The generation of magnetic fields by the Biermann battery and the interplay with the Weibel instability

An investigation of magnetic fields generated in an expanding bubble of plasma with misaligned temperature and density gradients (driving the Biermann battery mechanism) is performed. With gradient scales $L$, large-scale magnetic fields are generated by the Biermann battery mechanism with plasma $\beta \sim 1$, as long as $L$ is comparable to the ion inertial length $d_i$. For larger system sizes, $L/d_e > 100$ (where $d_e$ is the electron inertial length), the Weibel instability generates magnetic fields of similar magnitude but with wavenumber $k d_e \sim 0.2$. In both cases, the growth and saturation of these fields have a weak dependence on mass ratio $m_i/m_e$, indicating electron mediated physics. A scan in system size is performed at $m_i/m_e = 2000$, showing agreement with previous results with $m_i/m_e = 25$. In addition, the instability found at large system sizes is quantitatively demonstrated to be the Weibel instability. Furthermore, magnetic and electric energy spectra at scales below the electron Larmor radius are found to exhibit power law behavior with spectral indices $-16/3$ and $-7/3$, respectively.