Inverse Cascade Research Articles

Two-dimensional Rayleigh–Bénard convection without boundaries

We study the effects of Prandtl number $\mathit {Pr}$ and Rayleigh number $\mathit {Ra}$ in two-dimensional Rayleigh–Bénard convection without boundaries, i.e. with periodic boundary conditions. For Prandtl numbers in the range $10^{-3} \leqslant \mathit {Pr} \leqslant 10^2$ , the viscous dissipation scales as $\epsilon _\nu \propto \mathit {Pr}^{1/2}\mathit {Ra}^{-1/4}$ , which is based on the observation that enstrophy $\langle {\omega ^2}\rangle \propto \mathit {Pr}^0 \mathit {Ra}^{1/4}$ , and the Nusselt number tends to follow the ‘ultimate’ scaling $\mathit {Nu} \propto \mathit {Pr}^{1/2}\mathit {Ra}^{1/2}$ for all values of $\mathit {Pr}$ considered. The inverse cascade of kinetic energy forms the power-law spectrum $\hat {E}_u(k) \propto k^{-2.3}$ , which is close to $k^{-11/5}$ proposed by the Bolgiano–Obukhov (BO) scaling. The potential energy flux is not constant, in contrast to one of the main assumptions underlying the BO phenomenology. So, the direct cascade of potential energy forms the power-law spectrum $\hat {E}_\theta (k) \propto k^{-1.2}$ , which deviates from the expected $k^{-7/5}$ . Finally, at $\mathit {Pr} \to 0$ and $\infty$ , we find that the dynamics is dominated by vertically oriented elevator modes that grow without bound, even at high Rayleigh numbers and with large-scale dissipation present.

Subgrid-scale model considering the inverse energy cascade using an artificial neural network

For the closure of the subgrid-scale (SGS) stress tensor, an artificial neural network (ANN)-based SGS model that takes account of the inverse energy cascade in isotropic turbulence is developed. The data required for training this ANN-based SGS model are provided by direct numerical simulation of isotropic turbulence with an inverse energy cascade. Two input features, the root mean square of the rate-of-strain tensor and the product of the eigenvalues of the rate-of-strain tensor, are employed to characterize the inverse energy cascade. An a priori test reveals that the ANN-based model adequately predicts the SGS stress tensor in the backward energy transfer process, and the predictive capability of the gradient model is found to be slightly poorer than that of the ANN-based model, while that of the Smagorinsky model is not satisfactory. In comparison with the gradient model, the ANN-based model even predicts a few backward energy transfer events in the stage of excessive energy dissipation. In addition, the off-diagonal component of the SGS stress tensor, rather than the diagonal component, may be intimately associated with the inverse energy cascade. The ANN-based SGS model presented here is expected to provide inspiration for future investigations of the construction of SGS models that take account of the inverse energy cascade.

Flow patterns and energy spectra in forced quasi-two-dimensional turbulence: effect of system size and damping rate

An experimental investigation is conducted to study the flow patterns, spectral properties and energy fluxes in thin-layer turbulence with varying system sizes and damping rates. It is found that although a system-size vortex (an indicator of spectral condensation) occurs for small system sizes and does not for large ones, the spectra for different system sizes consistently exhibit a scaling close to $k^{-3}$ in inverse cascade (another indicator of spectral condensation). On the other hand, under a fixed system size larger than the friction-dominated length scale, the energy spectrum in the inverse cascade range changes from $k^{-3}$ to $k^{-5/3}$ as the damping rate increases, suggesting that the friction-dominated length scale may not be a suitable parameter for predicting spectral transition. At lower damping rates and large system sizes, turbulent structures grow larger via inverse cascade, manifesting as long streamers, and the small-scale vortices are suppressed. This suppression leads to a reduction of energy flux at intermediate scales and a change in the spectral shape. The dimensionless Taylor microscale is found to exhibit a monotonic dependence on the damping rate. With the reduction in the damping rate, the Taylor microscale increases to become comparable with the forcing scale, and the spectrum in inverse cascade transits to a steeper scaling, $k^{-3}$ , indicating that the dimensionless Taylor microscale may be used as a diagnostic parameter for spectral transition.

A transformer-based convolutional method to model inverse cascade in forced two-dimensional turbulence

The present work proposes a novel transformer-based convolutional neural network (TransCNN) method to effectively model the inverse energy cascade in two dimensional (2D) turbulence. The TransCNN structure combines large-scale features extracted by transformer with small-scale features from convolutional layers, thus is considered suitable for multi-scale modeling. The novel TransCNN method has been applied to model sub-grid scale (SGS) stress for large-eddy simulation (LES) of 2D turbulence, under the extremely challenging situation that the LES grid is too coarse to resolve the external forcing scale. The data-driven model trained by the novel TransCNN structure is compared to two deep CNN models with varying complexities. All models exhibit proficiency during a priori tests. Notably, TransCNN surpasses its counterparts in predictive accuracy and generalizability in a posteriori tests. An investigation into the receptive fields reveals that the TransCNN model can efficiently leverage global information with the transformer structure, which is key to its superior performance in representing the inverse energy cascade in the 2D turbulent simulations.

Author Correction: Onset of vortex clustering and inverse energy cascade in dissipative quantum fluids

Author Correction: Onset of vortex clustering and inverse energy cascade in dissipative quantum fluids

Direct simulation of vortex dynamics of multi-cellular Taylor–Green vortex by pseudo-spectral method

This study investigates the long-time vorticity dynamics of the multi-cellular configurations of the two-dimensional (2D) Taylor–Green vortex (TGV). The pseudo-spectral method is used to solve the incompressible Navier–Stokes equation to analyze the evolution of TGV arrays. The focus is on understanding vortex interactions leading to vortex filamentation and stripping (forward cascade) during primary instability; merger and reconnection (inverse cascade) among the TGV vortical cells subsequently. Here, consideration of multiple cells avoids imposing symmetries at the smallest periodic length scale, and thereby affecting disturbance growth. The initial condition is taken from the analytic solution of the TGV, and Fourier spectral method is employed to track the interactions of the initial doubly-periodic vortices. The full sequence of evolution from one equilibrium state to another for the TGV is not addressed before, as reported here to fill this gap for multiple TGV cells in both directions. By studying various vortical interactions in the ensemble, here we report the enstrophy and energy spectra for different number of TGV cells. This is crucial in understanding the very long-time evolution process, at post-critical Reynolds numbers for the 2D TGV problem in the same physical domain, (0≤(x,y)≤4π) having (4×4) and (6×6) cells. Reported results show the evolution of these vortical cells from original configurations to finally a (1×1) vortical cells — the universal state not demonstrated before.

Non-local gravity wave turbulence in presence of condensate

The $k^{-23/6}$ wave action spectrum with an inverse cascade is one of the fundamental Kolmogorov–Zakharov solutions for gravity wave turbulence, which is part of the citation for the Dirac Medal in 2003. Instead of confirming this solution, however, several existing simulations and experiments suggest a spectrum of $k^{-3}$ in set-ups corresponding to the inverse cascade. We provide a theoretical explanation for the latter, considering the condensate that naturally forms in finite domains of experiments/simulations. Our new theory hinges on: (1) derivation of a spectral diffusion equation when non-local interactions with the condensate become dominant, for the first time systematically formulated for quartet-interaction systems; and (2) careful analysis of the asymptotics of interaction coefficient with a remarkable cancellation of all leading-order terms.

Holographic turbulence from a random gravitational potential

We study the turbulent dynamics of a relativistic (2 + 1)-dimensional fluid placed in a stochastic gravitational potential. We demonstrate that the dynamics of the fluid can be obtained using a dual holographic description realized by an asymptotically Anti-de Sitter black brane driven by a random boundary metric. Using the holographic duality we study the energy power spectrum of a fluid with an inverse energy cascade and show that it is compatible with that of a compressible fluid flow. We calculate the local energy dissipation and the local fluid velocity distribution which provide other measures of the holographic fluid turbulence.

Numerical study of granulation in anelastic thermal convection in spherical shells

The present work investigates granulation or convective flow patterns in density-stratified (or anelastic) convection in spherical shells. The density-stratified thermal convection is typically present in astrophysical systems (such as solar convection); motivated by this, we performed a series of three-dimensional anelastic convection simulations in a spherical shell geometry using an in-house developed hybrid solver. We explored the effect of Rayleigh number and density scale height on the convective flow patterns. The granulation (or cell-like structures) are more prominent at higher density scale height and Rayleigh number. The granulation is further characterized by kinetic energy and helicity spectra. Our results support the argument that the convective flow patterns (or granulation) emerge due to inverse cascade owing to the presence of density stratification. Convective patterns (or granulation) are identified based on length scales, time scales, and flow velocity. The length scale of granules is further verified using a solar granulation model. Our analysis suggests the existence of inverse cascade and supergranulation on the spherical surface due to density-stratified thermal convection in spherical shells.

Thermodynamical Effects of Ocean Current Feedback in a Quasigeostrophic Coupled Model

Abstract Air–sea fluxes are the main drivers of ocean circulation, yet their representation in ocean-only models remains challenging. While a zeroth-order formulation accounting only for the state of the atmosphere is well adopted by the community, surface ocean feedback has gained attention over the last decades. In this paper, we focus on thermodynamical indirect feedback of surface ocean currents, which completes the “eddy killing” effect induced by the mechanical feedback. In this study, we quantify both the mechanical and thermodynamical contributions in the context of idealized, coupled quasigeostrophic simulations through sensitivity experiments on wind stress formulation. As compared to eddy killing which impacts kinetic energy levels, the indirect thermodynamical feedback induces significant changes in potential energy levels. The thermodynamical feedback also enhances by +27% the potential-to-kinetic turbulent energy conversion induced by relative wind stress formulation, as well as significant changes in both forward and inverse cascades of potential energy (PE). That is, accounting for ocean surface currents in the computation of wind stress significantly changes transfers of PE from the mean to the turbulent flow. These changes are mostly controlled by a reduced upscale energy flux rather than a more vigorous downscale flux, a process in line with results obtained for kinetic energy fluxes associated with the eddy killing effect.

Necessary and sufficient conditions for Kolmogorov’s flux laws on T2 and T3

Necessary and sufficient conditions for the third order Kolmogorov universal scaling flux laws are derived for the stochastically forced incompressible Navier Stokes equations on the torus in 2D and 3D. This paper rigorously generalises the result of (Bedrossian 2019 Commun. Math. Phys. 367 1045–75) to functions which are heavy-tailed in Fourier space or have local finite time singularities in the inviscid limit. In other words, we have rigorously derived the existence of the well known physical relationships, the direct and inverse cascades. Furthermore we show that the rate of the direct cascade is proportional to the amount of energy ‘escaping to infinity’ in spectral space as well as a measure of the total singularities within the solution. Similarly, an inverse cascade is proportional to the amount of energy that moves towards the k = 0 Fourier mode in the invisicid limit.

Relationship between the SST diurnal cycle over the Tropical Western Pacific Ocean and subseasonal/seasonal oscillations: Associations with wind speed and outgoing longwave radiation

In this study, the relationship between SST diurnal cycle,10 m wind speed (W10) and Outgoing Longwave Radiation (OLR) is investigated. A wavelet spectral analysis was applied to SST hourly data to identify the SST diurnal cycle over the Tropical Western Pacific Ocean (TWPO). The SST diurnal cycle was identified as a prominent spectrum peak in waves with an oscillation period of 1 day. An inverse energy cascade hypothesis suggests that the energy from the SST diurnal cycle propagates and gets absorbed by waves within the subseasonal timescale. Three windows were selected to represent the diurnal (0–2 days), subseasonal (15–60 days), and seasonal (80–200 days) timescales. A wavelet-filtered analysis was performed in these windows, revealing inverse SST/ wind speed and direct SST/ OLR correlations over TWPO. These findings are consistent with empirical parametric models. Additionally, this study demonstrates the rectification mechanism of SST through a wavelet-filtered approach, identifying statistically significant correlations at the 5 % level within the diurnal window (0–2 days), particularly in the central tropical Pacific. Wavelet-filtered anomalies of SST, W10, and OLR along 50–160°E reveal the alternating dry and wet phase propagation across the Indo-Pacific in the subseasonal window, which is associated with the Madden-Julian Oscillation (MJO). Furthermore, westward propagating anomalies in the Indian Ocean and eastward propagating anomalies east of the Maritime Continent (MC) and within the Pacific were identified in the seasonal window, resembling patterns of Rossby and equatorial Kelvin waves, respectively.

Turbulence scaling from deep learning diffusion generative models

Complex spatial and temporal structures are inherent characteristics of turbulent fluid flows and comprehending them poses a major challenge. This comprehension necessitates an understanding of the space of turbulent fluid flow configurations. We employ a diffusion-based generative model to learn the distribution of turbulent vorticity profiles and generate snapshots of turbulent solutions to the incompressible Navier-Stokes equations. We consider the inverse cascade in two spatial dimensions and generate diverse turbulent solutions that differ from those in the training dataset. We analyze the statistical scaling properties of the new turbulent profiles, calculate their structure functions, energy power spectrum, velocity probability distribution function and moments of local energy dissipation. All the learnt scaling exponents are consistent with the expected Kolmogorov scaling This agreement with established turbulence characteristics provides strong evidence of the model's capability to capture essential features of real-world turbulence.

Global climate modeling of the Jupiter troposphere and effect of dry and moist convection on jets

Aims. The atmosphere of Jupiter is characterized by banded jets, including an equatorial super-rotating jet, by an intense moist con-vective activity, and by perturbations exerted by vortices, waves, and turbulence. Even after space exploration missions to Jupiter and detailed numerical modeling of Jupiter, questions remain about the mechanisms underlying the banded jets and the role played by dry and moist convection in maintaining these jets. Methods. We report three-dimensional simulations of the Jupiter weather layer using a global climate model (GCM) called Jupiter-DYNAMICO, which couples hydrodynamical integrations on an icosahedral grid with detailed radiative transfer computations. We added a thermal plume model for Jupiter that emulates the effect of mixing of heat, momentum, and tracers by dry and moist convec-tive plumes that are left unresolved in the GCM mesh spacing with a physics-based approach. Results. Our Jupiter-DYNAMICO global climate simulations show that the large-scale Jovian flow, in particular the jet structure, could be highly sensitive to the water abundance in the troposphere and that an abundance threshold exists at which equatorial super-rotation develops. In contrast to our dry (or weakly moist) simulations, simulations that include the observed amount of tropospheric water exhibit a clear-cut super-rotating eastward jet at the equator and a dozen eastward mid-latitude jets that do not migrate poleward. The magnitudes agree with the observations. The convective activity simulated by our thermal plume model is weaker in the equatorial regions than in mid to high latitudes, as indicated by lightning observations. Regardless of whether they are dry or moist, our simulations exhibit the observed inverse energy cascade from small (eddies) to large scales (jets) in a zonostrophic regime.

Sign-Indefinite Invariants Shape Turbulent Cascades.

We highlight a noncanonical yet natural choice of variables for an efficient derivation of a kinetic equation for the energy density in nonisotropic systems, including internal gravity waves on a vertical plane, inertial, and Rossby waves. The existence of a second quadratic invariant simplifies the kinetic equation and leads to extra conservation laws for resonant interactions. We analytically determine the scaling of the radial turbulent energy spectrum. Our findings suggest the existence of an inverse energy cascade of internal gravity waves, from small to large scales, in practically relevant scenarios.

The Role of Surface Potential Vorticity in the Vertical Structure of Mesoscale Eddies in Wind-Driven Ocean Circulations

Abstract The vertical structure of ocean eddies is generally surface-intensified, commonly attributed to the dominant baroclinic modes arising from the boundary conditions (BCs). Conventional BC considerations mostly focus on either flat- or rough-bottom conditions. The impact of surface buoyancy anomalies—often represented by surface potential vorticity (PV) anomalies—has not been fully explored. Here, we study the role of the surface PV in setting the vertical distribution of eddy kinetic energy (EKE) in an idealized adiabatic ocean model driven by wind stress. The simulated EKE profile in the extratropical ocean tends to peak at the surface and have an e-folding depth typically smaller than half of the ocean depth. This vertical structure can be reasonably represented by a single surface quasigeostrophic (SQG) mode at the energy-containing scale resulting from the large-scale PV structure. Due to isopycnal outcropping and interior PV homogenization, the surface meridional PV gradient is substantially stronger than the interior PV gradient, yielding surface-trapped baroclinically unstable modes with horizontal scales comparable to or smaller than the deformation radius. These surface-trapped eddies then grow in size both horizontally and vertically through an inverse energy cascade up to the energy-containing scale, which dominates the vertical distribution of EKE. As for smaller horizontal scales, the EKE distribution decays faster with depth. Guided by this interpretation, an SQG-based scale-aware parameterization of the EKE profile is proposed. Preliminary offline diagnosis of a high-resolution simulation shows the proposed scheme successfully reproducing the dependence of the vertical structure of EKE on the horizontal grid resolution.

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.

Wavelet-based wavenumber spectral estimate of eddy kinetic energy: Application to the North Atlantic

An ensemble of eddy-rich North Atlantic simulations is analyzed, providing estimates of eddy kinetic energy (EKE) wavenumber spectra and spectral budgets below the mixed layer where energy input from surface convection and wind stress are negligible. A wavelet transform technique is used to estimate a spatially localized ‘pseudo-Fourier’ spectrum (Uchidaet al., 2023b), permitting comparisons to be made between spectra at different locations in a highly inhomogeneous and anisotropic environment. The EKE spectra tend to be stable in time but the spectral budgets are highly time dependent. We find evidence of a Gulf Stream imprint on the near Gulf Stream eddy field appearing as enhanced levels of EKE in the (nominally) North–South direction relative to the East–West direction. Surprisingly, this signature of anisotropy holds into the quiescent interior with a tendency of the orientation aligned with maximum EKE being associated with shallower spectral slopes and elevated levels of inverse EKE cascade. Conversely, the angle associated with minimum EKE is aligned with a steeper spectral slope and forward cascade of EKE. Our results also indicate that vertical motion non-negligibly affects the direction of EKE cascade. A summary conclusion is that the spectral characteristics of eddies in the wind-driven gyre below the mixed layer where submesoscale dynamics are expected to be weak tend to diverge from expectations built on inertial-range assumptions, which are stationary in time and horizontally isotropic in space.

Direct Measurement of Energy Transfer in Strongly Driven Rotating Turbulence.

A short, abrupt increase in energy injection rate into steady strongly driven rotating turbulent flow is used as a probe for energy transfer in the system. The injected excessive energy is localized in time and space and its spectra differ from those of the steady turbulent flow. This allows measuring energy transfer rates, in three different domains: In real space, the injected energy propagates within the turbulent field, as a wave packet of inertial waves. In the frequency domain, energy is transferred nonlocally to the low, quasigeostrophic modes. In wave number space, energy locally cascades toward small wave numbers, in a rate that is consistent with two-dimensional (2D) turbulence models. Surprisingly however, the inverse cascade of energy is mediated by inertial waves that propagate within the flow with small, but nonvanishing frequency. Our observations differ from measurements and theoretical predictions of weakly driven turbulence. Yet, they show that in strongly driven rotating turbulence, inertial waves play an important role in energy transfer, even in the vicinity of the 2D manifold.

Effect of thermal fluctuations on spectra and predictability in compressible decaying isotropic turbulence

This study investigates the impact of molecular thermal fluctuations on compressible decaying isotropic turbulence using the unified stochastic particle (USP) method, encompassing both two-dimensional (2-D) and three-dimensional (3-D) scenarios. The findings reveal that the turbulent spectra of velocity and thermodynamic variables follow the wavenumber (k) scaling law of ${k}^{(d-1)}$ for different spatial dimensions $d$ within the high wavenumber range, indicating the impact of thermal fluctuations on small-scale turbulent statistics. With the application of Helmholtz decomposition, it is found that the thermal fluctuation spectra of solenoidal and compressible velocity components ( ${\boldsymbol {u}}_{s}$ and ${\boldsymbol {u}}_{c}$ ) follow an energy ratio of 1 : 1 for 2-D cases, while the ratio changes to 2 : 1 for 3-D cases. Comparisons between 3-D turbulent spectra obtained through USP simulations and direct numerical simulations of the Navier–Stokes equations demonstrate that thermal fluctuations dominate the spectra at length scales comparable to the Kolmogorov length scale. Additionally, the effect of thermal fluctuations on the spectrum of ${\boldsymbol {u}}_{c}$ is significantly influenced by variations in the turbulent Mach number. We further study the impact of thermal fluctuations on the predictability of turbulence. With initial differences caused by thermal fluctuations, different flow realizations display significant disparities in velocity and thermodynamic fields at larger scales after a certain period of time, which can be characterized by ‘inverse error cascades’. Moreover, the results suggest a strong correlation between the predictabilities of thermodynamic fields and the predictability of ${\boldsymbol {u}}_{c}$ .