Exner equation

The Exner equation states that the change in bed elevation, ${\displaystyle \eta }$, over time, ${\displaystyle t}$, is equal to one over the grain packing density, ${\displaystyle {\epsilon }_o}$, times the negative divergence of sediment flux, ${\displaystyle {\mathbf{q}}_{\mathbf{s}}}$,

${\displaystyle \frac{\mathrm{\partial }\eta }{\mathrm{\partial }t}=-\frac{1}{{\epsilon }_o}\mathrm{\nabla }\cdot {\mathbf{q}}_{\mathbf{s}}}$

The packing density ${\displaystyle {\epsilon }_o}$ can also be expressed as ${\displaystyle (1-{\lambda }_p)}$, where ${\displaystyle {\lambda }_p}$ equals the bed porosity.

Good values of ${\displaystyle {\epsilon }_o}$ for natural systems range from 0.45 to 0.75. A typical 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 downstream direction ${\displaystyle x}$, as one is typically interested in the downstream distribution of erosion and deposition though a river reach

${\displaystyle \frac{\mathrm{\partial }\eta }{\mathrm{\partial }t}=-\frac{1}{{\epsilon }_o}\frac{\mathrm{\partial }{q}_s}{\mathrm{\partial }x}}$

where ${\displaystyle q_s}$ is scalar sediment flux in the downstream direction.