Turbulent Cascade Research Articles

Statistics of clustered vortices in the inverse energy cascade of two-dimensional turbulence

Strong, long-lived vortices play a prominent role in two-dimensional (2D) turbulent flows, and understanding their statistics is key to unravelling 2D turbulence. Here it is shown that clustered same-sign vortices in forced 2D turbulence have a scale-invariant distribution of areas and are described by a mean-field theory. Larger clusters come apart more quickly, and the flow surrounding the remaining clusters becomes progressively more randomized and neutral, while vortex merger concentrates coherent enstrophy into a smaller and smaller fraction of the domain.

A spectral inspection for turbulence amplification in oblique shock wave/turbulent boundary layer interaction

Turbulence amplification and the large-scale coherent structures in shock wave/turbulent boundary layer interaction flows have been studied at length in previous research, while the direct association between these two flow features is still lacking. In the present study, the transport equation of turbulent kinetic energy spectra is derived and utilized to analyse the scale-by-scale energy budget across the interaction zone, enabling us to reveal the association between the genesis of the large-scale motions and the turbulence amplification. For the presently considered flow with incipient shock-induced separation, we identified in turbulent kinetic energy spectra distribution that the most energetic motions are converted from the near-wall small-scale motions to large-scale motions consisting of velocity streaks and cross-stream circulations as they go through the interaction zone. The amplification of streamwise velocity fluctuation is triggered first, resulting in the emergence of large-scale velocity streaks, which is attributed to the adverse pressure gradient, as indicated by the spectra of the production term. The energy carried by large-scale velocity streaks is transferred to other velocity components by the pressure-strain term, producing large-scale cross-stream circulations. When large-scale motions are convected downstream, their energy is transferred via turbulent cascade to smaller scales and dissipated by viscosity. The spanwise uniform fluctuations, reminiscent of the unsteadiness of the separation bubble, are contributed primarily by the inter-scale energy transfer from the finite spanwise scale motions.

ANTI-SYMMETRY AND CAVITATION TOPOLOGICAL DISRUPTIONS OF COMPLEX SPACE AND INERT MASS ON THE BASIS OF EXCIMER SYSTEMS OF QUANTUM GENERATION OF EXCHANGE INERTIAL RADIATION. MONOMER EQUILIBRIUM AND MATTER FORMATION

The proposed mechanism of formation of inert mass, electric charge and gravitational mass through cavitational topological ruptures of complex space. Stabilization of gaps is carried out through the quantum generation of exchange inertial radiation. On the basis of the relations of non-relativistic quantum mechanics, such radiation was separated into two components: zeron radiation from the zero energy level of the quantum oscillator and heaton radiation from higher levels populated as a result of thermal excitations. Different signs of energy and temperature in two subspaces of the complex space form an antisymmetric excimer system of exchange inertial radiation generation, while the decomposition of these systems into monomers leads to the formation of stable and quasi-stable forms of matter. The parameters of such forms in elementary and cosmic systems are calculated. Quantitative relations in gravitational systems, obtained by the universal use of the constant fine structure and the extended analogue of the Planck constant, are consistent with actual data, in particular, regarding the jet expansion of the Metagalaxy, non-stationary processes in the interior and in the shells of celestial bodies, the mechanisms of thermoregulation of the Earth’s climate and the generation of the geomagnetic field. Turbulent energy cascades in tribocontact and collider systems, manifested in the emission spectra of excimer structures in these systems, are analyzed. The components of the inert mass of leptons, mesons, and hadrons were calculated, and the stabilization of the group velocity of heaton radiation quanta in the basis of giant nuclear resonances was found. The processes of tunnel breakdown of the energy barrier of excimers, which lead to the formation of energy vortices and jets in natural systems of different levels of organization, are analyzed. The negative consequences of mixing excimer structures of the inertial field with monomeric structures are shown on the example of well-known disasters in nuclear power and hydropower. It is shown that the widely reproduced Lorentz-invariant forms of the fields and the limitations of Einstein’s theories of relativity on the rate of propagation of interactions are adequate only in isolated monomeric systems where primitive conservation laws apply. The results obtained in the work are completely consistent with the fundamental theological concept of Creatio ex nihilo.

2D and Slab Turbulent Cascade Rates in the Inner Heliosphere

We present a theoretical and observational study of 2D and slab turbulence cascade (or heating) rates of transverse total turbulence energies, transverse cross helicity, transverse outward and inward Elsässer energy, transverse fluctuating magnetic energy density, and transverse fluctuating kinetic energy from the perihelion of the first Parker Solar Probe (PSP) orbit at ∼36.6 R ⊙ to Solar Orbiter (SolO) at ∼177 R ⊙. We use the Adhikari et al. (2021a) approach to calculate the observed transverse turbulence heating rate, and the nearly incompressible magnetohydrodynamic (NI MHD) turbulence transport theory to calculate the theoretical turbulence cascade rate. We find from the 1 day long PSP measurements at 66.5 R ⊙, and the SolO measurements at 176.3 R ⊙ that various transverse turbulent cascade rates increase with increasing angle, from 10° to 98°, between the mean solar wind speed and mean magnetic field (θ UB), indicating that the 2D heating rate is largest in the inner heliosphere. Similarly, we find from the theoretical and observed results that the 2D heating rate is larger than the slab heating rate as a function of heliocentric distance. We present a comparison between the theoretical and observed 2D and slab turbulence cascade rates as a function of heliocentric distance.

An open source package to perform basic and advanced statistical analysis of turbulence data and other complex systems

We present an open-source package that helps the user to perform a basic statistical analysis of a given turbulence dataset. We believe that this package is useful to the entire turbulence community. It can be used to estimate statistical quantities of turbulence such as the spectral density, turbulence intensity, integral length scale, Taylor microscale, Kolmogorov scale, and dissipation rate as well as velocity increments, structure functions, and their scaling exponents. Different well-known methods available in the literature were selected so that they can be compared. This package also performs an advanced analysis that includes the scale-dependent statistical description of turbulent cascade using the Fokker–Planck equation, consequently leading to the assessment of integral fluctuation theorems. As a precondition for the stochastic process approach, Markovian properties of the turbulent cascade in scale are tested. This is utilized to estimate the drift and diffusion coefficients of the Fokker–Planck equation. The total entropy production for each independent cascade trajectory can be determined using a Fokker–Planck equation. Estimating total entropy production admits a rigorous law of non-equilibrium stochastic thermodynamics, namely, the integral fluctuation theorem, which must be valid if Markov properties hold and the Fokker–Planck equation is correct. This approach to the turbulent cascade process has the potential for a new way to link the statistical description of turbulence, non-equilibrium stochastic thermodynamics, and local turbulent flow structures. Finally, we emphasize that the presented package can be used for the analysis of other data with turbulent-like complexity as well.

Energy Transport during 3D Small-scale Reconnection Driven by Anisotropic Plasma Turbulence

Energy dissipation in collisionless plasmas is a long-standing fundamental physics problem. Although it is well known that magnetic reconnection and turbulence are coupled and transport energy from system-size scales to subproton scales, the details of the energy distribution and energy dissipation channels remain poorly understood. Especially, the energy transfer and transport associated with 3D small-scale reconnection that occurs as a consequence of a turbulent cascade is unknown. We use an explicit fully kinetic particle-in-cell code to simulate 3D small-scale magnetic reconnection events forming in anisotropic and decaying Alfvénic turbulence. We identify a highly dynamic and asymmetric reconnection event that involves two reconnecting flux ropes. We use a two-fluid approach based on the Boltzmann equation to study the spatial energy transfer associated with the reconnection event and compare the power density terms in the two-fluid energy equations with standard energy-based damping, heating, and dissipation proxies. Our findings suggest that the electron bulk flow transports thermal energy density more efficiently than kinetic energy density. Moreover, in our turbulent reconnection event, the energy density transfer is dominated by plasma compression. This is consistent with turbulent current sheets and turbulent reconnection events, but not with laminar reconnection.

MHD turbulence: a biased review

This review of scaling theories of magnetohydrodynamic (MHD) turbulence aims to put the developments of the last few years in the context of the canonical time line (from Kolmogorov to Iroshnikov–Kraichnan to Goldreich–Sridhar to Boldyrev). It is argued that Beresnyak's (valid) objection that Boldyrev's alignment theory, at least in its original form, violates the Reduced-MHD rescaling symmetry can be reconciled with alignment if the latter is understood as an intermittency effect. Boldyrev's scalings, a version of which is recovered in this interpretation, and the concept of dynamic alignment (equivalently, local 3D anisotropy) are thus an example of a physical theory of intermittency in a turbulent system. The emergence of aligned structures naturally brings into play reconnection physics and thus the theory of MHD turbulence becomes intertwined with the physics of tearing, current-sheet disruption and plasmoid formation. Recent work on these subjects by Loureiro, Malletet al.is reviewed and it is argued that we may, as a result, finally have a reasonably complete picture of the MHD turbulent cascade (forced, balanced, and in the presence of a strong mean field) all the way to the dissipation scale. This picture appears to reconcile Beresnyak's advocacy of the Kolmogorov scaling of the dissipation cutoff (as$\mathrm {Re}^{3/4}$) with Boldyrev's aligned cascade. It turns out also that these ideas open the door to some progress in understanding MHD turbulence without a mean field – MHD dynamo – whose saturated state is argued to be controlled by reconnection and to contain, at small scales, a tearing-mediated cascade similar to its strong-mean-field counterpart (this is a new result). On the margins of this core narrative, standard weak-MHD-turbulence theory is argued to require some adjustment – and a new scheme for such an adjustment is proposed – to take account of the determining part that a spontaneously emergent 2D condensate plays in mediating the Alfvén-wave cascade from a weakly interacting state to a strongly turbulent (critically balanced) one. This completes the picture of the MHD cascade at large scales. A number of outstanding issues are surveyed: imbalanced turbulence (for which a new, tentative theory is proposed), residual energy, MHD turbulence at subviscous scales, and decaying MHD turbulence (where there has been dramatic progress recently, and reconnection again turned out to feature prominently). Finally, it is argued that the natural direction of research is now away from the fluid MHD theory and into kinetic territory – and then, possibly, back again. The review lays no claim to objectivity or completeness, focusing on topics and views that the author finds most appealing at the present moment.

Only two Betchov homogeneity constraints exist for isotropic turbulence

Statistically homogeneous flows obey exact kinematic relations. The Betchov homogeneity constraints (Betchov,J. Fluid Mech., vol. 1, 1956, pp. 497–504) for the average principal invariants of the velocity gradient are among the most well-known and extensively employed homogeneity relations. These homogeneity relations have far-reaching implications for the coupled dynamics of strain and vorticity, as well as for the turbulent energy cascade. Whether the Betchov homogeneity constraints are the only possible ones or whether additional homogeneity relations exist has not been proven yet. Here we show that the Betchov homogeneity constraints are the only homogeneity constraints for incompressible and statistically isotropic velocity gradient fields. Our analysis also applies to compressible/perceived velocity gradients, and it allows the derivation of homogeneity relations involving the velocity gradient and other dynamically relevant quantities, such as the pressure Hessian and viscous stresses.

Dropouts of Fully Stripped Ions in the Solar Wind: A Diagnostic for Wave Heating versus Reconnection

The SWICS instrument on board the ACE satellite has detected frequent intervals in the slow solar wind and interplanetary coronal mass ejections in which C6+ and other fully stripped ions are strongly depleted, though the ionization states of elements such as Si and Fe indicate that those ions should be present. It has been suggested that these “outlier” or “dropout” events can be explained by the resonant cyclotron heating process, because these ions all have the same cyclotron frequency as He2+. We investigate the region in the corona where these outlier events form. It must be above the ionization freeze-in height and the transition to collisionless plasma conditions, but low enough that the wind still feels the effects of solar gravity. We suggest that the dropout events correspond to relatively dense blobs of gas in which the heating is reduced because local variations in the Alfvén speed change the reflection of Alfvén waves and the turbulent cascade. As a result, the wave power at the cyclotron frequency of the fully stripped ions is absorbed by He2+ and may not be able to heat the other fully stripped ions enough to overcome solar gravity. If this picture is borne out, it may help to discriminate between resonant cyclotron heating and stochastic heating models of the solar wind.

Magnetospheric Multiscale Observations of Waves and Parallel Electric Fields in Reconnecting Current Sheets in the Turbulent Magnetosheath

AbstractDownstream of the Earth’s bow shock, the magnetosheath can exhibit strong turbulence. This turbulent cascade is associated with electron‐scale current sheets where magnetic reconnection can occur. We present data from NASA’s Magnetospheric Multiscale mission during a turbulent magnetosheath interval that includes two electron‐scale reconnecting current sheets, one with higher guide field and one that is closer to antiparallel. We examine the characteristics of energy conversion and the role of parallel electric fields and waves in that energy conversion. The high‐guide field reconnection event shows energy conversion dominated by electric fields and currents parallel to the background magnetic field, while the lower guide field event is perpendicular dominated, consistent with observations of laminar reconnecting current sheets in the magnetosheath. Both current sheets exhibit signs of a parallel electric field acceleration channel, and the higher guide field event shows whistler‐mode waves at the x‐line. Both current sheets show variation in dissipation along the n‐direction, with perpendicular dissipation on one side and parallel on the other. Between the two is a dispersion of pitch angles from 90° to 0° or 180°. This dispersion is likely caused by the acceleration channel and rapidly suppresses whistler waves, suggesting whistler waves do not contribute to the dissipation in guide field reconnection.

Scale‐Dependent Kurtosis of Magnetic Field Fluctuations in the Solar Wind: A Multi‐Scale Study With Cluster 2003–2015

AbstractDuring the lifetime of the Cluster mission, the inter‐spacecraft distances in the solar wind have changed from the large, fluid, scales (∼104 km), down to the scales of protons (∼102 km). As part of the guest investigator campaign, the mission achieved a formation where a pair of spacecraft were separated by ∼7 km. The small distances and the exceptional sensitivity of the search coil magnetometer provide an excellent data set for studying solar wind turbulence at electron scales. In this study, we investigate the intermittency of the magnetic field fluctuations in the slow solar wind. Using 20 time intervals with different constellation orientations of Cluster we cover spatial scales between 7 and 104 km. We compare time‐lagged increments from a single spacecraft with spatially lagged increments using multiple spacecraft. As the turbulent cascade proceeds to smaller scales in the inertial range, the deviation from Gaussian statistics is observed to increase in both temporal and spatial increments in the components transverse to the mean field direction. At ion scales, there is a maximum of kurtosis, and at sub‐ion scales, the fluctuations are only weakly non‐Gaussian. In the compressive component the deviation from Gaussian statistics is variable: it may increase throughout the inertial and sub‐ion ranges, but also, it may have a maximum at magnetohydrodynamic scales associated with large scale magnetic holes. The observations show differences in kurtosis of time and space increments when the spacecraft pairs are transverse to the flow, indicating its spatial anisotropy.

Analysis of tip leakage flow unsteadiness in a transonic turbine cascade using data-driven modal decomposition methods

The evolution of tip leakage flow and subsequent vortical structures is accompanied by inherent unsteadiness. This paper presents a novel characterization for the unsteady dynamics of turbine tip flow using data-driven, i.e., equation-free modal decomposition methods, which are applied to the hybrid Reynolds-averaged Navier–Stokes/large eddy simulation data at a transonic condition. By combining these techniques, the identified eigen-tuples (eigenvalues, eigenvectors, and time evolution) are well determined, and the differences between the obtained patterns (modes) are also pronounced. The snapshot proper orthogonal decomposition (POD) analysis can isolate the large-scale fluctuating structures that populate the rear part of the suction-side, which is mainly attributed to the shock-induced vortex instability. Similar to the turbulence cascade, macro-scale coherent structures that correspond to the tip leakage vortex shedding phenomena and the subsequently transitional and noisy parts closely related to the dissipation can be well derived by a quadruple reconstruction. Three dynamic mode decomposition (DMD) variants including the amplitude selecting-DMD method, the DMD with criterion method, and the sparsity promoting (SP)-DMD method are also compared in extracting dominant modes from the periodic tip flow, and the SP-DMD method which can distill modes of broadband frequencies and low dissipation is proved to be more conducive to representing and reconstructing the complex tip flow. Additional spectral-kernel-based POD (SPOD) analysis that can identify the similar primary unsteadiness frequencies as the DMD method is also encompassed in this study. Specifically, although it manifests that a physical resemblance of the pattern of pressure fluctuations to tip eddy unsteadiness can be captured by all these approaches, the behavior of small-scale vortical interaction downstream of the trailing edge can be clearly isolated with the intrinsic Karman-type vortex layer shedding process via DMD and SPOD approaches, which also demonstrates that these two techniques are more favorable to decomposing the complex tip flows into uncoupled single-frequency coherent structures compared to the conventional POD method. On this basis, resulting modes of velocity components have been accounted for verifying their contributions to the turbulent kinetic energy fields. The ensuing observations can offer a glimpse of the complex dynamics in the tip region, which also sheds light on features previously masked by conventional analysis approaches.

Whistler turbulence vs. whistler anisotropy instability: Particle-in-cell simulation and statistical analysis

Particle-in-Cell simulations and statistical analysis are carried out to study the dynamic evolution of a collisionless, magnetized plasma with co-existing whistler turbulence and electron temperature anisotropy as the initial condition, and the competing consequences of whistler turbulence cascade and whistler anisotropy instability growth. The results show that the operation of the whistler instability within whistler turbulence has almost no effects on the fluctuating magnetic field energy and intermittency generated by turbulence. However, it leads to a small reduction of the magnetic field wavevector anisotropy and a major reduction of the intermittency of electron temperature anisotropy. Hence, while the overall effect from whistler instability is minor as compared to that of whistler turbulence due to its much smaller field energy, the whistler instability may act as a regulation mechanism for kinetic-range turbulence through wave-particle interactions.

Turbulent cascades for a family of damped Szegő equations

In this paper, we study the transfer of energy from low to high frequencies for a family of damped Szegő equations. The cubic Szegő equation has been introduced as a toy model for a totally non-dispersive degenerate Hamiltonian equation. It is a completely integrable system which develops growth of high Sobolev norms, detecting transfer of energy and hence cascades phenomena. Here, we consider a two-parameter family of variants of the cubic Szegő equation and prove that, adding a damping term unexpectedly promotes the existence of turbulent cascades. Furthermore, we give a panorama of the dynamics for such equations on a six-dimensional submanifold.

Profile of a two-dimensional vortex condensate beyond the universal limit.

It is well known that an inverse turbulent cascade in a finite (2π×2π) two-dimensional periodic domain leads to the emergence of a system-sized coherent vortex dipole. We report a numerical hyperviscous study of the spatial vorticity profile inside one of the vortices. The exciting force was shortly correlated in time, random in space, and had a correlation length l_{f}=2π/k_{f} with k_{f} ranging from 100 to 12.5. Previously, it was found that in the asymptotic limit of small-scale forcing, the vorticity exhibits the power-law behavior Ω(r)=(3ε/α)^{1/2}r^{-1}, where r is the distance to the vortex center, α is the bottom friction coefficient, and ε is the inverse energy flux. Now we show that for a spatially homogeneous forcing with finite k_{f} the vorticity profile becomes steeper, with the difference increasing with the pumping scale but decreasing with the Reynolds number at the forcing scale. Qualitatively, this behavior is related to a decrease in the effective pumping of the coherent vortex with distance from its center. To support this statement, we perform an additional simulation with spatially localized forcing, in which the effective pumping of the coherent vortex, on the contrary, increases with r, and show that in this case the vorticity profile can be flatter than the asymptotic limit.

Spectral Analysis of the Martian Atmospheric Turbulence: InSight Observations

AbstractIn this study, we perform spectral analysis of Martian near‐surface turbulence using the in‐situ observations by NASA's InSight lander on Mars. A recent study on the daytime Martian boundary layer using the InSight observations showed that the conventional Kolmogorov energy cascade of inertial subrange fails to predict the spectral density of pressure (Banfield et al., 2020, https://doi.org/10.1038/s41561-020-0544-y). Here we extend this by investigating diurnal and seasonal variations in the spectral density of pressure, as the indicator of the Martian atmospheric turbulence. We show distinct spectral behaviors for the daytime and nighttime conditions. Moreover, we report the important effects of regional dust storms, gravity waves (GWs), bore, and solitary waves on the turbulent energy cascade. Our results show that the presence of a dust storm and GW activity can enhance the turbulence of the nighttime boundary layer of Mars despite the extreme stably stratified conditions compared to the terrestrial case.

Preferential Heating of Protons over Electrons from Coherent Structures during the First Perihelion of the Parker Solar Probe

The solar wind undergoes significant heating as it propagates away from the Sun; the exact mechanisms responsible for this heating remain unclear. Using data from the first perihelion of the Parker Solar Probe mission, we examine the properties of proton and electron heating occurring within magnetic coherent structures identified by means of the Partial Variance of Increments (PVI) method. Statistically, regions of space with strong gradients in the magnetic field, PVI ≥ 1, are associated with strongly enhanced proton but only slightly elevated electron temperatures. Our analysis indicates a heating mechanism in the nascent solar wind environment facilitated by a nonlinear turbulent cascade that preferentially heats protons over electrons.

Electromagnetic instabilities and plasma turbulence driven by electron-temperature gradient

Electromagnetic (EM) instabilities and turbulence driven by the electron-temperature gradient (ETG) are considered in a local slab model of a tokamak-like plasma. Derived in a low-beta asymptotic limit of gyrokinetics, the model describes perturbations at scales both larger and smaller than the electron inertial length$d_e$, but below the ion Larmor scale$\rho _i$, capturing both electrostatic and EM regimes of turbulence. The well-known electrostatic instabilities – slab and curvature-mediated ETG – are recovered, and a new instability is found in the EM regime, called the thermo-Alfvénic instability (TAI). It exists in both a slab version (sTAI, destabilising kinetic Alfvén waves) and a curvature-mediated version (cTAI), which is a cousin of the (electron-scale) kinetic ballooning mode. The cTAI turns out to be dominant at the largest scales covered by the model (greater than$d_e$but smaller than$\rho _i$), its physical mechanism hinging on the fast equalisation of the total temperature along perturbed magnetic field lines (in contrast to kinetic ballooning mode, which is pressure balanced). A turbulent cascade theory is then constructed, with two energy-injection scales:$d_e$, where the drivers are slab ETG and sTAI, and a larger (parallel system size dependent) scale, where the driver is cTAI. The latter dominates the turbulent transport if the temperature gradient is greater than a certain critical value, which scales inversely with the electron beta. The resulting heat flux scales more steeply with the temperature gradient than that due to electrostatic ETG turbulence, giving rise to stiffer transport. This can be viewed as a physical argument in favour of near-marginal steady-state in electron-transport-controlled plasmas (e.g. the pedestal). While the model is simplistic, the new physics that is revealed by it should be of interest to those attempting to model the effect of EM turbulence in tokamak-relevant configurations with high beta and large ETGs.

Statistical Equilibrium of Large Scales in Three-Dimensional Hydrodynamic Turbulence.

We investigate experimentally three-dimensional (3D) hydrodynamic turbulence at scales larger than the forcing scale. We manage to perform a scale separation between the forcing scale and the container size by injecting energy into the fluid using centimetric magnetic particles. We measure the statistics of the fluid velocity field at scales larger than the forcing scale (energy spectra, velocity distributions, and energy flux spectrum). In particular, we show that the large-scale dynamics are in statistical equilibrium and can be described with an effective temperature, although not isolated from the turbulent Kolmogorov cascade. In the large-scale domain, the energy flux is zero on average but exhibits intense temporal fluctuations. Our Letter paves the way to use equilibrium statistical mechanics to describe the large-scale properties of 3D turbulent flows.

Modeling the Solar Wind Turbulent Cascade Including Cross Helicity: With and Without Expansion

Simulations of the turbulent cascade forming in the solar wind, including cross helicity, commonly adopt a homogeneous setup, not taking into account wind expansion. Here we want to assess the predictions of decaying 3D compressible (low Mach number) MHD simulations, respectively homogeneous and with expansion, in order to examine which is the most fruitful approach to understanding the turbulent cascade in the solar wind. We follow turbulent evolution during 10 nonlinear turnover times, considering several initial values of the initial spectral slope and cross helicity. In the expanding case, the transverse sizes of the plasma volume are stretched by a factor of 5 during the simulation, corresponding to traveling from 0.2 up to 1 au. In homogeneous simulations, the relative cross helicity rises, and the Elsässer spectra E ± show “pinning,” with a steep dominant spectrum and flat subdominant spectrum, the final spectral indices depending on cross helicity but not initial indices. With expansion, the relative cross helicity decreases, and dominant and subdominant spectra share the same index, with the index relaxing to an asymptotic value that generally depends on the initial index. The absence of pinning, as well as the decrease of relative cross helicity, probably both rely on the permanent injection by expansion of an excess of magnetic energy at the largest scales, equivalent to injecting subdominant energy. Also, spectra generally steepen when initially starting flatter than k −5/3 but stop evolving at a finite time/distance.