Scaling law: Difference between revisions

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 The main performance parameter that is subjected to scaling law analysis is the [[Energy confinement time|energy confinement time]], &tau;<sub>E</sub>.
-The following are some of the most-used scalings for tokamaks:
+The energy confinement time is expressed in engineering variables:
-<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 = C I^{\alpha_I} B^{\alpha_B} \overline{n}^{\alpha_n} P^{\alpha_P} R^{\alpha_R} \kappa^{\alpha_\kappa} \epsilon^{\alpha_\epsilon} S_{cr}^{\alpha_S} M^{\alpha_M}</math>
-:<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 ''I<sub>M</sub>'' is given in MA, ''P<sub>M</sub>'' in MW, ''n<sub>20</sub>'' in 10<sup>20</sup> m<sup>-3</sup>, and ''B<sub>0</sub>'' in T.
+where
+* I (MA) is the plasma current
+* B (T) is the toroidal magnetic field
+* <math>\overline{n}</math> (10<sup>19</sup> m<sup>-3</sup>) is the central line averaged density
+* P (MW) is the absorbed power
+* R (m) is the major radius
+* &kappa; is the elongation
+* &epsilon; is the inverse aspect ratio
+* S<sub>cr</sub> is the cross sectional area
+* M is the hydrogen isotope mass
-For stellarators, a similar scaling has been obtained (ISS).
+The following tables shows some of the most generally used sets of scaling parameters for the ELMy [[H-mode]] and L-mode.
+<ref name="ITER">ITER Physics Expert Groups et al, ''ITER Physics Basis, Chapter 1'', [[doi:10.1088/0029-5515/39/12/301|Nucl. Fusion '''39''' (1999) 2137]] and
+[[doi:10.1088/0029-5515/39/12/302|Ibid., ''Chapter 2'']]</ref>
+<ref>J.G. Cordey, J.A. Snipes, M. Greenwald, et al., IAEA 20th Fusion Energy Conference, Vilamoura, Portugal, 2004, paper IAEA-CN-116/IT/P3-32, submitted to Nucl. Fusion.</ref>
+<ref>P.N. Yushmanov, T. Takizuka, K.S. Riedel, et al., [[doi:10.1088/0029-5515/30/10/001|Nucl. Fusion 30 (1990) 1999]]</ref>
+<ref>S.M. Kaye, et al., [[doi:10.1088/0029-5515/37/9/I10|Nucl. Fusion 37 (1997) 1303]]</ref>
+{| class="wikitable sortable" border="1" cellpadding="4" cellspacing="0"
+|+
+|- style="background:#FFDEAD;"
+!Scaling
+!<math>C\times10^3</math>
+!<math>\alpha_I</math>
+!<math>\alpha_B</math>
+!<math>\alpha_n</math>
+!<math>\alpha_P</math>
+!<math>\alpha_R</math>
+!<math>\alpha_\kappa</math>
+!<math>\alpha_\epsilon</math>
+!<math>\alpha_S</math>
+!<math>\alpha_M</math>
+|-
+|ITERH-98P(y,2)
+|56.2
+|0.93
+|0.15
+|0.41
+| -0.69
+|1.97
+|0.78
+|0.58
+| -
+|0.19
+|-
+|ITPAH-04P(y,1)
+|22.8
+|0.86
+|0.21
+|0.40
+| -0.65
+|0.32
+| -
+| -0.99
+|0.84
+|0.08
+|-
+|ITPAH-04P(y,2)
+|19.8
+|0.85
+|0.17
+|0.26
+| -0.45
+| -0.04
+| -
+| -1.25
+|0.82
+|0.11
+|-
+|ITPAH-04P(y,3)
+|88.0
+|0.90
+| -
+|0.30
+| -0.47
+|1.73
+| -
+|0.43
+| -
+| -
+|-
+|ITER-89P
+|48
+|0.85
+|0.20
+|0.10
+| -0.50
+|1.50
+|0.50
+|0.30
+| -
+|0.50
+|-
+|ITERL-96P(th)
+|23
+|0.96
+|0.03
+|0.40
+| -0.73
+|1.83
+|0.64
+| -0.06
+| -
+|0.20
+|-
+|}
+For [[stellarator]]s, 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>