Past exam of the collisionless plasma physics course of the University of Oxford / 2025 / 3

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.

Both mechanisms can extract free energy from an equilibrium pressure gradient in unfavorable magnetic field-line curvature, but the dynamics retained here is the macroscopic electromagnetic displacement of a flux tube. It depends on the total pressure gradient, including electron and ion contributions, and it does not require an ion temperature gradient specifically. Its stable branch has $\omega \sim k_{\parallel }{v}_A$; with low plasma beta, this is much faster than ion parallel streaming at the same $k_{\parallel }$.

The familiar ion-temperature-gradient instability is a slower drift-scale instability. In its usual electrostatic ordering the Boltzmann electron response is established by rapid electron parallel motion, while ion magnetic drifts, ion temperature evolution and polarization drift provide the instability dynamics. A representative ordering isThe last two expressions specify scales rather than exact geometric factors; $L_{T_i}^{-1}=-d\ln T_{0i}/dx$. The present leading kinetic magnetohydrodynamics system drops these relative species drifts and their finite-gyroradius corrections. It also retains the perturbed field direction, which enforces the advected-distribution cancellation in c; a rapid electron streaming rate alone cannot turn that cancellation into a Boltzmann electron response on an unperturbed field.

To recover ion-temperature-gradient instability physics, take the slow, nearly electrostatic limit $|\omega |/(k_{\parallel }{v}_A)\ll 1$, retain magnetic curvature drift, gradient-B drift and polarization drift in low-flow drift kinetics or gyrokinetics, and allow a sufficiently strong destabilizing ion temperature gradient. At long perpendicular wavelengths one may still have $k_{\perp }{\rho }_i\ll 1$, provided the leading nonzero ion polarization is kept; at $k_{\perp }{\rho }_i=O(1)$ full finite Larmor radius effects are needed. There is no contradiction between a drift-scale ion-temperature-gradient instability and the macroscopic interchange instability derived here: they use different asymptotic balances.