Past exam of the advanced fluid dynamics course of the University of Oxford / 2019 / 1 / d

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Solution

Take the trace of the statistically isotropic magnetic second-moment tensor. There are three spatial directions, so

⟨ B^{2} ⟩ = ⟨ B_{i} B_{i} ⟩ = v_{A}^{2} δ_{ii} = 3 v_{A}^{2} .

Choosing the nonnegative speed,

v_{A} = \frac{⟨ B ^{2} ⟩}{3} = \frac{⟨ B _{phys}^{2} ⟩}{3 μ _{0} ρ _{0}} .

Thus

v_{A}

is the root-mean-square value of any one Cartesian magnetic field component in Alfvén velocity units. It is not the magnitude of the mean Alfvén velocity, which is zero here. Statistical isotropy fixes a second moment even when the first moment vanishes.

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