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Plasma confinement time

Overview

Confinement time scalings are empirical relationships derived from experimental data across various fusion machines. These scalings help predict how changes in tokamak parameters (like size, magnetic field strength, and plasma density) will affect the confinement time and overall performance.

The energy confinement time \tau_E is calculated using one of a choice of empirical scalings. (\tau_E is defined below.)

Normally most confinement scalings will be of the form:

\tau_{\text{E}} = C I_{\text{p}}^{\alpha_{I}} B_{\text{T}}^{\alpha_{B}} \overline{n}_e^{\alpha_{n}} P_{\text{L}}^{\alpha_{P}} R^{\alpha_{R}} \kappa^{\alpha_{\kappa}} \epsilon^{\alpha_{\epsilon}} M^{\alpha_{M}}

Where \tau_{\text{E}} is the confinement time in seconds, C is a coefficient , I_{\text{p}} [MA] is the plasma current, B_{\text{T}} [T] is the toroidal magnetic field, \overline{n}_e [10^{19} \text{m}^{-3}] is the electron central line averaged density, P_{\text{L}} [MW] is the loss power, R [m] is the major radius, \kappa is the plasma elongation, \epsilon is the inverse aspect ratio and M is the average atomic mass of the plasma.

Classically the loss power, P_{\text{L}} is defined as:

P_{\text{L}} = \frac{W}{\tau_{\text{E}}}

where W is the total thermal energy of the plasma. We can look at it mainly as the difference in heating and loss powers in the plasma, as such we interpret it as power transported out from the “core” by charged particles. This leads to the classic definition of loss power for the scaling:

P_{\text{L}} = \underbrace{f_{\alpha}P_{\alpha} + P_{\text{c}} + P_{\text{OH}} + P_{\text{HCD}}}_{\text{Plasma heating}}

where f_{\alpha} is the fraction of alpha power that is coupled to the plasma, P_{\alpha} is the alpha power, P_{\text{c}} is the charged particle power, P_{\text{OH}} is the ohmic heating power, P_{\text{HCD}} is the plasma heating done by the external heating & current drive systems.

Calculating plasma confinement time | calculate_confinement_time()

The correspoding plasma confinement time is calculated by the calculate_confinement_time() function in physics.py with scalings taken from confinement_time.py.

A key definition of elongation is defined here and is used mainly in the ITER physics basis scalings 12:

\kappa_{\text{IPB}} = \frac{V_{\text{p}}}{2\pi R}\frac{1}{\pi a^2}

where V_{\text{p}} is the plasma volume, R is the plasma major radius and a is the plasma minor radius.

The loss power P_{\text{L}} [\mathtt{p\_plasma\_loss\_mw}] is calculated from above but may have a separate radiation term depending on the condition of i_rad_loss switch below.

Effect of radiation on energy confinement

Published confinement scalings are all based on low radiation pulses. A power plant will certainly be a high radiation machine, both in the core, due to bremsstrahlung and synchrotron radiation, and in the edge due to impurity seeding. The scaling data does not predict this radiation 24 25, that needs to be done by the radiation model. However, if the transport is very "stiff", as predicted by some models, then the additional radiation causes an almost equal drop in power transported by ions and electrons, leaving the confinement nearly unchanged.

To allow for these uncertainties, three options are available, using the switch i_rad_loss.

  • For i_rad_loss = 0 the total plasma radiation is taken from the loss power.
P_{\text{L}} = \underbrace{f_{\alpha}P_{\alpha} + P_{\text{c}} + P_{\text{OH}} + P_{\text{HCD}}}_{\text{Plasma heating}} - P_{\text{rad}}
  • For i_rad_loss = 1 the plasma radiation only from the "core" region is taken from the loss power.
P_{\text{L}} = \underbrace{f_{\alpha}P_{\alpha} + P_{\text{c}} + P_{\text{OH}} + P_{\text{HCD}}}_{\text{Plasma heating}} - P_{\text{rad,core}}
  • For i_rad_loss = 2 the plasma radiation is not taken from the loss power
P_{\text{L}} = \underbrace{f_{\alpha}P_{\alpha} + P_{\text{c}} + P_{\text{OH}} + P_{\text{HCD}}}_{\text{Plasma heating}}

Ignition

Switch ignite can be used to denote whether the plasma is ignited, i.e. fully self-sustaining without the need for any injected auxiliary power during the burn. If ignite = 1, the heating and current drive power P_{\text{HCD}}, does not contribute to the loss power term.

If ignite = 0, the plasma is not ignited, and the heating and current drive power P_{\text{HCD}}, does contribute to the loss power term. phase. An ignited plasma will be difficult to control and is unlikely to be practical. This option is not recommended.

Available confinement time scalings

Many energy confinement time scaling laws are available within PROCESS, for conventional aspect ratio tokamaks, spherical tokamaks, and stellarators. The value of i_confinement_time determines which of the scalings is used in the plasma energy balance calculation.

The scaling chosen with i_confinement_time is then calculated and multiplied with the H-factor [\mathtt{hfact}]. \mathtt{hfact} can be set as an interation variable by setting ixc = 10 in the IN.DAT input file.

0: User input confinement time

Is selected with i_confinement_time = 0

\tau_{\text{E}} = \mathtt{t\_electron\_confinement\_in}

1: Nec-Alcator scaling (Ohmic)

Is selected with i_confinement_time = 11

\tau_{\text{E}} = 0.07 n_{20}aRq_{\text{cyl}}

2: Mirnov scaling (H-mode)

Is selected with i_confinement_time = 21

\tau_{\text{E}} = 0.2 a \sqrt{\kappa_{95}}I_{\text{p}}

3: Merezhkin-Mukhovatov scaling (Ohmic / L-mode)

Is selected with i_confinement_time = 31

\tau_{\text{E}} = 0.0035 \overline{n}_{20}a^{0.25}R^{2.75}q_{\text{cyl}}\kappa_{95}^{0.125}M_i^{0.5}T_{10}^{0.5}

4: Shimomura scaling (H-mode)

Is selected with i_confinement_time = 41

\tau_{\text{E}} = 0.045 Ra B_{\text{T}}\sqrt{\kappa_{95}}\sqrt{M_{\text{i}}}

5: Kaye-Goldston scaling (L-mode)

Is selected with i_confinement_time = 51

\tau_{\text{E}} = 0.055 I_{\text{p}}^{1.24}P_{\text{L}}^{-0.58}R^{1.65}a^{-0.49}\kappa_{95}^{0.28}n_{20}^{0.26}B_{\text{T}}^{-0.09}\left(\frac{M_{\text{i}}}{1.5}\right)^{0.5}

6: ITER 89-P scaling (L-mode)

Is selected with i_confinement_time = 61 2

\tau_{\text{E}} = 0.048 I_{\text{p}}^{0.85}R^{1.2}a^{0.3}\kappa^{0.5}\overline{n}_{20}^{0.1}B_{\text{T}}^{0.2}M_{\text{i}}^{0.5} P_{\text{L}}^{-0.5}

7: ITER 89-0 scaling (L-mode)

Is selected with i_confinement_time = 7 2

\begin{aligned} \tau_E= & 0.04 I_{\text{p}}^{0.5} R^{0.3} a^{0.8} \kappa^{0.6} M_i^{0.5} \\ & +0.064 I_{\text{p}}^{0.8} R^{1.6} a^{0.6} \kappa^{0.2} \bar{n}_{20}^{0.6} B_0^{0.35} M_i^{0.2} / P_{\text{L}} \end{aligned}

8: Rebut-Lallia scaling (L-mode)

Is selected with i_confinement_time = 8 2

\begin{aligned} \tau_E= & 1.65\left[1.2 \times 10^{-5} I_{\text{p}} \ell^{1.5} Z_{e f f}^{-0.5}\right. \\ & \left.+0.146 \bar{n}_{20}^{0.75} I_{\text{p}}^{0.5} B_0^{0.5} \ell^{2.75} Z_{e f f}^{0.25} / P_{\text{L}}\right]\left(A_i / 2\right)^{0.5} \end{aligned}

where \ell = \left(a^2R\kappa\right)^{\frac{1}{3}}

9: Goldston scaling (L-mode)

Is selected with i_confinement_time = 9 1

\tau_{\text{E}} = 0.037 I_{\text{p}} P_{\text{L}}^{-0.5} R^{1.75}a^{-0.37}\kappa_{95}^{0.5} \left(\frac{M_i}{1.5}\right)^{0.5}

10: T-10 scaling (L-mode)

Is selected with i_confinement_time = 10 1

\tau_{\text{E}} = 0.095 a R B_{\text{T}} \kappa_{95}^{0.5} \frac{\overline{n}_{20}}{\overline{n}_{20*}}P_{\text{L}}^{-0.4} \left[\frac{Z_{\text{eff}}^2 I_{\text{p}}^4}{aRq_{\text{cyl}}^3\kappa_{95}^{1.5}} \right]^{0.08}

where \overline{n}_{20*} = 1.3\left(\frac{B_{\text{T}}}{Rq_{\text{cyl}}}\right) and \frac{\overline{n}_{20}}{\overline{n}_{20*}} \le 1

11: JAERI / Odajima-Shimomura scaling (L-mode)

Is selected with i_confinement_time = 11 1

\tau_{\text{E}} = \left[\frac{0.085\kappa_{95}a^2+0.069In_{20}^{0.6}B_{\text{T}}^{0.2}R^{1.6} a^{0.4} \kappa_{95}^{0.2} G\left(q_{\text{cyl}},Z_{\text{eff}}\right)}{P_{\text{L}}}\right]M_{\text{i}}^{0.5}

where G\left(q_{\text{cyl}},Z_{\text{eff}}\right) = Z_{\text{eff}}^{0.4}\left[\frac{\left(15 - Z_{\text{eff}}\right)}{20}\right]^{0.6}\left[3q_{\text{cyl}}\frac{q_{\text{cyl}}+5}{(q_{\text{cyl}}+2)(q_{\text{cyl}}+7)}\right]^{0.6}

12: Kaye "big" scaling (L-mode)

Is selected with i_confinement_time = 12 1

\tau_{\text{E}} = 0.1051 I_{\text{p}}^{0.85} P_{\text{L}}^{-0.5} R^{0.5} a^{0.3} \kappa^{0.25} n_{20}^{0.1}B_{\text{T}}^{0.3}M_{\text{i}}^{0.5}

13: ITER H90-P scaling (H-mode)

Is selected with i_confinement_time = 13 2

\tau_{\text{E}} = 0.064 I_{\text{p}}^{0.87} R^{1.82} a^{-0.12} \kappa_{95}^{0.35} \overline{n}_{20}^{0.09} B_{\text{T}}^{0.15} M_{\text{i}}^{0.5} P_{\text{L}}^{-0.5}

14: Minimum of ITER 89-P and ITER 89-O

Is selected with i_confinement_time = 14 1 2

Will return the value of ITER 89-P or ITER 89-O, whichever is smaller.

15: Riedel scaling (L-mode)

Is selected with i_confinement_time = 15 2

\tau_{\text{E}} = 0.044 I_{\text{p}}^{0.93} R^{1.37} a^{-0.049} \kappa_{95}^{0.588} \overline{n}_{20}^{0.078} B_{\text{T}}^{0.152} P_{\text{L}}^{-0.537}

16: Christiansen scaling (L-mode)

Is selected with i_confinement_time = 16 2

\tau_{\text{E}} = 0.24 I_{\text{p}}^{0.79} R^{0.56} a^{1.46} \kappa_{95}^{0.73} \overline{n}_{20}^{0.41} B_{\text{T}}^{0.29} P_{\text{L}}^{-0.79} M_{\text{i}}^{-0.02}

17: Lackner-Gottardi scaling (L-mode)

Is selected with i_confinement_time = 17 2

\tau_{\text{E}} = 0.12 I_{\text{p}}^{0.8} R^{1.8} a^{0.4} \left(\frac{\kappa_{95}}{\left(1+\kappa_{95}\right)^{0.8}}\right) \overline{n}_{20}^{0.6} \hat{q}^{0.4} P_{\text{L}}^{-0.6}

where \hat{q} = \frac{(1+\kappa_{95}a^2B_{\text{T}})}{0.4 I_{\text{p}} R}

18: Neo-Kaye scaling (L-mode)

Is selected with i_confinement_time = 18 2

\tau_{\text{E}} = 0.063 I_{\text{p}}^{1.12} R^{1.3} a^{-0.04} \kappa_{95}^{0.28} \overline{n}_{20}^{0.14} B_{\text{T}}^{0.04} P_{\text{L}}^{-0.59}

19: Riedel scaling (H-mode)

Is selected with i_confinement_time = 19 2

\tau_{\text{E}} = 0.1 M_{\text{i}}^{0.5} I_{\text{p}}^{0.884} R^{1.24} a^{-0.23} \kappa_{95}^{0.317} \overline{n}_{20}^{0.105} B_{\text{T}}^{0.207} P_{\text{L}}^{-0.486}

20: Amended ITER H90-P scaling (H-mode)

Is selected with i_confinement_time = 20 3

\tau_{\text{E}} = 0.082 M_{\text{i}}^{0.5} I_{\text{p}}^{1.02} R^{1.6} \kappa_{95}^{-0.19} B_{\text{T}}^{0.15} P_{\text{L}}^{-0.47}

21: Sudo et al. scaling (Stellarator)

Is selected with i_confinement_time = 21 4

\tau_{\text{E}} = 0.17 P_{\text{L}}^{-0.58} \overline{n}_{20}^{0.69} B^{0.84} a^{2.0} R^{0.75}

22: Gyro reduced Bohm scaling (Stellarator)

Is selected with i_confinement_time = 22 5

\tau_{\text{E}} = 0.25 P_{\text{L}}^{-0.6} \overline{n}_{20}^{0.6} B_{\text{T}}^{0.8} a^{2.4} R^{0.6}

23: Lackner-Gottardi scaling (Stellarator)

Is selected with i_confinement_time = 23 6

\tau_{\text{E}} = 0.17 P_{\text{L}}^{-0.6} \overline{n}_{20}^{0.6} B_{\text{T}}^{0.8} a^{2.0} R q_{95}^{0.4}

24: ITER H93 ELM-free scaling (H-mode)

Is selected with i_confinement_time = 24 7

\tau_{\text{E}} = 0.036 I_{\text{p}}^{1.06} B_{\text{T}}^{0.32} P_{\text{L}}^{-0.67} R^{1.79} \epsilon^{-0.11} \kappa^{0.66} \overline{n}_{20}^{0.17} M_{\text{i}}^{0.41}

25: TITAN Reversed-Field_Pinch scaling

Is selected with i_confinement_time = 25

Warning

This scaling has been removed

26: ITER H-97P ELM-free scaling (H-mode)

Is selected with i_confinement_time = 26 8

\tau_{\text{E}} = 0.031 M_{\text{i}}^{0.42} I_{\text{p}}^{0.95} R^{1.92} \epsilon^{0.08} \kappa_{95}^{0.63} \overline{n}_{19}^{0.35} B_{\text{T}}^{0.25} P_{\text{L}}^{-0.67}

27: ITER H-97P ELMy scaling (H-mode)

Is selected with i_confinement_time = 27 8 9

\tau_{\text{E}} = 0.029 M_{\text{i}}^{0.2} I_{\text{p}}^{0.9} R^{2.03} \epsilon^{-0.19} \kappa_{95}^{0.92} \overline{n}_{19}^{0.4} B_{\text{T}}^{0.20} P_{\text{L}}^{-0.66}

28: ITER-96P (ITER-97L) scaling (L-mode)

Is selected with i_confinement_time = 28 10

\tau_{\text{E}} = 0.023 M_{\text{i}}^{0.2} I_{\text{p}}^{0.96} R^{1.83} \epsilon^{-0.06} \kappa_{95}^{0.64} \overline{n}_{19}^{0.4} B_{\text{T}}^{0.03} P_{\text{L}}^{-0.73}

29: Valovic modified ELMy scaling (H-mode)

Is selected with i_confinement_time = 29

\tau_{\text{E}} = 0.067 M_{\text{i}}^{0.05} I_{\text{p}}^{0.9} R^{1.31} \kappa^{0.56} \overline{n}_{19}^{0.45} B_{\text{T}}^{0.17} P_{\text{L}}^{-0.68} a^{0.79}

Warning

The origin, name and values of this scaling cannot be confirmed.

30: Kaye 98 modified scaling (L-mode)

Is selected with i_confinement_time = 30

\tau_{\text{E}} = 0.021 M_{\text{i}}^{0.25} I_{\text{p}}^{0.81} R^{2.01} \kappa^{0.7} \overline{n}_{19}^{0.47} B_{\text{T}}^{0.14} P_{\text{L}}^{-0.73} \epsilon^{0.18}

Warning

The origin, name and values of this scaling cannot be confirmed.

31: ITERH-PB98P(y) scaling (H-mode)

Is selected with i_confinement_time = 31

\tau_{\text{E}} = 0.0615 M^{0.2} I_{\text{p}}^{0.9} R^{2.0} \kappa_{\text{IPB}}^{0.75} \overline{n}_{19}^{0.4} B_{\text{T}}^{0.1} P_{\text{L}}^{-0.66} \epsilon^{0.66}

Warning

The origin, name and values of this scaling cannot be confirmed.

32: IPB98(y) ELMy scaling (H-mode)

Is selected with i_confinement_time = 32 11 12

\tau_{\text{E}} = 0.0365 I_{\text{p}}^{0.97} B_{\text{T}}^{0.08} \overline{n}_{19}^{0.41} P_{\text{L}}^{-0.63} R^{1.93} \kappa^{0.67} \epsilon^{0.23} M^{0.2}

33: IPB98(y,1) ELMy scaling (H-mode)

Is selected with i_confinement_time = 33 11 12

\tau_{\text{E}} = 0.0503 I_{\text{p}}^{0.91} B_{\text{T}}^{0.15} \overline{n}_{19}^{0.44} P_{\text{L}}^{-0.65} R^{2.05} \kappa_{\text{IPB}}^{0.72} \epsilon^{0.57} M^{0.13}

34: IPB98(y,2) ELMy scaling (H-mode)

Is selected with i_confinement_time = 34 11 12

\tau_{\text{E}} = 0.0562 I_{\text{p}}^{0.93} B_{\text{T}}^{0.15} \overline{n}_{19}^{0.41} P_{\text{L}}^{-0.69} R^{1.97} \kappa_{\text{IPB}}^{0.78} \epsilon^{0.58} M^{0.19}

35: IPB98(y,3) ELMy scaling (H-mode)

Is selected with i_confinement_time = 35 11 12

\tau_{\text{E}} = 0.0564 I_{\text{p}}^{0.88} B_{\text{T}}^{0.07} \overline{n}_{19}^{0.4} P_{\text{L}}^{-0.69} R^{2.15} \kappa_{\text{IPB}}^{0.78} \epsilon^{0.64} M^{0.2}

36: IPB98(y,4) ELMy scaling (H-mode)

Is selected with i_confinement_time = 36 11 12

\tau_{\text{E}} = 0.0587 I_{\text{p}}^{0.85} B_{\text{T}}^{0.29} \overline{n}_{19}^{0.39} P_{\text{L}}^{-0.7} R^{2.08} \kappa_{\text{IPB}}^{0.76} \epsilon^{0.69} M^{0.17}

37: ISS95 scaling (Stellarator)

Is selected with i_confinement_time = 37 13

\tau_{\text{E}} = 0.079 a^{2.21} R^{0.65} P_{\text{L}}^{-0.59} \overline{n}_{19}^{0.51} B_{\text{T}}^{0.83} \iota_{2/3}^{0.4}

38: ISS04 scaling (Stellarator)

Is selected with i_confinement_time = 38 14

\tau_{\text{E}} = 0.134 a^{2.28} R^{0.64} P_{\text{L}}^{-0.61} \overline{n}_{19}^{0.54} B_{\text{T}}^{0.84} \iota_{2/3}^{0.41}

39: DS03 beta-independent scaling (H-mode)

Is selected with i_confinement_time = 39 15

\tau_{\text{E}} = 0.028 I_{\text{p}}^{0.83} B_{\text{T}}^{0.07} \overline{n}_{19}^{0.49} P_{\text{L}}^{-0.55} R^{2.11} \kappa_{95}^{0.75} \epsilon^{0.3} M^{0.14}

40: Murari "Non-power law" scaling (H-mode)

Is selected with i_confinement_time = 40 16

\tau_{\text{E}} = 0.0367 I_{\text{p}}^{1.006} R^{1.731} \kappa_{\text{IPB}}^{1.45} P_{\text{L}}^{-0.735} \\ \times \frac{\overline{n}_{19}^{0.49}}{1+e^\left({-9.403\left(\frac{\overline{n}_{19}^{0.49}}{B_{\text{T}}}\right)^{-1.365}}\right)}

41: Petty08 scaling (H-mode)

Is selected with i_confinement_time = 41 17

\tau_{\text{E}} = 0.052 I_{\text{p}}^{0.75} B_{\text{T}}^{0.3} \overline{n}_{19}^{0.32} P_{\text{L}}^{-0.47} R^{2.09} \kappa_{\text{IPB}}^{0.88} \epsilon^{0.84}

42: Lang high density scaling (H-mode)

Is selected with i_confinement_time = 42 18

\tau_{\text{E}} = 6.94\times 10^{-7} M^{0.2} \kappa_{\text{IPB}}^{0.37} \left(\frac{q_{95}}{q_{\text{cyl}}}\right)^{0.77} \\ \times A^{2.48205} \frac{I_{\text{p}}^{1.3678} B_{\text{T}}^{0.12} R^{1.2345} \overline{n}^{0.032236}}{A^{0.9\ln{A}}P_{\text{L}}^{0.74}} \left(\frac{\overline{n}_{e}}{n_{\text{GW}}}\right)^{-0.22 \ln{\left(\frac{\overline{n}_e}{n_{\text{GW}}}\right)}}

43: Hubbard nominal scaling (I-mode)

Is selected with i_confinement_time = 43 19

\tau_{\text{E}} = 0.014 I_{\text{p}}^{0.68} B_{\text{T}}^{0.77} \overline{n}_{20}^{0.02} P_{\text{L}}^{-0.29}

44: Hubbard lower scaling (I-mode)

Is selected with i_confinement_time = 44 19

\tau_{\text{E}} = 0.014 I_{\text{p}}^{0.6} B_{\text{T}}^{0.7} \overline{n}_{20}^{-0.03} P_{\text{L}}^{-0.33}

45: Hubbard upper scaling (I-mode)

Is selected with i_confinement_time = 45 19

\tau_{\text{E}} = 0.014 I_{\text{p}}^{0.76} B_{\text{T}}^{0.84} \overline{n}_{20}^{-0.07} P_{\text{L}}^{-0.25}

46: Menard NSTX scaling (H-mode)

Is selected with i_confinement_time = 46 20

\tau_{\text{E}} = 0.095 I_{\text{p}}^{0.75} B_{\text{T}}^{1.08} \overline{n}_{19}^{0.44} P_{\text{L}}^{-0.73} R^{1.97} \kappa_{\text{IPB}}^{0.78} \epsilon^{0.58} M^{0.19}

47: Menard NSTX-Petty08 hybrid scaling

Is selected with i_confinement_time = 47 20

  • If \epsilon \le 0.4 \ (A \ge 2.5) apply the Petty08 scaling
  • If \epsilon \ge 0.6 \ (A \le 1.7) apply the Menard NSTX scaling

Otherwise:

\tau_{\text{E}} = \frac{\epsilon - 0.4}{0.2}\tau_{\text{E,NSTX}}+ \frac{0.6-\epsilon}{0.2}\tau_{\text{E,Petty08}}

48: Buxton NSTX Gyro-Bohm scaling (H-mode)

Is selected with i_confinement_time = 48 21

\tau_{\text{E}} = 0.21 I_{\text{p}}^{0.54} B_{\text{T}}^{0.91} \overline{n}_{20}^{-0.05} P_{\text{L}}^{-0.38} R^{2.14}

49: ITPA20 scaling (H-mode)

Is selected with i_confinement_time = 49 22

\tau_{\text{E}} = 0.053 I_{\text{p}}^{0.98} B_{\text{T}}^{0.22} \overline{n}_{19}^{0.24} P_{\text{L}}^{-0.669} R^{1.71} \left(1+\delta \right)^{0.36} \kappa_{\text{IPB}}^{0.8} \epsilon^{0.35} M^{0.2}

50: ITPA20-IL scaling (H-mode)

Is selected with i_confinement_time = 50 23

\tau_{\text{E}} = 0.067 I_{\text{p}}^{1.29} B_{\text{T}}^{-0.13} P_{\text{L}}^{-0.644} \overline{n}_{19}^{0.15} M^{0.3} R^{1.19} \left(1+\delta \right)^{0.56} \kappa_{\text{IPB}}^{0.67}

Transport Powers

After the confinement time scaling with H-factor correction has been calculated, the ion and electron transport power densities are found. PROCESS assumes the scaling confinement time to be equal to the ion and electron energy confinement time.

This is simply the volume averaged thermal energy of the electron and ions divided by the H-factor corrected confinement time from the chosen scaling.

\mathtt{pden\_ion\_transport\_loss\_mw} = \frac{3}{2}\frac{n_{\text{i}} \langle T_{\text{i}} \rangle_{\text{n}}}{\tau_{\text{E}}}
\mathtt{pden\_electron\_transport\_loss\_mw} = \frac{3}{2}\frac{n_{\text{e}} \langle T_{\text{e}} \rangle_{\text{n}}}{\tau_{\text{E}}}

Here \langle T_{\text{i}} \rangle and \langle T_{\text{e}} \rangle are the ion and electron density weighted temperatures respectively.

Calculate the density and density weighted ratio:

\frac{n_{\text{i}}}{n_{\text{e}}}\frac{\langle T_{\text{i}} \rangle_{\text{n}}}{\langle T_{\text{e}} \rangle_{\text{n}}}

The density weighted global energy confinement time is then found in terms of this ratio:

\tau_{\text{E}} = \frac{\frac{n_{\text{i}}}{n_{\text{e}}}\frac{\langle T_{\text{i}} \rangle_{\text{n}}}{\langle T_{\text{e}} \rangle_{\text{n}}} + 1}{\left(\frac{\frac{n_{\text{i}}}{n_{\text{e}}}\frac{\langle T_{\text{i}} \rangle_{\text{n}}}{\langle T_{\text{e}} \rangle_{\text{n}}}}{\tau_{\text{i}}}+\frac{1}{\tau_{\text{e}}}\right)}

Key Constraints

Global plasma power balance

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

This constraint ensures self consistency between the the transport loss power used for the confinement scalings and the calculated confinement time in relation to the plasmas total thermal energy:

P_{\text{L}} = \frac{W}{\tau_{\text{E}}}
\underbrace{\frac{3}{2}\frac{n_{\text{i}} \langle T_{\text{i}} \rangle_{\text{n}}}{\tau_{\text{E}}} + \frac{3}{2}\frac{n_{\text{e}} \langle T_{\text{e}} \rangle_{\text{n}}}{\tau_{\text{E}}}}_{\frac{W}{\tau_{\text{E}}}} = \underbrace{\frac{f_{\alpha}P_{\alpha} + P_{\text{c}} + P_{\text{OH}} + P_{\text{HCD}}}{V_{\text{P}}} - \frac{P_{\text{rad}}}{V_{\text{p}}}}_{P_{\text{L}}}

The \frac{3}{2}n_{\text{i}} \langle T_{\text{i}} \rangle_{\text{n}} value is simply the volume averaged ion thermal energy density where \langle T_{\text{i}} \rangle_{\text{n}} is the density weighted temperature. The same goes for the \frac{3}{2}n_{\text{e}} \langle T_{\text{e}} \rangle_{\text{e}} electron thermal energy density term. \tau_{\text{E}} is the confinement time calculated from the chosen confinement scaling via i_confinement_time.

The constraint uses the loss power and thermal densities hence the inclusion of the V_{\text{p}} plasma volume term. The constraint is adapted depending on the condition of i_rad_loss which governs the radiation contribution to the loss power definition, see the radiation and energy confinement section for more info. The injected heating and current drive contribution P_{\text{HCD}} is also included or excluded depending if the plasma is deemed to be ignited with the ignite switch.

It is highly recommended to always have this constraint on as it is a global consistency checker

Lower limit on alpha particle confinement time ratio

This constraint can be activated by stating icc = 62 in the input file.

The value of f_alpha_energy_confinement_min can be set to the desired minimum total ratio between the alpha confinement and energy confinement times.

The scaling value falpha_energy_confinement can be varied also.

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  2. T.C. Hender et al., 'Physics Assessment of the European Reactor Study', AEA FUS 172, 1992. ↩↩↩↩↩↩↩↩↩↩

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