- Exner equation
Introduction
The Exner equation is a statement of
conservation of mass that applies tosediment in afluvial system such as ariver . [Paola, C., and V. R. Voller (2005), A generalized Exner equation for sediment mass balance, "J. Geophys. Res.", "110", F04014, doi:10.1029/2004JF000274.] It was developed by the Austrian meterologist and sedimentologistFelix Maria Exner , from whom it derives its name. [Parker, G. (2006), 1D Sediment Transport Morphodynamics with applications to Rivers and Turbidity Currents, Chapter 1, http://cee.uiuc.edu/people/parkerg/morphodynamics_e-book.htm.]The Equation
The Exner equation describes a
conservation of mass betweensediment in the bed andsediment in transport. It states that bed elevation increases (aggrades) proportionally to the amount of sediment that drops out of transport, and conversely decreases (degrades) proportionally to the amount of sediment that becomes entrained in the flow.Basic Equation
The equation states that the change in bed elevation, eta, over time, t, is equal to one over the grain packing density epsilon_o, times the negative
divergence of sedimentflux q_s through an infinitesimally smallcontrol volume .frac{partial eta}{partial t} = -frac{1}{epsilon_o} ablacdotmathbf{q_s}
Note that epsilon_o can also be expressed as 1-lambda_p), where lambda_p equals the bed
porosity .Good values of epsilon_o for natural systems range from 0.45 to 0.75. [Parker, G. (2006), 1D Sediment Transport Morphodynamics with applications to Rivers and Turbidity Currents, Chapter 4, http://cee.uiuc.edu/people/parkerg/morphodynamics_e-book.htm.] . 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 Wikipedia article
Spherical packing for more details).Often, for reasons of computational conveneience 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 x, as one is typically interested in the down-stream distribution of
erosion anddeposition though a river reach.frac{partial eta}{partial t} = -frac{1}{epsilon_o}frac{partialmathbf{q_s{partial x}
Including External Changes in Elevation
An additional form of the Exner equation adds a
subsidence term, sigma, to the mass-balance. This allows the absoluteelevation of the bed eta to be tracked over time in a situation in which it is being changed by outside influences, such astectonic or compression-related subsidence. In the convention of the following equation, sigma is positive with an increase in elevation over time and is negative with a decrease in elevation over time.frac{partial eta}{partial t} = -frac{1}{epsilon_o} ablacdotmathbf{q_s}+sigma
References
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