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

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.