Inertial Range Research Articles

Non-local dispersion and the reassessment of Richardson's t3-scaling law

The Richardson-scaling law states that the mean square separation of a fluid particle pair grows according to t3 within the inertial range and at intermediate times. The theories predicting this scaling regime assume that the pair separation is within the inertial range and that the dispersion is local, which means that only eddies at the scale of the separation contribute. These assumptions ignore the structural organization of the turbulent flow into large-scale shear layers, where the intense small-scale motions are bounded by the large-scale energetic motions. Therefore, the large scales contribute to the velocity difference across the small-scale structures. It is shown that, indeed, the pair dispersion inside these layers is highly non-local and approaches Taylor dispersion in a way that is fundamentally different from the Richardson-scaling law. Also, the layer's contribution to the overall mean square separation remains significant as the Reynolds number increases. This calls into question the validity of the theoretical assumptions. Moreover, a literature survey reveals that, so far, t3 scaling is not observed for initial separations within the inertial range. We propose that the intermediate pair dispersion regime is a transition region that connects the initial Batchelor- with the final Taylor-dispersion regime. Such a simple interpretation is shown to be consistent with observations and is able to explain why t3 scaling is found only for one specific initial separation outside the inertial range. Moreover, the model incorporates the observed non-local contribution to the dispersion, because it requires only small-time-scale properties and large-scale properties.

Geometry of Magnetic Fluctuations near the Sun from the Parker Solar Probe

Abstract Solar wind magnetic fluctuations exhibit anisotropy due to the presence of a mean magnetic field in the form of the Parker spiral. Close to the Sun, direct measurements were not available until the recently launched Parker Solar Probe (PSP) mission. The nature of the anisotropy and geometry of the magnetic fluctuations play a fundamental role in dissipation processes and in the transport of energetic particles in space. Using PSP data, we present measurements of the geometry and anisotropy of the inner heliosphere magnetic fluctuations, from fluid to kinetic scales. The results are surprising and different from 1 au observations. We find that fluctuations evolve characteristically with size scale. However, unlike 1 au solar wind, at the outer scale, the fluctuations are dominated by wavevectors quasi-parallel to the local magnetic field. In the inertial range, average wavevectors become less field aligned, but still remain more field aligned than near-Earth solar wind. In the dissipation range, the wavevectors become almost perpendicular to the local magnetic field in the dissipation range, to a much higher degree than those indicated by 1 au observations. We propose that this reduced degree of anisotropy in the outer scale and inertial range is due to the nature of large-scale forcing outside the solar corona.

Statistical study of electron density turbulence and ion-cyclotron waves in the inner heliosphere: Solar Orbiter observations

Context.The recently released spacecraft potential measured by the RPW instrument on board Solar Orbiter has been used to estimate the solar wind electron density in the inner heliosphere.Aims.The measurement of the solar wind’s electron density, taken in June 2020, has been analysed to obtain a thorough characterization of the turbulence and intermittency properties of the fluctuations. Magnetic field data have been used to describe the presence of ion-scale waves.Methods.To study and quantify the properties of turbulence, we extracted selected intervals. We used empirical mode decomposition to obtain the generalized marginal Hilbert spectrum, equivalent to the structure functions analysis, which additionally reduced issues typical of non-stationary, short time series. The presence of waves was quantitatively determined by introducing a parameter describing the time-dependent, frequency-filtered wave power.Results.A well-defined inertial range with power-law scalng was found almost everywhere in the sample studied. However, the Kolmogorov scaling and the typical intermittency effects are only present in fraction of the samples. Other intervals have shallower spectra and more irregular intermittency, which are not described by models of turbulence. These are observed predominantly during intervals of enhanced ion frequency wave activity. Comparisons with compressible magnetic field intermittency (from the MAG instrument) and with an estimate of the solar wind velocity (using electric and magnetic field) are also provided to give general context and help determine the cause of these anomalous fluctuations.

An Analysis of a Laminar-Turbulent Transition and Thermal Plumes Behavior in a Paramagnetic Fluid Subjected to an External Magnetic Field

Transition to turbulence and changes in the fluid flow structure are subjects of continuous analysis and research, especially for unique fields of research such as the thermo-magnetic convection of weakly magnetic fluids. Therefore, an experimental and numerical research of the influence of an external magnetic field on a natural convection’s fluid flow was conducted in the presented research. The experimental part was performed for an enclosure with a 0.5 aspect ratio, which was filled with a paramagnetic fluid and placed in a superconducting magnet in a position granting the enhancement of the flow. The process was recorded as temperature signals from the thermocouples placed in the analyzed fluid. The numerical research enabled an investigation based not only on temperature, but velocities as well. Experimental and numerical data were analyzed with the application of extended fast Fourier transform and wavelet analysis. The obtained results allowed the determination of changes in the nature of the flow and visualization of the influence of an imposed strong magnetic field on a magnetic fluid. It is proved that an applied magnetic field actuates the flow in Rayleigh-Benard convection and causes the change from laminar to turbulent flow for fairly low magnetic field inductions (2T and 3T for ΔT = 5 and 11 °C respectively). Fast Fourier transform allowed the definition of characteristic frequencies for oscillatory states in the flow, as well as an observation that the high values of magnetic field elongate the inertial range of the flow on the power spectrum density. Temperature maps obtained during numerical simulations granted visualizations of thermal plume formation and behavior with increasing magnetic field.

A Novel Method for Estimating the Intrinsic Magnetic Field Spectrum of Kinetic-Range Turbulence

Understanding plasma turbulence below the ion characteristic scales is one of the key open problems of solar wind physics. The bulk of our knowledge about the nature of the kinetic-scale fluctuations comes from the high-cadence measurements of the magnetic field. The spacecraft frame frequencies of the sub-ion scale fluctuations are frequently around the Nyquist frequencies of the magnetic field sampling rate. Thus, the resulting ‘measured’ time series may significantly differ from the ‘true’ ones. It follows that second-order moments (e.g., power spectral density, PSD) of the signal may also be highly affected in both their amplitude and their slope. In this paper, we focus on the estimation of the PSD slope for finitely sampled data and we unambiguously define a so-called local slope in the framework of Continuous Wavelet Transform. Employing Monte Carlo simulations, we derive an empirical formula that assesses the statistical error of the local slope estimation. We illustrate the theoretical results by analyzing measurements of the magnetic field instrument (MFI) on board the Wind spacecraft. Our analysis shows that the trace power spectra of magnetic field measurements of MFI can be modeled as the sum of PSD of an uncorrelated noise and an intrinsic signal. We show that the local slope strongly depends on the signal-to-noise (S/N) ratio, stressing that noise can significantly affect the slope even for S/N around 10. Furthermore, we show that the local slopes below the frequency corresponding to proton inertial length, 5≳kλpi>1, depend on the level of the magnetic field fluctuations in the inertial range (Pin), exhibiting a gradual flattening from about −11/3 for high Pin toward about −8/3 for low Pin.

Scale interactions and anisotropy in Rayleigh–Taylor turbulence

We study energy scale transfer in Rayleigh–Taylor (RT) flows by coarse graining in physical space without Fourier transforms, allowing scale analysis along the vertical direction. Two processes are responsible for kinetic energy flux across scales: baropycnal work $\varLambda$ , due to large-scale pressure gradients acting on small scales of density and velocity; and deformation work $\varPi$ , due to multiscale velocity. Our coarse-graining analysis shows how these fluxes exhibit self-similar evolution that is quadratic-in-time, similar to the RT mixing layer. We find that $\varLambda$ is a conduit for potential energy, transferring energy non-locally from the largest scales to smaller scales in the inertial range where $\varPi$ takes over. In three dimensions, $\varPi$ continues a persistent cascade to smaller scales, whereas in two dimensions $\varPi$ rechannels the energy back to larger scales despite the lack of vorticity conservation in two-dimensional (2-D) variable density flows. This gives rise to a positive feedback loop in 2-D RT (absent in three dimensions) in which mixing layer growth and the associated potential energy release are enhanced relative to 3-D RT, explaining the oft-observed larger $\alpha$ values in 2-D simulations. Despite higher bulk kinetic energy levels in two dimensions, small inertial scales are weaker than in three dimensions. Moreover, the net upscale cascade in two dimensions tends to isotropize the large-scale flow, in stark contrast to three dimensions. Our findings indicate the absence of net upscale energy transfer in three-dimensional RT as is often claimed; growth of large-scale bubbles and spikes is not due to ‘mergers’ but solely due to baropycnal work $\varLambda$ .

Erratum: “Dependence of 3D Self-correlation Level Contours on the Scales in the Inertial Range of Solar Wind Turbulence” (2019, ApJL, 883, L9)

Erratum: “Dependence of 3D Self-correlation Level Contours on the Scales in the Inertial Range of Solar Wind Turbulence” (2019, ApJL, 883, L9)

Comment on “Mathematical constraints on the scaling exponents in the inertial range of fluid turbulence” [Phys. Fluids 33(3), 031703 (2021)

The paper [Djenidi et al., Phys. Fluids 33(3), 031703 (2021)] considers a classical issue of an anomalous scaling of velocity structure functions in a high-Reynolds number turbulent flow. The paper offers a mathematical proof of the ground-breaking result: the intermittency is an artifact of the Reynolds number finiteness. However, the proof contains a technical error that makes this conclusion ungrounded.

Effect of wall temperature on the kinetic energy transfer in a hypersonic turbulent boundary layer

The effect of wall temperature on the transfer of kinetic energy in a hypersonic turbulent boundary layer for different Mach numbers and wall temperature ratios is studied by direct numerical simulation. A cold wall temperature can enhance the compressibility effect in the near-wall region through increasing the temperature gradient and wall heat flux. It is shown that the cold wall temperature enhances the local reverse transfer of kinetic energy from small scales to large scales, and suppresses the local direct transfer of kinetic energy from large scales to small scales. The average filtered spatial convection and average filtered viscous dissipation are dominant in the near-wall region, while the average subgrid-scale flux of kinetic energy achieves its peak value in the buffer layer. It is found that the wall can suppress the inter-scale transfer of kinetic energy, especially for the situation of a cold wall. A strong local reverse transfer of fluctuating kinetic energy is identified in the buffer layer in the inertial range. Helmholtz decomposition is applied to analyse the compressibility effect on the subgrid-scale flux of kinetic energy. A strong transfer of the solenoidal component of fluctuating kinetic energy is identified in the buffer layer, while a significant transfer of the dilatational component of fluctuating kinetic energy is observed in the near-wall region. It is also shown that compression motions have a major contribution to the direct transfer of fluctuating kinetic energy, while expansion motions play a marked role in the reverse transfer of fluctuating kinetic energy.

Toward autonomous large eddy simulations of turbulence based on interscale energy transfer among resolved scales

Is an autonomous large eddy simulation (LES) possible where a subgrid-scale (SGS) model is extracted in a bootstrapping way from LES fields? We show how the SGS energy transfer among resolved scales and its wave number distribution can be obtained from evolving LES fields. This information, supplemented by known asymptotic properties of energy flux in the inertial range, allows self-contained LES without prescribing extraneous SGS models. The method is tested in LES of isotropic turbulence at high Reynolds number (Re) where inertial range dynamics is expected and for lower Re decaying turbulence under conditions of the classical Comte-Bellot and Corrsin experiments (energy spectra in figure).

Spatial integration effect on velocity spectrum: Towards an interpretation of the − 11/3 power law observed in the spectra of turbine outputs

To improve the turbine operational life, the interaction between flow properties and turbine performance needs to be elucidated. We then propose to examine the physical origin of the power-law scaling in the inertial range of turbine power outputs by experimentally exploring the spectral content of a 1:20 scaled model of a three-bladed horizontal-axis turbine positioned in a 3D turbulent flow. First, measurements confirm that the turbine power frequency spectra exhibit a power law decay proportional to −11/3 in the inertial range. Knowing that the turbine power fluctuations are linearly dependent on the incoming velocity fluctuations, PIV measurements are carried out to study the effect of the spatially integrated velocity onto its resulted spectrum. It is demonstrated that in inhomogeneous anisotropic turbulent flow, the velocity spectrum of its spatial average along N direction(s) has an inertial slope of −5/3 − 2N/3. This information is used to physically interpret the power-law scaling in the inertial range of the turbine power spectra. The previously observed f−11/3 scaling results from a 2D-spatial average velocity field coupled with a spectral average over blades. This physical explanation confirms previous works in which a transfer function was developed between incoming turbulence and the turbine power outputs.

Numerical dispersion effects on the energy cascade in large-eddy simulation

Implicitly filtered large-eddy simulation (LES) is by nature numerically under-resolved. With the sole exception of Fourier-spectral methods, discrete numerical derivative operators cannot accurately represent the dynamics of all of the represented scales. Since the resolution scale in an LES usually lies in the inertial range, these poorly represented scales are dynamically significant and errors in their dynamics can affect all resolved scales. This Letter is focused on characterizing the effects of numerical dispersion error by studying the energy cascade in LES of convecting homogeneous isotropic turbulence. Numerical energy and transfer spectra reveal that energy is not transferred at the appropriate rate to wavemodes where significant dispersion error is present. This leads to a deficiency of energy in highly dispersive modes and an accompanying pile up of energy in the well resolved modes, since dissipation by the subgrid model is diminished. An asymptotic analysis indicates that dispersion error causes a phase decoherence between triad interacting wavemodes, leading to a reduction in the mean energy transfer rate for these scales. These findings are relevant to a wide range of LES, since turbulence commonly convects through the grid in practical simulations. Further, these results indicate that the resolved scales should be defined to not include the dispersive modes.

Fiber Tracking Velocimetry for Two-Point Statistics of Turbulence

We propose and validate a novel experimental technique to measure two-point statistics of turbulent flows. It consists in spreading rigid fibers in the flow and tracking their position and orientation in time and therefore been named ``Fiber Tracking Velocimetry'' (FTV). By choosing different fiber lengths, i.e. within the inertial or dissipative range of scales, the statistics of turbulence fluctuations at the selected lengthscale can be probed accurately by simply measuring the fiber velocity at its two ends, and projecting it along the transverse-to-the-fiber direction. By means of fully-resolved direct numerical simulations and experiments, we show that these fiber-based transverse velocity increments are statistically equivalent to the (unperturbed) flow transverse velocity increments. Moreover, we show that the turbulent energy dissipation rate can be accurately measured exploiting sufficiently short fibers. The technique has been tested against standard Particle Tracking Velocimetry (PTV) of flow tracers with excellent agreement. Our technique overcomes the well-known problem of PTV to probe two-point statistics reliably because of the fast relative diffusion in turbulence that prevents the mutual distance between particles to remain constant at the lengthscale of interest. This problem, making it difficult to obtain converged statistics for a fixed separation distance, is even more dramatic for natural flows in open domains. A prominent example are oceanic currents, where drifters (i.e.~the tracer-particle counterpart used in field measurements) disperse quickly, but at the same time their number has to be limited to save costs. Inspired by our laboratory experiments, we propose pairs of connected drifters as a viable option to solve the issue.

A New Approach to Decaying Homogeneous Isotropic Turbulence: Reynolds-Number Dependence of the Kolmogorov Constant (1941), Skewness of Velocity Difference and the Characteristic Function of Two-Point Velocity Probability Density

The non-dimensional prefactor of the Kolmogorov 2/3 law in the inertial range of homogeneous isotropic turbulence (HIT) has been believed to be a universal constant for many years, and then people ...

The Batchelor Spectrum of Passive Scalar Turbulence in Stochastic Fluid Mechanics at Fixed Reynolds Number

In 1959 Batchelor predicted that the stationary statistics of passive scalars advected in fluids with small diffusivity k should display a power spectrum along an inertial range contained in the viscous‐convective range of the fluid model. This prediction has been extensively tested, both experimentally and numerically, and is a core prediction of passive scalar turbulence.In this article we provide a rigorous proof of a version of Batchelor's prediction in the limit when the scalar is subjected to a spatially smooth, white‐in‐time stochastic source and is advected by the 2D Navier‐Stokes equations or 3D hyperviscous Navier‐Stokes equations in forced by sufficiently regular, nondegenerate stochastic forcing. Although our results hold for fluids at arbitrary Reynolds number, this value is fixed throughout. Our results rely on the quantitative understanding of Lagrangian chaos and passive scalar mixing established in our recent works. Additionally, in the limit, we obtain statistically stationary, weak solutions in to the stochastically forced advection problem without diffusivity. These solutions are almost‐surely not locally integrable distributions with nonvanishing average anomalous flux and satisfy the Batchelor spectrum at all sufficiently small scales. We also prove an Onsager‐type criticality result that shows that no such dissipative, weak solutions with a little more regularity can exist. © 2021 Wiley Periodicals LLC.

Multidimensional low-Mach number time-implicit hydrodynamic simulations of convective helium shell burning in a massive star

Context. A realistic parametrization of convection and convective boundary mixing in conventional stellar evolution codes is still the subject of ongoing research. To improve the current situation, multidimensional hydrodynamic simulations are used to study convection in stellar interiors. Such simulations are numerically challenging, especially for flows at low Mach numbers which are typical for convection during early evolutionary stages. Aims. We explore the benefits of using a low-Mach hydrodynamic flux solver and demonstrate its usability for simulations in the astrophysical context. Simulations of convection for a realistic stellar profile are analyzed regarding the properties of convective boundary mixing. Methods. The time-implicit Seven-League Hydro (SLH) code was used to perform multidimensional simulations of convective helium shell burning based on a 25 M⊙ star model. The results obtained with the low-Mach AUSM+-up solver were compared to results when using its non low-Mach variant AUSMB+-up. We applied well-balancing of the gravitational source term to maintain the initial hydrostatic background stratification. The computational grids have resolutions ranging from 180 × 902 to 810 × 5402 cells and the nuclear energy release was boosted by factors of 3 × 103, 1 × 104, and 3 × 104 to study the dependence of the results on these parameters. Results. The boosted energy input results in convection at Mach numbers in the range of 10−3–10−2. Standard mixing-length theory predicts convective velocities of about 1.6 × 10−4 if no boosting is applied. The simulations with AUSM+-up show a Kolmogorov-like inertial range in the kinetic energy spectrum that extends further toward smaller scales compared with its non low-Mach variant. The kinetic energy dissipation of the AUSM+-up solver already converges at a lower resolution compared to AUSMB+-up. The extracted entrainment rates at the boundaries of the convection zone are well represented by the bulk Richardson entrainment law and the corresponding fitting parameters are in agreement with published results for carbon shell burning. However, our study needs to be validated by simulations at higher resolution. Further, we find that a general increase in the entropy in the convection zone may significantly contribute to the measured entrainment of the top boundary. Conclusion. This study demonstrates the successful application of the AUSM+-up solver to a realistic astrophysical setup. Compressible simulations of convection in early phases at nominal stellar luminosity will benefit from its low-Mach capabilities. Similar to other studies, our extrapolated entrainment rate for the helium-burning shell would lead to an unrealistic growth of the convection zone if it is applied over the lifetime of the zone. Studies at nominal stellar luminosities and different phases of the same convection zone are needed to detect a possible evolution of the entrainment rate and the impact of radiation on convective boundary mixing.

Katabatic Winds over Steep Slopes: Overview of a Field Experiment Designed to Investigate Slope-Normal Velocity and Near-Surface Turbulence

We describe a new field campaign over a steep, snowy \(30^{\circ }\) alpine slope, designed to investigate three recurrent issues in experimental studies of steep-slope katabatic winds. (1) Entrainment is known to be present in katabatic jets and has been estimated at the interface between the jet and the boundary layer above it. However, to our knowledge, the slope-normal velocity component has never been measured in the katabatic jet. (2) It is hard to accurately measure turbulence in the first tens of centimetres above the surface using standard sonic anemometry due to the filtering effect of the long instrument path. The present field experiment used a three-dimensional multi-hole pitot-type probe with a high sampling frequency (1250 Hz) that was positioned as close to the surface as 3 cm. It provides three-dimensional mean velocity and Reynolds stress tensor from which dissipation can be estimated, as well as spectra for the turbulent quantities. Energy spectra reveal a well-developed inertial range and capture the inertial scales and some of the dissipative scales. (3) Measuring turbulence on a mast usually provides information about mean and turbulent quantities at certain discrete heights because the sensors are sparsely located inside the jet. We present the first measurements of well-developed katabatic flows where the full wind-speed and temperature profiles acquired, from tethered balloon are available at the location of the measurement mast, which comprises three-dimensional anemometry and thermometry.

Observation of Kolmogorov Turbulence in the Jovian Magnetosheath From JADE Data

AbstractPower spectra of magnetic‐field fluctuations near Jupiter have been analyzed since the early days of space exploration with Voyager and Ulysses flyby of Jupiter. Power spectra of velocity and density fluctuations, however, have not been as well studied, due to the lack of high‐resolution measurements required to resolve a significant fraction of the inertial range. Here, we investigate fluid‐scale turbulence in Jupiter's magnetosheath using measurements from the Jovian Auroral Distributions Experiment (JADE) instrument onboard Juno spacecraft. Both ion density and velocity spectra exhibit nearly Kolmogorov scaling ∼f−5/3. This indicates weakly compressible fluctuations and the presence of a scale‐invariant cascade through the inertial range. Although this is a specific case study and not necessarily representative of the overall behavior, our result shows that Alfvénic Kolmogorov‐type turbulence exists in at least some locations in Jupiter's magnetosheath.

Osnovnyye zakonomernosti vertikal'nogo turbulentnogo obmena v kvaziodnorodnykh i stratifitsirovannykh sloyakh Chernogo morya

Purpose. The purpose of the study is to assess the coefficient of vertical turbulent exchange for different layers of the Black Sea basin based on the experimental data on microstructure of the physical fields obtained for the period 2004–2019 in the Black Sea and using the semi-empirical models. Methods and Results. For the upper mixed layer, the turbulent energy dissipation rate ɛ and the exchange coefficient were calculated using the velocity fluctuation spectra based on the Kolmogorov hypotheses on the turbulence spectrum inertial range. In the stratified layers, the turbulence coefficient and the dissipation rate were experimentally determined both from the spectra of the velocity horizontal fluctuations’ gradients and the vertical spectra of temperature fluctuations using the concept of the effective scale of turbulent spots. Depending on the features of the hydrological regime and the prevailing energy contributors to turbulence generation, five layers were identified and described (including their characteristic power dependences of the vertical turbulent diffusion coefficients K on the buoyancy frequency N) using the 1.5D-model of vertical turbulent exchange for the basin under study. For the stratified layers, the 1.5D-model results were comparatively analyzed with those of the other semi-empirical and theoretical models describing the most probable hydrophysical processes in each specific layer; the relations for the vertical turbulent exchange coefficient were obtained depending on the buoyancy frequency. Conclusions. Comparison of the experimental data collected under different hydrometeorological conditions with the simulations resulted from the known turbulence models for the sea upper layer showed that the best agreement between the simulation and measurement data was provided by a multiscale model taking into account three basic mechanisms of turbulence generation: current velocity shear, instability of wave motions, and wave breaking. The turbulent exchange coefficient dependencies on depth are conditioned by the affect of the turbulence dominant source at a given level. In the stratified layers, the exchange coefficient dependence on buoyancy frequency is determined by the hydrophysical processes in each layer; the relations obtained for individual layers indicate intensity of the contributions of vertical advection, internal wave breakings, turbulence diffusion and geothermal flux.

Spectral Transfer and Kármán–Howarth–Monin Equations for Compressible Hall Magnetohydrodynamics

Abstract We derive two new forms of the Kármán–Howarth–Monin (KHM) equation for decaying compressible Hall magnetohydrodynamic (MHD) turbulence. We test them on results of a weakly compressible, 2D, moderate-Reynolds-number Hall MHD simulation and compare them with an isotropic spectral transfer (ST) equation. The KHM and ST equations are automatically satisfied during the whole simulation owing to the periodic boundary conditions and have complementary cumulative behavior. They are used here to analyze the onset of turbulence and its properties when it is fully developed. These approaches give equivalent results characterizing the decay of the kinetic + magnetic energy at large scales, the MHD and Hall cross-scale energy transfer/cascade, the pressure dilatation, and the dissipation. The Hall cascade appears when the MHD one brings the energy close to the ion inertial range and is related to the formation of reconnecting current sheets. At later times, the pressure dilatation energy exchange rate oscillates around zero, with no net effect on the cross-scale energy transfer when averaged over a period of its oscillations. A reduced 1D analysis suggests that all three methods may be useful to estimate the energy cascade rate from in situ observations.