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Heat pinch: Difference between revisions
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→Fick versus Fokker Planck
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== Fick versus Fokker Planck == | == Fick versus Fokker Planck == | ||
Part of the problem may be due to the use of a ''Fickian'' transport equation, | Part of the problem may be due to the use of a ''Fickian'' transport equation, the use of which is only recommended in homogeneous systems. | ||
In inhomogenous systems (such as fusion plasmas), the ''Fokker-Planck'' formulation seems more appropriate. | In inhomogenous systems (such as fusion plasmas), the ''Fokker-Planck'' formulation seems more appropriate. | ||
<ref>[[doi:10.1088/0741-3335/47/12B/S56|B.Ph. van Milligen, B.A. Carreras and R. | <ref>[[doi:10.1088/0741-3335/47/12B/S56|B.Ph. van Milligen, B.A. Carreras and R. Sánchez, ''The foundations of diffusion revisited'', Plasma Phys. Control. Fusion '''47''' (2005) B743–B754]]</ref> | ||
Within the Fokker-Planck formulation, the radial gradient of the heat conductivity produces a 'natural' heat pinch. | Within the Fokker-Planck formulation, the radial gradient of the heat conductivity produces a 'natural' heat pinch. | ||
By way of simplified example, one may write the Fokker-Planck heat transport equation | By way of simplified example, one may write the Fokker-Planck heat transport equation | ||
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I.e., the gradient of the heat conductivity produces a 'natural' pinch. | I.e., the gradient of the heat conductivity produces a 'natural' pinch. | ||
It should be noted that at the ''descriptive'' level, the Fick and Fokker-Planck equations are fully equivalent and capable of describing the same phenomena (with one exception<ref>[[doi:10.1088/0143-0807/26/5/023|B.Ph. van Milligen, P.D. Bons, B.A. Carreras and R. Sánchez, ''On the applicability of Fick’s law to diffusion in inhomogeneous systems'', Eur. J. Phys. '''26''' (2005) 913–925]]</ref>). | |||
It is only at the ''interpretative'' level that this difference appears: a 'pinch' may seem mysterious when using Fick's equation, while it appears 'natural' when using the Fokker-Planck equation. | |||
== Mesoscopic and microscopic mechanisms == | == Mesoscopic and microscopic mechanisms == | ||
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<ref>[[doi:10.1088/0029-5515/51/8/083006|Lu Wang and P.H. Diamond, ''Kinetic theory of the turbulent energy pinch in tokamak plasmas'', Nucl. Fusion '''51''' (2011) 083006]]</ref> | <ref>[[doi:10.1088/0029-5515/51/8/083006|Lu Wang and P.H. Diamond, ''Kinetic theory of the turbulent energy pinch in tokamak plasmas'', Nucl. Fusion '''51''' (2011) 083006]]</ref> | ||
At the mesoscopic level, critical gradients can provide strong inward transport | At the mesoscopic level, critical gradients can provide strong inward transport | ||
<ref>[[doi:10.1063/1.1763915|B.Ph. van Milligen, B.A. Carreras and R. | <ref>[[doi:10.1063/1.1763915|B.Ph. van Milligen, B.A. Carreras and R. Sánchez, ''Uphill transport and the probabilistic transport model'', Phys. Plasmas '''11''', 8 (2004) 3787]]</ref> | ||
== See also == | == See also == | ||