Parallel particle pressure can overcome magnetic tension. In the long-wavelength gyrotropic plasma limit, the Alfvén wave has , so the condition in the title makes it unstable. The detailed kinetic threshold at shorter scales requires additional physics.
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Firehose instability is a phenomenon that occurs in plasma physics, particularly in the context of magnetized plasmas, where the particles in the plasma can become unstable under certain conditions. This instability is named after the analogy of a fire hose, which can become unstable and whip around if water is flowing through it at a certain pressure.