 Solution

ID: past-exam-of-the-advanced-fluid-dynamics-course-of-the-university-of-oxford/2019/1/d/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.

 New to topics? Read the docs here!