= Solution
Assume the stated outward-decreasing <pressure>, $L_p>0$, and set
$$
\omega_I^2=\frac{2p_0}{\rho_0RL_p}=\frac{\beta v_A^2}{RL_p},\qquad
\boxed{k_c=\frac{\omega_I}{v_A}=\sqrt{\frac\beta{RL_p}}.}
$$
For $|k_\parallel|<k_c$, the <interchange instability> has
$$
\omega=\pm i\gamma,\qquad\gamma=\sqrt{\omega_I^2-k_\parallel^2v_A^2},\qquad
\xi_x=C_+e^{\gamma t}+C_-e^{-\gamma t}.
$$
An 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 $k_\parallel^2v_A^2$ 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 $\omega=\pm\sqrt{k_\parallel^2v_A^2-\omega_I^2}$: 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 $\delta p=-\xi_xp_0'-\Gamma p_0\nabla\cdot\boldsymbol\xi$; the compressive part is smaller than the steep-gradient advection by $O(L_p/R)$ or $O(\beta)$, giving the same leading $\delta p=-\xi_xp_0'$. 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.
\Image[/past-exam-of-the-collisionless-plasma-physics-course-of-the-university-of-oxford/original2025-interchange.png]
{title=<Magnetic tension> stabilizes the local <interchange instability> above its critical parallel <wavenumber>}
Back to article page
