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| [[Image:PaolaExnerEqnRock.jpg|thumb|right|This rock belonging to University of Minnesota Professor of Geology Chris Paola is inscribed with the Exner equation.]]
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| The '''Exner equation''' is a statement of [[conservation of mass]] that applies to [[sediment]] in a [[fluvial]] system such as a [[river]].<ref>{{cite doi|10.1029/2004JF000274}}</ref> It was developed by the Austrian meteorologist and sedimentologist [[Felix Maria Exner]], from whom it derives its name.<ref>Parker, G. (2006), 1D Sediment Transport Morphodynamics with applications to Rivers and Turbidity Currents, Chapter 1, http://vtchl.uiuc.edu/people/parkerg/_private/e-bookPowerPoint/RTe-bookCh1IntroMorphodynamics.ppt.</ref>
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| ==The equation==
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| The Exner equation describes [[conservation of mass]] between sediment in the bed of a channel and [[sediment transport|sediment that is being transported]]. It states that bed elevation increases (the bed [[aggradation|aggrades]]) proportionally to the amount of sediment that drops out of transport, and conversely decreases (the bed [[degradation (geology)|degrades]]) proportionally to the amount of sediment that becomes entrained by the flow.
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| ===Basic equation===
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| The equation states that the change in bed elevation, <math>\eta</math>, over time, <math>t</math>, is equal to one over the grain packing density, <math>\varepsilon_o</math>, times the negative [[divergence]] of sediment [[flux]], <math>q_s</math>.
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| : <math>\frac{\partial \eta}{\partial t} = -\frac{1}{\varepsilon_o}\nabla\cdot\mathbf{q_s}</math>
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| Note that <math>\varepsilon_o</math> can also be expressed as <math>(1-\lambda_p)</math>, where <math>\lambda_p</math> equals the bed [[porosity]].
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| Good values of <math>\varepsilon_o</math> for natural systems range from 0.45 to 0.75.<ref>Parker, G. (2006), 1D Sediment Transport Morphodynamics with applications to Rivers and Turbidity Currents, Chapter 4, http://vtchl.uiuc.edu/people/parkerg/_private/e-bookPowerPoint/RTe-bookCh4ConservationBedSed.ppt.</ref> A typical good value for spherical grains is 0.64, as given by [[random close pack]]ing. 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.
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| 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 <math>x</math>, as one is typically interested in the down-stream distribution of [[erosion]] and [[Deposition (sediment)|deposition]] though a river reach.
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| : <math>\frac{\partial \eta}{\partial t} = -\frac{1}{\varepsilon_o}\frac{\partial\mathbf{q_s}}{\partial x}</math>
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| ===Including external changes in elevation===
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| An additional form of the Exner equation adds a [[subsidence]] term, <math>\sigma</math>, to the mass-balance. This allows the absolute [[elevation]] of the bed <math>\eta</math> to be tracked over time in a situation in which it is being changed by outside influences, such as [[tectonic]] or compression-related subsidence ([[Isostasy|isostatic compression or rebound]]). In the convention of the following equation, <math>\sigma</math> is positive with an increase in elevation over time and is negative with a decrease in elevation over time.
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| : <math>\frac{\partial \eta}{\partial t} = -\frac{1}{\varepsilon_o}\nabla\cdot\mathbf{q_s}+\sigma</math>
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| ==References==
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| {{reflist|1}}
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| {{river morphology}}
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| {{DEFAULTSORT:Exner Equation}}
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| [[Category:Geomorphology]]
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| [[Category:Sedimentology]]
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| [[Category:Partial differential equations]]
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| [[Category:Conservation equations]]
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