Inverse Cascade In Two-dimensional Turbulence Research Articles

Inducing intermittency in the inverse cascade of two-dimensional turbulence by a fractal forcing

Can the inverse cascade of two-dimensional turbulence display intermittency? This work shows that if forced in a fractal set of dimension smaller than two, a two-dimensional turbulent flow develops clusters of different size vortices not uniformly spread in space as the ones shown in the figure, that display intermittent properties.

Coherent vortex in two-dimensional turbulence: Interplay of viscosity and bottom friction.

We examine coherent vortices appearing as a result of the inverse cascade of two-dimensional turbulence in a finite box in the case of pumping with arbitrary correlation time in the quasilinear regime. We demonstrate that the existence of the vortices depends on the ratio between the values of the bottom friction coefficient α and the viscous damping of the flow fluctuations at the pumping scale νk_{f}^{2} (ν is the kinematic viscosity coefficient and k_{f} is the characteristic wave vector at the pumping scale). The coherent vortices appear if νk_{f}^{2}≫α and cannot exist if νk_{f}^{2}≪α. Therefore there is a border value α∼νk_{f}^{2} separating the regions. In numerical simulations, νk_{f}^{2}/α can be arbitrary, whereas in a laboratory experiment νk_{f}^{2}/α≲1 and the coherent vortices can be observed solely near the border value of νk_{f}^{2}/α.

Dynamics of circular arrangements of vorticity in two dimensions.

The merger of two like-signed vortices is a well-studied problem, but in a turbulent flow, we may often have more than two like-signed vortices interacting. We study the merger of three or more identical corotating vortices initially arranged on the vertices of a regular polygon. At low to moderate Reynolds numbers, we find an additional stage in the merger process, absent in the merger of two vortices, where an annular vortical structure is formed and is long lived. Vortex merger is slowed down significantly due to this. Such annular vortices are known at far higher Reynolds numbers in studies of tropical cyclones, which have been noticed to a break down into individual vortices. In the preannular stage, vortical structures in a viscous flow are found here to tilt and realign in a manner similar to the inviscid case, but the pronounced filaments visible in the latter are practically absent in the former. Five or fewer vortices initially elongate radially, and then reorient their long axis closer to the azimuthal direction so as to form an annulus. With six or more vortices, the initial alignment is already azimuthal. Interestingly at higher Reynolds numbers, the merger of an odd number of vortices is found to proceed very differently from that of an even number. The former process is rapid and chaotic whereas the latter proceeds more slowly via pairing events. The annular vortex takes the form of a generalized Lamb-Oseen vortex (GLO), and diffuses inward until it forms a standard Lamb-Oseen vortex. For lower Reynolds number, the numerical (fully nonlinear) evolution of the GLO vortex follows exactly the analytical evolution until merger. At higher Reynolds numbers, the annulus goes through instabilities whose nonlinear stages show a pronounced difference between even and odd mode disturbances. Here again, the odd mode causes an early collapse of the annulus via decaying turbulence into a single central vortex, whereas the even mode disturbance causes a more orderly progression into a single vortex. Results from linear stability analysis agree with the nonlinear simulations, and predict the frequencies of the most unstable modes better than they predict the growth rates. It is hoped that the present findings, that multiple vortex merger is qualitatively different from the merger of two vortices, will motivate studies on how multiple vortex interactions affect the inverse cascade in two-dimensional turbulence.

Structure of coherent vortices generated by the inverse cascade of two-dimensional turbulence in a finite box.

We discuss the structure and geometrical characteristics of coherent vortices appearing as a result of the inverse cascade in two-dimensional turbulence in a finite box. We demonstrate that the universal velocity profile, established by J. Laurie et al. [Phys. Rev. Lett. 113, 254503 (2014)], corresponds to the passive regime of flow fluctuations. We find the vortex core radius and the vortex size, and we argue that the amount of vortices generated in the box depends on the system parameters.

Profile of coherent vortices in two-dimensional turbulence

The inverse cascade of two-dimensional turbulence in a restricted domain leads to creating a coherent flow containing a number of system-size vortices. We examine the case of forcing turbulence with zero bottom friction where the final statistically steady state is determined by viscosity. We analytically establish structure of the coherent vortices in the state.

Generalized vortex model for the inverse cascade of two-dimensional turbulence

We generalize Kirchhoff's point vortex model of two-dimensional fluid motion to a rotor model which exhibits an inverse cascade by the formation of rotor clusters. A rotor is composed of two vortices with like-signed circulations glued together by an overdamped spring. The model is motivated by a treatment of the vorticity equation representing the vorticity field as a superposition of vortices with elliptic Gaussian shapes of variable widths, augmented by a suitable forcing mechanism. The rotor model opens up the way to discuss the energy transport in the inverse cascade on the basis of dynamical systems theory.

Two-point vorticity statistics in the inverse cascade of two-dimensional turbulence

A statistical analysis of the two-point vorticity statistics in the inverse energy cascade of two-dimensional turbulence is presented in terms of probability density functions (PDFs). Evolution equations for the PDFs are derived in the framework of the Lundgren–Monin–Novikov hierarchy, and the unclosed terms are studied with the help of direct numerical simulations (DNS). Furthermore, the unclosed terms are evaluated in a Gaussian approximation and compared to the DNS results. It turns out that the statistical equations can be interpreted in terms of the dynamics of screened vortices. The two-point statistics is related to the dynamics of two point vortices with screened velocity field, where an effective relative motion of the two point vortices originating from the turbulent surroundings is identified to be a major characteristics of the dynamics underlying the inverse cascade.

Ratchet effect in the inverse turbulent cascade

We present a statistical analysis of the inverse cascade of two-dimensional turbulence in terms of screened vortices in the sense of Landau quasi particles. We emphasize that the effective behaviour of two screened vortices involves an interaction which favours the cluster formation of like-signed vortices. This interaction is of stochastic nature and allows one to consider the turbulent transport in the inverse cascade as a kind of ratchet process.

Field theory of the inverse cascade in two-dimensional turbulence

A two-dimensional fluid, stirred at high wave numbers and damped by both viscosity and linear friction, is modeled by a statistical field theory. The fluid's long-distance behavior is studied using renormalization-group (RG) methods, as begun by Forster, Nelson, and Stephen [Phys. Rev. A 16, 732 (1977)]. With friction, which dissipates energy at low wave numbers, one expects a stationary inverse energy cascade for strong enough stirring. While such developed turbulence is beyond the quantitative reach of perturbation theory, a combination of exact and perturbative results suggests a coherent picture of the inverse cascade. The zero-friction fluctuation-dissipation theorem (FDT) is derived from a generalized time-reversal symmetry and implies zero anomalous dimension for the velocity even when friction is present. Thus the Kolmogorov scaling of the inverse cascade cannot be explained by any RG fixed point. The beta function for the dimensionless coupling ĝ is computed through two loops; the ĝ3 term is positive, as already known, but the ĝ5 term is negative. An ideal cascade requires a linear beta function for large ĝ, consistent with a Padé approximant to the Borel transform. The conjecture that the Kolmogorov spectrum arises from an RG flow through large ĝ is compatible with other results, but the accurate k(-5/3) scaling is not explained and the Kolmogorov constant is not estimated. The lack of scale invariance should produce intermittency in high-order structure functions, as observed in some but not all numerical simulations of the inverse cascade. When analogous RG methods are applied to the one-dimensional Burgers equation using an FDT-preserving dimensional continuation, equipartition is obtained instead of a cascade--in agreement with simulations.

Robustness of the inverse cascade in two-dimensional turbulence.

We study quasisteady inverse cascades in unbounded and bounded two-dimensional turbulence driven by time-independent injection and dissipated by molecular viscosity. It is shown that an inverse cascade that carries only a fraction r of the energy input to the largest scales requires the enstrophy-range energy spectrum to be steeper than k(-5) (ruling out a direct cascade) unless 1-r<<1. A direct cascade requires the presence of an inverse cascade that carries virtually all energy to the largest scales (1-r<<1). These facts underlie the robustness of the Kolmogorov-Kraichnan k(-5/3) inverse cascade, which is readily observable in numerical simulations without an accompanying direct enstrophy cascade. We numerically demonstrate an instance where the k(-5/3) inverse-cascading range is realizable with 79% of the energy injection dissipated within the energy range and virtually all of the enstrophy dissipated in the vicinity of the forcing region. As equilibrium is approached, the respective logarithmic slopes -alpha and -beta of the ranges of wave numbers lower and higher than the forcing wave number satisfy alpha+beta approximately 8. These results are consistent with recent theoretical predictions.

The effects of quadratic drag on the inverse cascade of two-dimensional turbulence

We explore the effects of a quadratic drag, similar to that used in bulk aerodynamic formulas, on the inverse cascade of homogeneous two-dimensional turbulence. If a two-dimensional fluid is forced at a relatively small scale, then an inverse cascade of energy will be generated that may then be arrested by such a drag at large scales. Both scaling arguments and numerical experiments support the idea that in a statistically steady state the length scale of energy-containing eddies will not then depend on the energy input to the system; rather, the only external parameter that defines this scale is the quadratic drag coefficient itself. A universal form of the spectrum is suggested, and numerical experiments are in good agreement. Further, the turbulent transfer of a passive tracer in the presence of a uniform gradient is well predicted by scaling arguments based solely on the energy cascade rate and the nonlinear drag coefficient.

Detailed investigation of energy transfers in homogeneous stratified turbulence*

This paper investigates some irreversible mechanisms occurring in homogeneous stably stratified turbulent flows. In terms of the eigenmodes of the linear regime, the velocity-temperature field is decomposed into a vortex and two wavy components. Using an eddy-damped quasinormal Markovian (EDQNM) closure with the axisymmetry hypothesis, an analysis of the anisotropic energy transfers between the vortex kinetic energy, the wave kinetic and potential energy is made. Within the light of triadic exchanges, and by analogy of the resonance condition for three linearly interacting gravity waves, the closure model allows one to compute the detailed transfers for eight types of interactions. Results of the calculations include time evolution plots, for the isotropic closure model as well as two different types of the anisotropic closure. The pure vortical interactions are shown to be responsible for the irreversible anisotropic structure created by stable stratification, and this structure prevents the inverse cascade of two-dimensional turbulence.

Numerical simulation of the inverse cascade in two-dimensional turbulence

A numerical simulation of large-scale two-dimensional turbulence fed in a narrow band of wavenumbers is presented. The development of the inverse energy cascade predicted by Kraichnan [Phys. Fluids 10, 1417 (1967)] is observed.