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 ≃ − v × B − ∇ P e e n e {\displaystyle \mathbf {E} \simeq -\mathbf {v} \times \mathbf {B} -{\frac {\nabla P_{e}}{e\,n_{e}}}}
Using Faraday’s law,
∇ × E = − ∂ B ∂ t {\displaystyle \nabla \times \mathbf {E} =-\,{\frac {\partial \mathbf {B} }{\partial t}}}
one obtains the induction equation,
∂ B ∂ t = ∇ × ( v × B ) + ∇ T e × ∇ n e e n e {\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 P e = n e T e {\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 B {\displaystyle \mathbf {B} } , so magnetic fields can be generated from zero initial field provided non-collinear ∇ T e {\displaystyle \nabla T_{e}} and ∇ n e {\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. 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.
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