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Theory of critical balance in plasma turbulence

Critical balance, a statement of the approximate equality of a timescale of motion parallel to the magnetic field with a timescale of turbulent motion perpendicular to the field, is analyzed from a statistical description of the relative motion of two particles consistent with two-point correlation in a nonlinear gyrokinetic system with collisions. Critical balance is understood from a premise that the fastest time to any spatial scale that encounters decorrelated motions sets an overall correlation time for both perpendicular and parallel scales. For a regime of weak collisionality, this leads to the approximate equality of the parallel streaming time and the perpendicular turbulent correlation time, which is a standard statement of critical balance. These two timescales are only approximately equal because of effects in parallel streaming dynamics and collisions, the former producing deviations tied to variations of perpendicular and parallel wavenumbers. Analysis of a collisional regime shows that critical balance also applies in that limit with a modified relation between parallel and perpendicular spatial scales.