Non-diffusive transport - Revision history

Admin at 10:43, 26 January 2023

2023-01-26T10:43:45Z

← Older revision		Revision as of 12:43, 26 January 2023
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	Whatever the case, a well-established methodology exists to describe this deviation from standard diffusive transport (with characteristic scales): the [[Continuous Time Random Walk]] (CTRW) model.		Whatever the case, a well-established methodology exists to describe this deviation from standard diffusive transport (with characteristic scales): the [[Continuous Time Random Walk]] (CTRW) model.
	<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>		<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, {{ISBN\|9780750310307}}</ref>
	The CTRW model provides a mathematical framework for handling non-diffusive transport (arising as a generalisation of the diffusive transport when eliminating the stated characteristic scales), but it does not explain why such non-diffusive transport should arise: answering the latter requires detailed computer simulations of turbulence and experimental observations.		The CTRW model provides a mathematical framework for handling non-diffusive transport (arising as a generalisation of the diffusive transport when eliminating the stated characteristic scales), but it does not explain why such non-diffusive transport should arise: answering the latter requires detailed computer simulations of turbulence and experimental observations.

Admin: Updated reference links

2018-04-03T13:13:33Z

Updated reference links

← Older revision		Revision as of 15:13, 3 April 2018
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	In recent years, it has been suggested that the plasma may contain phenomena that invalidate this picture.		In recent years, it has been suggested that the plasma may contain phenomena that invalidate this picture.
	There may ''turbulent eddies'' in which particles become trapped for some time, and there certainly are ''transport barriers'', associated with rational magnetic surfaces, and ''stochastic regions'' of the magnetic field.		There may ''turbulent eddies'' in which particles become trapped for some time, and there certainly are ''transport barriers'', associated with rational magnetic surfaces, and ''stochastic regions'' of the magnetic field.
	<ref>[~~http~~:~~//dx.doi.org/~~10.1088/0741-3335/44/7/101 ~~J.H. Misguich at el.,~~ Plasma Phys. Controlled Fusion, '''44''', L29 (2002)]</ref>		<ref>J.H. Misguich at el., [[doi:10.1088/0741-3335/44/7/101\|Plasma Phys. Controlled Fusion, '''44''', L29 (2002)]]</ref>
	This could cause the waiting time distribution to become non-exponential; and thus the motion would be non-Markovian. Likewise, the phenomenon of ''streamers'', appearing in many models of plasma turbulence, could carry particles across long distances in the radial direction, and the distribution of particle steps could then also be deformed and develop ''long tails''. Consequently, the transport would then be non-local. Nobody knows exactly how important these phenomena are in the global transport picture.		This could cause the waiting time distribution to become non-exponential; and thus the motion would be non-Markovian. Likewise, the phenomenon of ''streamers'', appearing in many models of plasma turbulence, could carry particles across long distances in the radial direction, and the distribution of particle steps could then also be deformed and develop ''long tails''. Consequently, the transport would then be non-local. Nobody knows exactly how important these phenomena are in the global transport picture.

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	However, even without fully understanding the origin of the non-diffusive behaviour, it is possible to construct models based on these ideas, and see whether these models fare better in predicting the global transport properties of plasmas than the standard diffusive models.		However, even without fully understanding the origin of the non-diffusive behaviour, it is possible to construct models based on these ideas, and see whether these models fare better in predicting the global transport properties of plasmas than the standard diffusive models.
	<ref>~~[http://link.aip.org/link/?PHPAEN/11/2272/1~~ B.Ph. van Milligen, R. Sánchez, and B.A. Carreras, Phys. Plasmas '''11''', 2272 (2004)]</ref>		<ref>B.Ph. van Milligen, R. Sánchez, and B.A. Carreras, [[doi:10.1063/1.1701893\|Phys. Plasmas '''11''', 2272 (2004)]]</ref>
	<ref>~~[http://link.aip.org/link/?PHPAEN/11/3787/1~~ B.Ph. van Milligen, B.A. Carreras, and R. Sánchez, Phys. Plasmas '''11''', 3787 (2004)]</ref>		<ref>B.Ph. van Milligen, B.A. Carreras, and R. Sánchez, [[doi:10.1063/1.1763915\|Phys. Plasmas '''11''', 3787 (2004)]]</ref>
	Note that there is another ingredient that may be essential to explain deviations from the standard transport model: [[Self-Organised_Criticality\|self-organisation]]; we will not discuss this here.		Note that there is another ingredient that may be essential to explain deviations from the standard transport model: [[Self-Organised_Criticality\|self-organisation]]; we will not discuss this here.

	Another approach is to test whether non-diffusive transport phenomena actually occur in simulations and experiment. To do so, ''tracer particles'' are injected into the plasma fluid, and their evolution is followed in time. The tracer trajectories can be analyzed by means of several analysis techniques, e.g. by calculating the particle distribution probability function, or by detecting velocity correlations along trajectories. The application of this method has yielded clear indications that plasma turbulence induces non-diffusive transport in simulations.		Another approach is to test whether non-diffusive transport phenomena actually occur in simulations and experiment. To do so, ''tracer particles'' are injected into the plasma fluid, and their evolution is followed in time. The tracer trajectories can be analyzed by means of several analysis techniques, e.g. by calculating the particle distribution probability function, or by detecting velocity correlations along trajectories. The application of this method has yielded clear indications that plasma turbulence induces non-diffusive transport in simulations.
	<ref>~~[http://link.aip.org/link/?PHPAEN/8/5096/1~~ B. Carreras, V. Lynch, and G. Zaslavsky, Phys. Plasmas '''8''', 5096 (2001)] </ref>		<ref>B. Carreras, V. Lynch, and G. Zaslavsky, [[doi:10.1063/1.1416180\|Phys. Plasmas '''8''', 5096 (2001)]]</ref>
	<ref>~~[http://link.aip.org/link/?PHPAEN/13/022310/1~~ L. García and B. Carreras, Phys. Plasmas '''13''', 022310 (2006)]</ref>		<ref>L. García and B. Carreras, [[doi:10.1063/1.2172177\|Phys. Plasmas '''13''', 022310 (2006)]]</ref>
	<ref>~~[http://link.aip.org/link/?PHPAEN/11/3854/1~~ D. del Castillo-Negrete, B. Carreras, and V. Lynch, Phys. Plasmas '''11''', 3854 (2004)]</ref>		<ref>D. del Castillo-Negrete, B. Carreras, and V. Lynch, [[doi:10.1063/1.1767097\|Phys. Plasmas '''11''', 3854 (2004)]]</ref>
	<ref>~~[http://link.aip.org/link/?PHPAEN/15/112301/1~~ J. Mier, R. Sánchez, L. García, D. Newman, and B. Carreras, Phys. Plasmas '''15''', 112301 (2008)]</ref>		<ref>J. Mier, R. Sánchez, L. García, D. Newman, and B. Carreras, [[doi:10.1063/1.3006088\|Phys. Plasmas '''15''', 112301 (2008)]]</ref>
	<ref>~~[http://link.aip.org/link/?PHPAEN/16/042319/1~~ G. Sánchez Burillo, B.Ph. van Milligen, A. Thyagaraja, Phys. Plasmas '''16''', 042319 (2009)]</ref>		<ref>G. Sánchez Burillo, B.Ph. van Milligen, A. Thyagaraja, [[doi:10.1063/1.3118589\|Phys. Plasmas '''16''', 042319 (2009)]]</ref>
	No significant data are as yet available from actual experiments, due to the considerable experimental difficulty of injecting and following tracer particles, although some experiments in this sense are planned.		No significant data are as yet available from actual experiments, due to the considerable experimental difficulty of injecting and following tracer particles, although some experiments in this sense are planned.

	==References==		==References==
	<references />		<references />

Admin: Reverted edits by 61.56.198.89 (Talk) to last revision by Admin

2011-07-16T09:52:00Z

Reverted edits by 61.56.198.89 (Talk) to last revision by Admin

← Older revision		Revision as of 11:52, 16 July 2011
Line 1:		Line 1:
	~~Never~~ would ~~have thunk I~~ would ~~find~~ this so ~~indsipensbale~~.		It has long been known that the standard model for transport in magnetically confined plasmas ([[Neoclassical transport]]) often fails to provide an accurate description of experimental results: it tends to underestimate transport by one order of magnitude or more, typically; the non-Neoclassical component of transport is called "[[Anomalous transport\|anomalous]]". This is a very disappointing situation with a view to constructing a fusion reactor, since worse confinement means that an eventual reactor will need to be bigger and more expensive. Therefore, the search for the cause of this failure (and for methods to restore transport to its Neoclassical value) is one of the main issues of fusion research.

			The standard Neoclassical model is a collisional (diffusive) model, which means that transport is characterised by ''typical scale lengths'', both for space and time, so that the effective diffusion coefficient is essentially the ''mixing length'' value: <math>D = \Delta r^2 / \Delta t</math>, where <math>\Delta r</math> is the typical step size and <math>\Delta t</math> the typical waiting time.

			In recent years, it has been suggested that the plasma may contain phenomena that invalidate this picture.
			There may ''turbulent eddies'' in which particles become trapped for some time, and there certainly are ''transport barriers'', associated with rational magnetic surfaces, and ''stochastic regions'' of the magnetic field.
			<ref>[http://dx.doi.org/10.1088/0741-3335/44/7/101 J.H. Misguich at el., Plasma Phys. Controlled Fusion, '''44''', L29 (2002)]</ref>
			This could cause the waiting time distribution to become non-exponential; and thus the motion would be non-Markovian. Likewise, the phenomenon of ''streamers'', appearing in many models of plasma turbulence, could carry particles across long distances in the radial direction, and the distribution of particle steps could then also be deformed and develop ''long tails''. Consequently, the transport would then be non-local. Nobody knows exactly how important these phenomena are in the global transport picture.

			Whatever the case, a well-established methodology exists to describe this deviation from standard diffusive transport (with characteristic scales): the [[Continuous Time Random Walk]] (CTRW) model.
			<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>
			The CTRW model provides a mathematical framework for handling non-diffusive transport (arising as a generalisation of the diffusive transport when eliminating the stated characteristic scales), but it does not explain why such non-diffusive transport should arise: answering the latter requires detailed computer simulations of turbulence and experimental observations.

			However, even without fully understanding the origin of the non-diffusive behaviour, it is possible to construct models based on these ideas, and see whether these models fare better in predicting the global transport properties of plasmas than the standard diffusive models.
			<ref>[http://link.aip.org/link/?PHPAEN/11/2272/1 B.Ph. van Milligen, R. Sánchez, and B.A. Carreras, Phys. Plasmas '''11''', 2272 (2004)]</ref>
			<ref>[http://link.aip.org/link/?PHPAEN/11/3787/1 B.Ph. van Milligen, B.A. Carreras, and R. Sánchez, Phys. Plasmas '''11''', 3787 (2004)]</ref>
			Note that there is another ingredient that may be essential to explain deviations from the standard transport model: [[Self-Organised_Criticality\|self-organisation]]; we will not discuss this here.

			Another approach is to test whether non-diffusive transport phenomena actually occur in simulations and experiment. To do so, ''tracer particles'' are injected into the plasma fluid, and their evolution is followed in time. The tracer trajectories can be analyzed by means of several analysis techniques, e.g. by calculating the particle distribution probability function, or by detecting velocity correlations along trajectories. The application of this method has yielded clear indications that plasma turbulence induces non-diffusive transport in simulations.
			<ref>[http://link.aip.org/link/?PHPAEN/8/5096/1 B. Carreras, V. Lynch, and G. Zaslavsky, Phys. Plasmas '''8''', 5096 (2001)] </ref>
			<ref>[http://link.aip.org/link/?PHPAEN/13/022310/1 L. García and B. Carreras, Phys. Plasmas '''13''', 022310 (2006)]</ref>
			<ref>[http://link.aip.org/link/?PHPAEN/11/3854/1 D. del Castillo-Negrete, B. Carreras, and V. Lynch, Phys. Plasmas '''11''', 3854 (2004)]</ref>
			<ref>[http://link.aip.org/link/?PHPAEN/15/112301/1 J. Mier, R. Sánchez, L. García, D. Newman, and B. Carreras, Phys. Plasmas '''15''', 112301 (2008)]</ref>
			<ref>[http://link.aip.org/link/?PHPAEN/16/042319/1 G. Sánchez Burillo, B.Ph. van Milligen, A. Thyagaraja, Phys. Plasmas '''16''', 042319 (2009)]</ref>
			No significant data are as yet available from actual experiments, due to the considerable experimental difficulty of injecting and following tracer particles, although some experiments in this sense are planned.

			==References==
			<references />

61.56.198.89: ATApLJrYmjMkWenBqm

2011-07-16T05:49:13Z

ATApLJrYmjMkWenBqm

← Older revision		Revision as of 07:49, 16 July 2011
Line 1:		Line 1:
	It has long been known that the standard model for transport in magnetically confined plasmas ([[Neoclassical transport]]) often fails to provide an accurate description of experimental results: it tends to underestimate transport by one order of magnitude or more, typically; the non-Neoclassical component of transport is called "[[Anomalous transport\|anomalous]]". This is a very disappointing situation with a view to constructing a fusion reactor, since worse confinement means that an eventual reactor will need to be bigger and more expensive. Therefore, the search for the cause of this failure (and for methods to restore transport to its Neoclassical value) is one of the main issues of fusion research.		Never would have thunk I would find this so indsipensbale.

	The standard Neoclassical model is a collisional (diffusive) model, which means that transport is characterised by ''typical scale lengths'', both for space and time, so that the effective diffusion coefficient is essentially the ''mixing length'' value: <math>D = \Delta r^2 / \Delta t</math>, where <math>\Delta r</math> is the typical step size and <math>\Delta t</math> the typical waiting time.

	~~In recent years, it has been suggested that the plasma may contain phenomena that invalidate this picture.~~
	There may ''turbulent eddies'' in which particles become trapped for some time, and there certainly are ''transport barriers'', associated with rational magnetic surfaces, and ''stochastic regions'' of the magnetic field.
	~~<ref>[http://dx.doi.org/10.1088/0741-3335/44/7/101 J.H. Misguich at el., Plasma Phys. Controlled Fusion, '''44''', L29 (2002)]</ref>~~
	~~This could cause the waiting time distribution to become non-exponential; and thus the motion~~ would be non-Markovian. Likewise, the phenomenon of ''streamers'', appearing in many models of plasma turbulence, could carry particles across long distances in the radial direction, and the distribution of particle steps could then also be deformed and develop ''long tails''. Consequently, the transport would ~~then be non-local. Nobody knows exactly how important these phenomena are in the global transport picture.~~

	~~Whatever the case, a well-established methodology exists to describe this deviation from standard diffusive transport (with characteristic scales): the [[Continuous Time Random Walk]] (CTRW) model.~~
	~~<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>~~
	The CTRW model provides a mathematical framework for handling non-diffusive transport (arising as a generalisation of the diffusive transport when eliminating the stated characteristic scales), but it does not explain why such non-diffusive transport should arise: answering the latter requires detailed computer simulations of turbulence and experimental observations.

	However, even without fully understanding the origin of the non-diffusive behaviour, it is possible to construct models based on these ideas, and see whether these models fare better in predicting the global transport properties of plasmas than the standard diffusive models.
	~~<ref>[http://link.aip.org/link/?PHPAEN/11/2272/1 B.Ph. van Milligen, R. Sánchez, and B.A. Carreras, Phys. Plasmas '''11''', 2272 (2004)]</ref>~~
	~~<ref>[http://link.aip.org/link/?PHPAEN/11/3787/1 B.Ph. van Milligen, B.A. Carreras, and R. Sánchez, Phys. Plasmas '''11''', 3787 (2004)]</ref>~~
	~~Note that there is another ingredient that may be essential to explain deviations from the standard transport model: [[Self-Organised_Criticality\|self-organisation]]; we will not discuss~~ this ~~here.~~

	~~Another approach is to test whether non-diffusive transport phenomena actually occur in simulations and experiment. To do~~ so, ''tracer particles'' are injected into the plasma fluid, and their evolution is followed in time. The tracer trajectories can be analyzed by means of several analysis techniques, e.g. by calculating the particle distribution probability function, or by detecting velocity correlations along trajectories. The application of this method has yielded clear indications that plasma turbulence induces non-diffusive transport in simulations.
	~~<ref>[http://link.aip.org/link/?PHPAEN/8/5096/1 B. Carreras, V. Lynch, and G. Zaslavsky, Phys. Plasmas '''8''', 5096 (2001)] </ref>~~
	~~<ref>[http://link.aip.org/link/?PHPAEN/13/022310/1 L. García and B. Carreras, Phys. Plasmas '''13''', 022310 (2006)]</ref>~~
	~~<ref>[http://link.aip.org/link/?PHPAEN/11/3854/1 D. del Castillo-Negrete, B. Carreras, and V. Lynch, Phys. Plasmas '''11''', 3854 (2004)]</ref>~~
	~~<ref>[http://link.aip.org/link/?PHPAEN/15/112301/1 J. Mier, R. Sánchez, L. García, D. Newman, and B. Carreras, Phys. Plasmas '''15''', 112301 (2008)]</ref>~~
	~~<ref>[http://link.aip.org/link/?PHPAEN/16/042319/1 G. Sánchez Burillo, B.Ph. van Milligen, A. Thyagaraja, Phys. Plasmas '''16''', 042319 (2009)]</ref>~~
	No significant data are as yet available from actual experiments, due to the considerable experimental difficulty of injecting and following tracer particles, although some experiments in this sense are planned.

	~~==References==~~
	~~<references />~~

Admin at 07:01, 15 January 2011

2011-01-15T07:01:09Z

Admin: Reverted edits by Otihizuv (Talk) to last revision by Admin

2010-11-24T11:22:11Z

Reverted edits by Otihizuv (Talk) to last revision by Admin

Otihizuv at 06:58, 24 November 2010

2010-11-24T06:58:33Z

Admin at 06:44, 16 September 2009

2009-09-16T06:44:35Z

← Older revision		Revision as of 08:44, 16 September 2009
Line 1:		Line 1:
	It has long been known that the standard model for transport in magnetically confined plasmas ([[Neoclassical transport]]) often fails to provide an accurate description of experimental results: it tends to underestimate transport by one order of magnitude or more, typically. This is a very disappointing situation with a view to constructing a fusion reactor, since worse confinement means that an eventual reactor will need to be bigger and more expensive. Therefore, the search for the cause of this failure (and for methods to restore transport to its Neoclassical value) is one of the main issues of fusion research.		It has long been known that the standard model for transport in magnetically confined plasmas ([[Neoclassical transport]]) often fails to provide an accurate description of experimental results: it tends to underestimate transport by one order of magnitude or more, typically; the non-Neoclassical component of transport is called "[[Anomalous transport\|anomalous]]". This is a very disappointing situation with a view to constructing a fusion reactor, since worse confinement means that an eventual reactor will need to be bigger and more expensive. Therefore, the search for the cause of this failure (and for methods to restore transport to its Neoclassical value) is one of the main issues of fusion research.

	The standard Neoclassical model is a collisional (diffusive) model, which means that transport is characterised by ''typical scale lengths'', both for space and time, so that the effective diffusion coefficient is essentially the ''mixing length'' value: <math>D = \Delta r^2 / \Delta t</math>, where <math>\Delta r</math> is the typical step size and <math>\Delta t</math> the typical waiting time.		The standard Neoclassical model is a collisional (diffusive) model, which means that transport is characterised by ''typical scale lengths'', both for space and time, so that the effective diffusion coefficient is essentially the ''mixing length'' value: <math>D = \Delta r^2 / \Delta t</math>, where <math>\Delta r</math> is the typical step size and <math>\Delta t</math> the typical waiting time.

Admin at 13:28, 30 August 2009

2009-08-30T13:28:22Z

← Older revision		Revision as of 15:28, 30 August 2009
Line 9:		Line 9:

	Whatever the case, a well-established methodology exists to describe this deviation from standard diffusive transport (with characteristic scales): the [[Continuous Time Random Walk]] (CTRW) model.		Whatever the case, a well-established methodology exists to describe this deviation from standard diffusive transport (with characteristic scales): the [[Continuous Time Random Walk]] (CTRW) model.
	<ref>~~[http://books.google.es/books?id=Yaupom_qdKIC&lpg=PP1&ots=WmaABP3l92&dq=%22Balescu%22%20%22Aspects%20of%20anomalous%20transport%20in%20plasmas%22%20&lr=&pg=PP1~~ R. Balescu, Aspects of ~~anomalous transport~~ in ~~plasmas~~, ~~IOP Publishing (~~2005)]</ref>		<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>
	The CTRW model provides a mathematical framework for handling non-diffusive transport (arising as as generalisation of the diffusive transport when eliminating the stated characteristic scales), but it does not explain why such non-diffusive transport should arise: answering the latter requires detailed computer simulations of turbulence and experimental observations.		The CTRW model provides a mathematical framework for handling non-diffusive transport (arising as as generalisation of the diffusive transport when eliminating the stated characteristic scales), but it does not explain why such non-diffusive transport should arise: answering the latter requires detailed computer simulations of turbulence and experimental observations.

Admin at 15:39, 11 August 2009

2009-08-11T15:39:19Z

← Older revision		Revision as of 17:39, 11 August 2009
Line 8:		Line 8:
	This could cause the waiting time distribution to become non-exponential; and thus the motion would be non-Markovian. Likewise, the phenomenon of ''streamers'', appearing in many models of plasma turbulence, could carry particles across long distances in the radial direction, and the distribution of particle steps could then also be deformed and develop ''long tails''. Consequently, the transport would then be non-local. Nobody knows exactly how important these phenomena are in the global transport picture.		This could cause the waiting time distribution to become non-exponential; and thus the motion would be non-Markovian. Likewise, the phenomenon of ''streamers'', appearing in many models of plasma turbulence, could carry particles across long distances in the radial direction, and the distribution of particle steps could then also be deformed and develop ''long tails''. Consequently, the transport would then be non-local. Nobody knows exactly how important these phenomena are in the global transport picture.

	Whatever the case, a well-established methodology exists to describe this deviation from standard diffusive transport (with characteristic scales): the Continuous Time Random Walk (CTRW) model.		Whatever the case, a well-established methodology exists to describe this deviation from standard diffusive transport (with characteristic scales): the [[Continuous Time Random Walk]] (CTRW) model.
	<ref>[http://books.google.es/books?id=Yaupom_qdKIC&lpg=PP1&ots=WmaABP3l92&dq=%22Balescu%22%20%22Aspects%20of%20anomalous%20transport%20in%20plasmas%22%20&lr=&pg=PP1 R. Balescu, Aspects of anomalous transport in plasmas, IOP Publishing (2005)]</ref>		<ref>[http://books.google.es/books?id=Yaupom_qdKIC&lpg=PP1&ots=WmaABP3l92&dq=%22Balescu%22%20%22Aspects%20of%20anomalous%20transport%20in%20plasmas%22%20&lr=&pg=PP1 R. Balescu, Aspects of anomalous transport in plasmas, IOP Publishing (2005)]</ref>
	The CTRW model provides a mathematical framework for handling non-diffusive transport (arising as as generalisation of the diffusive transport when eliminating the stated characteristic scales), but it does not explain why such non-diffusive transport should arise: answering the latter requires detailed computer simulations of turbulence and experimental observations.		The CTRW model provides a mathematical framework for handling non-diffusive transport (arising as as generalisation of the diffusive transport when eliminating the stated characteristic scales), but it does not explain why such non-diffusive transport should arise: answering the latter requires detailed computer simulations of turbulence and experimental observations.

← Older revision		Revision as of 09:01, 15 January 2011
Line 10:		Line 10:
	Whatever the case, a well-established methodology exists to describe this deviation from standard diffusive transport (with characteristic scales): the [[Continuous Time Random Walk]] (CTRW) model.		Whatever the case, a well-established methodology exists to describe this deviation from standard diffusive transport (with characteristic scales): the [[Continuous Time Random Walk]] (CTRW) model.
	<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>		<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>
	The CTRW model provides a mathematical framework for handling non-diffusive transport (arising as as generalisation of the diffusive transport when eliminating the stated characteristic scales), but it does not explain why such non-diffusive transport should arise: answering the latter requires detailed computer simulations of turbulence and experimental observations.		The CTRW model provides a mathematical framework for handling non-diffusive transport (arising as a generalisation of the diffusive transport when eliminating the stated characteristic scales), but it does not explain why such non-diffusive transport should arise: answering the latter requires detailed computer simulations of turbulence and experimental observations.

	However, even without fully understanding the origin of the non-diffusive behaviour, it is possible to construct models based on these ideas, and see whether these models fare better in predicting the global transport properties of plasmas than the standard diffusive models.		However, even without fully understanding the origin of the non-diffusive behaviour, it is possible to construct models based on these ideas, and see whether these models fare better in predicting the global transport properties of plasmas than the standard diffusive models.
Line 17:		Line 17:
	Note that there is another ingredient that may be essential to explain deviations from the standard transport model: [[Self-Organised_Criticality\|self-organisation]]; we will not discuss this here.		Note that there is another ingredient that may be essential to explain deviations from the standard transport model: [[Self-Organised_Criticality\|self-organisation]]; we will not discuss this here.

	Another approach is to test whether non-diffusive transport phenomena actually occur in simulations and experiment. To do so, ''tracer particles'' are injected into the plasma fluid, and their evolution is followed in time. ~~Since the~~ tracer trajectories can be analyzed by means of several analysis techniques, e.g. by calculating the particle distribution probability function, or by detecting velocity correlations along trajectories. The application of this method has yielded clear indications that plasma turbulence induces non-diffusive transport in simulations.		Another approach is to test whether non-diffusive transport phenomena actually occur in simulations and experiment. To do so, ''tracer particles'' are injected into the plasma fluid, and their evolution is followed in time. The tracer trajectories can be analyzed by means of several analysis techniques, e.g. by calculating the particle distribution probability function, or by detecting velocity correlations along trajectories. The application of this method has yielded clear indications that plasma turbulence induces non-diffusive transport in simulations.
	<ref>[http://link.aip.org/link/?PHPAEN/8/5096/1 B. Carreras, V. Lynch, and G. Zaslavsky, Phys. Plasmas '''8''', 5096 (2001)] </ref>		<ref>[http://link.aip.org/link/?PHPAEN/8/5096/1 B. Carreras, V. Lynch, and G. Zaslavsky, Phys. Plasmas '''8''', 5096 (2001)] </ref>
	<ref>[http://link.aip.org/link/?PHPAEN/13/022310/1 L. García and B. Carreras, Phys. Plasmas '''13''', 022310 (2006)]</ref>		<ref>[http://link.aip.org/link/?PHPAEN/13/022310/1 L. García and B. Carreras, Phys. Plasmas '''13''', 022310 (2006)]</ref>
Line 23:		Line 23:
	<ref>[http://link.aip.org/link/?PHPAEN/15/112301/1 J. Mier, R. Sánchez, L. García, D. Newman, and B. Carreras, Phys. Plasmas '''15''', 112301 (2008)]</ref>		<ref>[http://link.aip.org/link/?PHPAEN/15/112301/1 J. Mier, R. Sánchez, L. García, D. Newman, and B. Carreras, Phys. Plasmas '''15''', 112301 (2008)]</ref>
	<ref>[http://link.aip.org/link/?PHPAEN/16/042319/1 G. Sánchez Burillo, B.Ph. van Milligen, A. Thyagaraja, Phys. Plasmas '''16''', 042319 (2009)]</ref>		<ref>[http://link.aip.org/link/?PHPAEN/16/042319/1 G. Sánchez Burillo, B.Ph. van Milligen, A. Thyagaraja, Phys. Plasmas '''16''', 042319 (2009)]</ref>
	No significant data are as yet available from actual experiments, due to the considerable experimental difficulty of ~~performing this task~~, although some experiments in this sense are planned.		No significant data are as yet available from actual experiments, due to the considerable experimental difficulty of injecting and following tracer particles, although some experiments in this sense are planned.

	==References==		==References==
	<references />		<references />

← Older revision		Revision as of 13:22, 24 November 2010
Line 1:		Line 1:
	~~=[http://aluxyxenud.co.cc This Page Is Currently Under Construction And Will Be Available Shortly, Please Visit Reserve Copy Page]=~~		It has long been known that the standard model for transport in magnetically confined plasmas ([[Neoclassical transport]]) often fails to provide an accurate description of experimental results: it tends to underestimate transport by one order of magnitude or more, typically; the non-Neoclassical component of transport is called "[[Anomalous transport\|anomalous]]". This is a very disappointing situation with a view to constructing a fusion reactor, since worse confinement means that an eventual reactor will need to be bigger and more expensive. Therefore, the search for the cause of this failure (and for methods to restore transport to its Neoclassical value) is one of the main issues of fusion research.
	It has long been known that the standard model for transport in magnetically confined plasmas ([[Neoclassical transport]]) often fails to provide an accurate description of experimental results: it tends to underestimate transport by one order of magnitude or more, typically; the non-Neoclassical component of transport is called ~~"~~[[Anomalous transport\|anomalous]]~~"~~. This is a very disappointing situation with a view to constructing a fusion reactor, since worse confinement means that an eventual reactor will need to be bigger and more expensive. Therefore, the search for the cause of this failure (and for methods to restore transport to its Neoclassical value) is one of the main issues of fusion research.

	The standard Neoclassical model is a collisional (diffusive) model, which means that transport is characterised by ''typical scale lengths'', both for space and time, so that the effective diffusion coefficient is essentially the ''mixing length'' value: ~~<~~math~~>~~D = \Delta r^2 / \Delta t~~<~~/math~~>~~, where ~~<~~math~~>~~\Delta r~~<~~/math~~>~~ is the typical step size and ~~<~~math~~>~~\Delta t~~<~~/math~~>~~ the typical waiting time.		The standard Neoclassical model is a collisional (diffusive) model, which means that transport is characterised by ''typical scale lengths'', both for space and time, so that the effective diffusion coefficient is essentially the ''mixing length'' value: <math>D = \Delta r^2 / \Delta t</math>, where <math>\Delta r</math> is the typical step size and <math>\Delta t</math> the typical waiting time.

	In recent years, it has been suggested that the plasma may contain phenomena that invalidate this picture.		In recent years, it has been suggested that the plasma may contain phenomena that invalidate this picture.
	There may ''turbulent eddies'' in which particles become trapped for some time, and there certainly are ''transport barriers'', associated with rational magnetic surfaces, and ''stochastic regions'' of the magnetic field.		There may ''turbulent eddies'' in which particles become trapped for some time, and there certainly are ''transport barriers'', associated with rational magnetic surfaces, and ''stochastic regions'' of the magnetic field.
	~~<~~ref~~>~~[http://dx.doi.org/10.1088/0741-3335/44/7/101 J.H. Misguich at el., Plasma Phys. Controlled Fusion, '''44''', L29 (2002)]~~<~~/ref~~>~~		<ref>[http://dx.doi.org/10.1088/0741-3335/44/7/101 J.H. Misguich at el., Plasma Phys. Controlled Fusion, '''44''', L29 (2002)]</ref>
	This could cause the waiting time distribution to become non-exponential; and thus the motion would be non-Markovian. Likewise, the phenomenon of ''streamers'', appearing in many models of plasma turbulence, could carry particles across long distances in the radial direction, and the distribution of particle steps could then also be deformed and develop ''long tails''. Consequently, the transport would then be non-local. Nobody knows exactly how important these phenomena are in the global transport picture.		This could cause the waiting time distribution to become non-exponential; and thus the motion would be non-Markovian. Likewise, the phenomenon of ''streamers'', appearing in many models of plasma turbulence, could carry particles across long distances in the radial direction, and the distribution of particle steps could then also be deformed and develop ''long tails''. Consequently, the transport would then be non-local. Nobody knows exactly how important these phenomena are in the global transport picture.

	Whatever the case, a well-established methodology exists to describe this deviation from standard diffusive transport (with characteristic scales): the [[Continuous Time Random Walk]] (CTRW) model.		Whatever the case, a well-established methodology exists to describe this deviation from standard diffusive transport (with characteristic scales): the [[Continuous Time Random Walk]] (CTRW) model.
	~~<~~ref~~>~~R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307~~<~~/ref~~>~~		<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>
	The CTRW model provides a mathematical framework for handling non-diffusive transport (arising as as generalisation of the diffusive transport when eliminating the stated characteristic scales), but it does not explain why such non-diffusive transport should arise: answering the latter requires detailed computer simulations of turbulence and experimental observations.		The CTRW model provides a mathematical framework for handling non-diffusive transport (arising as as generalisation of the diffusive transport when eliminating the stated characteristic scales), but it does not explain why such non-diffusive transport should arise: answering the latter requires detailed computer simulations of turbulence and experimental observations.

	However, even without fully understanding the origin of the non-diffusive behaviour, it is possible to construct models based on these ideas, and see whether these models fare better in predicting the global transport properties of plasmas than the standard diffusive models.		However, even without fully understanding the origin of the non-diffusive behaviour, it is possible to construct models based on these ideas, and see whether these models fare better in predicting the global transport properties of plasmas than the standard diffusive models.
	~~<~~ref~~>~~[http://link.aip.org/link/?PHPAEN/11/2272/1 B.Ph. van Milligen, R. Sánchez, and B.A. Carreras, Phys. Plasmas '''11''', 2272 (2004)]~~<~~/ref~~>~~		<ref>[http://link.aip.org/link/?PHPAEN/11/2272/1 B.Ph. van Milligen, R. Sánchez, and B.A. Carreras, Phys. Plasmas '''11''', 2272 (2004)]</ref>
	~~<~~ref~~>~~[http://link.aip.org/link/?PHPAEN/11/3787/1 B.Ph. van Milligen, B.A. Carreras, and R. Sánchez, Phys. Plasmas '''11''', 3787 (2004)]~~<~~/ref~~>~~		<ref>[http://link.aip.org/link/?PHPAEN/11/3787/1 B.Ph. van Milligen, B.A. Carreras, and R. Sánchez, Phys. Plasmas '''11''', 3787 (2004)]</ref>
	Note that there is another ingredient that may be essential to explain deviations from the standard transport model: [[Self-Organised_Criticality\|self-organisation]]; we will not discuss this here.		Note that there is another ingredient that may be essential to explain deviations from the standard transport model: [[Self-Organised_Criticality\|self-organisation]]; we will not discuss this here.

	Another approach is to test whether non-diffusive transport phenomena actually occur in simulations and experiment. To do so, ''tracer particles'' are injected into the plasma fluid, and their evolution is followed in time. Since the tracer trajectories can be analyzed by means of several analysis techniques, e.g. by calculating the particle distribution probability function, or by detecting velocity correlations along trajectories. The application of this method has yielded clear indications that plasma turbulence induces non-diffusive transport in simulations.		Another approach is to test whether non-diffusive transport phenomena actually occur in simulations and experiment. To do so, ''tracer particles'' are injected into the plasma fluid, and their evolution is followed in time. Since the tracer trajectories can be analyzed by means of several analysis techniques, e.g. by calculating the particle distribution probability function, or by detecting velocity correlations along trajectories. The application of this method has yielded clear indications that plasma turbulence induces non-diffusive transport in simulations.
	~~<~~ref~~>~~[http://link.aip.org/link/?PHPAEN/8/5096/1 B. Carreras, V. Lynch, and G. Zaslavsky, Phys. Plasmas '''8''', 5096 (2001)] ~~<~~/ref~~>~~		<ref>[http://link.aip.org/link/?PHPAEN/8/5096/1 B. Carreras, V. Lynch, and G. Zaslavsky, Phys. Plasmas '''8''', 5096 (2001)] </ref>
	~~<~~ref~~>~~[http://link.aip.org/link/?PHPAEN/13/022310/1 L. García and B. Carreras, Phys. Plasmas '''13''', 022310 (2006)]~~<~~/ref~~>~~		<ref>[http://link.aip.org/link/?PHPAEN/13/022310/1 L. García and B. Carreras, Phys. Plasmas '''13''', 022310 (2006)]</ref>
	~~<~~ref~~>~~[http://link.aip.org/link/?PHPAEN/11/3854/1 D. del Castillo-Negrete, B. Carreras, and V. Lynch, Phys. Plasmas '''11''', 3854 (2004)]~~<~~/ref~~>~~		<ref>[http://link.aip.org/link/?PHPAEN/11/3854/1 D. del Castillo-Negrete, B. Carreras, and V. Lynch, Phys. Plasmas '''11''', 3854 (2004)]</ref>
	~~<~~ref~~>~~[http://link.aip.org/link/?PHPAEN/15/112301/1 J. Mier, R. Sánchez, L. García, D. Newman, and B. Carreras, Phys. Plasmas '''15''', 112301 (2008)]~~<~~/ref~~>~~		<ref>[http://link.aip.org/link/?PHPAEN/15/112301/1 J. Mier, R. Sánchez, L. García, D. Newman, and B. Carreras, Phys. Plasmas '''15''', 112301 (2008)]</ref>
	~~<~~ref~~>~~[http://link.aip.org/link/?PHPAEN/16/042319/1 G. Sánchez Burillo, B.Ph. van Milligen, A. Thyagaraja, Phys. Plasmas '''16''', 042319 (2009)]~~<~~/ref~~>~~		<ref>[http://link.aip.org/link/?PHPAEN/16/042319/1 G. Sánchez Burillo, B.Ph. van Milligen, A. Thyagaraja, Phys. Plasmas '''16''', 042319 (2009)]</ref>
	No significant data are as yet available from actual experiments, due to the considerable experimental difficulty of performing this task, although some experiments in this sense are planned.		No significant data are as yet available from actual experiments, due to the considerable experimental difficulty of performing this task, although some experiments in this sense are planned.

	==References==		==References==
	~~<~~references /~~>~~		<references />

← Older revision		Revision as of 08:58, 24 November 2010
Line 1:		Line 1:
	It has long been known that the standard model for transport in magnetically confined plasmas ([[Neoclassical transport]]) often fails to provide an accurate description of experimental results: it tends to underestimate transport by one order of magnitude or more, typically; the non-Neoclassical component of transport is called "[[Anomalous transport\|anomalous]]". This is a very disappointing situation with a view to constructing a fusion reactor, since worse confinement means that an eventual reactor will need to be bigger and more expensive. Therefore, the search for the cause of this failure (and for methods to restore transport to its Neoclassical value) is one of the main issues of fusion research.		=[http://aluxyxenud.co.cc This Page Is Currently Under Construction And Will Be Available Shortly, Please Visit Reserve Copy Page]=
			It has long been known that the standard model for transport in magnetically confined plasmas ([[Neoclassical transport]]) often fails to provide an accurate description of experimental results: it tends to underestimate transport by one order of magnitude or more, typically; the non-Neoclassical component of transport is called "[[Anomalous transport\|anomalous]]". This is a very disappointing situation with a view to constructing a fusion reactor, since worse confinement means that an eventual reactor will need to be bigger and more expensive. Therefore, the search for the cause of this failure (and for methods to restore transport to its Neoclassical value) is one of the main issues of fusion research.

	The standard Neoclassical model is a collisional (diffusive) model, which means that transport is characterised by ''typical scale lengths'', both for space and time, so that the effective diffusion coefficient is essentially the ''mixing length'' value: <math>D = \Delta r^2 / \Delta t</math>, where <math>\Delta r</math> is the typical step size and <math>\Delta t</math> the typical waiting time.		The standard Neoclassical model is a collisional (diffusive) model, which means that transport is characterised by ''typical scale lengths'', both for space and time, so that the effective diffusion coefficient is essentially the ''mixing length'' value: <math>D = \Delta r^2 / \Delta t</math>, where <math>\Delta r</math> is the typical step size and <math>\Delta t</math> the typical waiting time.

	In recent years, it has been suggested that the plasma may contain phenomena that invalidate this picture.		In recent years, it has been suggested that the plasma may contain phenomena that invalidate this picture.
	There may ''turbulent eddies'' in which particles become trapped for some time, and there certainly are ''transport barriers'', associated with rational magnetic surfaces, and ''stochastic regions'' of the magnetic field.		There may ''turbulent eddies'' in which particles become trapped for some time, and there certainly are ''transport barriers'', associated with rational magnetic surfaces, and ''stochastic regions'' of the magnetic field.
	<ref>[http://dx.doi.org/10.1088/0741-3335/44/7/101 J.H. Misguich at el., Plasma Phys. Controlled Fusion, '''44''', L29 (2002)]</ref>		<ref>[http://dx.doi.org/10.1088/0741-3335/44/7/101 J.H. Misguich at el., Plasma Phys. Controlled Fusion, '''44''', L29 (2002)]</ref>
	This could cause the waiting time distribution to become non-exponential; and thus the motion would be non-Markovian. Likewise, the phenomenon of ''streamers'', appearing in many models of plasma turbulence, could carry particles across long distances in the radial direction, and the distribution of particle steps could then also be deformed and develop ''long tails''. Consequently, the transport would then be non-local. Nobody knows exactly how important these phenomena are in the global transport picture.		This could cause the waiting time distribution to become non-exponential; and thus the motion would be non-Markovian. Likewise, the phenomenon of ''streamers'', appearing in many models of plasma turbulence, could carry particles across long distances in the radial direction, and the distribution of particle steps could then also be deformed and develop ''long tails''. Consequently, the transport would then be non-local. Nobody knows exactly how important these phenomena are in the global transport picture.

	Whatever the case, a well-established methodology exists to describe this deviation from standard diffusive transport (with characteristic scales): the [[Continuous Time Random Walk]] (CTRW) model.		Whatever the case, a well-established methodology exists to describe this deviation from standard diffusive transport (with characteristic scales): the [[Continuous Time Random Walk]] (CTRW) model.
	<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>		<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>
	The CTRW model provides a mathematical framework for handling non-diffusive transport (arising as as generalisation of the diffusive transport when eliminating the stated characteristic scales), but it does not explain why such non-diffusive transport should arise: answering the latter requires detailed computer simulations of turbulence and experimental observations.		The CTRW model provides a mathematical framework for handling non-diffusive transport (arising as as generalisation of the diffusive transport when eliminating the stated characteristic scales), but it does not explain why such non-diffusive transport should arise: answering the latter requires detailed computer simulations of turbulence and experimental observations.

	However, even without fully understanding the origin of the non-diffusive behaviour, it is possible to construct models based on these ideas, and see whether these models fare better in predicting the global transport properties of plasmas than the standard diffusive models.		However, even without fully understanding the origin of the non-diffusive behaviour, it is possible to construct models based on these ideas, and see whether these models fare better in predicting the global transport properties of plasmas than the standard diffusive models.
	<ref>[http://link.aip.org/link/?PHPAEN/11/2272/1 B.Ph. van Milligen, R. Sánchez, and B.A. Carreras, Phys. Plasmas '''11''', 2272 (2004)]</ref>		<ref>[http://link.aip.org/link/?PHPAEN/11/2272/1 B.Ph. van Milligen, R. Sánchez, and B.A. Carreras, Phys. Plasmas '''11''', 2272 (2004)]</ref>
	<ref>[http://link.aip.org/link/?PHPAEN/11/3787/1 B.Ph. van Milligen, B.A. Carreras, and R. Sánchez, Phys. Plasmas '''11''', 3787 (2004)]</ref>		<ref>[http://link.aip.org/link/?PHPAEN/11/3787/1 B.Ph. van Milligen, B.A. Carreras, and R. Sánchez, Phys. Plasmas '''11''', 3787 (2004)]</ref>
	Note that there is another ingredient that may be essential to explain deviations from the standard transport model: [[Self-Organised_Criticality\|self-organisation]]; we will not discuss this here.		Note that there is another ingredient that may be essential to explain deviations from the standard transport model: [[Self-Organised_Criticality\|self-organisation]]; we will not discuss this here.

	Another approach is to test whether non-diffusive transport phenomena actually occur in simulations and experiment. To do so, ''tracer particles'' are injected into the plasma fluid, and their evolution is followed in time. Since the tracer trajectories can be analyzed by means of several analysis techniques, e.g. by calculating the particle distribution probability function, or by detecting velocity correlations along trajectories. The application of this method has yielded clear indications that plasma turbulence induces non-diffusive transport in simulations.		Another approach is to test whether non-diffusive transport phenomena actually occur in simulations and experiment. To do so, ''tracer particles'' are injected into the plasma fluid, and their evolution is followed in time. Since the tracer trajectories can be analyzed by means of several analysis techniques, e.g. by calculating the particle distribution probability function, or by detecting velocity correlations along trajectories. The application of this method has yielded clear indications that plasma turbulence induces non-diffusive transport in simulations.
	<ref>[http://link.aip.org/link/?PHPAEN/8/5096/1 B. Carreras, V. Lynch, and G. Zaslavsky, Phys. Plasmas '''8''', 5096 (2001)] </ref>		<ref>[http://link.aip.org/link/?PHPAEN/8/5096/1 B. Carreras, V. Lynch, and G. Zaslavsky, Phys. Plasmas '''8''', 5096 (2001)] </ref>
	<ref>[http://link.aip.org/link/?PHPAEN/13/022310/1 L. García and B. Carreras, Phys. Plasmas '''13''', 022310 (2006)]</ref>		<ref>[http://link.aip.org/link/?PHPAEN/13/022310/1 L. García and B. Carreras, Phys. Plasmas '''13''', 022310 (2006)]</ref>
	<ref>[http://link.aip.org/link/?PHPAEN/11/3854/1 D. del Castillo-Negrete, B. Carreras, and V. Lynch, Phys. Plasmas '''11''', 3854 (2004)]</ref>		<ref>[http://link.aip.org/link/?PHPAEN/11/3854/1 D. del Castillo-Negrete, B. Carreras, and V. Lynch, Phys. Plasmas '''11''', 3854 (2004)]</ref>
	<ref>[http://link.aip.org/link/?PHPAEN/15/112301/1 J. Mier, R. Sánchez, L. García, D. Newman, and B. Carreras, Phys. Plasmas '''15''', 112301 (2008)]</ref>		<ref>[http://link.aip.org/link/?PHPAEN/15/112301/1 J. Mier, R. Sánchez, L. García, D. Newman, and B. Carreras, Phys. Plasmas '''15''', 112301 (2008)]</ref>
	<ref>[http://link.aip.org/link/?PHPAEN/16/042319/1 G. Sánchez Burillo, B.Ph. van Milligen, A. Thyagaraja, Phys. Plasmas '''16''', 042319 (2009)]</ref>		<ref>[http://link.aip.org/link/?PHPAEN/16/042319/1 G. Sánchez Burillo, B.Ph. van Milligen, A. Thyagaraja, Phys. Plasmas '''16''', 042319 (2009)]</ref>
	No significant data are as yet available from actual experiments, due to the considerable experimental difficulty of performing this task, although some experiments in this sense are planned.		No significant data are as yet available from actual experiments, due to the considerable experimental difficulty of performing this task, although some experiments in this sense are planned.

	==References==		==References==
	<references />		<references />