Direct Enstrophy Cascade Research Articles

Self-similarity and the direct (enstrophy) cascade in forced two-dimensional fluid turbulence

In the absence of large-scale coherent structures, a widely used statistical theory of two-dimensional turbulence developed by Kraichnan, Leith, and Batchelor (KLB) predicts a power-law scaling for the energy, $E(k)\propto k^\alpha$ with an integral exponent $\alpha ={-3}$ , in the inertial range associated with the direct cascade. A power-law scaling is also observed in the presence of coherent structures, but the scaling exponent becomes fractal and often differs substantially from the value predicted by the KLB theory. Here we present a dynamical theory that sheds new light on the relationship between the spatial and temporal structure of the large-scale flow and the scaling of small-scale structures representing filamentary vorticity. Specifically, we find hyperbolic regions of the large-scale flow to play a key role in the flux of enstrophy between scales. Small-scale vorticity in these regions can be described by dynamically self-similar solutions of the Euler equation, which explains the power-law scaling. Furthermore, we find that correlations between different hyperbolic regions are responsible for the emergence of fractal scaling exponents.

Third‐Order Structure Functions of Zonal Winds in the Thermosphere Using CHAMP and GOCE Observations

AbstractWe use multi‐year observations of cross‐track winds (u) from the CHAllenging Minisatellite Payload (CHAMP) and the Gravity Field and Steady State Ocean Circulation Explorer (GOCE) to calculate third‐order structure functions in the thermosphere as a function of horizontal separation (s). They are computed using the mean (〈δu3〉) and the median and implemented over non‐polar satellite paths in both hemispheres. On height averages, 〈δu3〉 is shown to scale with s2 for s ≃ 80–1,000 km, in agreement with equivalent estimates in the lower atmosphere from aircraft observations. Conversely, follows an s3 power law for almost the whole s range, consistent with the two‐dimensional turbulence scaling law for a direct enstrophy cascade. These scaling laws appear independent of winds in distinct atmospheric regions. Furthermore, the functions are predominantly positive, indicating a preferential cyclonic motion for the wind.

Advancing Eddy Parameterizations: Dynamic Energy Backscatter and the Role of Subgrid Energy Advection and Stochastic Forcing

AbstractViscosity in the momentum equation is needed for numerical stability, as well as to arrest the direct cascade of enstrophy at grid scales. However, a viscous momentum closure tends to over‐dissipate eddy kinetic energy. To return excessively dissipated energy to the system, the viscous closure is equipped with what is called dynamic kinetic energy backscatter. The amplitude of backscatter is based on the amount of unresolved kinetic energy (UKE). This energy is tracked through space and time via a prognostic equation. Our study proposes to add advection of UKE by the resolved flow to that equation to explicitly consider the effects of nonlocality on the subgrid energy budget. UKE can consequently be advected by the resolved flow before it is reinjected via backscatter. Furthermore, we suggest incorporating a stochastic element into the UKE equation to account for missing small‐scale variability, which is not present in the purely deterministic approach. The implementations are tested on two intermediate complexity setups of the global ocean model FESOM2: an idealized channel setup and a double‐gyre setup. The impacts of these additional terms are analyzed, highlighting increased eddy activity and improved flow characteristics when advection and carefully tuned, stochastic sources are incorporated into the UKE budget. Additionally, we provide diagnostics to gain further insights into the effects of scale separation between the viscous dissipation operator and the backscatter operator responsible for the energy injection.

Filaments of uniform quasi-geostrophic potential vorticity in pure strain

Three-dimensional filaments of quasi-geostrophic potential vorticity are generic features of atmospheric and oceanic flows. They are often generated during the strong interactions between three-dimensional quasi-geostrophic vortices. They contribute to a direct cascade of enstrophy in spectral space. These filaments correspond to shear zones. Therefore they may be sensitive to shear instabilities akin to the Kelvin–Helmholtz instability of the classical two-dimensional vorticity strip. They are, however, often subjected to a straining flow induced by the surrounding vortices. This straining flow affects their robustness. This paper focuses on a simplified model of this situation. We consider the effect of a pure strain on a three-dimensional filament of uniform quasi-geostrophic potential vorticity. We first consider a quasi-static situation where the strain, assumed small, only affects the cross-sectional shape of the filament, but not the velocity field. We address the linear stability of the filament in that context and also show examples of the filament's nonlinear evolution. We then consider the linearised dynamics of the filament in pure strain. In particular we focus on the maximum perturbation amplification observed in the filament. We conclude that small to moderate strain rates are efficient at preventing a large perturbation growth. Nonlinear effects can nevertheless leads to the roll-up of weakly strained filaments.

The Interaction Between the Turbulence and Gravity Wave Observed in the Middle Stratosphere Based on the Round‐Trip Intelligent Sounding System

AbstractBased on the round‐trip intelligent sounding system, the structure function and spectrum analysis method were carried out to analyze the scale interactions between the small‐scale gravity wave and turbulence in the middle stratosphere. The result showed that the generation of turbulence was closely related to the Kelvin‐Helmholtz (KH) billows. An upscale energy cascade occurred on the spatial scale of KH billows, while a downscale energy cascade occurred on a larger spatial scale (gravity wave scale), which resulted in the breaking of the gravity wave and generation of turbulence. From the multilevel signal measurement, the obvious interactions between turbulence and gravity can be found. The two‐dimensional turbulence that the inverse energy cascade in 2‐D exists simultaneously with a direct enstrophy cascade to the small scales was found, which is considered to be the first‐hand observation result of relevant height.

Direct Numerical Simulations to Investigate Energy Transfer between Meso- and Synoptic Scales

Abstract Understanding the development of the atmospheric energy spectrum across scales is necessary to elucidate atmospheric predictability. In this manuscript, the authors investigate energy transfer between the synoptic scale and the mesoscale using direct numerical simulations (DNSs) of two-dimensional (2D) turbulence transfer under forcing applied at different scales. First, DNS results forced by a single kinetic energy source at large scales show that the energy spectra slopes of the direct enstrophy cascade are steeper than the theoretically predicted −3 slope. Second, the presence of two inertial ranges in 2D turbulence at intermediate scales is investigated by introducing a second energy source in the meso-α-scale range. The energy spectra for the DNS with two kinetic energy sources exhibit flatter slopes that are closer to −3, consistent with the observed kinetic energy spectra of horizontal winds in the atmosphere at synoptic scales. Further, the results are independent of model resolution and scale separation between the two energy sources, with a robust transition region between the lower synoptic and the upper meso-α scales in agreement with classical observations in the upper troposphere. These results suggest the existence of a mesoscale feedback on synoptic-scale predictability that emerges from the concurrence of the direct (downscale) enstrophy transfer in the synoptic scales and the inverse (upscale) kinetic energy transfer from the mesoscale to the synoptic scale in the troposphere.

Split energy cascade in turbulent thin fluid layers

We discuss the phenomenology of the split energy cascade in a three-dimensional thin fluid layer by mean of high resolution numerical simulations of the Navier-Stokes equations. We observe the presence of both an inverse energy cascade at large scales, as predicted for two-dimensional turbu- lence, and of a direct energy cascade at small scales, as in three-dimensional turbulence. The inverse energy cascade is associated with a direct cascade of enstrophy in the intermediate range of scales. Notably, we find that the inverse cascade of energy in this system is not a pure 2D phenomenon, as the coupling with the 3D velocity field is necessary to guarantee the constancy of fluxes.

Cascades and spectra of a turbulent spinodal decomposition in two-dimensional symmetric binary liquid mixtures

We study the fundamental physics of cascades and spectra in 2D Cahn-Hilliard-Navier-Stokes (CHNS) turbulence, and compare and contrast this system with 2D MagnetoHydroDynamic (MHD) turbulence. The important similarities include basic equations, ideal quadratic invariants, cascades and the role of linear elastic waves. Surface tension induces elasticity, and the balance between surface tension energy and turbulent kinetic energy determines a length scale (Hinze scale) of the system. The Hinze scale may be thought of as the scale of emergent critical balance between fluid straining and elastic restoring forces. The scales between the Hinze scale and dissipation scale constitute the elastic range of the 2D CHNS system. By direct numerical simulation, we find that in the elastic range, the mean square concentration spectrum $H^\psi_k$ of the 2D CHNS system exhibits the same power law ($-7/3$) as the mean square magnetic potential spectrum $H^A_k$ in the inverse cascade regime of 2D MHD. This power law is consistent with an inverse cascade of $H^\psi$, which is observed. The kinetic energy spectrum of the 2D CHNS system is $E^K_k\sim k^{-3}$ if forced at large scale, suggestive of the direct enstrophy cascade power law of 2D Navier-Stokes (NS) turbulence. The difference from the energy spectra of 2D MHD turbulence implies that the back reaction of the concentration field to fluid motion is limited. We suggest this is because the surface tension back reaction is significant only in the interfacial regions. The interfacial regions fill only a small portion of the 2D CHNS system, and their interface packing fraction is much smaller than that for 2D MHD.

SIMULTANEOUS OBSERVATION OF ENERGY AND ENSTROPHY CASCADES IN THIN-LAYER TURBULENCE

We report the simultaneous observation of the inverse energy and direct enstrophy cascade in thin-layer turbulence. The experiments are conducted in an electromagnetically driven flow with layers of stratified fluid. Recent questions regarding the two-dimensionality of electromagnetically driven turbulence in such experiments are addressed.

A SPH Model for Incompressible Turbulence

A coarse-grained particle model for incompressible Navier-Stokes (NS) equation is proposed based on spatial filtering by utilizing smoothed particle hydrodynamics (SPH) approximations. This model is similar to our previous developed SPH discretization of NS equation (Hu X.Y. & N.A. Adams, J. Comput. Physics, 227:264-278, 2007 and 228:2082-2091, 2009) and the Lagrangian averaged NS (LANS-α) turbulence model. Other than using smoothing approaches, this model obtains particle transport velocity by imposing constant σ which is associated with the particle density, and is called SPH-σ model. Numerical tests on two-dimensional decay and forced turbulences with high Reynolds number suggest that the model is able to reproduce both the inverse energy cascade and direct enstrophy cascade of the kinetic energy spectrum, the time scaling of enstrophy decay and the non-Guassian probability density function (PDF) of particle acceleration.

The direct enstrophy cascade of two-dimensional soap film flows

We investigate the direct enstrophy cascade of two-dimensional decaying turbulence in a flowing soap film channel. We use a coarse-graining approach that allows us to resolve the nonlinear dynamics and scale-coupling simultaneously in space and in scale. From our data, we verify an exact relation due to Eyink [“Local energy flux and the refined similarity hypothesis,” J. Stat. Phys. 78, 335–351 (1995); Eyink “Exact results on scaling exponents in the 2D enstrophy cascade,” Phys. Rev. Lett. 74, 3800–3803 (1995)] between traditional 3rd-order structure function and the enstrophy flux obtained by coarse-graining. We also present experimental evidence that enstrophy cascades to smaller (larger) scales with a 60% (40%) probability, in support of theoretical predictions by Merilees and Warn [“On energy and enstrophy exchanges in two-dimensional non-divergent flow,” J. Fluid Mech. 69, 625–630 (1975)] which appear to be valid in our flow owing to the ergodic nature of turbulence. We conjecture that their kinematic arguments break down in quasi-laminar 2D flows. We find some support for these ideas by using an Eulerian coherent structure identification technique, which allows us to determine the effect of flow topology on the enstrophy cascade. A key finding is that “centers” are inefficient at transferring enstrophy between scales, in contrast to “saddle” regions which transfer enstrophy to small scales with high efficiency.

Intermittency in 2D soap film turbulence

The Reynolds number dependency of intermittency for 2D turbulence is studied in a flowing soap film. The Reynolds number used here is the Taylor microscale Reynolds number Rλ, which ranges from 20 to 800. Strong intermittency is found for both the inverse energy and direct enstrophy cascades as measured by (a) the pdf of velocity differences P(δu(r)) at inertial scales r, (b) the kurtosis of P(∂xu), and (c) the scaling of the so-called intermittency exponent μ, which is zero if intermittency is absent. Measures (b) and (c) are quantitative, while (a) is qualitative. These measurements are in disagreement with some previous results but not all. The velocity derivatives are nongaussian at all Rλ but show signs of becoming gaussian as Rλ increases beyond the largest values that could be reached. The kurtosis of P(δu(r)) at various r indicates that the intermittency is scale dependent. The structure function scaling exponents also deviate strongly from the Kraichnan prediction. For the enstrophy cascade, the intermittency decreases as a power law in Rλ. This study suggests the need for a new look at the statistics of 2D turbulence.

Spectral non-locality, absolute equilibria and Kraichnan–Leith–Batchelor phenomenology in two-dimensional turbulent energy cascades

AbstractWe study the degree to which Kraichnan–Leith–Batchelor (KLB) phenomenology describes two-dimensional energy cascades in$\alpha $turbulence, governed by$\partial \theta / \partial t+ J(\psi , \theta )= \nu {\nabla }^{2} \theta + f$, where$\theta = {(- \Delta )}^{\alpha / 2} \psi $is generalized vorticity, and$\hat {\psi } (\boldsymbol{k})= {k}^{- \alpha } \hat {\theta } (\boldsymbol{k})$in Fourier space. These models differ in spectral non-locality, and include surface quasigeostrophic flow ($\alpha = 1$), regular two-dimensional flow ($\alpha = 2$) and rotating shallow flow ($\alpha = 3$), which is the isotropic limit of a mantle convection model. We re-examine arguments for dual inverse energy and direct enstrophy cascades, including Fjørtoft analysis, which we extend to general$\alpha $, and point out their limitations. Using an$\alpha $-dependent eddy-damped quasinormal Markovian (EDQNM) closure, we seek self-similar inertial range solutions and study their characteristics. Our present focus is not on coherent structures, which the EDQNM filters out, but on any self-similar and approximately Gaussian turbulent component that may exist in the flow and be described by KLB phenomenology. For this, the EDQNM is an appropriate tool. Non-local triads contribute increasingly to the energy flux as$\alpha $increases. More importantly, the energy cascade is downscale in the self-similar inertial range for$2. 5\lt \alpha \lt 10$. At$\alpha = 2. 5$and$\alpha = 10$, the KLB spectra correspond, respectively, to enstrophy and energy equipartition, and the triad energy transfers and flux vanish identically. Eddy turnover time and strain rate arguments suggest the inverse energy cascade should obey KLB phenomenology and be self-similar for$\alpha \lt 4$. However, downscale energy flux in the EDQNM self-similar inertial range for$\alpha \gt 2. 5$leads us to predict that any inverse cascade for$\alpha \geq 2. 5$will not exhibit KLB phenomenology, and specifically the KLB energy spectrum. Numerical simulations confirm this: the inverse cascade energy spectrum for$\alpha \geq 2. 5$is significantly steeper than the KLB prediction, while for$\alpha \lt 2. 5$we obtain the KLB spectrum.

Two-Dimensional Turbulence

In physical systems, a reduction in dimensionality often leads to exciting new phenomena. Here we discuss the novel effects arising from the consideration of fluid turbulence confined to two spatial dimensions. The additional conservation constraint on squared vorticity relative to three-dimensional (3D) turbulence leads to the dual-cascade scenario of Kraichnan and Batchelor with an inverse energy cascade to larger scales and a direct enstrophy cascade to smaller scales. Specific theoretical predictions of spectra, structure functions, probability distributions, and mechanisms are presented, and major experimental and numerical comparisons are reviewed. The introduction of 3D perturbations does not destroy the main features of the cascade picture, implying that 2D turbulence phenomenology establishes the general picture of turbulent fluid flows when one spatial direction is heavily constrained by geometry or by applied body forces. Such flows are common in geophysical and planetary contexts, are beautiful to observe, and reflect the impact of dimensionality on fluid turbulence.

Structure-function scaling of bounded two-dimensional turbulence

Statistical properties of forced two-dimensional turbulence generated in two different flow domains are investigated by numerical simulations. The considered geometries are the square domain and the periodic channel domain, both bounded by lateral no-slip sidewalls. The focus is on the direct enstrophy cascade range and how the statistical properties change in the presence of no-slip boundaries. The scaling exponents of the velocity and the vorticity structure functions are compared to the classical Kraichnan-Batchelor-Leith (KBL) theory, which assumes isotropy, homogeneity, and self-similarity for turbulence scales between the forcing and dissipation scale. Our investigation reveals that in the interior of the flow domain, turbulence can be considered statistically isotropic and locally homogeneous for the enstrophy cascade range, but it is weakly intermittent. However, the scaling of the vorticity structure function indicates a steeper slope for the energy spectrum than the KBL theory predicts. Near the walls the turbulence is strongly anisotropic at all flow scales.

Charney isotropy and equipartition in quasi-geostrophic turbulence

High-resolution simulations of forced quasi-geostrophic (QG) turbulence reveal that Charney isotropy develops under a wide range of conditions, and constitutes a preferred state also in β-plane and freely decaying turbulence. There is a clear analogy between two-dimensional and QG turbulence, with a direct enstrophy cascade that is governed by the prediction of Kraichnan (J. Fluid Mech., vol. 47, 1971, p. 525) and an inverse energy cascade following the classic k−5/3 scaling. Furthermore, we find that Charney's prediction of equipartition between the potential and kinetic energy in each of the two horizontal velocity components is approximately fulfilled in the inertial ranges.

Wavelet-based analysis of enstrophy transfers in two-dimensional turbulence

Two-dimensional turbulence admits two different ranges of scales: a direct enstrophy cascade from the injection scale to the small scales and an inverse energy cascade at large scales. It has already been shown in previous papers that vortical structures are responsible for the transfers of energy upscale while filamentary structures are responsible for the forward transfer of the enstrophy. Here we propose an original mathematical tool, the interaction function, for studying the space localization of the enstrophy fluxes. It is defined using an orthogonal two-dimensional wavelet decomposition.

Influence of the filtering tools on the analysis of two-dimensional turbulent flows

Numerous direct numerical simulations of soap film flows yield a large amount of data on two-dimensional turbulence. The analysis of such data is very sensitive to the analysis tools and the way they are used. On the one hand, some of the possible errors obtained by a misuse of the analysis tools are reported. On the other hand, a rigorous use of wavelet packets analysis reveals surprising results that slightly differ from the KLB theory for the flow considered. A careful wavelet packets filtering and a good computation of the energy and enstrophy fluxes show the role of the solid rotation vortices and the vorticity filaments to both the inverse energy cascade and the direct enstrophy cascade.

Cascades and fluxes in two-dimensional turbulence

We develop simple models of forced and unforced turbulence at large but finite Reynolds number. These models yield a number of predictions, such as (i) the form of the second- and third-order structure functions in the inertial range; (ii) the rate of generation of palinstrophy in forced and unforced turbulence; (iii) the rate of destruction of enstrophy in decaying turbulence; (iv) the rate of change of Loitsyansky’s integral in decaying turbulence; and, again for decaying turbulence; and (v) the existence of an inverse energy cascade, embedded within the direct enstrophy cascade. We show that these predictions are not mere artifacts of our simple models, but rather genuine features of two-dimensional turbulence. The most surprising findings relate to (iii), (iv), and (v) above, where we show that, once the turbulence is fully developed, the enstrophy dissipation rate scales as β∼⟨ω2⟩3∕2∕ln(Re), Loitsyansky’s integral grows at a rate proportional to βℓ6∕ln(Re), and the filamentation of vorticity fuels an inverse energy cascade with a flux of order βℓ2∕ln(Re). (Here, ω is the vorticity, ℓ is the integral scale, β the enstrophy dissipation rate, and Re the Reynolds number.) This suggests that, in the limit of Re→∞, the enstrophy dissipation rate tends to zero and the inverse energy cascade progressively shuts down. Loitsyansky’s integral is also an invariant in this limit, provided that ℓ remains finite as Re→∞.

Nonrobustness of the two-dimensional turbulent inverse cascade

The inverse energy cascade in two-dimensional Navier-Stokes turbulence is examined in the quasisteady regime, with small-scale, band-limited forcing at scale kf-1, with particular attention to the influence of forcing Reynolds number Re on the energy distribution at large scales. The strength of the inverse energy cascade, or fraction of energy input that is transferred to larger scales, increases monotonically toward unity with increasing Re proportional, variantkmax2kf2, where kmax is the maximum resolved wave number. Moreover, as Re increases beyond a critical value, for which a direct enstrophy cascade to small scales is first realized, the energy spectrum in the energy-cascading range steepens from a k-53 to k-2 dependence. The steepening is interpreted as the result of a greater tendency for coherent vortex formation in cases when forcing scales are adequately resolved. In spectral space, it is associated with nonlocality of the inverse energy transfer.