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

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

Codex  0 2026-10-25

Choose the drift-kinetic equation in

(r, μ, v_{∥})

with

v_{∥}

the laboratory parallel velocity along the exact

b

. To first order, before discarding terms small in the stated steep-gradient ordering, it is

(- iω + i k_{∥} v_{∥}) δ f_{α} + (u_{x} + v_{∥} δ b_{x}) \partial_{x} f_{0 α} + [\frac{q _{α}}{m _{α}} δ E_{∥} - μ (i k_{∥} δ B + δ b_{x} B_{0}^{'}) + v_{∥} u_{⊥} \cdot κ_{0}] \partial_{v_{∥}} f_{0 α} = 0, κ_{0} = - \frac{x ^}{R} .

The 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 ≪ δ p / p_{0}

and

L_{p} / L_{B}, L_{p} / R ≪ 1

. The field-direction perturbation multiplying the steep equilibrium gradient must remain:

δ (b \cdot \nabla f_{0 α}) = δ b_{x} \partial_{x} f_{0 α} .

Using

u_{x} = - iω ξ_{x}

and

δ b_{x} = i k_{∥} ξ_{x}

therefore gives the leading equation

(- iω + i k_{∥} v_{∥}) (δ f_{α} + ξ_{x} \partial_{x} f_{0 α}) + \frac{q _{α}}{m _{α}} δ E_{∥} \partial_{v_{∥}} f_{0 α} = 0.

The displaced branch satisfying quasineutrality has

δ E_{∥} = 0

to this order: its species number densities are

δ n_{α} = - ξ_{x} n_{0 α}^{'}

, so

\sum_{α} q_{α} δ n_{α} = - ξ_{x} \partial_{x} \sum_{α} q_{α} n_{0 α} = 0

. More explicitly, putting

D = - iω + i k_{∥} v_{∥}

gives

δ f_{α} = - ξ_{x} f_{0 α}^{'} - \frac{i q _{α} δ E _{∥}}{m _{α}} \frac{\partial _{v_{∥}} f _{0 α}}{ω - k _{∥} v _{∥}}, 0 = - i δ E_{∥} \sum_{α} \frac{q _{α}^{2}}{m _{α}} \int \frac{\partial _{v_{∥}} f _{0 α}}{ω - k _{∥} v _{∥}} d^{3} v .

Away from an independent longitudinal eigenfrequency, the summed susceptibility is nonzero, giving

δ E_{∥} = 0

. An independent longitudinal eigenmode is not forced by the interchange displacement. The required solution is thus

δ f_{α} = - ξ_{x} \partial_{x} f_{0 α} .

The apparent streaming denominator cancels identically, including at

ω = k_{∥} v_{∥}

; this response has no leading Landau damping singularity.

To see the temperature and density contributions explicitly, differentiate at fixed

μ, v_{∥}

and neglect

B_{0}^{'}

at this order:

\partial_{x} f_{0 α} = f_{0 α} [\frac{n _{0 α}^{'}}{n _{0 α}} + (\frac{m _{α} ( v _{∥}^{2} + 2 μ B _{0} )}{2 T _{0 α}} - \frac{3}{2}) \frac{T _{0 α}^{'}}{T _{0 α}}] .

The gyrotropic velocity-space measure is

d^{3} v = 2 π B_{0} d μ d v_{∥}

. The species perpendicular plasma pressure and parallel plasma pressure are the moments

p_{⊥ α} = m_{α} \int μ B f_{α} d^{3} v, p_{∥ α} = m_{α} \int v_{∥}^{2} f_{α} d^{3} v .

Explicit perturbations of the measure and of

μ B

produce terms proportional to

h

, which are subleading in these pressure moments; derivatives of the equilibrium measure produce the discarded

B_{0}^{'}

terms. The remaining Maxwell-Boltzmann distribution integrals give

δ p_{⊥ α} = δ p_{∥ α} = - ξ_{x} p_{0 α}^{'}, δ p_{⊥} = δ p_{∥} = \frac{p _{0}}{L _{p}} ξ_{x} .

Insertion into b proves

ω^{2} = k_{∥}^{2} v_{A}^{2} - \frac{2 p _{0}}{ρ _{0} R L _{p}} .

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