Heat pinch - Revision history

Admin: /* Fick versus Fokker Planck */

2012-07-23T15:21:04Z

Fick versus Fokker Planck

← Older revision		Revision as of 17:21, 23 July 2012
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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, ~~whose~~ use is only recommended in homogeneous systems.		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. Sánchez, ''The foundations of diffusion revisited'', Plasma Phys. Control. Fusion '''47''' (2005) B743–B754]]</ref>		<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>

Admin: /* Fick versus Fokker Planck */

2012-07-22T15:08:46Z

Fick versus Fokker Planck

← Older revision		Revision as of 17:08, 22 July 2012
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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 ==

Admin at 15:01, 22 July 2012

2012-07-22T15:01:00Z

← Older revision		Revision as of 17:01, 22 July 2012
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	Part of the problem may be due to the use of a ''Fickian'' transport equation, whose use is only recommended in homogeneous systems.		Part of the problem may be due to the use of a ''Fickian'' transport equation, whose use 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. ~~Sá́nchez~~, ''The foundations of diffusion revisited'', Plasma Phys. Control. Fusion '''47''' (2005) B743–B754]]</ref>		<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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	<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. ~~Sá́nchez~~, ''Uphill transport and the probabilistic transport model'', Phys. Plasmas '''11''', 8 (2004) 3787]]</ref>		<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 ==

Admin: /* Fick versus Fokker Planck */

2012-07-22T14:59:35Z

Fick versus Fokker Planck

← Older revision		Revision as of 16:59, 22 July 2012
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	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. Sá́nchez, ''The foundations of diffusion revisited'', Plasma Phys. Control. Fusion '''47''' (2005) B743–B754]]</ref>		<ref>[[doi:10.1088/0741-3335/47/12B/S56\|B.Ph. van Milligen, B.A. Carreras and R. Sá́nchez, ''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 ''V = -dχ/dr''.		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

			:<math>q_e = - \nabla(n_e \chi T_e) + n_eUT_e</math>

			Setting ''U = 0'' and assuming ''∇ n<sub>e</sub> = 0'', comparison with the above 'Fickian' heat transport equation shows that

			:<math>V = -\nabla \chi</math>

			I.e., the gradient of the heat conductivity produces a 'natural' pinch.

	== Mesoscopic and microscopic mechanisms ==		== Mesoscopic and microscopic mechanisms ==

Admin: Created page with 'The concept of heat pinch is related to the convective term proportional to ''V'' in the (electron) heat transport equation: : $q_e = -n_e\chi \nabla T_e + n_e V T_e$ …'

2012-07-22T14:32:34Z

Created page with 'The concept of heat pinch is related to the convective term proportional to ''V'' in the (electron) heat transport equation: :<math>q_e = -n_e\chi \nabla T_e + n_e V T_e</math> …'

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The concept of heat pinch is related to the convective term proportional to ''V'' in the (electron) heat transport equation:

:<math>q_e = -n_e\chi \nabla T_e + n_e V T_e</math>

Using modulation experiments, the contributions to (electron) transport due to the heat conductivity ''χ''
and the convection ''V'' can be distinguished.
<ref>[[doi:10.1103/PhysRevLett.95.185002|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'', Phys. Rev. Lett. '''95''' (2005) 185002]]</ref>
In such experiments, it was found that ''V'' is often large and directed inward (hence the term 'pinch'),
which raises the question which mechanism is responsible for this effect that drives heat up the gradient.

== Coupling between transport equations ==

To answer this question, first it should be noted that the heat transport equation only captures part of the transport problem in fusion plasmas.
A more complete description also considers ion heat transport and electron and ion density (particle) transport.
The mentioned set of transport equations is coupled (cf. [[Neoclassical transport]]), which may give rise to an ''apparent'' pinch.
<ref>[[doi:10.1088/0741-3335/37/8/001|N.J. Lopes Cardozo, ''Perturbative transport studies in fusion plasmas'', Plasma Phys. Control. Fusion '''37''' (1995) 799]]</ref>
This coupling between transport equations has successfully been invoked to explain some salient features of propagating heat pulses.
<ref>[[doi:10.1088/0741-3335/33/3/003|G.M.D. Hogeweij, J. O'Rourke and A.C.C. Sips, ''Evidence of coupling of thermal and particle transport from heat and density pulse measurements in JET'', Plasma Phys. Control. Fusion '''33''' (1991) 189]]</ref>
Even so, the coupling between transport equations by itself does not generally seem capable of explaining the observed heat pinch.

== Fick versus Fokker Planck ==

Part of the problem may be due to the use of a ''Fickian'' transport equation, whose use is only recommended in homogeneous systems.
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. Sá́nchez, ''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 ''V = -dχ/dr''.

== Mesoscopic and microscopic mechanisms ==

The preceding paragraphs only discuss the 'macroscopic' fluid level description.
But the fluid scale description should of course arise from underlying 'mesoscopic' or 'microscopic' level descriptions through an appropriate averaging procedure.
At the microscopic level, several mechanisms may be operative, such as Turbulent Equipartition.
<ref>[[doi:10.1063/1.2141396|V. Naulin, A.H. Nielsen, and J. Juul Rasmussen, ''Turbulence spreading, anomalous transport, and pinch effect'', Phys. Plasmas '''12''' (2005) 122306]]</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
<ref>[[doi:10.1063/1.1763915|B.Ph. van Milligen, B.A. Carreras and R. Sá́nchez, ''Uphill transport and the probabilistic transport model'', Phys. Plasmas '''11''', 8 (2004) 3787]]</ref>

== See also ==
* [[Profile consistency]]

== References ==
<references />

Heat pinch - Revision history

Admin: /* Fick versus Fokker Planck */

Admin: /* Fick versus Fokker Planck */

Admin at 15:01, 22 July 2012

Admin: /* Fick versus Fokker Planck */

Admin: Created page with 'The concept of heat pinch is related to the convective term proportional to ''V'' in the (electron) heat transport equation: :q_e = -n_e\chi \nabla T_e + n_e V T_e …'

Admin: Created page with 'The concept of heat pinch is related to the convective term proportional to ''V'' in the (electron) heat transport equation: : $q_e = -n_e\chi \nabla T_e + n_e V T_e$ …'