Magnetic field evolution for crystallization-driven dynamos in C/O white dwarfs

We investigate the evolution of magnetic fields generated by the crystallization-driven dynamo in carbon-oxygen white dwarfs (WDs) with masses $\lesssim1.05\ M_{\odot}$. We use scalings for the dynamo to demonstrate that the initial magnetic field strength ($B_{0}$) has an upper limit that depends on the initial convection zone size ($R_{\mathrm{out},0}$) and the WD mass. We solve the induction equation to follow the magnetic field evolution after the dynamo phase ends. We show that the predicted surface magnetic field strength ($B_{\mathrm{surf}}$) differs from $B_{0}$ by at least a factor of $\sim$0.3. This reduction depends on $R_{\mathrm{out},0}$, where values smaller than half of the star radius give $B_{\mathrm{surf}}\lesssim0.01\ B_{0}$. We implement electrical conductivities that account for the solid phase effect on the Ohmic diffusion. We observe that the conductivity increases as the solid core grows, freezing in the magnetic field at a certain point of the evolution and slowing its outwards transport. We study the effect of turbulent magnetic diffusivity induced by the convection and find that for a small $R_{\mathrm{out},0}$, $B_{\mathrm{surf}}$ is stronger than the non-turbulent diffusion cases because of the more rapid transport, but still orders of magnitude smaller than $B_{0}$. Given these limitations, the crystallization-driven dynamo theory could explain only magnetic C/O WDs with field strengths less than a few MG for the mass range 0.45-1.05 $M_{\odot}$. Our results also suggest that a buried fossil field must be at least 100 times stronger than observed surface fields if crystallization-driven convection is responsible for its transport to the surface.