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A low-frequency kinetic reduction averaging over fast cyclotron motion while allowing perpendicular wavelengths comparable with the gyroradius. It retains finite Larmor radius effects absent from the leading long-wavelength drift-kinetic equation.

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.

A magnetized long-wavelength plasma description with a common leading perpendicular flow, magnetic flux freezing and a gyrotropic pressure tensor. The kinetic magnetohydrodynamic momentum equation couples to a drift-kinetic equation for each species; parallel plasma pressure and perpendicular plasma pressure are calculated as velocity moments, rather than prescribed by an isotropic adiabatic closure. Relative species drifts enter at higher order in the ratio of gyroradius to macroscopic length.

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.

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

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.