最近的数值模拟表明螺旋度对\ADD{homogeneous}旋转流体动力学湍流有很大影响。主效应可以通过以下定律来概括，即$n+\tilde n = -4$，其中$n$和$\tilde n$分别是一维能量谱和螺旋度谱的幂律指数。我们研究了小Rossby数限制下的旋转湍流问题，使用的渐近弱湍流理论推导先前。我们表明，经验法是螺旋度方程的精确解，其中幂律指数对应于垂直（旋转轴）波数谱。结果表明，当螺旋度通量占主导地位时，小尺度叶栅的解趋于$\tilde n = -2$，而当能通量占主导地位时，小尺度叶栅的解趋于$\tilde n = -3/2$。后者的解决方案是兼容的所谓的最大螺旋度状态以前观察到的数值和理论推导时，仅使用能量方程的弱湍流制度，而前一种解决方案是受局部条件。
英文摘要：
Recent numerical simulations have shown the strong impact of helicity on \ADD{homogeneous} rotating hydrodynamic turbulence. The main effect can be summarized through the following law, $n+\tilde n = -4$, where $n$ and $\tilde n$ are respectively the power law indices of the one-dimensional energy and helicity spectra. We investigate this rotating turbulence problem in the small Rossby number limit by using the asymptotic weak turbulence theory derived previously. We show that the empirical law is an exact solution of the helicity equation where the power law indices correspond to perpendicular (to the rotation axis) wave number spectra. It is proposed that when the cascade towards small-scales tends to be dominated by the helicity flux the solution tends to $\tilde n = -2$, whereas it is $\tilde n = -3/2$ when the energy flux dominates. The latter solution is compatible with the so-called maximal helicity state previously observed numerically and derived theoretically in the weak turbulence regime when only the energy equation is used, whereas the former solution is constrained by a locality condition.
For helical isotropic turbulence, an improved two-term helical subgrid-scale (SGS) model is proposed and four types of dynamic methods are given to do large-eddy simulation (LES), which include the standard dynamic procedure, the least quatratic sum dynamic procedure, the dynamic procedure with single con- straint of helicity dissipation and the dynamic one with dual constraints of energy and helicity dissipation. Tested a priori and a posteriori in both steady and decaying helical isotropic turbulence, the four types of dynamic helical models and the dynamic Smogorinsky model are compared wit…
Using results of laboratory experiments, direct numerical simulations, geomagnetic and solar observations, it is shown that high moments of helicity distribution can dominate power spectra of the magnetic field generated by the magnetohydrodynamic (MHD) dynamo in swirling turbulence even for the cases with zero global helicity. The notion of helical distributed chaos has been used for this purpose.
It is shown that the correlation integrals (invariants) play an important, and at certain conditions a dominant, role in helical isotropic homogeneous turbulence both in the inertial and near-dissipation ranges. Results of direct numerical simulations and recent laboratory experiments with multiscale grids have been used for this purpose. A possibility of spontaneous breaking of reflection symmetry has been also briefly discussed in this context.
Following the exact decomposition in eigenstates of helicity for the Navier-Stokes equations in Fourier space [F. Waleffe, Phys. Fluids A 4, 350 (1992)] we introduce a modified version of helical shell models for turbulence with non-local triadic interactions. By using both an analytical argument and numerical simulation, we show that there exists a class of models, with a specific helical structure, that exhibits a statistically stable inverse energy cascade, in close analogy with that predicted for the Navier-Stokes equations restricted to the same helical interactions. We further support th…
While in classical turbulence helicity depletes nonlinearity and can alter the evolution of turbulent flows, in quantum turbulence its role is not fully understood. We present numerical simulations of the free decay of a helical quantum turbulent flow using the Gross-Pitaevskii equation at high spatial resolution. The evolution has remarkable similarities with classical flows, which go as far as displaying a dual transfer of incompressible kinetic energy and helicity to small scales. Spatio-temporal analysis indicates that both quantities are dissipated at small scales through non-linear excit…
Using direct numerical simulation of hydrodynamic turbulence with helicity forcing applied at all scales, a near-maximum helical turbulent state is obtained, with an inverse energy cascade at scales larger than the energy forcing scale and a forward helicity cascade at scales smaller than the energy forcing scale. In contrast to previous studies using decimated triads, our simulations contain all possible triads. By computing the shell-to-shell energy fluxes, we show that the inverse energy cascade results from weakly non-local interactions among homochiral triads. Varying the helicity injecti…
The role of the moments of helicity distribution in rotating turbulence has been studied using the notion of helical distributed chaos. Results of the direct numerical simulations, laboratory experiments and geophysical observations have been used in this investigation. It is shown, in particular, that even for the cases when the global helicity is equal to zero at least even moments are usually non-zero (due to the appearance of the local spatial regions with strong negative and positive helicity) and can play a significant role in the rotating turbulence as adiabatic invariants. Rotating buo…
We perform a study on the evolution of helical quantum turbulence at different temperatures by solving numerically the Gross-Pitaevskii and the Stochastic Ginzburg-Landau equations, using up to $4096^3$ grid points with a pseudospectral method. We show that for temperatures close to the critical the fluid described by these equations can act as a classical viscous flow, with the decay of the incompressible kinetic energy and the helicity becoming exponential. The transition from this behavior to the one observed at zero temperature is smooth as a function of temperature. Moreover, the presence…
A decomposition of the energy and helicity fluxes in a turbulent hydrodynamic flow is proposed. The decomposition is based on the projection of the flow to a helical basis that allows to investigate separately the role of interactions among modes of different helicity. The proposed formalism is then applied in large scale numerical simulations of a non-helical and a helical flow, where the decomposed fluxes are explicitly calculated. It is shown that the total energy flux can be split in to three fluxes that independently remain constant in the inertial range. One of these fluxes that correspo…
We study the effect of large-scale spectral forcing on the scale-dependent anisotropy of the velocity field in direct numerical simulations of homogeneous incompressible turbulence. Two forcing methods are considered: the steady ABC single wavenumber scheme and the unsteady non-helical or helical Euler scheme. The results are also compared with high resolution data obtained with the negative viscosity scheme. A fine-grained characterization of anisotropy, consisting in measuring some quantities related to the two-point velocity correlations, is used: we perform a modal decomposition of the spe…
The conventional approach to the turbulent energy cascade, based on Richardson-Kolmogorov phenomenology, ignores the topology of emerging vortices, which is related to the helicity of the turbulent flow. It is generally believed that helicity can play a significant role in turbulent systems, e.g., supporting the generation of large-scale magnetic fields, but its impact on the energy cascade to small scales has never been observed. We suggest for the first time a generalized phenomenology for isotropic turbulence with an arbitrary spectral distribution of the helicity. We discuss various scenar…
We use direct numerical simulations to compute structure functions, scaling exponents, probability density functions and turbulent transport coefficients of passive scalars in turbulent rotating helical and non-helical flows. We show that helicity affects the inertial range scaling of the velocity and of the passive scalar when rotation is present, with a spectral law consistent with $\sim k_{\perp}^{-1.4}$ for the passive scalar variance spectrum. This scaling law is consistent with the phenomenological argument presented in \cite{imazio2011} for rotating non-helical flows, wich states that i…
Ideas and theories of turbulence based on modifying the Navier-Stokes equation, to obtain equilibrium and non-equilibrium time-reversible dynamical ensembles relevant to helical turbulence, are presented. Discussions of controlling helicity to control the aerodynamic force, heat and noise are presented, together with the compressible turbulence relevant statistical mechanics analysis. A helical time-reversible system for nonequilibrium dynamical ensemble is constructed. Applications are also remarked.
Shell models of turbulence have a finite-time blowup in the inviscid limit, i.e., the enstrophy diverges while the single-shell velocities stay finite. The signature of this blowup is represented by self-similar instantonic structures traveling coherently through the inertial range. These solutions might influence the energy transfer and the anomalous scaling properties empirically observed for the forced and viscous models. In this paper we present a study of the instantonic solutions for a set of four shell models of turbulence based on the exact decomposition of the Navier-Stokes equations …
The dynamics of two-dimensional three-component (2D3C) flows is relevant to describe the long-time evolution of strongly rotating flows and/or of conducting fluids with a strong mean magnetic field. We show that in the presence of a strong helical forcing, the out-of-plane component ceases to behave as a passive advected quantity and develops a nontrivial dynamics which deeply changes its large-scale properties. We show that a small-scale helicity injection correlates the input on the 2D component with the one on the out-of-plane component. As a result, the third component develops a non-trivi…
We solve the Navier-Stokes equations with two simultaneous forcings. One forcing is applied at a given large-scale and it injects energy. The other forcing is applied at all scales belonging to the inertial range and it injects helicity. In this way we can vary the degree of turbulence helicity from non helical to maximally helical. We find that increasing the rate of helicity injection does not change the energy flux. On the other hand the level of total energy is strongly increased and the energy spectrum gets steeper. The energy spectrum spans from a Kolmogorov scaling law $k^{-5/3}$ for a …
In this paper, we study a new type of large-scale instability in obliquely rotating stratified fluids with small scale non-helical turbulence. The small-scale turbulence is generated by the external force with zero helicity and low Reynolds number. The theory uses the method of multiscale asymptotic developments. The nonlinear equations for large scale motions are obtained in the third order of the perturbation theory. In this paper, we consider the linear instability and the stationary nonlinear modes. We obtain solutions in the form of nonlinear Beltrami waves and localized vortex structures…
Helicity, a measure of the breakage of reflectional symmetry representing the topology of turbulent flows, contributes in a crucial way to their dynamics and to their fundamental statistical properties. We review several of their main features, both new and old, as the discovery of bi-directional cascades or the role of helical vortices in the enhancement of large-scale magnetic fields in the dynamo problem. The dynamical contribution in magnetohydrodynamic (MHD) of the cross-correlation between velocity and induction is discussed as well. We consider next how turbulent transport is affected b…
In this paper we study the enstrophy transers in helical turbulence using direct numerical simulation. We observe that the helicity injection does not have significant effects on the inertial-range energy and helicity spectra ($\sim k^{-5/3}$) and fluxes (constants). We also calculate the separate contributions to enstrophy transfers via velocity to vorticity and vorticity to vorticity channels. There are four different enstrophy fluxes associated with the former channel or vorticity stretching, and one flux associated with the latter channel or vorticity advection. In the inertial range, the …
Drift of interfaces in forced stably-stratified turbulence and the role of vertically-sheared helical structures
Department of Physics & INFN, University of Rome “Tor Vergata”（罗马第二大学物理系及意大利国家核物理研究所）；INFN Sezione di Lecce（意大利国家核物理研究所莱切分部）；CNR NANOTEC（意大利国家研究委员会纳米技术研究所）
We show that in decaying hydromagnetic turbulence with initial kinetic helicity, a weak magnetic field eventually becomes fully helical. The sign of magnetic helicity is opposite to that of the kinetic helicity - regardless of whether or not the initial magnetic field was helical. The magnetic field undergoes inverse cascading with the magnetic energy decaying approximately like t^{-1/2}. This is even slower than in the fully helical case, where it decays like t^{-2/3}. In this parameter range, the product of magnetic energy and correlation length raised to a certain power slightly larger than…
The role of the different helical components of the magnetic and velocity fields in the inverse spectral transfer of magnetic helicity is investigated through Fourier shell-to-shell transfer analysis. Both magnetic helicity and energetic transfer analysis are performed on chosen data from direct numerical simulations of homogeneous isothermal compressible magnetohydrodynamic turbulence, subject to both a large-scale mechanical forcing and a small-scale helical electromotive driving. The root mean square Mach number of the hydrodynamic turbulent steady-state taken as initial condition varies fr…
We present a numerical study of the statistical properties of three-dimensional dissipative turbulent flows at scales larger than the forcing scale. Our results indicate that the large scale flow can be described to a large degree by the truncated Euler equations with the predictions of the zero flux solutions given by absolute equilibrium theory, both for helical and non-helical flows. Thus, the functional shape of the large scale spectra can be predicted provided that scales sufficiently larger than the forcing length scale but also sufficiently smaller than the box size are examined. Deviat…
We study the evolution of kinetic and magnetic energy spectra in magnetohydrodynamic flows in the presence of strong cross helicity. For forced turbulence, we find weak inverse transfer of kinetic energy toward the smallest wavenumber. This is plausibly explained by the finiteness of scale separation between the injection wavenumber and the smallest wavenumber of the domain, which here is a factor of 15. In the decaying case, there is a slight increase at the smallest wavenumber, which is probably explained by the dominance of kinetic energy over magnetic energy at the smallest wavenumbers. Wi…
Spectral transfer processes in magnetohydrodynamic (MHD) turbulence are investigated analytically by decomposition of the velocity and magnetic fields in Fourier space into helical modes. Steady solutions of the dynamical system which governs the evolution of the helical modes are determined, and a stability analysis of these solutions is carried out. The interpretation of the analysis is that unstable solutions lead to energy transfer between the interacting modes while stable solutions do not. From this, a dependence of possible interscale energy and helicity transfers on the helicities of t…
We perform direct numerical simulations to compute the net circular polarization of gravitational waves from helical (chiral) turbulent sources in the early Universe for a variety of initial conditions, including driven (stationary) and decaying turbulence. We investigate the resulting gravitational wave signal assuming different turbulent geneses such as magnetically or kinetically driven cases. Under realistic physical conditions in the early Universe we compute numerically the wave number-dependent polarization degree of the gravitational waves. We find that the spectral polarization degree…
The effects of large scale mechanical forcing on the dynamics of rotating turbulent flows are studied by means of numerical simulations, varying systematically the nature of the mechanical force in time. We demonstrate that the statistically stationary solutions of these flows depend on the nature of the forcing mechanism. Rapidly enough rotating flows with a forcing that has a persistent direction relatively to the axis of rotation bifurcate from a non-helical state to a helical state despite the fact that the forcing is non-helical. We find that the nature of the mechanical force in time and…
The distributed chaos driven by Levich-Tsinober (helicity) integral: $I=\int \langle h({\bf x},t)~h({\bf x}+{\bf r}, t)\rangle d{\bf r}$ has been studied. It is shown that the helical distributed chaos can be considered as basis for complex turbulent flows with interplay between large-scale coherent structures and small-scale turbulence, such as Cuette-Taylor flow, wake behind cylinder and turbulent flow in the Large Plasma Device (LAPD) with inserted limiters. In the last case appearance of the helical distributed chaos, caused by the limiters, results in improvement of radial particle confin…
In an earlier paper we showed that the combination of azimuthal magnetic fields and super-rotation in Taylor-Couette flows of conducting fluids can be unstable against non-axisymmetric perturbations if the magnetic Prandtl number of the fluid is $Pm\neq 1$. Here we demonstrate that the addition of a weak axial field component may allow axisymmetric perturbation patterns for $Pm$ of order unity depending on the boundary conditions. The axisymmetric modes only occur for magnetic Mach numbers (of the azimuthal field) of order unity, while higher values are necessary for non-axisymmetric modes. Th…
Analytical theory of forced rotating sheared turbulence. The Parallel case
研究了旋转和剪切流作用下的强迫湍流。在以前的论文中[Leprovost和Kim，PRE in press（2008）]，我们考虑了剪切和旋转垂直的情况。在这里，我们考虑平行旋转和剪切的互补情况下，阐明旋转和流动剪切如何影响剪切流的产生（例如能量级联的方向），湍流水平，颗粒和动量的传输。与旋转和剪切垂直的情况相比，我们发现旋转仅对非常快速的旋转（$Ω\gg \A$）影响湍流振幅，在这种情况下，由于剪切流，旋转略微降低了各向异性。此外，关于湍流的输运性质，我们发现旋转只影响粒子的输运，并且只对快速旋转产生影响，导致几乎各向同性的输运（而在垂直旋转和剪切的情况下，即使对于缓慢旋转，旋转也有利于各向同性的输运）。
We study the statistical properties of helicity in direct numerical simulations of fully developed homogeneous and isotropic turbulence and in a class of turbulence shell models. We consider correlation functions based on combinations of vorticity and velocity increments that are not invariant under mirror symmetry. We also study the scaling properties of high-order structure functions based on the moments of the velocity increments projected on a subset of modes with either positive or negative helicity (chirality). We show that mirror symmetry is recovered at small-scales, i.e., chiral terms…
Spatial buoyancy-helical distributed chaos (turbulence) in the cumulus, stratiform, stratocumulus, cirrus and cirrus mammatus clouds have been studied using results of direct numerical simulations and measurements in cloudy atmosphere. It is shown that in the considered cases the moments of helicity distribution dominate the kinetic energy spectra both in the middle of the clouds and in the cloud-top regions.
Fully developed homogeneous isotropic turbulence in 2D is fundamentally different from 3D. In 2D, the simultaneous conservation of both energy and enstrophy in the inertial ranges of scales leads to a forward cascade of enstrophy and a reverse cascade of energy. In 3D, helicity, the integral of the scalar product of velocity and vorticity, is also an inviscid flow invariant along with kinetic energy. Unlike enstrophy, however, helicity does not block the cascade of energy to small scales. Energy and helicity are not only globally conserved but also conserved within each non-linear triadic inte…
The principles of restricted superposition of circularly polarized arbitrary-amplitude waves for several hydrodynamic type models are illustrated systematically with helical representation in a unified sense. It is shown that the only general modes satisfying arbitrary-amplitude superposition to kill the generic nonlinearity are the mono-wavelength homochiral Beltrami mode and the one-dimensional-two-component stratified vorticity mode, which we call the XYz flow/wave; while, there are other special superposition principles for some specific cases. We try to remark on the possible connections …
