Development of a Reduced Multi-Fluid Equilibrium Model and Its Application to Proton-Boron Spherical Tokamaks

TL;DR

This work develops a reduced multi-fluid equilibrium framework for proton-Boron plasmas in spherical tokamaks, retaining dominant toroidal inertia and self-consistent electrostatic coupling while neglecting poloidal inertia and pressure anisotropy. It couples a generalized Grad-Shafranov equation with species-specific Bernoulli relations under quasi-neutrality, enabling robust equilibrium solutions that reveal centrifugal separation and multi-kV electrostatic potentials as key physics drivers. Application to ENN's EHL-2 and EHL-3B demonstrates that multi-fluid effects are negligible below a boron Mach number of about $0.5$ but become dominant for $M_B>1$, leading to boron accumulation on the low-field side and significant shifts in $q$-profiles and $E_r$. The results establish a practical, computationally tractable baseline for p-$^{11}$B reactor design and provide a theoretical foundation for subsequent studies of stability, transport, and free-boundary dynamics, with potential applicability to other rotating multi-species fusion plasmas.

Abstract

Proton-Boron fusion requires extreme ion temperatures and robust confinement, making Spherical Tokamaks (ST) with high-power neutral beam injection primary candidates. In these devices, strong toroidal rotation and the large mass disparity between protons and boron ions drive complex multi-fluid effects - specifically centrifugal species separation and electrostatic polarization - that standard single-fluid magnetohydrodynamic (MHD) models fail to capture. While comprehensive multi-fluid models are often numerically stiff, we develop a reduced model balancing physical fidelity with computational robustness. By retaining dominant toroidal rotation and self-consistent potential while neglecting poloidal inertia and pressure anisotropy, the model couples a generalized Grad-Shafranov equation with species-specific Bernoulli relations and a quasi-neutrality constraint. The model is applied to two representative p-B ST configurations: the experimental EHL-2 and reactor-scale EHL-3B. Simulation results demonstrate that equilibrium modifications are governed by the ion Mach number ($M$). In the low-rotation regime ($M < 0.5$), multi-fluid effects are weak and solutions approach the single-fluid limit. However, at $M > 2$, strong centrifugal forces drive significant boron accumulation at the low-field side (LFS) and generate an internal electrostatic potential on the order of 10 kV. These findings confirm the necessity of multi-fluid modeling for accurate p-$^{11}$B reactor design and establish a theoretical foundation for future investigations into stability, transport, and free-boundary dynamics.

Figures (7)

• Figure 1: Multi-fluid equilibrium analysis of the EHL-2 configuration in the static limit ($M=0$). The panels display: (a) Toroidal current density $J_\phi$; (b) Electrostatic potential $\Phi$ (zero in static case); (c) Boron density $n_B$; (d) Poloidal magnetic flux $\psi$; (e) Midplane density profiles at $Z=0$; (f) Toroidal Mach number (zero); (g) Total magnetic field $B_{tot}$; (h)-(j) Flux-coordinate profiles of Temperature $T(\psi)$, Poloidal current function $F(\psi)$, and Safety factor $q(\psi)$; (k) Global parameters table. Note that at $M=0$, all species distributions are uniform on flux surfaces, and the solution recovers the standard single-fluid MHD result.
• Figure 2: EHL-2 equilibrium with moderate toroidal rotation ($M_B \sim 0.5$). Small centrifugal modifications are observed in the current density and pressure distributions, while the boron density $n_B$ begins to show a slight outboard shift. The self-consistent electrostatic potential $\Phi$ remains below $0.2$ kV, indicating that single-fluid approximations are still relatively accurate in this low-Mach regime.
• Figure 3: EHL-2 equilibrium under extreme toroidal rotation ($M_B \sim 5$, corresponding to $u_\phi \approx 1500$ km/s)Liang2025. This case tests the numerical robustness of the model in a high-Mach regime. (a) The toroidal current density $J_\phi$ becomes highly hollow and localized at the LFS. (c) The boron density $n_B$ is entirely expelled to the outboard edge, exhibit an extreme "crescent-shaped" distribution, leaving the core depleted. (b) A massive electrostatic potential $\Phi \approx 10$ kV is generated to shield electrons from the centrifugal force. Note that while $1500$ km/s is an extreme theoretical limit for EHL-2, it demonstrates the model's capability to capture strong shock-free separation physics.
• Figure 4: Baseline multi-fluid equilibrium for the reactor-scale EHL-3B device in the static case ($M=0$). Despite the high plasma beta ($\beta_t \approx 30\%$) and large Shafranov shift ($\sim 67$ cm), the species densities ($n_e, n_p, n_B$) remain perfectly aligned with the magnetic flux surfaces $\psi$ in the absence of rotation.
• Figure 5: EHL-3B equilibrium with moderate rotation ($M_B \sim 1.0$). At this reactor scale, the absolute rotation frequency is lower than in EHL-2 to achieve the same Mach number due to the larger radius. The centrifugal separation is clearly visible in the midplane density profiles (Panel e), and an electrostatic potential of around 1 kV is generated to maintain quasi-neutrality between protons and boron ions.