Kelvin's Theorem on conservation of circulations is an essential ingredient of G. I. Taylor's theory of turbulent energy dissipation by the process of vortex-line stretching. In previous work, we have proposed a nonlinear mechanism for the breakdown of Kelvin's Theorem in ideal turbulence at infinite Reynolds number. We develop here a detailed physical theory of this ``cascade of circulations''. Our analysis is based upon an effective equation for large-scale ``coarse-grained'' velocity, which contains a turbulent-induced ``vortex-force'' that can violate Kelvin's Theorem. We show that singularities of sufficient strength, which are observed to exist in turbulent flow, can lead to non-vanishing dissipation of circulation for an arbitrarily small filtering length in the effective equations. This result is an analogue for circulation of Onsager's theorem on energy dissipation for singular Euler solutions. The physical mechanism of the breakdown of Kelvin's Theorem is diffusion of lines of large-scale vorticity out of the advected loop. This phenomenon can be viewed as a classical analogue of the Josephson-Anderson phase-slip phenomenon in superfluids due to quantized vortex lines. We show that the circulation cascade is local in scale and use this locality to develop concrete expressions for the turbulent vortex-force by a multi-scale gradient-expansion. We discuss implications for Taylor's theory of turbulent dissipation and we point out some related cascade phenomena, in particular for magnetic-flux in magnetohydrodynamic (MHD) turbulence.
We discuss the role of particular velocity field configurations -- instantons, for short -- which are supposed to dominate the flow during the occurrence of extreme turbulent circulation events. Instanton equations, devised for the stochastic hydrodynamic setup of homogeneous and isotropic turbulence, are applied to the interpretation of direct numerical simulation results. We are able in this way to model the time evolution of extreme circulation events for a broad range of scales, through the combined use of eddy viscosity phenomenology and exact creeping instantons. While this approach work…
The velocity circulation, a measure of the rotation of a fluid within a closed path, is a fundamental observable in classical and quantum flows. It is indeed a Lagrangian invariant in inviscid classical fluids. In quantum flows, circulation is quantized, taking discrete values that are directly related to the number and the orientation of thin vortex filaments enclosed by the path. By varying the size of such closed loop, the circulation provides a measure of the dependence of the flow structure on the considered scale. Here we consider the scale dependence of circulation statistics in quantum…
We study intermittency of circulation moments in turbulent superfluid helium by using experimental grid turbulence and numerical simulations of the Hall-Vinen-Bekarevich-Khalatnikov model. More precisely, we compute the velocity circulation $Γ_r$ in loops of size $r$ laying in the inertial range. For both, experimental and numerical data, the circulation variance shows a clear Kolmogorov scaling $\langle Γ_r^2 \rangle \sim r^{8/3}$ in the inertial range, independently of the temperature. Scaling exponents of high-order moments are comparable, within error bars, to previously reported anomalous…
The turbulence problem at the level of scaling exponents is hard in part because of the multifractal scaling of small scales, which demands that each moment order be treated and understood independently. This conclusion derives from studies of velocity structure functions, energy dissipation, enstrophy density (that is, square of vorticity), etc. However, it is likely that there exist other physically pertinent quantities with uncomplicated structure in the inertial range, potentially resulting in huge simplifications in the turbulence theory. We show that velocity circulation around closed lo…
The understanding of turbulent flows is one of the biggest current challenges in physics, as no first-principles theory exists to explain their observed spatio-temporal intermittency. Turbulent flows may be regarded as an intricate collection of mutually-interacting vortices. This picture becomes accurate in quantum turbulence, which is built on tangles of discrete vortex filaments. Here, we study the statistics of velocity circulation in quantum and classical turbulence. We show that, in quantum flows, Kolmogorov turbulence emerges from the correlation of vortex orientations, while deviations…
An important idea underlying a plausible dynamical theory of circulation in three-dimensional turbulence is the so-called Area Rule, according to which the probability density function (PDF) of the circulation around closed loops depends only on the minimal area of the loop, not its shape. We assess the robustness of the Area Rule, for both planar and non-planar loops, using high-resolution data from Direct Numerical Simulations. For planar loops, the circulation moments for rectangular shapes match those for the square with only small differences, these differences being larger when the aspec…
The process of the kinetic energy and kinetic helicity transfer over the spectrum in an incompressible, rapidly rotating turbulent flow is considered. An analogue of the Fjortoft theorem for 3D rapidly rotating turbulence is proposed. It is shown that, similar to 2D turbulence, there are two cascades simultaneously: the inverse cascade of the kinetic energy and the direct cascade of the kinetic helicity, which in the case of 2D turbulence corresponds to the cascade of enstrophy. The proposed scenario is in agreement with our earlier calculations, some recent numerical simulations, and physical…
Area and Perimeter Rules of Velocity Circulation in Two-Dimensional Turbulence with Large-scale Absolute Equilibrium
School of Mechanics and Engineering Science and State Key Laboratory for Turbulence and Complex Systems, Peking University（北京大学力学与工程科学系及湍流与复杂系统国家重点实验室）
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…
This investigation presents evidence of the relation between the dynamics of intense events in small-scale turbulence and the energy cascade. We use the generalised (Hölder) means to track the temporal evolution of intense events of the enstrophy and the dissipation in direct numerical simulations of isotropic turbulence. We show that these events are modulated by large-scale fluctuations, and that their evolution is consistent with a local multiplicative cascade, as hypothesised by a broad class of intermittency models of turbulence.
In this Letter we show that a bifurcation cascade and fully sustained turbulence can share the phase space of a fluid flow system, resulting in the presence of competing stable attractors. We analyse the toroidal pipe flow, which undergoes subcritical transition to turbulence at low pipe curvatures and supercritical transition at high curvatures, as was previously documented. We provide decisive evidence that the nature of the supercritical transition is of Ruelle--Takens type and that, in a narrow range of intermediate curvatures, its dynamics competes with that of sustained turbulence emergi…
We investigate three-dimensional quantum turbulence as described by the Gross-Pitaevskii model using the analytical method exploited in the Onsager "ideal turbulence" theory. We derive the scale-independence of the scale-to-scale kinetic energy flux and establish a double-cascade scenario: at scales much larger than the mean intervortex $\ell_i$, the Richardson cascade becomes dominant, whereas at scales much smaller than $\ell_i$, another type of cascade is induced by quantum stress. We then evaluate the corresponding velocity power spectrum using a phenomenological argument. The relation bet…
We elaborate the statistical field theory of Turbulence suggested in the previous paper \cite{M20a}. We clarify and simplify the basic Energy pumping equation of that theory and study mathematical properties of singular field configuration (instanton) which determine the tails of PDF for the velocity circulation around large loop $C$ in isotropic turbulence at highest Reynolds numbers. Explicit analytic solution is found for the Clebsch instanton in an Euler equation for a planar loop circulation problem. This solution for vorticity is has a term proportional to a delta function in normal dire…
We re-visit the Area Law in Turbulence discovered many years ago \cite{M93} and verified recently in numerical experiments\cite{S19}. We derive this law in a simpler way, at the same time outlining the limits of its applicability. Using the PDF for velocity circulation as a functional of the loop in coordinate space, we obtain explicit formulas for vorticity correlations in presence of velocity circulation. These functions are related to the shape of the scaling function of the PDF as well as the shape of the minimal surface inside the loop. The background of velocity circulation does not elim…
We study the statistics of free-surface turbulence at large Reynolds numbers produced by direct numerical simulations in a fluid layer at different thickness with fixed characteristic forcing scale. We observe the production of a transient inverse cascade, with a duration which depends on the thickness of the layer, followed by a transition to three-dimensional turbulence initially produced close to the bottom, no-slip boundary. By switching off the forcing, we study the decaying turbulent regime and we find that it cannot be described by an exponential law. Our results show that boundary cond…
We study stationary solutions in the differential kinetic equation, which was introduced in for description of a local dual cascade wave turbulence. We give a full classification of single-cascade states in which there is a finite flux of only one conserved quantity. Analysis of the steady-state spectrum is based on a phase-space analysis of orbits of the underlying dynamical system. The orbits of the dynamical system demonstrate the blow-up behaviour which corresponds to a "sharp front" where the spectrum vanishes at a finite wave number. The roles of the KZ and thermodynamic scaling as inter…
We investigate a quite strong turbulence in a supercritical fluid near a gas-liquid critical point. Specifically, we consider a case in which the Kolmogorov scale is much smaller than the equilibrium correlation length $ξ$. Although equilibrium critical fluctuations are destroyed by turbulence, $ξ$ still provides a crossover length scale between two types of energy cascade. At scales much larger than $ξ$, the Richardson cascade becomes dominant, whereas at scales much smaller than $ξ$, another type of cascade, which we call the van der Waals cascade, is induced by density fluctuations. Experim…
We investigate the dynamics of energy cascades in quantum turbulence by directly observing the vorticity distributions in numerical simulations of the Gross-Pitaevskii equation. By Fourier filtering each scale of the vorticity distribution, we find that antiparallel vortex tubes at a large scale generate small-scale vortex tubes orthogonal to those at the large scale, which is a manifestation of the energy cascade from large to small scales. We reveal the dynamics of quantized vortex lines in these processes.
We refute the criticism expressed in a Comment by Krstulovic, L'vov, and Nazarenko [arXiv:2107.10598] on our paper [Phys. Rev. E 103, 023106 (2021)]. We first show that quantization of circulation is not ignored in our analysis. Then, we propose a more sophisticated analysis to avoid a subtle problem with the regularity of the velocity field. We thus defend the main results of our paper, which predicts the double-cascade scenario where the quantum stress cascade follows the Richardson cascade. We also provide a conjecture on the relation between the Kelvin-wave cascade and the quantum stress c…
We report on the observation of surface gravity wave turbulence at scales larger than the forcing ones in a large basin. In addition to the downscale transfer usually reported in gravity wave turbulence, an upscale transfer is observed, interpreted as the inverse cascade of weak turbulence theory. A steady state is achieved when the inverse cascade reaches a scale in between the forcing wavelength and the basin size, but far from the latter. This inverse cascade saturation, which depends on the wave steepness, is probably due to the emergence of nonlinear dissipative structures such as sharp-c…
We demonstrate that like in the forward cascade of three dimensional turbulence that displays intermittency (lack of self-similarity) due to the concentration of energy dissipation in a small set of fractal dimension less than three, the inverse cascade of two-dimensional turbulence can also display lack of self-similarity and intermittency if the energy injection is constrained in a fractal set of dimension less than two. A series of numerical simulations of two dimensional turbulence are examined, using different forcing functions of the same forcing length-scale but different fractal dimens…
Magnetic and kinetic energy in ideal incompressible MHD are not global invariants and, therefore, it had been justified to discuss only the cascade of their sum, total energy. We provide a physical argument based on scale-locality of the cascade, along with compelling evidence that at high Reynolds numbers, magnetic and kinetic energy budgets statistically decouple beyond a transitional "conversion" range. This arises because magnetic field-line stretching is a large-scale process which vanishes on average at intermediate and small scales within the inertial-inductive range, thereby allowing e…
We study forced, rapildy rotating and stably stratified turbulence in an elongated domain using an asymptotic expansion at simultaneously low Rossby number $Ro\ll1$ and large domain height compared to the energy injection scale, $h=H/\ell_{in}\gg1$. The resulting equations depend on the parameter $λ=(h Ro )^{-1}$ and the Froude number $Fr$. An extensive set of direct numerical simulations (DNS) is performed to explore the parameter space $(λ,Fr)$. We show that a forward energy cascade occurs in one region of this space, and a split energy cascade outside it. At weak stratification (large $Fr$)…
