On the edge of an inverse cascade

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.

Large-scale self-organization in dry turbulent atmospheres

How turbulent convective fluctuations organize to form larger-scale structures in planetary atmospheres remains a question that eludes quantitative answers. The assumption that this process is the result of an inverse cascade was suggested half a century ago in two-dimensional fluids, but its applicability to atmospheric and oceanic flows remains heavily debated, hampering our understanding of the energy balance in planetary systems. We show using direct numerical simulations with spatial resolutions of 122882 × 384 points that rotating and stratified flows can support a bidirectional cascade of energy, in three dimensions, with a ratio of Rossby to Froude numbers comparable to that of Earth's atmosphere. Our results establish that, in dry atmospheres, spontaneous order can arise through an inverse cascade to the largest spatial scales.

Equivalence of nonequilibrium ensembles: Two-dimensional turbulence with a dual cascade

We examine the conjecture of equivalence of nonequilibrium ensembles for turbulent flows in two dimensions in a dual-cascade setup. We construct a formally time-reversible Navier-Stokes equation in two dimensions by imposing global constraints of energy and enstrophy conservation. A comparative study of the statistical properties of its solutions with those obtained from the standard Navier-Stokes equations clearly shows that a formally time-reversible system is able to reproduce the features of a two-dimensional turbulent flow. Statistical quantities based on one- and two-point measurements show an excellent agreement between the two systems for the inverse- and direct-cascade regions. Moreover, we find that the conjecture holds very well for two-dimensional turbulent flows with both conserved energy and enstrophy at finite Reynolds number.

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)'.

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.

Inverse and Direct Energy Cascades in Three-Dimensional Magnetohydrodynamic Turbulence at Low Magnetic Reynolds Number

This experimental study analyzes the relationship between the dimensionality of turbulence and the upscale or downscale nature of its energy transfers. We do so by forcing low-Rm magnetohydrodynamic turbulence in a confined channel, while precisely controlling its dimensionality by means of an externally applied magnetic field. We first identify a specific length scale l[over ^]_{⊥}^{c} that separates smaller 3D structures from larger quasi-2D ones. We then show that an inverse energy cascade of horizontal kinetic energy along horizontal scales is always observable at large scales, and that it extends well into the region of 3D structures. At the same time, a direct energy cascade confined to the smallest and strongly 3D scales is observed. These dynamics therefore appear not to be simply determined by the dimensionality of individual scales, nor by the forcing scale, unlike in other studies. In fact, our findings suggest that the relationship between kinematics and dynamics is not universal and may strongly depend on the forcing and dissipating mechanisms at play.

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.

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.

Role of helicity for large- and small-scale turbulent fluctuations

The effects of the helicity on the dynamics of turbulent flows are investigated. The aim is to disentangle the role of helicity in fixing the direction, the intensity, and the fluctuations of the energy transfer across the inertial range of scales. We introduce an external parameter α that controls the mismatch between the number of positive and negative helically polarized Fourier modes. We present direct numerical simulations of Navier-Stokes equations from the fully symmetrical case, α=0, to the fully asymmetrical case, α=1, when only helical modes of one sign survive. We found a singular dependency of the direction of the energy cascade on α, measuring a positive forward flux as soon as only a few modes with different helical polarities are present. Small-scale fluctuations are also strongly sensitive to the degree of mode reduction, leading to a vanishing intermittency already for values of α∼0.1. If the analysis is restricted to sets of modes with the same helicity sign, intermittency is vanishing for the modes belonging to the minority set, and it is close to that measured on the original Navier-Stokes equations for the other set.

Disentangling the triadic interactions in Navier-Stokes equations

We study the role of helicity in the dynamics of energy transfer in a modified version of the Navier-Stokes equations with explicit breaking of the mirror symmetry. We select different set of triads participating in the dynamics on the basis of their helicity content. In particular, we remove the negative helically polarized Fourier modes at all wave numbers except for those falling on a localized shell of wave number, |k| ~ k(m). Changing k(m) to be above or below the forcing scale, k(f), we are able to assess the energy transfer of triads belonging to different interaction classes. We observe that when the negative helical modes are present only at a wave number smaller than the forced wave numbers, an inverse energy cascade develops with an accumulation of energy on a stationary helical condensate. Vice versa, when negative helical modes are present only at a wave number larger than the forced wave numbers, a transition from backward to forward energy transfer is observed in the regime when the minority modes become energetic enough.