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TJ-II:Instabilities

2010-11-24T08:46:25Z

Otihizuv:

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At [[TJ-II]], a number of studies have been performed to analyse and understand the various instabilities occurring in fusion-grade plasmas.

== MHD and Alfvén modes ==

The appearance of low-frequency Magneto-HydroDynamic (MHD) modes (some tens of kilohertz) in electron cyclotron heated plasmas depends on the rotational transform profile and the plasma density. In neutral beam injection plasmas, high-frequency modes (150- to 300-kHz) have been found in plasmas with line densities in the range 0.6 × 1019 m-3 to 3 × 1019 m-3 and heated with on/off-axis electron cyclotron heating. They are good candidates for global Alfvén eigenmodes related to the low-order resonance n/m = 3/2.
<ref>[http://www.new.ans.org/pubs/journals/fst/a_1283 R. Jiménez-Gómez et al., ''Analysis of Magnetohydrodynamic Instabilities in TJ-II Plasmas'', Fusion Science and Technology '''51''', 20 (2007)]</ref>

== ELM-like modes ==

ELM-like activity has been observed in plasmas with a stored energy above 1 kJ. The plasma is observed to develop bursts of magnetic activity (seen in Mirnov coil signals) which are followed by a large and distinct spike in the H&alpha; signal. An increase in electrostatic and magnetic fluctuations at the plasma edge and a cold pulse towards the plasma centre are also characteristic of these events. In addition, the electron temperature profile locally flattens at the plasma radius where the temperature is in the range 100-200 eV. This flattening can be explained in terms of enhanced electron heat conductivity. Between ELM-like events the electromagnetic turbulence at the edge decreases and the ''Te'' profiles recover their former shapes. This activity is probably triggered by a resonant m = 2, n = 3 mode.
<ref>[http://dx.doi.org/10.1088/0029-5515/40/11/306 I. García-Cortés et al, ''Edge-localized-mode-like events in the TJ-II stellarator'', Nucl. Fusion '''40''' (2000) 1867-1874]</ref>
<ref>[http://dx.doi.org/10.1088/0741-3335/48/5/002 J.A. Jiménez et al, ''Localized electromagnetic modes in MHD stable regime of the TJ-II Heliac'', Plasma Phys. Control. Fusion '''48''' (2006) 515-526]</ref>

== Influence of the magnetic well ==

MHD theory predicts that instabilities inside the plasma are stabilized by increasing the "[[Magnetic well|magnetic well]]". This idea is supported by magnetic well scan experiments in TJ-II.
The level of fluctuations, the degree of intermittency and the radial correlation of relevant quantities increase as the magnetic well is reduced.
<ref>[http://dx.doi.org/10.1088/0741-3335/43/12A/324 C. Hidalgo et al, ''On the radial scale of fluctuations in the TJ-II stellarator'', Plasma Phys. Control. Fusion '''43''' (2001) A313-A321]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/9/713/1 J. Castellano et al, ''Magnetic well and instability thresholds in the TJ-II stellarator'', Phys. Plasmas '''9''' (2002) 713]</ref>

== References ==
<references />

TJ-II:Turbulence

2010-11-24T08:39:08Z

Otihizuv:

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Transport in fusion-grade plasmas is often dominated by turbulent transport.
In contrast with [[Neoclassical transport]], turbulent transport (assumed to be the cause of the experimental so-called "[[Anomalous transport|anomalous]]" component of transport) is not well understood.
As a consequence, predictions of machine performance generally rely on rather crude [[Scaling law|scaling law techniques]], rather than first-principles calculations.
Improving our understanding of turbulence is hard, due to (1) the complexity of fusion-grade plasmas (the presence of charged particles and magnetic fields make this into a much harder topic than fluid turbulence), (2) the enormous variety of plasma instabilities, and (3) the difficulty of diagnosing the plasma due to the hostile conditions inside the plasma.

Our work on turbulence has focussed mainly on the analysis of edge [[TJ-II:Langmuir Probes|Langmuir probe]] data, although some analysis was done on other types of data (e.g., [[TJ-II:Reflectometry|reflectometry]] signals). Much effort was devoted to the development of new analysis techniques.

=== Bicoherence and wavelets ===

[[File:Bicoherence.png|300px|thumb|right|Auto-[[Bicoherence|bicoherence]] graph (''E&theta;'') during a spontaneous confinement transition at TJ-II, showing the coupling of high to low frequencies (horizontal and diagonal lines), i.e., a possible inverse spectral cascade. (from B.Ph. van Milligen et al, Nucl. Fusion 48 (2008) 115003)]]
Turbulence is essentially non-linear.
Non-linear interactions can be detected by means of higher-order spectra (e.g. quadratic interactions can be detected through the bi-spectrum). With Fourier analysis, however, in order to achieve statistically significant values for the bi-spectrum, very long time series are necessary. This fact has mostly precluded its use in fields like plasma turbulence, since long steady-state data series are not generally available. In our work, for the first time, the [[Bicoherence|bicoherence]] was calculated using wavelet transforms, thus making the detection of non-linear interactions with time resolution possible.
<ref>[http://link.aps.org/doi/10.1103/PhysRevLett.74.395 B.Ph. van Milligen et al, ''Nonlinear phenomena and intermittency in plasma turbulence'', Phys. Rev. Lett. '''74''', 3 (1995) 395]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/2/3017/1 B.Ph. van Milligen et al, ''Wavelet bicoherence: a new turbulence analysis tool'', Phys. Plasmas '''2''', 8 (1995) 3017]</ref>
<ref>[http://link.aip.org/link/?RSINAK/68/967/1 B.Ph. van Milligen et al, ''Statistically robust linear and non-linear wavelet analysis applied to plasma edge turbulence'', Rev. Sci. Instrum. '''68''' (1997) 967]</ref>
<ref>[http://link.aps.org/doi/10.1103/PhysRevLett.84.4842 P.H. Diamond et al, ''In search of the elusive zonal flow using cross-bicoherence analysis'', Phys. Rev. Lett. '''84''', 12 (2000) 4842]</ref>
A relation was found between confinement transitions and an increase of the bicoherence, as expected in the framework of [[TJ-II:Confinement transitions|shear/zonal flow]] models for turbulence stabilisation.
<ref>[http://dx.doi.org/10.1088/0029-5515/48/11/115003 B.Ph. van Milligen et al, ''Bicoherence during confinement transitions in the TJ-II stellarator'', Nucl. Fusion '''48''' (2008) 115003]</ref>

=== Self-similarity ===

Important transport phenomena such as profile stiffness (consistency),
<ref>[http://www.jspf.or.jp/PFR/PFR_articles/pfr2008S1/pfr2008_03-S1070.html B.Ph. van Milligen et al, ''Quantifying profile stiffness'', Plasma and Fusion Research, '''3''' (2008) S1070]</ref>
power degradation, the rapid propagation of perturbations,
<ref>[http://dx.doi.org/10.1088/0029-5515/47/3/004 B.Ph. van Milligen et al, ''Pulse propagation in a simple probabilistic transport model'', Nucl. Fusion '''47''' (2007) 189]</ref>
and the Bohm scaling of plasma confinement might be explained on the basis of profile self-regulation in the framework of the [[Self-Organised Criticality]] paradigm. This paradigm predicts that transport is regulated by avalanches, which would generate self-similar behaviour in space and time of the turbulent data.

In order to test this hypothesis, one can determine the shape of the autocorrelation function (ACF) of turbulent signals.
<ref>[http://link.aip.org/link/?PHPAEN/3/2664/1 B.A. Carreras et al, ''Fluctuation-induced flux at the plasma edge in toroidal devices'', Phys. Plasmas '''3''' (7) (1996) 2664]</ref>
<ref>[http://link.aps.org/doi/10.1103/PhysRevLett.80.4438 B.A. Carreras et al, ''Long-range time correlations in plasma edge turbulence'', Phys. Rev. Lett. '''80''', (1998) 4438]</ref>
<ref>[http://link.aps.org/doi/10.1103/PhysRevLett.83.3653 B.A. Carreras et al, ''Self-similarity properties of the probability distribution function of turbulence-induced particle fluxes at the plasma edge'', Phys. Rev. Lett. '''83''' (1999) 3653]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/7/3278/1 B.A. Carreras, ''Intermittency of plasma edge fluctuation data: Multifractal analysis'', Phys. Plasmas, '''7''', 8 (2000) 3278]</ref>
<ref>[http://dx.doi.org/10.1088/0741-3335/44/8/309 C. Hidalgo et al, ''Empirical similarity in the probability density function of turbulent transport in the edge plasma region in fusion plasmas'', Plasma Phys. Control. Fusion '''44''' (2002) 1557]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/12/052507/1 B.Ph. Van Milligen et al, ''Additional evidence for the universality of turbulent fluctuations and fluxes in the scrape-off layer region of fusion plasmas'', Phys. Plasmas '''12''' (2005) 052507]</ref>

Unfortunately, the most revealing information is present in the tail of the distribution (i.e., well beyond the correlation time), where statistics are generally poor.
Therefore, it is convenient to resort to the Rescaled-Range analysis technique and the determination of the Hurst exponent. We have shown that this type of analysis is far more robust with respect to random noise perturbations than the direct determination of the ACF or the Probability of Return.

The analysis of data from Langmuir probes taken at the plasma edge in a wide variety of fusion devices reveals the existence of self-similar behaviour or long-range correlations in all devices studied. The observed variation of the Hurst exponent in the plasma edge, 0.62 < H < 0.75, is small in spite of the variety of devices.
<ref>[http://link.aip.org/link/?PHPAEN/5/3632/1 B.A. Carreras et al, ''Self-similarity of the plasma edge fluctuations'', Phys. Plasmas '''5''', 10 (1998) 3632]</ref>
On the other hand, the variation of H in the scrape-off layer is much larger. In Wendelstein VII-AS, a slight decrease in H at the sheared flow layer was observed, possibly corresponding to a local decorrelation effect.

The repeated occurrence of values of H differing significantly from the value corresponding to random noise (H = 0.5) in all machines points to a universal aspect of the underlying turbulence. Further, the degree of self-similarity detected implies the existence of long-range correlations (with respect to the correlation time).
<ref>[http://link.aip.org/link/?PHPAEN/6/485/1 B.A. Carreras et al, ''Long-range time dependence in the cross-correlation function'', Phys. Plasmas '''6''', 2 (1999) 485]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/6/1885/1 B.A. Carreras et al, ''Experimental evidence of long-range correlation and self-similarity in plasma fluctuations'', Phys. Plasmas '''6''', 5 (1999) 1885]</ref>

In this framework, an important technique is the quiet-time analysis.
<ref>[http://link.aps.org/doi/10.1103/PhysRevE.66.036124 R. Sánchez et al, ''Quiet-time statistics: A tool to probe the dynamics of self-organized-criticality systems from within the strong overlapping regime'', Phys. Rev. E, '''66''' (2002) 036124]</ref>
<ref>[http://link.aps.org/doi/10.1103/PhysRevLett.90.185005 R. Sánchez et al, ''Quiet-time statistics of electrostatic turbulent fluxes from the JET tokamak and the W7-AS and TJ-II stellarators'', Phys. Rev. Lett. '''90''', 185005 (2003)]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/12/052304/1 V.E. Lynch et al, ''Determination of long-range correlation by quiet-time statistics'', Phys. Plasmas '''12''' (2005) 052304]</ref>

=== Turbulence classification ===

An important effort has also been made to identify and classify turbulence,
<ref>[http://link.aip.org/link/?PHPAEN/7/1408/1 E. Sánchez et al, ''Statistical characterization of fluctuation waveforms in the boundary region of fusion and non-fusion plasmas'', Phys. Plasmas '''7''', 5 (2000) 1408]</ref>
<ref>[http://dx.doi.org/10.1088/0741-3335/42/4/302 I. García-Cortés et al, ''Turbulent transport studies in the JET edge plasmas in limiter configuration'', Plasma Phys. Control. Fusion '''42''' (2000) 389]</ref>
<ref>[http://dx.doi.org/10.1016/j.crhy.2006.06.012 C. Hidalgo et al, ''Intermittency and structures in edge plasma turbulence'', Comptes Rendus Physique '''7''', 6 (2006) 679]</ref>
to analyse its spectra,
<ref>[http://link.aps.org/doi/10.1103/PhysRevLett.82.3621 M. A. Pedrosa et al, ''Empirical similarity of frequency spectra of the edge plasma fluctuations in toroidal magnetic confinement systems'', Phys. Rev. Lett. '''82''' (1999) 3621]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/6/4615/1 B.A. Carreras et al, ''Characterization of the frequency ranges of the plasma edge fluctuation spectra'', Phys. Plasmas '''6''', 12 (1999) 4615]</ref>
<ref>M. A. Pedrosa et al, ''Studies of spectra of the edge plasma fluctuations in toroidal magnetic confinement systems'', J. Plasma Fusion. Res. SERIES, '''2''' (1999) 77</ref>
and to determine its relation with local plasma parameters (rational surfaces, gradients, electric fields).
<ref>[http://dx.doi.org/10.1023/A:1022859915916 M.A. Pedrosa et al, ''Role of rational surfaces on fluctuations and transport in the plasma edge of the TJ-II stellarator'', Czechoslovak Journal of Physics, '''50''', 12 (2000) 1463]</ref>
<ref>[http://dx.doi.org/10.1088/0029-5515/42/10/305 B. Gonçalves et al, ''Experimental investigation of dynamical coupling between density gradients, radial electric fields and turbulent transport in the JET plasma boundary region'', Nucl. Fusion '''42''' (2002) 1205]</ref>
<ref>[http://dx.doi.org/10.1023/A:1026388305808 M.A. Pedrosa et al, ''Edge turbulence during limiter biasing experiments in the TJ-II stellarator'', Czechoslovak Journal of Physics, '''53''' (2003) 877]</ref>

=== Turbulence visualisation ===

Recently, much effort is being dedicated to the visualization of turbulent structures, and to the corresponding analysis techniques for extracting quantitative information from the images.
<ref>[http://www.new.ans.org/pubs/journals/fst/a_1250 J. A. Alonso et al, ''Two-Dimensional Turbulence Analysis Using High-Speed Visible Imaging in TJ-II Edge Plasmas'', Fusion Science and Technology '''50''', 2 (2006) 301]</ref>
<ref>[http://dx.doi.org/10.1088/0741-3335/48/12B/S44 J.A. Alonso et al, ''Impact of different confinement regimes on the two-dimensional structure of edge turbulence'', Plasma Phys. Control. Fusion '''48''' (2006) B465]</ref>
<ref>[http://dx.doi.org/10.1016/j.jnucmat.2009.01.140 D. Carralero et al, ''Turbulence studies by fast camera imaging experiments in the TJII stellarator'', J. Nucl. Mat. '''390-391''' (2009) 457]</ref>

== References ==
<references />

Non-diffusive transport

2010-11-24T06:58:33Z

Otihizuv:

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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 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.
<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. 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/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 performing this task, although some experiments in this sense are planned.

==References==
<references />

Hamada coordinates

2010-11-24T06:28:28Z

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Hamada coordinates are a set of [[Flux coordinates#Magnetic coordinates|magnetic coordinates]] in which the equilibrium current density <math>\mathbf{j}</math> lines are straight besides those of magnetic field <math>\mathbf{B}</math>. The periodic part of the [[Flux coordinates#Magnetic field representation in flux coordinates|stream functions]] of both <math>\mathbf{B}</math> and <math>\mathbf{j}</math> are flux functions (that can be chosen to be zero without loss of generality).

== Form of the Jacobian for Hamada coordinates ==
In this section, following D'haseleer et al we will translate the condition of straight current density lines into one for the '''Hamada''' coordinates Jacobian. For that we will make use of the equilibrium equation <math>\mathbf{j}\times\mathbf{B} = p'\nabla\psi </math>, which written in a magnetic coordinate system reads
:<math>
\frac{-I'_{tor}\Psi'_{pol} + I'_{pol}\Psi'_{tor}}{4\pi^2\sqrt{g_f}}
- \mathbf{B}\cdot\nabla\tilde{\eta} = p'~.
</math>
Taking the [[Flux coordinates#flux surface average|flux surface average]] <math>\langle\cdot\rangle</math> of this equation we find <math>(-{I}'_{tor}{\Psi}'_{pol} + {I}'_{pol}{\Psi}'_{tor})= 4\pi^2{p}'\langle(\sqrt{g_f})^{-1}\rangle^{-1}</math>, so that we have

:<math> \mathbf{B}\cdot\nabla\tilde{\eta} = {p}'\left(\frac{\langle(\sqrt{g_f})^{-1}\rangle^{-1}}{\sqrt{g_f}}-1\right)
</math>

In a coordinate system where <math>\mathbf{j}</math> is straight <math>\tilde{\eta}</math> is a function of <math>\psi</math> only, and therefore LHS of this equation must be zero in such a system. It therefore follows that the Jacobian of the '''Hamada''' system must satisfy
:<math>
\sqrt{g_H} = \langle\sqrt{g_H}^{-1}\rangle^{-1} = \frac{V'}{4\pi^2}~,
</math>
where the last identity follows from the [[Flux coordinates#Useful properties of the FSA|properties of the flux surface average]]. The Hamada angles are sometimes defined in 'turns' (i.e. <math>(\theta, \xi) \in [0,1)</math>) instead of radians (<math>(\theta, \xi) \in [0,2\pi)</math>)). This choice together with the choice of the volume <math>V</math> as radial coordinate makes the Jacobian equal to unity. Alternatively one can select <math>\psi = \frac{V}{4\pi^2}</math> as radial coordinate with the same effect.

== Magnetic field and current density expressions in a Hamada vector basis ==
With the form of the Hamada coordinates' Jacobian we can now write the explicit [[Flux coordinates#Contravariant Form|contravariant form]] of the magnetic field in terms of the '''Hamada''' basis vectors
:<math>
\mathbf{B} = 2\pi\Psi_{pol}'(V)\mathbf{e}_\theta + 2\pi\Psi_{tor}'(V)\mathbf{e}_\phi~.
</math>
This has the nice property of having flux constant contravariant coefficients (functions of the radial coordinate only). The current density contravariant looks alike
:<math>
\mu_0\mathbf{j} = 2\pi I_{pol}'(V)\mathbf{e}_\theta + 2\pi I_{tor}'(V)\mathbf{e}_\phi~.
</math>

The [[Flux coordinates#Convariant Form |covariant expression]] of the magnetic field is less clean
:<math>
\mathbf{B} = \frac{I_{tor}}{2\pi}\nabla\theta + \frac{I_{pol}^d}{2\pi}\nabla\phi + \nabla\tilde\chi~.
</math>
with contributions from the periodic part of the magnetic scalar potential <math>\tilde\chi</math> to all the covariant components. Nonetheless, the '''flux surface averaged Hamada covariant <math>B</math>-field angular components''' have simple expressions, i.e
:<math>
\langle B_\theta\rangle = \langle\mathbf{B}\cdot\mathbf{e}_\theta\rangle = \left\langle \frac{I_{tor}}{2\pi} + \frac{\partial \tilde\chi}{\partial \theta}\right\rangle = \frac{I_{tor}}{2\pi} + (V')^{-1}\int\partial_\theta\tilde\chi \sqrt{g} d\theta d\phi = \frac{I_{tor}}{2\pi}
</math>
where the integral over <math>\theta</math> is zero because the Jacobian in Hamada coordinates is not a function of this angle. Similarly
:<math>
\langle B_\phi\rangle = \langle\mathbf{B}\cdot\mathbf{e}_\phi\rangle = \frac{I^d_{pol}}{2\pi}~.
</math>

TJ-II:Thomson Scattering

2010-11-24T06:27:13Z

Otihizuv:

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[[File:Thomson.png|400px|thumb|right|The Thomson Scattering diagnostic at TJ-II]]

The high-resolution Thomson Scattering system
(located in [[TJ-II:Sectors|sector]] D2, &phi; = 14.5&deg;)
provides electron temperature, density, and pressure profiles at a single time in the discharge.
<ref>[http://link.aip.org/link/?RSINAK/70/763/1 C.J. Barth et al, ''High-resolution multiposition Thomson scattering for the TJ-II stellarator'', Rev. Sci. Instruments, Vol. '''70''', 1 (1999) 763-767]</ref>
<ref>[http://link.aps.org/doi/10.1103/PhysRevLett.85.4715 J. Herranz, I. Pastor, F. Castejón, E. de la Luna, I. García-cortés, C.J. Barth, E. Ascasíbar, J. Sánchez, and V. Tribaldos, ''Profile structures of TJ-II stellarator plamsas'', Phys. Rev. Lett. '''85''' (2000) 4715]</ref>
<ref>[http://link.aip.org/link/?RSINAK/72/3514/1 C.J. Barth et al, ''Calibration procedure and data processing for a TV Thomson scattering system'', Rev. Sci. Instruments, Vol. '''72''', 9 (2001) 3514-3527]</ref>
<ref>J. Herranz et al. ''Analysis of the Detection Process and Error Sources for the Thomson Scattering Profile Measurement in TJ-II''. Proc. issues of the 14th Topical Conf. on High Temperature Plasma Diagnostics, A.P.S., July 8, 2002, Madison, WI</ref>
<ref>[http://dx.doi.org/10.1016/S0920-3796(03)00387-9 J. Herranz et al, ''The spectrometer of the high-resolution multiposition Thomson scattering diagnostic for TJ-II'', Fus. Eng. And Design, '''65''' (2003) 525-536]</ref>
<ref>[http://link.aip.org/link/?RSINAK/74/3998/1 B.Ph. van Milligen et al, ''Revision of TV Thomson scattering data analysis and detection of profile structure'', Rev. Sci. Instrum. '''74''' (2003) 3998]</ref>
The laser chord is inclined by 17&deg; from the vertical.

For reliable operation of the diagnostic, the parameters of the plasma should lie in the design range.
E.g., very low density plasmas are unsuitable due to very low signal, intrinsic system noise, and stray light contamination, and very low temperature plasmas due to scattered spectra that lie mainly within the notch filter attenuation band.

== Calibration ==

'''Wavelength calibration''': The spectrometer is illuminated with light coming from Helium and Argon spectral lamps, both having well documented, intense lines in the range from 550 to 850 nm.

'''Position calibration''': A set of &sim; 20 carefully aligned pinholes, (diameter 300 microns, separation 20 mm), illuminated by a tungsten lamp, are imaged onto the spectrometer, to provide position calibration along the laser chord. In order to obtain the effective radius, the precise location of the laser chord and optical axis of the spectrometer with respect to TJII must be known. This measurement was performed when the system was first installed.

'''Relative calibration''': This procedure is used to calibrate the relative response of the spectrometer as a function of wavelength and position. A tungsten-halogen lamp with a known, broad spectrum (black-body like) is used to this effect. The current in the lamp is kept constant by using a regulated power supply.

'''Absolute calibration''': in order to provide absolute density calibration, Rayleigh scattering with N2, CO2 or other gases has been used. This method is hard to implement in TJII due to the large amount of stray light entering the spectrometer that makes the evaluation of Rayleigh scattering signal from the gas problematic. Other techniques, like evaluation of the absolute density factor in profiles at or near to an ECRH cut-off have been tried. Regular comparisons with data from the interferometer can be used as a method to get the absolute density factor on profiles. The shape of the ''ne'' profile is nevertheless robust, since it depends only on the relative calibration.

== Data description ==
The number of data points along the chord can vary from shot to shot depending on magnetic configuration, the shape of actual plasma, the influence of stray light level on data quality, etc; typically, it is of the order of 200. The estimated spatial resolution of diagnostic is approximately 2.4 mm.
Due to a delay between the software trigger and the firing of the laser, the Thomson Scattering measurement occurs 6.4 ms after the trigger time stored in the database.
Signal names in the [[TJ-II:Shot_database|TJ-II database]]:
'PerfilRho_', 'PerfilTe_', 'PerfildTe_', 'PerfilNe_', 'PerfildNe_', 'PerfilPe_', 'PerfildPe_'.

== References ==
<references />

Bayesian data analysis

2010-11-24T06:26:45Z

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The goal of [[:Wikipedia:Bayesian inference|Bayesian]]
<ref>D.S. Sivia, ''Data Analysis: A Bayesian Tutorial'', Oxford University Press, USA (1996) ISBN 0198518897</ref>
<ref>P. Gregory, ''Bayesian Logical Data Analysis for the Physical Sciences'', Cambridge University Press, Cambridge (2005) ISBN 052184150X</ref>
or integrated data analysis (IDA) is to combine the information from a set of diagnostics providing complementary information in order to recover the best possible reconstruction of the actual state of the system subjected to measurement.
<ref>[http://dx.doi.org/10.1088/0741-3335/44/8/306 R. Fischer, C. Wendland, A. Dinklage, et al, '' Thomson scattering analysis with the Bayesian probability theory'', Plasma Phys. Control. Fusion '''44''' (2002) 1501]</ref>
<ref>[http://dx.doi.org/10.1088/0741-3335/45/7/304 R. Fischer, A. Dinklage, and E. Pasch, ''Bayesian modelling of fusion diagnostics'', Plasma Phys. Control. Fusion '''45''' (2003) 1095-1111]</ref>
<ref>[http://link.aip.org/link/?RSINAK/75/4237/1 R. Fischer, A. Dinklage, ''Integrated data analysis of fusion diagnostics by means of the Bayesian probability theory'', Rev. Sci. Instrum. '''75''' (2004) 4237]</ref>
<ref>[http://www.new.ans.org/pubs/journals/fst/a_575 A. Dinklage, R. Fischer, and J. Svensson, ''Topics and Methods for Data Validation by Means of Bayesian Probability Theory'', Fusion Sci. Technol. '''46''' (2004) 355]</ref>
<ref>[http://dx.doi.org/10.1109/WISP.2007.4447579 J. Svensson, A. Werner, ''Large Scale Bayesian Data Analysis for Nuclear Fusion Experiments'', IEEE International Symposium on Intelligent Signal Processing (2007) 1]</ref>
<ref>[http://www.new.ans.org/pubs/journals/fst/a_10892 R. Fischer, C.J. Fuchs, B. Kurzan, et al., ''Integrated Data Analysis of Profile Diagnostics at ASDEX Upgrade'', Fusion Sci. Technol. '''58''' (2010) 675]</ref>
Like [[Function parametrization]] (FP), this technique requires having a ''forward model'' to predict the measurement readings for any given state of the physical system; however
* instead of computing an estimate of the inverse of the forward model (as with FP), IDA finds the best model state corresponding to a specific measurement by a maximization procedure (maximization of the likelihood);
* the handling of error propagation is more sophisticated within IDA, allowing non-Gaussian error distributions and absolutely general parameter interdependencies; and
* additionally, it provides a systematic way to include prior knowledge into the analysis.

== See also ==

* [[:Wikipedia:Markov chain Monte Carlo|Markov chain Monte Carlo]]

== References ==
<references />

TJ-II

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[[File:TJII_model.jpg|500px|thumb|right|TJ-II Model]]

TJ-II is a flexible Heliac installed at Spain's [[Laboratorio Nacional de Fusión|National Fusion Laboratory]].
It is one of Spain's [http://www.micinn.es/portal/site/MICINN/menuitem.6f2062042f6a5bc43b3f6810d14041a0/?vgnextoid=cb09846085f90210VgnVCM1000001034e20aRCRD Large Scientific Installations].
It is currently operational.

== History ==

[[File:Foto_grupo_Fusion_1996.jpg|300px|thumb|left|TJ-II and the TJ-II Team in 1996]]

The flexible Heliac TJ-II was designed on the basis of calculations performed by the team of physicists and engineers of [[CIEMAT]], in collaboration with the Oak Ridge National Laboratory ([http://en.wikipedia.org/wiki/ORNL ORNL], USA) and the Institut für PlasmaPhysik at Garching ([http://en.wikipedia.org/wiki/Max-Planck-Institut_f%C3%BCr_Plasmaphysik IPP], Germany).
<ref>[http://www.ornl.gov/info/reports/1987/3445602635147.pdf T.C. Hender et al, ''Studies of a flexible heliac configuration'', Report ORNL/TM-10374 (1987) OSTI ID: 6007697]</ref>
The TJ-II project received preferential support from [[Euratom]] for phase I (Physics) in 1986 and for phase II (Engineering) in 1990. The [[TJ-II:Construction|construction of this flexible Heliac]] was carried out in parts according to its constitutive elements, which were commissioned to various European companies, although 60% of the investments reverted back to Spanish companies.

The first plasma was produced in 1999.

== Precedents ==

TJ-II is the third magnetic confinement device in a series. In 1983, the device [[TJ-I]] was taken into operation.
The denomination of this device is due to the abbreviation of "Tokamak de la Junta de Energía Nuclear", this being the former denomination of [[CIEMAT]]. The abbreviation was maintained for successive devices for administrative reasons.

In 1994, the torsatron [[TJ-IU]] was taken into operation. This was the first magnetic confinement device entirely built in Spain. Currently, [[TJ-IU]] is located at the [http://www.ipf.uni-stuttgart.de/index_e.html University of Stuttgart] in Germany under the name of TJ-K.

== Description ==

[[File:TJ-II_3D_perspective.jpg|300px|thumb|right|TJ-II perspective view]]

In TJ-II, the magnetic trap is obtained by means of [[TJ-II:Coil system|various sets of coils]] that completely determine the magnetic surfaces before plasma initiation. The toroidal field is created by 32 coils. The three-dimensional twist of the central axis of the configuration is generated by means of two central coils: one circular and one helical. The horizontal position of the plasma is controlled by the vertical field coils. The combined action of these magnetic fields generate bean-shaped magnetic surfaces that guide the particles of the plasma so that they do not collide with the [[TJ-II:Vacuum system|vacuum vessel]] wall.

TJ-II discharges last around 0.25 s, with a repetition frequency of about 7 minutes.

== Goals and Research ==

The objective of the experimental program of TJ-II is to investigate the physics of a device with a helical magnetic axis having a great flexibility in its magnetic configuration, and to contribute to the international effort regarding the study of magnetic confinement devices for fusion.

Also refer to [[Plasma Physics at the LNF]].

== Operation ==

* A [[TJ-II:Vacuum system|vacuum system]] controls the pressure inside the vacuum vessel.
* The electric energy required for a TJ-II discharge is obtained from a [[TJ-II:Power supply|flywheel generator]].
* The [[TJ-II:Coil system|coils]] are cooled by means of a [[TJ-II:cooling system|cooling system]].
* An extensive set of systems is available to perform [[TJ-II:Plasma Wall Interaction|plasma wall conditioning]].
* Two movable [[TJ-II:Limiter|limiters]] can be used to limit the plasma.
* A [[TJ-II:Biasing probe|biasing probe]] can be used to apply a bias potential at the edge.

== Heating and fuelling ==

In order to fuel and heat the TJ-II plasma, the following systems are used:
* [[TJ-II:Gas puff|Gas puff]]
* [[TJ-II:Electron Cyclotron Resonant Heating|Electron Cyclotron Resonant Heating]] (ECRH)
* [[TJ-II:Neutral Beam Injection|Neutral Beam Injection]] (NBI)
* [[TJ-II:Electron Bernstein Wave Heating|Electron Bernstein Wave Heating]] (EBWH)
* [[TJ-II:Pellet injector|Pellet injector]] (PI)

== Control and data acquisition ==

The [[TJ-II:Control and data acquisition systems|Control and data acquisition systems]] were designed end developed at CIEMAT.

== Diagnostics ==

[[File:TJ-II_top_view_2009.jpg|400px|thumb|right|TJ-II in 2009; front left: Thomson Scattering; left and bottom right: NBI; top: HIBP]]

TJ-II is fitted with an extensive set of diagnostic systems installed in its 96 access [[TJ-II:Ports|ports]]. For information on the magnetic coordinate system (required for cross-diagnostic comparisons), see [[TJ-II:Magnetic_coordinates|TJ-II magnetic coordinates]].

''Passive diagnostics''
* [[TJ-II:Magnetics|Magnetics]]
* [[TJ-II:Halpha monitors|Halpha monitors]]
* [[TJ-II:Electron Cyclotron Emission|Electron Cyclotron Emission]]
* [[TJ-II:Soft X-rays|Soft X-rays]]
* [[TJ-II:Bolometry|Bolometry]]
* [[TJ-II:Spectroscopy|Spectroscopy]]
* [[TJ-II:Charge exchange spectroscopy|Charge exchange spectroscopy]]
* [[TJ-II:Compact Neutral Particle Analyzer|Compact Neutral Particle Analyzer]]
* [[TJ-II:Fast ion loss probe|Fast ion loss probe]]
* [[TJ-II:Fast camera|Fast camera]]

''Active diagnostics''
* [[TJ-II:Interferometry|Interferometry]]
* [[TJ-II:Reflectometry|Reflectometry]]
* [[TJ-II:Heavy Ion Beam Probe|Heavy Ion Beam Probe]]
* [[TJ-II:Langmuir Probes|Langmuir Probes]]
* [[TJ-II:Thomson Scattering|Thomson Scattering]]
* [[TJ-II:Diagnostic neutral beam|Diagnostic neutral beam]]
* [[TJ-II:Helium Beam|Helium Beam]]
* [[TJ-II:Lithium Beam|Lithium Beam]]

== Numerical resources ==

=== Simulation codes ===
* [[VMEC]] - 3D Plasma equilibrium, assuming nested flux surfaces
* [[PIES]] - 3D Plasma equilibrium
* [[ASTRA]] - Plasma transport
* [[PROCTR]] - Plasma transport
* [[EUTERPE]] - Gyrokinetic code
* [[EIRENE]] - A Monte Carlo neutral gas transport code
* [[FAFNER]]
* [[CUTIE]] - Full-tokamak fluid turbulence
* [[MOCA]] - Monte Carlo [[Neoclassical transport]] code
* [[TRECE]] - Microwave ray tracing
* [[Master]] - 1D Master Equation solver for [[Non-diffusive transport|non-diffusive transport]]
* [[TRUBA]]- Microwave beam/ray tracing including electron Bernstein wave calculations.

=== Data analysis ===
* [[Wave_ana]] - Linear and non-linear data analysis, spectral analysis using Fourier and Wavelets
* [[EBITA]] - Tomographic reconstruction
* [[TJ-II:Tomography|Tomography]] - Tomographic reconstruction based on mode decomposition in flux surface geometry
* [[TJ-II:FM|FM]] - Density reconstruction for the reflectometer

== References ==
<references />

Help:Contents

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Laboratorio Nacional de Fusión

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The National Fusion Laboratory is part of [[CIEMAT]].

The Laboratory is dedicated to the development of fusion by [[Magnetic confinement|magnetic confinement]] as a future energy generation option.
Research is mainly centered on the [[TJ-II|Flexible Heliac TJ-II]], and on materials studies.

== History ==

In 1975, a research group is created at the JEN (later to become [[CIEMAT]]) to study the subject of fusion.
In 1983, the small tokamak [[TJ-I]] is taken into operation, followed by the torsatron [[TJ-IU]] in 1994, and the flexible heliac [[TJ-II]] in 1999.

== Organization ==

[[LNF:Organization|Organization]] and personnel

== Projects and research ==

* [[TJ-II|The TJ-II Project]]
* [[Plasma Physics at the LNF|Plasma Physics at the LNF]]
* [[TECNO_FUS]]
* [[TechnoFusión]]

== Computer resources ==

Due to the large computational needs of the Laboratory, it makes use of both internal and external resources through collaborations:

* The CIEMAT computing centre, with the following computers:
** JEN50 (SGI Origin system, already phased out)
** Lince (HPC cluster)
** Fenix (SGI Altix system w/ Itanium processors, currently being phased out)
** Euler (Dell HPC cluster, 1152 Xeon cores, 13.8 Tflops)
* [http://www.bsc.es/ The Barcelona Supercomputing Centre]
* [http://www.res.es/ The Spanish Supercomputing Network]
* [http://www.computaex.es/ LUSITANIA]
* [http://bifi.unizar.es/ BIFI] (at the University of Zaragoza)
* [http://grid.bifi.unizar.es/egee/fusion-vo/ EGEE] (Enabling Grids for E-SciencE, a European computational grid)
* [http://www.ibercivis.es/ Ibercivis] (Spanish computational grid)

== Collaborations ==

The Laboratory participates in many international projects and collaborates with other institutions, such as:
* [http://www.jet.efda.org/ JET-EFDA]
* [http://en.wikipedia.org/wiki/ITER ITER]
* [http://www.ornl.gov/ Oak Ridge National Laboratory]
* [http://www.nifs.ac.jp/ NIFS]
* [http://www-jt60.naka.jaea.go.jp/english/index-e.html JT-60SA]
* [http://www.uc3m.es/ Universidad Carlos III], Madrid
* [http://www.tec.cr/sitios/vicerrectoria/vie/investigacion/plasma/Paginas/default.aspx Instituto Tecnológico de Costa Rica]
* [http://bifi.unizar.es/ Institute for Biocomputation and Physics of Complex Systems (BIFI)], Universidad de Zaragoza
* [http://www.bsc.es/ Barcelona Supercomputing Centre (BSC)]

== Events ==

The Laboratory has organised many events, among which:
* [http://linkinghub.elsevier.com/retrieve/pii/S0920379601005865 The 21st Symposium on Fusion Technology] (SOFT, 2000)
* [http://www-fusion.ciemat.es/ttf2002 The 9th EU-US Transport Task Force Meeting] (TTF, 2002)
* [http://eps2005.ciemat.es The 32nd European Physical Society Conference on Plasma Physics] (EPS, 2005)
* [http://www-fusion.ciemat.es/SW2005/ The 15th International Stellarator Workshop] (ISW, 2005)
* [http://plasma2.ulb.ac.be/EFTC/Documents_EFTC12_191206SHORT_MADRID/index.html The 12th European Fusion Theory Conference] (EFTC, 2008)
* [http://psi2008.ciemat.es/ The 18th Conference on Plasma Surface Interactions] (PSI, 2008)
* [http://www-fusion.ciemat.es/ttg2010/ The 3rd EFDA Transport Topical Group meeting and 15th EU-US Transport Task Force Meeting] (TTG-TTF, 2010)

== External Links ==

[http://www-fusion.ciemat.es Website of the Laboratorio Nacional de Fusión]

[http://www.ciemat.es/ Website of CIEMAT]

Edge Localized Modes

2010-11-24T01:17:49Z

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The steep edge gradients (of density and temperature) associated with an [[H-mode]] lead to quasi-periodic violent relaxation phenomena, known as Edge Localized Modes (ELMs), which have a strong impact on the surrounding vessel.
<ref>[http://dx.doi.org/10.1088/0741-3335/38/2/001 H. Zohm, ''Edge localized modes (ELMs)'', Plasma Phys. Control. Fusion '''38''' (1996) 105-128]</ref>
<ref>[http://dx.doi.org/10.1016/S0022-3115(97)80039-6 D.N. Hill, ''A review of ELMs in divertor tokamaks'', Journal of Nuclear Materials '''241-243''' (1997) 182-198]</ref>

== Physical mechanism ==

The physical mechanism of ELMs has not been fully clarified. Several possible explanations have been put forward:
* Nonlinear interchange modes <ref>[http://dx.doi.org/10.1088/0741-3335/38/8/046 A. Takayama and M. Wakatani, ''ELM modelling based on the nonlinear interchange mode in edge plasma'', Plasma Phys. Control. Fusion '''38''' (1996) 1411-1414]</ref>
* Coupled peeling-ballooning modes <ref>[http://link.aip.org/link/?PHPAEN/5/2687/1 J.W. Connor et al, ''Magnetohydrodynamic stability of tokamak edge plasmas'', Phys. Plasmas '''5''' (1998) 2687]</ref><ref>[http://link.aip.org/link/?PHPAEN/9/2037/1 P.B. Snyder et al, ''Edge localized modes and the pedestal: A model based on coupled peeling–ballooning modes'', Phys. Plasmas '''9''' (2002) 2037]</ref><ref>[http://dx.doi.org/10.1088/0741-3335/46/8/003 J.-S. Lönnroth et al, ''Predictive transport modelling of type I ELMy H-mode dynamics using a theory-motivated combined ballooning–peeling model'', Plasma Phys. Control. Fusion '''46''' (2004) 1197-1215]</ref><ref>[http://dx.doi.org/10.1088/0029-5515/49/9/095015 N. Hayashi et al, ''Integrated simulation of ELM energy loss and cycle in improved H-mode plasmas'', Nucl. Fusion '''49''' (2009) 095015]</ref>
* Peeling modes <ref>[http://link.aip.org/link/?APCPCS/871/87/1 C.G. Gimblett, ''Peeling mode relaxation ELM model'', AIP Conf. Proc. '''871''' (2006) 87-99]</ref>
* Flux surface peeling <ref>[http://dx.doi.org/10.1016/j.jnucmat.2004.09.067 E.R. Solano et al, ''ELMs and strike point jumps'', Journal of Nuclear Materials '''337-339''' (2005) 747-750 ]</ref>
* [[Criticality of MHD equilibrium]] <ref> [http://dx.doi.org/10.1088/0741-3335/46/3/L02 Emilia R. Solano, ''Criticality of the Grad–Shafranov equation: transport barriers and fragile equilibria'', Plasma Phys. Control. Fusion '''46''' (2004) L7-L13] </ref>
* [[Self-Organised Criticality]] <ref>[http://dx.doi.org/10.1088/0029-5515/43/10/003 R. Sánchez et al, ''Modelling of ELM-like phenomena via mixed SOC-diffusive dynamics'', Nucl. Fusion '''43''' (2003) 1031-1039 ]</ref>

== ELMs and machine operation ==

The occurrence of an ELM leads to a significant expulsion of heat and particles, with deleterious consequences for the vessel wall and machine operation.
Although Quiescent H-modes exist (without ELMs),
<ref>[http://link.aip.org/link/?PHPAEN/12/056121/1 K.H. Burrell et al, ''Advances in understanding quiescent H-mode plasmas in DIII-D'', Phys. Plasmas '''12''' (2005) 056121]</ref>
they are generally considered not convenient due to the accumulation of impurities.
To achieve steady state, an ELMy H-mode is preferred and this mode of operation is proposed as the standard operating scenario for [[ITER]], thus converting ELM mitigation into a priority.
<ref>[http://dx.doi.org/10.1016/j.fusengdes.2009.01.063 M.R. Wade, ''Physics and engineering issues associated with edge localized mode control in ITER'', Fusion Engineering and Design '''84''', Issues 2-6 (2009) 178-185]</ref>

== References ==
<references />

Continuous Time Random Walk

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The Continuous Time Random Walk (CTRW) provides a mathematical framework for the study of transport in heterogenous media. It is much more general than usual transport models based on (local, Markovian) Ordinary Differential Equations, and in particular can handle transport in systems without characteristic scales (such as systems in a state of [[Self-Organised Criticality]] or SOC).

== Motivation ==

Interestingly, the absence of local characteristic scales means that ''effective'' transport coefficients (the diffusivity etc.) become dependent on the system size, as is indeed suggested by experimental [[Scaling law|scaling laws]] for plasma confinement.

In the framework of transport in plasmas, it is believed that the presence of ''trapping regions'' (such as turbulent eddies, magnetic islands, internal transport barriers) may lead to [[Non-diffusive transport|sub-diffusion]], whereas the occurrence of ''streamers'' and profile self-regulation (via [[TJ-II:Turbulence|turbulence]]) may lead to [[Non-diffusive transport|super-diffusion]].
The goal of the CTRW approach is to model the effective transport in the presence of these complex phenomena.

== Starting point ==

CTRW theory
<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>
starts from the single-particle step distribution function (in one dimension)

:<math>\xi(\Delta x, \Delta t; x, t)</math>

giving the probability that a particle, located at position ''x'' at time ''t'', takes a step of size ''&Delta;x'' after waiting a time ''&Delta;t'' &gt; 0. All particles are assumed to be identical.

== The Master Equation ==

By making some suitable additional assumptions regarding the nature of this single-particle step distribution, it is possible to compute the average behaviour of the system in the limit of infinitely many particles, and to deduce an evolution equation for the particle (probability) density.
This equation is a (Generalized) Master Equation.

In standard CTRW theory, it is customary to assume that the single particle step distribution is ''separable'', i.e., that ''&Delta;x'' is independent from ''&Delta;t'', so that

: <math>\xi( \Delta x, \Delta t; x,t) = p(\Delta x; x,t+\Delta t) \psi(\Delta t; x,t)</math>

In addition, homogeneity in space and time is assumed (i.e., ''p'' and ''&psi;'' do not depend on ''x'' and ''t''). However, recently it was shown that a Master Equation can also be derived in the case that ''p'' depends on ''x'' and ''t'', while ''&psi;'' depends on ''x'' (but not ''t'').
<ref>[http://link.aip.org/link/?PHPAEN/11/2272/1 B.Ph. van Milligen, R. Sánchez, and B.A. Carreras, ''Probabilistic finite-size transport models for fusion: anomalous transport and scaling laws'', Phys. Plasmas '''11''', 5 (2004) 2272]</ref>
This significant extension of the standard CTRW model has led to the development of a model with very interesting properties from the point of view of plasma transport (see the cited reference and <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>).

The Generalized Master Equation (GME) can be written in the form

:<math>\frac{\partial n(x,t)}{\partial t} = \int_0^t \left ( \int{K(x-x',t-t',x',t')n(x',t')dx'} - n(x,t')\int{K(x-x',t-t',x',t')dx'}\right )dt'</math>

where ''n'' is the particle (probability) density, and ''K'' a kernel of the form

:<math>K( \Delta x, \Delta t; x,t) = p(\Delta x; x,t+\Delta t) \phi(\Delta t; x)</math>

The GME is an integro-differential equation, generalizing the usual (partial differential) equations for transport.
The particle flux at any point in space depends on the global distribution of the transported particle density field, and on its history (although the history dependence can be eliminated by choosing a Markovian waiting time distribution).

The treatment of boundary conditions in a GME is different from standard differential equations. <ref>[http://dx.doi.org/10.1088/1751-8113/41/21/215004 B.Ph. van Milligen, I. Calvo, and R. Sánchez, ''Continuous time random walks in finite domains and general boundary conditions: some formal considerations'', J. Phys. A: Math. Theor. '''41''' (2008) 215004]</ref>
The final (quasi) steady state of the system is a function of the balance between sources and sinks, rather than of imposed values or gradients at the system boundaries.

== Fractional Differential Equations ==

While the Master Equation corresponds to a CTRW in the limit of many particles, the
Fractional Differential Equation (FDE) corresponds to a Master Equation in the ''fluid limit''.
<ref>[http://link.aps.org/doi/10.1103/PhysRevE.71.011111 R. Sánchez, B.A. Carreras, and B.Ph. van Milligen, ''Fluid limit of nonintegrable continuous-time random walks in terms of fractional differential equations'', Phys. Rev. E '''71''' (2005) 011111]</ref>
The fluid limit is the limit in which only the part of the dynamics that is dominant for large scales and long times is retained, and is useful for understanding the steady state properties of a solution.

To obtain the fractional differential operators, it is necessary to make an assumption regarding the shape of the distributions appearing in the kernel ''K''. Invoking the Generalized Limit Theorem for the sums of random variables,
<ref>B. V. Gnedenko and A. N. Kolmogorov, ''Limit Distributions of Sums of Independent Random Variables'', Addison-Wesley, Reading, MA (1954)</ref>
these distributions are taken to be [[:Wikipedia:Stable_distribution|Lévy distributions]].
While the step distribution can be any Lévy distribution, the waiting time distribution must be ''positive extremal'', since &Delta;''t'' &gt; 0.
This choice allows modelling both
[[Non-diffusive transport|sub- and super-diffusive transport]], and in the appropriate limit, standard ("Fickian") transport is recovered.
If nothing else, this serves to show that all of the above constitute generalizations (on various levels) of the usual transport equations.

The main numerical advantage of the FDE approach over the GME is that the FDE allows constructing the final solution in the long-time limit by a single integration,
<ref>[http://dx.doi.org/10.1016/j.jcp.2003.07.008 V.E. Lynch et al, ''Numerical methods for the solution of partial differential equations of fractional order'', Journal of Computational Physics '''192''', 2 (2003) 406-421]</ref>
whereas the GME must be iterated in time.
The FDE approach can be used fruitfully to model transport in fusion plasmas, i.e., finite-size systems.
<ref>[http://link.aip.org/link/?PHPAEN/13/082308/1 D. del-Castillo-Negrete, ''Fractional diffusion models of nonlocal transport'', Phys. Plasmas '''13''' (2006) 082308]</ref>
On the other hand, the FDE approach does not capture some of the (interesting) dynamical behaviour inherent in the GME approach.
<ref>[http://dx.doi.org/10.1088/0029-5515/47/3/004 B.Ph. van Milligen, B.A. Carreras, V.E. Lynch and R. Sánchez, ''Pulse propagation in a simple probabilistic transport model'', Nucl. Fusion '''47''' (2007) 189]</ref>

== References ==
<references />

Effective plasma radius

2010-11-24T00:59:03Z

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The definition of the effective plasma radius is non-trivial for stellarators, yet needed for comparing the measurements of diagnostics.
Various alternative definitions are possible, some of which are discussed below.
The numerical value of the effective radius has hardly more significance than that of providing a flux surface label,
except when plasmas from different machines are compared.
Therefore, this issue is relevant for the elaboration of multi-machine [[Scaling law]]s.

== Normalized effective radius based on flux ==

If the [[Flux surface|flux surfaces]] are known (typically, by calculating the [[MHD equilibrium|Magneto-Hydrodynamic equilibrium]]), and assuming the existence of toroidally nested flux surfaces, the simplest procedure is to define the mean radius as a function of some flux quantity (i.e., any quantity that is constant on a flux surface).

E.g., at [[TJ-II]], magnetic equilibria can be obtained from the [[VMEC]] code (see [[TJ-II:Magnetic coordinates]]). It returns ''&psi;'', the toroidal magnetic flux. The normalized effective radius is defined by

:<math>\rho_{\rm eff} = \sqrt{\psi_N}</math>

where ''&psi;N'' is the normalized toroidal flux, such that it is 0 on the magnetic axis and 1 at the Last Closed Flux Surface (LCFS).

== Effective radius based on flux ==

To obtain the dimensional effective radius ''r''eff (in meters) of a flux surface, it is common to make the assumption that the shape of the flux surface does not deviate much from a [[:Wikipedia:Torus|torus]]. In this case, several possibilities exist to define a radius:
* Based on the volume ''V(&psi;)'' enclosed in a flux surface (using ''V'' = 2 &pi;2''Rr''eff2)
* Based on the surface area ''S(&psi;)'' of a flux surface (using ''S'' = 4 &pi;2''Rr''eff)
Here, ''R'' is the [[Toroidal coordinates|major radius]] of the [[:Wikipedia:Torus|torus]].
Particularly in helical systems, choosing a value of ''R'' may be inappropriate (since the magnetic axis is not a circle, and the shape of the flux surfaces deviates from that of a torus).
One can avoid making an (arbitrary) choice for ''R'' by defining
* ''r''eff = 2''V/S''
This still implicitly assumes the surfaces are very similar to a torus.

A different approach is offered by recognizing that the flux surfaces are topological toroids of a single parameter.
Then, the surface area and volume corresponding to such surfaces are related via a differential equation.
Assuming only that ''S'' is linear in ''r''eff (or ''V'' is cuadratic in ''r''eff), it follows that:
* ''r''eff = ''dV/dS''
This definition is more general, although its validity is subject to the mentioned assumption. A fully general definition follows from
* <math>r_{\rm eff} = \int_0^V{dV'/S(V')}</math>
but it requires knowledge of the full equilibrium in terms of the function ''S(V)''.

== Effective radius based on poloidal cross sections ==

A poloidal cross section is a cut of the flux surface with the plane ''&phi; = cst''.
The result of such a cut is a closed curve, of which its circumference and area are easily determined; an effective plasma radius can then be deduced, assuming the curve deviates only slightly from a circle.
The mean plasma radius can be determined by averaging the result over the angle ''&phi;''.

While the procedure is adequate for toroidally symmetric plasmas, it is not clear that this is also the case for non-axisymmetric systems, since the flux surface intersects the plane ''&phi; = cst'' obliquely, possibly leading to an over-estimate of the actual plasma size.
The intersection angle can be deduced from the inner product
:<math>\vec \nabla \psi \cdot \vec \nabla \phi</math>
which is zero for axisymmetic systems (since ''&psi;'' does not depend on ''&phi;''), but non-zero for stellarators.

== Effective radius based on field lines ==

If the flux surfaces are not known, the effective radius of a surface traced out by a field line can be found by following the field line and calculating the geometric mean of the distance between points on the field line and the magnetic axis. The mean should be weighed with ''1/B'' in order to account for the variation of the field strength along the flux surface.

This procedure, while general in principle, still assumes that the field lines lie on flux surfaces.
It can be used for magnetic configurations with [[Magnetic island|magnetic islands]], although this requires applying some special treatment for points inside the islands. It may be argued that assigning an effective radius to spatial points inside a magnetic island is not very useful, since such points are topologically disconnected from the main plasma volume. Similarly, the definition of an effective radius in ergodic magnetic zones is ambiguous, since the concept of flux surface has no meaning inside an ergodic zone.
<ref>[http://dx.doi.org/10.1016/j.jcp.2008.02.026 B. Seiwald et al, ''Optimization of energy confinement in the 1/&nu; regime for stellarators'', Journal of Computational Physics '''227''', 12 (2008) 6165-6183]</ref>

== Hybrid definitions ==

* Use the flux-based normalized effective radius &rho;eff defined above and multiply by the mean field-line based radius of the LCFS.

== See also ==

* [[:Media:Surf_vol.pdf|Comments on the use of the minor radius for stellarators]], where it is argued that it is preferable to use the Volume or Surface directly, instead of the effective radius, in view of the ambiguities in the definition of the latter - at least when making comparisons between different machines, e.g., in the framework of [[Scaling law]]s.

== References ==
<references />

Topology and transport

2010-11-24T00:38:46Z

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The aim of this page is to introduce and report progress on the activities included in the research project ''Influence of global flows and their topology on transport in turbulent plasmas'', funded by the Spanish [http://www.micinn.es Ministry of Science and Innovation] through the grant ENE2009-07247. The members of the research team are:

* Iván Calvo (Principal Investigator), [[Laboratorio Nacional de Fusión]], Asociación EURATOM-CIEMAT, Madrid,
* Benjamín Carreras, BACV Solutions Inc., Oak Ridge, Tennessee & University of Alaska,
* Irene Llerena, Universidad de Barcelona,
* Edilberto Sánchez, Laboratorio Nacional de Fusión, Asociación EURATOM-CIEMAT, Madrid,
* Guillermo Sánchez Burillo, Laboratorio Nacional de Fusión, Asociación EURATOM-CIEMAT, Madrid,
* Boudewijn van Milligen, Laboratorio Nacional de Fusión, Asociación EURATOM-CIEMAT, Madrid,

and this is a list of frequent collaborators (whose work is not directly funded by the above grant):

* Francisco Castejón, Laboratorio Nacional de Fusión, Asociación EURATOM-CIEMAT, Madrid,
* Fernando Falceto, Universidad de Zaragoza,
* Luis García, Universidad Carlos III, Madrid,
* Carlos Hidalgo, Laboratorio Nacional de Fusión, Asociación EURATOM-CIEMAT, Madrid,
* M. Ángeles Pedrosa, Laboratorio Nacional de Fusión, Asociación EURATOM-CIEMAT, Madrid,
* Raúl Sánchez, Oak Ridge National Laboratory, Oak Ridge, Tennessee & Universidad Carlos III, Madrid.

The aim of this proposal is to advance the understanding of turbulent transport in
magnetically confined fusion plasmas and study the impact of the topology and dynamics of global flows on it. We will
tackle concrete problems in three broad research areas of toroidally confined plasmas: topology of
structures in the presence of turbulence (or, in short, "the topology of flows"), non-diffusive transport,
and transitions to improved confinement regimes. Below, we summarize the concrete
objectives in each of the aforementioned research lines, as well as progress made to date.

== Topology of flows ==

Our long-term aim is to develop a set of diagnostics to characterize the
topological structures in fluid and particle models of a turbulent plasma. In the frame of this project,
and during the next three years, we plan to study the problem in pressure-gradient-driven turbulence,
with a possible extension to gyrokinetic models. For this we will make use of Computational
Homology. An important tool will be the software developed by the [http://chomp.rutgers.edu Computational Homology Project (CHomP)] which, in particular, computes some topological invariants
(the Betti numbers) of three-dimensional spaces. We will investigate the possibility of developing or
using other tools which allow to improve the numerical resolution of the calculations.

== Non-diffusive transport ==

Turbulent flows induce non-collisional transport in plasmas. The
final goal is to understand in a precise way the origin and implications of the non-Gaussian and non-
Markovian character of transport in certain dynamical and topological conditions of those turbulent
flows. We will try to make progress in the answer to questions such as: a) The relationship between
the topology of flows and the presence of non-diffusive transport. b) The connection between the
statistical properties of turbulence and the description by means of stochastic and kinetic equations of
particle transport. c) Find out whether there exists a mathematical framework in which one can derive
linear, fractional diffusion equations (in particular non-local) from non-linear, partial differential
equations (in particular local).

== Transitions to improved confinement regimes ==

The generation of global flows allows to
have regimes with enhanced confinement. We will study the importance of the interaction between
disparate spatio-temporal scales in the generation of sheared flows and the concomitant turbulence
reduction. Our work will be based on recent experimental results in TJ-II. We will use topological
techniques in phase space to gain insight into the dynamics of models with multiple states.

* '''Zonal Flows and long-range correlations in TJ-II'''

Recent experimental results in the TJ-II stellarator prove the existence of long-distance correlations
in the electrostatic potential around the bifurcation point for the emergence of the plasma edge sheared
flow layer <ref>[http://prl.aps.org/abstract/PRL/v100/i21/e215003 M.A. Pedrosa et al., Phys. Rev. Lett. '''100''', 215003 (2008)]</ref>. Trying
to understand these results from a theoretical point of view, we have formulated a phenomenological model
<ref>[http://www.iop.org/EJ/abstract/-search=68892777.7/0741-3335/51/6/065007 I. Calvo et al., Plasma Phys. Control. Fusion '''51''', 065007 (2009)]</ref> based on the
paradigm of flow shear generation by Reynolds stress and turbulence suppression by
shear. We suggest that the experimental results might be an indirect evidence of the development of poloidally asymmetric zonal flows.

== References ==
<references />

Nuclear fusion

2010-11-24T00:29:06Z

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Nuclear fusion is the process by which multiple like-charged atomic nuclei join together to form a heavier nucleus. It is accompanied by the release or absorption of energy.
See [[:Wikipedia:Nuclear_fusion|Wikipedia: Nuclear fusion]].

== Energy policy issues ==

There exist a wide consensus that the current methods for energy production are unsatisfactory in the long term, due to contamination, the greenhouse effect, diminishing resources, etc.
In order to decide what energy generation methods should be used, the pros and contras of each method should be considered carefully.
<ref>[http://en.wikipedia.org/wiki/Energy_development Energy development]</ref>
Thus, making a policy choice in favour of one or the other energy option requires defining one's stance on:
* The importance of climate change and the impact of the burning of fossil fuels<ref>[http://en.wikipedia.org/wiki/Intergovernmental_Panel_on_Climate_Change Intergovernmental Panel on Climate Change]</ref><ref>[http://www.newscientist.com/article/dn11462 Climate change: A guide for the perplexed]</ref><ref>[http://www.youtube.com/watch?v=2T4UF_Rmlio The American denial of Global Warming]</ref>
* Quantitative estimates of the energy generation potential of each of the available energy options
* Estimates of global population growth<ref>[http://en.wikipedia.org/wiki/Population_growth Population growth]</ref><ref>[http://www.populationconnection.org/ PopulationConnection.org]</ref> and expectations regarding future energy demand<ref>[http://www.worldenergyoutlook.org/ World Energy Outlook]</ref><ref>[http://www.eia.doe.gov/ U.S. Energy Information Administration]</ref>, taking into account the rapidly rising energy needs of emerging economies
* The relative importance of the environmental impact of each of the energy options
* Social threats associated with each energy option: e.g., nuclear proliferation, or the threats associated with politically unstable energy supply regions
* The social acceptability of each energy option
* The relative economic cost of each of the energy options (contemplating the complete energy generation trajectory, including environmental damage and clean-up)
* Opportunities offered by the energy options in terms of, e.g., economic stimulation and employment
Making the correct choice requires studying each of these complex issues and somehow balancing the risks and opportunities involved.
For some of them, the future evolution can be predicted with some confidence,
but for others the predictions are hotly debated.

In any case, the choice for any specific energy generation options should not be considered in isolation from other global issues, such as the exhaustion of natural resources, poverty, and overpopulation, but rather as an element in the general framework of sustainable development, since all these issues must be addressed to guarantee the establishment of a stable and livable society.
In particular, it is important to be aware that any technological solution to the energy problem (as well as to, e.g., agricultural production levels) will only mean temporary relief if population growth is not controlled.
The latter issue should therefore receive top priority in any effort to attain sustainable development.

Whatever the case may be, energy generation in the near future will most likely be based on a mix of many options, that will vary in accord with local economic, environmental, and social conditions.

== Energy generation in the future ==

As hinted at above, it appears desirable to significantly reduce our dependence on fossil fuels, due to the effect of the burning of fossil fuels on the global climate, contamination, and the accompanying loss of valuable resources (e.g., plastics).

What alternatives are available? A host of methods for energy generation exists that do not depend on fossil fuels. Among the primary such sources are wind energy, solar energy, and hydroelectric energy. Other sources, such as geothermal energy and wave energy, are and probably will remain quantitatively less important, or compete with food supply, as is the case with biofuels.

Hydroelectric energy is a potential 'base load' energy source, meaning that it is, or can be, available permanently and on demand. However, this energy source is only available at relatively few locations where the orography and climate is suitable.

Wind and solar energy are potentially plentiful, but strongly dependent on local weather. Therefore, they are not generally considered to be 'base load' energy sources, and can only serve to supplement other reliable energy sources. Energy generation should not be interrupted on a cloudy, windless winter's day! Combinations of various distinct power sources could mitigate this problem somewhat, but it is unlikely that it can be eliminated.

Nuclear fission power is an alternative that does not contribute to the greenhouse effect and serves as base load power supply, but suffers from problems associated with nuclear waste storage and processing, and public acceptance.

Therefore, the search for a base load power source for the future is still an unresolved issue.
The discovery of a method to store energy on a large scale would completely change the picture, and enhance the viability of solar and wind energy; but currently, no such method is available.<ref>[[:Wikipedia:Grid_energy_storage]]</ref>

== Fusion as an energy option ==

Fusion undoubtedly offers some important advantages. Once operative, energy supply would be virtually limitless; greenhouse gas exhaust would be zero; nuclear waste and the danger of nuclear accidents would be strongly reduced (with respect to fission power plants), and nuclear proliferation problems would be small or inexistent. On the other hand, there are complications due to the very complex technology required and the radioactive activation of the reactor vessel components.
A significant part of the latter complications are due to the projected use of D-T fuels (deuterium-tritium) in the first-generation fusion power plants, which is the fuel that is easiest to ignite, but which leads to intense neutron radiation. One may speculate that, if successful, a second generation of fusion power plants can be developed that runs on other fuel mixtures (such as D-D), leading to a reduction of the problems associated with radioactivity.

Differing from some other energy options, the implementation of energy generation by fusion is not immediate, and subject to the solution of a number of technical problems. The current consensus it that while the technical challenges are formidable, they can be overcome. Thus, the main discussion regarding fusion as an energy option is not about its technical feasibility, but about the timescales for implementation.
<ref>[http://dx.doi.org/10.1016/j.fusengdes.2005.08.015 C. LLewellyn Smith, Fusion Engineering and Design '''74''', Issues 1-4 (2005) 3-8]</ref>
While increased investment and improved focus of the current research efforts can certainly help to speed up progress, even under optimal conditions the time needed to achieve the first delivery of fusion-produced energy to the electricity grid is considerable, and it is unlikely that fusion can contribute to solving the short-term energy crisis (in the coming decades). Fusion must therefore be considered an energy option for the medium to long term.

== A fusion reactor ==

The fusion reaction that is easiest to obtain is the deuterium-tritium (DT) reaction.
A [[:Wikipedia:Fusion power|fusion power reactor]] delivering 1 GW of electric power to the network would approximately consume 200 kg of Tritium a year. The current world reserves are about 29 kg of tritium. Thus, a nuclear fusion reactor must provide its own fuel. This is achieved using so-called [[Breeding blanket|breeders]]. Tritium breeders capture the neutrons originating from nuclear fusion reactions, generating tritium that can be used as fuel for the reactor. For more information: [[TECNO_FUS]].

== The need for new materials for Fusion ==

A fusion power reactor delivering 1GW of electric power to the network would generate 1.3 &times; 1021 neutrons per second. This flux will make any conventional iron become brittle in less than a year. For this reason, a program for testing materials under intense neutron fluxes has been launched. The aim of the [[IFMIF]] program is to develop a fast neutron generation facility.

Also in Spain, a program for material testing has been launched recently:
[[TechnoFusión]].

==See also==

* [[:Wikipedia:Timeline of nuclear fusion|Timeline of nuclear fusion]]
* The [[ITER]] project

==References==
<references />

Biorthogonal decomposition

2010-11-24T00:23:00Z

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The Biorthogonal Decomposition (BOD, also known as Proper Orthogonal Decomposition, POD<ref>P. Holmes, J.L. Lumley, and G. Berkooz, ''Turbulence, Coherent Structures, Dynamical Systems and Symmetry'', Cambridge University Press (1996) ISBN 0521634199</ref>) applies to the analysis of multipoint measurements

:<math>Y(i,j)\,</math>

where ''i=1,...,N'' is a temporal index and ''j=1,...,M'' a spatial index (typically).
The time traces ''Y(i,j)'' for fixed ''j'' are usually sampled at a fixed rate (so ''t(i)'' is equidistant); however the measurement locations ''x(j)'' need not be ordered in any specific way.

The BOD decomposes the data matrix as follows:

:<math>Y(i,j) = \sum_k \lambda_k \psi_k(i) \phi_k(j),\,</math>

where &psi;k is a 'chrono' (a temporal function) and &phi;k a 'topo' (a spatial or detector-dependent function), such that the chronos and topos satisfy the following orthogonality relation

:<math>\sum_i{\psi_k(i)\psi_l(i)} = \sum_j{\phi_k(j)\phi_l(j)} = \delta_{kl}.\,</math>

The combination chrono/topo at a given ''k'', &psi;k(i) &phi;k(j),
is called a spatio-temporal 'mode' of the fluctuating system, and is constructed from the data matrix without any prejudice regarding the mode shape.
The &lambda;k are the eigenvalues (sorted in decreasing order), where ''k=1,...,min(N,M)'', and directly represent the square root of the fluctuation energy contained in the corresponding mode.
This decomposition is achieved using a standard [[:Wikipedia:Singular value decomposition|Singular value decomposition]] of the data matrix ''Y(i,j)'':

:<math>Y = U S V^T.\,</math>

where ''S'' is a diagonal ''N&times;M'' matrix and ''Skk'' = &lambda;k, the first min(''N,M'') columns of ''U'' (''N&times;N'') are the chronos and the first min(''N,M'') columns of ''V'' (''M&times;M'') are the topos. <ref>[[:Wikipedia:MATLAB|MATLAB]] code: <code>[U,S,V] = svd(Y,'econ');</code></ref>

Thus, the oscillations of the spatiotemporal fluctuating field are represented by means of a very small number of spatio-temporal modes that are constructed from the data themselves, without prejudice regarding the mode shape.
<ref>[http://link.aip.org/link/?PHPAEN/1/3288/1 T. Dudok de Wit et al., ''The biorthogonal decomposition as a tool for investigating fluctuations in plasmas'', Phys. Plasmas '''1''' (1994) 3288]</ref>

A limitation of the technique is that it assumes space-time separability.
This is not always the most appropriate assumption:
e.g., travelling waves have a structure such as ''cos(kx-&omega;t)''; however, most propagating waves can still be recognised clearly by their distinct footprint in the biorthogonal modes (provided there are not too many): a travelling wave will produce a pair of modes with similar amplitude and a 90&deg; phase difference.

== Relation with signal covariance ==

Assuming the signals ''Y(i,j)'' have zero mean (their temporal average is zero, or &Sigma;i ''Y(i,j)'' = 0), their [[:Wikipedia:Covariance|covariance]] is defined as:

:<math>C(j_1,j_2) = \sum_i {Y(i,j_1)Y(i,j_2)},\!</math>

Substituting the above expansion of ''Y'' and using the orthogonality relations, one obtains:

:<math>C(j_1,j_2) = \sum_k {\lambda_k^2 \phi_k(j_1)\phi_k(j_2)}</math>

The technique is therefore ideally suited to perform cross covariance analyses of multipoint measurements.

By multiplying this expression for the covariance matrix ''C'' with the vector &phi;k it is easy to show that the topos &phi;k are the eigenvectors of the covariance matrix ''C'', and &lambda;k2 the corresponding eigenvalues.

== See also ==

* [[:Wikipedia:Principal component analysis|Principal component analysis]]

== References ==
<references />

Flux coordinates

2010-11-24T00:22:48Z

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== General curvilinear coordinates ==
Here we briefly review the basic definitions of a general [[:Wikipedia:Curvilinear coordinates | curvilinear coordinate system]] for later convenience when discussing toroidal flux coordinates and magnetic coordinates.

=== Coordinates and basis vectors ===
Let <math>{\mathbf x}</math> be a set of euclidean coordinates on <math>{\mathbb R}^3</math> and let <math>(\psi(\mathbf{x}),\theta(\mathbf{x}),\phi(\mathbf{x}))</math> define a change of coordinates, arbitrary for the time being.
We can calculate the contravariant basis vectors as
:<math>
\mathbf{e}^i = \{\nabla\psi, \nabla\theta, \nabla\phi\}
</math>
and the dual covariant basis defined as
:<math>
\mathbf{e}_i= \frac{\partial\mathbf{x}}{\partial{u^i}}
\to
\mathbf{e}_i\cdot\mathbf{e}^j
= \delta_{i}^{j} \to \mathbf{e}_i
= \frac{\mathbf{e}^j\times\mathbf{e}^k}{|\mathbf{e}^i\cdot\mathbf{e}^j\times\mathbf{e}^k|}
= \sqrt{g}\;\mathbf{e}^j\times\mathbf{e}^k ~,
</math>
where <math>(i,j,k)</math> are cyclic permutations of <math>(1,2,3)</math> and we have used the notation <math>(u^1, u^2, u^3) = (\psi,\theta,\phi)</math>. The Jacobian <math>\sqrt{g}</math> is defined below.

Any vector field <math>\mathbf{B}</math> can be represented as
:<math>
\mathbf{B}
= (\mathbf{B}\cdot\mathbf{e}^i)\mathbf{e}_i
= B^i\mathbf{e}_i
</math>
or
:<math>
\mathbf{B}
= (\mathbf{B}\cdot\mathbf{e}_i)\mathbf{e}^i
= B_i\mathbf{e}^i ~.
</math>
In particular any basis vector <math>\mathbf{e}_i = (\mathbf{e}_i\cdot\mathbf{e}_j)\mathbf{e}^j</math>. The metric tensor is defined as
:<math>
g_{ij}
= \mathbf{e}_i\cdot\mathbf{e}_j
\; ; \;
g^{ij}
= \mathbf{e}^i\cdot\mathbf{e}^j
\; ; \;
g^j_i
= \mathbf{e}_i\cdot\mathbf{e}^j = \delta_i^j ~.
</math>

The metric tensors can be used to ''raise'' or ''lower'' indices. Take
:<math>
\mathbf{B}
= B_i\mathbf{e}^i = B_i g^{ij}\mathbf{e}_j = B^j\mathbf{e}_j~,
</math>
so that
:<math>
B^j = g^{ij} B_i~.
</math>

=== Jacobian ===
The Jacobian of the coordinate transformation <math>\mathbf{x}(\psi, \theta, \phi)</math> is defined as
:<math>
J = \det\left(\frac{\partial(x,y,z)}{\partial(\psi,\theta,\phi)}\right) = \frac{\partial\mathbf{x}}{\partial{\psi}}\cdot\frac{\partial\mathbf{x}}{\partial{\theta}} \times \frac{\partial\mathbf{x}}{\partial{\phi}}
</math>
and that of the inverse transformation
:<math>
J^{-1} = \det\left(\frac{\partial(\psi,\theta,\phi)}{\partial(x,y,z)}\right) = \nabla{\psi}\cdot\nabla{\theta} \times \nabla{\phi}
</math>
It can be seen that <ref name='Dhaeseleer'></ref> <math>g \equiv \det(g_{ij}) = J^2 \Rightarrow J = \sqrt{g}</math>

=== Some surface elements ===

Consider a surface defined by a constant value of <math>\phi</math>. Then, the surface element is
:<math>
d{\mathbf S}_\phi = \mathbf{e}_\psi\times\mathbf{e}_\theta d\psi d\theta = \sqrt{g}\, \nabla\phi d\psi d\theta .
</math>

As for a surface defined by a constant value of <math>\theta</math>:
:<math>
d{\mathbf S}_\theta = \mathbf{e}_\phi\times\mathbf{e}_\psi d\psi d\phi = \sqrt{g}\, \nabla\theta d\psi d\phi ,
</math>
or a constant <math>\psi</math> surface:
:<math>
d{\mathbf S}_\psi = \mathbf{e}_\theta\times\mathbf{e}_\phi d\theta d\phi = \sqrt{g}\, \nabla\psi d\theta d\phi .
</math>

=== Gradient, Divergence and Curl in curvilinear coordinates ===
The gradient of a function f is naturally given in the contravariant basis vectors:
:<math>
\nabla f = \frac{\partial f}{\partial u^i}\nabla u^i = \frac{\partial f}{\partial u^i}\mathbf{e}^i~.
</math>
The divergence of a vector <math>\mathbf{A}</math> is best expressed in terms of its contravariant components
:<math>
\nabla\cdot\mathbf{A} = \frac{1}{\sqrt{g}}\frac{\partial}{\partial u^i}(\sqrt{g}A^i)~,
</math>
while the curl is
:<math>
\nabla\times\mathbf{A} = \frac{\varepsilon_{ijk}}{\sqrt{g}}\frac{\partial}{\partial u^i}(\sqrt{g}A_j)\mathbf{e}_k
</math>
given in terms of the covariant base vectors, where <math>\varepsilon_{ijk}</math> is the [[::Wikipedia:Levi-Civita symbol| Levi-Civita]] symbol.

== Flux coordinates ==
A flux coordinate set is one that includes a [[Flux surface|flux surface]] label as a coordinate. A flux surface label is a function that is constant and single valued on each flux surface. In our naming of the general curvilinear coordinates we have already adopted the usual flux coordinate convention for toroidal equilibrium with nested flux surfaces, where <math>\psi</math> is the flux surface label and <math>\theta, \phi</math> are <math>2\pi</math>-periodic poloidal and toroidal-like angles.

Different flux surface labels can be chosen like toroidal <math>(\Psi_{tor})</math> or poloidal <math>(\Psi_{pol})</math> magnetic fluxes or the volume contained within the flux surface <math>V</math>. By single valued we mean to ensure that any flux label <math>\psi_1 = f(\psi_2)</math> is a monotonous function of any other flux label <math>\psi_2</math>, so that the function <math>f</math> is invertible at least in a volume containing the region of interest. We will denote a generic flux surface label by <math>\psi</math>.

To avoid ambiguity in the sign of line and surface integrals we impose <math>d\psi(V)/dV > 0</math>, the toroidal angle increases in the clockwise direction when seen from above and the poloidal angle increases such that <math> \nabla\psi\cdot\nabla\theta\times\nabla\phi > 0</math>.

=== Flux Surface Average ===
The Flux Surface Average (FSA) of a function <math>\Phi</math> is defined as the limit
:<math>
\langle\Phi\rangle = \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\int_{\delta \mathcal{V}} \Phi\; d\mathcal{V}
</math>
where <math>\delta \mathcal{V}</math> is the volume confined between two flux surfaces. It is therefore a ''volume average'' over an infinitesimal spatial region rather than a surface average. To avoid confusion, we denote volume elements or domains with the calligraphic <math>\mathcal{V}</math>. Capital <math>V</math> is reserved for the flux label (coordinate) defined as the volume within a flux surface.

Introducing the differential volume element <math>d\mathcal{V} = \sqrt{g} d\psi d\theta d\phi</math>
:<math>
\langle\Phi\rangle
= \lim_{\delta \mathcal{V} \to 0} \frac{1}{\delta \mathcal{V}}\int_{\delta \mathcal{V}} \Phi\; \sqrt{g} d\psi d\theta d\phi
= \frac{d\psi}{d V}\int_0^{2\pi}\int_0^{2\pi}\Phi\; \sqrt{g} d\theta d\phi
</math>
or, noting that <math>\langle 1\rangle = 1</math>, we have <math>\frac{dV}{d\psi} = \int_0^{2\pi}\int_0^{2\pi} \sqrt{g} d\theta d\phi</math> and
we get to a more practical form of the Flux Surface Average
:<math>
\langle\Phi\rangle
= \frac{\int_0^{2\pi}\int_0^{2\pi}\Phi\; \sqrt{g} d\theta d\phi}
{\int_0^{2\pi}\int_0^{2\pi} \sqrt{g} d\theta d\phi}
</math>

Note that <math>dS = |\nabla\psi|\sqrt{g}d\theta d\phi</math>, so the FSA is a surface integral ''weighted by'' <math>|\nabla V|^{-1}</math> :
:<math>
\langle\Phi\rangle
= \frac{d\psi}{d V}\int_0^{2\pi}\int_0^{2\pi}\Phi\; \sqrt{g} d\theta d\phi
= \frac{d\psi}{d V}\int_{S(\psi)}\frac{\Phi}{|\nabla\psi|}\; dS
= \int_{S(\psi)}\frac{\Phi}{|\nabla V|}\; dS
</math>

Applying Gauss' theorem to the definition of FSA we get to the identity
:<math>
\langle\nabla\cdot\Gamma\rangle
= \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\int_{\delta \mathcal{V}} \nabla\cdot\Gamma\; d\mathcal{V}
= \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\int_{S(\delta \mathcal{V})} \Gamma\cdot \frac{\nabla V}{|\nabla V|}dS
= \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\left(\langle\Gamma\cdot\nabla V\rangle_{S(V+\delta \mathcal{V})} - \langle\Gamma\cdot\nabla V\rangle_{S(V)} \right)
= \frac{d}{dV}\langle\Gamma\cdot\nabla V\rangle~.
</math>

==== Useful properties of FSA ====
Some useful properties of the FSA are

*<math> \langle\nabla\cdot\Gamma\rangle = \frac{d}{dV}\langle\Gamma\cdot\nabla V\rangle = \frac{1}{V'}\frac{d}{d\psi}V'\langle\Gamma\cdot\nabla \psi\rangle</math>

*<math> \int_{\mathcal{V}}\nabla\cdot\Gamma\; d\mathcal{V} = \langle\Gamma\cdot\nabla V\rangle = V'\langle\Gamma\cdot\nabla \psi\rangle \qquad \mathrm{where~} \mathcal{V} \mathrm{~is~the~volume~enclosed~by~a~flux~surface.}
</math>

*<math> \langle \mathbf{B}\cdot\nabla f \rangle = \langle \nabla\cdot(\mathbf{B} f) \rangle = 0~,\qquad \forall~ \mathrm{single~valued~} f(\mathbf{x}), ~ \mathrm{if}~ \nabla\cdot\mathbf{B} = 0 ~\mathrm{and}~ \nabla \psi\cdot\mathbf{B} = 0 </math>

*<math> \langle \nabla \psi\cdot\nabla\times \mathbf{A} \rangle = -\langle \nabla\cdot( \nabla\psi\times\mathbf{A}) \rangle = 0~.
</math>

*<math> \langle \mathbf{B}\cdot\nabla \theta\rangle =2\pi\frac{d\Psi_{pol}}{dV} \qquad \mathrm{for~any~poloidal~ angle~} \theta ~ (\mathrm{Note:}~ \theta(\mathbf{x})~\mathrm{is~not~single~valued})
</math>

*<math> \langle \mathbf{B}\cdot\nabla \phi\rangle =2\pi\frac{d\Psi_{tor}}{dV} \qquad \mathrm{for~any~toroidal~ angle~} \phi ~ (\mathrm{Note:}~ \phi(\mathbf{x})~\mathrm{is~not~single~valued})
</math>

*<math> \langle \sqrt{g}^{-1}\rangle = \frac{4\pi^2}{V'}
</math>

In the above <math>V' = \frac{dV}{d\psi}</math>. Some [[:Wikipedia: Vector calculus identities|vector identities]] are useful to derive the above identities.

=== Magnetic field representation in flux coordinates ===

==== Contravariant Form ====
Any [[:Wikipedia: solenoidal vector field| solenoidal vector field]] <math>\mathbf{B}</math> can be written as
<math> \mathbf{B} = \nabla\alpha\times\nabla\nu </math>
called its Clebsch representation. For a magnetic field with flux surfaces <math>(\psi = \mathrm{const}\; , \; \nabla\psi\cdot\mathbf{B} = 0)</math> we can choose, say, <math>\alpha</math> to be the flux surface label <math>\psi</math>
:<math>
\mathbf{B} = \nabla\psi\times\nabla\nu
</math>
Field lines are then given as the intersection of the constant-<math>\psi</math> and constant-<math>\nu</math> surfaces. This form provides a general expression for <math>\mathbf{B}</math> in terms of the covariant basis vectors of a flux coordinate system
:<math>
\mathbf{B} = \frac{\partial\nu}{\partial\theta}\nabla\psi\times\nabla\theta + \frac{\partial\nu}{\partial\phi}\nabla\psi\times\nabla\phi = \frac{1}{\sqrt{g}}\frac{\partial\nu}{\partial\theta}\mathbf{e}_\phi -\frac{1}{\sqrt{g}}\frac{\partial\nu}{\partial\phi}\mathbf{e}_\theta = B^\phi\mathbf{e}_\phi + B^\theta\mathbf{e}_\theta~.
</math>
in terms of the function <math>\nu</math>, sometimes referred to as the magnetic field's ''stream function''.

It is worthwhile to note that the Clebsch form of <math> \mathbf{B} </math> corresponds to a [[:Wikipedia: Magnetic potential|magnetic vector potential]]
<math> \mathbf{A} = \nu\nabla\psi </math> (or <math> \mathbf{A} = \psi\nabla\nu </math> as they differ only by the Gauge transformation <math> \mathbf{A} \to \mathbf{A} - \nabla (\psi\nu)</math>).

The general form of the stream function is
:<math>
\nu(\psi,\theta,\phi)
= \frac{1}{2\pi}(\Psi_{tor}'\theta
- \Psi_{pol}'\phi)
+ \tilde{\nu}(\psi,\theta,\phi)
</math>
where <math>\tilde{\nu}</math> is a differentiable function periodic in the two angles. This general form can be derived by using the fact that <math> \mathbf{B}</math> is a physical function (hence singe-valued). The specific form for the coefficients in front of the secular terms (i.e. the non-periodic terms) can be obtained from the [[Flux coordinates#Useful properties of FSA|FSA properties ]].

==== Covariant Form ====

If we consider an equilibrium magnetic field such that <math> \mathbf{j}\times\mathbf{B} \propto \nabla\psi</math>, where <math> \mathbf{j}</math> is the current density , then both <math> \mathbf{B}\cdot\nabla\psi = 0</math> and <math> \nabla\times\mathbf{B}\cdot\nabla\psi = 0</math> and the magnetic field can be written as
:<math>
\mathbf{B} = \nabla\chi -\eta\nabla\psi
</math>
where <math>\chi</math> is identified as the magnetic ''scalar'' potential. Its general form is
:<math>
\chi(\psi, \theta, \phi) = \frac{I_{tor}}{2\pi}\theta + \frac{I_{pol}^d}{2\pi}\phi + \tilde\chi(\psi, \theta, \phi)
</math>

[[Image:CurrentIntegrationCirtuits.png|256px|thumb|right|alt=Sample integration circuits for the definitions of currents.|Sample integration circuits for the current definitions.]]
[[Image:CurrentIntegrationCirtuitsPoloidalCurrent.png|256px|thumb|right|alt=Sample surface for the definition of the current though a disc.|Sample surface for the definition of the current though a disc. Note that only the current of more external surfaces contribute to the flux of charge through the surface.]]

Note that <math>I</math> is not the current but <math>\mu_0</math> times the current. The functional dependence on the angular variables is again motivated by the single-valuedness of the magnetic field. The particular form of the coefficients can be obtained noting that
:<math>
\int_S \mu_0\mathbf{j}\cdot d\mathbf{S}
= \int_{\partial S}\mathbf{B}\cdot d\mathbf{l}
= \oint(\nabla\chi-\eta\nabla\psi)\cdot d\mathbf{l}
= \oint(d\chi-\eta d\psi )
</math>
and choosing an integration circuit contained within a flux surface <math>(d\psi = 0)</math>. Then we get
:<math>
\int_S \mu_0\mathbf{j}\cdot d\mathbf{S}
= \Delta \chi = \frac{I_{tor}}{2\pi}\Delta\theta + \frac{I_{pol}^d}{2\pi}\Delta\phi~.
</math>

If we now choose a ''toroidal'' circuit <math>(\Delta\theta = 0, \Delta\phi = 2\pi)</math> we get
:<math>
I_{pol}^d = \int_S \mu_0\mathbf{j}\cdot d\mathbf{S}\; ; ~\mathrm{with}~ \partial S ~\mathrm{such~that}~ (\Delta\theta = 0, \Delta\phi = 2\pi)~.
</math>
here the superscript <math>d</math> is meant to indicate the flux is computed through a disc limited by the integration line, as opposed to the ribbon limited by the integration line on one side and the magnetic axis on the other that was used for the definition of poloidal magnetic flux <math>\Psi_{pol}</math> above these lines.
Similarly
:<math>
I_{tor} = \int_S \mu_0\mathbf{j}\cdot d\mathbf{S}\; ; ~\mathrm{with}~ \partial S ~\mathrm{such~that}~ (\Delta\theta = 2\pi, \Delta\phi = 0)~.
</math>

===== Contravariant Form of the current density =====

Taking the curl of the covariant form of <math>\mathbf{B}</math> the equilibrium current density <math>\mathbf{j}</math> can be written as
: <math>
\mu_0\mathbf{j} = \nabla\psi\times\nabla\eta~.
</math>
By very similar arguments as those used for <math>\mathbf{B}</math> (note that both <math>\mathbf{B}</math> and <math>\mathbf{j}</math> are solenoidal fields tangent to the flux surfaces) it can be shown that the general expression for <math>\eta</math> is
:<math>
\eta(\psi,\theta,\phi) = \frac{1}{2\pi}({I}_{tor}'\theta
- {I}_{pol}'\phi)
+ \tilde{\eta}(\psi,\theta,\phi)~.
</math>
Note that the poloidal current is now defined through a ribbon and not a disc. The two currents are related as <math>\nabla\cdot\mathbf{j} = 0</math> implies
:<math>
I_{pol} + I_{pol}^d = \oint_{\psi=0}\mathbf{B}\cdot d\mathbf{l} \Rightarrow I_{pol}' + (I_{pol}^d)' = 0 ~,
</math>
where the integral is performed along the magnetic axis and therefore does not depend on <math>\psi</math>. This can be used to show that a expanded version of <math>\mathbf{B}</math> is given as
:<math>
\mathbf{B} = -\tilde\eta\nabla\psi + \frac{I_{tor}}{2\pi}\nabla\theta + \frac{I_{pol}^d}{2\pi}\nabla\phi + \nabla\tilde\chi~.
</math>

== Magnetic coordinates ==
Magnetic coordinates are a particular type of flux coordinates in which the magnetic field lines are straight lines. In mathematical terms this implies that the periodic part of the magnetic field's stream function is zero in these coordinates so the magnetic field reads
:<math>
\mathbf{B} = \nabla\psi\times \nabla\left( \frac{\Psi_{tor}'}{2\pi}\theta_f
- \frac{\Psi_{pol}'}{2\pi}\phi_f \right)
= \frac{\Psi_{pol}'}{2\pi\sqrt{g}}\mathbf{e}_\theta + \frac{\Psi_{tor}'}{2\pi\sqrt{g}}\mathbf{e}_\phi~.
</math>
Now a field line is given by <math>\psi = \psi_0</math> and <math>\Psi_{tor}'\theta_f - \Psi_{pol}'\phi_f = 2\pi\nu_0</math>.

Note that, in general, the contravariant components of the magnetic field in a magnetic coordinate system
:<math>
B^{\theta_f} = \frac{\Psi_{pol}'}{2\pi\sqrt{g}}\; ;\quad B^{\phi_f} = \frac{\Psi_{tor}'}{2\pi\sqrt{g}}
</math>
are not flux functions, but their quotient is
:<math>
\frac{B^{\theta_f}}{B^{\phi_f}} = \frac{\Psi_{pol}'}{\Psi_{tor}'} \equiv \frac{\iota}{2\pi}~,
</math>
<math>\iota</math> being the [[rotational transform]]. In a magnetic coordinate system the ''poloidal'' <math> \mathbf{B}_P = B^\theta\mathbf{e}_\theta </math> and ''toroidal'' <math> \mathbf{B}_T = B^\phi\mathbf{e}_\phi</math> components of the magnetic field are individually divergence-less.

From the above general form of <math> \mathbf{B} </math> in magnetic coordinates it is easy to obtain the following identities valid for any magnetic coordinate system
:<math>
\mathbf{e}_\theta\times\mathbf{B} =\frac{1}{2\pi}\nabla\Psi_{tor}~,
</math>
:<math>
\mathbf{e}_\phi\times\mathbf{B} = -\frac{1}{2\pi}\nabla\Psi_{pol} ~.
</math>
=== Transforming between Magnetic coordinates systems ===
There are infinitely many systems of magnetic coordinates. Any transformation of the angles of the from
:<math>
\theta_F = \theta_f +\Psi_{pol}' G(\psi, \theta_f, \phi_f)\; ;\quad \phi_F = \phi_f +\Psi_{tor}' G(\psi, \theta_f, \phi_f)
</math>

where <math>G</math> is periodic in the angles, preserves the straightness of the field lines (as can be easily checked by direct substitution). The spatial function <math>G(\psi, \theta_f, \phi_f)</math>, is called the ''generating function''. It can be obtained from a [[magnetic differential equation]] if we know the Jacobians of the two magnetic coordinate systems <math> \sqrt{g_f}</math> and <math> \sqrt{g_F}</math>. In fact taking <math>\mathbf{B}\cdot\nabla</math> on any of the transformation of the angles and using the known expressions for the contravariant components of <math>\mathbf{B}</math> in magnetic coordinates we get
:<math>
2\pi\mathbf{B}\cdot\nabla G = \frac{1}{\sqrt{g_F}} - \frac{1}{\sqrt{g_f}}~.
</math>
The LHS of this equation has a particularly simple form when one uses a magnetic coordinate system. For instance, if we write <math>\mathbf{B}</math> in terms of the original magnetic coordinate system we get
:<math>
(\Psi_{pol}'\partial_{\theta_f} + \Psi_{tor}'\partial_{\phi_f}) G = \frac{\sqrt{g_f}}{\sqrt{g_F}} - 1~.
</math>
which can be turned into an algebraic equation on the Fourier components of <math>G</math>
:<math>
G_{nm} = \frac{-i}{\Psi_{pol}'n + \Psi_{tor}'m}\left(\frac{\sqrt{g_f}}{\sqrt{g_F}}\right)_{nm}~.
</math>
where
:<math>
G(\psi, \theta_f, \phi_f) = \sum_{n,m} G_{nm}(\psi) e^{i(n\theta_f + m\phi_f)}
</math>
and <math>G_{00} = 0 </math>.

Particular choices of G can be made so as to simplify the description of other fields. The most commonly used magnetic coordinate systems are:
<ref name='Dhaeseleer'>W.D. D'haeseleer, ''Flux coordinates and magnetic field structure: a guide to a fundamental tool of plasma theory'', Springer series in computational physics, Springer-Verlag (1991) ISBN 3540524193</ref>
* [[Hamada coordinates]]. <ref>S. Hamada, Nucl. Fusion '''2''' (1962) 23</ref><ref>[http://dx.doi.org/10.1063/1.1706651 J.M. Greene and J.L Johnson, ''Stability Criterion for Arbitrary Hydromagnetic Equilibria'', Phys. Fluids '''5''' (1962) 510]</ref> In these coordinates, both the magnetic field lines and current lines corresponding to the [[MHD equilibrium]] are straight. Referring to the definitions above, both <math>\tilde\nu</math> and <math>\tilde\eta</math> are zero in Hamada coordinates.
* [[Boozer coordinates]]. <ref>[http://dx.doi.org/10.1063/1.863297 A.H. Boozer, ''Plasma equilibrium with rational magnetic surfaces'', Phys. Fluids '''24''' (1981) 1999]</ref><ref>[http://dx.doi.org/10.1063/1.863765 A.H. Boozer, ''Establishment of magnetic coordinates for a given magnetic field'', Phys. Fluids '''25''' (1982) 520]</ref> In these coordinates, the magnetic field lines corresponding to the [[MHD equilibrium]] are straight and so are the ''diamagnetic lines '', i.e. the integral lines of <math>\nabla\psi\times\mathbf{B}</math>. Referring to the definitions above, both <math>\tilde\nu</math> and <math>\tilde\chi</math> are zero in Boozer coordinates.

== References ==
<references />

Function parametrization

2010-11-24T00:22:45Z

Otihizuv:

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Function Parametrization (also spelt Parameterization) or FP is a technique to provide fast (real-time) construction of system parameters from a set of diverse measurements. It consists of the numerical determination, by statistical regression on a database of simulated states, of simple functional representations
of parameters characterizing the state of a particular physical system, where the arguments of the functions are statistically independent combinations of diagnostic raw measurements of the system.
The technique, developed by H. Wind for the purpose of momentum determination from spark chamber data, <ref> Wind, H. `Function Parametrization'
in ``Proceedings of the 1972 CERN Computing and Data Processing School'', CERN 72--21, 1972, pp.~53--106. </ref> <ref>Wind, H.,
(a)`Principal component analysis and its application to track finding', (b) `interpolation and function representation'
in ``Formulae and Methods in Experimental Data Evaluation'',Vol. 3, European Physical Society, Geneva, 1984</ref> was introduced by B. Braams to plasma physics,
where it was first applied to the analysis of equilibrium magnetic measurements on the
circular cross-section ASDEX tokamak. <ref>B.J. Braams, W. Jilge, and K. Lackner, ''Fast determination of plasma parameters through function parametrization'', Nucl. Fusion '''26''' (1986) 699</ref> It was later extended to the non-circular cross-section ASDEX Upgrade tokamak<ref>[http://www.physics.ucc.ie/~pjm/people/trachtas.htm P.J. Mc Carthy, ''An Integrated Data Interpretation System for Tokamak Discharges'', PhD thesis, University College Cork, 1992]</ref>
and the Wendelstein 7-AS stellarator.
<ref>[http://iopscience.iop.org/0029-5515/39/4/308 H.P. Callaghan, P.J. Mc Carthy, J. Geiger "Fast equilibrium interpretation on the W7-AS stellarator using Function Parameterization", Nucl. Fusion "39" (1999) 509-523.]</ref>

== Method ==

The application of the technique requires that a model exists to compute the response of the measurements (''q'') to variations of the system parameters (''p''), i.e. the mapping ''q = M(p)'' is known.
In doing so, all functional dependencies are parametrized (hence the name of the technique),
e.g., spatially dependent functions ''f(r)'' are given in terms of an parametric expansion (such as a [[:Wikipedia:Polynomial|polynomial]]), and the corresponding parameters are included in the vector ''p''.

The fast reconstruction of the system parameters is obtained by computing the inverse of the mapping ''M''. To do so, the parameters ''p'' are varied over a range corresponding to the expected variation in actual experiments, the corresponding ''q'' are obtained, and the set of ''(p,q)'' data are stored in a database. This database is then subjected to a statistical analysis in order to recover the inverse of ''M''. This analysis is typically a [[:Wikipedia:Principal Component Analysis|Principal Component Analysis]]. This procedure is also amenable to a rather detailed error analysis, so that errors in the recovered parameters ''p'' for the interpretation of actual data ''q'' can be obtained.
<ref name=RTP>B.Ph. van Milligen, N.J. Lopes Cardozo, ''Function Parametrization: a fast inverse mapping method'', Comp. Phys. Commun. '''66''' (1991) 243</ref>

== Applications ==

* RTP <ref name=RTP></ref>
* TEXTOR <ref>B.Ph. van Milligen et al., ''Application of Function Parametrization to the analysis of polarimetry and interferometry data in TEXTOR'', Nucl. Fusion '''31''' (1991) 309</ref>
* ASDEX-UG <ref>[http://dx.doi.org/10.1016/S0920-3796(00)00109-5 W. Schneider, P.J. Mc Carthy, et al., ''ASDEX upgrade MHD equilibria reconstruction on distributed workstations'', Fusion Engineering and Design '''48''', Issues 1-2 (2000) 127-134]</ref>
* Wendelstein 7-X <ref>[http://dx.doi.org/10.1088/0029-5515/44/11/003 A. Sengupta, P.J. Mc Carthy, et al., ''Fast recovery of vacuum magnetic configuration of the W7-X stellarator using function parametrization and artificial neural networks'', Nucl. Fusion '''44''' (2004) 1176]</ref>

== Alternatives ==

* [[Bayesian data analysis]], which allows non-Gaussian error distributions.
* Neural networks.

== References ==
<references />

Anomalous transport

2010-11-24T00:22:24Z

Otihizuv:

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The best and most complete theory of transport in magnetically confined systems is the [[Neoclassical transport|Neoclassical]] theory.
However, it is found that transport often exceeds Neoclassical expectations by an order of magnitude or more (also see [[Non-diffusive transport]]).
<ref>[http://dx.doi.org/10.1063/1.859358 A.J.Wootton et al, ''Fluctuations and anomalous transport in tokamaks'', Phys. Fluids B ''2'' (1990) 2879]</ref>
The difference between actual transport and the Neoclassical expectation is called "[[:Wiktionary:anomaly|anomalous]]" transport.
It is generally assumed that the anomalous component of transport is generated by turbulence driven by micro-instabilities.
<ref name="Freidberg">J.P. Freidberg, ''Plasma physics and fusion energy'', Cambridge University Press (2007) ISBN 0521851076</ref>

== How important is turbulence? ==

In spite of lengthy studies into the subject, it is still controversial how important turbulent transport really is.
In part, this may be because turbulent transport gives a variable contribution to transport (depending on local and global parameters), whereas Neoclassical transport is always present.
And in part, because no complete theory for anomalous transport is available.
<ref>[http://dx.doi.org/10.1088/0741-3335/36/5/002 J.W. Conner and H.R. Wilson, ''Survey of theories of anomalous transport'', Plasma Phys. Control. Fusion '''36''' (1994) 719-795]</ref>

=== Arguments for ===

An important argument suggesting that anomalous transport is important to the degree that it often dominates the total transport is the [[Scaling law|scaling]] of transport with heating power and machine size.
<ref>[http://dx.doi.org/10.1109/27.650902 B.A. Carreras, ''Progress in anomalous transport research in toroidal magnetic confinement devices'', IEEE Trans. Plasma Science '''25''', 1281 (1997)]</ref>
The phenomenon of [[Scaling law|power degradation]], universally observed in all devices, is an indication that standard transport theories are inadequate to explain all transport, since these would not predict power degradation.
Following Freidberg,
<ref name="Freidberg" />
the cited [[Scaling law|scaling laws]] can be rewritten in terms of the temperature dependence (eliminating the heating power dependence).
Then, classical and neoclassical estimates would predict that the confinement increases with ''T'' (namely: ''&tau;E'' &prop; ''T0.5'', associated with [[Collisionality|collisionality]]).
However, the experimental scalings give a ''decrease'' with ''T''
(namely: ''&tau;E'' &prop; ''T&alpha;'' with '' &alpha;'' &lt; -1).
This unexpected behaviour is explained from increased turbulence levels (and enhanced transport) at higher values of (the gradients of) ''T''.

[[Profile consistency]] indicates that [[Self-Organised Criticality|self-organisation]] plays an important role in transport, and this can only be the case when instabilities or turbulence are able to regulate the profiles, i.e., when they carry an important fraction of transport.

The suppression of turbulence is possible, either actively (by imposing an external radial electric field), or spontaneously ([[H-mode]]s, [[Internal Transport Barrier]]s). As a consequence, transport is reduced significantly (to Neoclassical levels). This is a clear indication that turbulence is responsible for the main fraction of anomalous transport.

=== Arguments against ===

It has been argued that turbulence cannot be responsible for a significant fraction of the anomalous component of transport, since that would lead to high resistivity (due to collisions), which contradicts experimental observation.
<ref>L.C. Woods, ''Theory of tokamak transport: new aspects for nuclear fusion reactor design'', John Wiley and Sons (2006) ISBN 3527406255</ref>
However, this argument fails to note that anomalous transport may consist of collective events (e.g., ''streamers''), which does not require an enhanced collisionality.
As a side remark, this argument does show that the contribution of turbulence to transport is likely ''not'' of the diffusive type (see [[Non-diffusive transport]]).

== Physical mechanism ==

The physical mechanism behind anomalous transport has not been fully clarified.
However, it is generally assumed that anomalous transport is the consequence of microscopic instabilities.
The plasma potentially produces a plethora of such instabilities, due to the fact that it is in a state far from thermodynamic equilibrium, with steep density, temperature, and pressure gradients.
The most likely candidates involved in generating the observed anomalous transport are:
<ref>J. Weiland, ''Collective modes in inhomogeneous plasma: kinetic and advanced fluid theory'', Plasma physics series, CRC Press (2000) ISBN 0750305894</ref>
* Ion Temperature Gradient (ITG) instabilities
* Electron Temperature Gradient (ETG) instabilities
* Collisionless Trapped Electron Modes (TEM) <ref>[http://link.aps.org/doi/10.1103/PhysRevLett.33.1329 B. Coppi and G. Rewoldt, ''New Trapped-Electron Instability'', Phys. Rev. Lett. '''33''' (1974) 1329 - 1332]</ref> <ref>[http://link.aps.org/doi/10.1103/PhysRevLett.95.085001 F. Ryter et al, ''Experimental Study of Trapped-Electron-Mode Properties in Tokamaks: Threshold and Stabilization by Collisions'', Phys. Rev. Lett. '''95''' (2005) 085001]</ref>
* Dissipative Trapped Electron Modes (DTEM)
''(to be completed; references needed)''

== Can anomalous transport be modelled? ==

There are several answers to this question. Since all equations describing the motion of charged particles in fields are known, as well as the effects of collisions, detailed numerical (gyrokinetic) [[Plasma simulation|simulations]] are possible.
<ref>[http://link.aps.org/doi/10.1103/PhysRevLett.77.71 A.M. Dimits et al, ''Scalings of Ion-Temperature-Gradient-Driven Anomalous Transport in Tokamaks'', Phys. Rev. Lett. '''77''' (1996) 71 - 74]</ref>
However, due to the enormous disparity between the minimum and maximum scales involved (gyration times vs. transport times, and the gyroradius vs. the machine size), this is a major challenge.

An alternative approach is to model the net effect of turbulence without simulating the fine detail.
In doing so, it is not sufficient to introduce a simple additional "turbulent diffusivity", as this cannot possibly reproduce the observed global transport scaling behaviour.
It is probably necessary to use a [[Non-diffusive transport|non-diffusive]] description,
<ref>[http://dx.doi.org/10.1016/S0370-1573(02)00331-9 G. M. Zaslavsky, ''Chaos, fractional kinetics, and anomalous transport'', Physics Reports '''371''', Issue 6 (2002) 461-580]</ref>
and include non-linear phenomena such as [[Self-Organised Criticality|critical gradients]].

== Can anomalous transport be controlled? ==

Yes.
The impression is that anomalous transport is more difficult to control in tokamaks than in stellarators. However, limited control in tokamaks is possible by making use of edge transport barriers (cf. [[H-mode]]) and [[Internal Transport Barrier]]s (ITBs). This reduces transport to Neoclassical levels, at least transiently and locally.

Particularly in optimised stellarators (W7-AS), transport can be close to Neoclassical levels.
<ref>[http://dx.doi.org/10.1088/0741-3335/50/5/053001 M. Hirsch et al, ''Major results from the stellarator Wendelstein 7-AS'', Plasma Phys. Control. Fusion '''50''' (2008) 053001]</ref>

== References ==
<references />

Error propagation

2010-11-24T00:22:06Z

Otihizuv:

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Proper reporting of experimental measurements requires the calculation of error bars or "confidence intervals". The appropriate and satisfactory calibration of data and analysis of errors is essential to be able to judge the relevance of observed trends. Below, a brief definition of the main concepts and a discussion of generic ways to obtain error estimates is provided.
<ref>[http://www.nrbook.com/ W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in FORTRAN (Cambridge University Press, 1992), 2nd ed.]</ref>
<ref>P. Bevington and D. Robinson, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, UK, 2003), 3rd ed. ISBN 978-0072472271</ref>
Of course, any particular measuring device generally requires specific techniques.

== The measurement process==

The measuring device performs [[Data analysis techniques|measurements]] on a physical system ''P''.
As a result, it produces estimates of a set of physical parameters ''{p}''.
One may think of ''p'' as loose numbers (e.g., a confinement time), data along a spatial chord at a single time (e.g., a Thomson scattering profile), data at a point in space with time resolution (e.g., magnetic field fluctuations from a Mirnov coil), or data having both time and space resolution (e.g., tomographic data from Soft X-Ray arrays).
The actual measurement hardware does not deliver the parameters ''{p}'' directly, but produces a set of numbers ''{s}'', usually expressed in Volts, Amperes, or pixels.

== Calibration ==

The first task of the experimentalist is to translate the measured signals ''{s}'' into the corresponding physical parameters ''{p}''.
The second task is to provide error estimates (discussed below).
Generally, the translation of ''{s}'' into ''{p}'' requires having a (basic) model for the experiment studied and its interaction with the measuring device.
In the simplest cases, the relation between ''{s}'' and ''{p}'' is linear (e.g. conversion of the measured voltages from Mirnov coils to magnetic fields). Taking ''s'' and ''p'' to be vectors, such a conversion can be written as
:<math> p = A \cdot(s - b),</math>
where ''A'' is a (possibly diagonal) calibration matrix and ''b'' a vector for offset correction.
However, in fusion science it is more common that the conversion from ''s'' to ''p'' involves some (non-linear) numerical modelling of the physical (and measurement) system.
In this case, rather than assuming a linear relation, one assumes a non-linear map ''Mp'' between ''s'' and ''p'': ''p = Mp(s)''.
The subscript ''p'' indicates that ''Mp'' may depend on ''p''.
In principle, determining ''p'' from ''s'' now requires an iterative numerical approach.
The map ''Mp'' should be tested to check that it is not ill-conditioned (i.e. small variations in ''s'' produce large variations in ''p''), since that would render the measurements useless; ill-conditioning leads to error amplification.
Should this be the case, then the measurement set-up should be changed until the undesired ill-conditioning is removed.

== Error estimate (experimental error known) ==

When the error level in ''s'' is known (from experimental measurements performed on the measuring device itself), some techniques are available to calculate the error in ''p''.
In the linear case of Eq. (1), and even in slightly more complex situations, standard error propagation techniques can be used to compute the error in ''p'' from the error in ''s''.
Standard error propagation proceeds as follows:
:<math>z = f(x, y, ...)\,</math>
:<math>(\Delta z)^2 = \left ( \frac{\partial f}{\partial x}\right )^2 \Delta x^2 + \left ( \frac{\partial f}{\partial y}\right )^2 \Delta y^2 + ... </math>
This formula holds exclusively for a Gaussian (normal) distribution of errors (assuming the errors are small and that the independent variables ''x'', ''y'', ... are indeed independent).
<ref>[http://mathworld.wolfram.com/ErrorPropagation.html Error Propagation (MathWorld)]</ref>
One should be aware that many situations exist where error distributions are not normal (see below).
One can easily check whether the error distribution is normal by doing repeated experiments under the same conditions and observing the resulting distribution of ''s''.
When propagating errors, one should be aware that the calibration matrix ''A'' and the offset correction ''b'' may also contain errors, which should therefore also be propagated.
An important topic, and cause of many errors in the calculation of error propagation (and parameter fitting) is the issue of collinearity (linear dependencies between elements of ''s'' and/or ''p'').
The presence of collinearity may affect error levels enormously.
A quick check of possible problems in this sense can be made using the [[:Wikipedia:Monte Carlo method|Monte Carlo approach]] (see below).
Several techniques are available to handle collinearity, such as [[:Wikipedia:Principal component analysis|Principal component analysis]] (basically, by orthogonalization of the correlation matrix of ''s'').
The Monte Carlo approach also provides a simple method for error estimation for the much more difficult problem of a non-linear mapping ''Mp''.
This technique proceeds as follows.
To compute the error bar of ''p'', simulated measurements ''s'' are varied randomly within their (known) error bars, using the (known) error distribution, and the standard deviation of the resulting ''p'' is determined, the latter being equal to the error estimate.
This technique also provides a quick method to check for possible problems such as ill-conditioning, cited above.
When the model relating ''s'' and ''p'' is known, as well as the error distributions (and the latter may either be Gaussian or not), a more systematic approach to error propagation is provided by a technique known as the [[:Wikipedia:Maximum likelihood|maximum likelihood method]].
<ref>[http://dx.doi.org/10.1007/s10052-998-0104-x Particle Data Group, Eur. Phys. J. C 3, 1 (1998)]</ref>
This technique is simply the generalisation of standard error propagation to general error distributions (i.e. not limited to Gaussians).

== Systematic and random errors ==

It is important to distinguish systematic and random errors.
For example, the Monte Carlo technique discussed above will only yield random errors.
The estimations of ''p'' may, however, be systematically wrong because the model ''Mp'' used is wrong or based on a flawed hypothesis.
To check this, it is vital to cross-check the obtained values of ''p'' against the parameters obtained from another, independent measurement device.
If this is not possible, the data should be reinterpreted using an independent analysis code.
There is no alternative to determining systematic errors, except these two techniques (cross-checking between diagnostics and/or using independent models). Careful analysis of possible hidden or tacit assumptions is always recommendable.

== Error estimate (experimental error unknown) ==

Unfortunately, and although this should never occur, often the error in the original signals ''s'' is not even known.
We note, in any case, that a lower limit can always be set on ''s'', namely, the bit resolution of the ADC acquiring the signal, or the pixel size.
If there is no reasonable way to estimate the error in ''s'', then a comparison with other diagnostics is even more important than before.
A final recourse to error estimation is provided by performing repeated measurements in identical discharges.
Repeating the measurement ''s'' on experiments that have carefully been prepared in the same state (''p'') will provide a set of values ''s'' that varies across the experiments.
The standard deviation of ''s'' is then equal to the error bar of ''s''.
This technique can also be applied to a single experiment for time-resolved measurements.
A steady-state discharge can be used for this purpose.
The fluctuation amplitude of the signal ''s'' will then be equal to its error bar.
We note, however, that this poor man's approach to error estimation will always provide an upper limit of the error bars, since the actual (physical) variability of the signal is added to the random error, whereas it provides no indication of the systematic error.

== Test of statistical validity of the model ==

If a model is characterised by a number of free (fit) parameters ''&alpha;i, i = 1, ..., n'' and used to predict (or fit) some measurements, then, once error estimates are available, it can (and should) be subjected to a [[:Wikipedia:Chi-square test|&chi;2-test]].
The value of &chi;2 obtained should be close to the number of free parameters; if it isn't, the number of free parameters ''n'' should be modified until it is.

== Fluctuations and noise ==

The separation of noise and fluctuations is a highly non-trivial topic.
The simplest case is when the physically interesting phenomenon is slowly varying in time.
Random noise is usually characterised by a high frequency, so that a filter in frequency space can then separate signal and noise neatly.
<ref>D. Newland, An Introduction to Random Vibrations, Spectral and Wavelet Analysis (Dover, New York, 1993) ISBN 0486442748</ref>
However, when the physically interesting information is fluctuating, this signal-noise separation by frequency is not feasible, and much care is needed when analysing data.
The application of a set of techniques is required to understand such signals (cross correlation, conditional averaging, spectral analysis, bi-spectral analysis,
<ref>J. van den Berg, ed., Wavelets in Physics (Cambridge University Press, 1999) ISBN 978-0521593113</ref>, [[Biorthogonal decomposition]],
determination of fractal dimension, mutual information, reconstruction of chaotic attractor,
<ref>[http://link.aps.org/doi/10.1103/RevModPhys.65.1331 H. Abarbanel, R. Brown, J. Sidorowich, and L. S. Tsimring, Rev. Mod. Phys. 65, 1331 (1993)]</ref> ...).

== Non-Gaussian statistics ==

The distribution of random variations of a signal ''s'' around its mean value need not be Gaussian.
E.g., photon statistics are typically of the [[:Wikipedia:Poisson distribution|Poisson]] type, which is especially important for low signal levels.
<ref>[http://link.aip.org/link/?RSINAK/74/3998/1 B. van Milligen, I. Classen, and C. Barth, Rev. Sci. Instrum. 74, 3998 (2003)]</ref>
In other cases, the random component of the signal ''s'' is simply a non-linear function of a (Gaussian) noise source, causing the distribution to be skewed or distorted.
Or the random component of the measured signal could correspond to the maximum or minimum value of a (Gaussian) random number, leading to extremal ([[:Wikipedia:Gumbel distribution|Gumbel]]) distributions.
<ref>[http://link.aip.org/link/?PHPAEN/12/052507/1 B. van Milligen, R. Sánchez, B. Carreras, V. Lynch, B. LaBombard, M. Pedrosa, C. Hidalgo, B. Gonçalves, and R. Balbín, Phys. Plasmas 12, 052507 (2005)]</ref>
The [[:Wikipedia:Log-normal distribution|log-normal distribution]] is also quite common (e.g. in potential fluctuations).
<ref>[http://link.aip.org/link/?PHPAEN/11/5032/1 F. Sattin, N. Vianello, and M. Valisa, Phys. Plasmas 11, 5032 (2004)]</ref>
However, all the previous distributions can be obtained by suitable manipulations of Gaussian random variables.
A totally different class of statistics is known as [[:Wikipedia:Lévy distribution|Lévy distributions]] (of which the Gaussian distribution is only a special case), which is the class of distributions satisfying the requirement that the sum of independent random variables with a distribution ''P'' again has a distribution ''P'' (generalisation of the Central Limit Theorem).
Such distributions are expected to appear in [[Self-Organised Criticality|self-organised systems]] (such as plasmas).
In general, the detection of this type of non-Gaussian statistics is difficult. Some techniques are however available, such as renormalisation, [[:Wikipedia:Rescaled range|rescaled-range]] analysis,
<ref>[http://link.aip.org/link/?PHPAEN/6/1885/1 B. Carreras, B. van Milligen, M. Pedrosa, R. Balbín, C. Hidalgo, D. Newman, E. Sánchez, R. Bravenec, G. McKee, I. García-Cortés, et al., Phys. Plasmas 6, 1885 (1999)]</ref>
the detection of long-range time dependence,
<ref>[http://link.aip.org/link/?PHPAEN/6/485/1 B. Carreras, D. Newman, B. van Milligen, and C. Hidalgo, Phys. Plasmas 6, 485 (1999)]</ref>
finite-size [[:Wikipedia:Lyapunov exponent|Lyapunov exponents]],
<ref>[http://link.aip.org/link/?PHPAEN/8/5096/1 B. Carreras, V. Lynch, and G. Zaslavski, Phys. Plasmas 8, 5096 (2001)]</ref> etc.
Sometimes it is possible to obtain information on the nature of the errors by averaging experimental data (in space or time) - this is the renormalisation technique referred to above.
When averaging over ''N'' samples, the variation of the ''N''-averaged (or smoothed) data is less than that of the original data.
The way in which the variance (and other statistical moments) decreases with ''N'' provides information both on the type of statistics involved (Gaussian or otherwise) and on the random or non-random nature of the data variability (random contributions decaying to zero as ''N'' &rarr; &infin;).

== Integrated data analysis ==

Often, various different diagnostics provide information on the same physical parameter (e.g., in a typical fusion plasma experiment, the electron temperature ''Te'' is possibly measured by Thomson Scattering, ECE, and a HIBP, and indirectly also by SXR, although mixed with information on the electron density ''ne'' and ''Zeff''. The electron density is measured directly by Thomson Scattering, the HIBP, reflectometry, and interferometry, and indirectly by SXR).
Part of this information is local, and part is line-integrated. Instead of cross-checking these diagnostics for one or a few discharges, one could decide to make an integrated analysis of data.
This means using all information available to make the best possible reconstruction of, e.g., the electron density and temperature that is compatible with all diagnostics simultaneously.
To do this, the following conditions must apply: 1) The data should not contradict each other mutually. This requires a previous study concerning the mutual compatibility, i.e. data validation. 2) The data should be available with proper calibration and independent error estimates in a routine fashion. This means regular calibrations of the measuring device and crosschecks. 3) A suitably detailed model of the physical system should be available, capable of modelling all experimental conditions and all corresponding measurement data. Techniques based on e.g. [[Bayesian data analysis|Bayesian statistics]] then allow finding the most probable value of all physical parameters in the model, compatible with all measured signals.

== Summary ==

A proper analysis of error propagation requires having a reasonable model ''Mp'' that relates the measured signals ''s'' to the corresponding physical parameters ''p''.
Such a model can initially be rudimentary, implying the probable existence of large systematic errors.
The systematic observation and analysis of the results ''p'' and their properly propagated random errors ''&Delta;p'', and their comparison with similar results from other diagnostics should allow improvement of the model, thus reducing the systematic error and improving the agreement between independent measurements (and/or models).
This gradual improvement of the physics model is the basis of scientific progress.

== References ==
<references />

Scaling law

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A scaling law is an engineering tool to predict the value of a system variable as a function of some other significant variables.
<ref>O.J.W.F. Kardaun, ''Classical methods of statistics: with applications in fusion-oriented plasma physics'', Springer Science & Business (2005) ISBN 3540211152</ref>
Their extended use in magnetic confinement physics reflects the fact that detailed transport calculations or predictions on the basis of first principles are difficult in this field. In the latter context, they are mainly used to
* predict the performance of new (larger) devices, such as [[ITER]]
* summarize large amounts of experimental data
* make performance comparisons between devices
* make educated guesses at local transport mechanisms

== General method ==

The typical scaling law expression for a (dependent) variable ''y'' as a function of some (independent) system variables ''x1'', ''x2'',... is:
:<math>y = e^{\alpha_0} x_1^{\alpha_1} x_2^{\alpha_2} ...</math>
Here, the &alpha;i are the scaling parameters.
By taking the logarithm of this expression, it becomes linear in the parameters and simple (multivariate) linear regression tools can be used to determine the parameters from a set of data.
However, a proper analysis requires:
* using ''dimensionless'' variables (easily achieved by normalizing all quantities appropriately)<ref>[http://dx.doi.org/10.1088/0741-3335/50/4/043001 T.C. Luce, C.C. Petty, and J.G. Cordey,''Application of dimensionless parameter scaling techniques to the design and interpretation of magnetic fusion experiments'', Plasma Phys. Control. Fusion, '''50''', 4 (2008) 043001]</ref>
* guaranteeing the (linear) ''statistical independence'' of the independent variables (applying, e.g., [[:Wikipedia:Principal component analysis|Principal Component Analysis]])

== Dimensionless parameters ==

In the magnetic confinement context, and assuming quasi-neutrality, the relevant scaling laws (mainly, for the [[Energy confinement time|energy confinement time]]) can be cast into dimensionless forms that involve only three plasma parameters (apart from geometrical factors):
<ref name="ITER"/>
<ref>B.B. Kadomtsev, Sov. J. Plasma Phys. '''1''' (1975) 295</ref>

:<math>\rho* = \frac{\rho_i}{a}</math>

:<math>\beta = \frac{\left \langle p \right \rangle}{B^2/2\mu_0}</math>

:<math>\nu* \propto \nu_{ii}</math>

Here, &rho;i is the ion [[Larmor radius]] and &nu;ii the ion-ion collision frequency. Also see [[Beta|beta]] and [[Collisionality|collisionality]].

In dimensionless form, the diffusivities can be written as:
:<math>D = c_s \rho_s (\rho^*)^\alpha F(\nu^*,\beta,q, ...)\,</math>
When &alpha; = 0, the scaling is said to be of the Bohm type, and when &alpha; = 1, of the gyro-Bohm type.

== Confinement time scaling ==

The main performance parameter that is subjected to scaling law analysis is the [[Energy confinement time|energy confinement time]], &tau;E.
The following are some of the most-used scalings for tokamaks:
<ref name="ITER">[http://dx.doi.org/10.1088/0029-5515/39/12/301 ITER Physics Expert Groups et al, ''ITER Physics Basis, Chapter 1'', Nucl. Fusion '''39''' (1999) 2137] and [http://dx.doi.org/10.1088/0029-5515/39/12/302 Ibid., ''Chapter 2'']</ref>
* L-mode scaling (ITER89-P)
:<math>\tau_E^L = 0.048 I_M^{0.85} R_0^{1.2} a^{0.3} \kappa^{0.5} \bar n_{20}^{0.1} B_0^{0.2} A^{0.5} P_M^{-0.5}</math>

* ELMy [[H-mode]] scaling (IPB98(y,2))
:<math>\tau_E^H = 0.145 I_M^{0.93} R_0^{1.39} a^{0.58} \kappa^{0.78} \bar n_{20}^{0.41} B_0^{0.15} A^{0.19} P_M^{-0.69}</math>

where ''IM'' is given in MA, ''PM'' in MW, ''n20'' in 1020 m-3, and ''B0'' in T.

For stellarators, a similar scaling has been obtained (ISS).
<ref>[http://www.ipp.mpg.de/ISS ISS-IPP] and [http://iscdb.nifs.ac.jp/ ISS-NIFS] homepages</ref>
<ref>[http://dx.doi.org/10.1088/0029-5515/47/9/025 A. Dinklage et al, ''Physical model assessment of the energy confinement time scaling in stellarators'', Nuclear Fusion '''47''', 9 (2007) 1265-1273]</ref>
* ISS04v3
:<math>\tau_E = 0.148 R^{0.64} a^{2.33} \bar n_{20}^{0.55} B^{0.85} \iota_{2/3}^{0.41} P^{-0.61}</math>
There is an ongoing discussion on whether to replace the plasma size quantifiers ''(a,R)'' by ''(S,V)'' (surface and volume), which might be more appropriate for stellarator flux surfaces, deviating strongly from a [[:Wikipedia:Torus|torus]] (see [[Effective plasma radius]]).

=== Power degradation ===

One of the remarkable and initially unexpected properties of magnetically confined plasmas is the reduction of the [[Energy confinement time|energy confinement time]] &tau;E as the heating power ''P'' is increased. Typically:

:<math>\tau_E \propto P^{-\alpha}</math>

where &alpha; has a value of 0.6 &plusmn; 0.1. The reason for this behaviour has not been fully clarified. Qualitatively, it seems obvious that an increase of ''P'' will lead to an increase of (temperature and density) gradients, and thus an increase of "free energy" available to instabilities and turbulence. These instabilities may grow by feeding on the "free energy", which may lead to an increase of ([[Anomalous transport|anomalous]]) transport (i.e., more than the expected -diffusive- increase due to the increased gradient alone), producing the observed confinement degradation.
This phenomenon is therefore a form of plasma [[Self-Organised Criticality|self-organisation]].

=== Size scaling ===

The ELMy [[H-mode]] scaling is of the gyro-Bohm type (&alpha; = 1).
Gyro-Bohm scaling is what one would expect for diffusive transport based on a diffusive scale length proportional to &rho;i (the ion gyroradius).

By contrast, the L-mode scaling is of the Bohm type (&alpha; = 0), which suggests that transport may [[Non-diffusive transport|not be diffusive]] and not characterized by a typical scale length, i.e., it is dominated by the scale length corresponding to the machine size (non-locality).
<ref>A. Dinklage, ''Plasma physics: confinement, transport and collective effects'', Vol. 670 of Lecture notes in physics, Springer (2005) ISBN 3540252746</ref>
One possible explanation of this behaviour is [[Self-Organised Criticality]], i.e., the self-regulation of transport by turbulence, triggered when a critical value of the gradient is exceeded. As a corollary, this mechanism might also explain the phenomenon of [[Profile consistency|profile consistency]].

== References ==
<references />

Magnetic island

2010-11-23T23:58:27Z

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A magnetic island is a closed magnetic flux tube (cf. [[Flux surface]]), bounded by a [[Separatrix|separatrix]], isolating it from the rest of space.
Its topology is toroidal.

In the context of magnetic confinement fusion, the basic magnetic field configuration consists of toroidally nested [[Flux surface|flux surfaces]], while each flux surface is characterised by a certain value of the [[Rotational transform|rotational transform]] or safety factor ''q''. Magnetic islands can appear at flux surfaces with a rational value of the safety factor ''q = m/n''.
<ref>[http://dx.doi.org/10.1051/anphys:2004001 J.H. Misguich, J.-D. Reuss, D. Constantinescu, G. Steinbrecher, M. Vlad, F. Spineanu, B. Weyssow, R. Balescu, ''Noble internal transport barriers and radial subdiffusion of toroidal magnetic lines'', Ann. Phys. Fr. '''28''' (2003) 1]</ref>
Subsidiary islands can appear within an island.

== Island birth ==

The rupture of the assumed initial topology of toroidally nested flux surfaces needed to produce the island requires the reconnection of magnetic field lines, which can only occur with finite resistivity.
<ref>[http://dx.doi.org/10.1088/0029-5515/49/10/104025 F.L. Waelbroeck, ''Theory and observations of magnetic islands'', Nucl. Fusion '''49''' (2009) 104025]</ref>

== Island growth and saturation ==

The prediction of the non-linear saturated state of islands is the goal of [[Neoclassical transport|Neoclassical]] Tearing Mode (NTM) theory.
This theory has been developed to a considerable level of sophistication, although discrepancies with experimental observations remain.
<ref>[http://link.aip.org/link/?PHPAEN/12/080703/1 H. Lütjens and J.-F. Luciani, ''Saturation levels of neoclassical tearing modes in International Thermonuclear Experimental Reactor plasmas'', Phys. Plasmas '''12''' (2005) 080703]</ref>

== Island rotation ==

Islands can rotate within and/or with respect to the ambient plasma.
The observation of such rotating 'MHD modes' is ubiquitous in fusion plasmas with typical frequencies of the order of several tens of kHz.
The detection of such modes is possible by measuring perturbations of the magnetic field, or the electron density, temperature, or pressure.
If the ambient magnetic field (produced by external coils) has an appropriate structure, the island can also lock onto that structure.
<ref>[http://link.aps.org/doi/10.1103/PhysRevLett.78.1703 F.L. Waelbroeck and R. Fitzpatrick, ''Rotation and Locking of Magnetic Islands'', Phys. Rev. Lett. '''78''' (1997) 1703–1706]</ref>
Locked islands often lead to a [[Disruption|disruption]] (complete loss of confinement) in [[Tokamak|tokamaks]].

== Transport effects ==

It is generally assumed that the temperature is rapidly equilibrated along the magnetic field lines inside the island, so that radial transport is effectively short-circuited across the islands, decreasing the effective size of the main plasma.
<ref>[http://dx.doi.org/10.1088/0029-5515/39/12/302 ITER Physics Expert Group on Confinement and Transport et al, ''Chapter 2: Plasma confinement and transport'', Nucl. Fusion '''39''' (1999) 2175-2249 ]</ref>
However, is is possible to qualify this statement somewhat by taking into account the ratio between parallel and perpendicular transport within an island.
<ref>[http://dx.doi.org/10.1088/0029-5515/33/8/I03 B.Ph. van Milligen, A.C.A.P. van Lammeren, N.J. Lopes Cardozo, F.C. Schüller, and M. Verreck, ''Gradients of electron temperature and density across m=2 islands in RTP'', Nucl. Fusion '''33''' (1993) 1119]</ref>

The interaction of neighbouring island chains causes the magnetic field to become stochastic (according to the Chirikov criterion <ref>[http://dx.doi.org/10.1016/0370-1573(79)90023-1 B.V. Chirikov, ''A universal instability of many-dimensional oscillator systems'', Phys. Rep. '''52''', Issue 5 (1979) 263]</ref>), resulting in enhanced (anomalous) radial transport.
<ref>C.W. Horton, Y.H. Ichikawa, ''Chaos and structures in nonlinear plasmas'', World Scientific, 1996 ISBN 9789810226367</ref>

== Island control ==

Island control is possible by tailoring the ''q''-profile, external magnetic fields,
<ref>[http://dx.doi.org/10.1088/0741-3335/44/7/323 S.R. Hudson et al, ''Free-boundary full-pressure island healing in stellarator equilibria: coil-healing'', Plasma Phys. Control. Fusion '''44''' (2002) 1377]</ref>
and the pressure profile, or by spinning up the plasma.
<ref>[http://dx.doi.org/10.1088/0741-3335/45/12A/012 H. Zohm et al,''MHD limits to tokamak operation and their control'', Plasma Phys. Control. Fusion '''45''' (2003) A163]</ref>
Pressure effects can lead to 'island healing'.
<ref>[http://dx.doi.org/10.1088/0029-5515/45/7/006 R. Kanno et al, ''Formation and healing of n = 1 magnetic islands in LHD equilibrium'', Nucl. Fusion '''45''' (2005) 588]</ref>
Active control of islands by external means - in particular, Electron Cyclotron Heating and Current Drive - is also possible (cf. the 'Seek and destroy' technique, under development <ref>[http://www.rijnhuizen.nl/en/node/195 Seek and Destroy System for magnetic island control]</ref>).

== See also ==
* [[MHD equilibrium]]

== References ==
<references />

TechnoFusión

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[[File:Logo TF15-09.png|400px|right|]]

The TechnoFusión project, currently in a preparatory study phase, involves the construction of a Singular Scientific-Technical Facility (National Centre for Fusion Technologies - TechnoFusión) in the Region of Madrid, Spain, creating the required infrastructure for the development of the technologies required for future commercial fusion reactors, and assuring participation by Spanish research groups and companies.

The Spanish scientific community already possesses a critical amount of expertise on the science and technology that is needed for the success of this ambitious project, as is evident from the results obtained by Spanish researchers in the fusion field over the past few decades. TechnoFusión intends to take advantage of the existing expertise of university research groups, public research organisations (Organismo Público de Investigación, OPI) and private companies, by focussing on priority areas for the development, testing and analysis of materials that are needed for the construction of a commercial thermonuclear fusion reactor or complex remote handing systems.

The behaviour of components under the extreme conditions of a reactor is largely unknown, and this is precisely what TechoFusión pretends to explore. For this purpose, facilities are required for the manufacture, testing and analysis of critical materials, as well as facilities for the development and exploitation of numerical codes for the simulation of the behaviour of materials under extreme conditions.

In summary, TechnoFusión will focus on the creation of infrastructures for the following research areas: 1) material production and processing, 2) material irradiation, 3) plasma-wall interaction (thermal loads and the mechanism of atomic damage), 4) liquid metal technology, 5) material characterization techniques, 6) remote handling and 7) computer simulation.

The Singular Scientific-Technical Facility TechnoFusión will thus consist of a complex of seven large experimentation areas, many of which are unique in the world, with the following main technical objectives:

== Material Production and Processing ==
There are still some uncertainties about the materials that will be used to construct future fusion reactors, partly because it has not yet been possible to reproduce the extreme conditions to which such materials will be subjected. Therefore, it is of utmost importance to dispose of installations capable of manufacturing new materials on a semi-industrial scale and fabricating prototypes. Top priority materials include metals such as reinforced low activation ODS type steels ([[:Wikipedia:Oxide dispersion strengthened alloy|Oxide Dispersion Strengthened steels]]) and tungsten alloys. To manufacture such materials, equipment is required that currently is scarce or inexistent in Spain, such as a Vacuum Induction Furnace (VIM), a Hot Isostatic Pressing Furnace (HIP), a Furnace for Sintering assisted by a Pulsed Plasma Current (SPS), or a Vacuum Plasma Projection System (VPS).

== Material Irradiation ==
Even though the exact reactor conditions are only reproduced inside a reactor, it is possible to simulate the effects of neutrons and gamma radiation on materials by irradiating by ion and electron accelerators.
The effect of neutronic radiation will be characterized by combining three ion accelerators: one light ion accelerator of the tandem type for irradiating with He, with an energy of 6 MV, one light ion accelerator of the tandem type for irradiating with H (or D), with an energy of 5-6 MV, and a heavy ion accelerator of the cyclotron type, with k = 110, to implant heavy ions (Fe, W, Si, C) or high energy protons.
Additionally, a high magnetic field, between 5 and 10 T, must be incorporated into this facility in order to study the simultaneous effect of radiation and magnetic fields on materials.
The effects of ionizing gamma radiation will be studied using a Rhodotron® electron accelerator with a fixed energy of 10 MeV that will be shared with other TechnoFusión areas.

== Plasma Wall Interaction ==
Inside a fusion reactor, some materials will not be subjected only to radiation, but also to enormous heat loads in the case of plasma disruptions. In view of this, both: i) stationary conditions due to the intrinsic reactor properties: high density, low temperature and high power and ii) violent transient events (known as [[Edge Localized Modes|ELMs]] in plasma physics literature) must be reproduced. Therefore, it is essential to dispose of a device (a so-called “plasma gun”) to study plasma-material interactions simultaneously in steady state and transient regimes, thereby allowing an analysis of the modification of the materials and their properties in fusion reactors.
The mentioned plasma gun would consist of two main elements: i) a linear plasma device capable of generating hydrogen plasmas with steady state particle fluxes of up to 1024 m-2s-1 (i.e., of the order of the expected ITER fluxes) and impact energies in the range of 1-10 eV, and ii) a device of the quasi-stationary plasma accelerators (QSPA) type, providing pulses lasting 0.1-1.0 ms and energy fluxes in the 0.1-20 MJm-2 range, in a longitudinal magnetic field of the order of 1 T or greater.
These devices are connected by a common vacuum chamber, allowing the exchange of samples, and their simultaneous or consecutive exposure to the steady state and transient plasma flows under controlled conditions. Both devices will operate with hydrogen, deuterium, helium, and argon.

== Liquid Metal Technology ==
A number of [[ITER]], DEMO and [[IFMIF]] components will use liquid metals as refrigerants, tritium generators, neutron reproducers, moderators, etc., all of them under extreme conditions. Therefore, these applications need further research to be finally implemented in such installations.
The basic working scheme for this area in TechnoFusión facility is an arrangement of two liquid lithium loops, one of them coupled to the Rhodotron® electron accelerator to investigate the effects of gamma radiation on different conditions of the liquid lithium.
The main goals of this area are the studies of i) the free surface of liquid metals under conditions of internal energy deposition, and ii) the compatibility of structural materials and liquid metals in the presence of radiation. In addition, it will be possible to study the influence of magnetic fields on the cited phenomena as well as the development of methods for i) purification of liquid metals, ii) enrichment of lithium, iii) extraction of tritium, and iv) development of safety protocols for liquid metal handling.

== Characterization Techniques ==
The implementation of a wide range of techniques for the detailed characterisation of commercial or locally developed materials is proposed, applied before, during, and after their exposure to radiation or heat loads. The characterisation techniques include mechanical techniques (electromechanical devices, miniature mechanical testing devices, thermal fluency testing devices, nano-indenting techniques, etc.), compositional techniques ([[:Wikipedia:Secondary_ion_mass_spectrometry|Secondary Ions Mass Spectrometry]] (SIMS) and [[:Wikipedia:Atom-probe_tomography|Atomic Probe Tomography]] (APT)), structural and microstructural techniques ([[:Wikipedia:High_Resolution_Transmission_Electron_Microscopy|High Resolution Transmission Electron Microscopy]] (HRTEM) and [[:Wikipedia:X-Ray_Diffraction|X-Ray Diffraction]] (XRD)), and material processing techniques ([[:Wikipedia:Focused_ion_beam|Focused Ion Beam]] Systems coupled to a [[:Wikipedia:Scanning electron microscope|Scanning Electron Microscope]] (FIB/SEM)). Various systems will be used to characterise physical properties (electrical, dielectric, optical, etc.). TechnoFusión aspires to become the national materials characterisation laboratory of reference, in view of the fact that some of the cited techniques, such as SIMS or APT, are not readily available in Spain.

== Remote Handling Techniques ==
The conditions inside a fusion reactor are incompatible with the manual repair or replacement of parts, so that remote handling is indispensable. New robotic techniques need to be developed that are compatible with the hostile conditions, and existing techniques need to be certified for application at installations such as [[ITER]] or [[IFMIF]]. The size of the components that will be manipulated and the complications associated with their spatial location will require developing new remote handling techniques. Prototypes will be tested in an installation that is connected to the electron accelerator in order to simulate working conditions with gamma radiation, similar to those experienced during maintenance operations inside a reactor. Some prototypes considered for demonstrating remote handling capabilities are: the diagnostic Port Plugs (PP) and the Test [[Breeding blanket|Blanket]] Modules (TBM) for ITER, or the irradiation modules of IFMIF.

== Computer Simulation ==
In order to study conditions that cannot be reached in experiment and to accelerate the development of novel systems for a future commercial fusion power plant, TechnoFusión will stimulate an ambitious programme of computer simulations, combining the existing experience in the fusion field with resources from the National Supercomputation Network. Its goals include the implementation of the global simulation of a commercial fusion reactor, the interpretation of results, the validation of numerical tools, and the development of new tools. Another indispensable goal is the creation of a data acquisition system and the visualisation of results.

== See also ==

* [http://www.technofusion.es TechnoFusión website]

Neoclassical transport

2010-11-23T23:33:41Z

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The Neoclassical Transport Model is one of the pillars of the physics of magnetically confined plasmas.
<ref>[http://link.aps.org/doi/10.1103/RevModPhys.48.239 F.L. Hinton and R.D. Hazeltine, Rev. Mod. Phys. '''48''', 239 (1976)]</ref>
<ref>P. Helander and D.J. Sigmar, ''Collisional Transport in Magnetized Plasmas'', Cambridge University Press (2001) ISBN 0521807980</ref>
It provides a model for the transport of particles, momentum, and heat in complex magnetic geometries.
The difference between the Neoclassical and the Classical models lies in the incorporation of geometrical effects, which give rise to complex particle orbits and drifts that were ignored in the latter.

== Brief summary of the theory ==

The theory starts from the (Markovian) [http://en.wikipedia.org/wiki/Fokker-planck Fokker-Planck Equation] for the particle distribution function <math>f_\alpha(x,v,t)</math>:

:<math>
\frac{\partial f_\alpha}{\partial t} + v\cdot \nabla f_\alpha + F \frac{\partial f_\alpha}{\partial v} = C_\alpha(f)
</math>

where <math>\alpha</math> indicates the particle species, <math>v</math> is the velocity,
<math>F</math> is a force (the [http://en.wikipedia.org/wiki/Lorentz_force Lorentz force] acting on the particle) and <math>C_\alpha</math> the Fokker-Planck collision operator.
The derivation of this collision operator is highly non-trivial and requires making specific assumptions;
in particular it must be assumed that a single collision has a small random effect on the particle velocity,
and that the collisions are sufficiently frequent for the resulting particle trajectory to be described as a random walk.
The collision operator must also satisfy some obvious conservation laws (conservation of particles, momentum, and energy).

Once the collision operator is decided, the moments of the Fokker-Planck equation can be computed. First, some definitions are needed:

:<math>
n = \int{f d^3v}
</math>

(particle density)

:<math>
n u = \int{v f d^3v}
</math>

(particle flux)

:<math>
P = \int{m v \cdot v f d^3v}
</math>

(stress tensor)

:<math>
Q = \int{\frac{m v^2}{2} v f d^3v}
</math>

(energy flux)

:<math>
P' = \int{m (v-u) \cdot (v-u) f d^3v}
</math>

(pressure tensor)

:<math>
q = \int{\frac{m (v-u)^2}{2} (v-u) f d^3v}
</math>

(heat flux)

The main goal of Neoclassical transport theory is to provide a closed set of equations for the time evolution of these moments, for each particle species. Since the determination of any moment requieres knowledge of the next order moment, this requires truncating the set of moments (''closure'' of the set of equations).
<ref>T.J.M. Boyd and J.J. Sanderson, ''The physics of plasmas'', Cambridge University Press (2003) ISBN 0521459125</ref>

The theory takes account of all particle motion associated with toroidal geometry; specifically, ''&nabla; B'' and curvature drifts, and passing and trapped particles (banana orbits).
The theory is valid for all [[Collisionality|collisionality]] regimes, and includes effects due to resistivity and viscosity. An important prediction of the theory is the [[Bootstrap current|bootstrap current]].

''(Further detail needed)''

== Achievements ==
Neoclassical models have been used with success to predict transport under certain specific conditions.
''(Citation needed)''
The [[Bootstrap current|bootstrap current]] predicted by the theory is confirmed experimentally, both qualitatively and quantitatively.
''(Citation needed)''
In experimental studies, Neoclassical transport estimates are often used as a "baseline" transport level -
even though experimental values often exceed Neoclassical estimates by an order of magnitude or more.
In any case, this "baseline" level facilitates the comparison between devices.
Neoclassical theory is also used in the process of machine design and optimisation.
<ref>[http://dx.doi.org/10.1088/0741-3335/50/5/053001 M. Hirsch et al. ''Major results from the stellarator Wendelstein 7-AS'', Plasma Phys. Control. Fusion '''50''', 5 (2008) 053001]</ref>

== Limitations ==
Neoclassical theory is based on a set of assumptions that limit its range of applicability and explain why it is not capable of predicting transport in all magnetic confinement devices and under all circumstances.
These are:
* Maxwellianity. This assumption implies that a certain minimum level of [[Collisionality|collisionality]] is needed in order to ensure that Maxwellianisation is effective. The strong drives and resulting gradients that characterise fusion-grade plasmas often lead to a violation of this assumption.
* A fixed geometry. Neoclassical transport is calculated in a static magnetic geometry. In actual experiments (especially Tokamaks), the magnetic field evolves along with the plasma itself, leading to a modification of net transport. While a slow evolution (with respect to typical transport time scales) should not be problematic, rapid changes (such as magnetic reconnections) are outside of the scope of the theory.
* The linearity of the model. Neoclassical theory is a linear theory in which profiles are computed from sources, boundary conditions, and transport coefficients (that depend linearly on the profiles). No non-linear feedback of the profiles on the transport coefficients is contemplated. However, there are many experimental studies that show that the profiles feed back non-linearly on transport (via [[TJ-II:Turbulence|turbulence]]), leading to some degree of [[Self-Organised Criticality|self-organisation]].
* Locality. Neoclassical theory is a theory of diffusion, and therefore it assumes that particle motion between collisions is small with respect to any other relevant spatial scales. This assumption then allows writing down differential equations, expressing the fluxes in terms of ''local'' gradients. This basic assumption is violated under specific conditions, which may include: (a) the low-collisionality limit, (b) any situation in which the gradient scale length is very small (e.g., [[Internal Transport Barrier]]s), (c) locations close to the plasma edge<ref>[http://link.aip.org/link/?PHPAEN/8/3305/1 T. Fülöp, P. Helander, Phys. Plasmas 8, 3305 (2001)]</ref><ref>[http://dx.doi.org/10.1088/0741-3335/47/3/010 V. Tribaldos and J. Guasp, ''Neoclassical global flux simulations in stellarators'', Plasma Phys. Control. Fusion '''47''' (2005) 545]</ref>, and (d) particles transported in ''streamers''. Such phenomena could give rise to [[Non-diffusive transport|super-diffusion]]. Points (a) through (c) can be handled by using a Monte Carlo or Master Equation approach instead of deriving differential equations.
* Markovianity. A second assumption underlying diffusive models (including Neoclassics) is Markovianity, implying that the motion of any particle is only determined by its current velocity and position. However, there are situations, such as stochastic magnetic field regions, persistent turbulent eddies, or transport barriers, where this assumption may be violated (due to trapping effects, so that the preceding history of the particle trajectory becomes important). Typically, this would then give rise to [[Non-diffusive transport|sub-diffusion]].

== References ==
<references />

European Physical Society Conference on Plasma Physics

2010-11-23T23:27:30Z

Otihizuv:

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[[File:Eps_logo.jpg|168px|right|alt text]]

The scope of the [http://www.eps.org/ European Physical Society] (EPS) Conference on Plasma Physics encompasses the fields of fusion research, laser-plasma interaction and inertial confinement fusion, as well as dusty and low temperature plasmas.

Initially, the conference was denominated "European Conference on Controlled Fusion and Plasma Physics". Later, it was called the "European Physical Society Conference on Controlled Fusion and Plasma Physics". From 2004 onwards, this was changed to the "European Physical Society Conference on Plasma Physics".

== List of conferences ==

{| class="wikitable" align="center" border="1"
!''Conference'' !!''Proceedings''<ref>Proceedings are published in the Europhysics Conference Abstracts (ECA) series. Conference proceedings from 1998 through 2009 are available online from [http://epsppd.epfl.ch/ ELISE].</ref>!!''Publication''
|-
|1st, 1966 (Munich, Germany)|| ||
|-
|2nd, 1967 (Stockholm, Sweden)|| || [http://dx.doi.org/10.1088/0032-1028/10/4/308 Plasma Phys. '''10''' (1968) 421-476]
|-
|3rd, 1969 (Utrecht, The Netherlands)|| || [http://dx.doi.org/10.1007/BF01118673 Atomic Energy '''27''', 6 (1969) 1385-1386]
|-
|4th, 1970 (Rome, Italy)|| || [http://dx.doi.org/10.1007/BF01164689 Atomic Energy '''30''', 3 (1971) 393-396]
|-
|5th, 1972 (Grenoble, France)|| || [http://dx.doi.org/10.1007/BF01404530 Atomic Energy '''34''', 1 (1973) 88-91]
|-
|6th, 1973 (Moscow, USSR)|| || [http://dx.doi.org/10.1007/BF01161908 Atomic Energy '''35''', 6 (1973) 1157-1159]
|-
|7th, 1975 (Lausanne, Switzerland)|| || [http://dx.doi.org/10.1007/BF01118724 Atomic Energy '''40''', 2 (1976) 227-228]
|-
|8th, 1977 (Prague, Czechoslovakia)|| || [http://dx.doi.org/10.1007/BF01124442 Atomic Energy '''44''', 4 (1978) 445-446]
|-
|9th, 1979 (Geneva, Switzerland)|| ||
|-
|10th, 1981 (Moscow, USSR)|| ||
|-
|11th, 1984 (Aachen, Germany)||ECA Vol. 7D||
|-
|12th, 1985 (Budapest, Hungary)||ECA Vol. 9F||
|-
|13th, 1986 (Schliersee, Germany)||ECA Vol. 10C||
|-
|14th, 1987 (Madrid, Spain)||ECA Vol. 11D||
|-
|15th, 1988 (Dubrovnik, Yugoslavia)||ECA Vol. 12B||
|-
|16th, 1989 (Venice, Italy)||ECA Vol. 13B||
|-
|17th, 1990 (Amsterdam, The Netherlands)||ECA Vol. 14B||[http://iopscience.iop.org/0741-3335/32/11 Plasma Phys. Control. Fusion '''32''', 11 (1990)]

|-
|18th, 1991 (Berlin, Germany)||ECA Vol. 15C|| [http://www.iop.org/EJ/toc/0741-3335/33/13 Plasma Phys. Control. Fusion '''33''', 13 (1991)]
|-
|19th, 1992 (Innsbruck, Austria) <ref>Combined with the 9th Kiev International Conference on Plasma Theory and the 9th International Congress on Waves and Instabilities in Plasmas</ref>||ECA Vol. 16C|| [http://www.iop.org/EJ/toc/0741-3335/34/13 Plasma Phys. Control. Fusion '''34''', 13 (1992)]
|-
|20th, 1993 (Lisbon, Portugal)||ECA Vol. 17C||[http://www.iop.org/EJ/toc/0741-3335/35/SB Plasma Phys. Control. Fusion '''35''', B (1993)]
|-
|21st, 1994 (Montpellier, France)||ECA Vol. 18B||[http://www.iop.org/EJ/toc/0741-3335/36/12B Plasma Phys. Control. Fusion '''36''', 12B (1994)]
|-
|22nd, 1995 (Bournemouth, UK)||ECA Vol. 19C||[http://www.iop.org/EJ/toc/0741-3335/37/11A Plasma Phys. Control. Fusion '''37''', 11A (1995)]
|-
|23rd, 1996 (Kiev, Ukraine)||ECA Vol. 20C||[http://www.iop.org/EJ/toc/0741-3335/38/12A Plasma Phys. Control. Fusion '''38''', 12A (1996)]
|-
|24th, 1997 (Berchtesgaden, Germany)|| ECA Vol. 21A||[http://dx.doi.org/10.1088/0741-3335/39/12B/001 Plasma Phys. Control. Fusion '''39''', 12B (1997)]
|-
|25th, 1998 (Praha, Czech Republic) <ref>Combined with the 9th [[International Congress on Plasma Physics]]</ref>||[http://epsppd.epfl.ch/Praha/start.htm ECA Vol. 22C]|| [http://www.iop.org/EJ/toc/0741-3335/41/3A Plasma Phys. Control. Fusion '''41''', 3A (1999)]
|-
|26th, 1999 (Maastricht, The Netherlands)||[http://epsppd.epfl.ch/Maas/web/index.htm ECA Vol. 23J]||[http://www.iop.org/EJ/toc/0741-3335/41/12B Plasma Phys. Control. Fusion '''41''', 12B (1999)]
|-
|27th, 2000 (Budapest, Hungaria)||[http://epsppd.epfl.ch/Buda/start.htm ECA Vol. 24B]||[http://www.iop.org/EJ/toc/0741-3335/42/12B Plasma Phys. Control. Fusion '''42''', 12B (2000)]
|-
|28th, 2001 (Madeira, Portugal)||[http://epsppd.epfl.ch/Madeira/html/index.html ECA Vol. 25A]||[http://www.iop.org/EJ/toc/0741-3335/43/12A Plasma Phys. Control. Fusion '''43''', 12A (2001)]
|-
|29th, 2002 (Montreux, Switzerland)||[http://epsppd.epfl.ch/Montreux/start.htm ECA Vol. 26B]||[http://www.iop.org/EJ/toc/0741-3335/44/12B Plasma Phys. Control. Fusion '''44''', 12B (2002)]
|-
|[http://www.edu.ioffe.ru/conf/eps2003/ 30th, 2003 (St. Petersburg, Russia)]||[http://epsppd.epfl.ch/StPetersburg/start.html ECA Vol. 27A]||[http://www.iop.org/EJ/toc/0741-3335/45/12A Plasma Phys. Control. Fusion '''45''', 12A (2003)]
|-
|[http://fusion.org.uk/eps2004/ 31st, 2004 (London, UK)]||[http://epsppd.epfl.ch/London/start.htm ECA Vol. 28G]||[http://www.iop.org/EJ/toc/0741-3335/46/12B Plasma Phys. Control. Fusion '''46''', 12B (2004)]
|-
|[http://eps2005.ciemat.es/ 32nd, 2005 (Tarragona, Spain)]||[http://epsppd.epfl.ch/Tarragona/start.htm ECA Vol. 29C]||[http://www.iop.org/EJ/toc/0741-3335/47/12B Plasma Phys. Control. Fusion '''47''', 12B (2005)]
|-
|[http://eps2006.frascati.enea.it/ 33rd, 2006 (Rome, Italy)]||[http://epsppd.epfl.ch/Roma/start.htm ECA Vol. 30I]||[http://www.iop.org/EJ/toc/0741-3335/48/12B Plasma Phys. Control. Fusion '''48''', 12B (2006)]
|-
|[http://www.eps2007.ifpilm.waw.pl/ 34th, 2007 (Warsaw, Poland)]||[http://epsppd.epfl.ch/Warsaw/start.htm ECA Vol. 31F]||[http://www.iop.org/EJ/toc/0741-3335/49/12B Plasma Phys. Control. Fusion '''49''', 12B (2007)]
|-
|[http://eps2008.iesl.forth.gr/ 35th, 2008 (Hersonissos, Greece)]||[http://epsppd.epfl.ch/Hersonissos/start.htm ECA Vol. 32D]||[http://www.iop.org/EJ/toc/0741-3335/50/12 Plasma Phys. Control. Fusion '''50''', 12 (2008)]
|-
|[http://eps2009.uni-sofia.bg/ 36th, 2009 (Sofia, Bulgaria)] ||[http://epsppd.epfl.ch/Sofia/start.htm ECA Vol. 33E] || [http://www.iop.org/EJ/toc/0741-3335/51/12 Plasma Phys. Control. Fusion '''51''', 12 (2009)]
|-
|[http://www.eps2010.com/ 37th, 2010 (Dublin, Ireland)]|| [http://ocs.ciemat.es/EPS2010PAP/html/ ECA Vol. 34A] ||
|-
|[http://www-fusion-magnetique.cea.fr/eps2011/index.html 38th, 2011 (Strasbourg, France)]|| ||
|}

== Notes ==
<references />

== See also ==

* [[Media:Eps_logo_highquality2.png|High quality EPS conference logo template]]
* [http://plasma.ciemat.es European Physical Society Plasma Physics Division website] (EPS-PPD)