Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S16 / 1

Landau damping is the collisionless decay of a plasma wave through resonant energy exchange with particles whose velocity is close to the wave's phase velocity $v_{\varphi }=\omega /k$. Particles slightly slower than the wave tend to gain energy from it; particles slightly faster can give energy to it. For the decreasing resonant part of a Maxwell-Boltzmann distribution, the net transfer is from the wave to the electrons, so the wave amplitude decays.

In the linear Vlasov equation description this is collisionless phase mixing, not a requirement for frequent particle collisions. Fine-scale structure develops in the velocity dependence of the perturbation even while the macroscopic electric field decreases. The resonant contribution to the growth rate has the sign of $\partial f/\partial v$ at a positive wave phase velocity: a negative slope produces damping, whereas a sufficiently positive slope can drive an instability. A population comparison must be local to the resonance; merely counting all particles faster and slower than the wave is not the criterion.

Define the one-dimensional thermal widths ${\sigma }_p=\sqrt{k_{\mathrm{B}}{T}_p/m_e}$ and ${\sigma }_b=\sqrt{k_{\mathrm{B}}{T}_b/m_e}$. Normalize each velocity distribution to its electron number density:The total is $f=f_p+f_b$. It consists of a broad peak centred at zero and a much narrower displaced beam near $V$, with $V\gg {\sigma }_p\gg {\sigma }_b$. Their areas, rather than their peak heights, are $n_p$ and $n_b$. A weak narrow beam can make a positive-slope region on its low-velocity side even while contributing little to the total density.

For the shifted beam Maxwell-Boltzmann distribution,On the rising flank put $u=(V-v)/{\sigma }_b>0$. The positive slope is proportional to $u{e}^{-u^2/2}$, whose derivative is $(1-u^2)e^{-u^2/2}$. Its unique positive maximum is at $u=1$, givingThis is the region where beam drive can most readily overcome the negative plasma slope when that background changes little over the narrow beam width. It is an estimate for the first resonant unstable wave, not a proof that every beam-plasma dispersion relation has its exact marginal phase velocity at this value. Both the total slope and the real dispersion relation determine the exact marginal mode.

At $v_{\ast }=V-{\sigma }_b$, the beam's positive slope is $n_b{e}^{-1/2}/(\sqrt{2\pi }{\sigma }_b^2)$, while the background slope has magnitude $n_p{v}_{\ast }{e}^{-v_{\ast }^2/(2{\sigma }_p^2)}/(\sqrt{2\pi }{\sigma }_p^3)$. Thus a nonnegative total slope at this velocity requiresApproximating the background slope by its value at $V$ yieldsEquality is the local marginal-slope condition; strict inequality gives positive resonant drive in this approximation. The numerical $C$ is of order unity and can change when the resonant velocity is optimized or the full wave dispersion is retained.

Replacing $v_{\ast }$ by $V$ in the exponential is controlled only if $V{\sigma }_b/{\sigma }_p^2\ll 1$, in addition to ${\sigma }_b\ll {\sigma }_p$. The two width inequalities alone do not guarantee that extra condition far out on a Maxwellian tail. The preceding unapproximated slope inequality remains available when it is not satisfied.

Use $k_{\mathrm{B}}{T}_p=1000$ eV and $k_{\mathrm{B}}{T}_b=100$ eV. The nonrelativistic beam speed and cross-sectional area areHere $V/{\sigma }_p=\sqrt{20}$, ${\sigma }_b^2/{\sigma }_p^2=0.1$, and the exponential in the requested threshold is $e^{-10}$. With $C=\sqrt{e}$,Converting beam density to the magnitude of its electric current givesThis is an order-of-magnitude estimate from the source's asymptotic formula. The numerical hierarchy is only moderate: $V{\sigma }_b/{\sigma }_p^2=\sqrt{2}$. Retaining the background slope at $V-{\sigma }_b$ instead gives about $9.1\times 10^4$ A as a sufficient positive-slope current at that chosen velocity. The difference quantifies the sensitivity of the exponential-tail approximation; an exact kinetic instability threshold requires the dispersion relation. The requested estimate is therefore on the scale of tens of kiloamperes, rather than a precisely determined current.