Rosensweig instability - 2D spike pattern

Threshold and spacing

For a ferrofluid layer in a normal field the surface wave obeys ρ ω² / k = ρ g + σ k² − μ₀ M² k /(μ_r+1) (Cowley-Rosensweig; Zelazo-Melcher). The magnetic term wins when μ₀M²/(μ_r+1) exceeds 2√(ρ g σ); the first unstable wave has k_c = √(ρ g/σ), so the spike spacing is λ_c = 2π√(σ/(ρ g)). The magnetisation follows the Langevin law M = M_s(coth ξ − 1/ξ) with ξ = 3χH/M_s, and the applied flux density is B = μ₀(H + M). In the linear limit the threshold is B_c² = 2μ₀μ_r²(μ_r+1)/(μ_r−1)² √(ρ g σ).

Hexagons and hysteresis

Three waves at 120 degrees add up to a hexagonal array. Their amplitude obeys dA/dt = εA + g₂A² − g₃A³ with ε the relative distance to threshold. The quadratic term makes the bifurcation subcritical: spikes appear at ε = 0 but survive down to ε_sn = −g₂²/(4g₃), so the loop in the middle graph is traced when the field is swept up and down.

Honest simplifications: infinitely deep layer, secant permeability used for the threshold, coefficients g₂ = 0.45 and g₃ = 1 chosen to illustrate the shape (the real values depend on the fluid), schematic spike profiles and peak heights, no field gradients or finite layer depth. The threshold, spacing and magnetisation are physical; the amplitude is not calibrated against a measurement.

• Press Sweep field up and down and watch the spikes appear at Bc and vanish later at B_sn.
• Press Lunar gravity or Near weightlessness: the spikes get much wider and the threshold drops.
• Press Water-based fluid: stronger surface tension gives a larger spacing and a higher threshold.