TheExner equation describesconservation of mass between sediment in the bed of a channel andsediment that is being transported.^[1] It states that bed elevation increases (the bedaggrades) proportionally to the amount of sediment that drops out of transport, and conversely decreases (the beddegrades) proportionally to the amount of sediment that becomes entrained by the flow.It was developed by the Austrian meteorologist and sedimentologistFelix Maria Exner, from whom it derives its name.^[2]It is typically applied tosediment in afluvial system such as ariver.

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 negativedivergence of sedimentflux,${\displaystyle {\mathbf{q}}_{\mathbf{s}}}$,

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

Note that${\displaystyle {\epsilon }_o}$ can also be expressed as${\displaystyle (1-{\lambda }_p)}$, where${\displaystyle {\lambda }_p}$ equals the bedporosity.

Good values of${\displaystyle {\epsilon }_o}$ for natural systems range from 0.45 to 0.75.^[3] A typical value for spherical grains is 0.64, as given byrandom close packing. An upper bound for close-packed spherical grains is 0.74048 (seesphere 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 oferosion anddeposition 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.

• Geomorphology
• Sedimentology
• Partial differential equations
• Conservation equations

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