Shear Dynamo Research Articles

Compressible Test-field Method and Its Application to Shear Dynamos

In this study, we present a compressible test-field method (CTFM) for computing α-effect and turbulent magnetic diffusivity tensors, as well as those relevant for the mean ponderomotive force and mass source, applied to the full MHD equations. We describe the theoretical background of the method and compare it to the quasi-kinematic test-field method and to the previously studied variant working in simplified MHD (SMHD). We present several test cases using velocity and magnetic fields of the Roberts geometry and also compare with the imposed-field method. We show that, for moderate imposed-field strengths, the nonlinear CTFM (nCTFM) gives results in agreement with the imposed-field method. A comparison of different flavors of the nCTFM in the shear dynamo case also yields agreement up to equipartition field strengths. Some deviations between the CTFM and SMHD variants exist. As a relevant physical application, we study nonhelically forced shear flows, which exhibit large-scale dynamo action, and present a reanalysis of low-Reynolds-number, moderate shear systems, where we previously ignored the pressure gradient in the momentum equation and found no coherent shear-current effect. Another key difference is that in the earlier study we used magnetic forcing to mimic small-scale dynamo action, while here it is self-consistently driven by purely kinetic forcing. The kinematic CTFM with general validity forms the core of our analysis. We still find no coherent shear-current effect, but do recover strong large-scale dynamo action that, according to our analysis, is driven by incoherent effects.

Quasi-cyclic behaviour in non-linear simulations of the shear dynamo

This work was supported by the Science and Technology Facilities Council, grant ST/L000636/1.

Destruction of large-scale magnetic field in non-linear simulations of the shear dynamo

The Sun's magnetic field exhibits coherence in space and time on much larger scales than the turbulent convection that ultimately powers the dynamo. In the past the α-effect (mean-field) concept has been used to model the solar cycle, but recent work has cast doubt on the validity of the mean-field ansatz under solar conditions. This indicates that one should seek an alternative mechanism for generating large-scale structure. One possibility is the recently proposed ‘shear dynamo’ mechanism where large-scale magnetic fields are generated in the presence of a simple shear. Further investigation of this proposition is required, however, because work has been focused on the linear regime with a uniform shear profile thus far. In this paper we report results of the extension of the original shear dynamo model into the non-linear regime. We find that whilst large-scale structure can initially persist into the saturated regime, in several of our simulations it is destroyed via large increase in kinetic energy. This result casts doubt on the ability of the simple uniform shear dynamo mechanism to act as an alternative to the α-effect in solar conditions.

COHERENT NONHELICAL SHEAR DYNAMOS DRIVEN BY MAGNETIC FLUCTUATIONS AT LOW REYNOLDS NUMBERS

Nonhelical shear dynamos are studied with a particular focus on the possibility of coherent dynamo action. The primary results -- serving as a follow up to the results of Squire & Bhattacharjee [arXiv:1506.04109 (2015)] -- pertain to the "magnetic shear-current effect" as a viable mechanism to drive large-scale magnetic field generation. This effect raises the interesting possibility that the saturated state of the small-scale dynamo could drive large-scale dynamo action, and is likely to be important in the unstratified regions of accretion disk turbulence. In this paper, the effect is studied at low Reynolds numbers, removing the complications of small-scale dynamo excitation and aiding analysis by enabling the use of quasi-linear statistical simulation methods. In addition to the magnetically driven dynamo, new results on the kinematic nonhelical shear dynamo are presented. These illustrate the relationship between coherent and incoherent driving in such dynamos, demonstrating the importance of rotation in determining the relative dominance of each mechanism.

Large-scale dynamo action due to α fluctuations in a linear shear flow

We present a model of large-scale dynamo action in a shear flow that has stochastic, zero-mean fluctuations of the $\alpha$ parameter. This is based on a minimal extension of the Kraichnan-Moffatt model, to include a background linear shear and Galilean-invariant $\alpha$-statistics. Using the first order smoothing approximation we derive a linear integro-differential equation for the large-scale magnetic field, which is non perturbative in the shearing rate $S\,$, and the $\alpha$-correlation time $\tau_\alpha\,$. The white-noise case, $\tau_\alpha = 0\,$, is solved exactly, and it is concluded that the necessary condition for dynamo action is identical to the Kraichnan-Moffatt model without shear; this is because white-noise does not allow for memory effects, whereas shear needs time to act. To explore memory effects we reduce the integro-differential equation to a partial differential equation, valid for slowly varying fields when $\tau_\alpha$ is small but non zero. Seeking exponential modal solutions, we solve the modal dispersion relation and obtain an explicit expression for the growth rate as a function of the six independent parameters of the problem. A non zero $\tau_\alpha$ gives rise to new physical scales, and dynamo action is completely different from the white-noise case; e.g. even weak $\alpha$ fluctuations can give rise to a dynamo. We argue that, at any wavenumber, both Moffatt drift and Shear always contribute to increasing the growth rate. Two examples are presented: (a) a Moffatt drift dynamo in the absence of shear; (b) a Shear dynamo in the absence of Moffatt drift.

The radial gradient of the near-surface shear layer of the Sun

Helioseismology has provided unprecedented information about the internal rotation of the Sun. One of the important achievements was the discovery of two radial shear layers: one near the bottom of the convection zone (the tachocline) and one near the surface. These shear layers may be important ingredients for explaining the magnetic cycle of the Sun. We measure the logarithmic radial gradient of the rotation rate ($\rm d\ln\Omega/\rm d\ln r$) near the surface of the Sun using 15 years of f mode rotational frequency splittings from the Michelson Doppler Imager (MDI) and four years of data from the Helioseismic and Magnetic Imager (HMI). We model the angular velocity of the Sun in the upper $\sim 10$ Mm as changing linearly with depth and use a multiplicative optimally localized averaging inversion to infer the gradient of the rotation rate as a function of latitude. Both the MDI and HMI data show that $\rm d\ln\Omega/\rm d\ln r$ is close to $-1$ from the equator to 60$^{\circ}$ latitude and stays negative up to 75$^{\circ}$ latitude. However, the value of the gradient is different for MDI and HMI for latitudes above $60^{\circ}$. Additionally, there is a significant difference between the value of $d\ln\Omega/d\ln r$ using an older and recently reprocessed MDI data for latitudes above $30^\circ$. We could reliably infer the value of $\rm d\ln\Omega/\rm d\ln r$ up to 60$^{\circ}$, but not above this latitude, which will hopefully constrain theories of the near-surface shear layer and dynamo. Furthermore, the recently reprocessed MDI splitting data are more reliable than the older versions which contained clear systematic errors in the high degree f modes.

The elemental shear dynamo

AbstractA quasi-linear theory is presented for how randomly forced, barotropic velocity fluctuations cause an exponentially growing, large-scale (mean) magnetic dynamo in the presence of a uniform parallel shear flow. It is a ‘kinematic’ theory for the growth of the mean magnetic energy from a small initial seed, neglecting the saturation effects of the Lorentz force. The quasi-linear approximation is most broadly justifiable by its correspondence with computational solutions of nonlinear magnetohydrodynamics, and it is rigorously derived in the limit of small magnetic Reynolds number, ${\mathit{Re}}_{\eta } \ll 1$. Dynamo action occurs even without mean helicity in the forcing or flow, but random helicity variance is then essential. In a sufficiently large domain and with a small seed wavenumber in the direction perpendicular to the mean shearing plane, a positive exponential growth rate $\gamma $ can occur for arbitrary values of ${\mathit{Re}}_{\eta } $, viscous Reynolds number ${\mathit{Re}}_{\nu } $, and random-force correlation time ${t}_{f} $ and orientation angle ${\theta }_{f} $ in the shearing plane. The value of $\gamma $ is independent of the domain size. The shear dynamo is ‘fast’, with finite $\gamma \gt 0$ in the limit of ${\mathit{Re}}_{\eta } \gg 1$. Averaged over random realizations of the forcing history, the ensemble-mean magnetic field grows more slowly, if at all, compared to the r.m.s. field (magnetic energy). In the limit of small ${\mathit{Re}}_{\eta } $ and ${\mathit{Re}}_{\nu } $, the dynamo behaviour is related to the well-known alpha–omega ansatz when the force is slowly varying ($\gamma {t}_{f} \gg 1$) and to the ‘incoherent’ alpha–omega ansatz when the force is more rapidly fluctuating.

Large-Scale Magnetic Field Generation by Randomly Forced Shearing Waves

A rigorous theory for the generation of a large-scale magnetic field by random nonhelically forced motions of a conducting fluid combined with a linear shear is presented in the analytically tractable limit of low magnetic Reynolds number (Rm) and weak shear. The dynamo is kinematic and due to fluctuations in the net (volume-averaged) electromotive force. This is a minimal proof-of-concept quasilinear calculation aiming to put the shear dynamo, a new effect recently found in numerical experiments, on a firm theoretical footing. Numerically observed scalings of the wave number and growth rate of the fastest-growing mode, previously not understood, are derived analytically. The simplicity of the model suggests that shear dynamo action may be a generic property of sheared magnetohydrodynamic turbulence.

Numerical experiments on dynamo action in sheared and rotating turbulence

AbstractNumerical simulations of forced turbulence in elongated shearing boxes are carried out to demonstrate that a nonhelical turbulence in conjunction with a linear shear can give rise to a mean‐field dynamo. Exponential growth of magnetic field at scales larger than the outer (forcing) scale of the turbulence is found. Over a range of values of the shearing rate S spanning approximately two orders of magnitude, the growth rate of the magnetic field is proportional to the imposed shear, γ ∝ S, while the characteristic spatial scale of the field is lB ∝ S–1/2. The effect is quite general: earlier results for the nonrotating case by Yousef et al. (2008) are extended to shearing boxes with Keplerian rotation; it is also shown that the shear dynamo mechanism operates both below and above the threshold for the fluctuation dynamo. The apparently generic nature of the shear dynamo effect makes it an attractive object of study for the purpose of understanding the generation of magnetic fields in astrophysical systems. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Generation of Magnetic Field by Combined Action of Turbulence and Shear

The feasibility of a mean-field dynamo in nonhelical turbulence with a superimposed linear shear is studied numerically in elongated shearing boxes. Exponential growth of the magnetic field at scales much larger than the outer scale of the turbulence is found. The characteristic scale of the field is lB proportional S(-1/2) and the growth rate is gamma proportional S, where S is the shearing rate. This newly discovered shear dynamo effect potentially represents a very generic mechanism for generating large-scale magnetic fields in a broad class of astrophysical systems with spatially coherent mean flows.

Magnetic Diffusivity Tensor and Dynamo Effects in Rotating and Shearing Turbulence

The turbulent magnetic diffusivity tensor is determined in the presence of rotation or shear. The question is addressed whether dynamo action from the shear-current effect can explain large-scale magnetic field generation found in simulations with shear. For this purpose a set of evolution equations for the response to imposed test fields is solved with turbulent and mean motions calculated from the momentum and continuity equations. The corresponding results for the electromotive force are used to calculate turbulent transport coefficients. The diagonal components of the turbulent magnetic diffusivity tensor are found to be very close together, but their values increase slightly with increasing shear and decrease with increasing rotation rate. In the presence of shear, the sign of the two off-diagonal components of the turbulent magnetic diffusion tensor is the same and opposite to the sign of the shear. This implies that dynamo action from the shear--current effect is impossible, except perhaps for high magnetic Reynolds numbers. However, even though there is no alpha effect on the average, the components of the alpha tensor display Gaussian fluctuations around zero. These fluctuations are strong enough to drive an incoherent alpha--shear dynamo. The incoherent shear--current effect, on the other hand, is found to be subdominant.