplet, or mound-pit-mound complex.

Tree uprooting is an observable and consequential process that suddenly moves soil downslope, inverts the soil column, and roughens the surface with pit-mound topography. Quantifying fluxes due to tree throw is complicated by its stochastic nature and estimation requires averaging over a large area or long time. Here, we develop theory that leads to a dimensionless metric directly measurable from high resolution topographic data. The theory explains the flux and topographic roughness as a function of tree throw production and decay rate by creep-like processes. We then form a dimensionless variable that is the ratio of fluxes due to three throw versus creep-like processes. Applying the theory to hillslopes in Southern Indiana, we find that tree throw accounts for 10 to 20\% of the hillslope sediment flux. The theoretical and observational findings provide a framework and important constraints on quantifying Critical Zone function from topographic parameters such as roughness.