Plasma Inductance

Setting the normalised internal inductance

The value of the normalised internal inductance $l_i$ can be set with the i_ind_plasma_internal_norm switch.

User input

The user can specify the value of $l_i$ directly by stating i_ind_plasma_internal_norm = 0 in the input file.

IN.DAT

i_ind_plasma_internal_norm = 0
ind_plasma_internal_norm = 1.0

Wesson relation

This can be activated by stating i_ind_plasma_internal_normx = 1 in the input file.

ind_plasma_internal_norm is set to ind_plasma_internal_norm_wesson using:

$\texttt{ind_plasma_internal_norm_wesson} = \ln{\left(1.65+0.89\alpha_{\text{J}}\right)}$

This relation is based off of data taken from DIII-D shots¹.

This is only recommended for high aspect ratio tokamaks².

It is recommended to use this switch with i_alphaj = 1 and i_beta_norm_max = 1 as they are self-consistent with each other.

Menard Inductance Relation

This can be activated by stating ind_plasma_internal_norm = 2 in the input file.

ind_plasma_internal_norm is set to ind_plasma_internal_norm_menard using³:

$\texttt{ind_plasma_internal_norm_menard} = 3.4 - \kappa$

This relation is based off of data from NSTX for $l_i$ in the range of 0.4-0.85. This model should be used for $\kappa \ge 2.5$

This is only recommended for spherical tokamaks

It is recommended to use this switch with i_beta_norm_max = 3 as they are self-consistent with each other.

ITER Definitions

The generally agreed definition of $l_i$ is:

$l_i = \frac{\langle B_{\text{p}}^2 \rangle}{B_{\text{p}}(a)^2}$

where $B_{\text{p}}(a)^2$ is the square of the average poloidal field at the plasma edge. In the second term, the average on the boundary is made explicit, with Lp the line integral of the poloidal circumference of the last closed flux surface.

In the unpublished ITER Physics Guidelines document⁴, different approximations for the denominator are used.

Although the motivation for these approximations is not explicitly stated, it is likely due to the challenges involved in generating free-boundary equilibria that include an X-point at that time⁵⁶.

ITER Version 3 | `calculate_normalised_internal_inductance_iter_3()`

$\overbrace{l_i(3)}^{\texttt{ind_plasma_internal_norm_iter_3}} = \frac{2V\langle B_{\text{p}}^2 \rangle}{\mu_0^2I_{\text{p}}^2R_0}$

T. T. S et al., “Profile Optimization and High Beta Discharges and Stability of High Elongation Plasmas in the DIII-D Tokamak,” Osti.gov, Oct. 1990. https://www.osti.gov/biblio/6194284 (accessed Dec. 19, 2024). ↩
Tokamaks 4^th Edition, Wesson, page 116 ↩
J. E. Menard et al., “Fusion nuclear science facilities and pilot plants based on the spherical tokamak,” Nuclear Fusion, vol. 56, no. 10, p. 106023, Aug. 2016, doi: https://doi.org/10.1088/0029-5515/56/10/106023. ↩
N. A. Uckan, International Atomic Energy Agency, Vienna (Austria) and ITER Physics Group, "ITER physics design guidelines: 1989", no. No. 10. Feb. 1990 ↩
T. C. Luce, D. A. Humphreys, G. L. Jackson, and W. M. Solomon, “Inductive flux usage and its optimization in tokamak operation,” Nuclear Fusion, vol. 54, no. 9, p. 093005, Jul. 2014, doi: https://doi.org/10.1088/0029-5515/54/9/093005. ↩
G. L. Jackson et al., “ITER startup studies in the DIII-D tokamak,” Nuclear Fusion, vol. 48, no. 12, p. 125002, Nov. 2008, doi: https://doi.org/10.1088/0029-5515/48/12/125002. ↩