Two-Dimensional Turbulence with Local Interactions: Statistics of the Condensate.

Two-dimensional turbulence self-organizes through a process of energy accumulation at large scales, forming a coherent flow termed a condensate. We study the condensate in a model with local dynamics, the large-scale quasigeostrophic equation, observed here for the first time. We obtain analytical results for the mean flow and the two-point, second-order correlation functions, and validate them numerically. The condensate state requires partiy+time-reversal symmetry breaking. We demonstrate distinct universal mechanisms for the even and odd correlators under this symmetry. We find that the model locality is imprinted in the small scale dynamics, which the condensate spatially confines.

Genetic Variation in Toll-Like Receptor 5 and Colonization with Flagellated Bacterial Vaginosis-Associated Bacteria.

Bacterial vaginosis (BV) is a vaginal dysbiotic condition linked to negative gynecological and reproductive sequelae. Flagellated bacteria have been identified in women with BV, including Mobiluncus spp. and BV-associated bacterium-1 (BVAB1), an uncultivated, putatively flagellated species. The host response to flagellin mediated through Toll-like receptor 5 (TLR5) has not been explored in BV. Using independent discovery and validation cohorts, we examined the hypothesis that TLR5 deficiency-defined by a dominant negative stop codon polymorphism, rs5744168-is associated with an increased risk for BV and increased colonization with flagellated bacteria associated with BV (BVAB1, Mobiluncus curtisii, and Mobiluncus mulieris). TLR5 deficiency was not associated with BV status, and TLR5-deficient women had decreased colonization with BVAB1 in both cohorts. We stimulated HEK-hTLR5-overexpressing NF-κB reporter cells with whole, heat-killed M. mulieris or M. curtisii and with partially purified flagellin from these species; as BVAB1 is uncultivated, we used cervicovaginal lavage (CVL) fluid supernatant from women colonized with BVAB1 for stimulation. While heat-killed M. mulieris and CVL fluid from women colonized with BVAB1 stimulate a TLR5-mediated response, heat-killed M. curtisii did not. In contrast, partially purified flagellin from both Mobiluncus species stimulated a TLR5-mediated response in vitro We observed no correlation between vaginal interleukin 8 (IL-8) and flagellated BVAB concentrations among TLR5-sufficient women. Interspecies variation in accessibility of flagellin recognition domains may be responsible for these observations, as reflected in the potentially novel flagellin products encoded by Mobiluncus species versus those encoded by BVAB1.

Turbulence Statistics in a Two-Dimensional Vortex Condensate.

Disentangling the evolution of a coherent mean-flow and turbulent fluctuations, interacting through the nonlinearity of the Navier-Stokes equations, is a central issue in fluid mechanics. It affects a wide range of flows, such as planetary atmospheres, plasmas, or wall-bounded flows, and hampers turbulence models. We consider the special case of a two-dimensional flow in a periodic box, for which the mean flow, a pair of box-size vortices called "condensate," emerges from turbulence. As was recently shown, a perturbative closure describes correctly the condensate when turbulence is excited at small scales. In this context, we obtain explicit results for the statistics of turbulence, encoded in the Reynolds stress tensor. We demonstrate that the two components of the Reynolds stress, the momentum flux and the turbulent energy, are determined by different mechanisms. It was suggested previously that the momentum flux is fixed by a balance between forcing and mean-flow advection: using unprecedently long numerical simulations, we provide the first direct evidence supporting this prediction. By contrast, combining analytical computations with numerical simulations, we show that the turbulent energy is determined only by mean-flow advection and obtain for the first time a formula describing its profile in the vortex.

New type of anomaly in turbulence.

The turbulent energy flux through scales, ε̅, remains constant and nonvanishing in the limit of zero viscosity, which results in the fundamental anomaly of time irreversibility. It was considered straightforward to deduce from this the Lagrangian velocity anomaly, ⟨du(2)/dt⟩=-4ε̅ at t=0, where u[over →] is the velocity difference of a pair of particles, initially separated by a fixed distance. Here we demonstrate that this assumed first taking the limit t→0 and then ν→0, while a zero-friction anomaly requires taking viscosity to zero first. We find that the limits t→0 and ν→0 do not commute if particles deplete (accumulate) in shocks backward (forward) in time on the viscous time scale. We compute analytically the resultant Lagrangian anomaly for one-dimensional Burgers turbulence and find it completely altered: ⟨du(2)/dt⟩ has different values forward and backward in time. For incompressible flows, on the other hand, we show that the limits commute and the Lagrangian anomaly is still induced by the flux law, apparently due to a homogeneous distribution of fluid particles at all times.

Single flow snapshot reveals the future and the past of pairs of particles in turbulence.

We develop an analytic formalism and derive new exact relations that express the short-time dispersion of fluid particles via the single-time velocity correlation functions in homogeneous isotropic and incompressible turbulence. The formalism establishes a bridge between single-time Eulerian and long-time Lagrangian pictures of turbulent flows. In particular, we derive an exact formula for a short-term counterpart of the long-time Richardson law, and we identify a conservation law of turbulent dispersion which is true even in nonstationary turbulence.

Statistics of inhomogeneous turbulence in large-scale quasigeostrophic dynamics.

A remarkable feature of two-dimensional turbulence is the transfer of energy from small to large scales. This process can result in the self-organization of the flow into large, coherent structures due to energy condensation at the largest scales. We investigate the formation of this condensate in a quasigeostropic flow in the limit of small Rossby deformation radius, namely the large-scale quasigeostrophic model. In this model potential energy is transferred up-scale while kinetic energy is transferred down-scale in a direct cascade. We focus on a jet mean flow and carry out a thorough investigation of the second-order statistics for this flow, combining a quasilinear analytical approach with direct numerical simulations. We show that the quasilinear approach applies in regions where jets are strong and is able to capture all second-order correlators in that region, including those related to the kinetic energy. This is a consequence of the blocking of the direct cascade by the mean flow in jet regions, suppressing fluctuation-fluctuation interactions. The suppression of the direct cascade is demonstrated using a local coarse-graining approach allowing us to measure space dependent interscale kinetic energy fluxes, which we show are concentrated in between jets in our simulations. We comment on the possibility of a similar direct cascade arrest in other two-dimensional flows, arguing that it is a special feature of flows in which the fluid element interactions are local in space.

Dynamical landscape of transitional pipe flow.

The transition to turbulence in pipes is characterized by a coexistence of laminar and turbulent states. At the lower end of the transition, localized turbulent pulses, called puffs, can be excited. Puffs can decay when rare fluctuations drive them close to an edge state lying at the phase-space boundary with laminar flow. At higher Reynolds numbers, homogeneous turbulence can be sustained, and dominates over laminar flow. Here we complete this landscape of localized states, placing it within a unified bifurcation picture. We demonstrate our claims within the Barkley model, and motivate them generally. Specifically, we suggest the existence of an antipuff and a gap-edge-states which mirror the puff and related edge state. Previously observed laminar gaps forming within homogeneous turbulence are then naturally identified as antipuffs nucleating and decaying through the gap edge. We also discuss alternatives to the suggested bifurcation diagram, which could be relevant for wall-bounded flows other than straight pipes.

Time irreversibility of the statistics of a single particle in compressible turbulence.

We investigate time irreversibility from the point of view of a single particle in Burgers turbulence. Inspired by the recent work for incompressible flows [Xu et al., Proc. Natl. Acad. Sci. USA 111, 7558 (2014)], we analyze the evolution of the kinetic energy for fluid markers and use the fluctuations of the instantaneous power as a measure of time irreversibility. For short times, starting from a uniform distribution of markers, we find the scaling 〈[E(t)-E(0)](n)〉∝t and 〈p(n)〉∝Re(n-1) for the power as a function of the Reynolds number. Both observations can be explained using the "flight-crash" model, suggested by Xu et al. Furthermore, we use a simple model for shocks that reproduces the moments of the energy difference, including the pre-factor for 〈E(t)-E(0)〉. To complete the single-particle picture for Burgers we compute the moments of the Lagrangian velocity difference and show that they are bifractal. This arises in a similar manner to the bifractality of Eulerian velocity differences. In the above setting, time irreversibility is directly manifest as particles eventually end up in shocks. We additionally investigate time irreversibility in the long-time limit when all particles are located inside shocks and the Lagrangian velocity statistics are stationary. We find the same scalings for the power and energy differences as at short times and argue that this is also a consequence of rare "flight-crash" events related to shock collisions.

Statistical conservation law in two- and three-dimensional turbulent flows.

Particles in turbulence live complicated lives. It is nonetheless sometimes possible to find order in this complexity. It was proposed in Falkovich et al. [Phys. Rev. Lett. 110, 214502 (2013)] that pairs of Lagrangian tracers at small scales, in an incompressible isotropic turbulent flow, have a statistical conservation law. More specifically, in a d-dimensional flow the distance R(t) between two neutrally buoyant particles, raised to the power -d and averaged over velocity realizations, remains at all times equal to the initial, fixed, separation raised to the same power. In this work we present evidence from direct numerical simulations of two- and three-dimensional turbulence for this conservation. In both cases the conservation is lost when particles exit the linear flow regime. In two dimensions we show that, as an extension of the conservation law, an Evans-Cohen-Morriss or Gallavotti-Cohen type fluctuation relation exists. We also analyze data from a 3D laboratory experiment [Liberzon et al., Physica D 241, 208 (2012)], finding that although it probes small scales they are not in the smooth regime. Thus instead of 〈R-3〉, we look for a similar, power-law-in-separation conservation law. We show that the existence of an initially slowly varying function of this form can be predicted but that it does not turn into a conservation law. We suggest that the conservation of 〈R-d〉, demonstrated here, can be used as a check of isotropy, incompressibility, and flow dimensionality in numerical and laboratory experiments that focus on small scales.