Phase mixing in magnetic flux expulsion ($t_{\exp }\sim (L/U){\mathrm{Rm}}^{1/3}$)

Differential circulation of a weak magnetic field winds its flux function into progressively smaller radial scales. In a smooth eddy with angular-velocity gradient of order $U/L^2$, its radial wavenumber grows as $k_r\sim Ut/L^2$. Resistive damping becomes substantial when $\eta {\int }_0^t{k}_r^{2}dt\sim 1$, giving $t\sim (L^4/(\eta U^2){)}^{1/3}=(L/U){\mathrm{Rm}}^{1/3}$. This estimate presumes differential rotation and negligible magnetic feedback; a rigidly rotating eddy does not create this phase mixing. Finite magnetic diffusivity then permits weak interior field and boundary concentration, unlike exact ideal flux freezing.

In a smooth interior region with locally constant shear ${\Omega }^{\prime }$, a wound azimuthal Fourier mode has radial wavenumber $k_s\simeq -m{\Omega }^{\prime }t$. Integrating its resistive damping rate $\eta k_s^2$ gives the exponential decay in the title. The characteristic time is ${\tau }_c=[3/(\eta m^2{\Omega }^{\prime 2}){]}^{1/3}$. For $m=1$ and ${\Omega }^{\prime }=-{\Omega }_0/d$, ${\Omega }_0{\tau }_c=(3\mathrm{Rm}{)}^{1/3}$, where $\mathrm{Rm}={\Omega }_0{d}^2/\eta$. This is a high-magnetic Reynolds number interior transient on a time long compared with the rotation time but short compared with $d^2/\eta$. Physical magnetic field components may have additional algebraic factors from taking derivatives of the wound magnetic vector potential. A maintained exterior field, an axial core and a circulation-edge boundary layer prevent interpreting this estimate as an exact, uniform, infinite-time solution.