Magnetic Mirrors: the effects of $\nabla B$ $\parallel$ ${\bf B}$

This section addresses the mirroring properties of a longitudinal gradient in the magnetic field; i.e., the effects of gradients in magnetic field strength parallel to ${\bf B}$. Two ways of doing this are: using conservation of $\mu$ and the total kinetic energy, while the second looks at forces.

Figure 2.6 shows a cylindrically symmetric situation with $\nabla B \parallel {\bf B}$ (half of a so-called magnetic bottle). Note the field lines becoming closer together as $\vert{\bf B}\vert$ increases. For slow gradients and time-independent fields $\mu = v_{\perp}^{2} / 2B$ is constant and there is no electric field ${\bf E}$, so that the kinetic energy $m( v_{\parallel}^{2} + v_{\perp}^{2} )/2$ is constant. Thus

$$v_{\parallel}^{2}(z) = v_{\parallel}^{2}(0) - v_{\perp}^{2}(0) \left[ \frac{B(z)}{B(0)} - 1 \right] \ .$$ (2.41)

Thus the particle's parallel speed decreases as it moves into the region with increased B, and may actually vanish at some point (the magnetic mirror point). This point depends on the initial parallel and perpendicular speeds and the fractional increase in B. A particle reaching its magnetic mirror point is reflected and retraces its trajectory - note that Eq. (2.41) is for $v_{\parallel}(z)^{2}$.

The pitch angle $\alpha$ of a particle is defined by

$$\tan \alpha = \frac{v_{\perp}}{v_{\parallel}} \ .$$ (2.42)

Exercise 2.4: Show that a particle will be reflected from a magnetic field gradient if

$$\sin^{2} \alpha \ge \frac{B(0)}{B(z)} \ .$$ (2.43)

The other approach toward magnetic mirroring is to directly study the forces on a particle. In the case of axisymmetric fields, ${\bf B}_{\theta} = 0$ and ${\bf B}_{z}$ and ${\bf B}_{r}$ are related by $\nabla . {\bf B} = 0$. When $\partial B_{z} / \partial z$ is slowly varying one may integrate the equation $\nabla . {\bf B} = 0$ to obtain

$$B_{r} = - \frac{r}{2} \frac{\partial B}{\partial z} \ .$$ (2.44)

Then

$$F_{\parallel} = q v_{\perp} B_{r} = - \mu \frac{\partial B}{\partial z} \ .$$ (2.45)

This equation shows that the particle experiences a decrease in $v_{\parallel}$, is potentially reflected, and (for constant $\mu$ and kinetic energy) increases in $v_{\perp}$ when it enters a region with larger B.

Note that not all particles entering a region with increased B will be reflected. Instead, those with $sin^{2} \alpha < B(0) / B(z)$ will pass through the magnetic enhancement. Define B_m, the maximum field in the trap, to be the mirror field: then particles with

$$sin^{2} \alpha_{0} = sin^{2} \left( \tan^{-1}[ v_{\perp}(0) / v_{\parallel}(0) ] \right) < \frac{B(0)}{B_{m}}$$ (2.46)

will not be mirrored. This leads to loss cone anisotropies in the particle distribution function. Importantly, these loss cone anisotropies can drive plasma waves and radio emissions which are observable, including Earth's Auroral Kilometric Radiation and various solar emissions. Moreover, generation of the waves and scattering by the wave fields drive the plasma particles into the loss cone, leading to loss of plasma. Magnetic mirroring can also lead to particle energisation, especially at shock waves.