Seven models run on this page, each in the visitor's browser: the induction equation in a differentially rotating conductor, a Faraday disc with the conductivity of mercury, the tearing modes of a reconnecting current sheet, the Ω effect in circular Couette flow, WKB tunnelling with electron screening for the D-D reaction, the spin-torsion coupling of Einstein-Cartan theory, and lattice-Boltzmann flow past a cylinder. A tokamak rendering closes the page for scale. The equations each model solves sit beside it, and every number in a readout is computed from that model.
Mercury is liquid at room temperature with an electrical conductivity of 1.04 × 10⁶ S/m (CRC Handbook of Chemistry and Physics, 20 °C). Its magnetic diffusivity \eta = 1/(\mu_0\sigma) is 0.765 m²/s, so a field soaks through a mercury volume of size L in a time of order L²/η, about a third of a second across half a metre. Flow at speed U works against that diffusion, and the magnetic Reynolds number Rm = UL/η says which wins.
The induction equation holds the whole contest. Its first term carries field lines with the flow and stretches them where the flow shears; the second smooths them at the rate η sets. Differential rotation, where the angular velocity changes with radius, shears a field that crosses the axis into an azimuthal one. That is the Ω effect, and it is the half of a dynamo that liquid-metal experiments reach most easily.
The first figure solves the equation in two dimensions for a prescribed rotation\;\omega(r) = \omega_0/(1 + (r/a)^4) in a uniform applied field. The lines wind, the azimuthal component grows with the shear, and at large Rm the flux is expelled from the rotating core, the result Weiss (1966) found for the same problem. The Faraday disc after it takes the same conductivity into a geometry with a closed-form answer.
\begin{gathered} \frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{v} \times \mathbf{B}) + \eta \nabla^2 \mathbf{B} \\[4pt] \eta = \frac{1}{\mu_0 \sigma} \end{gathered}The induction equation: advection and stretching by the flow against resistive diffusion.
R_m = \frac{UL}{\eta} = \mu_0 \sigma U LMagnetic Reynolds number. For mercury at 1 m/s over 0.5 m, Rm = 0.65; the rotating-conductor figure spans 2 to 2000.
\mathbf{B} = \nabla \times (A\,\hat{\mathbf{z}}), \qquad \frac{\partial A}{\partial t} + \mathbf{v}\cdot\nabla A = \eta\,\nabla^2 AIn two dimensions the field is a flux function A whose contours are field lines; this is the equation the first figure integrates.
\mathbf{J} = \sigma\,(\mathbf{E} + \mathbf{v}\times\mathbf{B}), \qquad \mathbf{F} = \mathbf{J}\times\mathbf{B}Ohm's law in a moving conductor and the Lorentz force density that couples the field back to the flow.
A disc turning in an axial field is the oldest electrical machine. Every element of the conductor at radius r moves at ωr across B and sees a motional field ωrB directed along the radius, so the potential climbs as ½ωBr² from the axle to the rim. Contacts at the axle and the rim collect the difference.
\varepsilon = \int_{r_i}^{R} \omega r B\, dr = \tfrac{1}{2}\,\omega B\,(R^2 - r_i^2)The homopolar EMF. The current the disc can drive is set by its own radial resistance ln(R/rᵢ)/(2πσh).
\delta = \sqrt{\frac{2}{\mu_0 \sigma \omega_f}} = \sqrt{\frac{\eta}{\pi f}}Skin depth: how far an alternating field at frequency f penetrates the conductor. In mercury it is 70 mm at 50 Hz and 16 mm at 1 kHz.
| σ (Hg, 20 °C) | 1.04×10⁶S/m |
| η = 1/(μ₀σ) | 0.765m²/s |
| Disc thickness h | 10mm |
| Axle contact r_i | 5mm |
| δ = h at | 2440Hz |
Resistivity also lets field lines break and rejoin. Where a flow presses oppositely directed field together into a thin current sheet, the sheet tears into a chain of magnetic islands at a rate set by the Lundquist number S = Lv_A/η, the same ratio as Rm with the Alfvén speed as the velocity. The figure below draws that process from the tearing-mode growth rates; the sheared flows of the dual-disc confinement concept act on exactly this.
Between two coaxial cylinders turning at different rates the steady flow is the circular Couette profile Ω(r) = A + B/r², with A and B fixed by the two wall speeds. A radial field B_r = B₀r₁/r threading the annulus is divergence-free, is a steady solution of the diffusion operator, and is left alone by the azimuthal flow. The shear r dΩ/dr is not so gentle with the azimuthal component: it feeds B_φ at the rate rB_r dΩ/dr, so B_φ grows linearly in time until diffusion, working against it over the gap width, holds it at a level of order Rm B₀.
The figure integrates that equation on a radial grid with B_φ = 0 at both walls. The wound lines on the left are field lines traced through the computed field, with dφ/dr = B_φ/(rB_r), so the winding is the toroidal field. The dashed white profile is the steady state from a direct solve of the same equation; the dotted amber one is the ideal-MHD growth with no diffusion. The Ω-effect rate in the readout is the peak of |rB_r dΩ/dr|/B₀, which for this profile sits at the inner wall.
\begin{gathered} \Omega(r) = A + \frac{B}{r^2} \\[4pt] A = \frac{\Omega_2 r_2^2 - \Omega_1 r_1^2}{r_2^2 - r_1^2} \\[4pt] B = \frac{(\Omega_1 - \Omega_2)\, r_1^2 r_2^2}{r_2^2 - r_1^2} \end{gathered}Circular Couette flow between cylinders at r₁ and r₂; the shear is r dΩ/dr = −2B/r².
\frac{\partial B_\phi}{\partial t} = r B_r \frac{d\Omega}{dr} + \eta\left(\nabla^2 - \frac{1}{r^2}\right) B_\phiThe azimuthal component of the induction equation for an axisymmetric field in an azimuthal flow. B_r = B₀r₁/r is unchanged by it.
Two deuterons at a relative energy E face a Coulomb barrier that stands near 480 keV at the nuclear radius. Below that they react by tunnelling. For a pure Coulomb potential the WKB probability is the Gamow factor exp(−2πη), with the Sommerfeld parameter η = Z₁Z₂α/(v/c); for D-D, 2πη = √(E_G/E) with E_G = 986 keV. The cross-section is that factor times S(E)/E, where the astrophysical S-factor varies slowly. The figure uses the Bosch and Hale (1992) parametrisation of S(E) for both D-D branches.
Electrons around the nuclei lower the barrier. In the adiabatic picture of Assenbaum, Langanke and Rolfs (1987) their effect is a constant drop U_e in the potential inside the turning point, so the reaction at energy E proceeds as the bare one would at E + U_e, an enhancement f(E) = exp(πηU_e/E) when U_e ≪ E. Greife et al. (1995) measured U_e = 25 ± 5 eV for a D₂ gas target. Raiola et al. (2002) measured several hundred electronvolts in deuterated metals, with the largest values near 800 eV, and the same group's survey of the periodic table (Raiola et al. 2004) modelled the difference with the Debye length of the metal's quasi-free electrons.
The slider sets U_e and the presets carry those measured values. The readouts give the enhancement at the probe energy, the cross-sections, and the Debye and Thomas-Fermi lengths for the chosen electron density together with the U_e each implies. The last readout evaluates the Gamow factor at E = U_e, the effective energy of a deuteron pair with no kinetic energy of its own: at 300 eV it is 10⁻²⁵.
\begin{gathered} \eta = \frac{Z_1 Z_2 e^2}{4\pi\varepsilon_0 \hbar v} \\[4pt] 2\pi\eta = \sqrt{\frac{E_G}{E}} \\[4pt] E_G = 2\mu c^2 (\pi\alpha Z_1 Z_2)^2 \end{gathered}Sommerfeld parameter and Gamow energy; μ is the reduced mass, 937.8 MeV/c² for D-D.
\begin{gathered} \sigma(E) = \frac{S(E)}{E}\, e^{-2\pi\eta(E)} \\[4pt] P_{\mathrm{WKB}} = \exp\!\left[-\frac{2}{\hbar}\int_{r_n}^{r_c} \sqrt{2\mu\,(V(r) - E)}\; dr\right] \end{gathered}Cross-section through the S-factor, and the WKB integral from the nuclear radius to the classical turning point r_c = e²/(4πε₀E).
\begin{gathered} f(E) = \frac{\sigma_{\mathrm{s}}(E)}{\sigma_{\mathrm{b}}(E)} = \frac{\sigma_{\mathrm{b}}(E + U_e)}{\sigma_{\mathrm{b}}(E)} \\[4pt] f(E) \approx \exp\!\left(\frac{\pi\eta(E)\,U_e}{E}\right) \end{gathered}Screening enhancement, Assenbaum, Langanke and Rolfs (1987); the exponential form holds for U_e ≪ E.
\begin{gathered} \lambda_D = \sqrt{\frac{\varepsilon_0 k_B T}{n_e e^2}} \\[4pt] \lambda_{TF} = \sqrt{\frac{2\varepsilon_0 E_F}{3 n_e e^2}} \\[4pt] U_e = \frac{e^2}{4\pi\varepsilon_0\,\lambda} \end{gathered}Debye length of a classical electron gas and Thomas-Fermi length of a degenerate one, each with the screening energy it implies.
| E_G = B_G² (D-D) | 985.8keV |
| S(0), both branches | 109keV·b |
| r_n = 1.2(2^⅓ + 2^⅓) | 3.02fm |
| V_c at r_n | 476keV |
| T for λ_D | 293K |
Einstein-Cartan theory lets the connection carry an antisymmetric part, the torsion tensor, and ties it to the spin density of matter by an algebraic equation. Torsion does not propagate: where the spin density vanishes, the theory is general relativity. Cartan's equation and the modified Einstein equation are set out here in the form given by Hehl, von der Heyde, Kerlick and Nester (1976).
Because the coupling is algebraic, the size of the effect follows from the spin density alone. For a fully polarised fermion gas the spin density is nħ/2, the torsion is 4πℓ_P²n, and its contribution to the effective energy density relative to the rest mass is (π/2)ℓ_P²ƛ_C n. The figure plots both against mass density. At nuclear density the ratio is 1.4 × 10⁻⁴¹; it reaches one near 2 × 10⁵⁸ kg/m³, forty orders of magnitude above the densest matter in a neutron star.
T^{\lambda}{}_{\mu\nu} = \Gamma^{\lambda}{}_{\mu\nu} - \Gamma^{\lambda}{}_{\nu\mu}The torsion tensor: the antisymmetric part of the affine connection.
\begin{gathered} T^{\lambda}{}_{\mu\nu} + \delta^{\lambda}_{\mu}\, T^{\sigma}{}_{\nu\sigma} - \delta^{\lambda}_{\nu}\, T^{\sigma}{}_{\mu\sigma} = \kappa\, s^{\lambda}{}_{\mu\nu} \\[4pt] \kappa = \frac{8\pi G}{c^4} \end{gathered}Cartan's equation: the modified torsion equals the spin tensor with the gravitational coupling. For a totally antisymmetric spin tensor the trace terms vanish and T = κs.
\begin{gathered} G_{\mu\nu}(\{\}) = \kappa\, T^{\mathrm{eff}}_{\mu\nu} \\[4pt] \rho_{\mathrm{eff}} = \rho - \frac{2\pi G}{c^4}\, s^2 \\[4pt] p_{\mathrm{eff}} = p - \frac{2\pi G}{c^4}\, s^2 \end{gathered}The Einstein equation on the Riemannian part of the connection, with the spin-squared terms of a Weyssenhoff fluid in mass-density units; s² = ½ s_{μν}s^{μν}.
\begin{gathered} s = \tfrac{1}{2}\,\xi n\hbar \\[4pt] |T| \sim \kappa c\, s = 4\pi\,\xi\,\ell_P^2\, n \\[4pt] \frac{2\pi G s^2/c^4}{n m} = \frac{\pi}{2}\,\xi^2\,\ell_P^2\,\bar\lambda_C\, n \end{gathered}Polarised fermions of number density n and polarisation ξ; ℓ_P² = ħG/c³ and ƛ_C = ħ/(mc). These are the curves the figure draws.
The Navier-Stokes equations balance a fluid element's inertia against pressure and viscous stress. One number, the Reynolds number Re = UD/ν, fixes the character of flow past a body of size D: steady and symmetric at low Re, then a pair of standing eddies, then, above Re ≈ 47 for a cylinder, a wake that sheds vortices alternately from each side at a frequency the Strouhal number St = fD/U reports.
The figure does not integrate Navier-Stokes directly. It runs a lattice-Boltzmann scheme, nine discrete velocities on a square grid relaxing toward local equilibrium, whose long-wavelength limit is the same incompressible equations with a viscosity set by the relaxation time. The Reynolds number is the slider; the Strouhal number is measured from the transverse velocity at the amber probe as the wake passes it.
\begin{gathered} \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{u} = -\frac{1}{\rho}\nabla p + \nu\,\nabla^2 \mathbf{u} \\[4pt] \nabla\cdot\mathbf{u} = 0 \end{gathered}Incompressible Navier-Stokes: u is velocity, p pressure, ρ density and ν kinematic viscosity.
Re = \frac{UD}{\nu}, \qquad St = \frac{fD}{U}, \qquad \nu = \frac{1}{3}\left(\tau - \tfrac{1}{2}\right)Reynolds and Strouhal numbers, and the lattice viscosity the BGK relaxation time τ gives in lattice units.
A tokamak holds a D-shaped plasma with a strong toroidal field from a ring of coils and a poloidal field from the plasma's own current; the two twist into helical field lines that keep the plasma off the wall. The rendering shows the coil set, the central solenoid, the vessel in cutaway, the plasma cross-section with its pressure map, and field lines wound with the safety factor rising from 1 on axis to 3.6 at the edge. The magnetic confinement page carries the Grad-Shafranov solver and the dual-disc concept this rendering is drawn against.
| Major radius R₀ | 1.85m |
| Minor radius a | 0.97m |
| Aspect ratio | 1.9 |
| B on axis | 12.2T |
| Plasma current | 8.7MA |
| Elongation κ | 1.8 |
| Triangularity δ | 0.5 |
| TF coils | 18 |
Each figure is a TypeScript model drawn on a canvas in the visitor's browser; nothing on this page is a recording. The induction solver works on a 104 × 104 grid with an exact rotational back-trace and explicit diffusion; the Couette solver on 160 radial points with a tridiagonal solve for its steady state; the lattice-Boltzmann channel is 240 × 96 cells; the reconnection model carries twelve tearing modes; the tunnelling and torsion figures are closed-form curves evaluated on the fly.
Every canvas stops when it leaves the screen or the tab is hidden, caps its resolution at 1.5 device pixels per CSS pixel, budgets its stepping to a few milliseconds per frame, and renders a single still when the visitor has asked for reduced motion. The constants come from the CRC Handbook for mercury, Bosch and Hale (1992) for the D-D S-factors, and CODATA values elsewhere.