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In the steep-gradient, low-plasma beta limit of kinetic magnetohydrodynamics, magnetic induction gives $\delta b_x=i{k}_{\parallel }{\xi }_x$ and $u_x=-i\omega {\xi }_x$. Therefore the forcing of the drift-kinetic equation is $(u_x+v_{\parallel }\delta b_x)f_0^{\prime }=(-i\omega +i{k}_{\parallel }{v}_{\parallel }){\xi }_x{f}_0^{\prime }$. It cancels the streaming operator on $\delta f=-{\xi }_x{f}_0^{\prime }$, leaving no leading kinetic phase mixing on this displacement branch. Freezing the parallel derivative to the equilibrium direction would lose this cancellation.

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.

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.