Inertial Range Research Articles

Dual scaling and the n-thirds law in grid turbulence

A dual scaling of the turbulent longitudinal velocity structure function $\overline {{(\delta u)}^n}$ , i.e. a scaling based on the Kolmogorov scales ( $u_K$ , $\eta$ ) and another based on ( $u'$ , $L$ ) representative of the large scale motion, is examined in the context of both the Kármán–Howarth equation and experimental grid turbulence data over a significant range of the Taylor microscale Reynolds number $Re_\lambda$ . As $Re_\lambda$ increases, the scaling based on ( $u'$ , $L$ ) extends to increasingly smaller values of $r/L$ while the scaling based on ( $u_K$ , $\eta$ ) extends to increasingly larger values of $r/\eta$ . The implication is that both scalings should eventually overlap in the so-called inertial range as $Re_\lambda$ continues to increase, thus leading to a power-law relation $\overline {{(\delta u)}^n} \sim r^{n/3}$ when the inertial range is rigorously established. The latter is likely to occur only when $Re_\lambda \to \infty$ . The use of an empirical model for $\overline {{(\delta u)}^n}$ , which complies with $\overline {{(\delta u)}^n} \sim r^{n/3}$ as $Re_\lambda \to \infty$ , shows that the finite Reynolds number effect may differ between even- and odd-orders of $\overline {{(\delta u)}^n}$ . This suggests that different values of $Re_\lambda$ may be required between even and odd values of $n$ for compliance with $\overline {{(\delta u)}^n} \sim r^{n/3}$ . The model describes adequately the dependence on $Re_\lambda$ of the available experimental data for $\overline {{(\delta u)}^n}$ and supports indirectly the extrapolation of these data to infinitely large $Re_\lambda$ .

Advanced detached-eddy simulation of the MD 30P-30N three-element airfoil

An experimental version in the Detached-Eddy Simulation (DES) family (called Advanced DES or ADES) is introduced and tested on a geometry that is fairly complex but two-dimensional. The essential change in ADES is that the user is given control of the regions treated with full turbulence modelling (RANS) and those treated with Large-Eddy Simulation (LES). This zonal character makes the approach more powerful, but less practical, so that in its current state it is not ready for industrial CFD. The grid requirements of the two regions are very different, and multi-block grid structure is natural. Another key feature is a Volumetric Synthetic Turbulence Generator (VSTG), installed to feed the LES region with viable resolved turbulence, so that the resolved Reynolds stresses rapidly substitute for the modelled Reynolds stresses present in the RANS region. The VSTG operates in a volume, rather than on a surface and can be active in attached boundary layers, at a trailing edge, or after separation. The well-known McDonnell-Douglas 30P-30N airfoil is simulated with periodic lateral boundary conditions. The VSTG is successful, and the desired nature of simulation is obtained in each region. ADES involves zonal decisions, but appears robust. An inertial range is clearly indicated in frequency spectra. A grid-refinement study is included, as well as variations in lateral domain size and STG positions; this led to a matrix of 11 simulations. Cases are shown at four angles of attack and with three RANS models in addition to ADES. Pressure and friction distributions and velocity and shear stress profiles are compared in detail. The prospects for an evolution of ADES into a practical routine approach in the long term are discussed.

Compressible two-dimensional turbulence: cascade reversal and sensitivity to imposed magnetic field

We study the impact of compressibility on two-dimensional turbulent flows, such as those modeling astrophysical disks. We demonstrate that the direction of cascade undergoes continuous transition as the Mach number Ma increases, from inverse at Ma = 0, to direct at . Thus, at comparable amounts of energy flow from the pumping scale to large and small scales, in accord with previous data. For supersonic turbulence with the cascade is direct, as in three dimensions, which results in multifractal density field. For that regime () we derive a Kolmogorov-type law for potential forcing and obtain an explicit expression for the third order correlation tensor of the velocity. We further show that all third order structure functions are zero up to first order in the inertial range scales, which is in sharp contrast with incompressible turbulence where the third order structure function, that describes the energy flux associated with the energy cascade is non-zero. The properties of compressible turbulence have significant implications on the amplification of magnetic fields in conducting fluids. We thus demonstrate that imposing external magnetic field on compressible flows of conducting fluids allows to manipulate the flow producing possibly large changes even at small Mach numbers. Thus Zeldovich’s antidynamo theorem, by which at Ma = 0 the magnetic field is zero in the steady state, must be used with caution. Real flows have finite Ma and, however small it is, for large enough values of I, the magnetic flux through the disk, the magnetic field changes the flow appreciably, or rearranges it completely. This renders the limit Ma → 0 singular for non-zero values of I. Of particular interest is the effect of the density multifractality, at which is relevant for astrophysical disks. We demonstrate that in that regime, in the presence of non-zero I the magnetic field energy is enhanced by a large factor as compared to its estimates based on the mean field. Finally, based on the insights described above, we propose a novel two-dimensional Burgers’ turbulence, whose three-dimensional counterpart is used for studies of the large-scale structure of the Universe, as a model for supersonic two-dimensional magnetohydrodynamic flows.

Decomposition of Vertical Velocity and Its Zonal Wavenumber Kinetic Energy Spectra in the Hydrostatic Atmosphere

Abstract The spectrum of kinetic energy of vertical motions (VKE) is less well understood compared to the kinetic energy spectrum of horizontal motions (HKE). One challenge that has limited progress in describing the VKE spectrum is a lack of a unified approach to the decomposition of vertical velocities associated with the Rossby motions and inertia–gravity (IG) wave flows. This paper presents such a unified approach using a linear Rossby–IG vertical velocity normal-mode decomposition appropriate for a spherical, hydrostatic atmosphere. New theoretical developments show that for every zonal wavenumber k, the limit VKE is proportional to the total mechanical energy and to the square of the frequency of the normal mode. The theory predicts a VKE ∝ k−5 and a VKE ∝ k1/3 power law for the Rossby and IG waves, assuming a k−3 and a k−5/3 power law for the Rossby and IG HKE spectra, respectively. The Kelvin and mixed Rossby–gravity wave VKE spectra are predicted to follow k−1 and k−5 power laws, respectively. The VKE spectra for ERA5 data from August 2018 show that the Rossby VKE spectra approximately follow the predicted a k−5 power law. The expected k1/3 power law for the gravity wave VKE spectrum is found only in the SH midlatitude stratosphere for k ≈ 10–60. The inertial range IG VKE spectra in the tropical and midlatitude troposphere reflect a mixture of ageostrophic and convection-coupled dynamics and have slopes between −1 and −1/3, likely associated with too steep IG HKE spectra. The forcing by quasigeostrophic ageostrophic motions is seen as an IG VKE peak at synoptic scales in the SH upper troposphere, which gradually moves to planetary scales in the stratosphere. Significance Statement The spectrum of kinetic energy of vertical motions (VKE) is less well understood compared to the kinetic energy spectrum of horizontal motions. One challenge is a lack of a unified approach to the decomposition of vertical velocities associated with the Rossby motions and inertia–gravity (IG) wave flows. This paper presents such a unified approach using a linear Rossby–IG vertical velocity normal-mode decomposition appropriate for a spherical, hydrostatic atmosphere. It is shown that for every zonal wavenumber, the limit VKE is proportional to the total mechanical energy and to the square of the frequency of the normal mode. The theory is successfully applied to the ERA5 data. It leads the way for a more accurate computation of momentum fluxes.

Turbulent dynamo action and its effects on the mixing at the convective boundary of an idealized oxygen-burning shell

Convection is one of the most important mixing processes in stellar interiors. Hydrodynamic mass entrainment can bring fresh fuel from neighboring stable layers into a convection zone, modifying the structure and evolution of the star. Because flows in stellar convection zones are highly turbulent, multidimensional hydrodynamic simulations are fundamental to accurately capture the physics of mixing processes. Under some conditions, strong magnetic fields can be sustained by the action of a turbulent dynamo, adding another layer of complexity and possibly altering the dynamics in the convection zone and at its boundaries. In this study, we used our fully compressible SEVEN-LEAGUE HYDRO code to run detailed and highly resolved three-dimensional magnetohydrodynamic simulations of turbulent convection, dynamo amplification, and convective boundary mixing in a simplified setup whose stratification is similar to that of an oxygen-burning shell in a star with an initial mass of 25 M⊙. We find that the random stretching of magnetic field lines by fluid motions in the inertial range of the turbulent spectrum (i.e., a small-scale dynamo) naturally amplifies the seed field by several orders of magnitude in a few convective turnover timescales. During the subsequent saturated regime, the magnetic-to-kinetic energy ratio inside the convective shell reaches values as high as 0.33, and the average magnetic field strength is ∼1010 G. Such strong fields efficiently suppress shear instabilities, which feed the turbulent cascade of kinetic energy, on a wide range of spatial scales. The resulting convective flows are characterized by thread-like structures that extend over a large fraction of the convective shell. The reduced flow speeds and the presence of magnetic fields with strengths up to 60% of the equipartition value at the upper convective boundary diminish the rate of mass entrainment from the stable layer by ≈20% as compared to the purely hydrodynamic case.

A multiple time scale approach for anisotropic inertial wave turbulence

Wave turbulence is the study of the long-time statistical behaviour of equations describing a set of weakly nonlinear interacting waves. Such a theory, which has a natural asymptotic closure, allows us to probe the nature of turbulence more deeply than the exact Kolmogorov laws by rigorously proving the direction of the cascade and the existence of an inertial range, predicting stationary spectra for conserved quantities, or evaluating the Kolmogorov constant. An emblematic example is given by fast rotating fluids for which a wave turbulence theory has been derived by Galtier (Phys. Rev. E, vol. 68, issue 1, 2003, p. 015301). This work involves non-trivial analytical developments for a problem that is anisotropic by nature. We propose here a new path for the derivation of the kinetic equation by using the anisotropy at the beginning of the analysis. We show that the helicity basis is not necessary to obtain the wave amplitude equation for the canonical variables that involve a combination of poloidal and toroidal fields. The multiple time scale method adapted to this anisotropic problem is then used to derive the kinetic equation that is the same as the original work when anisotropy is eventually taken into account. This result proves the commutativity between asymptotic closure and anisotropy. In addition, the multiple time scale method informs us that the kinetic equation can be derived without imposing restrictions on the probability distribution of the wave amplitude such as quasi-Gaussianity, or on the phase such as random phase approximation that naturally occurs dynamically.

Energy transfer and third-order law in forced anisotropic magneto-hydrodynamic turbulence with hyper-viscosity

The third-order law links energy transfer rates in the inertial range of magneto- hydrodynamic (MHD) turbulence with third-order structure functions. Anisotropy, a typical property in the solar wind, challenges the applicability of the third-order law with the isotropic assumption. To shed light on the energy transfer process in the presence of anisotropy, we conducted direct numerical simulations of forced MHD turbulence with normal and hyper-viscosity under various strengths of the external magnetic field ( $B_0$ ), and calculated three forms of third-order structure function with or without averaging of the azimuthal or polar angles with respect to $B_0$ direction. Correspondingly, three estimated energy transfer rates were obtained. The result shows that the peak of normalized third-order structure function occurs at larger scales closer to the $B_0$ direction, and the maximum of longitudinal transfer rates shifts away from the $B_0$ direction at larger $B_0$ . Compared with normal viscous cases, hyper-viscous cases can attain better separated inertial range, thus facilitating the estimation of the energy cascade rates. We find that the widespread use of the isotropic form of the third-order law in estimating the energy transfer rates is questionable in some cases, especially when the anisotropy arising from the mean magnetic field is inevitable. In contrast, the direction-averaged third-order structure function properly accounts for the effect of anisotropy and predicts the energy transfer rates and inertial range accurately, even at very high $B_0$ . With limited statistics, the third-order structure function shows a stronger dependence on averaging of azimuthal angles than the time, especially for high $B_0$ cases. These findings provide insights into the anisotropic effect on the estimation of energy transfer rates.

Performance Analysis of Visual–Inertial–Range Cooperative Localization for Unmanned Autonomous Vehicle Swarm

The swarm of small UAVs is an emerging technology that will enable abundant cooperative tasks. To tackle the positioning problem for the UAV swarm, cooperative localization (CL) has been intensively studied since it uses relative measurement to improve the positioning availability and accuracy for the swarm in GPS-denied environments. Besides relying on inter-UAV range measurement, traditional CL algorithms need to place anchors as location references, which limits their applicability. To implement an infrastructure-less swarm navigation system, a consumer-grade camera together with an inertial device can provide rich environment information, which can be recognized as a kind of local location reference. This paper aims to analyze the fundamental performance of visual–inertial–range CL, which is also a popular metric for UAV planning and sensing optimizing, especially for resource-limited environments. Specifically, a closed-form Fisher information matrix (FIM) of visual–inertial–range CL is constructed in Rn×SO(n) manifold. By introducing an equivalent FIM and utilizing of the sparsity of the FIM, the performance of pose estimation can be efficiently calculated. A series of numerical simulations validate its effectiveness for analyzing the CL performance.

Entropy and fluctuation relations in isotropic turbulence

Based on a generalized local Kolmogorov–Hill equation expressing the evolution of kinetic energy integrated over spheres of size $\ell$ in the inertial range of fluid turbulence, we examine a possible definition of entropy and entropy generation for turbulence. Its measurement from direct numerical simulations in isotropic turbulence leads to confirmation of the validity of the fluctuation relation (FR) from non-equilibrium thermodynamics in the inertial range of turbulent flows. Specifically, the ratio of probability densities of forward and inverse cascade at scale $\ell$ is shown to follow exponential behaviour with the entropy generation rate if the latter is defined by including an appropriately defined notion of ‘temperature of turbulence’ proportional to the kinetic energy at scale $\ell$ .

Are there wall effects on pressure drop through randomly packed beds of spherical catalyst particles?

AbstractFor packed beds of low tube‐to‐particle diameter ratio (), pressure drop has been found to increase at low Re <100 and decrease at high Re >1000, compared to that in unbounded beds. Particle‐resolved CFD (PRCFD) in the low Re viscous region shows that widely accepted corrections overestimate the wall effect, and an improved correction formula is given. In the inertial region at high Re it is important to use accurate and consistent values for void fraction, which strongly affects pressure drop. Literature data show reasonable agreement with a correlation developed for unbounded beds (Dixon AG., AIChE J. 2023;69(6):e18035), when the void fractions for the bounded beds are used. PRCFD simulations also confirm this agreement, and only a small further correction is suggested for the remaining wall effects on fixed bed pressure drop in the inertial range, since the correlation uses the correct limit for as Re increases.

Wavelet determination of magnetohydrodynamic-range power spectral exponents in solar wind turbulence seen by Parker Solar Probe

Context. The high Reynolds number solar wind flow provides a natural laboratory for the study of turbulence in situ. Parker Solar Probe samples the solar wind between 0.17 AU and 1 AU, providing an opportunity to study how turbulence evolves in the expanding solar wind. Aims. We aim to obtain estimates of the scaling exponents and scale breaks of the power spectra of magnetohydrodynamic (MHD) turbulence at sufficient precision to discriminate between Kolmogorov and Iroshnikov-Kraichnan (IK) turbulence, both within each spectrum and across multiple samples at different distances from the Sun and at different plasma β. Methods. We identified multiple long-duration intervals of uniform solar wind turbulence, sampled by PSP/FIELDS and selected to exclude coherent structures, such as pressure pulses and current sheets, and in which the primary proton population velocity varies by less than 20% of its mean value. The local value of the plasma β for these datasets spans the range 0.14 < β < 4. All selected events span spectral scales from the approximately ‘1/f’ range at low frequencies, through the MHD inertial range (IR) of turbulence, and into the kinetic range, below the ion gyrofrequency. We estimated the power spectral density (PSD) using a discrete Haar wavelet decomposition, which provides accurate estimates of the IR exponents. Results. Within 0.3 AU of the Sun, the IR exhibits two distinct ranges of scaling. The inner, high-frequency range has an exponent consistent with that of IK turbulence within uncertainties. The outer, low-frequency range is shallower, with exponents in the range from –1.44 to –1.23. Between 0.3 and 0.5 AU, the IR exponents are closer to, but steeper than, that of IK turbulence and do not coincide with the value –3/2 within uncertainties. At distances beyond 0.5 AU from the Sun, the exponents are close to, but mostly steeper than, that of Kolmogorov turbulence, –5/3: uncertainties inherent in the observed exponents exclude the value –5/3. Between these groups of spectra we find examples, at 0.26 AU and 0.61 AU, of two distinct ranges of scaling within the IR with an inner, high-frequency range with exponents ∼ − 1.4, and a low-frequency range with exponents close to the Kolmogorov value of –5/3. Conclusions. Since the PSD-estimated scaling exponents are a central predictor in turbulence theories, these results provide new insights into our understanding of the evolution of turbulence in the solar wind.

Turbulence Properties of Interplanetary Coronal Mass Ejections in the Inner Heliosphere: Dependence on Proton Beta and Flux Rope Structure

Interplanetary coronal mass ejections (ICMEs) have low proton beta across a broad range of heliocentric distances and a magnetic flux rope structure at large scales, making them a unique environment for studying solar wind fluctuations. Power spectra of magnetic field fluctuations in 28 ICMEs observed between 0.25 and 0.95 au by Solar Orbiter and Parker Solar Probe have been examined. At large scales, the spectra were dominated by power contained in the flux ropes. Subtraction of the background flux rope fields increased the mean spectral index from −5/3 to −3/2 at kd i ≤ 10−3. Rope subtraction also revealed shorter correlation lengths in the magnetic field. The spectral index was typically near −5/3 in the inertial range at all radial distances regardless of rope subtraction and steepened to values consistently below −3 with transition to kinetic scales. The high-frequency break point terminating the inertial range evolved approximately linearly with radial distance and was closer in scale to the proton inertial length than the proton gyroscale, as expected for plasma at low proton beta. Magnetic compressibility at inertial scales did not show any significant correlation with radial distance, in contrast to the solar wind generally. In ICMEs, the distinctive spectral properties at injection scales appear mostly determined by the global flux rope structure while transition-kinetic properties are more influenced by the low proton beta; the intervening inertial range appears independent of both ICME features, indicative of a system-independent scaling of the turbulence.

Non-universality of the Turbulent Spectra at Sub-ion Scales in the Solar Wind: Dispersive Effects versus the Doppler Shift

Large surveys of the power spectral density of the magnetic fluctuations in the solar wind have reported different slope distributions at MHD, sub-ion and sub-electron scales: the smaller the scale, the broader the distribution. Here, we review briefly some of the most relevant explanations of the broadening of the slopes at sub-ion scales. Then, we present a new one that has been overlooked in the literature, which is based on the relative importance of the dispersive effects with respect to the Doppler shift due to the mean flow speed. We build a toy model based on a dispersion relation of a linear mode that matches at high frequency (ω ≳ ω ci) the Alfvén (respectively whistler) mode at high oblique (respectively quasi-parallel) propagation angles θ kB . Starting with a double power-law spectrum of turbulence in the inertial range and at the sub-ion scales, the transformed spectrum (in frequency f) as it would be measured in the spacecraft reference frame shows a broad range of slopes at the sub-ion scales that depend both on the angle θ kB and the flow speed V. Varying θ kB in the range 4°–106° and V in the range 400−800 km s−1 the resulting distribution of slopes at the sub-ion scales reproduces quite well the observed one in the solar wind. Fluctuations in the solar wind speed and the wavevector anisotropy of the turbulence may explain (or at least contribute to) the variability of the spectral slopes reported from spacecraft observations in the solar wind.

Spectral Theory of Turbulent Heat Transfer in the Presence of a Rough Wall.

Using the phenomenological theory of turbulence, a direct link between the Stanton number-a dimensionless number that represents the ratio of transferred heat to the thermal capacity of the fluid-and the scalar spectrum is established for both smooth wall and rough wall conditions. The effect of different scales of motion on heat transfer is demonstrated by investigating relevant limits of the scalar spectrum. It is shown that two important observations in literature-the lack of increase in heat transfer beyond a certain roughness size and the nonclassical Prandtl number scaling-are reproduced if only the viscous inertial and diffusive range of the scalar spectrum is accounted for.

Effect of Schmidt Number on Forced Isotropic Turbulence with Passive Scalars

Traditionally, Fourier spectra have been employed to gain a deeper understanding of turbulence flow structures. The investigation of isotropic forced turbulence with passive scalars offers a straightforward means to examine the disparities between velocity and passive scalar spectra. This flow configuration has been extensively studied in the past, encompassing a range of Reynolds and Schmidt numbers. In this present study, direct numerical simulations (DNS) of this flow are conducted at sufficiently high Reynolds numbers, enabling the formation of a wide inertial range. The primary focus of this investigation is to quantitatively assess the variations in scalar spectra with the Schmidt number (Sc). The spectra exhibit a transition from a k−5/3 scaling for low Sc to a k−4/3 scaling for high Sc. The emergence of the latter power law becomes evident at Sc = 2, with its width expanding as Sc increases. To gain further insights into the underlying flow structures, a statistical analysis is performed by evaluating quantities aligned with the principal axes of the strain field. The study reveals that enstrophy is primarily influenced by the vorticity aligned with the intermediate principal strain axis, while the scalar gradient variance is predominantly controlled by the compressive strain. To provide a clearer understanding of the differences between enstrophy and scalar gradient variance, joint probability density functions (PDFs) and visualizations of the budget terms for both quantities are presented. These visualizations serve to elucidate the distinctions between the two and offer insights into their respective behaviors.

Statistical analysis of stochastic magnetic fluctuations in space plasma based on theMMSmission

ABSTRACTBased on the Magnetospheric Multiscale (MMS) mission we look at magnetic field fluctuations in the Earth’s magnetosheath. We apply the statistical analysis using a Fokker–Planck equation to investigate processes responsible for stochastic fluctuations in space plasmas. As already known, turbulence in the inertial range of hydromagnetic scales exhibits Markovian features. We have extended the statistical approach to much smaller scales in space, where kinetic theory should be applied. Here we study in detail and compare the characteristics of magnetic fluctuations behind the bow shock, inside the magnetosheath, and near the magnetopause. It appears that the first Kramers–Moyal coefficient is linear and the second term is a quadratic function of magnetic increments, which describe drift and diffusion, correspondingly, in the entire magnetosheath. This should correspond to a generalization of Ornstein–Uhlenbeck process. We demonstrate that the second-order approximation of the Fokker–Planck equation leads to non-Gaussian kappa distributions of the probability density functions. In all cases in the magnetosheath, the approximate power-law distributions are recovered. For some moderate scales, we have the kappa distributions described by various peaked shapes with heavy tails. In particular, for large values of the kappa parameter this shape is reduced to the normal Gaussian distribution. It is worth noting that for smaller kinetic scales the rescaled distributions exhibit a universal global scale invariance, consistently with the stationary solution of the Fokker–Planck equation. These results, especially on kinetic scales, could be important for a better understanding of the physical mechanism governing turbulent systems in space and astrophysical plasmas.

A parameter-free LES model for anisotropic mesh adaptivity

Balancing accuracy and computational cost is a challenge in the modelling of turbulent flows. A widely used method for turbulence modelling is large–eddy simulation (LES). LES allows one to describe large scale flow features at a reasonable computational cost compared to the more accurate direct numerical simulation (DNS), making it a popular choice for engineering applications. One strategy to balance accuracy and cost with LES is through the use of mesh adaptivity, which allows the degrees of freedom in a problem to be reduced by changing spatial discretisation. However, mesh adaptivity can affect accuracy when using the standard Smagorinsky LES model with an implicit filter, considering that the parameter Cs is highly dependent on the filter width, which depends on mesh resolution. This work is aimed to develop an LES model that does not require any user–defined parameters and is suitable for mesh adaptivity with implicit filter. In this study we introduce a parameter–free LES model incorporating an anisotropic eddy–viscosity formulation combined with anisotropic mesh adaptivity. In our model, the parameter Cs in the eddy–viscosity formulation of the Smagorinsky model, is replaced by a function that evaluates the relative location of turbulence fluctuations in each element with respect to the turbulence spectrum inertial range. The anisotropic formulation of the eddy–viscosity allows for the application of an appropriate filter width in different directions, improving accuracy. Additionally, the mesh adaptivity algorithm assesses the local turbulence fluctuations via local Reynolds number and vortex identification criteria. This assessment leads to the refinement of regions with higher turbulence fluctuations down to the smallest scale limit in the inertial range in the corresponding direction, and also leads to the coarsening of regions without turbulence fluctuation up to largest scale limit in the inertial range. This method is tested using a flow past a sphere test case. The results are compared both qualitatively and quantitatively to results obtained with the standard Smagorinsky model, and demonstrate the better performance of our method with lower computational cost. This allows us to simulate large Reynolds number cases and compare our quantitative results to experimental results found in the literature, emphasising that our method produces good results at reasonable computational cost.

Exact Intermittent Solutions in a Turbulence Multi-Branch Shell Model

Reproducing complex phenomena with simple models marks our understanding of the phenomena themselves, and this is what Jack Herring’s work demonstrated multiple times. In that spirit, this work studies a turbulence shell model consisting of a hierarchy of structures of different scales ℓn such that each structure transfers its energy to two substructures of scale ℓn+1=ℓn/λ. For this model, we construct exact inertial range solutions that display intermittency, i.e., absence of self-similarity. Using a large ensemble of these solutions, we investigate how the probability distributions of the velocity modes change with scale. It is demonstrated that, while velocity amplitudes are not scale-invariant, their ratios are. Furthermore, using large deviation theory, we show how the probability distributions of the velocity modes can be re-scaled to collapse in a scale-independent form. Finally, we discuss the implications the present results have for real turbulent flows.

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.

Scaling Analysis of Time-Reversal Asymmetries in Fully Developed Turbulence

In fully developed turbulence, there is a flux of energy from large to small scales in the inertial range until the dissipation at small scales. It is associated with irreversibility, i.e., a breaking of the time reversal symmetry. Such turbulent flows are characterized by scaling properties, and we consider here how irreversibility depends on the scale. Indicators of time-reversal symmetry for time series are tested involving triple correlations in a non-symmetric way. These indicators are built so that they are zero for a time-reversal symmetric time series, and a departure from zero is an indicator of irreversibility. We study these indicators applied to two fully developed turbulence time series, from flume tank and wind tunnel databases. It is found that irreversibility occurs in the inertial range and has scaling properties with slopes close to one. A maximum value is found around the injection scale. This confirms that the irreversibility is associated with the turbulent cascade in the inertial range and shows that the irreversibility is maximal at the injection scale, the largest scale of the turbulent cascade.