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The science
Rosensweig (normal-field) instability
A magnetisable liquid under a vertical field stays flat until the magnetic pressure beats gravity plus surface tension. For a deep layer, linearising the interface gives the dispersion relation
ρ ω² = F k² − ρ g k − σ k³ with the magnetic coefficient F = μ0 M² / (1 + 1/μr) (Cowley & Rosensweig 1967). A mode grows when ω² > 0.
The weakest field that destabilises any wavenumber is Mc² = (1 + 1/μr) · 2√(ρ g σ) / μ0, and the first unstable wavelength is the capillary length λc = 2π √(σ / (ρ g)) (about 10 mm for ρ = 1200 kg/m³, σ = 30 mN/m), independent of the field.
Above onset the amplitude is limited by nonlinearity. The pattern here is evolved with the Swift-Hohenberg equation ut = εu − (1+∇²)²u + a u² − u³ with ε = ½(M²/Mc² − 1), a = 0.8; its quadratic term selects hexagons and makes the transition subcritical, so spikes appear abruptly and persist when the field is lowered again (hysteresis, run "Sweep field up and down").
Honest simplifications
The pattern equation is a generic model of the instability, not a solution of the full magnetostatic free-boundary problem. The spike height in millimetres is a rough scale (0.35 λc per unit u), and the critical field of a real fluid also depends on layer depth and the nonlinear magnetisation curve. The 3D page is a decorative shader; this page uses the physical parameters instead.
What to try
- Raise M slowly through Mc: the flat surface becomes hexagonal spikes with the wavelength lambda_c.
- Run the field sweep: spikes appear at Mc but vanish only below it (hysteresis loop in the third graph).
- Make the fluid heavier or the surface tension larger: Mc rises, lambda_c changes.