$\nabla B$ drift, with $\nabla B$ $\perp$ B

Figure 2.3 shows the path of a positively charged particle in this case. Assuming that the gradient is on scale lengths long compared with the gyroradius, i.e.,

$$r_{L} \ll L = \left( \vert\nabla B\vert / \vert{\bf B}\vert \right)^{-1} \ ,$$ (2.17)

the orbit is almost circular but does not quite close. Since $r_{L} \propto B^{-1}$, r_L will be slightly smaller where B is larger and slightly larger where B is smaller. This causes the orbit to drift in the direction of its gyromotion where r_L is larger, perpendicular to both $\nabla B$ and to ${\bf B}$. A negatively charged particle drifts in the opposite direction. The physics is thus clear.

Quantitatively we write

$${\bf B}({\bf r} = {\bf x}_{c}) = {\bf b} \left( B(0) + {\bf x}_{c} . \nabla B(0) \right) \ ,$$ (2.18)

where ${\bf x}_{c}$ represents the unperturbed gyromotion and ${\bf b}$ is the unit vector of the magnetic field. The perpendicular equation of motion is

$$\frac{d {\bf v}_{\perp}}{d t} = \Omega_{c} {\bf v}_{\perp} \t... ...b} \left( 1 + \frac{{\bf x}_{c} . \nabla B}{B(0)} \right) \ .$$ (2.19)

The particle velocity is now written as ${\bf v}_{\perp} = {\bf v}_{c} + {\bf v}_{\perp,1}$, where ${\bf v}_{c}$ represents the unperturbed gyromotion and ${\bf v}_{\perp, 1}$ is the sum of the drift velocity and any first order perturbations to the gyromotion. Substituting into (2.19), grouping the zeroth order terms and deleting them, and ignoring the second order term ${\bf v}_{\perp,1} \times {\bf b} {\bf x}_{c} . \nabla B(0)$, the first order equation becomes

$$\frac{d {\bf v}_{\perp, 1}}{d t} = \Omega_{c} \left( {\bf v}_... ...bla B / B_(0) + {\bf v}_{\perp, 1} \times {\bf b} \right) \ .$$ (2.20)

This equation is next averaged over a gyroperiod and ${\bf v}_{\perp, 1}$ is identified as the constant drift velocity ${\bf v}_{D}$, so that the time derivative becomes zero and the drift velocity obeys the equation

$${\bf v}_{D} \times {\bf b} = - < {\bf v}_{c} \times {\bf b}\ \frac{ {\bf x}_{c} . \nabla B}{B(0)} > \ .$$ (2.21)

Now ${\bf v}_{c} = - {\bf x}_{c} \times {\bf b} \Omega_{c}$ and the time-average of the righthand term simplifies considerably since the x component of the term $< {\bf x}_{c} {\bf x}_{c} . \nabla B >$becomes

$$< x_{c, x} {\bf x}_{c} . \nabla B > < x_{c,x} ( x_{c,x} \left( \frac{\partial B}{\partial x} + x_{c,y} \frac{\partial B}{\partial y} \right) >$$ = (2.22)

$$< x_{c,x}^{2} > \frac{\partial B}{\partial x}$$ =

$$r_{L}^{2} / 2 \frac{\partial B}{\partial x} \ ,$$ = (2.23)

with a similar result for the y component. That is,

$${\bf V}_{D} \times {\bf b} = \frac{ r_{L}^{2} \Omega_{c}}{2}\ \frac{\nabla B}{B} \ .$$ (2.24)

Rearranging, the final result is

$${\bf v}_{\nabla B} \frac{1}{2} \frac{m v_{\perp}^{2}}{q B} \frac{ {\bf B} \times \nabla B}{B^{2}}$$ = (2.25)

$$\frac{1}{2} v_{\perp} r_{L} \frac{ {\bf B} \times \nabla B}{B^{2}} \ .$$ = (2.26)

The $\nabla B$ drift speed therefore depends on the charge, mass, and perpendicular energy of the particle, as well as on the magnetic field strength and the scale length of the gradient. This drift can therefore cause currents and charge separations in the plasma. Moreover, as seen below, the combination of a convection electric field and a $\nabla B$ plasma drift can lead to particle acceleration.