Three-Dimensional MHD Simulation of Z-Pinch Sausage and Kink Modes: A Proof of Concept

# Z-pinch MHD Simulation – v0.1 (February 09, 2026) This is the first public version of an open-source, pure-NumPy 3D cylindrical MHD Z-pinch simulation. Key features:- 64×64×64 Eulerian cylindrical grid (r-θ-z)- Exact Bennett equilibrium (density, current, azimuthal B-field, pressure balance)- Physical m=0 sausage and m=1 kink mode seeding (density + velocity perturbations)- Full continuity equation (cylindrical divergence with 1/r term)- Momentum equations in r, θ, z (pressure gradients + full J×B Lorentz force + centrifugal term)- Adiabatic energy equation + ohmic heating from resistivity- Frozen-in advection (ideal induction: ∇×(v×B))- Resistivity (magnetic diffusion + η J² heating)- Simple div B cleaning (hyperbolic + diffusive) to control monopole errors- Axis singularity treatment (averaging B_θ and v_r at r=0)- Artificial viscosity for shock smoothing Diagnostics & outputs (10 plots + 1 animation):1. Initial density profile vs r (mid-z)2. Pinch radius evolution vs time3. m=0 & m=1 mode amplitude growth (semilog)4. Final density r-z slice (mid-θ)5. Total energy conservation (E / E₀)6. Mass conservation (M / M₀)7. Momentum conservation (p_x, p_y, p_z)8. Angular momentum conservation (L_z / L_z0)9. Magnetic energy conservation (E_mag / E_mag0)10. Reconnection proxy (max J² vs time)+ Animated GIF: density r-z slice evolution (smooth fade, 15 fps) This version focuses on early linear dynamics and basic numerical stability. Higher resolution, nonlinear saturation, and more physics (viscosity, full div-cleaning, etc.) planned for v0.2+. License: MIT Code is public domain / open for anyone to use, fork, extend. Feedback, forks, collaborations welcome! Run on Colab / local Python with NumPy + matplotlib + tqdm.