On the shear-current effect: toward understanding why theories and simulations have mutually and separately conflicted. (arXiv:2104.11112v2 [physics.flu-dyn] UPDATED)

<a href="http://arxiv.org/find/physics/1/au:+Zhou_H/0/1/0/all/0/1">Hongzhe Zhou</a>, <a href="http://arxiv.org/find/physics/1/au:+Blackman_E/0/1/0/all/0/1">Eric G. Blackman</a>

The shear-current effect (SCE) of mean-field dynamo theory refers to the

combination of a shear flow and a turbulent coefficient $beta_{21}$ with a

favorable negative sign for exponential mean-field growth, rather than positive

for diffusion. There have been long standing disagreements among theoretical

calculations and comparisons of theory with numerical experiments as to the

sign of kinetic ($beta^u_{21}$) and magnetic ($beta^b_{21}$) contributions.

To resolve these discrepancies, we combine an analytical approach with

simulations, and show that unlike $beta^b_{21}$, the kinetic SCE

$beta^u_{21}$ has a strong dependence on the kinetic energy spectral index and

can transit from positive to negative values at $mathcal{O}(10)$ Reynolds

numbers if the spectrum is not too steep. Conversely, $beta^b_{21}$ is always

negative regardless of the spectral index and Reynolds numbers. For very steep

energy spectra, the positive $beta^u_{21}$ can dominate even at energy

equipartition $u_text{rms}simeq b_text{rms}$, resulting in a positive total

$beta_{21}$ even though $beta^b_{21}<0$. Our findings bridge the gap between

the seemingly contradictory results from the second-order-correlation

approximation (SOCA) versus the spectral-$tau$ closure (STC), for which

opposite signs for $beta^u_{21}$ have been reported, with the same sign for

$beta^b_{21}<0$. The results also offer an explanation for the simulations

that find $beta^u_{21}>0$ and an inconclusive overall sign of $beta_{21}$ for

$mathcal{O}(10)$ Reynolds numbers. The transient behavior of $beta^u_{21}$ is

demonstrated using the kinematic test-field method. We compute dynamo growth

rates for cases with or without rotation, and discuss opportunities for further

work.

The shear-current effect (SCE) of mean-field dynamo theory refers to the

combination of a shear flow and a turbulent coefficient $beta_{21}$ with a

favorable negative sign for exponential mean-field growth, rather than positive

for diffusion. There have been long standing disagreements among theoretical

calculations and comparisons of theory with numerical experiments as to the

sign of kinetic ($beta^u_{21}$) and magnetic ($beta^b_{21}$) contributions.

To resolve these discrepancies, we combine an analytical approach with

simulations, and show that unlike $beta^b_{21}$, the kinetic SCE

$beta^u_{21}$ has a strong dependence on the kinetic energy spectral index and

can transit from positive to negative values at $mathcal{O}(10)$ Reynolds

numbers if the spectrum is not too steep. Conversely, $beta^b_{21}$ is always

negative regardless of the spectral index and Reynolds numbers. For very steep

energy spectra, the positive $beta^u_{21}$ can dominate even at energy

equipartition $u_text{rms}simeq b_text{rms}$, resulting in a positive total

$beta_{21}$ even though $beta^b_{21}<0$. Our findings bridge the gap between

the seemingly contradictory results from the second-order-correlation

approximation (SOCA) versus the spectral-$tau$ closure (STC), for which

opposite signs for $beta^u_{21}$ have been reported, with the same sign for

$beta^b_{21}<0$. The results also offer an explanation for the simulations

that find $beta^u_{21}>0$ and an inconclusive overall sign of $beta_{21}$ for

$mathcal{O}(10)$ Reynolds numbers. The transient behavior of $beta^u_{21}$ is

demonstrated using the kinematic test-field method. We compute dynamo growth

rates for cases with or without rotation, and discuss opportunities for further

work.

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