--- title: "Guiding center theory" sort_title: "Guiding center theory" date: 2021-09-21 categories: - Physics - Electromagnetism - Plasma physics layout: "concept" --- When discussing the [Lorentz force](/know/concept/lorentz-force/), we introduced the concept of *gyration*: a particle in a uniform [magnetic field](/know/concept/magnetic-field/) $$\vb{B}$$ *gyrates* in a circular orbit around a **guiding center**. Here, we will generalize this result to more complicated situations, for example involving [electric fields](/know/concept/electric-field/). The particle's equation of motion combines the Lorentz force $$\vb{F}$$ with Newton's second law: $$\begin{aligned} \vb{F} = m \dv{\vb{u}}{t} = q \big( \vb{E} + \vb{u} \cross \vb{B} \big) \end{aligned}$$ We now allow the fields vary slowly in time and space. We thus add deviations $$\delta\vb{E}$$ and $$\delta\vb{B}$$: $$\begin{aligned} \vb{E} \to \vb{E} + \delta\vb{E}(\vb{x}, t) \qquad \quad \vb{B} \to \vb{B} + \delta\vb{B}(\vb{x}, t) \end{aligned}$$ Meanwhile, the velocity $$\vb{u}$$ can be split into the guiding center's motion $$\vb{u}_{gc}$$ and the *known* Larmor gyration $$\vb{u}_L$$ around the guiding center, such that $$\vb{u} = \vb{u}_{gc} + \vb{u}_L$$. Inserting: $$\begin{aligned} m \dv{}{t}\big( \vb{u}_{gc} + \vb{u}_L \big) = q \big( \vb{E} + \delta\vb{E} + (\vb{u}_{gc} + \vb{u}_L) \cross (\vb{B} + \delta\vb{B}) \big) \end{aligned}$$ We already know that $$m \: \idv{\vb{u}_L}{t} = q \vb{u}_L \cross \vb{B}$$, which we subtract from the total to get: $$\begin{aligned} m \dv{\vb{u}_{gc}}{t} = q \big( \vb{E} + \delta\vb{E} + \vb{u}_{gc} \cross (\vb{B} + \delta\vb{B}) + \vb{u}_L \cross \delta\vb{B} \big) \end{aligned}$$ This will be our starting point. Before proceeding, we also define the average of $$\Expval{f}$$ of a function $$f$$ over a single gyroperiod, where $$\omega_c$$ is the cyclotron frequency: $$\begin{aligned} \Expval{f} \equiv \int_0^{2 \pi / \omega_c} f(t) \dd{t} \end{aligned}$$ Assuming that gyration is much faster than the guiding center's motion, we can use this average to approximately remove the finer dynamics, and focus only on the guiding center. ## Uniform electric and magnetic field Consider the case where $$\vb{E}$$ and $$\vb{B}$$ are both uniform, such that $$\delta\vb{B} = 0$$ and $$\delta\vb{E} = 0$$: $$\begin{aligned} m \dv{\vb{u}_{gc}}{t} = q \big( \vb{E} + \vb{u}_{gc} \cross \vb{B} \big) \end{aligned}$$ Dotting this with the unit vector $$\vu{b} \equiv \vb{B} / |\vb{B}|$$ makes all components perpendicular to $$\vb{B}$$ vanish, including the cross product, leaving only the (scalar) parallel components $$u_{gc\parallel}$$ and $$E_\parallel$$: $$\begin{aligned} m \dv{u_{gc\parallel}}{t} = \frac{q}{m} E_{\parallel} \end{aligned}$$ This simply describes a constant acceleration, and is easy to integrate. Next, the equation for $$\vb{u}_{gc\perp}$$ is found by subtracting $$u_{gc\parallel}$$'s equation from the original: $$\begin{aligned} m \dv{\vb{u}_{gc\perp}}{t} = q (\vb{E} + \vb{u}_{gc} \cross \vb{B}) - q E_\parallel \vu{b} = q (\vb{E}_\perp + \vb{u}_{gc\perp} \cross \vb{B}) \end{aligned}$$ Keep in mind that $$\vb{u}_{gc\perp}$$ explicitly excludes gyration. If we try to split $$\vb{u}_{gc\perp}$$ into a constant and a time-dependent part, and choose the most convenient constant, we notice that the only way to exclude gyration is to demand that $$\vb{u}_{gc\perp}$$ does not depend on time. Therefore: $$\begin{aligned} 0 = \vb{E}_\perp + \vb{u}_{gc\perp} \cross \vb{B} \end{aligned}$$ To find $$\vb{u}_{gc\perp}$$, we take the cross product with $$\vb{B}$$, and use the fact that $$\vb{B} \cross \vb{E}_\perp = \vb{B} \cross \vb{E}$$: $$\begin{aligned} 0 = \vb{B} \cross (\vb{E}_\perp + \vb{u}_{gc\perp} \cross \vb{B}) = \vb{B} \cross \vb{E} + \vb{u}_{gc\perp} B^2 \end{aligned}$$ Rearranging this shows that $$\vb{u}_{gc\perp}$$ is constant. The guiding center drifts sideways at this speed, hence it is called a **drift velocity** $$\vb{v}_E$$. Curiously, $$\vb{v}_E$$ is independent of $$q$$: $$\begin{aligned} \boxed{ \vb{v}_E = \frac{\vb{E} \cross \vb{B}}{B^2} } \end{aligned}$$ Drift is not specific to an electric field: $$\vb{E}$$ can be replaced by a general force $$\vb{F}/q$$ without issues. In that case, the resulting drift velocity $$\vb{v}_F$$ does depend on $$q$$: $$\begin{aligned} \boxed{ \vb{v}_F = \frac{\vb{F} \cross \vb{B}}{q B^2} } \end{aligned}$$ ## Non-uniform magnetic field Next, consider a more general case, where $$\vb{B}$$ is non-uniform, but $$\vb{E}$$ is still uniform: $$\begin{aligned} m \dv{\vb{u}_{gc}}{t} = q \big( \vb{E} + \vb{u}_{gc} \cross (\vb{B} + \delta\vb{B}) + \vb{u}_L \cross \delta\vb{B} \big) \end{aligned}$$ Assuming the gyroradius $$r_L$$ is small compared to the variation of $$\vb{B}$$, we set $$\delta\vb{B}$$ to the first-order term of a Taylor expansion of $$\vb{B}$$ around $$\vb{x}_{gc}$$, that is, $$\delta\vb{B} = (\vb{x}_L \cdot \nabla) \vb{B}$$. We thus have: $$\begin{aligned} m \dv{\vb{u}_{gc}}{t} = q \big( \vb{E} + \vb{u}_{gc} \cross \vb{B} + \vb{u}_{gc} \cross (\vb{x}_L \cdot \nabla) \vb{B} + \vb{u}_L \cross (\vb{x}_L \cdot \nabla) \vb{B} \big) \end{aligned}$$ We approximate this by taking the average over a single gyration, as defined earlier: $$\begin{aligned} m \dv{\vb{u}_{gc}}{t} = q \big( \vb{E} + \vb{u}_{gc} \cross \vb{B} + \vb{u}_{gc} \cross \Expval{ (\vb{x}_L \cdot \nabla) \vb{B} } + \Expval{ \vb{u}_L \cross (\vb{x}_L \cdot \nabla) \vb{B} } \big) \end{aligned}$$ Where we have used that $$\Expval{\vb{u}_{gc}} = \vb{u}_{gc}$$. The two averaged expressions turn out to be: $$\begin{aligned} \Expval{ (\vb{x}_L \cdot \nabla) \vb{B} } = 0 \qquad \quad \Expval{ \vb{u}_L \cross (\vb{x}_L \cdot \nabla) \vb{B} } \approx - \frac{u_L^2}{2 \omega_c} \nabla B \end{aligned}$$