The Exner equation is a statement of conservation of mass

Conservation of mass

The law of conservation of mass, also known as the principle of mass/matter conservation, states that the mass of an isolated system will remain constant over time...

 that applies to sediment

Sediment

Sediment is naturally occurring material that is broken down by processes of weathering and erosion, and is subsequently transported by the action of fluids such as wind, water, or ice, and/or by the force of gravity acting on the particle itself....

 in a fluvial

Fluvial

Fluvial is used in geography and Earth science to refer to the processes associated with rivers and streams and the deposits and landforms created by them...

 system such as a river

River

A river is a natural watercourse, usually freshwater, flowing towards an ocean, a lake, a sea, or another river. In a few cases, a river simply flows into the ground or dries up completely before reaching another body of water. Small rivers may also be called by several other names, including...

. It was developed by the Austrian meteorologist and sedimentologist Felix Maria Exner, from whom it derives its name.

The equation

The Exner equation describes conservation of mass

Conservation of mass

The law of conservation of mass, also known as the principle of mass/matter conservation, states that the mass of an isolated system will remain constant over time...

 between sediment in the bed of a channel and sediment that is being transported

Sediment transport

Sediment transport is the movement of solid particles , typically due to a combination of the force of gravity acting on the sediment, and/or the movement of the fluid in which the sediment is entrained...

. It states that bed elevation increases (the bed aggrades

Aggradation

Aggradation is the term used in geology for the increase in land elevation due to the deposition of sediment. Aggradation occurs in areas in which the supply of sediment is greater than the amount of material that the system is able to transport...

) proportionally to the amount of sediment that drops out of transport, and conversely decreases (the bed degrades

Degradation (geology)

In geology, degradation refers to the lowering of a fluvial surface, such as a stream bed or floodplain, through erosional processes. It is the opposite of aggradation. Degradation is characteristic of channel networks in which either bedrock erosion is taking place, or in systems that are...

) proportionally to the amount of sediment that becomes entrained by the flow.

Basic equation

The equation states that the change in bed elevation, , over time, , is equal to one over the grain packing density, , times the negative divergence

Divergence

In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...

 of sediment flux

Flux

In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...

, .

Note that can also be expressed as , where equals the bed porosity

Porosity

Porosity or void fraction is a measure of the void spaces in a material, and is a fraction of the volume of voids over the total volume, between 0–1, or as a percentage between 0–100%...

.

Good values of for natural systems range from 0.45 to 0.75. A typical good value for spherical grains is 0.64, as given by random close pack

Random close pack

Random close packing is an empirical parameter used to characterize the maximum volume fraction of solid objects obtained when they are packed randomly. For example, when a solid container is filled with grain, shaking the container will reduce the volume taken up by the objects, thus allowing...

ing. An upper bound for close-packed spherical grains is 0.74048. (See sphere packing

Sphere packing

In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space...

 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 , as one is typically interested in the down-stream distribution of erosion

Erosion

Erosion is when materials are removed from the surface and changed into something else. It only works by hydraulic actions and transport of solids in the natural environment, and leads to the deposition of these materials elsewhere...

 and deposition though a river reach.

Including external changes in elevation

An additional form of the Exner equation adds a subsidence

Subsidence

Subsidence is the motion of a surface as it shifts downward relative to a datum such as sea-level. The opposite of subsidence is uplift, which results in an increase in elevation...

 term, , to the mass-balance. This allows the absolute elevation

Elevation

The elevation of a geographic location is its height above a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface ....

 of the bed 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

Isostasy

Isostasy is a term used in geology to refer to the state of gravitational equilibrium between the earth's lithosphere and asthenosphere such that the tectonic plates "float" at an elevation which depends on their thickness and density. This concept is invoked to explain how different topographic...

). In the convention of the following equation, is positive with an increase in elevation over time and is negative with a decrease in elevation over time.