Exner equation

The equation states that the change in bed elevation, ${\displaystyle \eta }$, over time, ${\displaystyle t}$, is equal to one over the grain packing density, ${\displaystyle \varepsilon _{o}}$, times the negative divergence of sediment flux, ${\displaystyle q_{s}}$.

${\displaystyle {\frac {\partial \eta }{\partial t}}=-{\frac {1}{\varepsilon _{o}}}\nabla \cdot \mathbf {q_{s}} }$

Note that ${\displaystyle \varepsilon _{o}}$ can also be expressed as ${\displaystyle (1-\lambda _{p})}$, where ${\displaystyle \lambda _{p}}$ equals the bed porosity.

Good values of ${\displaystyle \varepsilon _{o}}$ for natural systems range from 0.45 to 0.75.[3] A typical good value for spherical grains is 0.64, as given by random close packing. An upper bound for close-packed spherical grains is 0.74048. (See sphere packing for more details); this degree of packing is extremely improbable in natural systems, making random close packing the more realistic upper bound on grain packing density.

Often, for reasons of computational convenience and/or lack of data, the Exner equation is used in its one-dimensional form. This is generally done with respect to the down-stream direction ${\displaystyle x}$, as one is typically interested in the down-stream distribution of erosion and deposition though a river reach.

${\displaystyle {\frac {\partial \eta }{\partial t}}=-{\frac {1}{\varepsilon _{o}}}{\frac {\partial \mathbf {q_{s}} }{\partial x}}}$

An additional form of the Exner equation adds a subsidence term, ${\displaystyle \sigma }$, to the mass-balance. This allows the absolute elevation of the bed ${\displaystyle \eta }$ to be tracked over time in a situation in which it is being changed by outside influences, such as tectonic or compression-related subsidence (isostatic compression or rebound). In the convention of the following equation, ${\displaystyle \sigma }$ is positive with an increase in elevation over time and is negative with a decrease in elevation over time.

${\displaystyle {\frac {\partial \eta }{\partial t}}=-{\frac {1}{\varepsilon _{o}}}\nabla \cdot \mathbf {q_{s}} +\sigma }$