FusionWiki - User contributions [en]

Loss power

2023-10-13T21:51:07Z

Eldond: Created page with "LNF Nationally Funded Projects"

The loss power or power flowing through the [[separatrix|last closed flux surface]] into the [[Scrape-Off Layer]] is defined as

:<math>P_{SOL} = P_{Ohm} + P_{alpha} + P_{aux} - P_{rad,core} - \frac{dW}{dt}</math>

or the sum of Ohmic, alpha (fusion), and auxiliary heating power minus power radiated out of the core and changes in stored energy.
Auxiliary power can be delivered by [[neutral beams]] or [[RF heating]] sources.

H-mode power threshold

2023-10-13T21:46:03Z

Eldond: Created page with "== LNF - Nationally funded project == '''Title''': '''Enter Title here''' '''Reference''': Referencia Plan Nacional '''Area/subarea''': Energía y Transporte / Energía '''Principal Investigator(s)''': [https://orcid.org/0000-0000-0000-0000 John Doe] '''Project type''': (si aplica: Proyecto coordinado) '''Start-end dates''': dd/mm/yyyy - dd/mm/yyyy '''Financing granted (direct costs)''': 100000 € == Description of the project == Enter text here <!-- If applic..."

Tokamak plasmas [[L-H transition|transition]] from [[L-mode]] to [[H-mode]] when the [[loss power]] exceeds a threshold.<ref>[[doi:10.1088/0029-5515/51/10/103020|P. Gohil, et al., Nucl. Fusion 51, 103020 (2011)]]</ref> The threshold varies depending on many factors, including electron density <math>n_e</math> and main ion species (hydrogen, deuterium, deuterium/tritium mix, helium).

[[File:Gohil_2011_nf_fig1_lh_threshold.png]]

File:Gohil 2011 nf fig1 lh threshold.png

2023-10-13T21:42:57Z

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== Summary ==
LH power threshold vs density for various ion species and heating sources, from figure 1 of [[doi:10.1088/0029-5515/51/10/103020|P. Gohil, et al., Nucl. Fusion 51, 103020 (2011)]]

L-mode

2023-10-13T21:40:03Z

Eldond: /* Instructions */

L-mode, or low confinement mode, is a regime of tokamak operation. In contrast to [[H-mode]], there is no prominent [[pedestal|transport barrier at the edge of the plasma]] and [[energy confinement time]] <math>\tau_E</math> is typically about half of what can be achieved in H-mode. L-mode occurs at low input power; increasing power above a [[H-mode power threshold|threshold]]<ref>[[doi:10.1088/0029-5515/51/10/103020|P. Gohil, et al., Nucl. Fusion 51, 103020 (2011)]]</ref> prompts a [[L-H transition|transition]]<ref>[[doi:10.1103/PhysRevLett.108.155002|L. Schmitz, et al., Phys. Rev. Lett. 108, 155002 (2012)]]</ref> into H-mode.

Super H-mode

2023-10-13T21:34:02Z

Eldond: /* Book */

Super H-mode is a regime of tokamak operation wherein the pedestal height (and therefore fusion performance) is substantially higher than in standard H-mode operation. It was predicted using the [[EPED]] model for the pedestal height and then experimentally confirmed on the [[:Wikipedia:DIII-D (tokamak)|DIII-D tokamak]],<ref name=snyder_2015_nf>[[doi:10.1088/0029-5515/55/8/083026|P. B. Snyder, et al., Nucl. Fusion 55, 083026 (2015)]]</ref> followed by reproduction on Alcator C-Mod.<ref name=snyder_2019_nf>[[doi:10.1088/1741-4326/ab235b|P. B. Snyder, et al., Nucl. Fusion 59, 086017 (2019)]]</ref>

[[File:Snyder_2015_nf_fig6_superh.png]]

File:Snyder 2015 nf fig6 superh.png

2023-10-13T21:29:25Z

Eldond: Created page with "This page was created by User:Eldond|. It had to be recreated due to database corruption."

== Summary ==
Experimental measurements vs. prediction for pedestal height vs. density (discovery of super H-mode).
From figure 6 of [[doi:10.1088/0029-5515/55/8/083026|P. B. Snyder, et al., Nucl. Fusion 55, 083026 (2015)]]

H98

2023-10-13T21:22:14Z

Eldond:

<math>H_{98}</math> is a metric for plasma confinement quality. It is defined as the ratio of the [[energy confinement time]] to the confinement time predicted by the IPB98(y,2) [[scaling law]].<ref name=iter_ch2>[[doi:10.1088/0029-5515/39/12/302|ITER Physics Basis, Chapter 2, (equation 11 and table 5), Nuclear Fusion 39, 2175 (1999)]]</ref>

:<math>H_{98} = \frac{\tau_E}{\tau_{E,98y2}}</math>

where

:<math>\tau_{E,98y2} = 0.0562 \; I_p^{0.93} \; B_T^{0.15} \; \langle n_e \rangle^{0.41} \; P_{SOL}^{-0.69} \; R_{geo}^{1.97} \; \kappa_a^{0.78} \; \epsilon^{0.58} \; M^{0.19} </math>

:<math>\tau_E = \frac{W}{P_{input}-dW/dt}</math>

and <math>I_p</math> is the plasma current, <math>B_T</math> is the toroidal magnetic field at <math>R_{geo}</math>, <math>\langle n_e \rangle</math> is the average density, <math>P_{SOL}</math> is the loss power across the [[Separatrix|LCFS]] into the [[Scrape-Off Layer|SOL]], <math>R_{geo}</math> is the geometric major radius (average of maximum and minimum <math>R</math> of the LCFS) of the plasma, <math>\kappa_a</math> is the elongation, defined unusually in this case as <math>\kappa_a=Area_{CX}/\pi a^2</math>, <math>\epsilon=a/R_{geo}</math> is the inverse aspect ratio, <math>M</math> is the ion mass, <math>a</math> is the minor radius, <math>W</math> is stored energy, and <math>P_{input}</math> is the input heating power.

For a basic, predictable [[H-mode]] scenario, <math>H_{98}\approx1.0</math>. [[L-mode]] will have <math>H_{98}</math> significantly below 1, and some regimes such as [[super H-mode]] can have <math>H_{98}</math> significantly above 1.0.

H98

2023-10-10T21:36:40Z

Eldond:

<math>H_{98}</math> is a metric for plasma confinement quality. It is defined as the ratio of the [[energy confinement time]] to the confinement time predicted by the IPB98(y,2) [[scaling law]].<ref name=iter_ch2>[[doi:10.1088/0029-5515/39/12/302|ITER Physics Basis, Chapter 2, (equation 11 and table 5), Nuclear Fusion 39, 2175 (1999)]]</ref>

:<math>H_{98} = \frac{\tau_E}{\tau_{E,98y2}}</math>

where

:<math>\tau_{E,98y2} = 0.0562 \; I_p^{0.93} \; B_T^{0.15} \; \langle n_e \rangle^{0.41} \; P_{SOL}^{-0.69} \; R_{geo}^{1.97} \; \kappa_a^{0.78} \; \epsilon^{0.58} \; M^{0.19} </math>

:<math>\tau_E = \frac{W}{P-dW/dt}</math>

and <math>I_p</math> is the plasma current, <math>B_T</math> is the toroidal magnetic field at <math>R_{geo}</math>, <math>\langle n_e \rangle</math> is the average density, <math>P_{SOL}</math> is the loss power across the [[Separatrix|LCFS]] into the [[Scrape-Off Layer|SOL]], <math>R_{geo}</math> is the geometric major radius (average of maximum and minimum <math>R</math> of the LCFS) of the plasma, <math>\kappa_a</math> is the elongation, defined unusually in this case as <math>\kappa_a=Area_{CX}/\pi\;a^2</math>, <math>\epsilon=a/R_{geo}</math> is the inverse aspect ratio, <math>M</math> is the ion mass, and <math>a</math> is the minor radius.

For a basic, predictable [[H-mode]] scenario, <math>H_{98}\approx1.0</math>. [[L-mode]] will have <math>H_{98}</math> significantly below 1, and some regimes such as [[super H-mode]] can have <math>H_{98}</math> significantly above 1.0.

H98

2023-10-10T21:18:45Z

Eldond: /* > 2020 */

<math>H_{98}</math> is a metric for plasma confinement quality. It is defined as the ratio of the [[energy confinement time]] to the confinement time predicted by the IPB98(y,2) [[scaling law]].<ref name=iter_ch2>[[doi:10.1088/0029-5515/39/12/302|ITER Physics Basis, Chapter 2, (equation 11 and table 5), Nuclear Fusion 39, 2175 (1999)]]</ref>

:<math>H_{98} = \frac{\tau_E}{\tau_{E,98y2}}</math>

where

:<math>\tau_{E,98y2} = 0.0562 \; I_p^{0.93} \; B_T^{0.15} \; \langle n_e \rangle^{0.41} \; P_{SOL}^{-0.69} \; R_{geo}^{1.97} \; \kappa_a^{0.78} \; \epsilon^{0.58} \; M^{0.19} </math>

:<math>\tau_E = \frac{W}{P-dW/dt}</math>

and <math>I_p</math> is the plasma current, <math>B_T</math> is the toroidal magnetic field at <math>R_{geo}</math>, <math>P_{SOL}</math> is the loss power across the [[Separatrix|LCFS]] into the [[Scrape-Off Layer|SOL]], <math>R_{geo}</math> is the geometric major radius (average of maximum and minimum <math>R</math> of the LCFS) of the plasma, <math>\kappa_a</math> is the elongation, defined unusually in this case as <math>\kappa=Area_{CX}/\pi\;a^2</math>, <math>\epsilon=a/R_{geo}</math> is the inverse aspect ratio, <math>M</math> is the ion mass, and <math>a</math> is the minor radius.

Strongly radiating

2023-10-10T20:34:47Z

Eldond: Created page with "== Summary == The plasma is said to be detached from the divertor target plate when the primary plasma-neutral interaction takes place away from the plate because a region of high neutral density buffers the plate from the plasma. Detachment happens when physical processes (e.g. radiation, charge exchange, recombination, ...) in the Scrape-Off Layer (SOL) dissipate enough energy and momentum upstream of the divertor target plate. These processes can be activated at..."

The strongly radiating regime of [[Scrape-Off Layer]]/[[divertor]] operation is one where the divertor <math>T_e</math> is low enough that a significant fraction of power flowing out of the core plasma is dissipated by hydrogen or impurity radiation.<ref name=stangeby_book_p184>[[doi:10.1201/9780367801489|P. Stangeby, "The Plasma Boundary of Magnetic Fusion Devices" (2000) p.184]]</ref> Although power conducted to the [[divertor]] target will be reduced, the radiated power can still produce a high heat load on the target if the radiation source is close to the target.<ref name=stangeby_book_p184></ref><ref>[[doi:10.1016/j.nme.2019.01.010|D. Eldon, et al., Nucl. Mater. Energy 18, 285 (2019)]]</ref>

Compare with [[sheath limited]], [[conduction limited]], [[high recycling]], and [[Detachment|detached divertor]] regimes.

High recycling

2023-10-10T20:29:33Z

Eldond: Admin moved page TJ.II:The study of edge turbulence in the presence of ECRH and NBI heating to TJ-II:The study of edge turbulence in the presence of ECRH and NBI heating without leaving a redirect

The high recycling regime occurs when low <math>T_e</math> at the [[divertor]] targets leads to a high particle flux reaching the targets. Hydrogen is released from or reflected off of the target plates once they are saturated, and this "recycling" fuel source balances losses due to the high particle flux to the targets. The flux does not all have to come from the core plasma. The advantage of the high recycling regime is only that low <math>T_e</math> reduces sputtering, but high recycling alone does not prevent any power from reaching the divertor.<ref name=stangeby_book_p183>[[doi:10.1201/9780367801489|P. Stangeby, "The Plasma Boundary of Magnetic Fusion Devices" (2000) p.183]]</ref>

"[[Conduction limited]]" and "high recycling" are often used interchangably.

Compare with [[sheath limited]], [[conduction limited]], [[strongly radiating]], and [[Detachment|detached divertor]] regimes.

Sheath limited

2023-10-10T20:20:59Z

Eldond: /* Personal Information */ Removed obsolete web page link due to webs server cleanup

The sheath-limited regime is a state in which the [[Scrape-Off Layer]] is approximately isothermal along each flux tube (poloidal <math>T_e</math> variation is insignficant), such that the [[plasma sheath]] is the only important part of the edge plasma with respect to the transport of particles and power between the confined plasma and the solid surface of the [[divertor]] target plate.<ref name=stangeby_book_p52>[[doi:10.1201/9780367801489|P. Stangeby, "The Plasma Boundary of Magnetic Fusion Devices" (2000) p.52]]</ref> This is a strongly [[Detachment|attached]] plasma state.

The "limited" in "sheath limited" does not set any limit on power reaching the divertor; it is the state of the SOL that is limited or determined.<ref name=stangeby_book_p52></ref>

Compare with [[conduction limited]], [[high recycling]], [[strongly radiating]], and [[Detachment|detached divertor]] regimes.

Conduction limited

2023-10-10T20:20:47Z

Eldond: Admin moved page Conference on: Plasma Surface Interactions to Conference on Plasma Surface Interactions without leaving a redirect

The conduction limited regime of [[Scrape-Off Layer]] operation is where <math>T_e</math> gradients arise in the SOL due to finite plasma thermal conductivity.<ref name=stangeby_book_p52>[[doi:10.1201/9780367801489|P. Stangeby, "The Plasma Boundary of Magnetic Fusion Devices" (2000) p.52]]</ref>

Compare with [[sheath limited]], [[high recycling]], [[strongly radiating]], and [[Detachment|detached divertor]] regimes.

Sheath limited

2023-10-10T20:18:48Z

Eldond: Created page with "== Topics == * Plasma facing materials, coatings & conditioning techniques * Erosion and Deposition * Plasma fuelling, recycling and tritium inventory * Plasma surface interactions in future fusion devices * Plasma edge diagnostics * Plasma boundary physics * Plasma properties near surfaces * Plasma impurities, their transport & control * Control of the plasma boundary, e.g. by divertors, pumping and biasing == List of conferences == {| class="wikitable" align="cent..."

Sheath limited

2023-10-10T20:18:05Z

Eldond: /* Experimental proposals, Spring 2024 */

The sheath-limited regime is a state in which the [[Scrape-Off Layer]] is approximately isothermal along each flux tube (poloidal <math>T_e</math> variation is insignficant), such that the [[plasma sheath]] is the only important part of the edge plasma with respect to the transport of particles and power between the confined plasma and the solid surface of the [[divertor]] target plate.<ref name=stangeby_book_p52>[[doi:10.1201/9780367801489|P. Stangeby, "The Plasma Boundary of Magnetic Fusion Devices" (2000) p.52]]</ref> This is a strongly [[Detachment|attached]] plasma state.

The "limited" in "sheath limited" does not set any limit on power reaching the divertor; it is the state of the SOL that is limited or determined.<ref stangeby_book_p52></ref>

Compare with [[conduction limited]], [[high recycling]], [[strongly radiating]], and [[Detachment|detached divertor]] regmines.

Plasma sheath

2023-10-09T16:18:00Z

Eldond: Cambio de teléfono

At the boundary of the plasma (such as between plasma and divertor target), an electrostatic sheath forms and impacts energy flux leaving the plasma as it filters most electrons while attracting ions. <ref name=marki_2007_jnm>[[doi:10.1016/j.jnucmat.2007.01.197|J. Marki, et al., J. Nucl. Mater. 363, 382 (2007)]]</ref> Heat flux <math>q_{se}</math> at the sheath edge is related to the particle flux at the sheath edge <math>\Gamma_{se}</math>, electron temperature <math>T_e</math>, and the sheath heat transmission coefficient <math>\gamma</math> by:
:<math>q_{se}=\gamma T_e \Gamma_{se}</math>

Sheath theory gives
:<math>\gamma = 2.5 \frac{T_i}{T_e} + \frac{2}{1-\delta_e} - 0.5 \ln\left[\left(2 \pi \frac{m_e}{m_i}\right)\left(1+\frac{T_i}{T_e}\right)\frac{2}{\left(1-\delta_e\right)^2}\right]</math>
where <math>T_i</math> is the ion temperature and <math>\delta_e</math> is the secondary electron emission coefficient, both of which are difficult or impossible to measure at the divertor plate.<ref name=marki_2007_jnm></ref><ref name=stangeby_2000_book_p649>[[doi:10.1201/9780367801489|P. Stangeby, "The Plasma Boundary of Magnetic Fusion Devices" (2000) p.649]]</ref>
Setting <math>T_i=T_e</math> and <math>\delta_e=0</math> gives <math>\gamma=7</math>, which is often used due to the simplicity of the assumptions.<ref name=stangeby_2000_book_p400>[[doi:10.1201/9780367801489|P. Stangeby, "The Plasma Boundary of Magnetic Fusion Devices" (2000) p.224]]</ref><ref name=stangeby_2000_book_p224>[[doi:10.1201/9780367801489|P. Stangeby, "The Plasma Boundary of Magnetic Fusion Devices" (2000) p.400]]</ref>

Magnetic strike point

2023-03-31T19:28:26Z

Eldond: Add links

[[File:Strike_points.png|350px|right|thumb|Cross section of a tokamak plasma (<math>R</math> increases to the right) with strike points and related features labeled.]]

Magnetic strike points are where the [[separatrix]] of a [[divertor|diverted]] [[tokamak]] plasma intersects the wall.
Typically, the highest heat loads are deposited at the outer (large <math>R</math>) strike point of the primary separatrix.
Whether or not the strike points of a secondary separatrix receive significant fluxes is sensitive to <math>dR_{sep}</math>, the separation distance between the separatrices at the outboard midplane.
As separation shrinks, the secondary outer strike point will start to receive higher flux, and the primary inner strike point will receive less.
In a double null configuration with very small separation between the primary and secondary separatrices, the strike point in the direction of the [[:wiki:Guiding_center#Grad-B_drift|<math>\vec{B} \times \vec{\nabla} B</math> drift]] will tend to receive more heat flux, breaking the symmetry.

The strike points are important references for profiles of various quantities along the surface of the divertor target plates.
Many quantities, like heat flux, particle flux, and electron temperature tend to have peaks near the strike points.
It is often useful to cast plots vs. <math>x-x_{strike}</math> instead of just <math>x</math> (where ''x'' might be ''R'', ''Z'', or distance along the plate).

Magnetic strike point

2023-03-31T04:37:08Z

Eldond: Created page with "Cross section of a tokamak plasma (<math>R</math> increases to the right) with strike points and related features labeled. Magnet..."

[[File:Strike_points.png|350px|right|thumb|Cross section of a tokamak plasma (<math>R</math> increases to the right) with strike points and related features labeled.]]

Magnetic strike points are where the separatrix of a diverted tokamak plasma intersects the wall.
Typically, the highest heat loads are deposited at the outer (large <math>R</math>) strike point of the primary separatrix.
Whether or not the strike points of a secondary separatrix receive significant fluxes is sensitive to <math>dR_{sep}</math>, the separation distance between the separatrices at the outboard midplane.
As separation shrinks the secondary outer strike point will start to receive higher flux, and the primary inner strike point will receive less.
In a double null configuration with very small separation between the primary and secondary separatrices, the strike point in the direction of the <math>\vec{B} \times \vec{\nabla} B</math> drift will tend to receive more heat flux, breaking the symmetry.

The strike points are important references for profiles of various quantities along the surface of the divertor target plates.
Many quantities, like heat flux, particle flux, and electron temperature tend to have peaks near the strike points.
It is often useful to cast plots vs. <math>x-x_{strike}</math> instead of just <math>x</math> (where ''x'' might be ''R'', ''Z'', or distance along the plate).

File:Strike points.png

2023-03-31T04:28:18Z

Eldond: Figure with annotations calling out magnetic strike points and related features.

== Summary ==
Figure with annotations calling out magnetic strike points and related features.

Triangularity

2023-03-30T21:50:12Z

Eldond: Add simple illustration of shift, elongation, triangularity, and squareness

[[File:Geometry.png|400px|thumb|right|Sketch of tokamak geometry, including separatrix]]
[[File:Cross_section_1shift_2elong_3triang_4square.png|400px|thumb|right|Illustration of the m=1,2,3, and 4 perturbations to a tokamak plasma cross section. Triangularity is the m=3 perturbation.]]
The triangularity refers to the shape of the poloidal cross section of the Last Closed [[Flux surface]] (LCFS) or [[separatrix]] of a [[tokamak]].
Assuming<ref>T.C. Luce, [[doi:10.1088/0741-3335/55/9/095009|Plasma Phys. Control. Fusion '''55''' (2013) 095009 ]]</ref>:
* ''Rmax'' is the maximum value of ''R'' along the LCFS or separatrix.
* ''Rmin'' is the minimum value of ''R'' along the LCFS or separatrix.
* ''Rgeo'' is the geometric major radius, defined as ''(Rmax + Rmin)/2''.
* ''a'' is the minor radius of the plasma, defined as ''(Rmax - Rmin)/2''.
* ''Rupper'' is the major radius of the highest vertical point of the LCFS or separatrix.
* ''Rlower'' is the major radius of the lowest vertical point of the LCFS or separatrix.
The upper triangularity is then defined as follows:
:<math> \delta_{upper} = (R_{geo}-R_{upper})/a</math>
and similar for δlower.
The overall triangularity is defined as the mean of δupper and δlower.

Triangularity, especially the triangularity opposite the dominant X-point (so upper triangularity for a lower null plasma), influences the stability and character of the [[pedestal]] and [[Edge Localized Modes|ELMs]].<ref>[[doi:10.1088/0741-3335/42/5A/319|T.H. Osborne, et al., Plasma Phys. Control. Fusion '''42''' (2000) A175]]</ref>

Some devices (TCV and DIII-D) can form plasma cross sections with negative triangularity (the X-points are pushed to larger <math>R</math> than the center of the plasma), which makes H-mode difficult or impossible to access but improves performance of the L-mode.<ref>[[doi:10.1088/1741-4326/abdb95|W. Han, et al., Nucl. Fusion '''61''' (2021) 034003]]</ref>

== See also ==

* [[Ellipticity]]
* [[Toroidal coordinates]]
* [[Effective plasma radius]]

== References ==
<references />

File:Cross section 1shift 2elong 3triang 4square.png

2023-03-30T21:48:58Z

Eldond: Eldond uploaded a new version of File:Cross section 1shift 2elong 3triang 4square.png

== Summary ==
Illustration of the m=1,2,3, and 4 tokamak plasma cross section shape perturbations: m=1 is the Shafranov shift, which displaces the center of a flux surface from the nominal geometric center of the LCFS, m=2 is elongation, which stretches the plasma vertically, m=3 is triangularity, which is increased by pulling the X-points to smaller R, and m=4 is squareness.

Ellipticity

2023-03-30T21:47:45Z

Eldond: Add a figure with simple illustrations of Shafranov shift, elongation, triangularity, and squareness.

[[File:Geometry.png|400px|thumb|right|Sketch of tokamak geometry]]
[[File:Cross_section_1shift_2elong_3triang_4square.png|400px|thumb|right|Illustration of the m=1,2,3, and 4 perturbations to a tokamak plasma cross section. Ellipticity/elongation is the m=2 perturbation (second from the left).]]
The ellipticity (also referred to as elongation<ref name="Luce2013">T.C. Luce, [[doi:10.1088/0741-3335/55/9/095009|Plasma Phys. Control. Fusion '''55''' (2013) 095009 ]]</ref>) refers to the shape of the poloidal cross section of the Last Closed [[Flux surface]] or [[separatrix]] of a [[tokamak]].

Assuming<ref name="Luce2013" />:
* ''Rmax'' is the maximum value of ''R'' along the LCFS or separatrix.
* ''Rmin'' is the minimum value of ''R'' along the LCFS or separatrix.
* ''Zmax'' is the maximum value of ''Z'' along the LCFS or separatrix.
* ''Zmin'' is the minimum value of ''Z'' along the LCFS or separatrix.
* ''a'' is the minor radius of the plasma, defined as ''(Rmax - Rmin)/2''.
The ellipticity is then defined as follows:
:<math> \kappa = (Z_{max}-Z_{min})/2a</math>

Higher elongation is beneficial for fusion performance, but comes with increased vertical instability growth rate and thus increased risk of vertical displacement event (VDE) type disruptions.<ref>D.A. Humphreys, et al., ''Experimental vertical stability studies for ITER performance and design guidance'' [[doi:10.1088/0029-5515/49/11/115003|Nucl. Fusion '''49''' (2009) 115003]]</ref>
Because of vertical stability constraints, <math>\kappa</math> is usually limited to a value close to about 1.8.

== See also ==

* [[Triangularity]]
* [[Toroidal coordinates]]
* [[Effective plasma radius]]

== References ==
<references />

File:Cross section 1shift 2elong 3triang 4square.png

2023-03-30T21:44:18Z

Eldond: Illustration of the m=1,2,3, and 4 tokamak plasma cross section shape perturbations: m=1 is the Shafranov shift, which displaces the center of a flux surface from the nominal geometric center of the LCFS, m=2 is elongation, which stretches the plasma vertically, m=3 is triangularity, which is increased by pulling the X-points to smaller R, and m=4 is squareness.

User:Eldond

2023-03-30T16:49:36Z

Eldond: Created page with "David Eldon researches detachment and heat flux control systems at General Atomics."

David Eldon researches detachment and heat flux control systems at General Atomics.

Detachment control

2023-03-30T16:44:52Z

Eldond: Created page with "== Summary and motivation == Trade off between problems for the divertor and problems for the core. Detachment and heat e..."

== Summary and motivation ==

[[file:1_problems_schematic.png|400px|thumb|right|Trade off between problems for the divertor and problems for the core.]]

Detachment and heat exhaust control systems aim to meet the requirements of prolonging the lifetime of plasma facing components, particularly in the divertor, while avoiding excessive use of the actuators used to achieve and maintain detachment, which can have harmful side effects.
That is, there is an optimal degree of [[detachment]] or heat dissipation that protects the plasma facing components while minimizing problems in the core.

'''Possible problems associated with insufficient detachment and heat dissipation:'''
* Thermal stress on plasma facing components due to high heat flux
* Melting due to high heat flux
* Sputtering of wall material due to high electron temperature <math>T_e</math> next to wall

'''Possible problems associated with excessive detachment / impurity content:'''
* Reduced performance due to suboptimal scenario properties / excess density
* Reduced [[energy confinement time]] due to excess core radiation
* Fuel dilution due to extrinsic impurity seeding used to promote detachment
* Excitation of various MHD instabilities, reducing fusion performance further
* Increased [[effective charge state|effective charge state <math>Z_{eff}</math>]] and resistivity and therefore more difficult current drive and potentially shorter pulse length
* H-L back transitions due to higher H-mode power threshold at high density and/or power loss via core radiation
* [[Greenwald limit|Density limit]] [[disruption|disruptions]]
* [[MARFE]]s
* Radiative collapse [[disruption|disruptions]]

== Basic technique ==
The problem with an attached plasma with low radiation is that heat and particle exhaust out of the core plasma becomes concentrated in a narrow part of the chamber wall, usually in the divertor.
In a tokamak, this takes the form of a narrow annulus next to the [[magnetic strike point]].
To avoid this concentration and distribute the heat exhaust load over a larger area, the flow of energy and particles through the [[Scrape-Off Layer]] (SOL) is interrupted by activating dissipation processes like radiation and charge exchange.
Low <math>Z</math> impurities like neon, nitrogen, and carbon are efficient radiators at low <math>T_e</math> but less so at high <math>T_e</math>, which reduces their ability to cool the core plasma.<ref name="kallenbach_2013_ppcf">[[doi:10.1088/0741-3335/55/12/124041|A. Kallenbach, et al., Plasma Phys. Control. Fusion '''55''' (2013) 124041]]</ref>
To ensure that the SOL is cold enough for low Z impurities to radiate, edge density can be increased by puffing in additional hydrogenic (H,D, or T) gas.
With an appropriate combination of density and impurity content, [[detachment]] can begin.

The plasma state is measured with some set of sensors connected to the Plasma Control System (PCS) to transmit data in real-time.
For example, Langmuir probes can be used to estimate [[detachment|degree of detachment]],<ref name=eldon_2022_ppcf>[[doi:10.1088/1361-6587/ac6ff9|D. Eldon, et al., Plasma Phys. Control. Fusion '''64''' (2022) 075002]]</ref> triple-tipped Langmuir probes<ref name=eldon_2021_nme>[[doi:10.1016/j.nme.2021.100963|D. Eldon, et al., Nucl. Mater. Energy '''27''' (2021) 100963]]</ref> or divertor Thomson scattering<ref name=eldon_2017_nf>[[doi:doi.org/10.1088/1741-4326/aa6b16|D. Eldon, et al., Nucl. Fusion '''57''' (2017) 066039]]</ref> can be used to measure <math>T_e</math>, or bolometers can measure radiated power.<ref name=kallenbach_2012_nf>[[doi:10.1088/0029-5515/52/12/122003|A. Kallenbach, et al., Nucl. Fusion '''52''' (2012) 122003]]</ref><ref name="eldon_2019_nme">[[doi:10.1016/j.nme.2019.01.010|D. Eldon, et al., Nucl. Mater. Energy '''18''' (2019) 285]]</ref>
Data from the chosen sensor(s) is formulated into one or more control variables.
A target or reference value is set for each control variable in the PCS, and a control policy such as [[:Wikipedia:PID_controller|PID]] compares the measurement to the reference to decide on a command to one more actuators.

Possible actuators are gas valves for adding fuel or impurities, impurity powder droppers, or pellet launchers. Tests so far have used gas valves.

== Advanced techniques ==

Detachment control is fundamentally a tool for integrating core and edge scenarios.
Thus, it is natural to try to combine basic detachment control with other requirements of an integrated scenario, such as ELM removal and wall conditioning.

At ASDEX-Upgrade, a detachment control system to also control impurity-induced ELM suppression.<ref name="bernert_2021_nf">[[doi:10.1088/1741-4326/abc936|M. Bernert, et al., Nucl. Fusion '''61''' (2021) 024001]]</ref> In this case, the control variable is the the height of a local radiation centroid above (in a lower null plasma) the magnetic X-point.
It was found that this is first of all a viable control variable that is useful even when measurements at the divertor plate are saturated at low levels in deep detachment, and furthermore that positioning the radiator a specific distance above the X-point results in ELM suppression.
Somewhat related work at DIII-D has achieved ELM mitigation by impurity seeding, but without the sophisticated X-point radiator height controller.<ref name=eldon_2023_nme>[[doi:10.1016/j.nme.2022.101332|D. Eldon, et al., Nucl. Mater. Energy '''34''' (2023) 101332]]</ref>

Another promising discovery is that dropping boron nitride powder is not only useful for boron wall conditioning, but also can result in ELM removal.<ref name=gilson_2021_nme>[[doi:10.1016/j.nme.2021.101043|E.P. Gilson, et al., Nucl. Mater. Energy '''28''' (2021) 101043]]</ref>

== History ==

Radiated power control is the most commonly deployed control system related to dissipation and heat exhaust handling. These systems use foil or UV photodiode bolometers to measure radiated power. The first prototype was demonstrated and published in 1995 at ASDEX Upgrade,<ref>[[doi:10.1088/0029-5515/35/10/I07|A. Kallenbach, et al., Nucl. Fusion '''35''' (1995) 1231]]</ref>
with a demonstration at DIII-D reported in 1997.<ref>[[doi:10.1016/S0022-3115(97)80110-9|G.L. Jackson, et al., J. Nucl. Mater. '''241''' (1997) 618]]</ref>
Radiated power controllers have been demonstrated on
CMOD<ref>[[doi:10.1063/1.873447|J.A. Goetz, et al., Phys. Plasmas '''6''' (1999) 1899]]</ref>
JT-60U<ref>[[doi:10.1088/0029-5515/49/11/115010|N. Asakura, et al., Nucl. Fusion '''49''' (2009) 115010]]</ref>
JET,<ref>[[doi:10.1088/0029-5515/51/8/082001|G.P. Maddison, et al., Nucl. Fusion '''51''' (2011) 082001]]</ref>
and EAST,<ref>[[doi:10.1088/1741-4326/aab506|K. Wu, et al., Nucl. Fusion '''58''' (2018) 056019]]</ref>
and progress has continued on ASDEX Upgrade<ref name="kallenbach_2012_nf" /><ref name="kallenbach_2013_ppcf" /><ref name="kallenbach_2015_nf">[[doi:10.1088/0029-5515/55/5/053026|A. Kallenbach, et al., Nucl. Fusion '''55''' (2015) 053026]]</ref><ref name="kallenbach_2016_ppcf">[[doi:10.1088/0741-3335/58/4/045013|A. Kallenbach, Plasma Phys. Control. Fusion '''58''' (2016) 045013]]</ref> and DIII-D<ref name="eldon_2019_nme" />.

Divertor power loads assessed with shunt resistors have been used as a control variable at ASDEX Upgrade and reported in 2010.<ref>[[doi:10.1088/0741-3335/52/5/055002|A. Kallenbach, et al., Plasma Phys. Control. Fusion '''52''' (2010) 055002]]</ref>

Electron temperature <math>T_e</math> measured with divertor Thomson scattering was used as a control variable at DIII-D,<ref>[[doi:10.1016/j.jnucmat.2014.11.099|E. Kolemen, et al., J. Nucl. Mater. '''463''' (2015) 1186]]</ref><ref name="eldon_2017_nf" />
and <math>T_e</math> from triple-tipped Langmuir probes was used for detachment control at EAST.<ref name="eldon_2021_nme" />

Heat flux from surface thermocouples was used for feedback control at Alcator CMOD,<ref>[[doi:10.1088/1741-4326/aa7923|D. Brunner, et al., Nucl. Fusion '''57''' (2017) 086030]]</ref>
heat flux as calculated from Langmuir probes was used at COMPASS,<ref>[[doi:10.1088/1361-6587/abf03e|I. Khodunov, et al., Plasma Phys. Control. Fusion '''63''' (2021) 065012]]</ref>
and a model for heatflux was used for control at DIII-D.<ref>[[doi:10.1016/j.fusengdes.2021.112560|H. Anand, et al., Fus. Eng. Design '''171''' (2021) 112560]]</ref>

Attachment fraction, based on ion saturation current <math>I_{sat}</math> measurements from Langmuir probes, has been used as a control variable at
JET,<ref>[[doi:10.1088/1361-6587/aa5951|C. Guillemaut, et al., Plasma Phys. Control. Fusion '''59''' (2017) 045001]]</ref>
EAST,<ref>[[doi:10.1016/j.fusengdes.2020.111557|Q.P. Yuan, Fus. Eng. Design '''154''' (2020) 111557]]</ref>
DIII-D,<ref name="eldon_2021_nme" />
and KSTAR.<ref name="eldon_2022_ppcf" />
While many other control systems have developed semi-independently, the JET design was the direct basis for the successors at EAST and DIII-D. The KSTAR implementation was also a result of this lineage, but with modifications resulting from lessons learned while operating with the JET design.

The position of the detachment front along the divertor leg (between the X-point and the divertor target plate) has been controlled on TCV using the MANTIS camera to view C-III emission (peaks at about 8-10 eV).<ref>[[doi:10.1038/s41467-021-21268-3|T. Ravensbergen, et al., Nature Communications '''12''' (2021) 1105]]</ref>

The position of a radiation centroid relative to the magnetic X-point has been controlled at ASDEX Upgrade.<ref name="bernert_2021_nf" />

In 2020, ITPA DSOL 43 was formed to coordinate global efforts to develop detachment control systems.

== References ==
</references>

File:1 problems schematic.png

2023-03-30T15:44:49Z

Eldond: Simple schematic illustrating the tension between protecting the divertor via impurity seeding and high density to reach detachment versus the negative side effects these actuators have on the core plasma. A detachment control system should try to maintain an optimal state where both divertor and core solutions are acceptable (if such a state exists for a given scenario).

== Summary ==
Simple schematic illustrating the tension between protecting the divertor via impurity seeding and high density to reach detachment versus the negative side effects these actuators have on the core plasma. A detachment control system should try to maintain an optimal state where both divertor and core solutions are acceptable (if such a state exists for a given scenario).

Effective charge state

2023-03-29T20:22:01Z

Eldond: Created page with "The effective charge state of ions in a plasma <math>Z_{eff}</math> is important to many physical processes. :<math>Z_{eff} = \frac{\sum_i n_i Z^2_i}{\sum_i n_i Z_i}</math> wh..."

The effective charge state of ions in a plasma <math>Z_{eff}</math> is important to many physical processes.
:<math>Z_{eff} = \frac{\sum_i n_i Z^2_i}{\sum_i n_i Z_i}</math>
where <math>n_i</math> is ion density for species <math>i</math> and <math>Z_i</math> is the charge state of species <math>i</math>.

<math>\sum_i n_i Z_i = n_e </math> by quasi-neutrality, so
:<math>Z_{eff} = \frac{\sum_i n_i Z^2_i}{n_e}</math>
<math>Z_{eff}</math> would be 1 for a plasma composed purely of hydrogen isotopes, but in practice is always higher due to impurities from plasma facing components.
<math>Z_{eff}=2</math> is a common baseline assumption for carbon-walled tokamaks,<ref>[[doi:10.1088/0029-5515/39/11Y/338|T.N. Carlstrom, et al., Nucl. Fusion '''39''' (1999) 1941]]</ref><ref>[[doi:10.1103/PhysRevLett.107.215001|T. Eich, et al., Phys. Rev. Lett. '''107''' (2011) 215001]]</ref> and some multi-device databases are restricted to cases with low-ish <math>Z_{eff}</math><ref>[[doi:10.1063/1.866892|ITER Physics Basis Chapter 2: Plasma confinement and transport, Nucl. Fusion '''39''' (1999) 2175]]</ref>, making their scaling laws valid for <math>Z_{eff}<3</math> and questionable for higher <math>Z_{eff}</math>.
The 1999 ITER physics basis estimates <math>Z_{eff}=1.8-1.9</math>;<ref>[[doi:10.1088/0029-5515/39/12/301|ITER Physics Basis Chapter 1: Overview and summary, Nucl. Fusion '''39''' (1999) 2137]]</ref><ref>[[doi:10.1088/0029-5515/39/12/304|ITER Physics Basis Chapter 4: Power and particle control, Nucl. Fusion '''39''' (1999) 2391]]</ref> higher than this will dilute the fuel too much.

== References ==
<references />

Triangularity

2023-03-28T05:35:51Z

Eldond: Mention how triangularity affects the plasma and mention negative triangularity.

[[File:Geometry.png|400px|thumb|right|Sketch of tokamak geometry, including separatrix]]
The triangularity refers to the shape of the poloidal cross section of the Last Closed [[Flux surface]] (LCFS) or [[separatrix]] of a [[tokamak]].
Assuming<ref>T.C. Luce, [[doi:10.1088/0741-3335/55/9/095009|Plasma Phys. Control. Fusion '''55''' (2013) 095009 ]]</ref>:
* ''Rmax'' is the maximum value of ''R'' along the LCFS or separatrix.
* ''Rmin'' is the minimum value of ''R'' along the LCFS or separatrix.
* ''Rgeo'' is the geometric major radius, defined as ''(Rmax + Rmin)/2''.
* ''a'' is the minor radius of the plasma, defined as ''(Rmax - Rmin)/2''.
* ''Rupper'' is the major radius of the highest vertical point of the LCFS or separatrix.
* ''Rlower'' is the major radius of the lowest vertical point of the LCFS or separatrix.
The upper triangularity is then defined as follows:
:<math> \delta_{upper} = (R_{geo}-R_{upper})/a</math>
and similar for δlower.
The overall triangularity is defined as the mean of δupper and δlower.

Triangularity, especially the triangularity opposite the dominant X-point (so upper triangularity for a lower null plasma), influences the stability and character of the [[pedestal]] and [[Edge Localized Modes|ELMs]].<ref>[[doi:10.1088/0741-3335/42/5A/319|T.H. Osborne, et al., Plasma Phys. Control. Fusion '''42''' (2000) A175]]</ref>

Some devices (TCV and DIII-D) can form plasma cross sections with negative triangularity (the X-points are pushed to larger <math>R</math> than the center of the plasma), which makes H-mode difficult or impossible to access but improves performance of the L-mode.<ref>[[doi:10.1088/1741-4326/abdb95|W. Han, et al., Nucl. Fusion '''61''' (2021) 034003]]</ref>

== See also ==

* [[Ellipticity]]
* [[Toroidal coordinates]]
* [[Effective plasma radius]]

== References ==
<references />

Heat flux width

2023-03-28T05:24:55Z

Eldond: Add page describing the scrape off layer heat flux width

The [[Scrape-Off Layer]] (SOL) heat flux width <math>\lambda_q</math> is the length scale of the decaying exponential heat flux profile on the open flux surfaces.
Here, there is a competition between heat transport parallel to the field, which conducts heat to the divertors, and perpendicular diffusion.
Since parallel conduction is much faster than perpendicular diffusion, heat flux widths (<math>\lambda_q</math>) are fairly narrow—usually a few mm to a cm.
<math>\lambda_q</math> is assumed to be set at or near the outboard midplane,<ref name="eich_2013">[[doi:10.1016/j.jnucmat.2013.01.011|T. Eich, et al., J. Nucl. Mater. '''438''' (2013) S72-S77]]</ref> which is where the dominant heat source from the core into the SOL is located.
So when <math>\lambda_q</math> is quoted, it should be understood that the value at the outboard midplane is given unless otherwise stated.

As heat flows through the SOL to the divertor, the profile is broadened by [[magnetic flux expansion]].
After passing the [[Divertor|X-point]], perpendicular diffusion can go in both directions; outboard and deeper into the SOL as before, and inward to the [[Divertor|private flux region]] (PFR).<ref name=eich_2011>[[doi:10.1103/PhysRevLett.107.215001|T. Eich, et al., Phys. Rev. Lett. '''107''' (2011) 215001]]</ref>
This has the effect of smoothing the profile.

The heat flux <math>q</math> profile at the divertor (neglecting dissipation in the SOL, such as in the case of detachment) is then<ref name="eich_2013" />
:<math>q(\bar{s})=\frac{q_0}{2} \exp\left(\left(\frac{S}{2\lambda_q}\right)^2-\frac{\bar{s}}{\lambda_q f_x}\right) \cdot \mathrm{erfc}\left(\frac{S}{2\lambda_q}-\frac{\bar{s}}{S f_x}\right) +q_{BG}</math>
where <math>\bar{s}=s-s_0</math>, <math>s</math> is distance along the divertor plate, <math>s_0</math> is the [[magnetic strike point]] position, <math>S</math> is the width of the Gaussian blur effect that is convoluted with the exponential profile, <math>f_x</math> is the flux expansion (distance between flux surfaces at the divertor / distance between the same surfaces at the midplane), and <math>q_{BG}</math> is a background heat flux (which could come from radiated heat, for example).
An equation for the heat flux at the divertor is useful because the divertor heat flux profile can be measured by infrared thermography, Langmuir probes, or surface thermocouples.

This functional form was fit to heat flux profiles from several devices, and the resulting <math>\lambda_q</math> values were regressed versus several important parameters, such as field, power, safety factor, and device size.<ref name=eich_2013_nf>[[doi:10.1088/0029-5515/53/9/093031|T. Eich, et al., Nucl. Fusion '''53''' (2013) 093031]]</ref>
The regression analysis had several variants with different subsets of available devices and different parameters.
An example is
:<math>\lambda_q = C B_T^{\alpha_B} q_{cyl}^{\alpha_q} P_{SOL}^{\alpha_P} R_{geo}^{\alpha_R}</math>
with a fit to [[:Wikipedia:Joint European Torus|JET]], [[:Wikipedia:DIII-D (fusion reactor)|DIII-D]], and [[:Wikipedia:ASDEX Upgrade|ASDEX Upgrade]] resulting in

<math>C=0.86\pm 0.25</math> mm, <math>\alpha_B=-0.80\pm 0.21</math>, <math>\alpha_q=1.11\pm 0.15</math>, <math>\alpha_P=0.11 \pm 0.09</math>, <math>\alpha_R=-0.13 \pm 0.16</math>,

where <math>B_T</math> is the toroidal magnetic field, <math>q_cyl</math> is the cylindrical safety factor, <math>P_{SOL}</math> is the power flowing into the SOL, and <math>R_{geo}</math> is the geometric major radius of the plasma.

There have been other regression fits by different researchers using different subsets of devices and different parameters.<ref>[[doi:10.1016/j.jnucmat.2013.01.028|M.A. Makowski, et al., J. Nucl. Mater. '''438''' (2013) S208-S211]]</ref>

Ellipticity

2023-03-27T17:42:47Z

Eldond: Mention typical max kappa value

[[File:Geometry.png|400px|thumb|right|Sketch of tokamak geometry]]
The ellipticity (also referred to as elongation<ref name="Luce2013">T.C. Luce, [[doi:10.1088/0741-3335/55/9/095009|Plasma Phys. Control. Fusion '''55''' (2013) 095009 ]]</ref>) refers to the shape of the poloidal cross section of the Last Closed [[Flux surface]] or [[separatrix]] of a [[tokamak]].

Assuming<ref name="Luce2013" />:
* ''Rmax'' is the maximum value of ''R'' along the LCFS or separatrix.
* ''Rmin'' is the minimum value of ''R'' along the LCFS or separatrix.
* ''Zmax'' is the maximum value of ''Z'' along the LCFS or separatrix.
* ''Zmin'' is the minimum value of ''Z'' along the LCFS or separatrix.
* ''a'' is the minor radius of the plasma, defined as ''(Rmax - Rmin)/2''.
The ellipticity is then defined as follows:
:<math> \kappa = (Z_{max}-Z_{min})/2a</math>

Higher elongation is beneficial for fusion performance, but comes with increased vertical instability growth rate and thus increased risk of vertical displacement event (VDE) type disruptions.<ref>D.A. Humphreys, et al., ''Experimental vertical stability studies for ITER performance and design guidance'' [[doi:10.1088/0029-5515/49/11/115003|Nucl. Fusion '''49''' (2009) 115003]]</ref>
Because of vertical stability constraints, <math>\kappa</math> is usually limited to a value close to about 1.8.

== See also ==

* [[Triangularity]]
* [[Toroidal coordinates]]
* [[Effective plasma radius]]

== References ==
<references />

Ellipticity

2023-03-27T17:40:24Z

Eldond: Mention the effects of increasing elongation

[[File:Geometry.png|400px|thumb|right|Sketch of tokamak geometry]]
The ellipticity (also referred to as elongation<ref name="Luce2013">T.C. Luce, [[doi:10.1088/0741-3335/55/9/095009|Plasma Phys. Control. Fusion '''55''' (2013) 095009 ]]</ref>) refers to the shape of the poloidal cross section of the Last Closed [[Flux surface]] or [[separatrix]] of a [[tokamak]].

Assuming<ref name="Luce2013" />:
* ''Rmax'' is the maximum value of ''R'' along the LCFS or separatrix.
* ''Rmin'' is the minimum value of ''R'' along the LCFS or separatrix.
* ''Zmax'' is the maximum value of ''Z'' along the LCFS or separatrix.
* ''Zmin'' is the minimum value of ''Z'' along the LCFS or separatrix.
* ''a'' is the minor radius of the plasma, defined as ''(Rmax - Rmin)/2''.
The ellipticity is then defined as follows:
:<math> \kappa = (Z_{max}-Z_{min})/2a</math>

Higher elongation is beneficial for fusion performance, but comes with increased vertical instability growth rate and thus increased risk of vertical displacement event (VDE) type disruptions.<ref>D.A. Humphreys, et al., ''Experimental vertical stability studies for ITER performance and design guidance'' [[doi:10.1088/0029-5515/49/11/115003|Nucl. Fusion '''49''' (2009) 115003]]</ref>

== See also ==

* [[Triangularity]]
* [[Toroidal coordinates]]
* [[Effective plasma radius]]

== References ==
<references />

Toroidal coordinates

2023-03-27T16:53:33Z

Eldond: /* Simple toroidal coordinates */ add COCOS warning: the directions of theta and phi vary in different conventions

[[File:Toroidal coordinates.png|400px|thumb|right|A simple toroidal coordinate system]]

Coordinate systems used in toroidal systems:

== Cartesian coordinates ==

(''X'', ''Y'', ''Z'')
<ref>[[:Wikipedia:Cartesian coordinate system]]</ref>

== Cylindrical coordinates ==

<math>(R, \phi, Z)</math>, where
<ref>[[:Wikipedia:Cylindrical coordinate system]]</ref>
* <math>R^2 = X^2 + Y^2</math>, and
* <math>\tan \phi = Y/X</math>.

<math>\phi</math> is called the ''toroidal angle'' (and not the ''cylindrical'' angle, at least not in the context of magnetic confinement).

Cylindrical symmetry (symmetry under rotation over <math>\phi</math>) is referred to as ''[[axisymmetry]]''.

== Simple toroidal coordinates ==

<math>(r, \phi, \theta)</math>, where
* <math>R = R_0 + r \cos \theta</math>, and
* <math>Z = r \sin \theta</math>
<math>R_0</math> corresponds to the torus axis and is called the ''major radius'', while <math>0 \le r \le a</math> is called the ''minor radius'', and ''θ'' the ''poloidal angle''.
The ratio <math>R_0/a</math> is called the ''aspect ratio'' of the torus.

In order to adapt this simple system better to the [[Flux surface|magnetic surfaces]] of an axisymmetric [[MHD equilibrium]], it may be enhanced by
<ref>R.L. Miller et al, ''Noncircular, finite aspect ratio, local equilibrium model'', [[doi:10.1063/1.872666|Phys. Plasmas '''5''' (1998) 973]]</ref>
* letting <math>R_0/a</math> depend on <math>r</math> (to account for the [[Shafranov shift]] of flux surfaces) <ref>R.D. Hazeltine, J.D. Meiss, ''Plasma confinement'', Courier Dover Publications (2003) {{ISBN|0486432424}}</ref>
* adding [[ellipticity]] (<math>\kappa</math>), [[triangularity]] (<math>\delta</math>), and squareness (<math>\zeta</math>) to account for non-circular flux surface cross sections. A popular simple expression for shaped flux surfaces is: <ref> R.L. Miller, M.S. Chu, J.M. Greene, Y.R. Lin-Liu and R.E. Waltz, ''Noncircular, finite aspect ratio, local equilibrium model'', [[doi:10.1063/1.872666|Phys. Plasmas '''5''' (1998) 973]]</ref>

:<math>R(r,\theta) = R_0(r) + r \cos(\theta + \arcsin \delta \sin \theta)</math>
:<math>Z(r,\theta) = Z_0(r) + \kappa(r) r \sin(\theta + \zeta \sin 2 \theta) </math>

Warning: there are varying conventions for the directions of \theta and \phi. Which convention is used can depend on the local facility, the software being used, or other context. To help reduce confusion, the different tokamak coordinate conventions have been described and codified in the COCOS system.<ref>O. Sauter and S.Yu. Medvedev, ''Tokamak coordinate conventions: COCOS'', [[doi:10.1016/j.cpc.2012.09.010|Computer Physics Communications '''184''', (2013) 293-302]]</ref>

== Toroidal coordinates ==

<math>(\zeta, \eta, \phi)</math>, where
<ref>Morse and Feshbach, ''Methods of theoretical physics'', McGraw-Hill, New York, 1953 {{ISBN|007043316X}}</ref>
<ref>[[:Wikipedia:Toroidal coordinates]]</ref>

:<math>R = R_p \frac{\sinh \zeta}{\cosh \zeta - \cos \eta}</math>

:<math>Z = R_p \frac{\sin \eta}{\cosh \zeta - \cos \eta}</math>

where <math>R_p</math> is the pole of the coordinate system.
Surfaces of constant <math>\zeta</math> are tori with major radii <math>R = R_p/\tanh \zeta</math> and minor radii <math>r = R_p/\sinh \zeta</math>.
At <math>R = R_p</math>, <math>\zeta = \infty</math>, while at infinity and at <math>R = 0, \zeta = 0</math>.
The coordinate <math>\eta</math> is a poloidal angle and runs from 0 to <math>2\pi</math>.
This system is orthogonal.

The Laplace equation separates in this system of coordinates, thus allowing an expansion of the vacuum magnetic field in toroidal harmonics.
<ref>F. Alladio, F. Crisanti, ''Analysis of MHD equilibria by toroidal multipolar expansions'', [[doi:10.1088/0029-5515/26/9/002|Nucl. Fusion '''26''' (1986) 1143]]</ref>
<ref>B.Ph. van Milligen and A. Lopez Fraguas, ''Expansion of vacuum magnetic fields in toroidal harmonics'', [[doi:10.1016/0010-4655(94)90112-0|Computer Physics Communications '''81''', Issues 1-2 (1994) 74-90]]</ref>

== Magnetic coordinates ==
Magnetic coordinates are a particular type of [[flux coordinates]] in which the magnetic field lines are straight lines. Magnetic coordinates adapt to the magnetic field, and therefore to the [[MHD equilibrium]] (also see [[Flux surface]]).
Magnetic coordinates simplify the description of the magnetic field.
In 3 dimensions (not assuming axisymmetry), the most commonly used 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, [[doi:10.1088/0029-5515/2/1-2/005|Nucl. Fusion '''2''' (1962) 23]]</ref><ref>J.M. Greene and J.L Johnson, ''Stability Criterion for Arbitrary Hydromagnetic Equilibria'', [[doi:10.1063/1.1706651|Phys. Fluids '''5''' (1962) 510]]</ref> In these coordinates, both the field lines and current lines corresponding to the [[MHD equilibrium]] are straight.
* [[Boozer coordinates]]. <ref>A.H. Boozer, ''Plasma equilibrium with rational magnetic surfaces'', [[doi:10.1063/1.863297|Phys. Fluids '''24''' (1981) 1999]]</ref><ref>A.H. Boozer, ''Establishment of magnetic coordinates for a given magnetic field'', [[doi:10.1063/1.863765|Phys. Fluids '''25''' (1982) 520]]</ref> In these coordinates, the 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>.

These two coordinate systems are related.
<ref>K. Miyamoto, ''Controlled fusion and plasma physics'', Vol. 21 of Series in
Plasma Physics, CRC Press (2007) {{ISBN|1584887095}}</ref>

== References ==
<references />