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<ref>Yu.N. Dnestrovsky et al, Sov. J. Plasma Phys. '''16''' (1990) 120</ref> | <ref>Yu.N. Dnestrovsky et al, Sov. J. Plasma Phys. '''16''' (1990) 120</ref> | ||
The resulting (stiff) profiles are known as ''canonical'' profiles. | The resulting (stiff) profiles are known as ''canonical'' profiles. | ||
<ref> | <ref>Yu.N. Dnestrovsky et al, ''Canonical profiles in tokamak plasmas with an arbitrary cross section'', [[doi:10.1134/1.1520282|Plasma Physics Reports '''28''', 11 (2002) 887-899]]</ref> | ||
This phenomenology is due to plasma [[Self-Organised Criticality|self-organisation]], | This phenomenology is due to plasma [[Self-Organised Criticality|self-organisation]], | ||
<ref> | <ref>Yu.N. Dnestrovsky et al, ''Self-organization of plasma in tokamaks'', [[doi:10.1134/1.1992581|Plasma Physics Reports '''31''', 7 (2005) 529-553]]</ref> | ||
i.e., the feedback mechanism regulating the profiles (by turbulence) is often dominant over the various source terms. | i.e., the feedback mechanism regulating the profiles (by turbulence) is often dominant over the various source terms. | ||
<ref>[http://link.aip.org/link/?PHPAEN/8/4096/1 | <ref>F. Jenko et al, ''Critical gradient formula for toroidal electron temperature gradient modes'', [http://link.aip.org/link/?PHPAEN/8/4096/1 Phys. Plasmas '''8''' (2001) 4096]</ref> | ||
== Observations == | == Observations == | ||
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=== [[Tokamak]]s === | === [[Tokamak]]s === | ||
* TFTR <ref> | * TFTR <ref>E.D. Fredrickson, J.D. Callen, et al., ''Heat pulse propagation studies in TFTR'', [[doi:10.1088/0029-5515/26/7/002|Nucl. Fusion '''26''' (1986) 849]]</ref> | ||
* ASDEX <ref> | * ASDEX <ref>G. Becker, ''Electron temperature profile invariance in OH, L- and H-mode plasmas and consequences for the anomalous transport'', [[doi:10.1088/0029-5515/32/1/I07|Nucl. Fusion '''32''' (1992) 81]]</ref> | ||
* Various devices <ref> | * Various devices <ref>F. Wagner and U. Stroth, ''Transport in toroidal devices-the experimentalist's view'', [[doi:10.1088/0741-3335/35/10/002|Plasma Phys. Control. Fusion '''35''' (1993) 1321]]</ref><ref>F. Ryter, C. Angioni, et al., ''Experimental studies of electron transport'', [[doi:10.1088/0741-3335/43/12A/325|Plasma Phys. Control. Fusion '''43''' (2001) A323]]</ref> | ||
=== [[Stellarator]]s === | === [[Stellarator]]s === | ||
* W7-AS <ref> | * W7-AS <ref>U. Stroth, ''A comparative study of transport in stellarators and tokamaks'', [[doi:10.1088/0741-3335/40/1/002|Plasma Phys. Control. Fusion '''40''' (1998) 9]]</ref> | ||
* Various devices <ref>[http://www-pub.iaea.org/MTCD/Meetings/FEC2008/th_p8-24.pdf | * Various devices <ref>Yu.N. Dnestrovsky et al, [http://www-pub.iaea.org/MTCD/Meetings/FEC2008/th_p8-24.pdf IAEA Fusion Energy Conference, Geneva (2008) TH/P8-24]</ref> | ||
== Proposed explanations == | == Proposed explanations == | ||
The phenomenon is not fully understood, but attempts at explanation have been made. These include: | The phenomenon is not fully understood, but attempts at explanation have been made. These include: | ||
* Marginal stability <ref> | * Marginal stability <ref>P.H. Diamond and T.S. Hahm, ''On the dynamics of turbulent transport near marginal stability'', [[doi:10.1063/1.871063|Phys. Plasmas '''2''' (1995) 3640]]</ref> | ||
* Critical gradient models <ref> | * Critical gradient models <ref>F. Imbeaux and X. Garbet, ''Analytical solutions for the propagation of heat pulses with temperature gradient length-dependent diffusion coefficient'', [[doi:10.1088/0741-3335/44/8/301|Plasma Phys. Control. Fusion '''44''' (2002) 1425]]</ref> | ||
* [[Heat pinch|Energy pinch]] <ref> | * [[Heat pinch|Energy pinch]] <ref>P. Mantica, G. Gorini, G.M.D. Hogeweij, N.J. Lopes Cardozo, and A.M.R. Schilham, ''Heat Convection and Transport Barriers in Low-Magnetic-Shear Rijnhuizen Tokamak Project Plasmas'', [[doi:10.1103/PhysRevLett.85.4534|Phys. Rev. Lett. '''85''' (2000) 4534–4537]]</ref><ref>X. Garbet, N. Dubuit, E. Asp, Y. Sarazin, C. Bourdelle, P. Ghendrih, and G.T. Hoang, ''Turbulent fluxes and entropy production rate'', [[doi:10.1063/1.1951667|Phys. Plasmas '''12''' (2005) 082511]]</ref><ref>P. Mantica, A. Thyagaraja, J. Weiland, G.M.D. Hogeweij, and P.J. Knight, ''Heat Pinches in Electron-Heated Tokamak Plasmas: Theoretical Turbulence Models versus Experiments'', [[doi:10.1103/PhysRevLett.95.185002|Phys. Rev. Lett. '''95''' (2005) 185002]]</ref> | ||
* [[Self-Organised Criticality]] | * [[Self-Organised Criticality]] | ||
* Turbulent equipartition <ref> | * Turbulent equipartition <ref>V. Naulin, A.H. Nielsen, and J. Juul Rasmussen, ''Turbulence spreading, anomalous transport, and pinch effect'', [[doi:10.1063/1.2141396|Phys. Plasmas '''12''' (2005) 122306]]</ref><ref>Lu Wang and P.H. Diamond, ''Kinetic theory of the turbulent energy pinch in tokamak plasmas'', [[doi:10.1088/0029-5515/51/8/083006|Nucl. Fusion '''51''' (2011) 083006]]</ref> | ||
== Quantification methods == | == Quantification methods == | ||
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It is customary to introduce an ad-hoc transport model with a critical gradient (sharply enhanced transport above a critical value of the local gradient) to attempt to quantify the 'criticality' of transport: | It is customary to introduce an ad-hoc transport model with a critical gradient (sharply enhanced transport above a critical value of the local gradient) to attempt to quantify the 'criticality' of transport: | ||
<ref> | <ref>F. Imbeaux, F. Ryter, and X. Garbet, ''Modelling of ECH modulation experiments in ASDEX Upgrade with an empirical critical temperature gradient length transport model'', [[doi:10.1088/0741-3335/43/11/306|Plasma Phys. Control. Fusion '''43''' (2001) 1503]]</ref> | ||
<ref> | <ref>X. Garbet, P. Mantica, F. Ryter, et al., ''Profile stiffness and global confinement'', [[doi:10.1088/0741-3335/46/9/002|Plasma Phys. Control. Fusion '''46''' (2004) 1351]]</ref> | ||
:<math>\chi = \chi_0 + \chi_1 \xi \left ( \frac{R}{L_T}-\frac{R}{L_{T,crit}}\right )^\alpha H\left ( \frac{R}{L_T}-\frac{R}{L_{T,crit}}\right )</math> | :<math>\chi = \chi_0 + \chi_1 \xi \left ( \frac{R}{L_T}-\frac{R}{L_{T,crit}}\right )^\alpha H\left ( \frac{R}{L_T}-\frac{R}{L_{T,crit}}\right )</math> | ||
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However, it is possible to devise methods for the objective quantification of profile stiffness that do not depend so much on the introduction of any ad-hoc model, simply by making this idea of stiffness explicit (i.e., by measuring the response of the gradient to a change in drive or heat source). | However, it is possible to devise methods for the objective quantification of profile stiffness that do not depend so much on the introduction of any ad-hoc model, simply by making this idea of stiffness explicit (i.e., by measuring the response of the gradient to a change in drive or heat source). | ||
<ref>[http://www.jspf.or.jp/PFR/PFR_articles/pfr2008S1/pfr2008_03-S1070.html | <ref>B.Ph. van Milligen et al, ''Quantifying profile stiffness'', [http://www.jspf.or.jp/PFR/PFR_articles/pfr2008S1/pfr2008_03-S1070.html Plasma and Fusion Research, '''3''' (2008) S1070]</ref> | ||
The [[:Wikipedia:Stiffness|general definition of stiffness of a system]] is | The [[:Wikipedia:Stiffness|general definition of stiffness of a system]] is | ||