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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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Wilber Berryhill is the title his parents gave him and he completely digs that name. Distributing production is exactly where her primary income comes from. Ohio is where her home is. As a lady what she truly likes is fashion and she's been doing it for fairly a whilst.
My homepage; online psychic