Magnetic field amplification by chaotic flow — Roberts & Ponomarenko dynamos
—Growth rate γ
~10Rm_critical
—Final |B|²
Kinematic dynamo problem: Given a prescribed flow u, does the magnetic field B grow? Governed by the induction equation ∂B/∂t = ∇×(u×B) + η∇²B, where η=1/Rm. Roberts dynamo: Periodic helical flow; proven dynamo for all Rm above threshold. Ponomarenko: Simplest known smooth dynamo — helical flow in a cylinder. Growth requires Rm > Rm_c. Field shown as Bz component on a 2D cross-section.