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

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.

💡 Research Summary

The paper addresses a critical gap in the modeling of proton‑boron (p‑¹¹B) fusion plasmas confined in spherical tokamaks (STs). In such devices, high‑power neutral beam injection (NBI) drives strong toroidal rotation, and the large mass disparity between protons (≈ 1 amu) and boron ions (≈ 11 amu) leads to pronounced centrifugal effects. Heavy boron ions are pushed toward the low‑field side (LFS) of magnetic surfaces, creating charge separation that must be balanced by a self‑consistent electrostatic potential Φ. Conventional single‑fluid magnetohydrodynamic (MHD) equilibrium models, based on the Grad‑Shafranov (GS) equation, assume a single pressure surface and cannot represent these multi‑species phenomena. Full multi‑fluid formulations exist, but they incorporate poloidal flows, pressure anisotropy, finite‑orbit‑width corrections, and relativistic electron dynamics, resulting in stiff, often trans‑sonic equations that are numerically intractable for routine design work.

Reduced Multi‑Fluid Model
The authors propose a reduced multi‑fluid equilibrium model that retains only the dominant physics: toroidal rotation and electrostatic coupling. The key simplifications are: (i) neglect of poloidal flow inertia, thereby preserving the elliptic nature of the GS operator; (ii) assumption of isothermal flux‑surface temperature for each species, which eliminates pressure anisotropy; (iii) massless electron approximation, setting electron centrifugal potential to zero. Under axisymmetry (∂/∂φ = 0) the steady‑state momentum balance for each species s reads

 mₛ nₛ (uₛ·∇)uₛ = −∇pₛ + nₛ qₛ(E + uₛ×B).

With uₛ = R Ωₛ(ψ) ê_φ, integration along magnetic field lines yields a generalized Bernoulli (Boltzmann) relation

 nₛ(R,ψ) = Nₛ(ψ) exp