Kolmogorov Spectrum Research Articles

Quantum turbulence in superfluid helium: Decay and energy spectrum

The paper presents a comprehensive numerical study of the free decay of vortex tangle in superfluid helium. The initial vortex tangle represents one of the stationary configurations of loops in a counterflow with a laminar normal fluid component. The calculations are carried out in the framework of the vortex line method using the full Biot–Savart law over a wide range of temperatures. The aim of the study is to identify the role of various factors introduced into the numerical procedures (removal of small loops and segments during reconnections, the addition of and exclusion of vortex points on loops) and to determine the evolution of energy spectrum during the decay of quantum turbulence. A statistical approach is used to calculate the kinetic energy distribution on length scales. The calculations are carried out using periodic boundary conditions in a cube. The results show that, in agreement with the Feynman–Vinen theory, initially the rates of reduction in vortex line density at different temperatures are the same. However, when the vortex structure becomes rarefied, the influence of the mutual friction force becomes apparent, in agreement with Schwarz's theory. Statistical method for determining the energy spectrum is used. The Kolmogorov spectrum is not observed during decays at any temperature.

Simulation of a temporally dynamic atmospheric wavefront using the modified von-Kármán spectrum and Zernike-based method

In the simulation of the atmospheric wavefront, the Kolmogorov spectrum facilitates the mathematical treatments, but this model gives inaccurate predictions in cases where the effects of outer- and inner-scale can not be neglected. Therefore, we derive the covariance formula of Zernike coefficients based on the modified von-Kármán turbulence spectrum to improve the Zernike-based method. Further, using the frozen flow hypothesis, the temporal power spectra of Zernike coefficients are derived, and the effects of wind velocity and direction, as well as the outer and inner scales, are also taken into account. Then, we introduce a method for generating the temporally dynamic atmospheric wavefront using the inverse Fourier transform to draw random time-evolving coefficients from the temporal spectra. We evaluate the performance on simulated atmospheric wavefront, and the results confirm that accurate spatial and temporal statistics are obtained.

A theoretical study of a radiofrequency wave propagation through a realistic turbulent marine atmospheric boundary layer based on large eddy simulation

This study aims at modeling and investigating the impact of realistic turbulence inhomogeneity on radiowave propagation by utilizing large eddy simulations (LES). An up-to-date version of the X-LES method for generating realistic turbulent phase screens is first introduced. It combines atmospheric simulations with the classical Tatarski statistical modeling. This method naturally incorporates vertical turbulence inhomogeneity into phase screens at scales resolved by LES. It is then extended to sub-grid scales by weighting statistically generated phase variations with the vertical profile of the turbulent structure constant extracted from atmospheric data. This method is applied to replicate turbulence of a classical tropical marine atmospheric boundary layer. The impact of the generated medium on the propagation of a 10 GHz spherical wave emanating from a Gaussian aperture is analyzed through a statistical study of log-amplitude profiles performed for three different source altitudes. Results first show that contrary to a classical homogeneous turbulence modeling, log-amplitude profiles resulting from the propagation into inhomogeneous turbulence exhibit a statistical heterogeneity strongly dependent of source altitude. Furthermore, classical stochastic phase screen generation from a homogeneous Von-Kàrmàn Kolmogorov spectrum seems to give a statistically significant underestimation of the actual impact of turbulence compared to the X-LES method.

Probing the Interstellar Medium from Scintillation of the Swooshing Pulsar B0919+06

Radio observations of pulsars offer a potential method to probe the intricate microstructure in the turbulent interstellar medium. Here we report on a high-resolution dynamic spectral analysis of the “swooshing pulsar” B0919+06 observed with the Five-hundred-meter Aperture Spherical radio Telescope over multiple epochs and with the ultrawideband receiver on the Parkes radio telescope. For all observations, the dynamic scintillation spectra, two-dimensional autocovariance functions, and secondary spectra are presented. At 1250 MHz, the decorrelation bandwidth, diffraction timescale, and the drift rate are determined to be Δν d = 25.89 ± 7.55 MHz, Δτd=14.42±3.98 minutes, and dt/dν=0.07±0.14minutes MHz−1, respectively. The frequency dependencies of the scintillation parameters exhibit single power-law spectral behaviors, indicating that the electron density fluctuations in the interstellar medium approximately follow the Kolmogorov spectrum. The secondary spectra exhibit two distinct parabolic arcs with well-determined curvatures of 0.002 and 0.02 s3 for the outer and inner arcs, respectively. The locations of the scattering screens are approximately determined to be 157.3 and 726.0 pc, respectively, from the pulsar for isotropic scattering. The inner scintillation arc is present contemporaneously over a wide frequency range, indicating that the scintillation arc is a broadband phenomenon. The arc curvature scales with observing frequency as a power law with an index of −2.05 ± 0.05, which implies that the scattering screen spans a physical distance from 689.7 to 883.3 pc from the pulsar.

Numerical investigation of turbulence generation using Zakharov-like model equation

This study investigates the turbulence generation behavior with a Zakharov-like (ZL) equation in a fluid system. The model equation is derived using conservation equations (mass and momentum conservation), and the source of nonlinearity is the high amplitude of the acoustic wave. The Zakharov-like equation has been derived and then solved numerically, then turned into a modified nonlinear Schrödinger equation. Furthermore, modulation instability, or Benjamin–Feir instability, of the model equations, which leads to the emergence of Akhmediev breathers, is discussed. The numerical simulation uses a finite difference method for temporal evolution and a pseudo-spectral approach to determine spatial regimes. The outcomes indicate that the situation involving the nonlinear Schrödinger equation case displays a periodic pattern in space and time. The findings also demonstrate that the localization of structure and the Fermi, Pasta, and Ulam (FPU) recurrences are disrupted for the modified nonlinear Schrödinger equation and Zakharov-like equation cases. The energy spectrum exhibits a power law behavior that approximately follows k−1.65 in the ZL model equation case, and it is steeper than Kolmogorov's spectrum within the inertial sub-range.

Damping of MHD turbulence in a partially ionized medium

ABSTRACT The coupling state between ions and neutrals in the interstellar medium plays a key role in the dynamics of magnetohydrodynamic (MHD) turbulence, but is challenging to study numerically. In this work, we investigate the damping of MHD turbulence in a partially ionized medium using 3D two-fluid (ions + neutrals) simulations generated with the athenak code. Specifically, we examine the velocity, density, and magnetic field statistics of the two-fluid MHD turbulence in different regimes of neutral-ion coupling. Our results demonstrate that when ions and neutrals are strongly coupled, the velocity statistics resemble those of single-fluid MHD turbulence. Both the velocity structures and kinetic energy spectra of ions and neutrals are similar, while their density structures can be significantly different. With an excess of small-scale sharp density fluctuations in ions, the density spectrum in ions is shallower than that of neutrals. When ions and neutrals are weakly coupled, the turbulence in ions is more severely damped due to the ion-neutral collisional friction than that in neutrals, resulting in a steep kinetic energy spectrum and density spectrum in ions compared to the Kolmogorov spectrum. We also find that the magnetic energy spectrum basically follows the shape of the kinetic energy spectrum of ions, irrespective of the coupling regime. In addition, we find large density fluctuations in ions and neutrals and thus spatially inhomogeneous ionization fractions. As a result, the neutral-ion decoupling and damping of MHD turbulence take place over a range of length-scales.

Direct numerical simulation for lid-driven cavity under various Reynolds numbers in fully staggered grid

Finite difference method in a fully staggered grid is applied to solve the incompressible Navier–Stokes equation with direct numerical simulations. Without a turbulent or transient model, lid-driven cavity simulations are conducted with various Reynolds numbers from 102 to 106. The fluid property is fixed, and a lid velocity is changed to set the Reynolds number condition. Time steps are adjusted to keep the consistency of Courant number conditions. Simulation results are compared with the experimental measurements for a Reynolds number of 104 condition, in which the result shows relatively larger values of non-dimensional root mean square (RMS) compared to the other Reynolds number conditions. Vertical and horizontal velocity components show comparably higher RMS distributions around a downstream eddy region and above a bottom surface region, respectively, when the Reynolds number is 104. Time-averaged and RMS distributions show reasonable agreement with the experimental results, and a velocity spectral analysis shows the Kolmogorov spectrum of −5/3 slope for all velocity components. Taylor–Görtler-like (TGL) vortices are observed clearly in the downstream jet region. When the Reynolds number increases, the size of the TGL vortical structure in the spanwise direction decreases and numerous small-scale vortices occur.

Wave structure function, spatial coherence radius, and Fried parameter of optical wave propagating through ocean turbulence with a slant path.

The analytical formula for characteristic parameters of optical wave (wave structure function, spatial coherence radius, and Fried parameter) in the slant path of ocean turbulence are derived and analyzed. Under the Rytov approximation, the wave structure function derived by the oceanic power spectrum of the refractive index of optical turbulent fluctuations in the slant path still complies with the five-thirds power law of the Kolmogorov spectrum in the inertial subregion, and the relationship between spatial coherence radius and Fried parameters satisfies 2.1 times. The correctness of analytical formulation of the wave structure function is demonstrated by comparing the numerical results of the original integral formula with the analytical formula of the derived in this paper.

Atmospheric Turbulence with Kolmogorov Spectra: Software Simulation, Real-Time Reconstruction and Compensation by Means of Adaptive Optical System with Bimorph and Stacked-Actuator Deformable Mirrors

Atmospheric turbulence causes refractive index fluctuations, which in turn introduce extra distortions to the wavefront of the propagated radiation. It ultimately degrades telescope resolution (in imaging applications) and reduces radiation power density (in focusing applications). One of the possible ways of researching the impact of turbulence is to numerically simulate the spectrum of refractive index fluctuations, to reproduce it using a wavefront corrector and to measure the resultant wavefront using, for example, a Shack–Hartmann sensor. In this paper, we developed turbulence simulator software that generates phase screens with Kolmogorov spectra. We reconstructed the generated set of phase screens using a stacked-actuator deformable mirror and then compensated for the introduced wavefront distortions using a bimorph deformable mirror. The residual amplitude of the wavefront reconstructed by the 19-channel stacked-actuator mirror was 0.26 λ, while the residual amplitude of the wavefront compensated for by the 32-channel bimorph mirror was 0.08 λ.

Metallicity Dependence of Molecular Cloud Hierarchical Structure at Early Evolutionary Stages

The formation of molecular clouds out of H i gas is the first step toward star formation. Its metallicity dependence plays a key role in determining star formation throughout cosmic history. Previous theoretical studies with detailed chemical networks calculate thermal equilibrium states and/or thermal evolution under one-zone collapsing background. The molecular cloud formation in reality, however, involves supersonic flows, and thus resolving the cloud internal turbulence/density structure in three dimensions is still essential. We here perform magnetohydrodynamics simulations of 20 km s−1 converging flows of warm neutral medium (WNM) with 1 μG mean magnetic field in the metallicity range from the solar (1.0 Z ⊙) to 0.2 Z ⊙ environment. The cold neutral medium (CNM) clumps form faster with higher metallicity due to more efficient cooling. Meanwhile, their mass functions commonly follow at three cooling times regardless of the metallicity. Their total turbulence power also commonly shows the Kolmogorov spectrum with its 80% in the solenoidal mode, while the CNM volume alone indicates the transition toward Larson’s law. These similarities measured at the same time in units of the cooling time suggest that the molecular cloud formation directly from the WNM alone requires a longer physical time in a lower-metallicity environment in the 1.0–0.2 Z ⊙ range. To explain the rapid formation of molecular clouds and subsequent massive star formation possibly within ≲10 Myr as observed in the Large/Small Magellanic Clouds, the H i gas already contains CNM volume instead of pure WNM.

Observations of Kolmogorov Turbulence in Saturn's Magnetosphere

AbstractThe Kolmogorov scaling in the inertial range of scales is a distinct characteristic of fully developed turbulence, and studying it offers valuable insights into the evolution of turbulence. In this work, we perform a statistical survey of the power spectra with the Kolmogorov scaling in Saturn's magnetosphere using Cassini measurements. Two cases study show that both magnetic‐field and electron density spectra exhibit f −5/3 at the MHD scales. The statistical analysis reveals a wide‐ranging and abundant presence of Kolmogorov spectra throughout magnetosphere, observed across all local times. Interestingly, the occurrence rate of these Kolmogorov‐like events within Saturn's magnetosphere surpasses that observed in the planetary magnetosheath. The measurements of magnetic compressibility for the Kolmogorov‐like events show the dominance of incompressible Alfvénic turbulence (44.64%) with respect to magnetosonic‐like one (6.94%). In addition, the source and evolution of the turbulent fluctuations are further discussed.

Jackson R. Herring and the Statistical Closure Problem of Turbulence: A Review of Renormalized Perturbation Theories

The pioneering applications of the methods of theoretical physics to the turbulence statistical closure problem are summarised. These are: the direct-interaction approximation (DIA) of Kraichnan, the self-consistent-field theory of Edwards, and the self-consistent-field theory of Herring. Particular attention is given to the latter, in terms of its elegance and its pedagogical value. We then concentrate on the assessment of these theories and take the historical route of Kraichnan’s diagnosis of the failure of DIA, followed by Edwards’s analysis of the failure of his self-consistent theory, when compared to the Kolmogorov spectrum. As all three theories are closely related, these analyses also shed light on Herring’s theory. The second-generation theories that grew out of this assessment are then discussed. First, there were the Lagrangian theories, initially stemming from the work of Kraichnan and Herring, and later the purely Eulerian local energy-transfer (LET) theory. The latter is significant because its development exposes the underlying problems with the pioneering theories in terms of the basic physics of the inertial energy transfer. In particular, later work allows us to assign a unified explanation of the incompatibility of all three pioneering theories with the Kolmororov spectrum, in that they are all Markovian approximations (in wavenumber) to the non-Markovian phenomenon of fluid turbulence. In the interests of completeness, we briefly review the formalisms of Wyld and Martin, Siggia, and Rose. More recent developments are also discussed, in order to bring the subject up to the present day.

Studies on Quantum Turbulence with Vinen

In my research career in the field of quantum turbulence, I have been encouraged by Vinen. In this article, I review my works motivated by him and my joint collaborations with him. Vinen encouraged me in studies on quantum turbulence at zero temperature, Kolmogorov spectrum of a vortex tangle without mutual friction, and fully coupled dynamics of quantized vortices and normal fluid. Joint works include studies on diffusion of a vortex tangle, Kelvin wave cascade, quantum turbulence created by an oscillating object, and coupled dynamics of tracer particles and quantized vortices.

Performance analysis of coherent DPSK SIMO laser-based satellite-to-ground communication link over weak-to-strong turbulence channels considering Kolmogorov and non-Kolmogorov spectrum models

The performance of satellite-to-ground laser-based communication links is highly affected by atmospheric turbulence. Coherent detection with spatial diversity at the ground station receiver can mitigate the scintillation effects caused by atmospheric turbulence. Traditionally, the scintillation effects are modeled based on the Kolmogorov spectrum model. However, the experiments have indicated that scintillation effects on the laser beam propagation have non-Kolmogorov properties. Our goal in the present work is to analyze the average bit error rate (BER), outage probability (OP), and ergodic capacity of the satellite-to-ground heterodyne optical communication system with receiver spatial diversity. A differential phase-shift keying modulation technique is considered in this work. The propagated laser signal from the satellite to the ground station is assumed to be subjected to Málaga-distributed atmospheric turbulence. The atmospheric turbulence statistics are carried out based on the conventional Kolmogorov spectrum model and the three-layer altitude (TLA) non-Kolmogorov spectrum model. The performance of popular diversity combining techniques, namely, maximum ratio combining (MRC) and equal gain combining (EGC) techniques are analyzed. The statistical models of the MRC technique under the Málaga-distributed atmospheric channel model are obtained in analytical form expressions. The statistical models of the EGC technique under the Málaga-distributed atmospheric channel model are obtained via the fast Fourier transform representation of the characteristic function method. Based on these statistical models, average BER, OP, and ergodic capacity expressions for each type of diversity combining technique are derived. For the communication system under investigation, the performance of MRC and EGC multiple aperture receiver systems are compared to a single aperture receiver with the same total aperture area. These comparisons are carried out under the same conditions in terms of zenith angle and signal-to-noise ratio. The obtained results show that the performance of the optical communication system under investigation with MRC and EGC receivers can be improved by increasing the order of diversity. In addition, it is found that the difference in the performance between Kolmogorov and TLA non-Kolmogorov spectrum models is not significant at low zenith angles, while this difference increases as the zenith angle increases. All numerical results are verified by Monte-Carlo simulations.

Theory of magnetic turbulence and shock formation induced by a collisionless plasma instability

Magnetic fields are ubiquitous in universe, space, and laboratory plasmas. Especially, self-generated magnetic fields are important to know the mind of nature. The formation of Weibel-mediated collisionless shock is studied theoretically as a structure formation by the linear plasma wave growth, nonlinear saturation, and mode–mode coupling. Following a series of computer simulations and experimental studies of the physics, a simple model equation is proposed here to describe the time evolution of magnetic turbulence. Weibel instability is saturated by magnetic pressure, and thicker filaments continue to be generated by current coalescence (magnetic reconnection) mechanism. The model equation concludes the fact that the filament spacing increases linearly in time, and the magnetic energy power spectrum is given as Bk2 ∝ k−2. The time evolution of the turbulence is characterized with the cascade toward smaller k. Such inverse cascade is well-known in 2D hydrodynamic turbulence such as a typhoon or hurricane formation and is known to have Kolmogorov spectrum k−5/3. Although only a small difference in power, the physics of inverse cascades is very different as shown in the present paper. With use of Alfvén current limit condition, the criteria of collisionless shock formation are evaluated. The present theory is compared to corresponding experiments done with Omega and NIF lasers and a variety of PIC simulations. The theory is also applied to evaluate the strength of magnetic field near the shock front of the supernova remnant SN1006. The enhancement of magnetic field of about 25 μG is concluded in the present theory. Finally, a universality of the model equation is shown by applying the theory to the turbulent mixing due to Rayleigh–Taylor instability at the contact surface of two fluids in a gravitational or inertial force, which is very important in compressing plasma such as inertial confinement fusion by implosion. It is shown that the well-known evolution physics, mixing layer of the two fluids grows in proportion to (time)2, can be explained with the same model equation.

Propagation effects in the synthesis of wind turbine aerodynamic noise

The sound field radiated by a wind turbine changes significantly with propagation distance, depending on the meteorological conditions and on the type of ground. In this article, we present a wind turbine noise synthesis model which is based on theoretical source and propagation models. The source model is based on Amietâ’s theory for the prediction of the trailing edge noise and the turbulent inflow noise. The trailing edge noise uses the wall pressure spectrum calculated with Leeâ’s model for the suction side and Goodyâ’s model for the pressure side. The Kolmogorov spectrum is used for the prediction of the turbulent inflow noise. To account for the propagation effects associated with atmospheric refraction and ground reflection, a wide angle parabolic equation in inhomogeneous moving medium is considered. The scattering due to the turbulence in the atmosphere is accounted for using the Harmonoise model. The synthesis method is based on the moving monopole model to accurately predict the amplitude modulations at the receiver, and uses cross-fading between overlapping grains to obtain the time signals from the frequency-domain prediction model. Finally, audio signals are provided for a few test cases to emphasize various propagation phenomena associated with wind turbine noise.

Turbulent Magnetic Field Amplification by the Interaction of a Shock Wave and Inhomogeneous Medium

Magnetic fields of the order of 100 μG observed in young supernova remnants cannot be amplified by shock compression alone. To investigate the amplification caused by a turbulent dynamo, we perform three-dimensional MHD simulations of the interaction between a shock wave and an inhomogeneous density distribution with a shallow spectrum in the preshock medium. The postshock turbulence is mainly driven by the strongest preshock density contrast and follows the Kolmogorov scaling. The resulting turbulence amplifies the postshock magnetic field. The time evolution of the magnetic fields agrees with the prediction of the nonlinear turbulent dynamo theory of Xu & Lazarian. When the initially weak magnetic field is perpendicular to the shock normal, the maximum amplification of the field’s strength reaches a factor of ≈200, which is twice as large as that for a parallel shock. We find that the perpendicular shock exhibits a smaller turbulent Alfvén Mach number in the vicinity of the shock front than the parallel shock. However, the strongest magnetic field has a low volume filling factor and is limited by the turbulent energy due to the reconnection diffusion taking place in a turbulent and magnetized fluid. The magnetic field strength averaged along the z-axis is reduced by a factor ≳10. We decompose the turbulent velocity and magnetic field into solenoidal and compressive modes. The solenoidal mode is dominant and evolves to follow the Kolmogorov scaling, even though the preshock density distribution has a shallow spectrum. When the preshock density distribution has a Kolmogorov spectrum, the turbulent velocity’s compressive component increases.

Electron Density Variations in the Interstellar Medium and the Average Frequency Profile of a Scintle from Pulsar Scintillation Spectra

We observed the scintillation pattern of nine bright pulsars at 324 MHz and three at 1.68 GHz and analyzed the wavenumber spectrum, which is related to electron density variations of the plasma turbulence of the interstellar medium (ISM). For all pulsars the frequency section of the autocorrelation function (ACF) of the dynamic spectra to at least 45% of the maximum corresponds to predictions of scattering theories with a range of power-law exponents of the wavenumber spectrum of 3.56 ≤ α ≤ 3.97 with errors ≤0.05 and a mean with standard deviation of 3.76 ± 0.13. The range includes α = 3.67 for the Kolmogorov spectrum. Similar results, although with larger errors, were found from the Fourier transform of the ACFs down to ∼10−3 of the maximum. No clear case of a distinction between thin-screen and extended-medium scattering models was found. The average frequency profile of the scintles can be characterized for steep wavenumber spectra with α ≲ 4 by a cusp with a somewhat rounded peak. For flatter spectra, down to at least α ∼ 3.56 the cusp with its peak becomes more pronounced and its decay steepens. We discuss our findings in the context of the scattering characteristics of the ISM.

Method of wavefront phase retrieval from wavefront curvature sensing using membrane modes.

Wavefront phase retrieval is one of the most critical problems in adaptive optics. Here, phase retrieval by solving the transport of intensity equation using membrane vibration modes is proposed. Our study shows that the wavefront curvature sensing signal on the pupil can be expanded as a set of corresponding membrane vibration modes. The analytic expressions of the reconstructed phase are given. The coefficients of the functions are obtained by the integral over the pupil and boundary. Several representative Zernike circular and annular polynomials are respectively fitted by eigenfunctions and membrane modes in the absence of noise. In addition, wavefront recovery from noisy curvature data of the simulated atmospheric turbulence phase based on Zernike modes and Kolmogorov spectrum is demonstrated to verify the accuracy and robustness of the proposed method.

Plasma noise in TianQin time-delay interferometry

TianQin is a proposed geocentric space-based gravitational wave observatory mission, which requires time-delay interferometry (TDI) to cancel laser frequency noise. With high demands for precision, solar-wind plasma environment at $\sim 10^5$ km above the Earth may constitute a non-negligible noise source to laser interferometric measurements between satellites, as charged particles perturb the refractivity along light paths. In this paper, we first assess the plasma noises along single links from space-weather models and numerical orbits, and analyze the time and frequency domain characteristics. Particularly, to capture the plasma noise in the entire measurement band of $10^{-4} - 1$ Hz, we have performed additional space-weather magnetohydrodynamic simulations in finer spatial and temporal resolutions and utilized Kolmogorov spectra in high-frequency data generation. Then we evaluate the residual plasma noises of the first- and second-generation TDI combinations. Both analytical and numerical estimations have shown that under normal solar conditions the plasma noise after TDI is less than the secondary noise requirement. Moreover, TDI is shown to exhibit moderate suppression on the plasma noise below $\sim 10^{-2}$ Hz due to noise correlation between different arms, when compared with the secondary noise before and after TDI.