Ferrofluids under oscillatory magnetic fields

Ferrofluids exhibit two canonical interfacial instabilities, a static Rosensweig (normal-field) instability that produces a lattice of peaks and a dynamical Faraday instability that produces parametrically excited standing waves. Here we present a systematic phase diagram of ferrofluid surface states driven by a purely AC vertical magnetic field with zero mean. Scanning a broad range of frequencies and field amplitudes, we resolve two robust branches: a Faraday-wave regime that includes a stable square lattice and a Rosensweig-like peak--valley regime indistinguishable in morphology from Rosensweig peaks. The Faraday-onset boundary is well described by a power law close to $\sqrt{f}$, while the Rosensweig-like peak onset becomes essentially frequency independent at low viscosity. The wave vector of the square lattice grows linearly with frequency over our accessible band. We present a surface-wave theory that captures the full phenomenology, including the emergence of Rosensweig peaks under zero-mean AC driving, the near-$\sqrt{f}$ scaling of the phase boundaries, the linear growth of the selected wave vector with frequency, and the preference for square over hexagonal lattices.