Monday, 18 December 2023

Turbulence as a subject area does not comprise a single problem whose solution will justify victory to be claimed for all those who have serious interest in it: mathematicians, physicists and practising engineers. Some synthesis is essential. On that basis, one can generate a short list of important problems whose solution represents serious progress. This is the attempt to which I will set myself up---more as a target for criticism.

The one-dimensional ($1D$) Galerkin-truncated Burgers equation, with both dissipation and noise terms included, is studied using spectral methods. When the truncation-scale Reynolds number $R_{\rm min}$ is varied, from very small values to order $1$ values, the scale-dependent correlation time $\tau(k)$ is shown to follow the expected crossover from the short-distance $\tau(k) \sim k^{-2}$ Edwards-Wilkinson scaling to the universal long-distance Kardar-Parisi-Zhang scaling $\tau(k) \sim k^{-3/2}$. In the inviscid limit: $R_{\rm min}\to \infty$, we show that the system displays {\it another} crossover to the Galerkin-truncated inviscid-Burgers regime that admits thermalised solutions with $\tau(k) \sim k^{-1}$. The scaling form of the time-correlation functions are shown to follow the known analytical laws and the skewness and excess kurtosis of the interface increments distributions are characterised.

I will give an overview of the Cahn-Hilliard-Navier-Stokes framework for turbulence in multi-phase flows. This will be based on the work I have been doing with several students and coworkers, most recently with my student Nadia Bihari Padhan.

There are challenges in the field theory of hydrodynamic turbulence. For example, whether renormalised viscosity of two-dimensional turbulence is negative or positive? Does turbulent energy flux get suppressed with the increase of space dimension? Earlier, Fournier and Frisch [Phys. Rev. A, 17, 747, 1978] performed field-theoretic and eddy-damped quasi-normal Markovian calculation of d-dimensional turbulence and showed that the energy cascade changes sign from negative to positive at d = dc ≈ 2.06. In this paper, we revisit d-dimensional turbulence and perform recursive renormalization and energy transfer calculations. We employ Craya-Herring basis that provides separate renormalized viscosities and energy transfers for its two components. We test the stability of nonequilibrium energy spectrum and equilibrium spectrum . We also investigate how the energy cascade rate is suppressed with the increase of space dimension.

It is well known that fully developed turbulence displays robust scaling laws in the inertial range at adequately high Reynolds numbers. The scaling exponents that govern such laws display a nontrivial dependence on the order of the moment – a phenomenon known as anomalous scaling. This scaling anomaly exists in moments of different quantities such as the velocity circulation around closed loops and velocity differences around balls. The consensus largely is that the different sets of exponents change character around moment-order three. This change in character occurring around order three is of crucial importance to some open problems in turbulence research such as the phenomenon of anomalous dissipation. In this talk we will motivate the importance of scaling characteristics of turbulence using data from both experiments and simulations of homogeneous and isotropic turbulence.

I demonstrate that black hole dynamics simplifies - without trivializing - in the limit in which the number of spacetime dimensions D in which the black holes live is taken to infinity. In the strict large D limit and under certain conditions I show the equations that govern black hole dynamics reduce to the equations describing the dynamics of a non gravitational membrane propagating in an unperturbed spacetime (e.g. flat space). In the stationary limit black hole thermodynamics maps to membrane thermodynamics, which we formulate in a precise manner. We also demonstrate that the large D black hole membrane agrees with the fluid gravity map in the appropriate regime.

Tuesday, 19 December 2023

I will describe an attempt to describe turbulence using the methods of quantum field theory. We consider waves that interact via four-wave scattering (such as sea waves, plasma waves, spin waves, and many others). By summing the series of the most UV-divergent terms in the perturbation theory, we show that the true dimensionless coupling is different from the naive estimate, and find that the effective interaction either decays or grows explosively with the cascade extent, depending on the sign of the new coupling. The explosive growth possibly signals the appearance of a multi-wave bound state (solitons, shocks, cusps) similar to confinement in quantum chromodynamics.

We find that IR divergence in the effective coupling could be responsible for an anomalous scaling in wave turbulence.

Turbulent fluctuations of circulation have several statistical properties that have come to light in recent years. Among them are the existence of a stable energy cascade in the asymptotic limit of high Reynolds numbers, the connection between the statistical description of vortex structures and multifractal inertial range scaling, the shape of probability distribution functions (PDFs), the scale-dependent behavior of the circulation flatness, and the scaling exponents of circulation moments, etc. Some of them have been accurately addressed by Direct Numerical Simulations (DNS) of Navier Stokes equations and some using a vortex gas model of homogeneous and isotropic turbulence. In this talk, we report possible extensions of the vortex gas model to the case of non-planar circulation contours. Theoretical and numerical investigations have suggested that the minimal surfaces bounded by such contours determine the shape of properly rescaled circulation PDFs. Unbiased by these indications, we look for the bounded surfaces which lead to consistent vortex gas model evaluations of the circulation PDFs. Our computational strategy relies on a variation of the two-dimensional Gaussian Multiplicative Chaos model, suitable for application for curved surfaces. The predictions are compared to the DNS results. We are thus able to critically revisit the role (and mathematical necessity) of minimal surfaces within the context of the vortex gas model of circulation statistics

I will discuss progress in generating and analysing steady state turbulent flow via holography

We discuss recent results on scaling and multifractal fluctuations in active and classical turbulence.

TBA

Wednesday, 20 December 2023

We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension $d >2$. This solution family is equivalent to a fractal curve in complex space $\mathbb C^d$ with random steps parametrized by $N\to \infty$ Ising variables $\sigma_i=\pm 1$, in addition to a rational number $\frac{p}{q}$ and an integer winding number $r$, related by $\sum \sigma_i = q r$. The energy dissipation rate grows as $\nu/\mu^2$ in the continuum limit when chemical potential $\mu \rightarrow 0$, leading to anomalous dissipation at $\mu \propto \sqrt{\nu} \to 0$. The same method is used to compute the local vorticity distribution. The small perturbation of the fixed manifold satisfies the linear equation we solved in a general form. This perturbation decays as $t^{-\lambda}$, with a continuous spectrum of indexes $\lambda$ in the local limit $\mu \to 0$.

We will discuss the non-equilibrium dynamics of a low temperature 2-dim black hole coupled to an external bath, using the dual SYK model. Tracing over the bath degrees of freedom, in the Schwinger-Keldysh formalism, leads to solving non-local stochastic differential equations for the BH degrees of freedom. We describe the combination of SYK operators that enable complete evaporation of the BH, and the fluctuations that characterise the end point of the evaporation process. This talk is based on JHEP 08 (2023) 171 (arXiv:2210.15579).

Mesoscale convection comprises buoyancy-driven turbulence at an intermediate range of scales. These processes are typically characterized by regular convection cell patterns and can be found in atmospheric flows or stellar interiors. We will discuss possible dynamical origins of these patterns, the related turbulent transport of heat and momentum, and data-driven approaches for the modeling of unresolved scales in these systems. The related numerical simulation studies start from Rayleigh-Bénard systems and are extended to non-Boussinesq cases.

Natural convective flows are highly turbulent. For example, flows in the Earth's outer core, Jupiter's atmosphere, and solar interior are driven vigorously, and the Rayleigh number, which signifies the strength of thermal forcing, is extremely large in these settings. Direct numerical simulations of convective flows are limited to Rayleigh numbers that are way lower than in natural setting due to extreme computations required to resolve all the relevant scales. One way to achieve high Rayleigh numbers at affordable costs is to use slender computational domains, though the influences due to confining boundaries need to be properly accounted for. Moving in this direction, we study thermal convection in a slender cylinder, which is ten times as high as it is wide, for a vast range of governing parameters. We simulate flows covering more than four orders of magnitude in Prandtl and Rayleigh numbers reaching up to low Prandtl number of liquid sodium, and observe that the flow properties in the slender cell are essentially the same as those in wider domains when Rayleigh numbers are larger than 10^10. We also estimate the turbulent Prandtl number and find that it increases with decreasing molecular Prandtl number. Further, to mimic the influence of rotation---as present in almost all natural flows---we simulate rotating convection and find that the way rotation alters the flow properties in the slender domain is similar to that observed in wider convection domains. Our findings suggest that exploring convection in confined domains is a viable approach towards our quest to better understanding the turbulent flows in nature.

We investigate the spectral properties of buoyancy-driven bubbly flows. Using high-resolution numerical simulations and phenomenology of homogeneous turbulence, we identify the relevant energy transfer mechanisms. We find (a) at a high enough Galilei number (ratio of the buoyancy to viscous forces) the velocity power spectrum shows the Kolmogorov scaling for the range of scales between the bubble diameter and the dissipation scale. (b) For scales smaller than the dissipation scale, the physics of pseudo-turbulence is recovered.

When a short pulse high-intensity laser pulse falls on a target it ionizes it to form a plasma medium. For an overdense plasma medium, the laser pulse is unable to propagate beyond the critical density surface. It dumps its energy to the lighter electron species of the plasma at the critical density layer and may also get partially reflected from the surface. The energetic electrons flowing inside the plasma constitute a forward current. This current is neutralized by a return shielding currents from the background plasma. The combination of spatially overlapping forward and return electron beams is fraught with a host of instabilities. The prominent amongst them is the Weibel and filamentation instabilities which are responsible for the spatial separation of forward and return currents leading to magnetic field generation. Both simulations and experiments have shown that this results in the development of turbulence in the magnetic field. The talk will cover the detailed description of the turbulent characteristics observed experimentally, and its understanding gleaned from simulations and analysis. A puzzling behavior in the observed spectral characteristics of the turbulent state will be outlined and a preliminary understanding of the same will be provided.

Thursday, 21 December 2023

I will describe our recent efforts to generalize the effective field theories for thermal systems (especially the effective field theory of hydrodynamics) to nonequilibrium matter, including active solids and fluids. Our approach is organized around a generalized time-reversal transformation, which can provide strong constraints on the effective field theory that seem to go beyond the standard Landau paradigm, even at leading non-trivial order. Our approach will be illustrated using three examples: (1) active solids and fluids; (2) kinematically-constrained "fracton" hydrodynamic universality classes, with and without microscopic time-reversal symmetry; (3) a new universality class of "flocking spins" on a lattice, which seems to have qualitatively distinct behavior than the conventional theory of "Malthusian" flocking birds in active matter.

Recently non-linear phenomena in gravity has been understood and identified in connection with non-linear phenomena in hydrodynamics. This has motivated new developments in our understanding of gravity and potential impact on observations. As well, it has shed new light on fluid phenomena.

This talk will discuss particular aspects of this exciting/intriguing intersection of fronts.

Quantum hydrodynamics and turbulence is an important research topic in low temperature physics. Quantum condensed systems such as superfluid helium and atomic Bose-Einstein condensates (BECs) have order parameters. Thus hydrodynamics and turbulence of these systems are severely restricted by the order parameters and quantum mechanics. The typical example is quantized vortex; any rotational motion of superfluid is sustained only by quantized vortices. Quantum hydrodynamics and turbulence [1] have been long studied in superfluid helium since 1950’s[2], and recently in atomic Bose-Einstein condensates (BECs) too [3]. In this presentation, I would review the characteristics and motivation of this field and discuss the recent novel topics.

1. Fully coupled two-fluid dynamics in superfluid 4He

Hydrodynamics of superfluid 4He is well described by the two-fluid model. The most characteristic phenomenon of the two-fluid model is thermal counterflow, which has been studied for more than a half century. However, the three-dimensional coupled dynamic of the two-fluid model is seldom addressed. The recent visualization experiments show that the profile of the normal fluid flow is seriously modified by the development of superfluid turbulence [4] and anisotropic velocity fluctuation in the laminar normal fluid [5]. We performed numerically the three-dimensional coupled dynamics of the two-fluid mode, l and found the serious change of the profile of the normal fluid flow [6] and the anisotropic velocity fluctuation [7].

2. Hydrodynamics and turbulence of atomic BECs

Most experiments of atomic BECs have been performed for systems trapped by a harmonic potential. The recent realization of the box potential has made this system more attractive. Navon et al. observed the clear statistical law of quantum turbulence in a BEC trapped by a box potential [8] and the cascade flux sustaining the turbulence [9]. It is known that the restoration of symmetries is one of the most fascinating properties of classical turbulence [10]. This phenomenon is reported in Ref. 8. We performed the simulation of the Gross-Pitaevskii model and found the particle distribution in momentum space became isotropic as the turbulence develops [11].

[1] C. F. Barenghi, L. Skrbek, K. R. Sreenivasan, Quantum Turbulence (Cambridge Univ. Press) (2023): M. Tsubota, M. Kobayashi, H. Takeuchi, Phys. Rep. 522, 191 (2013).; M. Tsubota, K. Fujimoto, S. Yui, J. Low Temp. Phys. 188, 119 (2017)

[2] J. T. Tough, Superfluid turbulence, in Prog. in Low Temp. Phys., edited by D. F. Brewer (North-Holland, Amsterdam,1982), Vol. 8, Chap. 3.

[3] M. C. Tsatsos, P. E. S. Tavares, A. Cidrim, A. R. Fritsch, M. A. Caracanhas, F. E. A. dos Santos, C. F. Barenghi, V. S. Bagnato,Phys. Rep., 622 (2016) 1.

[4] A. Marakov, J. Gao, W. Guo, S.W. Van Sciver, G. G. Ihas,, D. N. McKinsey, and W. F. Vinen, Phys. Rev. B 91, 094503 (2015).

[5] B. Mastracci, S. Bao, W. Guo, and W. F. Vinen, Phys. Rev. Fluids 4, 083305 (2019).

[6] S. Yui, M. Tsubota, H. Kobayashi, Phys. Rev. Lett. 120, 155301 (2018).

[7] S. Yui, H. Kobayashi, M. Tsubota, W. Guo, Phys. Rev. Lett. 124, 155301 (2020).

[8] N. Navon, A. L. Gaunt, R. P. Smith, Z. Hadzibabic, Nature 539, 72 (2016).

[9] N. Navon, C. Eigen, J. Zhang, R. Lopes, A. L. Gaunt, K. Fujimoto, M. Tsubota, R. P. Smith and Z. Hadzibabic, Science 366, 1267 (2019).

[10] U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge University Press) 1995.

[11] Y. Sano, N. Navon, M. Tsubota, EPL140, 66002 (2022).

In this talk we will present numerical results on the scaling of energy cascade in a harmonically trapped, turbulent, rotating Bose-Einstein condensate in two dimensions. We achieve turbulence by injecting a localized perturbation into the condensate and gradually increasing its rotation frequency from an initial value to a maximum. The main characteristics of the turbulent state depend on the initial conditions, rotation frequency, and ramp-up time. We will discuss the energy spectra and the fluxes of kinetic energy by considering initial profiles without vortices and with vortex lattices. In the former case, we find the presence of Kolmogorov-like scaling ($k^{-5/3}$) of the incompressible kinetic energy in the inertial range, with a forward cascade of energy at smaller scales for a high rotation rate. However, in the latter case, the energy spectrum follows Vinen scaling ($k^{-1}$) at transient iterations and for cases of high rotating frequencies shows Kolmogorov-like scaling at longer durations, with positive kinetic energy fluxes implying a forward cascade of the energy.

One of the most impressive phenomena in atmospheric flows is the organization of turbulent convective motions into large-scale structures. Turbulence theory has proposed an inverse cascade in two-dimensional flows as the mechanism behind this process, but its applicability to realistic atmospheric and oceanic flows remains heavily debated. In this talk I will discuss recent advances in our understanding of three-dimensional flows under specific geometrical constraints, the possibility of phase transitions giving rise to spontaneous self-organization even when these flows remain three-dimensional, and present direct numerical simulations with unprecedented spatial resolution that support the development of a bi-directional cascade of energy for parameters comparable to that of the Earth’s atmosphere.

Despite exceptional theoretical, numerical, and experimental efforts conducted over the past thirty years, no existing models are capable of faithfully reproducing all multi-scale statistical and topological properties exhibited by particle trajectories in turbulence. We propose a machine learning approach, based on a state-of-the-art Diffusion Model, to generate single-particle trajectories in three-dimensional turbulence at high Reynolds numbers, thereby bypassing the need for direct numerical simulations or experiments to obtain reliable Lagrangian data. Our model demonstrates the ability to quantitatively reproduce all relevant statistical benchmarks over the entire range of time scales, including the presence of fat tails distribution for the velocity increments, anomalous power law, and enhancement of intermittency around the dissipative scale.

Heavy particles in turbulent flow are expected to centrifuge out of vortical regions and agglomerate in strain regions. We show that in a rotating system, particles can permanently reside in the vicinity of vortices. In addition, the clustering is extreme, into low-dimensional objects such as fixed points, limit cycles or chaotic orbits. We will discuss the Basset history force, and how it changes the dynamics.

TBA

Friday, 22 December 2023

Fluid turbulence is a major unsolved problem of physics exhibiting an emergent complex structure from simple rules. We will briefly review the problem and discuss three avenues towards its solution: field theory, gravity, and deep learning.