Past exam of the collisionless plasma physics course of the University of Oxford / 2025 / 2

Write the ordinary plasma wave asThe eikonal function $S$ is dimensionless; it varies by order ${\epsilon }^{-1}$ across a background scale $L_{\mathrm{bg}}$, where $\epsilon \sim \lambda /L_{\mathrm{bg}}\ll 1$. The amplitudes and equilibrium plasma parameters vary on $L_{\mathrm{bg}}$, so $|\nabla {\mathbf{a}}_j|\ll k|{\mathbf{a}}_j|$ and $|\nabla k|\ll k^2$. The equilibrium is stationary, so $\omega$ is constant along propagation. Assume infinitesimal fields, negligible collisions, $k{v}_{\mathrm{th},s}\ll \omega$, negligible ion response, and a fixed ordinary plasma wave polarization with ${\mathbf{B}}_0\parallel \hat{\mathbf{z}}$ and spatial variation along $x$. The WKB approximation is used away from wave turning points and plasma wave resonances.

Replacing derivatives on the fast factor by $i\mathbf{k}$ and $-i\omega$ in the Maxwell equations reproduces the local magnetized cold-plasma dielectric tensor equation of b. In this one-dimensional ordinary plasma wave geometry, the Maxwell equations and the local cold parallel current density in fact give the scalar equation directly:For $\delta E_z=a(x)e^{iS(x)}$, substitution givesThe leading eikonal equation is $(S^{\prime }{)}^2=k^2(x)$, and the next order gives $2k{a}^{\prime }+k^{\prime }a=0$, hence the flux-normalized WKB amplitude $a\propto k^{-1/2}$. In particular, require $|k^{\prime }|/|k{|}^2\ll 1$ and $|k^{\prime \prime }|/|k{|}^3\ll 1$; these fail at the wave turning points.

A wave turning point satisfies $k^2(x)=0$, equivalently ${\omega }_{pe}^2(x)={\omega }^2$. ThereforeFor $0<\omega <{\omega }_{pe0}$ the two wave turning points lie inside the column and are simple. Defining $a=-x_T$ and $b=x_T$, the local dispersion relation becomes

Use positive real local wavenumberswithin $|x|\le L$. If the exterior is vacuum, continue $k_1,k_3$ as $\omega /c$ outside the column. Define the positive phase integrals and barrier integralThe three WKB approximations, with the common $e^{-i\omega t}$ suppressed, areThe flux-normalized WKB amplitudes make squared travelling-wave coefficients proportional to directed Poynting flux. Since ${\Phi }^{\prime }=-k_1$ and ${\Psi }^{\prime }=k_3$, $A_{\mathrm{in}}$ and $A_{\mathrm{t}}$ travel to the right, while $A_{\mathrm{r}}$ and $A_{\mathrm{back}}$ travel to the left. The constants $\pi /4$ are a convenient Airy turning-point connection formula convention.

At $a=-x_T$, defineThe first-order Taylor expansion is $k^2(x)=-\alpha (x-a)+O((x-a{)}^2)$, so the local wave equation becomesThis is the Airy ordinary differential equation. Its $z<0$ side is oscillatory and its $z>0$ side is evanescent. A matching overlap requires $\ell \ll |x-a|\ll x_T$, hence well-separated wave turning points with $\ell \ll x_T$. At $b=x_T$, the coordinate $z_R=(b-x)/\ell$ gives the same Airy ordinary differential equation, with its evanescent side again at positive argument.

The real Airy function asymptotics used for the Airy turning-point connection formula are, for $z,s\to +\infty$,These real-axis asymptotics fix both normalization and phase.^[ref] Near $a$, $I\sim 2z^{3/2}/3$, $\Phi \sim 2(-z{)}^{3/2}/3$, and ${\kappa }^{-1/2}\sim {\ell }^{1/2}{z}^{-1/4}$. Matching to the constants of c gives $C_A=2\sqrt{\pi }{\ell }^{1/2}D$ and $C_B=\sqrt{\pi }{\ell }^{1/2}G$. ThereforeIn particular, the decaying component matches the cosine with the requested $-\pi /4$ shift, with a factor two in its oscillatory coefficient. The growing component matches a sine with the indicated minus sign. Expanding into travelling waves givesThe factor two is essential when converting the real oscillatory components to incident and reflected travelling waves.

Let $J(x)={\int }_x^b\kappa (s)ds=\eta -I(x)$. The WKB approximation in the barrier can be rewritten asThe right Airy turning-point connection formula givesso $A_{\mathrm{t}}=D_R+i{G}_R/2$ and $A_{\mathrm{back}}=D_R-i{G}_R/2$. Impose no incident wave from the right, $A_{\mathrm{back}}=0$. ThenConsequently, within the leading Airy turning-point connection formula calculation,For an opaque barrier, $\eta \gg 1$,The transmission coefficient is a ratio of Poynting fluxes; the $k^{-1/2}$ normalization already includes their local wavenumber factors. The displayed denominator is useful for consistent matching and Wronskian flux conservation, but does not claim accuracy for subleading terms beyond the WKB approximation, or for merging wave turning points.

For real $k^2(x)$, the wave equation has the conserved Wronskian fluxFor this ordinary plasma wave, the time-averaged Poynting flux is $⟨S_x⟩=c^2\mathcal{J}/(8\pi \omega )$. With no incident wave from the right, evaluating the same Wronskian flux on both sides provesThe coefficients in f also satisfy this identity algebraically, before the opaque-barrier expansion.

A plasma wave resonance makes the ideal cold description singular and requires an appropriate causal regularization. A passive resonant layer can absorb wave energy into particle motion, or convert it into another branch. For the two measured outgoing channels one then haswith $A>0$ when resonant absorption or uncounted mode conversion occurs. A numerical value of $A$ requires the layer physics; a resonance by itself does not specify a universal loss fraction. Counting all converted and absorbed energy preserves conservation of energy.

The barrier wavenumber is $\kappa ={\omega }_{pe0}\sqrt{x_T^2-x^2}/(cL)$. The substitution $x=x_T\sin u$ givesThus the leading transmission coefficient isAt fixed $\omega /{\omega }_{pe0}$, increasing the column width or peak plasma frequency suppresses transmission exponentially. Raising $\omega$ towards ${\omega }_{pe0}$ narrows the evanescent region and increases transmission. However, when $\eta$ becomes order one or smaller the two wave turning points interact; the opaque-barrier WKB approximation cannot be extrapolated to claim $T\to 1$ at $\omega ={\omega }_{pe0}$. Smooth matching to an exterior is also implicit in the leading result; at $|x|=L$ the stated profile joins continuously to vacuum.

The shaded area in the lower panel is $\eta$ after restoring the factor ${\omega }_{pe0}L/c$.