============================== Transport and Stability ============================== SCPN-Fusion-Core couples the equilibrium solver with 1.5D radial transport, multiple stability analysis tools, and AI-based turbulence surrogates. 1.5D Radial Transport ---------------------- The integrated transport solver (``integrated_transport_solver.py``) solves the coupled energy and particle diffusion equations on the normalised radial coordinate :math:`\rho`: .. math:: :label: transport-particle \frac{\partial n}{\partial t} = \frac{1}{V'}\frac{\partial}{\partial \rho} \!\left[V'\!\left(D\frac{\partial n}{\partial \rho}\right)\right] + S_n .. math:: :label: transport-energy \frac{3}{2}\frac{\partial (n T)}{\partial t} = \frac{1}{V'}\frac{\partial}{\partial \rho} \!\left[V'\!\left(\chi\frac{\partial T}{\partial \rho}\right)\right] + P_\text{heat} - P_\text{rad} - P_\text{loss} where :math:`D` is the particle diffusion coefficient, :math:`\chi` is the thermal diffusivity, :math:`V'(\rho)` is the flux-surface volume derivative, and :math:`S_n` is the particle source. The anomalous transport coefficients :math:`D` and :math:`\chi` are derived from the turbulence oracle (see below) or from prescribed scaling laws. Auxiliary Heating Source Normalisation ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The auxiliary-heating source in ``TransportSolver.evolve_profiles`` is power-normalised in physical units: 1. input auxiliary power :math:`P_\mathrm{aux}` [MW] is converted to W, 2. a radial deposition shape is volume-normalised with :math:`dV(\rho)`, 3. volumetric power density is mapped to temperature source terms using :math:`\frac{3}{2} n \frac{dT}{dt} = P`. This enforces an explicit MW->keV/s consistency path and provides per-step telemetry in ``_last_aux_heating_balance`` with target and reconstructed injected powers for ion/electron channels. A reproducible benchmark lane for this contract is available at: ``validation/benchmark_transport_power_balance.py``. IPB98(y,2) Confinement Scaling ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The global energy confinement time is cross-checked against the IPB98(y,2) scaling law (ITER Physics Basis, Nuclear Fusion 39, 1999): .. math:: :label: ipb98 \tau_E = 0.0562 \; I_p^{0.93} \; B_T^{0.15} \; \bar{n}_{e,19}^{0.41} \; P_\text{loss}^{-0.69} \; R^{1.97} \; \kappa^{0.78} \; \varepsilon^{0.58} \; M^{0.19} where :math:`I_p` is in MA, :math:`B_T` in T, :math:`\bar{n}_{e,19}` in :math:`10^{19}\,\text{m}^{-3}`, :math:`P_\text{loss}` in MW, :math:`R` in m, :math:`\kappa` is the elongation, :math:`\varepsilon = a/R` is the inverse aspect ratio, and :math:`M` is the effective ion mass in amu. The uncertainty quantification follows the Bayesian regression framework of Verdoolaege et al. (Nuclear Fusion 61, 2021) using the 20-shot ITPA H-mode confinement database. RF Heating Models ------------------ The ``rf_heating`` module simulates auxiliary heating deposition profiles using simplified ray-tracing: **ICRH** -- Ion Cyclotron Resonance Heating Power is deposited at the ion cyclotron resonance layer :math:`\omega = n \omega_{ci}` where :math:`\omega_{ci} = eB / m_i`. **ECRH** -- Electron Cyclotron Resonance Heating Power deposited at the electron cyclotron resonance :math:`\omega = n \omega_{ce}` where :math:`\omega_{ce} = eB / m_e`. **LHCD** -- Lower Hybrid Current Drive Non-inductive current drive via lower-hybrid wave absorption. Turbulence Models ------------------ FNO Turbulence Suppressor ^^^^^^^^^^^^^^^^^^^^^^^^^^ The Fourier Neural Operator (FNO) turbulence model (``fno_turbulence_suppressor.py``) provides real-time spectral turbulence prediction using 12 Fourier modes. The FNO architecture (Li et al. 2021) learns the solution operator of the turbulence PDE directly in Fourier space: .. math:: u_{l+1}(x) = \sigma\!\left(W_l\,u_l(x) + \mathcal{F}^{-1} \!\left[R_l \cdot \mathcal{F}(u_l)\right](x)\right) where :math:`\mathcal{F}` is the Fourier transform, :math:`R_l` is a learnable spectral filter, and :math:`\sigma` is a nonlinear activation. Multi-regime training data is generated from a modified Hasegawa-Wakatani model with regime-dependent parameters (ITG, TEM, ETG). Turbulence Oracle ^^^^^^^^^^^^^^^^^^ The turbulence oracle (``turbulence_oracle.py``) predicts the dominant instability regime (ITG, TEM, or ETG) from local plasma parameters and provides anomalous transport coefficient estimates for the transport solver. MHD Stability -------------- Sawtooth Oscillations ^^^^^^^^^^^^^^^^^^^^^^ The ``mhd_sawtooth`` module models sawtooth crash dynamics using a modified Kadomtsev reconnection model (Kadomtsev 1975). The sawtooth period is estimated from the resistive evolution of the :math:`q = 1` surface, and the crash amplitude is computed from the flux reconnection geometry. Hall-MHD Effects ^^^^^^^^^^^^^^^^^ The ``hall_mhd_discovery`` module incorporates two-fluid Hall-MHD effects (Huba, NRL Plasma Formulary 2019) that become significant in compact, high-field tokamaks where the ion skin depth :math:`d_i = c/\omega_{pi}` is comparable to the equilibrium scale length. Hall-MHD modifications include: - Whistler-wave dispersion at frequencies above :math:`\omega_{ci}` - Hall-term correction to the magnetic induction equation: :math:`\partial \mathbf{B}/\partial t = \nabla \times [(\mathbf{v} - d_i \mathbf{J}/ne) \times \mathbf{B}]` - Modified reconnection rates relevant to sawtooth crash timing Stability Analysis ^^^^^^^^^^^^^^^^^^^ The ``stability_analyzer`` module provides: - **Nyquist stability** analysis for closed-loop feedback systems - **Lyapunov stability** margins for nonlinear dynamics - **Vertical stability** index (decay index :math:`n = -R/B_z \cdot \partial B_z/\partial R`) for positional control - **Beta limits** (:math:`\beta_N` Troyon limit, :math:`\beta_p` critical) Self-Organised Criticality (Legacy Research Lane) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The ``sandpile_fusion_reactor`` module is retained as a legacy SOC research lane for reproducibility and exploratory avalanche studies. It is not part of the release-gated transport validation path (which is based on Gyro-Bohm + Chang-Hinton + EPED-like boundary scaling). Fusion Ignition and Burn Physics ---------------------------------- The ``fusion_ignition_sim`` module computes the Lawson criterion and ignition margins using the Bosch-Hale D-T fusion reactivity parametrisation: .. math:: \langle\sigma v\rangle_{\text{DT}} = C_1\,\theta\, \sqrt{\frac{\xi}{m_r c^2 T^3}} \exp(-3\xi) where :math:`\theta`, :math:`\xi`, and :math:`C_1` are fit parameters (Bosch & Hale, Nuclear Fusion 32, 1992). The ignition condition :math:`Q \to \infty` requires that alpha-particle heating alone sustains the plasma temperature: .. math:: P_\alpha = \frac{1}{4} n_D n_T \langle\sigma v\rangle E_\alpha \;\geq\; P_\text{loss} where :math:`E_\alpha = 3.52\,\text{MeV}` per fusion event. Warm Dense Matter EOS ^^^^^^^^^^^^^^^^^^^^^^ The ``wdm_engine`` module provides a reduced equation-of-state model for warm dense matter conditions relevant to inertial confinement scenarios and pellet ablation physics. Related Modules ---------------- - :mod:`scpn_fusion.core.integrated_transport_solver` -- coupled transport - :mod:`scpn_fusion.core.rf_heating` -- ICRH/ECRH/LHCD heating - :mod:`scpn_fusion.core.fno_turbulence_suppressor` -- FNO model - :mod:`scpn_fusion.core.turbulence_oracle` -- turbulence regime predictor - :mod:`scpn_fusion.core.mhd_sawtooth` -- sawtooth crash model - :mod:`scpn_fusion.core.hall_mhd_discovery` -- Hall-MHD effects - :mod:`scpn_fusion.core.stability_analyzer` -- stability margins - :mod:`scpn_fusion.core.sandpile_fusion_reactor` -- SOC criticality (legacy lane) - :mod:`scpn_fusion.core.fusion_ignition_sim` -- ignition/burn physics - :mod:`scpn_fusion.core.wdm_engine` -- warm dense matter EOS