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In Gaussian units, the equation in the title uses the gyrotropic pressure tensor $\mathbf{P}=p_{\perp }\mathbf{I}+(p_{\parallel }-p_{\perp })\mathbf{bb}$. This allows pressure anisotropy. Its magnetic force contains both magnetic pressure and magnetic tension.

Use the source's Gaussian units and neglect the displacement current, as in nonrelativistic magnetohydrodynamics. Write $\Delta p=p_{\perp }-p_{\parallel }$ for the pressure anisotropy. The pressure tensor is $\mathsf{P}=p_{\perp }\mathsf{I}-\Delta p\mathbf{bb}$, so its force density isAmpère's law and the solenoidal magnetic-field constraint give the Lorentz force decompositionAdding these two force densities and using $\mathbf{BB}=B^2\mathbf{bb}$ yieldsThe first term is the gradient of the modified magnetohydrodynamic total pressure; the second is the anisotropic Maxwell stress tensor contribution together with the directional pressure force. Their effective coefficient is the pressure-anisotropy-modified magnetic tension, $\mathcal{T}=B^2/(4\pi )+\Delta p$. At fixed magnetic field, positive pressure anisotropy increases resistance to bending, while negative pressure anisotropy reduces it. When $\mathcal{T}$ becomes negative, bending experiences an amplifying force rather than a restoring force. Gradients of $\Delta p$ can also drive motion, so the curvature interpretation refers specifically to the bending contribution.

For $k_{\parallel }\ne 0$ the Alfvén wave becomes a firehose instability when its squared angular frequency is negative:Equality is marginal, rather than exponential growth. The pressure anisotropy must have negative sign; its absolute magnitude alone is insufficient.

For the intergalactic plasma, a negative relative pressure anisotropy of $-10^{-2}$ reaches this fluid threshold only for $\beta >200$. In particular, the literal pair $\beta =100$ and relative pressure anisotropy $-0.01$ is stable because the threshold is $-0.02$. Since the quoted numbers are approximate and allow larger plasma beta, this weakly magnetized plasma is susceptible, rather than necessarily unstable at every quoted value. For example, $\beta =300$ and relative pressure anisotropy $-0.01$ are unstable. Positive pressure anisotropy does not excite this branch.

For the fusion-device plasma, $\beta \sim 10^{-2}$ requires a negative relative pressure anisotropy below roughly $-200$, whereas its magnitude is only about $0.1$. The stabilizing magnetic tension dominates, so it is not susceptible to this fluid firehose instability under the stated conditions. Indeed, nonnegative perpendicular pressure requires the relative pressure anisotropy to be no smaller than $-1$; at such a small plasma beta even this extreme cannot satisfy the threshold.

For a static equilibrium, linearization of magnetic induction givesThe rotating local basis must be differentiated. In particular, ${\partial }_{\parallel }\hat{\mathbf{x}}={\mathbf{b}}_0/R$ and ${\partial }_{\parallel }{\mathbf{b}}_0=-\hat{\mathbf{x}}/R$. Their contributions proportional to ${\xi }_{\parallel }$ cancel in the radial component, yieldingWriting $h=\delta B/B_0$, the magnitude component iswhere $\nabla \cdot \mathbf{\xi }={\xi }_x/R+i{k}_y{\xi }_y+i{k}_{\parallel }{\xi }_{\parallel }$ when ${\partial }_x{\xi }_x=0$.

Set $M=B_0^2/(4\pi )={\rho }_0{v}_A^2$ and $\delta \Pi =\delta p_{\perp }+Mh$. The transverse $y$ component of the linear kinetic magnetohydrodynamic momentum equation isSince $k_y\gg k_{\parallel }\sim \omega /v_A$, perpendicular magnetic pressure enforces the leading pressure-balanced plasma perturbation condition $\delta \Pi =0$, givingIn the radial component, the curvature part of the perturbed magnetic force is $-2Mh/R$: one contribution comes from $({\mathbf{B}}_0\cdot \nabla )\delta \mathbf{B}$ and one from $(\delta \mathbf{B}\cdot \nabla ){\mathbf{B}}_0$. The perturbed pressure anisotropy contributes $-(\delta p_{\perp }-\delta p_{\parallel })/R$. After the leading total-pressure constraint has removed ${\partial }_x\delta \Pi$, the radial equation isSubstituting $Mh=-\delta p_{\perp }$ givesAlthough $h$ is small at low plasma beta, its contribution $Mh$ to the force balance is of the same order as $\delta p_{\perp }$ and must be kept.

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.