Inverse Cascade Research Articles

Ocean 2D eddy energy fluxes from small mesoscale processes with SWOT

Abstract. We investigate ocean dynamics at different scales in the Agulhas Current system, a region of important interocean exchange of heat and energy. While ocean observations and some of the most advanced climate models capture the larger mesoscale dynamics (> 100 km), the smaller-scale fronts and eddies are underrepresented. The recently launched NASA–CNES Surface Water and Ocean Topography (SWOT) wide-swath altimeter mission observes the smaller ocean geostrophic scales down to 15 km in wavelength globally. Here we will analyse different eddy diagnostics in the Agulhas Current region and quantify the contributions from the larger mesoscales observable today and the smaller scales to be observed with SWOT. Surface geostrophic diagnostics of eddy kinetic energy, strain, and energy cascades are estimated from modelled sea surface height (SSH) fields of the Massachusetts Institute of Technology general circulation model (MITgcm) latitude–longitude polar cap (LLC4320) simulation subsampled at 1/10∘. In this region, the smaller scales (<150 km) have a strong signature on the horizontal geostrophic strain rate and for all eddy diagnostics in the Western Boundary Current and along the meandering Agulhas Extension. We investigate the horizontal cascade of energy using a coarse-graining technique, and we observe that the wavelength range where the inverse cascade occurs is biased towards larger mesoscale wavelengths with today’s altimetric sampling. We also calculate the projected sampling of the eddy diagnostics under the SWOT swaths built with the NASA–CNES simulator to include the satellite position and realistic noise. For the swaths, a neural network noise mitigation method is implemented to reduce the residual SWOT random error before calculating eddy diagnostics. In terms of SSH, observable wavelengths of 15 to 20 km are retrieved after neural network noise mitigation, as opposed to wavelengths larger than 40 km before the noise reduction.

Dark matter halo mass functions and density profiles from mass and energy cascade

Halo abundance and structure play a central role for modeling structure formation and evolution. Without relying on a spherical or ellipsoidal collapse model, we analytically derive the halo mass function and cuspy halo density (inner slope of −4/3) based on the mass and energy cascade theory in dark matter flow. The hierarchical halo structure formation leads to halo or particle random walk with a position-dependent waiting time tau _g. First, the inverse mass cascade from small to large scales leads to the halo random walk in mass space with tau _gpropto m_h^{-lambda }, where m_h is the halo mass and lambda is a halo geometry parameter with predicted value of 2/3. The corresponding Fokker-Planck solution for halo random walk in mass space gives rise to the halo mass function with a power-law behavior on small scale and exponential decay on large scale. This can be further improved by considering two different lambda for haloes below and above a critical mass scale m_h^*, i.e. a double-lambda halo mass function. Second, a double-gamma density profile can be derived based on the particle random walk in 3D space with a position-dependent waiting time tau _g propto Phi (r)^{-1} propto r^{-gamma }, where Phi is the gravitational potential and r is the particle distance to halo center. Theory predicts gamma =2/3 that leads to a cuspy density profile with an inner slope of −4/3, consistent with the predicted scaling laws from energy cascade. The Press-Schechter mass function and Einasto density profile are just special cases of proposed models. The small scale permanence can be identified due to the scale-independent rate of mass and energy cascade, where density profiles of different halo masses and redshifts converge to the -4/3 scaling law (rho _h propto r^{-4/3}) on small scales. Theory predicts the halo number density scales with halo mass as propto m_h^{-1.9}, while the halo mass density scales as propto m_h^{4/9}. Results were compared against the Illustris simulations. This new perspective provides a theory for nearly universal halo mass functions and density profiles.

A Physical Model for the Observed Inverse Energy Cascade in Typhoon Boundary Layers

AbstractThe knowledge of the observed inverse energy cascade in typhoon boundary layers holds significant implications for understanding the formation, enhancement, and deformation of typhoons. This study reveals that the observed inverse energy cascade originates from the rapid rotation of typhoons. The transition from the direct energy cascade regime to the inverse energy cascade regime can be characterized by the Zeman length scale. The turbulence structures behave as two dimensional above the Zeman length scale. Through symmetry analysis, it is discovered that the ratio of the inverse energy cascade flux to the direct energy cascade flux is proportional to the turbulence Rossby number with a power of −2. These findings offer a framework for incorporating the inverse energy cascade into the turbulence parameterizations and improving typhoon modeling.

Supply mechanisms of the geostrophic mode in rotating turbulence: interactions with self, waves and eddies

Direct numerical simulations are performed in rotating turbulence for different regimes at various Rossby and inertial Reynolds numbers ( $\textit {Re}_I$ ). A new algorithm, adapted from stratified turbulence (Lam et al., J. Fluid Mech., vol. 923, 2021, A31) to rotating turbulence, permits to separate the three-dimensional velocity field into three parts: inertial waves (IWs), eddies and a geostrophic mode (GM). It uses the space–time properties of waves and their advection by the GM to filter the IWs from the rest of the motion. We obtain balance equations for the separate energies of waves, eddies and the GM. Their mutual interactions are evaluated and analysed via Sankey diagrams that provide a global picture of energy exchanges. When the flow is forced at large scale, it mainly feeds the wave part and the multiple interactions lead to energy dissipation in eddy and GM motion. We also show that, in addition to the wave/wave interaction that feeds the GM, corresponding to different mechanisms described in the literature, other non-documented interactions feed it, as the eddy/wave interaction or the eddy/eddy interaction at moderate $\textit {Re}_I$ . We propose a scale-by-scale analysis of the transfer to the GM: we show that transfers from wave or eddy occur at large scale, that they either inject or remove energy, and that this occurs with or without direct cascade depending on the kind of interaction, wave/wave, eddy/wave or eddy/eddy. The self-interaction of the GM is an inverse cascade for its horizontal component, shaping it into a very large-scale flow.

Multi-scale reconstruction of turbulent rotating flows with proper orthogonal decomposition and generative adversarial networks

Data reconstruction of rotating turbulent snapshots is investigated utilizing data-driven tools. This problem is crucial for numerous geophysical applications and fundamental aspects, given the concurrent effects of direct and inverse energy cascades. Additionally, benchmarking of various reconstruction techniques is essential to assess the trade-off between quantitative supremacy, implementation complexity and explicability. In this study, we use linear and nonlinear tools based on the proper orthogonal decomposition (POD) and generative adversarial network (GAN) for reconstructing rotating turbulence snapshots with spatial damages (inpainting). We focus on accurately reproducing both statistical properties and instantaneous velocity fields. Different gap sizes and gap geometries are investigated in order to assess the importance of coherency and multi-scale properties of the missing information. Surprisingly enough, concerning point-wise reconstruction, the nonlinear GAN does not outperform one of the linear POD techniques. On the other hand, the supremacy of the GAN approach is shown when the statistical multi-scale properties are compared. Similarly, extreme events in the gap region are better predicted when using GAN. The balance between point-wise error and statistical properties is controlled by the adversarial ratio, which determines the relative importance of the generator and the discriminator in the GAN training.

The Impact of Observed Drag Reduction Over Land on Typhoon Forecasting

AbstractThe surface drag coefficient plays a crucial role in determining the momentum exchange between the Earth's surface and the atmosphere. Previous studies have observed a tendency for the drag coefficient to decrease at high wind speeds over land, primarily due to the inverse energy cascade. This study aims to examine the impacts of the reduced drag coefficient on the typhoon forecasting skills in terms of the track, the central pressure, and the 10‐m maximum wind speed. Drag reduction is approached through the logarithmic wind profile and a new formula that considers the reduction in roughness length. The results show that the track and the 10‐m maximum wind speed forecasts are improved, but these improvements are primarily evident over plain areas, as the typhoon structure is significantly disrupted by the topography over mountainous areas. Additionally, the reduced drag coefficient results in less dissipation of the typhoon's wind fields over land. As the wind speeds at coastal areas are enhanced compared with those in operational model, the changes in the Coriolis force are responsible for the track prediction improvement. These findings highlight the importance of parameterizing coherent processes in current numerical models.

A data-assimilative modeling investigation of Gulf Stream variability

An advanced data-assimilative ocean circulation model is used to investigate Gulf Stream (GS) variability during 2017–2018. The modeling system applies a strong-constraint, 4D variational data assimilation algorithm. It assimilates satellite-based sea surface height and sea surface temperature measurements and in situ temperature and salinity profiles. Model skill assessment metrics along with comparisons of GS position and GS's three-dimensional mean kinetic energy with historical observations are applied to validate the data-assimilative model. The resulting time- and space-continuous ocean state estimates are used to diagnose eddy kinetic energy conversion and cross-stream eddy heat and salt fluxes over the two-year study period. The processes leading to kinetic energy conversion are primarily due to GS meanders. Significant inverse energy cascading (EKE→MKE and EKE→EPE) can occur during GS-eddy interactions, particularly during onshore intrusions or offshore meanderings of the GS. Throughout the two-year study period, the cross-stream eddy heat and salt fluxes off Cape Hatteras were predominantly positive (onshore). Both GS offshore meandering (occurring 44% of the time and associated with shelf/slope water export) and GS intrusion (occurring 56% of the time) contribute to onshore heat and salt transport. Improved understanding of these processes and dynamics requires strong integration of an advanced observational infrastructure that combines remote sensing; fixed, mobile, and shore-based observing components; and high-resolution data assimilative models.

Quasi-geostrophic vortex vertical alignment in near collapse interactions

We consider a three-vortex interaction which leads to the vertical alignment of two like-signed quasi-geostrophic vortices in a continuously stratified, rotating fluid. The interaction is close to the classical collapse interaction of three co-planar vortices except that the vortices centres move on close but different horizontal planes. The vertical alignment of vortices helps create larger structures and contributes in physical space to the inverse energy cascade observed in spectral space in geostrophic turbulence.

Outer-layer similarity and energy transfer in a rough-wall turbulent channel flow

Direct numerical simulations (DNSs) are performed to investigate the roughness effects on the statistical properties and the large-scale coherent structures in the turbulent channel flow over three-dimensional sinusoidal rough walls. The outer-layer similarities of mean streamwise velocity and Reynolds stresses are examined by systematically varying the roughness Reynolds number $k^{+}$ and the ratio of the roughness height to the half-channel height $k / \delta$ . The energy transfer mechanism of turbulent motions in the presence of roughness elements with different sizes is explored through spectral analysis of the transport equation of the two-point velocity correlation and the scale-energy path display of the generalized Kolmogorov equation. The results show that, with increasing $k^+$ , the downward shift of the mean streamwise velocity profile in the logarithmic region increases and the peak intensities of turbulent Reynolds stresses decrease. At an intermediate Reynolds number ( $Re_{\tau }= 1080$ ), the length scale and intensity of the large-scale coherent structures increase for a small roughness ( $k^{+}=10$ ), which leads to failure of the outer-layer similarity in rough-wall turbulence, and decrease for a large roughness ( $k^{+}=60$ ), as compared with the smooth-wall case. The existence of the small roughness ( $k^{+}=10$ ) enhances the mechanism of inverse energy cascade from the inner-layer small-scale structures to the outer-layer large-scale structures. Correspondingly, the self-sustaining processes of the outer-layer large-scale coherent structures, including turbulent production, interscale transport, pressure transport and spatial turbulent transport, are all enhanced, whereas the large roughness ( $k^{+}=60$ ) weakens the energy transfer between the inner and outer regions.

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.

Observed Oceanic Surface Modes in the Northern South China Sea

Abstract Using observations and theoretical models, a substantial topographic modulation on the quasigeostrophic (QG) dynamics, which results in a primary surface mode distinct from the classic first baroclinic (BC1) mode with a flat bottom, is revealed in the northern South China Sea (NSCS). In contrast to open oceans, the surface-intensified modes decay downward more rapidly over the continental slope of the NSCS, with a mean e-folding scale of approximately 1/5 of water depth. The subinertial flow variability appears to be vertically incoherent, with planetary and topographic Rossby waves dominating in the upper and deep layers, respectively. With a larger deformation radius (Rd), the surface-mode Rossby waves propagate at a phase speed ∼1.5 times of that of the BC1 mode. Moreover, the modal structures can be substantially modified by seasonal NSCS circulation, which is significantly enhanced over continental slopes. Analysis of the triad interactions further implies that the short waves tend to transfer energy to larger scales via the inverse cascade and only those with wavelengths larger than Rd ≈ 70 km in the NSCS can persist because of a slower unstable growth rate but a higher fraction of upscale energy transfer. The present theory excludes the bottom-trapped mode, which is closely associated with topographic Rossby waves and is observed to be significant in the abyssal NSCS. Hence, a complete normal-mode basis for any QG state is required for a study that focuses on flow variability throughout the water column. Our findings provide an insight into the vertical partition of horizontal kinetic energy for QG motions, as well as the relevant oceanic variation in marginal seas.

Inverse cascade based on nonlinear Schrödinger equation analysis with nonlinear feedback control

This paper focuses on the wave inverse cascade instability analysis with self-regulating feedback control for a fixed external potential field and a highly localized finite-amplitude initial pulse. The wave inverse cascade instability analysis is carried out by solving the corresponding two-dimensional generalized nonlinear Schrödinger equation. The wave field firstly suffers from the modulation instability, followed by collapse into turbulence containing the shortest-wavelength modes in the system. This is followed by inverse cascade of the shortest wavelength modes back to the longer-wavelength ones, until a statistical stationary turbulent state is reached. It is found that the inverse cascade is limited to the shorter-wavelength modes with the wavenumber $\left |k\right |\geq 100$ . This shows that the viscous damping $p_i$ acts like a control switch to the inverse cascade, and the feedback control can also regulate the intensity of the inverse cascade mode.

Effective Chiral Magnetic Effect from Neutrino Radiation.

We develop an approach to chiral kinetic theories for electrons close to equilibrium and neutrinos away from equilibrium based on a systematic power counting scheme for different timescales of electromagnetic and weak interactions. Under this framework, we derive electric and energy currents along magnetic fields induced by neutrino radiation in general nonequilibrium states. This may be regarded as an effective chiral magnetic effect (CME), which is present without a chiral chemical potential, unlike the conventional CME. We also consider the so-called gain region of core-collapse supernovae as an example and find that the effective CME enhanced by persistent neutrino emission in time is sufficiently large to lead to the inverse cascade of magnetic and fluid kinetic energies and observed magnitudes of pulsar kicks. Our framework may also be applicable to other dense-matter systems involving nonequilibrium neutrinos.

Annular self-organization of the two-dimensional vorticity condensate

Self-organization of the vorticity condensate of forced two-dimensional turbulence is examined in analogy with the mixing of a background planetary vorticity gradient in geophysical flows. Starting from the theoretical vorticity profile of the condensate, different scenarios are illustrated by the construction of idealized angular momentum conserving rearrangements of vorticity into a staircase profile similar to the potential vorticity staircase of $\beta$ -plane turbulence. Two sets of numerical experiments are then presented that illustrate a similar self-organization of the background vorticity gradient in fully turbulent flows. In the first set of experiments, the flow is initialized with a laminar vortex dipole corresponding to the theoretical condensate vorticity profile. A fluctuation vorticity field is then induced by a stochastic forcing at smaller scales, which induces an azimuthally symmetric self-organization of the laminar flow into distinct annular bands. In the second set of experiments, the flow is initialized from rest with stochastic forcing generating a turbulent inverse energy cascade, out of which emerges the condensate in a self-consistent evolution as the turbulent energy accumulates at the domain scale. A self-organization of the condensate is again observed, giving a distinct annular structure on top of the theoretically predicted condensate profile. A major difference from the potential vorticity staircase of geophysical flows is the emergence in many cases of significantly non-monotonic radial vorticity profiles.

Confronting the Neutron Star Population with Inverse Cascades

The origin and evolution of magnetic fields of neutron stars from birth have long been a source of debate. Here, motivated by recent simulations of the Hall cascade with magnetic helicity, we invoke a model where the large-scale magnetic field of neutron stars grows as a product of small-scale turbulence through an inverse cascade. We apply this model to a simulated population of neutron stars at birth and show how this model can account for the evolution of such objects across the diagram, explaining both pulsar and magnetar observations. Under the assumption that small-scale turbulence is responsible for large-scale magnetic fields, we place a lower limit on the spherical harmonic degree of the energy-carrying magnetic eddies of ≈40. Our results favor the presence of a highly resistive pasta layer at the base of the neutron star crust. We further discuss the implications of this paradigm on direct observables, such as the nominal age and braking index of pulsars.

Topographically Generated Submesoscale Shear Instabilities Associated with Brazil Current Meanders

Abstract The western boundary current system off southeastern Brazil is composed of the poleward-flowing Brazil Current (BC) in the upper 300 m and the equatorward flowing Intermediate Western Boundary Current (IWBC) underneath it, forming a first-baroclinic mode structure in the mean. Between 22° and 23°S, the BC-IWBC jet develops recurrent cyclonic meanders that grow quasi-stationarily via baroclinic instability, though their triggering mechanisms are not yet well understood. Our study, thus, aims to propose a mechanism that could initiate the formation of these mesoscale eddies by adding the submesoscale component to the hydrodynamic scenario. To address this, we perform a regional 1/50° (∼2 km) resolution numerical simulation using CROCO (Coastal and Regional Ocean Community model). Our results indicate that incoming anticyclones reach the slope upstream of separation regions and generate barotropic instability that can trigger the meanders’ formation. Subsequently, this process generates submesoscale cyclones that contribute, along with baroclinic instability, to the meanders’ growth, resulting in a submesoscale-to-mesoscale inverse cascade. Last, as the mesoscale cyclones grow, they interact with the slope, generating inertially and symmetrically unstable anticyclonic submesoscale vortices and filaments. Significance Statement Off southeastern Brazil, the Brazil Current develops recurrent cyclonic meanders. Such meanders enhance the open-ocean primary productivity and are of societal importance as they are located in a region rich in oil and gas where oil-spill accidents have already happened. This study aims to explore the processes responsible for triggering the formation of these mesoscale eddies. We find that incoming anticyclones reach the slope upstream of separation regions and generate barotropic instabilities that eject submesoscale filaments and vortices and can trigger the meanders’ formation. Such results show that topographically generated submesoscale instabilities can play an important role in the dynamics of mesoscale meanders off southeastern Brazil. Moreover, this may indicate that resolving the submesoscale dynamics in operational numerical models may contribute to an increase in the predictability of the regional eddies.

Optimization of Vehicle Powertrain Mounting System Based on Generalized Inverse Cascade Method under Uncertainty

This paper presents a summary of the optimization design process for a multi-objective, two-level engineering problem, utilizing the generalized inverse cascade method under uncertainty. The primary objective is to enhance the vibration isolation performance of a mounting system, considering the influence of uncertain factors on its stiffness. The focus is on determining the value range of the design variables at the bottom layer, ensuring that the design goal is met with a specified confidence level. To illustrate the application of this methodology, the optimization design of a powertrain mount is used as a case study. A data-driven approach is adopted, establishing a quantitative mapping relationship between mount stiffness, force transmission rate, modal decoupling rate, and other design indicators. This is achieved through the development of a CRBM-DBN approximate model, which combines Conditional Restricted Boltzmann Machines (CRBMs) and a Deep Belief Network (DBN). Additionally, an intelligent optimization algorithm and interval search technology are employed to determine the optimal design interval for the mount stiffness. Simulation and experimental verification are conducted using selected parameter combinations. The results demonstrate notable improvements in the vibration isolation performance, modal decoupling rate, and vehicle NVH performance when compared to the original state. These findings provide valuable insights for the interval optimization design of similar multi-objective, as well as two-level engineering problems, serving as useful references for future research and applications.

Inverse Cascade Spectrum of Gravity Waves in the Presence of a Condensate: A Direct Numerical Simulation.

During the set of direct numerical simulations of the forced isotropic turbulence of surface gravity waves in the framework of primordial dynamical equations, the universal inverse cascade spectrum was observed. The slope of the spectrum is the same (in the margin of error) for different levels of pumping and nonlinearity as well as dissipation present in the system. In all simulation runs formation of the inverse cascade spectrum was accompanied by the appearance of a strong long wave background (condensate). The observed slope of the spectrum ∼k^{-3.07} is different from the constant wave action flux solution predicted by the wave turbulence theory ∼k^{-23/6}.

Rayleigh–Jeans thermalization vs beam cleaning in multimode optical fibers

Classical nonlinear waves exhibit, as a general rule, an irreversible process of thermalization toward the Rayleigh–Jeans equilibrium distribution. On the other hand, several recent experiments revealed a remarkable effect of spatial organization of an optical beam that propagates through a graded-index multimode optical fiber (MMF), a phenomenon termed beam self-cleaning. Our aim here is to evidence the qualitative impact of disorder (weak random mode coupling) on the process of Rayleigh–Jeans thermalization by considering two different experimental configurations. In a first experiment, we launch speckle beams in a relatively long MMF. Our results report a clear and definite experimental demonstration of Rayleigh–Jeans thermalization through light propagation in MMFs, over a broad range of kinetic energy (i.e., degree of spatial coherence) of the injected speckle beam. In particular, the property of energy equipartition among the modes is clearly observed in the condensed regime. The experimental results also evidence the double turbulence cascade process: while the power flows toward the fundamental mode (inverse cascade), the energy flows toward the higher-order modes (direct cascade). In a 2nd experiment, a coherent laser beam is launched into a relatively short MMF length. It reveals an effect of beam cleaning driven by an incipient process of Rayleigh–Jeans thermalization. As discussed through numerical simulations, the fast process of Rayleigh–Jeans thermalization observed in the 1st experiment can be attributed due to a random phase dynamics among the modes, which is favored by the injection of a speckle beam and the increased impact of disorder in the long fiber system.

Korrelyatsii zavikhrennosti vnutri kogerentnogo vikhrya

We investigate fluctuations of vorticity inside a coherent vortex generated by the inverse energy cascade in two-dimensional turbulence. Temporal and spatial correlations can be characterized by the pair correlation function. The interaction of fluctuations leads to a nonzero third moment of vorticity. We analyze the pair correlation function and the third moment using a model in which the pumping is short-correlated in time and derive explicit expressions for the Gaussian spatial correlation function for the pumping force.