Biermann battery

In plasma physics, the Biermann battery is a thermoelectric effect that produces a magnetic field when the electron density and temperature gradients have non-collinear (perpendicular) components, and was first identified by Ludwig Biermann in 1950.

Physics

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

$$𝐄\simeq -𝐯\times 𝐁-\frac{\mathrm{\nabla }{P}_e}{e{n}_e}$$

Using Faraday’s law,

$$\mathrm{\nabla }\times 𝐄=-\frac{\partial 𝐁}{\partial t}$$

one obtains the induction equation,

$$\frac{\partial 𝐁}{\partial t}=\mathrm{\nabla }\times (𝐯\times 𝐁)+\frac{\mathrm{\nabla }{T}_e\times \mathrm{\nabla }{n}_e}{e{n}_e}$$

where we used $P_e=n_e{T}_e$. 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 $𝐁$, so magnetic fields can be generated from zero initial field provided non-collinear $\mathrm{\nabla }{T}_e$ and $\mathrm{\nabla }{n}_e$ are present.

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