Shell model for rotating turbulence

Dynamic scaling in rotating turbulence: A shell model study

We investigate the scaling form of appropriate timescales extracted from time-dependent correlation functions in rotating turbulent flows. In particular, we obtain precise estimates of the dynamic exponents z_{p}, associated with the timescales, and their relation with the more commonly measured equal-time exponents ζ_{p}. These theoretical predictions, obtained by using the multifractal formalism, are validated through extensive numerical simulations of a shell model for such rotating flows.

Statistics of Lagrangian trajectories in a rotating turbulent flow

We investigate the Lagrangian statistics of three-dimensional rotating turbulent flows through direct numerical simulations. We find that the emergence of coherent vortical structures because of the Coriolis force leads to a suppression of the "flight-crash" events reported by Xu et al. [Proc. Natl. Acad. Sci. (USA) 111, 7558 (2014)PNASA60027-842410.1073/pnas.1321682111]. We perform systematic studies to trace the origins of this suppression in the emergent geometry of the flow and show why such a Lagrangian measure of irreversibility may fail in the presence of rotation.

Dynamic multiscaling in magnetohydrodynamic turbulence

We present a study of the multiscaling of time-dependent velocity and magnetic-field structure functions in homogeneous, isotropic magnetohydrodynamic (MHD) turbulence in three dimensions. We generalize the formalism that has been developed for analogous studies of time-dependent structure functions in fluid turbulence to MHD. By carrying out detailed numerical studies of such time-dependent structure functions in a shell model for three-dimensional MHD turbulence, we obtain both equal-time and dynamic scaling exponents.

Logarithmic discretization and systematic derivation of shell models in two-dimensional turbulence

A detailed systematic derivation of a logarithmically discretized model for two-dimensional turbulence is given, starting from the basic fluid equations and proceeding with a particular form of discretization of the wave-number space. We show that it is possible to keep all or a subset of the interactions, either local or disparate scale, and recover various limiting forms of shell models used in plasma and geophysical turbulence studies. The method makes no use of the conservation laws even though it respects the underlying conservation properties of the fluid equations. It gives a family of models ranging from shell models with nonlocal interactions to anisotropic shell models depending on the way the shells are constructed. Numerical integration of the model shows that energy and enstrophy equipartition seem to dominate over the dual cascade, which is a common problem of two-dimensional shell models.

Multiscaling in superfluid turbulence: A shell-model study

We examine the multiscaling behavior of the normal- and superfluid-velocity structure functions in three-dimensional superfluid turbulence by using a shell model for the three-dimensional (3D) Hall-Vinen-Bekharevich-Khalatnikov (HVBK) equations. Our 3D-HVBK shell model is based on the Gledzer-Okhitani-Yamada shell model. We examine the dependence of the multiscaling exponents on the normal-fluid fraction and the mutual-friction coefficients. Our extensive study of the 3D-HVBK shell model shows that the multiscaling behavior of the velocity structure functions in superfluid turbulence is more complicated than it is in fluid turbulence.

Inverse energy cascade in nonlocal helical shell models of turbulence

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 nonlocal 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 the idea that turbulent energy transfer is the result of a strong entanglement among triads possessing different transfer properties.

Fluctuating hydrodynamics and turbulence in a rotating fluid: universal properties

We analyze the statistical properties of three-dimensional (3D) turbulence in a rotating fluid. To this end we introduce a generating functional to study the statistical properties of the velocity field v. We obtain the master equation from the Navier-Stokes equation in a rotating frame and thence a set of exact hierarchical equations for the velocity structure functions for arbitrary angular velocity Ω. In particular we obtain the differential forms for the analogs of the well-known von Karman-Howarth relation for 3D fluid turbulence. We examine their behavior in the limit of large rotation. Our results clearly suggest dissimilar statistical behavior and scaling along directions parallel and perpendicular to Ω. The hierarchical relations yield strong evidence that the nature of the flows for large rotation is not identical to pure two-dimensional flows. To complement these results, by using an effective model in the small-Ω limit, within a one-loop approximation, we show that the equal-time correlation of the velocity components parallel to Ω displays Kolmogorov scaling q(-5/3) wherein as for all other components, the equal-time correlators scale as q(-3) in the inertial range where q is a wave vector in 3D. Our results are generally testable in experiments and/or direct numerical simulations of the Navier-Stokes equation in a rotating frame.

Cascades and dissipation ratio in rotating magnetohydrodynamic turbulence at low magnetic Prandtl number

A phenomenology of isotropic magnetohydrodynamic (MHD) turbulence subject to both rotation and applied magnetic field is presented. It is assumed that the triple correlation decay time is the shortest between the eddy turn-over time and the ones associated to the rotating frequency and the Alfvén wave period. For Pm=1 it leads to four kinds of piecewise spectra, depending on four parameters: injection rate of energy, magnetic diffusivity, rotation rate, and applied field. With a shell model of MHD turbulence (including rotation and applied magnetic field), spectra for Pm ≤ 1 are presented, together with the ratio between magnetic and viscous dissipations.

Third-order structure function for rotating three-dimensional homogeneous turbulent flow

A form for the two-point third-order structure function has been calculated for three-dimensional (3D) homogeneous incompressible slowly rotating turbulent fluid. It has been argued that it may possibly hint at the initiation of the phenomenon of two-dimensionalization of the 3D incompressible turbulence owing to rotation.

Large- and small-scale interactions and quenching in an alpha2-dynamo

The evolution of the large-scale magnetic field in a turbulent flow of conducting fluid is considered in the framework of a multiscale alpha2-dynamo model, which includes the poloidal and the toroidal components for the large-scale magnetic field and a shell model for the small-scale magnetohydrodynamical turbulence. The conjugation of the mean-field description for the large-scale field and the shell formalism for the small-scale turbulence is based on strict conformity to the conservation laws. The model displays a substantial magnetic contribution to the alpha effect. It was shown that a large-scale magnetic field can be generated by current helicity even solely. The alpha quenching and the role of the magnetic Prandtl number (Pm) are studied. We have determined the dynamic nature of the saturation mechanism of dynamo action. Any simultaneous cross correlation of alpha and large-scale magnetic field energy EB is negligible, whereas coupling between alpha and EB becomes substantial for moderate time lags. An unexpected result is the behavior of the large-scale magnetic energy with variation of the magnetic Prandtl number. Diminishing of Pm does not have an inevitable ill effect on the magnetic field generation. The most efficient large-scale dynamo operates under relatively low Prandtl numbers--then the small-scale dynamo is suppressed and the decrease of Pm can lead even to superequipartition of the large-scale magnetic field (i.e., EB>Eu). In contrast, the growth of Pm does not promote the large-scale magnetic field generation. A growing counteraction of the magnetic alpha effect reduces the level of mean large-scale magnetic energy at the saturated state.