Spectral Condensation Research Articles

Flow patterns and energy spectra in forced quasi-two-dimensional turbulence: effect of system size and damping rate

An experimental investigation is conducted to study the flow patterns, spectral properties and energy fluxes in thin-layer turbulence with varying system sizes and damping rates. It is found that although a system-size vortex (an indicator of spectral condensation) occurs for small system sizes and does not for large ones, the spectra for different system sizes consistently exhibit a scaling close to $k^{-3}$ in inverse cascade (another indicator of spectral condensation). On the other hand, under a fixed system size larger than the friction-dominated length scale, the energy spectrum in the inverse cascade range changes from $k^{-3}$ to $k^{-5/3}$ as the damping rate increases, suggesting that the friction-dominated length scale may not be a suitable parameter for predicting spectral transition. At lower damping rates and large system sizes, turbulent structures grow larger via inverse cascade, manifesting as long streamers, and the small-scale vortices are suppressed. This suppression leads to a reduction of energy flux at intermediate scales and a change in the spectral shape. The dimensionless Taylor microscale is found to exhibit a monotonic dependence on the damping rate. With the reduction in the damping rate, the Taylor microscale increases to become comparable with the forcing scale, and the spectrum in inverse cascade transits to a steeper scaling, $k^{-3}$ , indicating that the dimensionless Taylor microscale may be used as a diagnostic parameter for spectral transition.

Spectral condensation and bidirectional energy transfer in quasi-geostrophic turbulence above small-scale topography

Spectral condensation and bidirectional energy transfer in quasi-geostrophic turbulence above small-scale topography

Spectral condensation in laboratory two-dimensional turbulence

Due to the inverse energy cascade, when the energy dissipation length scale in two-dimensional turbulence is larger than the domain size, energy is expected to condense into the lowest allowed mode. Thus, the appearance of large-scale structures is often taken as a proxy for spectral condensation, particularly in laboratory realizations of two-dimensional turbulence. We show that large-scale flows alone, however, are not robust indicators of spectral condensation, because small domains may both weaken the inverse cascade and introduce new dissipation mechanisms.

Computing the shape gradient of stellarator coil complexity with respect to the plasma boundary

Coil complexity is a critical consideration in stellarator design. The traditional two-step optimization approach, in which the plasma boundary is optimized for physics properties and the coils are subsequently optimized to be consistent with this boundary, can result in plasma shapes which cannot be produced with sufficiently simple coils. To address this challenge, we propose a method to incorporate considerations of coil complexity in the optimization of the plasma boundary. Coil complexity metrics are computed from the current potential solution obtained with the REGCOIL code (Landreman,Nucl. Fusion, vol. 57, 2017, 046003). While such metrics have previously been included in derivative-free fixed-boundary optimization (Drevlaket al.,Nucl. Fusion, vol. 59, 2018, 016010), we compute the local sensitivity of these metrics with respect to perturbations of the plasma boundary using the shape gradient (Landreman & Paul,Nucl. Fusion, vol. 58, 2018, 076023). We extend REGCOIL to compute derivatives of these metrics with respect to parameters describing the plasma boundary. In keeping with previous research on winding surface optimization (Paulet al.,Nucl. Fusion, vol. 58, 2018, 076015), the shape derivatives are computed with a discrete adjoint method. In contrast with the previous work, derivatives are computed with respect to the plasma surface parameters rather than the winding surface parameters. To further reduce the resolution required to compute the shape gradient, we present a more efficient representation of the plasma surface which uses a single Fourier series to describe the radial distance from a coordinate axis and a spectrally condensed poloidal angle. This representation is advantageous over the standard cylindrical representation used in the VMEC code (Hirshman & Whitson,Phys. Fluids, vol. 26, 1983, pp. 3553–3568), as it provides a uniquely defined poloidal angle, eliminating a null space in the optimization of the plasma surface. In comparison with previous spectral condensation methods (Hirshman & Breslau,Phys. Plasmas, vol. 5, 1998, p. 2664), the modified poloidal angle is obtained algebraically rather than through the solution of a nonlinear optimization problem. The resulting shape gradient highlights features of the plasma boundary that are consistent with simple coils and can be used to couple coil and fixed-boundary optimization.

Universality in spectral condensation

Self-organization is the spontaneous formation of spatial, temporal, or spatiotemporal patterns in complex systems far from equilibrium. During such self-organization, energy distributed in a broadband of frequencies gets condensed into a dominant mode, analogous to a condensation phenomenon. We call this phenomenon spectral condensation and study its occurrence in fluid mechanical, optical and electronic systems. We define a set of spectral measures to quantify this condensation spanning several dynamical systems. Further, we uncover an inverse power law behaviour of spectral measures with the power corresponding to the dominant peak in the power spectrum in all the aforementioned systems.

A block moment method to handle spectral condensation phenomenon in parabolic control problems

This article is devoted to the characterization of the minimal null control time for abstract linear control problem. More precisely we aim at giving a precise answer to the following question: what is the minimal time needed to drive the solution of the system starting from any initial condition in a given subspace to zero? Our setting will encompass a wide variety of systems of coupled one dimensional linear parabolic equations with a scalar control.

Rare transitions to thin-layer turbulent condensates

Turbulent flows in a thin layer can develop an inverse energy cascade leading to spectral condensation of energy when the layer height is smaller than a certain threshold. These spectral condensates take the form of large-scale vortices in physical space. Recently, evidence for bistability was found in this system close to the critical height: depending on the initial conditions, the flow is either in a condensate state with most of the energy in the two-dimensional (2-D) large-scale modes, or in a three-dimensional (3-D) flow state with most of the energy in the small-scale modes. This bistable regime is characterised by the statistical properties of random and rare transitions between these two locally stable states. Here, we examine these statistical properties in thin-layer turbulent flows, where the energy is injected by either stochastic or deterministic forcing. To this end, by using a large number of direct numerical simulations (DNS), we measure the decay time $\unicode[STIX]{x1D70F}_{d}$ of the 2-D condensate to 3-D flow state and the build-up time $\unicode[STIX]{x1D70F}_{b}$ of the 2-D condensate. We show that both of these times $\unicode[STIX]{x1D70F}_{d},\unicode[STIX]{x1D70F}_{b}$ follow an exponential distribution with mean values increasing faster than exponentially as the layer height approaches the threshold. We further show that the dynamics of large-scale kinetic energy may be modelled by a stochastic Langevin equation. From time-series analysis of DNS data, we determine the effective potential that shows two minima corresponding to the 2-D and 3-D states when the layer height is close to the threshold.

Interaction of Phase-Modulated Femtosecond Pulses with an Optically Dense Quasi-Resonant Medium of Rubidium Vapors

For the first time, it is demonstrated that the magnitude and sign of the effect of “spectral condensation” of a laser pulse at the resonant-transition frequency of a dense medium can be controlled by changing the driving-pulse parameters (chirp, pulse width, and pulse amplitude). In the process of this, importantly, the driving-pulse energy and spectrum remain unchanged. Direct time-resolved measurements revealed an oscillatory character of the induced superradiance of rubidium vapors representing a long train of decaying short pulses. The width and repetition rate of the pulses in the train are determined by atomic density N0 of the medium, while the width of an entire superradiance pulse (10 ps) is considerably larger than that of the driving laser pulse (50 fs).

Thermalization of Photons in a Microcavity

A theoretical interpretation of the spectral condensation of photons in a microcavity based on solution of kinetic equations describing photon-gas thermalization as a result of Raman scattering from an atomic-gas thermostat is proposed.

Long-range ordering of topological excitations in a two-dimensional superfluid far from equilibrium

We study the relaxation of a 2D ultracold Bose-gas from a nonequilibrium initial state containing vortex excitations in experimentally realizable square and rectangular traps. We show that the subsystem of vortex gas excitations results in the spontaneous emergence of a coherent superfluid flow with a non-zero coarse-grained vorticity field. The streamfunction of this emergent quasi-classical 2D flow is governed by a Boltzmann-Poisson equation. This equation reveals that maximum entropy states of a neutral vortex gas that describe the spectral condensation of energy can be classified into types of flow depending on whether or not the flow spontaneously acquires angular momentum. Numerical simulations of a neutral point vortex model and a Bose gas governed by the 2D Gross-Pitaevskii equation in a square reveal that a large scale monopole flow field with net angular momentum emerges that is consistent with predictions of the Boltzmann-Poisson equation. The results allow us to characterise the spectral energy condensate in a 2D quantum fluid that bears striking similarity with similar flows observed in experiments of 2D classical turbulence. By deforming the square into a rectangular region, the resulting maximum entropy state switches to a dipolar flow field with zero net angular momentum.

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.

Experimental confirmation of self-regulating turbulence paradigm in two-dimensional spectral condensation.

Turbulent transport in magnetic fusion plasmas can be significantly suppressed by Reynolds-stress-induced zonal flows, allowing effective plasma confinement. We present experimental evidence of spatiotemporal correlation between small-scale turbulence-induced Reynolds stress and large-scale zonal flow production in the E×B driven hydrodynamic spectral condensation. We show that Reynolds stress is generated effectively by anisotropic vorticity structures possessing collective tilt angle. The maximum amplitude of the tilt, the Reynolds stress, and the mean zonal flow production coincide with the transition time of the velocity field, indicating a key role of turbulence-induced Reynolds stress in the condensation of the flow. The analysis of the energy transfer between turbulence and zonal flow shows coherent oscillations with π/2 phase delay, thus indicating a predator-prey-like interaction between zonal flow and turbulence.

Upscale energy transfer in three-dimensional rapidly rotating turbulent convection.

Rotating Rayleigh-Bénard convection exhibits, in the limit of rapid rotation, a turbulent state known as geostrophic turbulence. This state is present for sufficiently large Rayleigh numbers representing the thermal forcing of the system, and is characterized by a leading order balance between the Coriolis force and pressure gradient. This turbulent state is itself unstable to the generation of depth-independent or barotropic vortex structures of ever larger scale through a process known as spectral condensation. This process involves an inverse cascade mechanism with a positive feedback loop whereby large-scale barotropic vortices organize small scale convective eddies. In turn, these eddies provide a dynamically evolving energy source for the large-scale barotropic component. Kinetic energy spectra for the barotropic dynamics are consistent with a k-3 downscale enstrophy cascade and an upscale cascade that steepens to k-3 as the box-scale condensate forms. At the same time the flow maintains a baroclinic convective component with an inertial range consistent with a k-5/3 spectrum. The condensation process resembles a similar process in two dimensions but is fully three-dimensional.

Order, Criticality, and Excitations in the Extended Falicov-Kimball Model

Using exact numerical techniques, we investigate the nature of excitonic (electron-hole) bound states and the development of exciton coherence in the one-dimensional half-filled extended Falicov-Kimball model. The ground-state phase diagram of the model exhibits, besides band-insulator and staggered orbital ordered phases, an excitonic insulator (EI) with power-law correlations. The criticality of the EI state shows up in the von Neumann entropy. The anomalous spectral function and condensation amplitude provide the binding energy and coherence length of the electron-hole pairs which, on their part, point towards a Coulomb interaction driven crossover from BCS-like electron-hole pairing fluctuations to tightly bound excitons. We show that while a mass imbalance between electrons and holes does not affect the location of the BCS-BEC crossover regime, it favors staggered orbital ordering to the disadvantage of the EI. Within the Bose-Einstein condensation (BEC) regime, the quasiparticle dispersion develops a flat valence-band top, in accord with the experimental finding for Ta2NiSe5.

Inverse Energy Cascade in Forced Two-Dimensional Quantum Turbulence

We demonstrate an inverse energy cascade in a minimal model of forced 2D quantum vortex turbulence. We simulate the Gross-Pitaevskii equation for a moving superfluid subject to forcing by a stationary grid of obstacle potentials, and damping by a stationary thermal cloud. The forcing injects large amounts of vortex energy into the system at the scale of a few healing lengths. A regime of forcing and damping is identified where vortex energy is efficiently transported to large length scales via an inverse energy cascade associated with the growth of clusters of same-circulation vortices, a Kolmogorov scaling law in the kinetic energy spectrum over a substantial inertial range, and spectral condensation of kinetic energy at the scale of the system size. Our results provide clear evidence that the inverse energy cascade phenomenon, previously observed in a diverse range of classical systems, can also occur in quantum fluids.

Inverse energy cascade and turbulent transport in a quasi-two-dimensional magnetized electrolyte system: An experimental study

We present an experimental study of the inverse energy cascade, spectral condensation, and turbulent particle transport in an electromagnetically driven thin layer of NaCl electrolyte. The presence of the bottom friction provides an energy sink at large scales for the turbulent flow. This energy sink crucially contributes to the balance of the forcing and dissipation which makes the inverse cascade steady. The present work provides an estimation of the linear dissipation rate on an experimental basis. We also show how the dissipation rate affects the characteristic features of the velocity spectrum and the dynamics of the spectral condensation. A quantitative study of the turbulent diffusion shows a significant decrease of the radial transport during the spectral condensation process.

STRUCTURE FORMATION IN SPECTRALLY CONDENSED TURBULENCE

Two-dimensional (2D) turbulence supports the inverse energy cascade and the spectral condensate generation, which are studied in laboratory experiments. The generation of the spectral condensation via the inverse energy cascade dramatically reduces the radial transport in 2D flows. In this paper we report experimental results related to the formation of spectral condensates. The dynamics of the structural formation is reposted.

Turbulence in fluid layers

Flows in natural fluid layers are often forced simultaneously at scales smaller and much larger than the depth. For example, Earth atmospheric flows are powered by gradients of solar heating: vertical gradients cause three-dimensional (3D) convection while horizontal gradients drive planetary scale flows. Nonlinear interactions spread energy over scales. The question is whether intermediate scales obtain their energy from a large-scale 2D flow or from a small-scale 3D turbulence. The paradox is that 2D flows do not transfer energy downscale while 3D turbulence does not support an upscale transfer. Here we demonstrate experimentally how a large-scale vortex and a small-scale turbulence conspire to provide for an upscale energy cascade in thick layers. We show that a strong planar vortex suppresses vertical motions thus facilitating an upscale energy cascade. In a bounded system, spectral condensation into a box-size vortex provides for a self-organized planar flow which secures an upscale energy transfer.

Robust inverse energy cascade and turbulence structure in three-dimensional layers of fluid

Here, we report the first evidence of the inverse energy cascade in a flow dominated by 3D motions. Experiments are performed in thick fluid layers where turbulence is driven electromagnetically. It is shown that if the free surface of the layer is not perturbed, the top part of the layer behaves as quasi-2D and supports the inverse energy cascade regardless of the layer thickness. Well below the surface the cascade survives even in the presence of strong 3D eddies developing when the layer depth exceeds half the forcing scale. In a bounded flow at low bottom dissipation, the inverse energy cascade leads to the generation of a spectral condensate below the free surface. Such coherent flow can destroy 3D eddies in the bulk of the layer and enforce the flow planarity over the entire layer thickness.

Upscale energy transfer in thick turbulent fluid layers

Flows in natural fluid layers are often forced simultaneously at scales smaller and much larger than the depth. For example, the Earth’s atmospheric flows are powered by gradients of solar heating: vertical gradients cause three-dimensional (3D) convection whereas horizontal gradients drive planetary scale flows. Nonlinear interactions spread energy over scales1,2. The question is whether intermediate scales obtain their energy from a large-scale 2D flow or from a small-scale 3D turbulence. The paradox is that 2D flows do not transfer energy downscale whereas 3D turbulence does not support an upscale transfer. Here we demonstrate experimentally how a large-scale vortex and small-scale turbulence conspire to provide for an upscale energy cascade in thick layers. We show that a strong planar vortex suppresses vertical motions, thus facilitating an upscale energy cascade. In a bounded system, spectral condensation into a box-size vortex provides for a self-organized planar flow that secures an upscale energy transfer.