Dimensional transition in rotating turbulence

Effective Transport by 2D Turbulence: Vortex-Gas Theory vs Scale-Invariant Inverse Cascade

The scale-invariant inverse energy cascade is a hallmark of 2D turbulence, with its theoretical energy spectrum observed in both direct numerical simulations (DNS) and laboratory experiments. Under this scale-invariance assumption, the effective diffusivity of a 2D turbulent flow is dimensionally controlled by the energy flux and the friction coefficient only. Surprisingly, however, we show that such scaling predictions are invalidated by numerical solutions of the 2D Navier-Stokes equation forced at intermediate wave number and damped by weak linear or quadratic drag. We derive alternate scaling-laws for the effective diffusivity based on the emergence of intense, isolated vortices causing spatially inhomogeneous frictional dissipation localized within the small vortex cores. The predictions quantitatively match DNS data. This study points to a universal large-scale organization of 2D turbulent flows in physical space, bridging standard 2D Navier-Stokes turbulence with large-scale geophysical turbulence.

Phase transitions in anisotropic turbulence

Turbulence is a widely observed state of fluid flows, characterized by complex, nonlinear interactions between motions across a broad spectrum of length and time scales. While turbulence is ubiquitous, from teacups to planetary atmospheres, oceans, and stars, its manifestations can vary considerably between different physical systems. For instance, three-dimensional turbulent flows display a forward energy cascade from large to small scales, while in two-dimensional turbulence, energy cascades from small to large scales. In a given physical system, a transition between such disparate regimes of turbulence can occur when a control parameter reaches a critical value. The behavior of flows close to such transition points, which separate qualitatively distinct phases of turbulence, has been found to be unexpectedly rich. Here, we survey recent findings on such transitions in highly anisotropic turbulent fluid flows, including turbulence in thin layers and under the influence of rapid rotation. We also review recent work on transitions induced by turbulent fluctuations, such as random reversals and transitions between large-scale vortices and jets, among others. The relevance of these results and their ramifications for future investigations are discussed.

Odd Viscosity Suppresses Intermittency in Direct Turbulent Cascades

Intermittency refers to the broken self-similarity of turbulent flows caused by anomalous spatiotemporal fluctuations. In this Letter, we ask how intermittency is affected by a nondissipative viscosity, known as odd viscosity (also Hall viscosity or gyroviscosity), which appears in parity-breaking fluids such as magnetized polyatomic gases, electron fluids under magnetic field, and spinning colloids or grains. Using a combination of Navier-Stokes simulations and theory, we show that intermittency is suppressed by odd viscosity at small scales. This effect is caused by parity-breaking waves, induced by odd viscosity, that break the multiple scale invariances of the Navier-Stokes equations. Building on this insight, we construct a two-channel helical shell model that reproduces the basic phenomenology of turbulent odd-viscous fluids including the suppression of anomalous scaling. Our findings illustrate how a fully developed direct cascade that is entirely self-similar can emerge below a tunable length scale, paving the way for designing turbulent flows with adjustable levels of intermittency.

Pattern formation by turbulent cascades

Fully developed turbulence is a universal and scale-invariant chaotic state characterized by an energy cascade from large to small scales at which the cascade is eventually arrested by dissipation1-6. Here we show how to harness these seemingly structureless turbulent cascades to generate patterns. Pattern formation entails a process of wavelength selection, which can usually be traced to the linear instability of a homogeneous state7. By contrast, the mechanism we propose here is fully nonlinear. It is triggered by the non-dissipative arrest of turbulent cascades: energy piles up at an intermediate scale, which is neither the system size nor the smallest scales at which energy is usually dissipated. Using a combination of theory and large-scale simulations, we show that the tunable wavelength of these cascade-induced patterns can be set by a non-dissipative transport coefficient called odd viscosity, ubiquitous in chiral fluids ranging from bioactive to quantum systems8-12. Odd viscosity, which acts as a scale-dependent Coriolis-like force, leads to a two-dimensionalization of the flow at small scales, in contrast with rotating fluids in which a two-dimensionalization occurs at large scales4. Apart from odd viscosity fluids, we discuss how cascade-induced patterns can arise in natural systems, including atmospheric flows13-19, stellar plasma such as the solar wind20-22, or the pulverization and coagulation of objects or droplets in which mass rather than energy cascades23-25.

Inferring turbulent environments via machine learning

The problem of classifying turbulent environments from partial observation is key for some theoretical and applied fields, from engineering to earth observation and astrophysics, e.g., to precondition searching of optimal control policies in different turbulent backgrounds, to predict the probability of rare events and/or to infer physical parameters labeling different turbulent setups. To achieve such goal one can use different tools depending on the system's knowledge and on the quality and quantity of the accessible data. In this context, we assume to work in a model-free setup completely blind to all dynamical laws, but with a large quantity of (good quality) data for training. As a prototype of complex flows with different attractors, and different multi-scale statistical properties we selected 10 turbulent 'ensembles' by changing the rotation frequency of the frame of reference of the 3d domain and we suppose to have access to a set of partial observations limited to the instantaneous kinetic energy distribution in a 2d plane, as it is often the case in geophysics and astrophysics. We compare results obtained by a machine learning (ML) approach consisting of a state-of-the-art deep convolutional neural network (DCNN) against Bayesian inference which exploits the information on velocity and entropy moments. First, we discuss the supremacy of the ML approach, presenting also results at changing the number of training data and of the hyper-parameters. Second, we present an ablation study on the input data aimed to perform a ranking on the importance of the flow features used by the DCNN, helping to identify the main physical contents used by the classifier. Finally, we discuss the main limitations of such data-driven methods and potential interesting applications.

Geometric microcanonical theory of two-dimensional truncated Euler flows

This paper presents a geometric microcanonical ensemble perspective on two-dimensional truncated Euler flows, which contain a finite number of (Fourier) modes and conserve energy and enstrophy. We explicitly perform phase space volume integrals over shells of constant energy and enstrophy. Two applications are considered. In the first part, we determine the average energy spectrum for highly condensed flow configurations and show that the result is consistent with Kraichnan's canonical ensemble description, despite the fact that no thermodynamic limit is invoked. In the second part, we compute the probability density for the largest-scale mode of a free-slip flow in a square, which displays reversals. We test the results against numerical simulations of a minimal model and find excellent agreement with the microcanonical theory, unlike the canonical theory, which fails to describe the bimodal statistics. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.

Intermittency of three-dimensional perturbations in a point-vortex model

Three-dimensional (3D) instabilities on a (potentially turbulent) two-dimensional (2D) flow are still incompletely understood, despite recent progress. Here, based on known physical properties of such 3D instabilities, we propose a simple, energy-conserving model describing this situation. It consists of a regularized 2D point-vortex flow coupled to localized 3D perturbations ("ergophages"), such that ergophages can gain energy by altering vortex-vortex distances through an induced divergent velocity field, thus decreasing point-vortex energy. We investigate the model in three distinct stages of evolution: (i) The linear regime, where the amplitude of the ergophages grows or decays exponentially on average, with an instantaneous growth rate that fluctuates randomly in time. The instantaneous growth rate has a small auto-correlation time, and a probability distribution featuring a power-law tail with exponent between -2 and -5/3 (up to a cutoff) depending on the point-vortex base flow. Consequently, the logarithm of the ergophage amplitude performs a Lévy flight. (ii) The passive-nonlinear regime of the model, where the 2D flow evolves independently of the ergophage amplitudes, which saturate by non-linear self-interactions without affecting the 2D flow. In this regime the system exhibits a new type of on-off intermittency that we name Lévy on-off intermittency, which we define and study in a companion paper [van Kan et al., Phys. Rev. E 103, 052115 (2021)1063-651X10.1103/PhysRevE.103.052115]. We compute the bifurcation diagram for the mean and variance of the perturbation amplitude, as well as the probability density of the perturbation amplitude. (iii) Finally, we characterize the fully nonlinear regime, where ergophages feed back on the 2D flow, and study how the vortex temperature is altered by the interaction with ergophages. It is shown that when the amplitude of the ergophages is sufficiently large, the condensate is disrupted and the 2D flow saturates to a zero-temperature state. Given the limitations of existing theories, our model provides a new perspective on 3D instabilities growing on 2D flows, which will be useful in analyzing and understanding the much more complex results of DNS and potentially guide further theoretical developments.

Counterflow-Induced Inverse Energy Cascade in Three-Dimensional Superfluid Turbulence

Finite-temperature quantum turbulence is often described in terms of two immiscible fluids that can flow with a nonzero-mean relative velocity. Such out-of-equilibrium state is known as counterflow superfluid turbulence. We report here the emergence of a counterflow-induced inverse energy cascade in three-dimensional superfluid flows by performing extensive numerical simulations of the Hall-Vinen-Bekarevich-Khalatnikov model. As the intensity of the mean counterflow is increased, an abrupt transition, from a fully three-dimensional turbulent flow to a quasi-two-dimensional system exhibiting a split cascade, is observed. The findings of this work could motivate new experimental settings to study quasi-two-dimensional superfluid turbulence in the bulk of three-dimensional experiments. They might also find applications beyond superfluids in systems described by more than one fluid component.

Quantitative Experimental Observation of Weak Inertial-Wave Turbulence

We report the quantitative experimental observation of the weak inertial-wave turbulence regime of rotating turbulence. We produce a statistically steady homogeneous turbulent flow that consists of nonlinearly interacting inertial waves, using rough top and bottom boundaries to prevent the emergence of a geostrophic flow. As the forcing amplitude increases, the temporal spectrum evolves from a discrete set of peaks to a continuous spectrum. Maps of the bicoherence of the velocity field confirm such a gradual transition between discrete wave interactions at weak forcing amplitude and the regime described by weak turbulence theory (WTT) for stronger forcing. In the former regime, the bicoherence maps display a near-zero background level, together with sharp localized peaks associated with discrete resonances. By contrast, in the latter regime, the bicoherence is a smooth function that takes values of the order of the Rossby number in line with the infinite-domain and random-phase assumptions of WTT. The spatial spectra then display a power-law behavior, both the spectral exponent and the spectral level being accurately predicted by WTT at high Reynolds number and low Rossby number.

Regime transition in the energy cascade of rotating turbulence

Transition from a split to a forward kinetic energy cascade system is explored in the context of rotating turbulence using direct numerical simulations with a three-dimensional isotropic random force uncorrelated with the velocity field. Our parametric study covers confinement effects in high-aspect-ratio domains and a broad range of rotation rates. The data presented here add substantially to previous works, which, in contrast, focused on smaller and shallower domains. Results indicate that for fixed geometrical dimensions the Rossby number acts as a control parameter, whereas for a fixed Rossby number the product of the domain size along the rotation axis and the forcing wave number governs the amount of energy that cascades inversely. The regime transition criterion hence depends on both control parameters.

Anatomical location of AICA loop in CPA as a prognostic factor for ISSNHL

The cerebellopontine angle (CPA) is a triangular-shaped space that lies at the junction of the pons and cerebellum. It contains cranial nerves and the anterior inferior cerebellar artery (AICA). The anatomical shape and location of the AICA is variable within the CPA and internal auditory canal (IAC). A possible etiology of idiopathic sudden sensorineural hearing loss (ISSNHL) is ischemia of the labyrinthine artery, which is a branch of the AICA. As such, the position of the AICA within the CPA and IAC may be related to the clinical development of ISSNHL. We adopted two methods to classify the anatomic position of the AICA, then analyzed whether these classifications affected the clinical features and prognosis of ISSNHL. We retrospectively reviewed patient data from January 2015 to March 2018. Two established classification methods designed by Cahvada and Gorrie et al. were used. Pure tone threshold at four different frequencies (0.5, 1, 4, and 8 kHz), at two different time points (at initial presentation and three months after treatment), were analyzed. We compared the affected and unaffected ears, and investigated whether there were any differences in hearing recovery and symptoms between the two classification types. There was no difference in AICA types between ears with and without ISSNHL. Patients who had combined symptoms such as tinnitus and vertigo did not show a different AICA distribution compared with patients who did not. There were differences in quantitative hearing improvement between AICA types, although without statistic significance (p = 0.09–0.13). At two frequencies, 1 and 4 kHz, there were differences in Chavda types between hearing improvement and no improvement (p < 0.05). Anatomical variances of the AICA loop position did not affect the incidence of ISSNHL or co-morbid symptoms including tinnitus and vertigo. In contrast, comparisons of hearing improvement based on Chavda type classification showed a statistical difference, with a higher proportion of Chavda type 1 showing improvements in hearing (AICA outside IAC).

Vortex Cascade Features of Turbulent Flow in Hydro-Turbine Blade Passage with Complex Geometry

A large-eddy simulation of three-dimensional turbulent flow for a hydro-turbine in the transitional process of decreasing load from rated power to no-load has been implemented by using ANSYS-Fluent in this paper. The survival space occupied by different scale flow structures for the different guide vane opening degrees was well captured. The flow characteristics in the transitional process were obtained. Different forms of the channel vortex were studied. The features of the vortex cascade and dissipation of the turbulent energy in blade passage were analyzed. The results show that the scales of the vortex structures have a large change in the transitional process of rejecting load, and the vortex distributions in the blade passage are significantly distinguished. The survival space of the different scale eddies in the blade passage is closely related to the scales of the vortex. The survival volume ratio of the adjacent scale vortex in the runner is about 1.2–1.6. The turbulent kinetic energy and eddy viscosity increase rapidly along the blade passage with the small-scale eddies going up, which implies that a dissipating path for the energy in the blade passage is formed.

Discontinuous Transition from Direct to Inverse Cascade in Three-Dimensional Turbulence

Inviscid invariants of flow equations are crucial in determining the direction of the turbulent energy cascade. In this work we investigate a variant of the three-dimensional Navier-Stokes equations that shares exactly the same ideal invariants (energy and helicity) and the same symmetries (under rotations, reflections, and scale transforms) as the original equations. It is demonstrated that the examined system displays a change in the direction of the energy cascade when varying the value of a free parameter which controls the relative weights of the triadic interactions between different helical Fourier modes. The transition from a forward to inverse cascade is shown to occur at a critical point in a discontinuous manner with diverging fluctuations close to criticality. Our work thus supports the observation that purely isotropic and three-dimensional flow configurations can support inverse energy transfer when interactions are altered and that inside all turbulent flows there is a competition among forward and backward transfer mechanisms which might lead to multiple energy-containing turbulent states.

Passive scalars: Mixing, diffusion, and intermittency in helical and nonhelical rotating turbulence

We use direct numerical simulations to compute structure functions, scaling exponents, probability density functions, and effective transport coefficients of passive scalars in turbulent rotating helical and nonhelical 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 ∼k_{⊥}^{-1.4} for the passive scalar variance spectrum. This scaling law is consistent with a phenomenological argument [P. Rodriguez Imazio and P. D. Mininni, Phys. Rev. E 83, 066309 (2011)PLEEE81539-375510.1103/PhysRevE.83.066309] for rotating nonhelical flows, which follows directly from Kolmogorov-Obukhov scaling and states that if energy follows a E(k)∼k^{-n} law, then the passive scalar variance follows a law V(k)∼k^{-n_{θ}} with n_{θ}=(5-n)/2. With the second-order scaling exponent obtained from this law, and using the Kraichnan model, we obtain anomalous scaling exponents for the passive scalar that are in good agreement with the numerical results. Multifractal intermittency models are also considered. Intermittency of the passive scalar is stronger than in the nonhelical rotating case, a result that is also confirmed by stronger non-Gaussian tails in the probability density functions of field increments. Finally, Fick's law is used to compute the effective diffusion coefficients in the directions parallel and perpendicular to rotation. Calculations indicate that horizontal diffusion decreases in the presence of helicity in rotating flows, while vertical diffusion increases. A simple mean field argument explains this behavior in terms of the amplitude of velocity fluctuations.

Variations of characteristic time scales in rotating stratified turbulence using a large parametric numerical study

We study rotating stratified turbulence (RST) making use of numerical data stemming from a large parametric study varying the Reynolds, Froude and Rossby numbers, Re, Fr and Ro in a broad range of values. The computations are performed using periodic boundary conditions on grids of 1024(3) points, with no modeling of the small scales, no forcing and with large-scale random initial conditions for the velocity field only, and there are altogether 65 runs analyzed in this paper. The buoyancy Reynolds number defined as R(B) = ReFr2 varies from negligible values to ≈ 10(5), approaching atmospheric or oceanic regimes. This preliminary analysis deals with the variation of characteristic time scales of RST with dimensionless parameters, focusing on the role played by the partition of energy between the kinetic and potential modes, as a key ingredient for modeling the dynamics of such flows. We find that neither rotation nor the ratio of the Brunt-Väisälä frequency to the inertial frequency seem to play a major role in the absence of forcing in the global dynamics of the small-scale kinetic and potential modes. Specifically, in these computations, mostly in regimes of wave turbulence, characteristic times based on the ratio of energy to dissipation of the velocity and temperature fluctuations, T(V) and T(P), vary substantially with parameters. Their ratio γ=T(V)/T(P) follows roughly a bell-shaped curve in terms of Richardson number Ri. It reaches a plateau - on which time scales become comparable, γ≈0.6 - when the turbulence has significantly strengthened, leading to numerous destabilization events together with a tendency towards an isotropization of the flow.

Critical behavior in the inverse to forward energy transition in two-dimensional magnetohydrodynamic flow

We investigate the critical transition from an inverse cascade of energy to a forward energy cascade in a two-dimensional magnetohydrodynamic flow as the ratio of magnetic to mechanical forcing amplitude is varied. It is found that the critical transition is the result of two competing processes. The first process is due to hydrodynamic interactions and cascades the energy to the large scales. The second process couples small-scale magnetic fields to large-scale flows, transferring the energy back to the small scales via a nonlocal mechanism. At marginality the two cascades are both present and cancel each other. The phase space diagram of the transition is sketched.

Disentangling inertial waves from eddy turbulence in a forced rotating-turbulence experiment

We present a spatiotemporal analysis of a statistically stationary rotating-turbulence experiment, aiming to extract a signature of inertial waves and to determine the scales and frequencies at which they can be detected. The analysis uses two-point spatial correlations of the temporal Fourier transform of velocity fields obtained from time-resolved stereoscopic particle image velocimetry measurements in the rotating frame. We quantify the degree of anisotropy of turbulence as a function of frequency and spatial scale. We show that this space-time-dependent anisotropy is well described by the dispersion relation of linear inertial waves at large scale, while smaller scales are dominated by the sweeping of the waves by fluid motion at larger scales. This sweeping effect is mostly due to the low-frequency quasi-two-dimensional component of the turbulent flow, a prominent feature of our experiment that is not accounted for by wave-turbulence theory. These results question the relevance of this theory for rotating turbulence at the moderate Rossby numbers accessible in laboratory experiments, which are relevant to most geophysical and astrophysical flows.

Resolving the paradox of oceanic large-scale balance and small-scale mixing

A puzzle of oceanic dynamics is the contrast between the observed geostrophic balance, involving gravity, pressure gradient, and Coriolis forces, and the necessary turbulent transport: in the former case, energy flows to large scales, leading to spectral condensation, whereas in the latter, it is transferred to small scales, where dissipation prevails. The known bidirectional constant-flux energy cascade maintaining both geostrophic balance and mixing tends towards flux equilibration as turbulence strengthens, contradicting models and recent observations which find a dominant large-scale flux. Analyzing a large ensemble of high-resolution direct numerical simulations of the Boussinesq equations in the presence of rotation and no salinity, we show that the ratio of the dual energy flux to large and to small scales agrees with observations, and we predict that it scales with the inverse of the Froude and Rossby numbers when stratification is (realistically) stronger than rotation. Furthermore, we show that the kinetic and potential energies separately undergo a bidirectional transfer to larger and smaller scales. Altogether, this allows for small-scale mixing which drives the global oceanic circulation and will thus potentially lead to more accurate modeling of climate dynamics.

On the edge of an inverse cascade

We demonstrate that systems with a parameter-controlled inverse cascade can exhibit critical behavior for which at the critical value of the control parameter the inverse cascade stops. In the vicinity of such a critical point, standard phenomenological estimates for the energy balance will fail since the energy flux towards large length scales becomes zero. We demonstrate this using the computationally tractable model of two-dimensional (2D) magnetohydrodynamics in a periodic box. In the absence of any external magnetic forcing, the system reduces to hydrodynamic fluid turbulence with an inverse energy cascade. In the presence of strong magnetic forcing, the system behaves as 2D magnetohydrodynamic turbulence with forward energy cascade. As the amplitude of the magnetic forcing is varied, a critical value is met for which the energy flux towards the large scales becomes zero. Close to this point, the energy flux scales as a power law with the departure from the critical point and the normalized amplitude of the fluctuations diverges. Similar behavior is observed for the flux of the square vector potential for which no inverse flux is observed for weak magnetic forcing, while a finite inverse flux is observed for magnetic forcing above the critical point. We conjecture that this behavior is generic for systems of variable inverse cascade.