Plasma Radiation

Overview

In PROCESS we have two distinct plasma radiation contributions that are calculated separately. These are the synchrotron radiation and then the line and Bremsstrahlung radiation combined.

By changing the input parameter radius_plasma_core_norm, the user may set the normalised radius defining the 'core' region of the plasma. Anything past this normalised radius is known as the plasma 'edge'. The synchrotron radiation loss is always deemed to come from the 'core' region.

Line & Bremsstrahlung Radiation

The radiation per unit volume is determined using loss functions computed by the ADAS405 code ¹. The effective collisional–radiative coefficients necessary to determine the ionization state and radiative losses of each ionic species, assuming equilibrium ionization balance in an optically thin plasma, were sourced from ADF11-derived data files ². These coefficients utilize the generalized collisional-radiative approach ³ for elements such as H, He, Be, C, N, O, Ne, and Si.

For Ni, the data is based on ⁴, for Fe it is derived from ⁵; and for W, the data is obtained from ⁶. The Ni and Fe rates incorporate a density dependence as described in ⁷. For Kr and Xe, data is taken from the ADAS baseline.

The computed loss functions exhibit a weak dependence on density but are evaluated at a fixed electron density of $10^{19} \text{m}^{-3}$ . This differs from strict coronal equilibrium, which assumes density independence. In practice, non-local effects arising from density and temperature gradients are significant but are not considered here. The loss functions account for Bremsstrahlung, line radiation, and recombination radiation, represented by:

$P_i = n_i n_e L_Z (Z_i, T)$

where $P_i$ is the radiation per unit volume (excluding synchrotron radiation), $L_Z (Z_i, T)$ is the loss function for ion species $i$ at temperature $T$ , and $n_i$ is the density of ion species $i$ .

The radiation emission is numerically integrated over the plasma profile, using the corresponding temperature and density distributions. Emission is only considered from within the separatrix, as the PROCESS model does not account for the scrape-off layer. The plasma inside the separatrix is divided into two regions: the “core” and the “edge,” separated by a normalized minor radius defined by the user. Radiation is calculated separately for the core and edge regions, except for synchrotron radiation, which is assumed to originate solely from the core.

Synchrotron radiation | `psync_albajar_fidone()`

The formula below is the current synchrotron radiation loss power implemented in PROCESS. It is a combination of the general form given by Albajar et al. ⁸ and a wall reflectivity correction given by Fidone et al. ⁹

$\begin{aligned} P_{\text{syn}}(\mathrm{MW})= & 3.84 \times 10^{-8}\times \frac{(1-f_{\text{reflect}})^{0.62}}{\left[1+0.12 \frac{T_{\text{e0}}}{p_{a_0}^{0.41}}\left(1.0 - f_{\text{reflect}}\right)^{0.41}\right]^{1.51}} \\ & \times R a^{1.38} \kappa^{0.79}B_{\text{T}}^{2.62} n_{\text{e0,20}}^{0.38} T_{\text{e0}}\left(16+T_{\text{e0}}\right)^{2.61} \\ & \times K\left(\alpha_n, \alpha_T, \beta_T\right) \\ & \times G(A) \end{aligned}$

$K(\alpha_n, \alpha_T, \beta_T) = \left(\alpha_n +3.87\alpha_T +1.46\right)^{-0.79} \\ \times \left(1.98+\alpha_T\right)^{1.36}\beta_T^{2.14} \\ \times \left(\beta_T^{1.53}+1.87\alpha_T-0.16\right)^{-1.33}$

$p_{a_0} = 6.04 \times 10^3 \frac{a n_{\text{e0(20)}}}{B_{\text{T}}}$

where $T_{\text{e0}}$ is the central electron temperature in keV, $R$ is the plasma major radius, $a$ is the plasma minor radius, $\kappa$ is the plasma separatrix elongation, $B_{\text{T}}$ is the on axis toroidal magnetic field, $n_{\text{e0,20}}$ is the central electron density in units of $10^{20} \text{m}^{-3}$ , $\alpha_n$ is the density profile peaking parameter, $\alpha_T$ is the temperature profile peaking parameter, $\beta_T$ is the secondary temperature profile peaking parameter and $A$ is the plasma aspect ratio.

The original form of the synchrotron radiation formula presented by Albajar et al. ⁸ is based off using multiple non-linear regression from a database consisting of 3000 complete computations of the analytical expression for synchrotron power derived by Albajar et al.⁸ to include aspect ratio and temperature profile dependence.

The fitting variable range is:

$10 < T_{\text{e0}} < 100 \ \text{keV}$
$10^2 < p_{\text{a0}} < 1 \times 10^4$
$1 < \kappa < 2.5$
$0 < \alpha_n < 2$
$0 < \alpha_T < 8$
$1 < \beta_T < 8$

The root mean square error of the fit is found to be 5.8%

The original form also uses a fair estimation of the effect of a wall with a reflection coefficient $f_{\text{reflect}}$ obtained from the Trubnikov approach ¹⁰.

$P_{\text{syn,r}} \propto \left(1 − f_{\text{reflect}}\right)^{1/2} P_{\text{syn}}$

It should also be noted that the wall reflection coefficient for the synchrotron radiation is poorly known.

The wall reflection factor ( $f_{\text{reflect}}$ ) may be set by the user by inputting f_sync_reflect = <value>.

The wall reflectivity correction presented later by Fidone et al. ⁹ is of the form:

$P_{\text{syn,r}} \propto \frac{\left(1 − f_{\text{reflect}}\right)^{0.62}}{\left[1+\left(1 − f_{\text{reflect}}\right)^{0.41}\right]^{1.51}} P_{\text{syn}}$

A comparison of the different reflection functions canbe seen in the graph below:

Normalised synchrotron radiation loss vs wall reflectivity

The power loss due to synchrotron radiation grows as the aspect ratio decreases. At high aspect ratios $(A > 6)$ , although $P_{\text{syn}}$ increases with $R$ , the normalized synchrotron loss saturates. This is due to the fact that the magnetic field inhomogeneity vanishes for large $A$ . For the above reason a $G$ correction factor is implemented.This gives an root mean square error of 6.2% with respect to a secondary dataset consisting of 640 complete computations for the same range of variables variables above and for an aspect ratio interval of $1.5 <A< 15$ .

$G\left(A\right) = 0.93\left[1+0.85 e^{-0.82A}\right]$

Impurity Radiation Class | `ImpurityRadiation`

Initialization | `init()`

Initialize the FusionReactionRate class with the given plasma profile.

Parameters:

plasma_profile (PlasmaProfile): The parameterized temperature and density profiles of the plasma. Taken from the plasma_profile object.

Attributes:

plasma_profile (PlasmaProfile): Plasma profile instance.
rho (numpy.ndarray): Density profile along the x-axis.
rhodx (numpy.ndarray): Density profile step size along the x-axis.
imp (numpy.ndarray): Indices of impurities with a fraction greater than 1.0e-30.
pimp_profile (numpy.ndarray): Impurity profile array initialized to zeros.
radtot_profile (numpy.ndarray): Total radiation profile array initialized to zeros.
radcore_profile (numpy.ndarray): Core radiation profile array initialized to zeros.
radtot (float): Total radiation initialized to 0.0.
radcore (float): Core radiation initialized to 0.0.

Key Constraints

Radiation power density upper limit

This constraint can be activated by stating icc = 17 in the input file.

$\frac{P_{\text{rad}}}{V_{\text{plasma}}} < \frac{P_{\text{inj}} + P_{\alpha}f_{\alpha, \text{plasma}} + P_{\text{charged}} + P_{\text{ohmic}} }{V_{\text{plasma}}}$

Ensures that the calculated total radiation power density does not exceed the total heating power to the plasma (i.e. the sum of the fusion alpha power coupled to the plasma, other charged particle fusion power, auxiliary injected power and the ohmic heating power). If the radiation power is higher than the heating power then the change in the stored plasma thermal energy is negative, $-\frac{\mathrm{d}W}{\mathrm{d}T}$ . This means the plasma is cooling and is thus not a viable plasma solution.

$f_{\alpha, \text{plasma}}$ is the fraction of alpha power that is coupled to the plasma (f_alpha_plasma).

The scaling value fradpwr can be varied also.

It is recommended to have this constraint on as it is a plasma stability model

Radiation wall load upper limit

This constraint can be activated by stating icc = 67 in the input file.

The limiting value of $q_{\text{fw,rad}}$ in $\mathrm {MWm^{-2}}$ is be set using input parameter pflux_fw_rad_max.

The scaling value fradwall can be varied also.

“ADAS: Docmentation,” Adas.ac.uk, 2024. https://www.adas.ac.uk/manual.php ↩
“OPEN-ADAS,” Adas.ac.uk, 2025. https://open.adas.ac.uk/adf11 (accessed Jan. 15, 2025). ↩
H. P. Summers et al., “Ionization state, excited populations and emission of impurities in dynamic finite density plasmas: I. The generalized collisional–radiative model for light elements,” Plasma Physics and Controlled Fusion, vol. 48, no. 2, pp. 263–293, Jan. 2006, doi: https://doi.org/10.1088/0741-3335/48/2/007. ↩
M. Arnaud and R. Rothenflug, “An updated evaluation of recombination and ionization rates,” Astronomy & Astrophysics Supplement Series, vol. 60, no. 3, pp. 425–457, Jun. 1985. ↩
M. Arnaud, R. Rothenflug, An updated evaluation of recombination and ionization rates, Astron. & Astrophys. Supp. Ser. 60 (1985) 425–457. ↩
T. Pütterich, R. Neu, R. Dux, A. D. Whiteford, and M. G. O’Mullane, “Modelling of measured tungsten spectra from ASDEX Upgrade and predictions for ITER,” Plasma Physics and Controlled Fusion, vol. 50, no. 8, p. 085016, Jun. 2008, doi: https://doi.org/10.1088/0741-3335/50/8/085016. ↩
Summers, H. P. (1974). Tables and Graphs of Collisional Dielectronic Recombination and Ionisation Coefficients and Ionisation Equilibria of H-like to A-like Ions of Elements. Appleton Laboratory. ↩
F. Albajar, J. Johner, and G. Granata, “Improved calculation of synchrotron radiation losses in realistic tokamak plasmas,"Nuclear Fusion, vol. 41, no. 6, pp. 665–678, Jun. 2001, doi: https://doi.org/10.1088/0029-5515/41/6/301. ↩↩↩
I. Fidone, G Giruzzi, and G. Granata, “Synchrotron radiation loss in tokamaks of arbitrary geometry,” Nuclear Fusion, vol. 41, no. 12, pp. 1755–1758, Dec. 2001, doi: https://doi.org/10.1088/0029-5515/41/12/102. ↩↩
Trubnikov, B.A., in Reviews of Plasma Physics, Vol. 7 (Leontovich, M.A., Ed.), Consultants Bureau, New York (1979) 345. ↩