Drift kinetics

The slow dynamics of charged-particle guiding centers after averaging over rapid cyclotron motion, assuming gyroradii much smaller than resolved perpendicular length scales. Depending on the ordering, the retained transverse motion may include just the common electric drift or also magnetic curvature drift, gradient-B drift and polarization drift. These choices give distinct limits, not interchangeable closures.

The inertial correction to the electric drift ${\mathbf{u}}_E=c\mathbf{E}\times \mathbf{B}/B^2$ in a slowly varying magnetic field. In the simplest uniform-field electrostatic case it is ${\mathbf{v}}_{\mathrm{pol}}=m_s{c}^2{\partial }_t{\mathbf{E}}_{\perp }/(q_s{B}^2)$. Its species-dependent current density supplies the perpendicular polarization response needed in many slow plasma instabilities.

The transverse guiding center drift produced by a magnetic field strength gradient, with first adiabatic invariant $\mu =v_{\perp }^2/(2B)$ and signed cyclotron frequency ${\Omega }_s$. Together with magnetic curvature drift, it produces velocity-dependent transverse transport.

The transverse guiding center drift associated with magnetic field-line curvature. The signed cyclotron frequency ${\Omega }_s=q_{s}B/(m_{s}c)$ makes oppositely charged species drift oppositely at fixed velocity. The resulting species-dependent response is retained in slow drift-scale instabilities.

The collisionless equation in the title transports a gyrophase-independent phase-space distribution function at fixed first adiabatic invariant. Its leading guiding center position has $\dot{\mathbf{R}}={\mathbf{u}}_{\perp }+v_{\parallel }\mathbf{b}$. The parallel acceleration includes $(q/m)E_{\parallel }-\mu \mathbf{b}\cdot \nabla B$ and, when required, terms from motion of the local frame. Both $v_{\parallel }$ and the parallel derivative refer to the exact perturbed magnetic field.