Pressure anisotropy ($p_{\perp }-p_{\parallel }$)

Directional dependence of particle pressure. In a gyrotropic plasma, only parallel plasma pressure and perpendicular plasma pressure are independent. Their difference changes the effective magnetic tension and can drive firehose instability or mirror instability.

In Gaussian units, the gyrotropic pressure tensor adds directional stress to the magnetic tension. A transverse displacement of a uniform magnetic field obeys $\rho {\partial }_t^2{\mathbf{\xi }}_{\perp }=\mathcal{T}{\partial }_{\parallel }^2{\mathbf{\xi }}_{\perp }$. Positive $\mathcal{T}$ gives a restoring force; negative $\mathcal{T}$ gives the fluid firehose instability. This follows by combining $-\nabla \cdot \mathsf{P}$ with the divergence of the Maxwell stress tensor. In SI replace $B^2/(4\pi )$ by $B^2/{\mu }_0$.

A compressive kinetic plasma instability generally driven by sufficiently large perpendicular plasma pressure relative to parallel plasma pressure. Perturbed field-strength minima trap particles and alter perpendicular pressure-balanced plasma perturbation. Its threshold depends on species distributions and plasma beta; isotropic equilibrium alone does not imply this instability.

Parallel particle pressure can overcome magnetic tension. In the long-wavelength gyrotropic plasma limit, the Alfvén wave has ${\omega }^2=k_{\parallel }^2[B^2/(4\pi )+p_{\perp }-p_{\parallel }]/\rho$, so the condition in the title makes it unstable. The detailed kinetic threshold at shorter scales requires additional physics.