Collisionless plasma physics course of the University of Oxford (MMathPhys/MSc paper)

Source PDF: mmathphys.physics.ox.ac.uk/sites/g/files/vzsiac2991/files/mmathphys/documents/media/a15278s1_collisionless_plasma_physics.pdf

Use Gaussian units, plane waves proportional to $e^{i\mathbf{k}\cdot \mathbf{r}-i\omega t}$, and the signed cyclotron frequency ${\Omega }_s=q_s{B}_0/(m_{s}c)$. Write ${\Omega }_e=e{B}_0/(m_{e}c)>0$ for its electron magnitude, so the signed electron value is $-{\Omega }_e$. Define the species plasma frequency by ${\omega }_{ps}^2=4\pi n_{0s}{q}_s^2/m_s$. The linearization of the cold Lorentz force equations givesSolving this linear system and using the species current density ${\mathbf{j}}_s=n_{0s}{q}_s{\mathbf{u}}_s$ gives the conductivity tensorThe magnetized cold-plasma dielectric tensor is $\mathbf{ϵ}=\mathbf{I}+(4\pi i/\omega ){\sum }_s{\mathbf{\sigma }}_s$. ThusAt the stated electron frequency scale, the ion contribution is smaller by $O(Z{m}_e/m_i)$ for $q_i=Ze$ and quasineutrality. To this accuracy,These expressions require negligible collisions and the cold ordering $k{v}_{\mathrm{th},s}\ll |\omega |$; sufficiently close to a cyclotron resonance, thermal broadening or collisions must also be retained.

The Faraday law and Ampere-Maxwell law give $\delta \mathbf{B}=(c/\omega )\mathbf{k}\times \delta \mathbf{E}$ andWith refractive index $n=kc/\omega$, the linear system is thereforeSetHere $R$ labels the cold-plasma right-handed wave and $L$ the cold-plasma left-handed wave, with their polarizations fixed explicitly in d. In the electron-only response,Writing $N=n^2$, expansion of the determinant yields the dispersion relationEquivalently,Division givesThe undivided determinant equation remains valid when this rational form has simultaneous zero factors.

A plasma wave resonance has $n^2\to \infty$. At finite coefficients of the magnetized cold-plasma dielectric tensor, its generic condition is the vanishing of the coefficient of $n^4$:A plasma wave cutoff has $n^2=0$, so the constant term of the dispersion relation vanishes:Both statements are branch conditions: check the undivided dispersion relation at special angles and simultaneous zero factors. In particular, the parallel plasma oscillation at $P=0$ is a longitudinal branch with arbitrary $k$ in the cold limit, while the transverse parallel branches have plasma wave cutoffs at $R=0$ or $L=0$. Poles of the magnetized cold-plasma dielectric tensor can give additional limiting plasma wave resonances, such as the parallel cold-plasma right-handed wave at $\omega ={\Omega }_e$; these are not obtained by assuming finite $S,P,g$.

For parallel propagation, use the undivided dispersion relation:The five requested cases are:

For the perpendicular cases the undivided dispersion relation factors as $(n^2-P)(S{n}^2-RL)=0$, up to an overall nonzero factor. This also identifies the extraordinary plasma wave plasma wave resonance at $S=0$.

An electrostatic plasma wave has $\delta \mathbf{E}=-i\mathbf{k}\varphi$. In the absence of externally imposed charge density, the Gauss law for the magnetized cold-plasma dielectric tensor is $\mathbf{k}\cdot \mathbf{ϵ}\cdot \delta \mathbf{E}=0$. Consequently,The antisymmetric terms of the magnetized cold-plasma dielectric tensor cancel in this scalar contraction. For oblique propagation this is the leading longitudinal condition in the large-refractive index approximation; a small transverse electric field is generally required by the full Maxwell equations at finite $n$. For parallel propagation an exactly longitudinal solution exists: $P=0$, hence ${\omega }^2\simeq {\omega }_{pe}^2$. It coincides with the $\theta \to 0$ limit of the finite-coefficient plasma wave resonance condition in c. At exactly $\theta =0$, $P$ factors out of the transverse dispersion relation, leaving the independent plasma oscillation rather than an isolated divergence of the transverse branches.

For the ordinary plasma wave, choose $\mathbf{k}=k\hat{\mathbf{x}}$ and $\delta \mathbf{E}=E_z\hat{\mathbf{z}}$. The equations in a give $u_{xs}=u_{ys}=0$ andThe driven particle motion is entirely parallel to ${\mathbf{B}}_0$: the parallel Lorentz force has no magnetic contribution. Nevertheless, the ordinary plasma wave is electromagnetic, since $\delta \mathbf{E}\perp \mathbf{k}$ and the Faraday law givesAn electrostatic plasma wave instead has $\delta \mathbf{E}\parallel \mathbf{k}$ and zero perturbed magnetic field. Alignment with the equilibrium magnetic field does not make a wave electrostatic. At the exact plasma wave cutoff $k=0$, this propagating-wave distinction becomes degenerate.

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 ${\omega }_{pe0}L/c$.

Use Gaussian units and the kinetic magnetohydrodynamics equationstogether with species continuity equations, quasineutrality and the drift-kinetic equation. Here $p_{\perp },p_{\parallel }$ sum the species pressures. Interpret the thermal speed denominator in the source as the species mass $m_{\alpha }$, so $v_{\mathrm{th}\alpha }^2=2T_{0\alpha }/m_{\alpha }$ and the first adiabatic invariant is $\mu =v_{\perp }^2/(2B)$.

The specified Maxwell-Boltzmann distribution has zero mean motion and isotropic pressure, $p_{\perp 0}=p_{\parallel 0}=p_0$. All equilibrium scalars depend only on $x$, while ${\mathbf{b}}_0\cdot \nabla x=0$ and ${\mathbf{b}}_0\cdot \nabla B_0=0$. Therefore the equilibrium streaming and mirror terms of the drift-kinetic equation vanish. The remaining terms vanish because ${\mathbf{u}}_0=0$ and $E_{\parallel 0}=0$; the parallel electric constraint is also satisfied because the equilibrium parallel pressure gradient is zero. Species continuity equations and the induction equation are stationary, and quasineutrality holds by assumption.

It remains to check the perpendicular force balance. The radial magnetic pressure and magnetic tension giveThis is preciselyThus the Z-pinch is a steady solution of the stated kinetic magnetohydrodynamics system. Its equilibrium electric current is supplied by the small species drifts implicit in this approximation; zero leading common flow does not require zero electric current.

For a static equilibrium, linearization of magnetic induction givesThe rotating local basis must be differentiated. In particular, ${\partial }_{\parallel }\hat{\mathbf{x}}={\mathbf{b}}_0/R$ and ${\partial }_{\parallel }{\mathbf{b}}_0=-\hat{\mathbf{x}}/R$. Their contributions proportional to ${\xi }_{\parallel }$ cancel in the radial component, yieldingWriting $h=\delta B/B_0$, the magnitude component iswhere $\nabla \cdot \mathbf{\xi }={\xi }_x/R+i{k}_y{\xi }_y+i{k}_{\parallel }{\xi }_{\parallel }$ when ${\partial }_x{\xi }_x=0$.

Set $M=B_0^2/(4\pi )={\rho }_0{v}_A^2$ and $\delta \Pi =\delta p_{\perp }+Mh$. The transverse $y$ component of the linear kinetic magnetohydrodynamic momentum equation isSince $k_y\gg k_{\parallel }\sim \omega /v_A$, perpendicular magnetic pressure enforces the leading pressure-balanced plasma perturbation condition $\delta \Pi =0$, givingIn the radial component, the curvature part of the perturbed magnetic force is $-2Mh/R$: one contribution comes from $({\mathbf{B}}_0\cdot \nabla )\delta \mathbf{B}$ and one from $(\delta \mathbf{B}\cdot \nabla ){\mathbf{B}}_0$. The perturbed pressure anisotropy contributes $-(\delta p_{\perp }-\delta p_{\parallel })/R$. After the leading total-pressure constraint has removed ${\partial }_x\delta \Pi$, the radial equation isSubstituting $Mh=-\delta p_{\perp }$ givesAlthough $h$ is small at low plasma beta, its contribution $Mh$ to the force balance is of the same order as $\delta p_{\perp }$ and must be kept.

Choose the drift-kinetic equation in $(\mathbf{r},\mu ,v_{\parallel })$ with $v_{\parallel }$ the laboratory parallel velocity along the exact $\mathbf{b}$. To first order, before discarding terms small in the stated steep-gradient ordering, it isThe last term is the linear acceleration associated with the moving curved field. Terms quadratic in the perturbations have already been removed. A formulation using parallel velocity relative to the common flow gives the same physical result after transforming back to these variables.

The two mirror terms and the curvature acceleration are subleading here. Their relative sizes are controlled by $h\ll \delta p/p_0$ and $L_p/L_B,L_p/R\ll 1$. The field-direction perturbation multiplying the steep equilibrium gradient must remain:Using $u_x=-i\omega {\xi }_x$ and $\delta b_x=i{k}_{\parallel }{\xi }_x$ therefore gives the leading equationThe displaced branch satisfying quasineutrality has $\delta E_{\parallel }=0$ to this order: its species number densities are $\delta n_{\alpha }=-{\xi }_x{n}_{0\alpha }^{\prime }$, so ${\sum }_{\alpha }{q}_{\alpha }\delta n_{\alpha }=-{\xi }_x{\partial }_x{\sum }_{\alpha }{q}_{\alpha }{n}_{0\alpha }=0$. More explicitly, putting $\mathcal{D}=-i\omega +i{k}_{\parallel }{v}_{\parallel }$ givesAway from an independent longitudinal eigenfrequency, the summed susceptibility is nonzero, giving $\delta E_{\parallel }=0$. An independent longitudinal eigenmode is not forced by the interchange displacement. The required solution is thusThe apparent streaming denominator cancels identically, including at $\omega =k_{\parallel }{v}_{\parallel }$; this response has no leading Landau damping singularity.

To see the temperature and density contributions explicitly, differentiate at fixed $\mu ,v_{\parallel }$ and neglect $B_0^{\prime }$ at this order:The gyrotropic velocity-space measure is $d^{3}v=2\pi B_{0}d\mu d{v}_{\parallel }$. The species perpendicular plasma pressure and parallel plasma pressure are the momentsExplicit perturbations of the measure and of $\mu B$ produce terms proportional to $h$, which are subleading in these pressure moments; derivatives of the equilibrium measure produce the discarded $B_0^{\prime }$ terms. The remaining Maxwell-Boltzmann distribution integrals giveInsertion into b proves

The result is a fluid Eulerian perturbation generated by translating the entire equilibrium phase-space distribution function through ${\xi }_x$:Here $\Delta$ follows the displaced flux tube. Magnetic flux freezing carries the tube and its particles across the equilibrium pressure gradient. To leading order the tube changes neither its field strength nor its length by enough to alter the particle phase-space distribution function: these changes are smaller by plasma beta or $L_p/R$. Parallel streaming remains within the same displaced tube, on which the original equilibrium phase-space distribution function was uniform. It therefore cannot erase the displacement-generated contrast between adjacent tubes. This is why the term $v_{\parallel }\delta b_x{\partial }_x{f}_{0\alpha }$ in c exactly cancels the apparent parallel relaxation of $-{\xi }_x{\partial }_x{f}_{0\alpha }$. No collisional restoration of a Maxwell-Boltzmann distribution is needed.

Assume the stated outward-decreasing pressure, $L_p>0$, and setFor $|k_{\parallel }|<k_c$, the interchange instability hasAn outward-displaced flux tube carries higher pressure than its new surroundings. In the unfavorable magnetic field-line curvature of the Z-pinch, the resulting imbalance drives it farther outward. Parallel variation bends field lines, and magnetic tension supplies the stabilizing $k_{\parallel }^2{v}_A^2$ term. The fastest local interchange is at $k_{\parallel }=0$, with $\gamma ={\omega }_I$; this endpoint is understood as the limit of the stated Alfvénic ordering. There is no leading $k_y$ dependence once $k_y\gg k_{\parallel }$.

For $|k_{\parallel }|>k_c$, the two real frequencies are $\omega =\pm \sqrt{k_{\parallel }^2{v}_A^2-{\omega }_I^2}$: the displacement oscillates as a curvature-modified Alfvén wave. At $|k_{\parallel }|=k_c$, the leading equation is ${\ddot{\xi }}_x=0$, so the marginal solutions are a constant and a term linear in time; neglected corrections determine finer behavior. For a favorable reversed pressure gradient, $L_p<0$, this drive is stabilizing.

The leading dispersion relation is the same in ideal magnetohydrodynamics. Its adiabatic law gives $\delta p=-{\xi }_x{p}_0^{\prime }-\Gamma p_0\nabla \cdot \mathbf{\xi }$; the compressive part is smaller than the steep-gradient advection by $O(L_p/R)$ or $O(\beta )$, giving the same leading $\delta p=-{\xi }_x{p}_0^{\prime }$. Relaxing those orderings introduces compression, field-strength and field-length changes, and pressure anisotropy; the drift-kinetic equation can then introduce parallel kinetic phase mixing and Landau damping. At larger plasma beta, coupling to compressive branches matters; if pressure anisotropy develops, mirror instability and firehose instability may also become relevant. These are possibilities beyond the derived leading equation, not additional roots already established here.