bluemira.plasma_physics.rules_of_thumb
A collection of simple 0-D rules of thumb for tokamak plasmas.
Functions
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A 0-D estimate of the loop voltage during burn |
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Estimate the normalised external plasma self-inductance. |
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Estimate the plasma mutual inductance. |
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Calculate the cylindrical safety factor. |
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Calculate the kink safety factor at the plasma edge |
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Calculate the cylindrical equivalent safety factor at the plasma edge |
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Estimate safety factor at the 95th percentile flux surface based on an empirical fit. |
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Estimate the normalised plasma internal inductance based on an empirical fit. |
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Estimate the vertical field to keep the plasma in equilibrium. |
Module Contents
- bluemira.plasma_physics.rules_of_thumb.estimate_loop_voltage(R_0: float, B_t: float, Z_eff: float, T_e: float, n_e: float, q_0: float) float
A 0-D estimate of the loop voltage during burn
- Parameters:
R_0 (float) – Major radius [m]
B_t (float) – Toroidal field on axis [T]
Z_eff (float) – Effective charge [dimensionless]
T_e (float) – Electron temperature on axis [keV]
n_e (float) – Electron density [1/m^3]
q_0 (float) – Safety factor on axis
- Returns:
Loop voltage during burn [V]
- Return type:
float
Notes
Zohm, W. Morris (2022)
\(v_{loop}=2\pi R_{0}\dfrac{2\pi B_{t}}{\mu_{0}q_{0}\sigma_{0}R_{0}}\)
- where \(\sigma_{0}\) is the Spitzer conductivity on axis:
\(\sigma_{0} = 1.92e4 (2-Z_{eff}^{-1/3}) \dfrac{T_{e}^{3/2}}{Z_{eff}ln\Lambda}\)
Assumes no non-inductive current on axis
Assumes a circular cross-section on axis
There is no neo-classical resistivity on axis because there are no trapped particles
- bluemira.plasma_physics.rules_of_thumb.estimate_Le(A: float, kappa: float) float
Estimate the normalised external plasma self-inductance.
- Parameters:
A (float) – Last closed flux surface aspect ratio
kappa (float) – Last closed flux surface elongation
- Returns:
Normalised plasma external inductance
- Return type:
float
Notes
Assuming a LCFS parameterisation as per:
flux_surface_hirshman()
- bluemira.plasma_physics.rules_of_thumb.estimate_M(A: float, kappa: float) float
Estimate the plasma mutual inductance.
- Parameters:
A (float) – Last closed flux surface aspect ratio
kappa (float) – Last closed flux surface elongation
- Returns:
Plasma mutual inductance
- Return type:
float
Notes
Assuming a LCFS parameterisation as per
flux_surface_hirshman()
- bluemira.plasma_physics.rules_of_thumb.calc_cyl_safety_factor(R_0: float, A: float, B_0: float, I_p: float) float
Calculate the cylindrical safety factor.
\[\dfrac{2\pi}{\mu_{0}} \dfrac{R_{0}}{A}**2 \dfrac{B_{0}}{R_{0} I_{p}}\]- Parameters:
R_0 (float) – Plasma major radius [m]
A (float) – Plasma aspect ratio
B_0 (float) – Toroidal field at major radius [T]
I_p (float) – Plasma current [A]
- Returns:
Cylindrical safety factor
- Return type:
float
Notes
Sometimes also written with \(\dfrac{2\pi}{\mu_{0}} = 5\) and I_p in [MA]
- bluemira.plasma_physics.rules_of_thumb.calc_qstar_freidberg(R_0: float, A: float, B_0: float, I_p: float, kappa: float) float
Calculate the kink safety factor at the plasma edge
\(q_{*}=\dfrac{2\pi a^2 B_0}{\mu_0 R_0 I_p}\) \(\bigg(\dfrac{1+\kappa^2}{2}\bigg)\)
- Parameters:
R_0 (float) – Plasma major radius [m]
A (float) – Plasma aspect ratio
B_0 (float) – Toroidal field at major radius [T]
I_p (float) – Plasma current [A]
kappa (float) – Plasma elongation
- Returns:
Kink safety factor
- Return type:
float
Notes
Freidberg, Ideal MHD, p 131
- bluemira.plasma_physics.rules_of_thumb.calc_qstar_uckan(R_0: float, A: float, B_0: float, I_p: float, kappa: float, delta: float) float
Calculate the cylindrical equivalent safety factor at the plasma edge
- Parameters:
R_0 (float) – Plasma major radius [m]
A (float) – Plasma aspect ratio
B_0 (float) – Toroidal field at major radius [T]
I_p (float) – Plasma current [A]
kappa (float) – Plasma elongation
delta (float) – Plasma triangularity
- Returns:
Cylindrical equivalent safety factor
- Return type:
float
Notes
Uckan et al., ITER Physics Design Guidelines, 1989, sec. 2.3 https://inis.iaea.org/search/search.aspx?orig_q=RN:21068960
- bluemira.plasma_physics.rules_of_thumb.estimate_q95_uckan(R_0: float, A: float, B_0: float, I_p: float, kappa: float, delta: float) float
Estimate safety factor at the 95th percentile flux surface based on an empirical fit.
- Parameters:
R_0 (float) – Plasma major radius [m]
A (float) – Plasma aspect ratio
B_0 (float) – Toroidal field at major radius [T]
I_p (float) – Plasma current [A]
kappa (float) – Plasma elongation
delta (float) – Plasma triangularity
- Returns:
Estimate of 95th percentile safety factor
- Return type:
float
Notes
Uckan et al., ITER Physics Design Guidelines, 1989, sec. 2.3 https://inis.iaea.org/search/search.aspx?orig_q=RN:21068960 Ref [11] in the above does not appear to include the geometry factor
- bluemira.plasma_physics.rules_of_thumb.estimate_li_wesson(q_star: float, q_0: float = 1.0) float
Estimate the normalised plasma internal inductance based on an empirical fit.
- Parameters:
q_star (float) – Cylindrical equivalent safety factor
q_0 (float) – Safety factor on axis
- Returns:
Normalised lasma internal inductance
- Return type:
float
Notes
Wesson, Tokamaks 3rd edition, page 120
This appears to give high values for li, even when using q* at rho=0.95
- bluemira.plasma_physics.rules_of_thumb.estimate_vertical_field(R_0: float, A: float, I_p: float, beta_p_th: float, l_i: float, kappa_95: float | None = None) float
Estimate the vertical field to keep the plasma in equilibrium.
- Parameters:
R_0 (float) – Plasma major radius [m]
A (float) – Plasma aspect ratio
I_p (float) – Plasma current [A]
beta_p_th (float) – Thermal poloidal beta
l_i (float) – Normalised internal inductance
kappa_95 (float | None) – Plasma elongation at the 95th percentile flux surface
- Returns:
Plasma equilibrium vertical field [T]
- Return type:
float
Notes
See e.g. Ferrara et al., “Alcasim simulation code for Alcator C-Mod” https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4178094
The kappa term is not always present in textbooks and the like, and is almost certainly irrelevant at the end of breakdown.