Curvature drifts

Curvature of magnetic field lines can also cause plasma particles to drift. Figure 2.4 shows this situation. As the particle moves along a curved magnetic field line it experiences a centrifugal force due to the field curvature, and therefore drifts perpendicular to both the centrifugal force and ${\bf B}$ as described in Section 2.3. Defining the radius of curvature ${\bf R}_{c}$ of the magnetic field lines as in Figure 2.4, then

$${\bf F}_{curv} = \frac{m v_{\parallel}^{2}}{R_{c}^{2}} {\bf R}_{c}$$ (2.27)

and so

$${\bf v}_{curv} \frac{m}{q} \frac{v_{\parallel}^{2}}{R_{c}^{2}} \frac{{\bf R}_{c} \times {\bf B}}{B^{2}}$$ = (2.28)

$$\frac{m}{q} \frac{v_{\parallel}^{2}}{B} \frac{{\bf B} \times \nabla B}{B^{2}} \ .$$ = (2.29)

This final form relates the radius of curvature to the magnetic field using Ampere's Law (assuming no plasma currents).

Gradients in a plasma's magnetic field are constrained by Ampere's Law $\nabla \times B = \mu_{0} {\bf j}$ (neglecting the displacement current). This means that plasma particles are almost always subject simultaneously to both the $\nabla B$ and curvature drifts, not just one or the other. Combining Eqs (2.25) and (2.29), the combined drift is

$${\bf v}_{B} = \frac{m}{2q B} ( v_{\perp}^{2} + 2 v_{\parallel}^{2} ) \frac{{\bf B} \times \nabla B}{B^{2}} \ .$$ (2.30)

This drift velocity naturally leads to charge separations and currents in plasmas, as well as dispersion of particles with different parallel & perpendicular energies, charges, and masses.