Upscale Energy Cascade Research Articles

Deep Learning Improves Global Satellite Observations of Ocean Eddy Dynamics

AbstractOcean eddies affect large‐scale circulation and induce a kinetic energy cascade through their non‐linear interactions. However, since global observations of eddy dynamics come from satellite altimetry maps that smooth eddies and distort their geometry, the strength of this cascade is underestimated. Here, we use deep learning to improve observational estimates of global surface geostrophic currents and explore the implications for the cascade. By synthesizing multi‐modal satellite observations of sea surface height (SSH) and temperature, we achieve up to a 30% improvement in spatial resolution over the community‐standard SSH product. This reveals numerous strongly interacting eddies that were previously obscured by smoothing. In many regions, these newly resolved eddies lead to nearly an order‐of‐magnitude increase in the upscale kinetic energy cascade that peaks in spring and is strong enough to drive the seasonality of large mesoscale eddies. Our study suggests that deep learning can be a powerful paradigm for satellite oceanography.

The Eastern Mediterranean Boundary Current: Seasonality, Stability, and Spiral Formation

Abstract Seasonal variability and the effect of bottom interaction on the dynamics of the along-slope boundary current flowing around the Levantine Basin are investigated using nested high-resolution simulations of the eastern Mediterranean Sea. The numerical solutions show a persistent boundary current year-round that is ≈60 km wide and ≈200 m deep. An enstrophy balance diagnostic reveals significant bottom-drag influence on the boundary current, leading to anticyclonic vorticity generation in thin regions along the coast, which in turn become unstable and roll into surface-intensified anticyclonic spirals characterized by O(1) Rossby numbers. An eddy kinetic energy generation analysis suggests that a mix of baroclinic and barotropic instabilities is likely responsible for the spiral formation. The boundary current and spirals play a crucial role in the cross-shore transport of materials. In winter, the anticyclonic spirals frequently interact and exchange material with the energetic offshore submesoscale flow field. In summer, when the offshore flow structures are relatively less energetic, the spirals remain confined to the boundary current region as they are advected by the boundary current and undergo an upscale kinetic energy (KE) cascade that is manifested in spiral merging and growth up to 100 km in diameter. In both seasons, a coarse-graining analysis demonstrates that the cross-scale KE fluxes are spatially localized in coherent structures. The upscale KE fluxes typically occur within the spirals, while the downscale KE fluxes are confined to fronts and filaments at spiral peripheries.

On Energy Cascades in General Flows: A Lagrangian Application

AbstractAn important characteristic of geophysically turbulent flows is the transfer of energy between scales. Balanced flows pass energy from smaller to larger scales as part of the well‐known upscale cascade, while submesoscale and smaller scale flows can transfer energy eventually to smaller, dissipative scales. Much effort has been put into quantifying these transfers, but a complicating factor in realistic settings is that the underlying flows are often strongly spatially heterogeneous and anisotropic. Furthermore, the flows may be embedded in irregularly shaped domains that can be multiply connected. As a result, straightforward approaches like computing Fourier spatial spectra of nonlinear terms suffer from a number of conceptual issues. In this paper, we develop a method to compute cross‐scale energy transfers in general settings, allowing for arbitrary flow structure, anisotropy, and inhomogeneity. We employ Green's function approach to the kinetic energy equation to relate kinetic energy at a point to its Lagrangian history. A spatial filtering of the resulting equation naturally decomposes kinetic energy into length‐scale‐dependent contributions and describes how the transfer of energy between those scalestakes place. The method is applied to a doubly periodic simulation of vortex merger, resulting in the demonstration of the expected upscale energy cascade. Somewhat novel results are that the energy transfers are dominated by pressure work, rather than kinetic energy exchange, and dissipation is a noticeable influence on the larger scale energy budgets. We also describe, but do not employ here, a technique for developing filters to use in complex domains.

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

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

Spectral structure of 5 year time series of horizontal wind speed at the Boulder Atmospheric Observatory

AbstractWe investigate the spectral structures of 5 year, 1 min time series of horizontal wind speeds at 100 and 10 m heights at the Boulder Atmospheric Observatory tower located in the eastern slope of the Rocky Mountains, USA. In the full‐scale spectra, the diurnal spectral peak, which is usually insignificant at a coastal or offshore site, is the most significant at both heights. The spectrum is enhanced on the low‐frequency side of the diurnal peak during winter, but on the high‐frequency side during summer, which indicates frequent synoptic weather events during winter supplanted by mesoscale events during summer. In terms of the spectral density in the spectral gap of Van der Hoven (1957), separating boundary layer turbulence from the synoptic‐scale fluctuations, at a frequency between 10−4 and 10−3 Hz, we rank the daily time series at 100 m height and sample the summer top and winter bottom 10 percentile cases. The winter cases of the reduced spectral density in the gap region present the f−3 spectrum (f is frequency) and negatively skewed velocity increment distributions, which are the signatures of enstrophy (the integral of squared vorticity) cascade of turbulent two‐dimensional (2‐D) flows. In contrast, the summer cases of the enhanced spectral density present the f−5/3 spectrum and positively skewed velocity increment distributions, which are the signatures of upscale energy cascade of 2‐D flows. In these mesoscale events that fill up the gap, the turbulence intensity‐wind speed relationship is very sensitive to the choice of the averaging period.

An experimental study of multiple zonal jet formation in rotating, thermally driven convective flows on a topographic beta-plane

A series of rotating, thermal convection experiments were carried out on the Coriolis platform in Grenoble, France, to investigate the formation and energetics of systems of zonal jets through nonlinear eddy/wave-zonal flow interactions on a topographic β-plane. The latter was produced by a combination of a rigid, conically sloping bottom and the rotational deformation of the free upper surface. Convection was driven by a system of electrical heaters laid under the (thermally conducting) sloping bottom and led to the production of intense, convective vortices. These were observed to grow in size as each experiment proceeded and led to the development of weak but clear azimuthal jet-like flows, with a radial scale that varied according to the rotation speed of the platform. Detailed analyses reveal that the kinetic energy-weighted radial wavenumber of the zonal jets, kJy, scales quite closely either with the Rhines wavenumber as kJy ≃ 2(βT/2urms)1/2, where urms is the rms total or eddy velocity and βT is the vorticity gradient produced by the sloping topography, or the anisotropy wavenumber as kJy≃1.25(βT3/ϵ)1/5, where ϵ is the upscale turbulent energy transfer rate. Jets are primarily produced by the direct quasi-linear action of horizontal Reynolds stresses produced by trains of topographic Rossby waves. The nonlinear production rate of zonal kinetic energy is found to be strongly unsteady, however, with fluctuations of order 10-100 times the amplitude of the mean production rate for all cases considered. The time scale of such fluctuations is found to scale consistently with either an inertial time scale, τp∼1./urmsβT, or the Ekman spin-down time scale. Kinetic energy spectra show some evidence for a k−5/3 inertial subrange in the isotropic component, suggestive of a classical Kolmogorov-Batchelor-Kraichnan upscale energy cascade and a steeper spectrum in the zonal mean flow, though not as steep as k−5, as anticipated for fully zonostrophic flow. This is consistent with a classification of all of these flows as marginally zonostrophic, as expected for values of the zonostrophy parameter Rβ ≃ 1.6–1.7, though a number of properties related to flow anisotropy were found to vary significantly and systematically within this range.

Applications of a Moist Nonhydrostatic Formulation of the Spectral Energy Budget to Baroclinic Waves. Part I: The Lower-Stratospheric Energy Spectra

Abstract The authors investigate the mesoscale dynamics that produce the lower-stratospheric energy spectra in idealized moist baroclinic waves, using the moist nonhydrostatic formulation of spectral energy budget of kinetic energy and available potential energy by J. Peng et al. The inclusion of moist processes energizes the lower-stratospheric mesoscale, helping to close the gap between observed and simulated energy spectra. In dry baroclinic waves, the lower-stratospheric mesoscale is mainly forced by weak downscale cascades of both horizontal kinetic energy (HKE) and available potential energy (APE) and by a weak conversion of APE to HKE. At wavelengths less than 1000 km, the pressure vertical flux divergence also has a significant positive contribution to the HKE; however, this positive contribution is largely counteracted by the negative HKE vertical flux divergence. In moist baroclinic waves, the lower-stratospheric mesoscale HKE is mainly generated by the pressure and HKE vertical flux divergences. This additional HKE is partly converted to APE and partly removed by diffusion. Another negative contribution to the mesoscale HKE is from the forcing of a visible upscale HKE cascade. Besides the conversion of HKE, however, the three-dimensional divergence also has a significant positive contribution to the mesoscale APE. With these two direct APE sources, the lower-stratospheric mesoscale also undergoes a much stronger upscale APE cascade. These results suggest that both downscale and upscale cascades through the mesoscale are permitted in the real atmosphere and the direct forcing of the mesoscale is available to feed the upscale energy cascade.

Numerical reproduction method of unsteady small-scale eddy field in the ocean

A new method for reproducing transient oceanic turbulence fields is proposed by combining spectral analysis of velocity time series and numerical simulation. The spatial distributions of velocity components were extracted from four sets of velocity time series by means of the cross-correlations of continuous wavelet transforms. The extracted eddies were used as forcing components in direct numerical simulations to generate eddies of smaller scales. However, the non-linear interactions in the simulation also produce larger-scale motions, which duplicate the forced low-wavenumber components. To eliminate this upscale energy cascade with a practical computational speed, we adopted a partial spectral filter, in which spectral method was used only for the interaction of low-wavenumber components. The proposed method was applied to reproduce a flow field in a small water patch at a depth of 2200m in the north Pacific Ocean. The estimated energy dissipation rate and turbulent diffusivity were 6.4×10−11m2s−3 and 9.0×10−6m2s−1, respectively, which are reasonable for a deep ocean far from bottom boundaries.

Turbulence in fluid layers

Flows in natural fluid layers are often forced simultaneously at scales smaller and much larger than the depth. For example, Earth atmospheric flows are powered by gradients of solar heating: vertical gradients cause three-dimensional (3D) convection while horizontal gradients drive planetary scale flows. Nonlinear interactions spread energy over scales. The question is whether intermediate scales obtain their energy from a large-scale 2D flow or from a small-scale 3D turbulence. The paradox is that 2D flows do not transfer energy downscale while 3D turbulence does not support an upscale transfer. Here we demonstrate experimentally how a large-scale vortex and a small-scale turbulence conspire to provide for an upscale energy cascade in thick layers. We show that a strong planar vortex suppresses vertical motions thus facilitating an upscale energy cascade. In a bounded system, spectral condensation into a box-size vortex provides for a self-organized planar flow which secures an upscale energy transfer.

Upscale energy transfer in thick turbulent fluid layers

Flows in natural fluid layers are often forced simultaneously at scales smaller and much larger than the depth. For example, the Earth’s atmospheric flows are powered by gradients of solar heating: vertical gradients cause three-dimensional (3D) convection whereas horizontal gradients drive planetary scale flows. Nonlinear interactions spread energy over scales1,2. The question is whether intermediate scales obtain their energy from a large-scale 2D flow or from a small-scale 3D turbulence. The paradox is that 2D flows do not transfer energy downscale whereas 3D turbulence does not support an upscale transfer. Here we demonstrate experimentally how a large-scale vortex and small-scale turbulence conspire to provide for an upscale energy cascade in thick layers. We show that a strong planar vortex suppresses vertical motions, thus facilitating an upscale energy cascade. In a bounded system, spectral condensation into a box-size vortex provides for a self-organized planar flow that secures an upscale energy transfer.

Global diagnostic energetics of five state-of-the-art climate models

In this study, the global energy cycle of five state-of-the-art climate models is evaluated in the wave number domain for all seasons. The energy cycle estimates are based on 30 years of 6-hourly data obtained at pressure levels of all models. The models energetics are compared to those obtained from three reanalysis datasets (ERA-40, JRA-25 and NCEP-R2). The results show that the distributions of the energetics integrands and the shape of the various wave number spectra are reasonably well simulated. Many important features can be found in most models, namely both the upscale and downscale energy cascade for the wave–wave interactions of kinetic energy, the downscale energy cascade for the wave–wave interactions of available potential energy and the downscale energy transfer for the zonal–wave interactions of kinetic energy. However, the magnitude in the integrands distributions is generally excessive, yielding too much energy and an overactive energy cycle in the models. Accordingly, this energy excess is also reflected in the various spectra, specially but not exclusively, at the synoptic scale wave numbers for the energy conversion/transfer rates. The well known cold pole bias and the too strong tropospheric jets, along with their dislocation in some cases, still persist in the climate models. These are some of the deficiencies in the models directly implicated in the energy cycle. Apparently, simply increasing the horizontal and vertical resolutions is not enough to eliminate these deficiencies due to somewhat opposite effects achieved by refining both spatial resolutions. Therefore, more accurate physics parameterisations as well as improved numerical schemes and resolution dependence of parameterisations seem to be essential for a significant improvement in the models energetics. Moreover, efforts should be made to improve the physical processes controlling the generation of zonal available potential energy and dissipation of eddy kinetic energy, in which the synoptic scale should be fundamental, as inferred from the excessive energy conversion/transfer rates in the models spectra.

A new proof on net upscale energy cascade in two-dimensional and quasi-geostrophic turbulence

A general proof that more energy flows upscale than downscale in two-dimensional turbulence and barotropic quasi-geostrophic (QG) turbulence is given. A proof is also given that in surface QG turbulence, the reverse is true. Though some of these results are known in restricted cases, the proofs given here are pedagogically simpler, require fewer assumptions and apply to both forced and unforced cases.

Horizontal Wavenumber Spectra of Vertical Vorticity and Horizontal Divergence in the Upper Troposphere and Lower Stratosphere

Abstract The author shows that the horizontal two-point correlations of vertical vorticity and the associated vorticity wavenumber spectrum can be constructed from previously measured velocity structure functions in the upper troposphere and lower stratosphere. The spectrum has a minimum around k = 10−2 cycles per kilometer (cpkm) corresponding to wavelengths of 100 km. For smaller wavenumbers it displays a k−1 range and for higher wavenumbers, corresponding to mesoscale motions, it grows as k1/3. The two-point correlation of horizontal divergence of horizontal velocity and the associated horizontal spectrum is also constructed. The horizontal divergence spectrum is of the same order of magnitude as the vorticity spectrum in the mesoscale range and show similar inertial range scaling. It is argued that these results show that the mesoscale motions are not dominated by internal gravity waves. Instead, the author suggests that the dynamic origin of the k1/3 range is stratified turbulence. However, in contrast to Lilly, the author finds that stratified turbulence is not a phenomenon associated with an upscale energy cascade, but with a downscale energy cascade.

Energy Spectrum and Energy Flow of the Arctic Oscillation in the Phase Speed Domain

In this study, energy spectrum and energy flow of the large-scale atmospheric motions are examined in reference to the Arctic Oscillation index. Attention is concentrated to the barotropic component of the atmosphere in the framework of the 3D normal mode decomposition.According to the result of the observational analysis in the phase speed domain, the Arctic Oscillation (AO) is characterized as the energy increase at the meridional index l=3 with simultaneous decrease at l=5 of the zonal field. The result is consistent with the theory of the AO by the singular eigenmode of the global atmosphere. The accumulation of energy at the eigenmode of the AO is explained by the enhanced energy flux FZi associated with the zonal-wave interaction.It is found in this study that the energy flux FZi comes from eddies at the spherical Rhines speed cR where the planetary-scale Rossby wave becomes stationary. The result suggests that the low-frequency variability associated with the AO is maintained by energy flux from cR, which is compensated by the up-scale energy cascade from synoptic eddies in addition to the forced stationary planetary waves by topography.

Multiresolution analysis for 2D turbulence. Part 1: Wavelets vs cosine packets, a comparative study

The widely accepted theory of two-dimensional turbulence predicts a direct downscale enstrophy cascade with an energy spectrum behaving like $k^{-3}$ and an inverse upscale energy cascade with a $k^{-5/3}$ decay. Nevertheless, this theory is in fact an idealization valid only in an infinite domain in the limit of infinite Reynolds numbers, and is almost impossible to reproduce numerically. A more complete theoretical framework for the two-dimensional turbulence has been recently proposed by Tung et al . This theory seems to be more consistent with experimental observations, and numerical simulations than the classical one developed by Kraichnan, Leith and Batchelor.Multiresolution methods like the wavelet packets or the cosine packets, well known in signal decomposition, can be used for the 2D turbulence analysis. Wavelet or cosine decompositions are more and more used in physical applications and in particular in fluid mechanics. Following the works of M. Farge et al , we present a numerical and qualitative study of a two-dimensional turbulence fluid using these methods. The decompositions allow to separate the fluid in two parts which are analyzed and the corresponding energy spectra are computed. In the first part of this paper, the methods are presented and the numerical results are briefly compared to the theoretical spectra predicted by the both theories. A more detailed study, using only wavelet packets decompositions and based on numerical and experimental data, will be carried out and the results will be reported in the second part of the paper. A tentative of physical interpretation of the different components of the flow will be also proposed.

On the double cascades of energy and enstrophy in two dimensional turbulence. Part 1. Theoretical formulation

The Kraichnan-Leith-Batchelor scenario of a dual cascade, consisting of an upscale pure energy cascade and a downscale pure enstrophy cascade, is an idealization valid only in an infi nite domain in the limit of in finite Reynolds number. In realistic situations there are double cascades of energy and enstrophy located both upscale and downscale of injection, as long as there are cascades. We outline the statistical theory governing the double cascades and predict the form of the energy spectrum. We show that in general the twin conservation of energy and enstrophy imply the presence of two constant fluxes in each inertial range. This gives rise to a more complicated energy spectrum, which cannot be predicted using dimensional arguments as in the classical theory.

Numerical simulations of convective equilibrium under prescribed forcing

AbstractSome characteristic properties of simulated moist convective equilibria are examined for different imposed cooling rates. These numerical simulations extend the work of Vallis et al. and were motivated by the need to show that their conclusions (concerning the upscale energy cascade) were not sensitive to vertical resolution. Although kinetic energy does cascade to large scales, much of the large‐scale motion in the model is associated with horizontally‐divergent winds, and the energy spectra may be better explained as a direct consequence of the generation of convective lines. These lines have a typical spacing of about 60 km which leads to a local maximum in the kinetic‐energy spectrum.In addition, the design of our experiments was found to match that envisaged in recent idealized ‘heatengine theories’ of radiative‐convective equilibria, thereby providing an opportunity to evaluate their utility. It is shown that a variant of these scaling theories appears to fit the statistical properties of our simulations quite well and provides expressions for the convective available potential energy, cloud mass flux and the fractional area occupied by convective updraughts. Scaling arguments also suggest that the convective line separation is of the order of the convecting layer depth divided by the square root of the updraught fractional area—consistent with the wavelength of the local maximum in the energy spectrum.

The coherent structures of shallow-water turbulence: Deformation-radius effects, cyclone/anticyclone asymmetry and gravity-wave generation.

Over a large range of Rossby and Froude numbers, we investigate the dynamics of initially balanced decaying turbulence in a shallow rotating fluid layer. As in the case of incompressible two-dimensional decaying turbulence, coherent vortex structures spontaneously emerge from the initially random flow. However, owing to the presence of a free surface, a wealth of new phenomena appear in the shallow-water system. The upscale energy cascade, common to strongly rotating flows, is arrested by the presence of a finite Rossby deformation radius. Moreover, in contrast to near-geostrophic dynamics, a strong asymmetry is observed to develop as the Froude number is increased, leading to a clear dominance of anticyclonic vortices over cyclonic ones, even though no beta effect is present in the system. Finally, we observe gravity waves to be generated around the vortex structures, and, in the strongest cases, they appear in the form of shocks. We briefly discuss the relevance of this study to the vortices observed in Jupiter's atmosphere.

A Numerical Simulation of Amplification of Low-Frequency Planetary Waves and Blocking Formations by the Upscale Energy Cascade

Abstract In this study, nonlinear numerical simulations of amplification of low-frequency planetary waves and concurrent blocking formations were performed. The simulations are conducted by a barotropic spectral model derived from three-dimensional spectral primitive equations with a basis of vertical structure functions and Hough harmonics. The model is truncated to include only barotropic Rossby components of the atmosphere with simple physics of biharmonic diffusion, topographic forcing, baroclinic instability, and zonal surface stress. These four physical processes are found to be sufficient to produce a realistic and persistent dipole blocking with a sharp transition from zonal to meridional flows on a sphere. Analyzing energetics of blocking formations in the model, we showed an amplification of the meridional dipole mode was confirmed by means of the upscale energy cascade from synoptic disturbances under an environment of persistently amplified wavenumber 2. When the persistent wavenumber 2 exists...

Symbiotic Relation between Planetary and Synoptic-Scale Waves

Abstract It is hypothesized that the low-frequency planetary scale waves and the high-frequency cyclone-scale waves in an equilibrated state of the atmosphere are symbiotically dependent upon one another. This is demonstrated with an analysis of a dissipative atmospheric model driven by a zonally symmetric forcing. Under a geophysically relevant parameter condition, the synoptic scale waves in the equilibrated state of this system intermittently extract a sufficient amount of energy from the modified instantaneous zonal flow to compensate not only for their own dissipative loss of energy but also for a net supply of energy to the planetary scale waves through the upscale energy cascade process. The planetary scale waves gain this energy in the barotropic form. The planetary scale waves, in turn, create localized strong baroclinic regions whereby the synoptic scale waves may preferentially intensify downstream of the model planetary jet streams. Such cyclone scale eddies collectively give rise to two model...