Inertial Range Research Articles

Markovian Features of the Solar Wind at Subproton Scales

Abstract The interplanetary magnetic field carried out from the Sun by the solar wind displays fluctuations over a wide range of scales. While at large scales, say at frequencies lower than 0.1–1 Hz, fluctuations display the universal character of fully developed turbulence with a well-defined Kolmogorov-like inertial range, the physical and dynamical properties of the small-scale regime as well as their connection with the large-scale ones are still a debated topic. In this work we investigate the near-Sun magnetic field fluctuations at subproton scales by analyzing the Markov property of fluctuations and recovering basic information about the nature of the energy transfer across different scales. By evaluating the Kramers–Moyal coefficients we find that fluctuations in the subproton range are well described as a Markovian process with Probability Density Functions (PDFs) modeled via a Fokker–Planck (FP) equation. Furthermore, we show that the shape of the PDFs is globally scale-invariant and similar to the one recovered for the stationary solution of the FP equation at different scales. The relevance of our results on the Markovian character of subproton scale fluctuations is also discussed in connection with the occurrence of turbulence in this domain.

Numerical study on the influence of pre-swirl angle on internal flow characteristics of centrifugal pumps

The effect of inlet pre-swirl on the performance of a centrifugal pump is studied by numerical simulation. The governing equations are Navier–Stokes equations and the shear stress transport k–ω turbulence model. The numerical results show that the optimal operating point moves from the low flow region to the high flow region as the pre-swirl angle shifts from positive to negative. It is found by contours of Omega–Liutex that the positive pre-swirl angle is able to weaken the vortex on the blade suction and reduce the energy dissipation. On both the 0.5Q0 and 1.2Q0 operating conditions, the proportion of entropy production loss in the impeller and volute is about 60% and 30%, respectively. As the pre-swirl angle changes from negative to positive, the entropy production loss in the inlet and outlet pipes increases slowly, and the entropy production loss in the volute and impeller shows a decreasing trend and the peak area of entropy loss moves toward the outlet. Under the four pre-swirl angles, the main frequency is always the passing frequency of the blade. The pre-swirl angle affects the pressure fluctuation at the main frequency but has little effect at the secondary frequency. The change in velocity pulsation amplitude in the impeller in the positive pre-swirl angle is smaller than that in the negative pre-swirl angle. As a result, for the positive pre-swirl angle, the turbulent kinetic energy density in the impeller is low, and the energy loss is low, compared with negative pre-swirl. Under the low flow condition (0.5Q0), the change in velocity pulsation amplitude in the inertial range of the energy spectrum under negative pre-swirl is greater than that under positive pre-swirl.

Dynamics of finite-size spheroids in turbulent flow: the roles of flow structures and particle boundary layers

We study the translational and rotational dynamics of neutrally buoyant finite-size spheroids in hydrodynamic turbulence by means of fully resolved numerical simulations. We examine axisymmetric shapes, from oblate to prolate, and the particle volume dependences. We show that the accelerations and rotations experienced by non-spherical inertial-scale particles result from volume filtered fluid forces and torques, similar to spherical particles. However, the particle orientations carry signatures of preferential alignments with the surrounding flow structures, which are reflected in distinct axial and lateral fluctuations for accelerations and rotation rates. The randomization of orientations does not occur even for particles with volume-equivalent diameter size in the inertial range, here up to $60$ dissipative units ( $\eta$ ) at Taylor-scale Reynolds number ${Re_{\lambda }=120}$ . Additionally, we demonstrate that the role of fluid boundary layers around the particles cannot be neglected in reaching a quantitative understanding of particle statistical dynamics, as they affect the intensities of the angular velocities and the relative importance of tumbling with respect to spinning rotations. This study brings to the fore the importance of inertial-scale flow structures in homogeneous and isotropic turbulence and their impacts on the transport of neutrally buoyant bodies with sizes in the inertial range.

Hydrodynamic and acoustic pressure fluctuations in water pipes due to an orifice: Comparison of measurements with Large Eddy Simulations

The present article investigates the turbulent wall pressure fluctuations due to flow through single-hole sharp-square-edged orifices (opening area ratios of 11%, 20% and 31%) in a water-filled pipe by means of measurements and incompressible Large Eddy Simulations (LES) for opening area ratios of 20% and 31%. The orifice plate thickness was half the orifice diameter. The static pressure measurements and simulations show that the vena-contracta factors of the orifices are dependent on Reynolds numbers, which is due to the finite thickness of the orifices. These LES simulations provide a good prediction of the near-field hydrodynamic wall pressure fluctuations, which dominate up to the first three pipe diameters downstream of the orifice. Upstream and far downstream wall pressure fluctuations are dominated by acoustic pressure fluctuations. The source of the radiated sound field was estimated from the surface integral of the fluctuations across the orifice in the LES. The sound field was calculated by implementing the sound source in a one dimensional acoustic model of the test section. The power spectrum density (PSD) of the wall pressure fluctuations for pipe Reynolds numbers varying from 4000 to 10000 collapsed when presented in dimensionless form, which uses the dynamic pressure in the orifice as reference pressure and the ratio of cross-sectional averaged orifice velocity and orifice diameter as characteristic frequency. In the inertial range of the turbulence a global decay of the pressure PSD and of the sound source show a power of the Strouhal number between −113 and −2.8. For higher frequencies the pressure fluctuations were dominated by acoustic resonances. For wider orifices the magnitude of these acoustic pressure fluctuations is predicted within a factor two by the acoustic model. For the narrowest orifice (opening ratio of 11%) the measured PSD of pressure fluctuations displays sharp peaks due to self-sustained oscillations (whistling). A simple power law of the sound source is not accurate due to the unstable hydrodynamic modes related to orifice thickness. However, using the simple model for the sound source the acoustic model predicts the acoustic pressure fluctuations reasonably well for all orifices. At low frequency one observes a peak in the PSD hydrodynamic wall-pressure fluctuations at one pipe diameter downstream of the orifice. It is probably due to an acoustically silent “edge-tone-like” sinusoidal self-sustained motion of the jet.

An In-depth Numerical Study of Exact Laws for Compressible Hall Magnetohydrodynamic Turbulence

Abstract Various exact laws governing compressible magnetohydrodynamic (MHD) and compressible Hall-MHD (CHMHD) turbulence have been derived in recent years. Other than their fundamental theoretical interest, these laws are generally used to estimate the energy dissipation rate from spacecraft observations in order to address diverse problems related, e.g., to heating of the solar wind and magnetospheric plasmas. Here we use various 10243 direct numerical simulation data of free-decay isothermal CHMHD turbulence obtained with the GHOST code (Geophysical High-Order Suite for Turbulence) to analyze two of the recently derived exact laws. The simulations reflect different intensities of the initial Mach number and the background magnetic field. The analysis demonstrates the equivalence of the two laws in the inertial range and relates the strength of the Hall effect to the amplitude of the cascade rate at sub-ion scales. When taken in their general form (i.e., not limited to the inertial range), some subtleties regarding the validity of the stationarity assumption or the absence of the forcing in the simulations are discussed. We show that the free-decay nature of the turbulence induces a shift from a large-scale forcing toward the presence of a scale-dependent reservoir of energy fueling the cascade or dissipation. The reduced form of the exact laws (valid in the inertial range) ultimately holds even if the stationarity assumption is not fully verified.

Intermittency in the Expanding Solar Wind: Observations from Parker Solar Probe (0.16 au), Helios 1 (0.3–1 au), and Voyager 1 (1–10 au)

Abstract We examine statistics of magnetic-field vector components to explore how intermittency evolves from near-Sun plasma to radial distances as large as 10 au. Statistics entering the analysis include autocorrelation, magnetic structure functions of the order of n (SF n ), and scale-dependent kurtosis (SDK), each grouped in ranges of heliocentric distance. The Goddard Space Flight Center Space Physics Data Facility provides magnetic-field measurements for resolutions of 6.8 ms for Parker Solar Probe, 6 s for Helios, and 1.92 s for Voyager 1. We compute SF2 to determine the scales encompassing the inertial range and examine SDK to investigate the degree of non-Gaussianity. Autocorrelations are used to resolve correlation scales. Correlation lengths and ion inertial lengths provide an estimate of effective Reynolds number (Re). Variation in Re allows us to examine for the first time the relationship between SDK and Re in an interplanetary plasma. A conclusion from this observed relationship is that regions with lower Re at a fixed physical scale have on average lower kurtosis, implying less intermittent behavior. Kolmogorov refined similarity hypothesis is applied to magnetic SF n and kurtosis to calculate intermittency parameters and fractal scaling in the inertial range. A refined Voyager 1 magnetic-field data set is generated.

The Yaglom Scaling of the Third-order Structure Functions in the Inner Heliosphere Observed by Helios 1 and 2

Abstract The third-order scaling law, Yaglom law, of Elsässer fluctuations in the solar wind is believed to reflect the inertial range energy cascade of the MHD turbulence and provides an approach to evaluate the cascade rate. However, the occurrence ratio with the Yaglom scaling law, the fraction of the intervals where the Yaglom linear scaling is observed, is reported to be low (0.05–0.30) in the high-latitude solar wind observed by the Ulysses spacecraft. Whether the occurrence ratio could be higher in other conditions remains unknown. Here, we analyze the occurrence of the third-order scaling in the inner heliosphere with the first 100 days of observation of the Helios 1 and Helios 2 spacecraft. We investigate 162 intervals in the leading edges and 323 intervals in the trailing edges of the high-speed streams, respectively. All of these intervals have a time duration of 9 hr. We find that in the inner heliosphere the occurrence ratio is relatively high in the leading edges (0.58) and moderate in the trailing edges (0.45). Among the data intervals with the Yaglom scaling in the leading edges, 94.7% of intervals give positive rates, while in the trailing edges 78.6% give negative rates. The variations of the occurrence ratio with various turbulence parameters are shown. The cascade rate is found to be higher than the proton heating rate calculated from the data, which have third-order scaling. These new results raise several questions related to the nature and origin of the third-order scaling law and may initiate new studies on solar wind turbulence.

Low-Altitude Sensing of Urban Atmospheric Turbulence with UAV

The capabilities of a quadcopter in the hover mode for low-altitude sensing of atmospheric turbulence with high spatial resolution in urban areas characterized by complex orography are investigated. The studies were carried out in different seasons (winter, spring, summer, and fall), and the quadcopter hovered in the immediate vicinity of ultrasonic weather stations. The DJI Phantom 4 Pro quadcopter and AMK-03 ultrasonic weather stations installed in different places of the studied territory were used in the experiment. The smoothing procedure was used to study the behavior of the longitudinal and lateral spectra of turbulence in the inertial and energy production ranges. The longitudinal and lateral turbulence scales were estimated by the least-square fit method with the von Karman model as a regression curve. It is shown that the turbulence spectra obtained with DJI Phantom 4 Pro and AMK-03 generally coincide, with minor differences observed in the high-frequency region of the spectrum. In the inertial range, the behavior of the turbulence spectra shows that they obey the Kolmogorov–Obukhov “5/3” law. In the energy production range, the longitudinal and lateral turbulence scales and their ratio measured by DJI Phantom 4 Pro and AMK-03 agree to a good accuracy. Discrepancies in the data obtained with the quadcopter and the ultrasonic weather stations at the territory with complex orography are explained by the partial correlation of the wind velocity series at different measurement points and the influence of the inhomogeneous surface.

Global expressions for high-order structure functions in Burgers turbulence

Since the famous work by Kolmogorov on incompressible turbulence, the structure-function theory has been a key foundation of modern turbulence study. Due to the simplicity of Burgers turbulence, structure functions are calculated to arbitrary orders, which provides numerous implications for other compressible turbulent systems. We present the derivation of exact forcing-scale resolving expressions for high-order structure functions of the burgers turbulence. Compared with the previous theories, where the structure functions are calculated in the inertial range based on the statistics of shocks, our expressions link high-order structure functions in different orders without extra information on the flow structure and are valid beyond the inertial range, therefore they are easily checked by numerical simulations.

Mathematical reformulation of the Kolmogorov–Richardson energy cascade in terms of vortex stretching

In this paper, with the aid of direct numerical simulations (DNS) of forced turbulence in a periodic domain, we mathematically reformulate the Kolmogorov–Richardson energy cascade in terms of vortex stretching. By using the description, we prove that if the Navier–Stokes flow satisfies a new regularity criterion in terms of the enstrophy production rate, then the flow does not blow up. Our DNS results seem to support this regularity criterion. Next, we mathematically construct the hierarchy of tubular vortices, which is statistically self-similar in the inertial range. Under the assumptions of the scale-locally of the vortex stretching/compressing (i.e. energy cascade) process and the statistical independence between vortices that are not directly stretched or compressed, we can derive the −5/3 power law of the energy spectrum of statistically stationary turbulence without directly using the Kolmogorov hypotheses.

Spectral Properties of the N Component of the Heliospheric Magnetic Field from IMP and ACE Observations for 1973–2020

Abstract We analyze the normal (N) component of the heliospheric magnetic field observed by the Interplanetary Monitoring Platform and the Advanced Composition Explorer spacecraft for the period 1973–2020. Parameters characterizing the frequency spectrum are calculated with a novel technique, which is based on calculating variances at incremental lags to yield the integral of a turbulence spectrum. We compare this technique with the standard second-order structure function to show their similarity in the inertial range, and use the latter to calculate correlation functions. We find that the yearly average for magnetic field magnitude and the variance attained their lowest values since spacecraft observations began for the period that includes the 2020 solar minimum, 4.2 nT and 3.3 nT2, respectively. The ratio of the magnitude of fluctuations of the N component to the field magnitude shows little variation, with an average value of 0.43 ± 0.04. The average value of the spectral index of the energy range for the whole data set is −1.0 ± 0.1, and shows some solar-cycle dependence. The average value for the inertial range is an almost constant −1.69 ± 0.04. While the break between the energy and the inertial range is difficult to determine accurately to search for a solar-cycle dependence, an indirect indication of such a dependence follows when the ratio of spectral levels in the energy and in the inertial range is calculated. The e-folding correlation length has an average value of 1.1 ± 0.3 Mkm, with a clear solar-cycle dependence.

The “Singular” Behavior of the Solar Wind Scaling Features during Parker Solar Probe–BepiColombo Radial Alignment

Abstract At the end of 2020 September, the Parker Solar Probe (PSP) and BepiColombo were radially aligned: PSP was orbiting near 0.17 au and BepiColombo near 0.6 au. This geometry is of particular interest for investigating the evolution of solar wind properties at different heliocentric distances by observing the same solar wind plasma parcels. In this work, we use the magnetic field observations from both spacecraft to characterize both the topology of the magnetic field at different heliocentric distances (scalings, high-order statistics, and multifractal features) and its evolution when moving from near-Sun to far-Sun locations. We observe a breakdown of the statistical self-similar nature of the solar wind plasma with an increase in the efficiency of the nonlinear energy cascade mechanism when moving away from the Sun. We find a complex organization of large field gradients to dissipate the excess of kinetic energy across the inertial range near the Sun, whereas the topological organization of small fluctuations is still primarily responsible for the energy transfer rate at 0.6 au. These results provide, for the first time, evidence of the different roles of dissipation mechanisms near and far away from the Sun.

High-resolution Simulations of the Inner Heliosphere in Search of the Kelvin–Helmholtz Waves

Abstract The Kelvin–Helmholtz instability (KHI) can be generated at velocity shears in plasmas. While shears are abundant in the solar wind, whether they generate KHI in situ remains an open question, because of the lack of models that can simultaneously resolve the global structure of the expanding solar wind and the local structure of much smaller-scale velocity shears. In this paper, we use the Grid Agnostic MHD for Extended Research Applications model whose high resolving power, in combination with a highly refined spatial grid, allowed us to extend the simulation from global scales roughly into the first decade of the inertial range (∼1.5 × 105 km, which we refer to as mesoscale). We employ this computational capability to extract from the simulation the local properties of radial and azimuthal solar wind velocity shears and investigate their KH stability using a linear dispersion relation, which includes both the finite width of the shear and plasma compressibility. We find that radial shears, which dominate the global structure of the inner heliosphere, are stabilized by compressibility. However, depending on the local Alfvén speed, sound speed, shear thickness, and the strength of the stabilizing azimuthal magnetic field, the azimuthal shears generated inside stream interaction regions could be KH-unstable. While our highly resolved simulation allowed us to analyze the local properties of the velocity shears, its resolution was still insufficient to confirm the instability. We argue that even higher resolution simulations are required to reproduce in situ generation of KHI at velocity shears in the solar wind.

Energy spectrum and energy budget of superfluid turbulence using two-fluid shell model

Using a two-fluid shell model, we numerically investigate the energy spectrum and the scale by scale energy budget for superfluid turbulence in the temperature range of T = 0.9–2.175 K. Both the normal and superfluid components exhibit the Kolmogorov-like spectrum (Ek ∼ k−5/3) in the inertial range for all the temperatures. The scale to scale energy budget analysis in the wavenumber space indicates a strong coupling between the normal and superfluid components at low temperature (T = 0.9 K), which is quite evident from the observation of balancing between the viscous dissipation and the energy transfer due to the mutual friction between the components at low temperature for the normal component. We also compute the temporal probability distribution function of the energy induced by the mutual friction to a particular component, which indicates the positive energy supply at low temperature (T = 0.9 K) and negative energy supply at high temperature (T = 1.9 K) for the normal component. The results are consistent with the previous numerical and experimental studies.

A randomly stirred model for Bolgiano-Obukhov scaling in turbulence in a stably stratified fluid.

A randomly stirred model, akin to the one used by DeDominicis and Martin for homogeneous isotropic turbulence, is introduced to study Bolgiano-Obukhov scaling in fully developed turbulence in a stably stratified fluid. The energy spectrum E(k), where k is a wavevector in the inertial range, is expected to show the Bolgiano-Obukhov scaling at a large Richardson number Ri (a measure of the stratification). We find that the energy spectrum is anisotropic. Averaging over the directions of the wavevector, we find [Formula: see text], where εθ is the constant energy transfer rate across wavenumbers with very little contribution coming from the kinetic energy flux. The constant K0 is estimated to be of O(0.1) as opposed to the Kolmogorov constant, which is O(1). Further for a pure Bolgiano-Obukhov scaling, the model requires that the large distance 'stirring' effects dominate in the heat diffusion and be small in the velocity dynamics. These could be reasons why the Bolgiano-Obukhov scaling is difficult to observe both numerically and experimentally. This article is part of the theme issue 'Scaling the turbulence edifice (part 2)'.

Puff turbulence in the limit of strong buoyancy.

We provide a numerical validation of a recently proposed phenomenological theory to characterize the space–time statistical properties of a turbulent puff, both in terms of bulk properties, such as the mean velocity, temperature and size, and scaling laws for velocity and temperature differences both in the viscous and in the inertial range of scales. In particular, apart from the more classical shear-dominated puff turbulence, our main focus is on the recently discovered new regime where turbulent fluctuations are dominated by buoyancy. The theory is based on an adiabaticity hypothesis which assumes that small-scale turbulent fluctuations rapidly relax to the slower large-scale dynamics, leading to a generalization of the classical Kolmogorov and Kolmogorov–Obukhov–Corrsin theories for a turbulent puff hosting a scalar field. We validate our theory by means of massive direct numerical simulations finding excellent agreement.This article is part of the theme issue ‘Scaling the turbulence edifice (part 2)’.

Scaling Laws for Partially Developed Turbulence

We formulate multifractal models for velocity differences and gradients which describe the full range of length scales in turbulent flow, namely: laminar, dissipation, inertial, and stirring ranges. The models subsume existing models of inertial range turbulence. In the localized ranges of length scales in which the turbulence is only partially developed, we propose multifractal scaling laws with scaling exponents modified from their inertial range values. In local regions, even within a fully developed turbulent flow, the turbulence is not isotropic nor scale invariant due to the influence of larger turbulent structures (or their absence). For this reason, turbulence that is not fully developed is an important issue which inertial range study can not address. In the ranges of partially developed turbulence, the flow can be far from universal, so that standard inertial range turbulence scaling models become inapplicable. The model proposed here serves as a replacement. Details of the fitting of the parameters for the τp and ζp models in the dissipation range are discussed. Some of the behavior of ζp for larger p is unexplained. The theories are verified by comparing to high resolution simulation data.

Characteristics of Turbulence Driven by Transient Magnetic Reconnection in the Terrestrial Magnetotail

Abstract This paper investigates the evolution of turbulence within the magnetotail reconnection exhaust observed by the Magnetospheric Multiscale spacecraft. The reconnection was in an unsteady state that caused significant temporal variations in the outflow speed. By dividing the exhaust into nine fast flows, we analyzed and compared the characteristics of turbulence in these nine flows. We find that the strength of the intermittency has a good relationship with the peak speed of the fast flows. The higher-order analysis of magnetic field fluctuations reveals that the turbulence is multifractal in the inertial range for these flows except one with the highest peak speed. Moreover, the turbulence is monofractal on kinetic scales in all of these fast flows. The magnetic energy was intermittently dissipated in these turbulent flows, predominantly occurred in the coherent structures. Since the coherent structures with the largest energy dissipation in these flows are different, we suggest that the mechanism of energy dissipation may be different among these flows.

Inertial-range Magnetic-fluctuation Anisotropy Observed from Parker Solar Probe’s First Seven Orbits

Abstract Solar wind turbulence is anisotropic with respect to the mean magnetic field. Anisotropy leads to ambiguity when interpreting in situ turbulence observations in the solar wind because an apparent change in the measurements could be due to either the change of intrinsic turbulence properties or to a simple change of the spacecraft sampling direction. We demonstrate the ambiguity using the spectral index and magnetic compressibility in the inertial range observed by the Parker Solar Probe during its first seven orbits ranging from 0.1 to 0.6 au. To unravel the effects of the sampling direction, we assess whether the wave-vector anisotropy is consistent with a two-dimensional (2D) plus slab turbulence transport model and determine the fraction of power in the 2D versus slab component. Our results confirm that the 2D plus slab model is consistent with the data and the power ratio between 2D and slab components depends on radial distance, with the relative power in 2D fluctuations becoming smaller closer to the Sun.

Relative dispersion with finite inertial ranges

Relative dispersion experiments are often analysed using theoretical predictions from two- and three-dimensional turbulence. These apply to infinite inertial ranges, assuming the same dispersive behaviour over all scales. With finite inertial ranges, the metrics are less conclusive. We examine this using pair separation probability density functions (PDFs), obtained by integrating a Fokker–Planck equation with different diffusivity profiles. We consider time-based metrics, such as the relative dispersion, and separation-based metrics, such as the finite scale Lyapunov exponent (FSLE). As the latter cannot be calculated from a PDF, we introduce a new measure, the cumulative inverse separation time (CIST), which can. This behaves like the FSLE, but advantageously has analytical solutions in the inertial ranges. This allows the establishment of consistency between the time- and space-based metrics, something which has been lacking previously. We focus on three dispersion regimes: non-local spreading (as in a two-dimensional enstrophy inertial range), Richardson dispersion (as in an energy inertial range) and diffusion (for uncorrelated pair motion). The time-based metrics are more successful with non-local dispersion, as the corresponding PDF applies from the initial time. Richardson dispersion is barely observed, because the self-similar PDF applies only asymptotically in time. In contrast, the separation-based CIST correctly captures the dependencies, even with a short (one decade) inertial range, and is superior to the traditional FSLE at large scales. Nevertheless, it is advantageous to use all measures together, to seek consistent indications of the dispersion.