Two-dimensional Inverse Energy Cascade Research Articles

Two-dimensional inverse energy cascade in a laboratory surf zone for varying wave directional spread

Surfzone eddies enhance the dispersion and transport of contaminants, bacteria, and larvae across the nearshore, altering coastal water quality and ecosystem health. During directionally spread wave conditions, vertical vortices (horizontal eddies) are injected near the ends of breaking crests. Energy associated with these eddies may be transferred to larger-scale, low-frequency rotational motions through an inverse energy cascade, consistent with two-dimensional turbulence. However, our understanding of the relationships between the wave conditions and the dynamics and energetics of low-frequency surfzone eddies are largely based on numerical modeling. Here, we test these relationships with remotely sensed and in situ observations from large-scale directional wave basin experiments with varying wave conditions over alongshore-uniform barred bathymetry. Surface velocities derived with particle image velocimetry were employed to assess the spatial scales of low-frequency surfzone eddies and compute structure functions with alongshore velocities. Second-order structure functions for directionally spread waves (σθ≥10°) are consistent with energy flux to larger or smaller length scales, while normally incident, unidirectional waves do not display this behavior. Third-order structure functions suggest that the surfzone flows exhibit a bidirectional energy cascade—a direct cascade to smaller and inverse cascade to larger length scales—during large directional spreads waves (σθ≥18°). However, there is not decisive evidence of an inverse energy cascade for moderate directional spreads (σθ=10°). Energy flux varies by cross-shore location and increases with increasing directional spread and wave height. Eddy decorrelation length scales weakly depend on wave directional spread. These findings advance our understanding of the dynamics linking wave breaking to large-scale rotational motions that enhance mixing and lead to rip currents, important conduits for cross-shore material exchange.

Observations of two-dimensional turbulence in the surfzone

Low-frequency, many-minute-period horizontal surfzone eddies are an important mechanism for the dispersion of material, transporting larvae, pollutants, sediment, and swimmers both across and along the nearshore. Previous numerical, laboratory, and field observations on alongshore uniform bathymetry with no or roughly uniform mean background flows suggest that the low-frequency eddies may be the result of a two-dimensional inverse energy cascade that transfers energy from relatively small spatial-scale vorticity injected by depth limited breaking waves to larger and larger spatial scales. Here, using remotely sensed high-spatial resolution estimates of currents, those results are extended to surfzones with strong complex mean circulation patterns [flows O(1 m/s)] owing to nonuniform bathymetry. Similar to previous results, wavenumber spectra and second-order structure functions calculated from the observations are consistent with a two-dimensional inverse energy cascade. The size of the largest eddies is shown to depend on the surfzone width and the spatial scales of the mean currents. Third-order structure functions also are consistent with an inverse cascade for spatial scales greater than ∼50 m. At smaller scales, the third-order structure functions suggest a mixture of inverse and forward cascades.

Inverse cascade suppression and shear-layer formation in magnetohydrodynamic turbulence subject to a guide field and misaligned rotation

Astrophysical flows are often subject to both rotation and large-scale background magnetic fields. Individually, each is known to two-dimensionalize the flow in the perpendicular plane. In realistic settings, both of these effects are simultaneously present and, importantly, need not be aligned. In this work, we numerically investigate three-dimensional forced magnetohydrodynamic turbulence subject to the competing effects of global rotation and a perpendicular background magnetic field. We focus on the case of a strong background field and find that increasing the rotation rate from zero produces significant changes in the structure of the turbulent flow. Starting with a two-dimensional inverse energy cascade at zero rotation, the flow first transitions to a forward cascade of kinetic energy, then to a shear-layer dominated regime and finally to a second shear-layer regime where the kinetic energy flux is strongly suppressed and the energy transfer is mediated by the induced magnetic field. We show that the first two transitions occur at distinct values of the Rossby number, and the third occurs at a distinct value of the Lehnert number. The three-dimensional results are confirmed using an asymptotic two-dimensional, three-component model, which allows us to extend our results to the planetary-relevant case of an arbitrary angle between the rotation vector and guide field. More generally, our results demonstrate that, when considering the simultaneous limits of strong rotation and a strong guide field, the order in which those limits are taken matters in the misaligned case.

Field Evidence of Inverse Energy Cascades in the Surfzone

AbstractLow-frequency currents and eddies transport sediment, pathogens, larvae, and heat along the coast and between the shoreline and deeper water. Here, low-frequency currents (between 0.1 and 4.0 mHz) observed in shallow surfzone waters for 120 days during a wide range of wave conditions are compared with theories for generation by instabilities of alongshore currents, by ocean-wave-induced sea surface modulations, and by a nonlinear transfer of energy from breaking waves to low-frequency motions via a two-dimensional inverse energy cascade. For these data, the low-frequency currents are not strongly correlated with shear of the alongshore current, with the strength of the alongshore current, or with wave-group statistics. In contrast, on many occasions, the low-frequency currents are consistent with an inverse energy cascade from breaking waves. The energy of the low-frequency surfzone currents increases with the directional spread of the wave field, consistent with vorticity injection by short-crested breaking waves, and structure functions increase with spatial lags, consistent with a cascade of energy from few-meter-scale vortices to larger-scale motions. These results include the first field evidence for the inverse energy cascade in the surfzone and suggest that breaking waves and nonlinear energy transfers should be considered when estimating nearshore transport processes across and along the coast.

Robustness of vortex populations in the two-dimensional inverse energy cascade

We study how the properties of forcing and dissipation affect the scaling behaviour of the vortex population in the two-dimensional turbulent inverse energy cascade. When the flow is forced at scales intermediate between the domain and dissipation scales, the growth rates of the largest vortex area and the spectral peak length scale are robust to all simulation parameters. For white-in-time forcing the number density distribution of vortex areas follows the scaling theory predictions of Burgess & Scott (J. Fluid Mech., vol. 811, 2017, pp. 742–756) and shows little sensitivity either to the forcing bandwidth or to the nature of the small-scale dissipation: both narrowband and broadband forcing generate nearly identical vortex populations, as do Laplacian diffusion and hyperdiffusion. The greatest differences arise in comparing simulations with correlated forcing to those with white-in-time forcing: in flows with correlated forcing the intermediate range in the vortex number density steepens significantly past the predicted scale-invariant $A^{-1}$ scaling. We also study the impact of the forcing Reynolds number $Re_{f}$, a measure of the relative importance of nonlinear terms and dissipation at the forcing scale, on vortex formation and the scaling of the number density. As $Re_{f}$ decreases, the flow changes from one dominated by intense circular vortices surrounded by filaments to a less structured flow in which vortex formation becomes progressively more suppressed and the filamentary nature of the surrounding vorticity field is lost. However, even at very small $Re_{f}$, and in the absence of intense coherent vortex formation, regions of anomalously high vorticity merge and grow in area as predicted by the scaling theory, generating a three-part number density similar to that found at higher $Re_{f}$. At late enough stages the aggregation process results in the formation of long-lived circular vortices, demonstrating a strong tendency to vortex formation, and via a route distinct from the axisymmetrization of forcing extrema seen at higher $Re_{f}$. Our results establish coherent vortices as a robust feature of the two-dimensional inverse energy cascade, and provide clues as to the dynamical mechanisms shaping their statistics.

Scaling theory for vortices in the two-dimensional inverse energy cascade

We propose a new similarity theory for the two-dimensional inverse energy cascade and the coherent vortex population it contains when forced at intermediate scales. Similarity arguments taking into account enstrophy conservation and a prescribed constant energy injection rate such that $E\sim t$ yield three length scales, $l_{\unicode[STIX]{x1D714}}$, $l_{E}$ and $l_{\unicode[STIX]{x1D713}}$, associated with the vorticity field, energy peak and streamfunction, and predictions for their temporal evolutions, $t^{1/2}$, $t$ and $t^{3/2}$, respectively. We thus predict that vortex areas grow linearly in time, $A\sim l_{\unicode[STIX]{x1D714}}^{2}\sim t$, while the spectral peak wavenumber $k_{E}\equiv 2\unicode[STIX]{x03C0}l_{E}^{-1}\sim t^{-1}$. We construct a theoretical framework involving a three-part, time-evolving vortex number density distribution, $n(A)\sim t^{\unicode[STIX]{x1D6FC}_{i}}A^{-r_{i}},~i\in 1,2,3$. Just above the forcing scale ($i=1$) there is a forcing-equilibrated scaling range in which the number of vortices at fixed $A$ is constant and vortex ‘self-energy’ $E_{v}^{cm}=(2{\mathcal{D}})^{-1}\int \overline{\unicode[STIX]{x1D714}_{v}^{2}}A^{2}n(A)\,\text{d}A$ is conserved in $A$-space intervals $[\unicode[STIX]{x1D707}A_{0}(t),A_{0}(t)]$ comoving with the growth in vortex area, $A_{0}(t)\sim t$. In this range, $\unicode[STIX]{x1D6FC}_{1}=0$ and $n(A)\sim A^{-3}$. At intermediate scales ($i=2$) sufficiently far from the forcing and the largest vortex, there is a range with a scale-invariant vortex size distribution. We predict that in this range the vortex enstrophy $Z_{v}^{cm}=(2{\mathcal{D}})^{-1}\int \overline{\unicode[STIX]{x1D714}_{v}^{2}}An(A)\,\text{d}A$ is conserved and $n(A)\sim t^{-1}A^{-1}$. The final range ($i=3$), which extends over the largest vortex-containing scales, conserves $\unicode[STIX]{x1D70E}_{v}^{cm}=(2{\mathcal{D}})^{-1}\int \overline{\unicode[STIX]{x1D714}_{v}^{2}}n(A)\,\text{d}A$. If $\overline{\unicode[STIX]{x1D714}_{v}^{2}}$ is constant in time, this is equivalent to conservation of vortex number $N_{v}^{cm}=\int n(A)\,\text{d}A$. This regime represents a ‘front’ of sparse vortices, which are effectively point-like; in this range we predict $n(A)\sim t^{r_{3}-1}A^{-r_{3}}$. Allowing for time-varying $\overline{\unicode[STIX]{x1D714}_{v}^{2}}$ results in a small but significant correction to these temporal dependences. High-resolution numerical simulations verify the predicted vortex and spectral peak growth rates, as well as the theoretical picture of the three scaling ranges in the vortex population. Vortices steepen the energy spectrum $E(k)$ past the classical $k^{-5/3}$ scaling in the range $k\in [k_{f},k_{v}]$, where $k_{v}$ is the wavenumber associated with the largest vortex, while at larger scales the slope approaches $-5/3$. Though vortices disrupt the classical scaling, their number density distribution and evolution reveal deeper and more complex scale invariance, and suggest an effective theory of the inverse cascade in terms of vortex interactions.

The generation of zonal jets by large-scale mixing

The development of zonal flows on a midlatitude β-plane subject to a time-varying topographic forcing is investigated in a series of numerical integrations in which the forcing is concentrated at large scales, and in which the usual two-dimensional inverse energy cascade is absent. In contrast to the case of small-scale forcing, where mixing of potential vorticity occurs largely through the action of small-scale eddies, mixing of potential vorticity in this case occurs predominantly in latitudinally localized Rossby wave critical layer regions, whose width grows continuously in time due to the entrainment of background fluid. The potential vorticity is found to organize into a piecewise constant staircase-like profile, monotonic in latitude, provided the ratio \documentclass[12pt]{minimal}\begin{document}$L_\text{Rh}/L_f\gtrsim 1$\end{document}LRh/Lf≳1, where \documentclass[12pt]{minimal}\begin{document}$L_\text{Rh}$\end{document}LRh is the usual Rhines scale and Lf is the scale of the forcing; this may be regarded as supplemental to the condition \documentclass[12pt]{minimal}\begin{document}$L_\text{Rh}/L_{\varepsilon }\gtrsim 6$\end{document}LRh/Lɛ≳6, where Lɛ = (ɛ/β3)1/5 and ɛ is the rate of energy input, obtained recently [R. K. Scott and D. G. Dritschel, “The structure of zonal jets in geostrophic turbulence,” J. Fluid Mech. 711, 576–598 (2012)10.1017/jfm.2012.410] for the case of small-scale forcing. The numerical results further suggest that the nature of the potential vorticity mixing is controlled by the ratio Lɛ/Lf, and occurs predominantly in critical layers when Lɛ/Lf ≲ 1/6. A combined condition for staircase formation may therefore be expressed as \documentclass[12pt]{minimal}\begin{document}$L_\text{Rh}/L_{\varepsilon }\gtrsim \max \lbrace 6,L_f/L_{\varepsilon }\rbrace$\end{document}LRh/Lɛ≳max{6,Lf/Lɛ}. Finally, in a separate set of experiments it is shown that when forcing is represented by an additive source term in the evolution equation, as is common practice in numerical investigations of two-dimensional turbulence, the effect of non-conservation of potential vorticity may obscure the development of the staircase profile in the critical layer mixing dominated regime.

Infrared Reynolds number dependency of the two-dimensional inverse energy cascade

High-resolution simulations of forced two-dimensional turbulence reveal that the inverse cascade range is sensitive to an infrared Reynolds number, Reα = kf/kα, where kf is the forcing wavenumber and kα is a frictional wavenumber based on linear friction. In the limit of high Reα, the classic k−5/3 scaling is lost and we obtain steeper energy spectra. The sensitivity is traced to the formation of vortices in the inverse energy cascade range. Thus, it is hypothesized that the dual limit Reα → ∞ and Reν = kd/kf → ∞, where kd is the small-scale dissipation wavenumber, will lead to a steeper energy spectrum than k−5/3 in the inverse energy cascade range. It is also found that the inverse energy cascade is maintained by non-local triad interactions.

Physical mechanism of the inverse energy cascade of two-dimensional turbulence: a numerical investigation

We report an investigation of inverse energy cascade in steady-state two-dimensional turbulence by direct numerical simulation (DNS) of the two-dimensional Navier–Stokes equation, with small-scale forcing and large-scale damping. We employed several types of damping and dissipation mechanisms in simulations up to 20482resolution. For all these simulations we obtained a wavenumber range for which the mean spectral energy flux is a negative constant and the energy spectrum scales ask−5/3, consistent with the predictions of Kraichnan (Phys. Fluids, vol. 439, 1967, p. 1417). To gain further insight, we investigated the energy cascade in physical space, employing a local energy flux defined by smooth filtering. We found that the inverse energy cascade is scale local, but that the strongly local contribution vanishes identically, as argued by Kraichnan (J. Fluid Mech., vol. 47, 1971, p. 525). The mean flux across a length scale ℓ was shown to be due mainly to interactions with modes two to eight times smaller. A major part of our investigation was devoted to identifying the physical mechanism of the two-dimensional inverse energy cascade. One popular idea is that inverse energy cascade proceeds via merger of like-sign vortices. We made a quantitative study employing a precise topological criterion of merger events. Our statistical analysis showed that vortex mergers play a negligible direct role in producing mean inverse energy flux in our simulations. Instead, we obtained with the help of other works considerable evidence in favour of a ‘vortex thinning’ mechanism, according to which the large-scale strains do negative work against turbulent stress as they stretch out the isolines of small-scale vorticity. In particular, we studied a multi-scale gradient (MSG) expansion developed by Eyink (J. Fluid Mech., vol. 549, 2006a, p. 159) for the turbulent stress, whose contributions to inverse cascade can all be explained by ‘thinning’. The MSG expansion up to second order in space gradients was found to predict well the magnitude, spatial structure and scale distribution of the local energy flux. The majority of mean flux was found to be due to the relative rotation of strain matrices at different length scales, a first-order effect of ‘thinning’. The remainder arose from two second-order effects, differential strain rotation and vorticity gradient stretching. Our findings give strong support to vortex thinning as the fundamental mechanism of two-dimensional inverse energy cascade.

Physical Mechanism of the Two-Dimensional Inverse Energy Cascade

We study the physical mechanisms of the two-dimensional inverse energy cascade using theory, numerics, and experiment. Kraichnan's prediction of a -5/3 spectrum with constant, negative energy flux is verified in our simulations of 2D Navier-Stokes equations. We observe a similar but shorter range of inverse cascade in laboratory experiments. Our theory predicts, and the data confirm, that inverse cascade results mainly from turbulent stress proportional to small-scale strain rotated by 45 degrees. This "skew-Newtonian" stress is explained by the elongation and thinning of small-scale vortices by large-scale strain which weakens their velocity and transfers their energy upscale.

A turbulent constitutive law for the two-dimensional inverse energy cascade

We develop a fundamental approach to a turbulent constitutive law for the 2D inverse cascade, based upon a convergent multi-scale gradient (MSG) expansion. To first order in gradients we find that the turbulent stress generated by small-scale eddies is proportional not to strain but instead to `skew-strain,' i.e. the strain tensor rotated by $45^\circ.$ The skew-strain from a given scale of motion makes no contribution to energy flux across eddies at that scale, so that the inverse cascade cannot be strongly scale-local. We show that this conclusion extends a result of Kraichnan for spectral transfer and is due to absence of vortex-stretching in 2D. This `weakly local' mechanism of inverse cascade requires a relative rotation between the principal directions of strain at different scales and we argue for this using both the dynamical equations of motion and also a heuristic model of `thinning' of small-scale vortices by an imposed large-scale strain. Carrying out our expansion to second-order in gradients, we find two additional terms in the stress that can contribute to energy cascade. The first is a Newtonian stress with an `eddy-viscosity' due to differential strain-rotation, and the second is a tensile stress exerted along vorticity contour-lines. The latter was anticipated by Kraichnan for a very special model situation of small-scale vortex wave-packets in a uniform strain field. We prove a proportionality in 2D between the mean rates of differential strain-rotation and of vorticity-gradient stretching, analogous to a similar relation of Betchov for 3D. According to this result the second-order stresses will also contribute to inverse cascade when, as is plausible, vorticity contour-lines lengthen on average by turbulent advection.

On the contribution of coherent vortices to the two-dimensional inverse energy cascade

We investigate theoretically and numerically the inverse energy cascade in statistically steady two-dimensional turbulence. We perform a numerical testing of the analytical results proposed by Lindborg (J. Fluid Mech. vol. 338, 1999, p. 259), and we show that these predictions are quantitatively verified in the inverse energy cascade provided that the quantities involved in the spatial average computation are also averaged over all directions. Then, we define a simple measurable criterion based on the kinetic energy induced by coherent vortices in physical space. Using this criterion, we introduce more selective analyses of the energy cascade that reveal spatial properties of energy transfers which are concealed by global spatial averages. We conclude that there exist convective fluxes in both physical and scale space that feed the energy cascade processes in strongly energetic regions. In two dimensions, these regions are mostly localized around coherent structures. In the turbulent background, this mechanism manifests itself as a deficit of the kinetic energy and weaker inverse energy transfers.

Intermittency and coherent structures in the two-dimensional inverse energy cascade: comparing numerical and laboratory experiments.

We study the internal intermittency in the inverse energy cascade and in the condensation regime of two-dimensional turbulence, using physical and numerical experimental approaches. The analysis confirms that the velocity increments have nearly Gaussian distributions at all scales in the inverse cascade regime; it moreover shows that, in the condensation regime, the probability distribution functions of the velocity increments are non-Gaussian but do not significantly vary with the scale; it follows that one may consider that there is essentially no intermittency (in the usual sense), in the condensation regime. In both regimes, we emphasize that coherent structures (i.e., long-lived vortices) are clearly visible on the vorticity field, and we suggest the non-Gaussianity of the distributions in the condensation regime is due to the presence of a large-scale long-lived structures. The study is supplemented by the analysis of the distribution of energy transfers at various scales.

Intermittency of a passive tracer in the inverse energy cascade.

We report an experimental study of the dispersion of a passive tracer in the two-dimensional inverse energy cascade, which shows that a nonintermittent velocity field can sustain a strongly intermittent concentration field. The experiment suggests the exponents of the intermittent concentration field saturate at large orders towards xi(infinity) approximately 1.2. These observations are in excellent agreement with a recent numerical work [A. Celani, A. Lanotte, A. Mazzino, and M. Vergassola, Phys. Rev. Lett. 84, 2385 (2000)] and theoretical expectations [E. Balkovsky and V. Lebedev, Phys. Rev. E 58, 5776 (1998); V. Yakhot, ibid. 55, 329 (1997)].

Universality and saturation of intermittency in passive scalar turbulence

The statistical properties of a scalar field advected by the nonintermittent Navier-Stokes flow arising from a two-dimensional inverse energy cascade are investigated. The universality properties of the scalar field are probed by comparing the results obtained with two types of injection mechanisms. Scaling properties are shown to be universal, even though anisotropies injected at large scales persist down to the smallest scales and local isotropy is not fully restored. Scalar statistics is strongly intermittent and scaling exponents saturate to a constant for sufficiently high orders. This is observed also for the advection by a velocity field rapidly changing in time, pointing to the genericity of the phenomenon.

Intermittency in the two-dimensional inverse cascade of energy: Experimental observations

An extensive experimental study of the two-dimensional inverse energy cascade is presented. The experiments are performed in electromagnetically driven flows, using thin, stably-stratified layers. Complete instantaneous velocity fields are measured using particle imaging velocimetry techniques. Depending on the bottom-wall friction, two different regimes are observed: when the friction is low, the energy transferred from the forcing scale towards large scales accumulates in the lowest accessible mode, leading to a mean rotation of the flow and to an energy spectrum displaying a sharp peak at the minimum wave-number. This condensation is accompanied by the emergence of a very strong vortex around which the rotation is organized. At higher frictions, the inverse energy cascade conjectured by Kraichnan [Phys. Fluids 10, 1417 (1967)] is observed and is found to be stationary, homogeneous and isotropic, with a Kolmogorov constant consistent with numerical estimates. This inverse cascade does not appear to be characterized by the presence of strong coherent vortices. The characteristic size of the latter is of the order of the injection scale. Their statistical properties tend to show that the cascade is rather driven by a clustering mechanism involving same sign vortices rather than a sequence of merging events producing larger and larger vortices. Intermittency effects are also investigated for the inverse cascade range. It is found that, within experimental errors, there is no intermittency in the inverse cascade range of two-dimensional turbulence and that the statistics of velocity increments, either longitudinal or transverse, are close to Gaussian. These results constitute the first experimental study of intermittency in two-dimensional turbulence as well as the first observation of normal scaling in a field of research which has been increasingly concerned with anomalous exponents.

Experimental Observation of the Two-Dimensional Inverse Energy Cascade

We report the first clear experimental evidence of a stationary two-dimensional inverse energy cascade. The experiments are performed in electromagnetically driven flows, using thin, stably stratified layers; measurements of the instantaneous velocity fields show that the fluctuations are homogeneous and isotropic in the inertial range. The energy spectrum displays the expected ${k}^{\ensuremath{-}5/3}$ law with a Kolmogorov constant lying in the range 5.5--7, which is consistent with the current numerical estimates.

Some properties of two-dimensional inverse energy cascade dynamics

Some properties of two-dimensional inverse energy cascade dynamics

Experimental study of the two-dimensional inverse energy cascade in a square box

A quantitative experimental study of the two-dimensional inverse energy cascade is presented. The flow is electrically driven in a horizontal layer of mercury and three-dimensional perturbations are suppressed by means of a uniform magnetic field, so that the flow can be well approximated by a two-dimensional Navier–Stokes equation with a steady forcing term and a linear friction due to the Hartmann layer. Turbulence is produced by the instability of a periodic square network of 36 electrically driven alternating vortices. The inverse cascade is limited at large scales, either by the linear friction or by the finite size of the domain, depending on the experimental parameters. In the first case, $k^{-\frac{5}{3}$ spectra are measured and the corresponding two-dimensional Kolmogorov constant is in the range 3–7. In the second case, a condensation of the turbulent energy in the lowest mode, corresponding to a spontaneous mean global rotation, is observed. Such a condensation was predicted by Kraichnan (1967) from statistical thermodynamics arguments, but without the symmetry breaking. Random reversals of the rotation sense, owing to turbulent fluctuations, are more and more sparse as friction is decreased. The lowest mode fluctuations and the small scales are statistically independent.