Enstrophy Transfer Research Articles

Unsteady wake and heat transfer characteristics of three tandem circular cylinders in forced and mixed convection flows

In natural convection (high Richardson number Ri), a high Prandtl number (Pr) leads to thinner thermal boundary layers, enlarging the thermal gradient and hence the enhancement of buoyancy effect. In forced convection (low Ri), a high Pr introduces thicker velocity boundary layers. In mixed convection scenarios, where both forced and natural convection are significant, the interaction between Pr and Ri determines the resultant flow pattern and heat transfer characteristic. Three tandem circular cylinders with an identical spacing ratio of 4.0 in both forced and mixed convection flows were numerically investigated by using finite element method. The computations were carried out in the range of Pr = 5–50 and Ri = 0–2 at a low Reynolds number of Re = 150. The results of the squared strain rate and the vorticity shed light on the enstrophy transfer process. Thermal plume structures in the far wake originate from the upper dispersed vortices due to the high superimposed buoyancy at low Pr, while they are suppressed at high Pr. The increase in Pr plays a role as the flow stabilization, while the growth of Ri plays the reverse role. The time-averaged velocity, pressure coefficient, and temperature become more asymmetrical at high Ri. The Nusselt number of the upstream cylinder is approximately equal to the empirical result without the consideration of thermal buoyancy. Due to the thermal buoyancy, the migration of shear layers along the cylinder surface leads to the frequency alteration and harmonic frequency in the drag, lift, and Nusselt coefficients.

Eddy–Internal Wave Interactions: Stimulated Cascades in Cross-Scale Kinetic Energy and Enstrophy Fluxes

Abstract The interactions between oceanic mesoscale eddies, submesoscale currents, and internal gravity waves (IWs) are investigated in submesoscale-resolving realistic simulations in the North Atlantic Ocean. Using a novel analysis framework that couples the coarse-graining method in space with temporal filtering and a Helmholtz decomposition, we quantify the effects of the interactions on the cross-scale kinetic energy (KE) and enstrophy fluxes. By systematically comparing solutions with and without IW forcing, we show that externally forced IWs stimulate a reduction in the KE inverse cascade associated with mesoscale rotational motions and an enhancement in the KE forward cascade associated with divergent submesoscale currents, i.e., a “stimulated cascade” process. The corresponding IW effects on the enstrophy fluxes are seasonally dependent, with a stimulated reduction (enhancement) in the forward enstrophy cascade during summer (winter). Direct KE and enstrophy transfers from currents to IWs are also found, albeit with weaker magnitudes compared with the stimulated cascades. We further find that the forward KE and enstrophy fluxes associated with IW motions are almost entirely driven by the scattering of the waves by the rotational eddy field, rather than by wave–wave interactions. This process is investigated in detail in a companion manuscript. Finally, we demonstrate that the stimulated cascades are spatially localized in coherent structures. Specifically, the magnitude and direction of the bidirectional KE fluxes at submesoscales are highly correlated with, and inversely proportional to, divergence-dominated circulations, and the inverse KE fluxes at mesoscales are highly correlated with strain-dominated circulations. The predominantly forward enstrophy fluxes in both seasons are also correlated with strain-dominated flow structures.

Cross-Component Energy Transfer in Superfluid Helium-4

The reciprocal energy and enstrophy transfers between normal fluid and superfluid components dictate the overall dynamics of superfluid 4He including the generation, evolution and coupling of coherent structures, the distribution of energy among lengthscales, and the decay of turbulence. To better understand the essential ingredients of this interaction, we employ a numerical two-way model which self-consistently accounts for the back-reaction of the superfluid vortex lines onto the normal fluid. Here we focus on a prototypical laminar (non-turbulent) vortex configuration which is simple enough to clearly relate the geometry of the vortex line to energy injection and dissipation to/from the normal fluid: a Kelvin wave excitation on two vortex anti-vortex pairs evolving in (a) an initially quiescent normal fluid, and (b) an imposed counterflow. In (a), the superfluid injects energy and vorticity in the normal fluid. In (b), the superfluid gains energy from the normal fluid via the Donnelly–Glaberson instability.

Subgrid Parameterizations of Ocean Mesoscale Eddies Based on Germano Decomposition

AbstractOcean models at intermediate resolution (1/4°), which partially resolve mesoscale eddies, can be seen as Large eddy simulations of the primitive equations, in which the effect of unresolved eddies must be parameterized. In this work, we propose new subgrid models that are consistent with the physics of two‐dimensional flows. We analyze subgrid fluxes in barotropic decaying turbulence using Germano (1986, https://doi.org/10.1063/1.865568) decomposition. We show that Leonard and Cross stresses are responsible for the enstrophy dissipation, while the Reynolds stress is responsible for additional kinetic energy (KE) backscatter. We utilize these findings to propose a new model, consisting of three parts, that is compared to a baseline dynamic Smagorinsky model. The three‐component model accurately simulates the spectral transfer of energy and enstrophy and improves the representation of KE spectrum, resolved KE and enstrophy decay in a posteriori experiments. The backscattering component of the new model (Reynolds stress) is implemented both in quasi‐geostrophic and primitive equation ocean models and improves statistical characteristics, such as the vertical profile of eddy KE, meridional overturning circulation and cascades of kinetic and potential energy.

A data-driven approach to model enstrophy transfers in large eddy simulation of forced two-dimensional turbulence

Contemporary research on the application of data-driven techniques to modeling subgrid closure in two-dimensional turbulence has been limited. Furthermore, the incorporation of the enstrophy cascade and other two-dimensional turbulence-specific physics has received insufficient attention. To address these gaps, a novel physics-based shallow feed-forward neural network framework was designed in this study to model subgrid closure in three selected cases of forced two-dimensional turbulence with a forcing that provides energy and enstrophy at a particular wavenumber. As a novel approach, we trained our framework to learn the subgrid vorticity transport vector from a set of appropriate resolved flow variables. Another framework used in recent works which directly learned the subgrid forcing field was also investigated. Both frameworks were assessed using a priori and a posteriori tests for two selected filter widths. Both frameworks performed accurately for the lower filter width but less accurately for the higher filter width. However, we demonstrate that our new framework has wider usefulness for model diagnosis. Ad hoc clipping procedures were used to make the models more generalizable to higher filter widths, and stable and consistent a posteriori tests were observed for all test cases and filter widths when the subgrid forcing field was modified to enhance the model's subgrid dissipative characteristics. In contrast, modifying the enstrophy fluxes did not perform as consistently. These findings demonstrate the potential of the novel physics-based framework for improving subgrid modeling in two-dimensional turbulence.

Efficiency of energy and enstrophy transfers in periodical flows

We apply a coarse-graining technique to understand the efficiency of scale-to-scale transport of energy and enstrophy in a quasi-two-dimensional weakly turbulent periodic flow. The investigated periodic flow resembles the propagation of a monochromatic tide in a tidal channel, connected to open sea through an inlet. The interaction of the periodic flow with the inlet mouth generates vortical structures in a wide spectrum of scales, and recently, how the corresponding energy and enstrophy fluxes change their signs depending on the tidal phase has been shown. In the present study, we are interested to extend the analysis to the efficiency of the nonlinear transfer rates by analyzing the geometric alignment between the turbulent stresses and the strain rates for the energy, and the vorticity stress and large-scale vorticity gradient for the enstrophy. Our results suggest that, depending on the phase of the period, energy is efficiently transferred to larger scales (inverse cascade) in a finite range of scales, whereas the observed direct energy cascade for very small and very large scales is much less efficient. Enstrophy shows similar behaviors in terms of transitions between direct and inverse cascading; however, all transfers seem to be relatively inefficient.

On the simultaneous cascades of energy, helicity, and enstrophy in incompressible homogeneous turbulence

We characterise the incompressible turbulence cascade in terms of the concurrent inter-scale and inter-space exchanges of the scale-by-scale energy, helicity and enstrophy. The governing equations for the scale-by-scale helicity and enstrophy are derived in a similar fashion to that of the second order structure function following Hill (J. Fluid Mech., vol. 468, 2002, pp. 317–326). We examine the instantaneous dynamics, applying these equations to forced periodic turbulence and a von Kármán flow focusing on scales in the dissipative range$r=2.5\eta$, the near-dissipative range$r=0.5\lambda$and the onset inertial range$r=\lambda$(where$\eta$and$\lambda$are the Kolmogorov and Taylor length scales, respectively). The signature of the random sweeping effect is observed in all three individual budgets and between the energy and enstrophy transfers. As in the energy cascade, the anti correlation of the pressure transport and non-linear transfer is identified also in the helicity cascade. Owing to its lack of positive definiteness, the helicity transfers are found to be decorrelated from the others. However a connection between the energy cascade and helicity is identified kinematically. This connection reveals the large-scale sweeping motions are a key element in the overall energy cascade and underpins previous observations of large-scale intermittency. Taken together, this work extends a classic framework to gain novel insight on turbulence dynamics that underlay the statistically steady state, and demonstrates how transfers are interconnected.

Stability and Perturbation Enstrophy Transfer Mechanism of Flow in Slightly Divergent Channels

The global linear stability of a 2-D viscous incompressible flow through a slightly divergent finite length channel is investigated using the stream function–vorticity formulation. The flow problem is studied under inlet–outlet nonperiodic boundary conditions using a stabilized finite element method. The perturbation eigenmodes, growth rates, as well as perturbation enstrophy transfer mechanisms are analyzed to demonstrate the inlet–outlet interactions of the base flow and the perturbation enstrophy. Results show that slightly divergent channel brings the critical Reynolds number to a low value. The eigenmode is different from the classical normal mode of perturbations. It is also found that slight divergence moves the maximum envelope of the least-stable eigenmode upstream in the flow bulk. Exchange of stability is eventually triggered above the critical Reynolds number as analyzed in the transfer mechanism for the perturbation enstrophy.

Role of Mixed‐Layer Instabilities in the Seasonal Evolution of Eddy Kinetic Energy Spectra in a Global Submesoscale Permitting Simulation

AbstractA submesoscale‐permitting global ocean simulation is used to study the upper ocean turbulence in high kinetic energy (KE) regions. Submesoscale processes peak in winter so that the geostrophic KE spectra tend to be relatively shallow in winter () with steeper spectra in summer (). This transition in KE spectral scaling has two phases. In the first phase (late autumn), KE spectra show the presence of two spectral regimes: power‐law in mesoscales and power‐law in submesoscales. The first phase appears with the onset of mixed‐layer instabilities, which convert available potential energy into KE, and this process results in a flattening of KE spectra at submesoscales. However, KE spectra at longer wavelengths follow scaling associated with a forward enstrophy transfer. In the second phase (late winter), KE produced through mixed‐layer instabilities is transferred to larger scales, and power‐law also develops in mesoscales.

Advective structure functions in anisotropic two-dimensional turbulence

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Kinetic energy and enstrophy transfer in compressible Rayleigh–Taylor turbulence

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Near-Surface Ocean Kinetic Energy Spectra and Small-Scale Intermittency from Ship-Based ADCP Data in the Bay of Bengal

AbstractHorizontal currents in the Bay of Bengal were measured on eight cruises covering a total of 8600 km using a 300-kHz acoustic Doppler current profiler (ADCP). The cruises are distributed over multiple seasons and regions of the Bay. Horizontal wavenumber spectra of these currents over depths of 12–54 m and wavelengths from 2 to 400 km were decomposed into rotational and divergent components assuming isotropy. An average of across- and along-track spectra over all cruises shows that the spectral slope of horizontal kinetic energy for wavelengths of 10–80-km scales with an exponent of −1.7 ± 0.05, which transitions to a steeper slope for wavelengths above 80 km. The rotational component is significantly larger than the divergent component at scales greater than 80 km, while the ratio of the two is nearly constant with a mean of 1.16 ± 0.4 between 10 and 80 km. The measurements show a fair amount of variability and spectral levels vary between cruises by about a factor of 5 over 10–100 km. Velocity differences over 10–80 km show probability density functions and structure functions with stretched exponential behavior and anomalous scaling. Comparisons with the Garrett–Munk internal wave spectrum indicate that inertia–gravity waves account for only a modest fraction of the kinetic energy between 10 and 80 km. These constraints suggest that the near-surface flow in the Bay is primarily balanced and follows a forward enstrophy transfer quasigeostrophic regime for wavelengths greater than approximately 80 km, with a larger role for unbalanced rotating stratified turbulence and internal waves at smaller scales.

Scale locality of helicity cascade in physical space

Scale locality is a key concept in turbulent cascade theory and is also associated with reflection symmetry. Vortex stretching is proven to participate in the helicity cascade process while destroying the conservative characteristic of enstrophy transfer in three-dimensional flows. Numerical evidence indicates that a turbulent structure with scale L will also largely transfer its helicity to structures with scales of around 0.3L. However, the scale locality of the helicity cascade is slightly weaker than that of the energy cascade in physical space. The weaker scale locality suggests that more scales should be involved for turbulent modeling of helical turbulence.

Vorticity gradient stretching in the direct enstrophy transfer process of two-dimensional turbulence

In two-dimensional turbulence enstrophy is injected at the forcing scale and transferred down to the dissipative scale where viscosity dominates. On intermediate scales the forward enstrophy cascade is accompanied by a stretching of the vorticity gradient (VG) at the strain rate (SR), resulting in a preferential orthogonality between the VG and the stretching direction. In a coordinate system formed by the SR eigenvectors at a fixed time, we find that the VG statistically turns toward the direction perpendicular to the eigenvector corresponding to the stretching, as shown in the figure.

Enstrophy transfers in helical turbulence

In this paper we study the enstrophy transers in helical turbulence using direct numerical simulation. We observe that the helicity injection does not have significant effects on the inertial-range energy and helicity spectra ($\sim k^{-5/3}$) and fluxes (constants). We also calculate the separate contributions to enstrophy transfers via velocity to vorticity and vorticity to vorticity channels. There are four different enstrophy fluxes associated with the former channel or vorticity stretching, and one flux associated with the latter channel or vorticity advection. In the inertial range, the fluxes due to vorticity stretching are larger than that due to advection. These transfers too are insensitive to helicity injection.

Observed Storm Track Dynamics in Drake Passage

AbstractThe dynamics of an oceanic storm track—where energy and enstrophy transfer between the mean flow and eddies—are investigated using observations from an eddy-rich region of the Antarctic Circumpolar Current downstream of the Shackleton Fracture Zone (SFZ) in Drake Passage. Four years of measurements by an array of current- and pressure-recording inverted echo sounders deployed between November 2007 and November 2011 are used to diagnose eddy–mean flow interactions and provide insight into physical mechanisms for these transfers. Averaged within the upper to mid-water column (400–1000-m depth) and over the 4-yr-record mean field, eddy potential energy is highest in the western part of the storm track and maximum eddy kinetic energy occurs farther away from the SFZ, shifting the proportion of eddy energies from to about 1 along the storm track. There are enhanced mean 3D wave activity fluxes immediately downstream of SFZ with strong horizontal flux vectors emanating northeast from this region. Similar patterns across composites of Polar Front and Subantarctic Front meander intrusions suggest the dynamics are set more so by the presence of the SFZ than by the eddy’s sign. A case study showing the evolution of a single eddy event, from 15 to 23 July 2010, highlights the storm-track dynamics in a series of snapshots. Consistently, explaining the eddy energetics pattern requires both horizontal and vertical components of W, implying the importance of barotropic and baroclinic processes and instabilities in controlling storm-track dynamics in Drake Passage.

Scale locality of the energy cascade using real space quantities

The classical energy cascade in turbulence as described by Richardson and Kolmogorov is predominantly a conjecture relying on the locality of interactions between scales of turbulence. This picture is generally accepted and assumes that energy and enstrophy transfers occur between neighboring scales of turbulence and that vortex stretching plays a major role in the dynamics of this energy cascade. Direct numerical simulation data for Reλ ranging from 37 to 1131 is used to gather evidence for the cascade by investigating the energy and enstrophy fluxes between scales and the interplay between vorticity at one scale and strain at an adjacent scale. This is achieved by using a bandpass filter to educe the turbulent structures at various length scales, allowing one to determine the fluxes between these scales and to interrogate the role of nonlocal (in physical space) vortex stretching. It is shown that the structures of a length scale L mostly transfer their energy to structures of size 0.3L and that most of the enstrophy flux goes from structures of scale L to 0.3L. Furthermore, vortical structures of a length scale Lω are stretched mostly by straining structures of size 3Lω to 5Lω, and the stretching by eddies of sizes larger than 10Lω is negligible. The stretching is dominated by the most extensive principal strain rate of the straining structures. These observations are found to be independent of Reλ for the range investigated in this study. These results provide strong evidence for the classical view of an energy cascade transferring energy from large to small scales through a hierarchy of steps, each step consisting of the stretching of vortices by somewhat larger structures.

Surface Ocean Enstrophy, Kinetic Energy Fluxes, and Spectra From Satellite Altimetry

AbstractSpectra and fluxes of enstrophy and kinetic energy (KE) are estimated in different parts of the midlatitudinal oceans using geostrophic currents derived from altimetry data. The presence of a strong inverse flux of surface KE is confirmed at scales larger than approximately 200 km, whereas a robust enstrophy cascading regime, accompanied by an approximate KE spectrum, is observed from about 200 to 100 km. The character of fluxes and spectra is shown to compare favorably with those from a comprehensive Earth system model. In addition, as gridded altimeter data are affected by smoothening and interpolation, the qualitative robustness of the results is verified by sensitivity experiments using space and time‐filtered output from the Earth system model. Given the rotational character of the flow, this large‐scale inverse KE and smaller‐scale forward enstrophy transfer scenario is consistent with expectations from three‐dimensional rapidly rotating and strongly stratified turbulence studies as well as detailed analyses of spectra and fluxes in the upper‐level midlatitude troposphere. In further accord with results from the atmosphere, decomposing the currents into stationary and eddy components (demarcated here by variability greater and less than 100 days, respectively), it is seen that, in addition to the eddy‐eddy contribution, the stationary‐eddy and stationary‐stationary fluxes play a significant role in the inverse (forward) flux of KE (enstrophy). Thus, it is quite possible that, from about 200 to 100 km, the altimeter is capturing the rotationally dominated portion of a surface oceanic counterpart of the upper tropospheric Nastrom‐Gage spectrum.

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.

Turbulence in an Auto-Oscillating Complex Plasma

We present an experiment on the development of turbulence in a microparticle cloud in a low-temperature plasma laboratory onboard the International Space Station. First, the complex plasma cloud was stabilized with an alternating electric field. Once the stabilization was switched OFF, a so-called heartbeat instability developed in which the microparticles are oscillating radially. We show that the kinetic energy spectra develop a double cascade once the cloud becomes unstable. A double cascade has been predicted by Kraichnan and Leith for forced 2-D turbulence, but has only seldomly been observed in experiments. We also show that the reduced rates of energy and enstrophy transfer can be used to infer the excitation wavenumber $k_{\text {exc}}$ , in good agreement with $k_{\text {exc}}$ obtained from the energy spectrum.