 Solution

ID: past-exam-of-the-collisionless-plasma-physics-course-of-the-university-of-oxford/2025/3/f/solution

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

ω \sim k_{∥} v_{A}

; with low plasma beta, this is much faster than ion parallel streaming at the same

k_{∥}

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 is

k_{∥} v_{th, i} ≲ ∣ ω ∣ ≪ k_{∥} v_{th, e}, ∣ ω ∣ ≪ k_{∥} v_{A}, ω_{* T_{i}} \sim \frac{k _{y} v _{th, i}^{2}}{Ω _{i} L _{T_{i}}}, ω_{d i} \sim \frac{k _{y} v _{th, i}^{2}}{Ω _{i} R} .

The last two expressions specify scales rather than exact geometric factors;

L_{T_{i}}^{- 1} = - d ln T_{0 i} / d x

. 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

∣ ω ∣/ (k_{∥} v_{A}) ≪ 1

k_{⊥} ρ_{i} ≪ 1

, provided the leading nonzero ion polarization is kept; at

k_{⊥} ρ_{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.

 New to topics? Read the docs here!