Biermann battery

The Biermann source term may be derived by starting from a single-fluid Ohm’s law that retains the electron pressure term,

$${\displaystyle \mathbf {E} \simeq -\mathbf {v} \times \mathbf {B} -{\frac {\nabla P_{e}}{e\,n_{e}}}}$$

Using Faraday’s law,

$${\displaystyle \nabla \times \mathbf {E} =-\,{\frac {\partial \mathbf {B} }{\partial t}}}$$

one obtains the induction equation,

$${\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=\nabla \times (\mathbf {v} \times \mathbf {B} )+{\frac {\nabla T_{e}\times \nabla n_{e}}{e\,n_{e}}}}$$

where we used ${\displaystyle P_{e}=n_{e}T_{e}}$. The first term on the right-hand side describes advection of magnetic field by the bulk flow, and the last term is the Biermann battery source term. Physically, the associated currents are driven by pressure-force-induced rotational motion between electrons and ions. Notably, the Biermann term contains no explicit dependence on ${\displaystyle \mathbf {B} }$, so magnetic fields can be generated from zero initial field provided non-collinear ${\displaystyle \nabla T_{e}}$ and ${\displaystyle \nabla n_{e}}$ are present.

In astrophysics, the Biermann battery effect is a candidate mechanism for the source of seed fields in protogalactic environments.^[2] It is also important in laser-produced plasmas, where perpendicular temperature gradients may arise from laser-imposed temperature profiles.^[3] See also cosmological simulations demonstrating Biermann battery field generation in protostellar disks.^[4]