Scale Separation Ratios Research Articles

Large-scale circulations in a shear-free convective turbulence: Mean-field simulations

It has been previously shown [Elperin et al., “Formation of large-scale semi-organized structures in turbulent convection,” Phys. Rev. E 66, 066305 (2002)] that a non-rotating turbulent convection with nonuniform large-scale flows contributes to the turbulent heat flux. As a result, the turbulent heat flux depends explicitly not only on the gradients of the large-scale temperature, but also on the gradients of the large-scale velocity. This is because the nonuniform large-scale flows produce anisotropic velocity fluctuations, which modify the turbulent heat flux. This effect causes an excitation of a convective-wind instability and formation of large-scale semi-organized coherent structures (large-scale convective cells). In the present study, we perform mean-field numerical simulations of shear-free convection, which take into account the modification of the turbulent heat flux by nonuniform large-scale flows. We use periodic boundary conditions in horizontal direction as well as stress-free or no-slip boundary conditions in vertical direction. We show that the redistribution of the turbulent heat flux by the nonuniform large-scale motions in turbulent convection plays a crucial role in the formation of the large-scale semi-organized coherent structures. In particular, this effect results in a strong reduction of the critical effective Rayleigh number (based on the eddy viscosity and turbulent temperature diffusivity) required for the formation of the large-scale convective cells. We demonstrate that the convective-wind instability is excited when the scale separation ratio between the height of the convective layer and the integral turbulence scale is large. The level of the mean kinetic energy at saturation increases with the scale separation ratio. We also show that inside the large-scale convective cells, there are local regions with the positive vertical gradient of the potential temperature, which implies that these regions are stably stratified.

Turbulent viscosity and magnetic Prandtl number from simulations of isotropically forced turbulence

Context.Turbulent diffusion of large-scale flows and magnetic fields plays a major role in many astrophysical systems, such as stellar convection zones and accretion discs.Aims.Our goal is to compute turbulent viscosity and magnetic diffusivity which are relevant for diffusing large-scale flows and magnetic fields, respectively. We also aim to compute their ratio, which is the turbulent magnetic Prandtl number, Pmt, for isotropically forced homogeneous turbulence.Methods.We used simulations of forced turbulence in fully periodic cubes composed of isothermal gas with an imposed large-scale sinusoidal shear flow. Turbulent viscosity was computed either from the resulting Reynolds stress or from the decay rate of the large-scale flow. Turbulent magnetic diffusivity was computed using the test-field method for a microphysical magnetic Prandtl number of unity. The scale dependence of the coefficients was studied by varying the wavenumber of the imposed sinusoidal shear and test fields.Results.We find that turbulent viscosity and magnetic diffusivity are in general of the same order of magnitude. Furthermore, the turbulent viscosity depends on the fluid Reynolds number (Re) and scale separation ratio of turbulence. The scale dependence of the turbulent viscosity is found to be well approximated by a Lorentzian. These results are similar to those obtained earlier for the turbulent magnetic diffusivity. The results for the turbulent transport coefficients appear to converge at sufficiently high values of Re and the scale separation ratio. However, a weak trend is found even at the largest values of Re, suggesting that the turbulence is not in the fully developed regime. The turbulent magnetic Prandtl number converges to a value that is slightly below unity for large Re. For small Re we find values between 0.5 and 0.6 but the data are insufficient to draw conclusions regarding asymptotics. We demonstrate that our results are independent of the correlation time of the forcing function.Conclusions.The turbulent magnetic diffusivity is, in general, consistently higher than the turbulent viscosity, which is in qualitative agreement with analytic theories. However, the actual value of Pmtfound from the simulations (≈0.9−0.95) at large Re and large scale separation ratio is higher than any of the analytic predictions (0.4−0.8).

An experimental evidence of the failure of Cauchy elasticity for the overall modeling of a non-centro-symmetric lattice under static loading

Materials with coarse inner architecture being easily made with modern additive or folding processes, the question of their overall behavior rises. Do they behave like classical elastic continua, or do they exhibit additional higher-order effects ? Further, if present are those effects stable with respect to imperfections (geometry, constitutive material, ...) ? In this view, the current work is an experimental investigation for the need, in static, of a higher-order overall description. It comes from noticing that such behaviors are up to now nearly exclusively studied from a theoretical and numerical point of view. In the present study a non-centro-symmetric sample has been manufactured, based on an industrial honeycomb geometry used for aeronautic/aerospace composite materials. The geometrical anisotropy of the elementary cell and the scale separation ratio have been chosen in order to detect non-classical couplings. Samples are obtained by Fused Deposition Modeling (FDM), one of the most widespread 3D printing techniques. Simple experiments based on load controlled tests with full-field kinematic measurement have been performed. A distributed load control reveals that the overall behavior of the architectured material cannot be described within the realm of Cauchy elasticity.

Turbulent reconnection of magnetic bipoles in stratified turbulence

We consider strongly stratified forced turbulence in a plane-parallel layer with helicity and corresponding large-scale dynamo action in the lower part and non-helical turbulence in the upper. The magnetic field is found to develop strongly concentrated bipolar structures near the surface. They form elongated bands with a sharp interface between opposite polarities. Unlike earlier experiments with imposed magnetic field, the inclusion of rotation does not strongly suppress the formation of these structures. We perform a systematic numerical study of this phenomenon by varying magnetic Reynolds number, scale separation ratio, and Coriolis number. We focus on the formation of a current sheet between bipolar regions where reconnection of oppositely oriented field lines occurs. We determine the reconnection rate by measuring either the inflow velocity in the vicinity of the current sheet or by measuring the electric field in the reconnection region. We demonstrate that for large Lundquist numbers, S>10^3, the reconnection rate is nearly independent of S in agreement with results of recent numerical simulations performed by other groups in simpler settings.

CHARACTERIZING THE MEAN-FIELD DYNAMO IN TURBULENT ACCRETION DISKS

The formation and evolution of a wide class of astrophysical objects is governed by turbulent, magnetized accretion disks. Understanding their secular dynamics is of primary importance. Apart from enabling mass accretion via the transport of angular momentum, the turbulence affects the long-term evolution of the embedded magnetic flux, which in turn regulates the efficiency of the transport. In this paper, we take a comprehensive next step towards an effective mean-field model for turbulent astrophysical disks by systematically studying the key properties of magnetorotational turbulence in vertically-stratified, isothermal shearing boxes. This allows us to infer emergent properties of the ensuing chaotic flow as a function of the shear parameter as well as the amount of net-vertical flux. Using the test-field method, we furthermore characterize the mean-field dynamo coefficients that describe the long-term evolution of large-scale fields. We simultaneously infer the vertical shape and the spectral scale dependence of these closure coefficients, with the latter describing non-local contributions to the turbulent electromotive force. Based on this, we infer a scale-separation ratio of about ten for the large-scale dynamo. We finally determine scaling properties of the mean-field dynamo coefficients. The relevant component of the dynamo {\alpha} effect is found to scale linearly with the shear rate, as is the corresponding turbulent diffusion, {\eta}. Together, these scalings allow us to predict, in a quantitative manner, the cycle period of the well-known butterfly diagram. This lends new support to the importance of the {\alpha}{\Omega} mechanism in determining the evolution of large-scale magnetic fields in turbulent accretion disks.

Range of validity of an extended WKB theory for atmospheric gravity waves: one-dimensional and two-dimensional case

AbstractA computational model of the pseudo-incompressible equations is used to probe the range of validity of an extended Wentzel–Kramers–Brillouin theory (XWKB), previously derived through a distinguished limit of a multiple-scale asymptotic analysis of the Euler or pseudo-incompressible equations of motion, for gravity-wave packets at large amplitudes. The governing parameter of this analysis had been the scale-separation ratio$\varepsilon $between the gravity wave and both the large-scale potential-temperature stratification and the large-scale wave-induced mean flow. A novel feature of the theory had been the non-resonant forcing of higher harmonics of an initial wave packet, predominantly by the large-scale gradients in the gravity-wave fluxes. In the test cases considered a gravity-wave packet is propagating upwards in a uniformly stratified atmosphere. Large-scale winds are induced by the wave packet, and possibly exert a feedback on the latter. In the limit$\varepsilon \ll 1$all predictions of the theory can be validated. The larger$\varepsilon $is the more the transfer of wave energy to the mean flow is underestimated by the theory. The numerical results quantify this behaviour but also show that, qualitatively, XWKB remains valid even when the gravity-wave wavelength approaches the spatial scale of the wave-packet amplitude. This includes the prevalence of first and second harmonics and the smallness of harmonics with wave number higher than two. Furthermore, XWKB predicts for the vertical momentum balance an additional leading-order buoyancy term in Euler and pseudo-incompressible theory, compared with the anelastic theory. Numerical tests show that this term is relatively large with up to$30\hspace{0.167em} \% $of the total balance. The practical relevance of this deviation remains to be assessed in future work.

Spontaneous Formation of Magnetic Flux Concentrations in Stratified Turbulence

The negative effective magnetic pressure instability discovered recently in direct numerical simulations (DNS) may play a crucial role in the formation of sunspots and active regions in the Sun and stars. This instability is caused by a negative contribution of turbulence to the effective mean Lorentz force (the sum of turbulent and non-turbulent contributions) and results in formation of large-scale inhomogeneous magnetic structures from initial uniform magnetic field. Earlier investigations of this instability in DNS of stably stratified, externally forced, isothermal hydromagnetic turbulence in the regime of large plasma beta are now extended into the regime of larger scale separation ratios where the number of turbulent eddies in the computational domain is about 30. Strong spontaneous formation of large-scale magnetic structures is seen even without performing any spatial averaging. These structures encompass many turbulent eddies. The characteristic time of the instability is comparable to the turbulent diffusion time, L^2/eta_t, where eta_t is the turbulent diffusivity and L is the scale of the domain. DNS are used to confirm that the effective magnetic pressure does indeed become negative for magnetic field strengths below the equipartition field. The dependence of the effective magnetic pressure on the field strength is characterized by fit parameters that seem to show convergence for larger values of the magnetic Reynolds number.

Cycles and cycle modulations

AbstractSome selected concepts of the solar activity cycle are reviewed. Cycle modulations through a stochastic α effect are being identified with limited scale separation ratios. Three-dimensional turbulence simulations with helicity and shear are compared at two different scale separation ratios. In both cases the level of fluctuations shows relatively little variation with the dynamo cycle. Prospects for a shallow origin of sunspots are discussed in terms of the negative effective magnetic pressure instability. Tilt angles of bipolar active regions are discussed as a consequence of shear rather than the Coriolis force.

Two distinct mechanisms of coherence in randomly perturbed dynamical systems

We carefully examine two mechanisms--coherence resonance and self-induced stochastic resonance--by which small random perturbations of excitable systems with large time scale separation may lead to the emergence of new coherent behaviors in the form of limit cycles. We analyze what controls the degree of coherence in these two mechanisms and classify their very different properties. In particular we show that coherence resonance arises only at the onset of bifurcation and is rather insensitive against variations in the noise amplitude and the time scale separation ratio. In contrast, self-induced stochastic resonance may arise away from bifurcations and the properties of the limit cycle it induces are controlled by both the noise amplitude and the time scale separation ratio.