Erodability

Geological and experimental studies have shown that the erosion of bedrock by rivers follows in first approach the following expression[7] known as the shear stress model of stream power erosion:

${\displaystyle {dz \over dt}=-K_{\tau }(\tau \ -\tau \ _{c})^{a}}$

where z is the riverbed elevation, t is time, K_\tau is the erodability, ${\displaystyle \tau }$ is the basal shear stress of the water flow, and a is an exponent. For a river channel with a slope S and a water depth D, ${\displaystyle \tau }$ can be expressed as:

${\displaystyle \tau \ =\rho \ gDS}$

Note that ${\displaystyle K_{\tau }}$ embeds not only mechanical properties inherent to the rock but also other factors unaccounted in the previous two equations, such as the availability of river tools (pebbles being dragged by the current) that actually produce the abrasion of the riverbed.

${\displaystyle K_{\tau }}$ can be measured in the lab for weak rocks, but river erosion rates in natural geological scenarios are often slower than 0.1 mm/yr, and therefore the river incision must be dated over periods longer than a few thousand years to make accurate measurements. K_e values range between 10⁻⁶ to 10⁺² m yr⁻¹ Pa^−1.5 for a=1.5 and 10⁻⁴ to 10⁺⁴ m yr⁻¹ Pa⁻¹ for a=1.[8] However, the hydrological conditions in these time scales are usually poorly constrained, impeding a good the quantification of D.

An alternative model for bedrock erosion is the unit stream power, which assumes that erosion rates are proportional to the potential energy loss of the water per unit area:

${\displaystyle {dz \over dt}=-K_{\omega }\ \omega \ }$

where ${\displaystyle K_{\omega }}$ is the erodability, and ${\displaystyle \omega }$ is the unit stream power, which is easily calculated as:

${\displaystyle \omega \ =\rho \ gQS/W}$

where Q is the water discharge of the river [m³/s], and W is the width of the river channel [m].

Relative differences in long-term erodability can be estimated by quantifying the erosion response under similar climatic and topographic conditions with different rock lithology.[9]