Physics Methods Reference¶

This document catalogs the primary physics models, equations, and scaling laws implemented in scpn-control.

1. Equilibrium (Grad-Shafranov)¶

Grad-Shafranov Equation¶

\[\Delta^* \psi = R \frac{\partial}{\partial R}\left(\frac{1}{R}\frac{\partial \psi}{\partial R}\right) + \frac{\partial^2 \psi}{\partial Z^2} = -\mu_0 R^2 p'(\psi) - F(\psi) F'(\psi)\]

Source: Grad & Rubin (1958), Shafranov (1966).
Implementation: src/scpn_control/core/fusion_kernel.py:380 (Picard + SOR).
Simplifications: Axisymmetric assumption. Poloidal current $F(\psi)$ and pressure $p(\psi)$ are specified as polynomial or pedestal functions of $\psi$.

Green's Function for Coil Flux¶

$$\psi_{coil}(R,Z) = \frac{\mu_0 I}{\pi k} \sqrt{R R_c} [(1-k^2/2) K(k^2) - E(k^2)]$$ where $k^2 = \frac{4 R R_c}{(R+R_c)^2 + (Z-Z_c)^2}$.

Source: Jackson, Classical Electrodynamics (1999) Ch. 5 Eq. 5.37.
Implementation: src/scpn_control/core/fusion_kernel.py:1250.
Simplifications: Assumes filamentary circular coils.

2. Transport (1.5D)¶

Cylindrical Heat Diffusion¶

\[\frac{3}{2} n \frac{\partial T}{\partial t} = \frac{1}{r} \frac{\partial}{\partial r} \left( r n \chi \frac{\partial T}{\partial r} \right) + P_{heat} - P_{rad}\]

Source: Wesson, Tokamaks (2011) Ch. 3.
Implementation: src/scpn_control/core/integrated_transport_solver.py:850.
Simplifications: 1D radial approximation on flux surfaces.

Chang-Hinton Neoclassical Transport¶

\[\chi_i = 0.66 (1 + 1.54 \epsilon) q^2 \rho_i^2 \nu_{ii} / (\epsilon^{3/2} (1 + 0.74 \nu_*^{2/3}))\]

Source: Chang & Hinton, Phys. Fluids 25, 1493 (1982) Eq. 10.
Implementation: src/scpn_control/core/integrated_transport_solver.py:85.
Simplifications: Simplified aspect ratio and collisionality dependence.

Sauter Bootstrap Current¶

Source: Sauter et al., Phys. Plasmas 6, 2834 (1999).
Implementation: src/scpn_control/core/integrated_transport_solver.py:125.
Simplifications: Uses analytic fit for trapped particle fraction $f_t$.

3. Radiation and Sinks¶

Bremsstrahlung Radiation¶

\[P_{br} = 5.35 \times 10^{-37} n_e^2 Z_{eff} T_e^{1/2} \quad [W/m^3]\]

Source: Wesson, Tokamaks (2011) Ch. 14.5.1.
Implementation: src/scpn_control/core/integrated_transport_solver.py:785, src/scpn_control/control/tokamak_digital_twin.py:175.
Simplifications: Pure Bremsstrahlung; assumes Maxwellian distribution.

Tungsten Line Radiation¶

Piecewise power-law fit to ADAS data for tungsten in coronal equilibrium.

Source: Pütterich et al., Nucl. Fusion 50, 025012 (2010).
Implementation: src/scpn_control/core/integrated_transport_solver.py:755.
Simplifications: Coronal equilibrium (no transport effects on charge states).

4. Scaling Laws¶

IPB98(y,2) Confinement Scaling¶

\[\tau_E = 0.0562 \cdot I_p^{0.93} \cdot B_T^{0.15} \cdot \bar{n}_e^{0.41} \cdot M^{0.19} \cdot R^{1.97} \cdot \varepsilon^{0.58} \cdot \kappa^{0.78} \cdot P^{-0.69}\]

Source: ITER Physics Basis, Nucl. Fusion 39, 2175 (1999).
Implementation: src/scpn_control/core/scaling_laws.py:45.
Simplifications: Global fit; does not capture local profile effects.

Greenwald Density Limit¶

\[n_G = \frac{I_p}{\pi a^2} \quad [10^{20} m^{-3}]\]

Source: Greenwald, Plasma Phys. Control. Fusion 44, R27 (2002).
Implementation: src/scpn_control/control/disruption_predictor.py:150.

5. Control and Dynamics¶

Vertical Stability Growth Rate¶

Estimated from Naydon instability timescale for elongated plasmas.

Source: Naydon et al., Phys. Plasmas (2005).
Implementation: src/scpn_control/control/h_infinity_controller.py:115.

Kuramoto-Sakaguchi Phase Dynamics¶

\[\frac{d\theta_i}{dt} = \omega_i + K R \sin(\psi - \theta_i - \alpha) + \zeta \sin(\Psi - \theta_i)\]

Source: Kuramoto (1975), Sakaguchi & Kuramoto (1986).
Implementation: src/scpn_control/phase/kuramoto.py:80.
Simplifications: Mean-field coupling.