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Advanced fluid dynamics course of the University of Oxford

Past exam of the advanced fluid dynamics course of the University of Oxford

2019

Take the <trace> of the statistically isotropic <magnetic second-moment tensor>. There are three spatial directions, so
$$
\langle B^2\rangle
=\langle B_iB_i\rangle
=v_A^2\delta_{ii}=3v_A^2.
$$
Choosing the nonnegative <speed>,
$$
\boxed{v_A=\sqrt{\frac{\langle B^2\rangle}{3}}
=\sqrt{\frac{\langle B_{\rm phys}^2\rangle}{3\mu_0\rho_0}}.}
$$
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.

Solution

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

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

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

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