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Abstract
One of the leading goals of rogue wave research is to develop a robust rogue wave warning system to mitigate the danger they pose. One such system has been developed by the European Centre for Medium-Range Weather Forecasts (ECMWF), called the freak wave warning system (FWWS), based on nonlinear wave effects. The FWWS predicts maximum expected wave envelope height as a risk parameter for forecasts. Recently, a data-driven alternative has been proposed by Häfner et al., which was distilled from a neural network using wave buoy observations. However, it has yet to be evaluated by a spectral wave model for application to operational wave forecasting. The data-driven, learned model emphasizes bandwidth-controlled linear superposition as the predominant mechanism in crest-to-trough rogue wave generation, while nonlinear effects are a secondary term. The present work evaluates the performance of the empirical model using output from an ECMWF global wave hindcast. We find that the prediction models based on bandwidth effects have the highest log likelihood scores, with the empirical model outperforming all other tested models. In contrast, the expected maximum envelope wave height from the FWWS does not predict the occurrence of rogue waves. These results indicate that the empirical model with wave model input is a skillful predictor and should be considered for operational implementation to improve rogue wave forecasting.
Significance Statement
Rogue waves are unexpectedly large and unpredictable waves. Encounters with rogue waves can result in damage to marine vessels and offshore infrastructures. Research is devoted to developing systems to predict the risk of rogue wave events. A novel predictive model has been shown to perform well in predicting rogue wave occurrences based on wave buoy observations. This model is a symbolic expression distilled from an artificial neural network that incorporates known rogue wave dynamics and that can be evaluated alongside traditional risk estimates with minor adjustments to operational forecasting systems. This study evaluates the newly proposed empirical model in a forecasting setting. Our work demonstrates the efficacy of the empirical model for operational forecasting purposes.
Abstract
One of the leading goals of rogue wave research is to develop a robust rogue wave warning system to mitigate the danger they pose. One such system has been developed by the European Centre for Medium-Range Weather Forecasts (ECMWF), called the freak wave warning system (FWWS), based on nonlinear wave effects. The FWWS predicts maximum expected wave envelope height as a risk parameter for forecasts. Recently, a data-driven alternative has been proposed by Häfner et al., which was distilled from a neural network using wave buoy observations. However, it has yet to be evaluated by a spectral wave model for application to operational wave forecasting. The data-driven, learned model emphasizes bandwidth-controlled linear superposition as the predominant mechanism in crest-to-trough rogue wave generation, while nonlinear effects are a secondary term. The present work evaluates the performance of the empirical model using output from an ECMWF global wave hindcast. We find that the prediction models based on bandwidth effects have the highest log likelihood scores, with the empirical model outperforming all other tested models. In contrast, the expected maximum envelope wave height from the FWWS does not predict the occurrence of rogue waves. These results indicate that the empirical model with wave model input is a skillful predictor and should be considered for operational implementation to improve rogue wave forecasting.
Significance Statement
Rogue waves are unexpectedly large and unpredictable waves. Encounters with rogue waves can result in damage to marine vessels and offshore infrastructures. Research is devoted to developing systems to predict the risk of rogue wave events. A novel predictive model has been shown to perform well in predicting rogue wave occurrences based on wave buoy observations. This model is a symbolic expression distilled from an artificial neural network that incorporates known rogue wave dynamics and that can be evaluated alongside traditional risk estimates with minor adjustments to operational forecasting systems. This study evaluates the newly proposed empirical model in a forecasting setting. Our work demonstrates the efficacy of the empirical model for operational forecasting purposes.
Abstract
The characteristics of modulated internal solitary waves (ISWs) under the influence of one mesoscale eddy pair in the Luzon Strait, involving one anticyclonic eddy (AE) and one cyclonic eddy (CE) induced by the Kuroshio intrusion, were investigated using a nested high-resolution numerical model in the northeastern South China Sea (SCS). The presence of mesoscale eddies greatly impacts the nonlinear evolution of type-a and type-b ISWs. The eddy pair contributes to distinct wave properties and energy evolutions. Compared to type-b waves, type-a waves display more pronounced modulatory characteristics with a larger spatial scale. CE currents and horizontal inhomogeneous stratification are crucial in modulating the wave behaviors, which induce extremely large-amplitude depression ISWs. The AE thereafter yields retardation effects on the wave energy evolution. The average depth-integrated available potential and kinetic energy showed relative growth rates of −66.12% and −46.07%, respectively, for type-a waves, and −24.26% and −20.15%, respectively, for type-b waves along the propagation path up to the AE core. The deformed and distorted ISW crest lines propagating further northward exhibit a more dramatic shoaling evolution. The maximum total energies of type-a and type-b waves at the north station are approximately 13.5 and 3.5 times, respectively, greater than those at the south station on the continental shelf of the Dongsha Atoll. This work provides essential insights into modulated ISW dynamics under the mesoscale eddy pair within the northeastern SCS deep basin.
Abstract
The characteristics of modulated internal solitary waves (ISWs) under the influence of one mesoscale eddy pair in the Luzon Strait, involving one anticyclonic eddy (AE) and one cyclonic eddy (CE) induced by the Kuroshio intrusion, were investigated using a nested high-resolution numerical model in the northeastern South China Sea (SCS). The presence of mesoscale eddies greatly impacts the nonlinear evolution of type-a and type-b ISWs. The eddy pair contributes to distinct wave properties and energy evolutions. Compared to type-b waves, type-a waves display more pronounced modulatory characteristics with a larger spatial scale. CE currents and horizontal inhomogeneous stratification are crucial in modulating the wave behaviors, which induce extremely large-amplitude depression ISWs. The AE thereafter yields retardation effects on the wave energy evolution. The average depth-integrated available potential and kinetic energy showed relative growth rates of −66.12% and −46.07%, respectively, for type-a waves, and −24.26% and −20.15%, respectively, for type-b waves along the propagation path up to the AE core. The deformed and distorted ISW crest lines propagating further northward exhibit a more dramatic shoaling evolution. The maximum total energies of type-a and type-b waves at the north station are approximately 13.5 and 3.5 times, respectively, greater than those at the south station on the continental shelf of the Dongsha Atoll. This work provides essential insights into modulated ISW dynamics under the mesoscale eddy pair within the northeastern SCS deep basin.
Abstract
Westward-propagating Caribbean Current eddies modify the volume-integrated potential vorticity (PV) balance in the western Caribbean Sea, influencing the circulation of the Panamá–Colombia Gyre (PCG) and coastal currents hundreds of kilometers to the south of the eddies’ mean trajectory. Using 22 years of output from the Hybrid Coordinate Ocean Model, we apply a volume-integrated eddy phase-averaged 1.5-layer PV balance, showing that PV fluxes into the PCG region are balanced by frictional PV dissipation represented by linear drag along the coastline. Coastal currents in the PCG region vary by a factor of 2 in phase with the passage of a Caribbean Current eddy over the 116-day average eddy period. Flow separation at the Isthmus of Panamá results in a vortex shed from the Darien Gulf, which slows the coastal currents in the gyre region from their maximum during eddy events. An annual ensemble average PV balance in the gyre region shows that the mean PV influx to this region is higher from August to October. Correspondingly, the range of coastal currents in the gyre region over an eddy event is modestly influenced by the PV influx magnitude. Eddy-influenced reversals in the coastal current can occur between November and July at Bocas del Toro and year-round at Colón. Such coastal current reversals are important for the alongshore transport of larvae, freshwater, and chemical tracers.
Abstract
Westward-propagating Caribbean Current eddies modify the volume-integrated potential vorticity (PV) balance in the western Caribbean Sea, influencing the circulation of the Panamá–Colombia Gyre (PCG) and coastal currents hundreds of kilometers to the south of the eddies’ mean trajectory. Using 22 years of output from the Hybrid Coordinate Ocean Model, we apply a volume-integrated eddy phase-averaged 1.5-layer PV balance, showing that PV fluxes into the PCG region are balanced by frictional PV dissipation represented by linear drag along the coastline. Coastal currents in the PCG region vary by a factor of 2 in phase with the passage of a Caribbean Current eddy over the 116-day average eddy period. Flow separation at the Isthmus of Panamá results in a vortex shed from the Darien Gulf, which slows the coastal currents in the gyre region from their maximum during eddy events. An annual ensemble average PV balance in the gyre region shows that the mean PV influx to this region is higher from August to October. Correspondingly, the range of coastal currents in the gyre region over an eddy event is modestly influenced by the PV influx magnitude. Eddy-influenced reversals in the coastal current can occur between November and July at Bocas del Toro and year-round at Colón. Such coastal current reversals are important for the alongshore transport of larvae, freshwater, and chemical tracers.
Abstract
Several models are presented for the sea surface height (SSH) signature of the interior-ocean internal-wave continuum. Most are based on the Garrett–Munk internal-wave model. One is derived from the frequency spectrum of dynamic height from mooring observations. The different models are all plausibly consistent with accepted dynamical and semiempirical spectral descriptions of the climatological interval-wave field in the interior ocean, but they result in different proportionalities between interior and SSH spectral energy levels. The differences arise in part from differences in the treatment of near-surface stratification, and a major source of uncertainty for all the models comes from inadequately constrained assumptions about the energy in the low-vertical-mode internal-wave field. Most of these models suggest that the SSH signature of the internal-wave continuum will be visible in SSH measurements from the Surface Water and Ocean Topography (SWOT) wide-swath satellite altimeter. Temporal variability of internal-wave energy levels and the internal-wave directional spectrum are less well characterized but will also be consequential for the observability of internal-wave signals in SWOT data.
Abstract
Several models are presented for the sea surface height (SSH) signature of the interior-ocean internal-wave continuum. Most are based on the Garrett–Munk internal-wave model. One is derived from the frequency spectrum of dynamic height from mooring observations. The different models are all plausibly consistent with accepted dynamical and semiempirical spectral descriptions of the climatological interval-wave field in the interior ocean, but they result in different proportionalities between interior and SSH spectral energy levels. The differences arise in part from differences in the treatment of near-surface stratification, and a major source of uncertainty for all the models comes from inadequately constrained assumptions about the energy in the low-vertical-mode internal-wave field. Most of these models suggest that the SSH signature of the internal-wave continuum will be visible in SSH measurements from the Surface Water and Ocean Topography (SWOT) wide-swath satellite altimeter. Temporal variability of internal-wave energy levels and the internal-wave directional spectrum are less well characterized but will also be consequential for the observability of internal-wave signals in SWOT data.
Abstract
A series of idealized numerical simulations is used to examine the generation of mode-one superinertial coastally trapped waves (CTWs). In the first set of simulations, CTWs are resonantly generated when freely propagating mode-one internal tides are incident on the coast such that the angle of incidence of the internal wave causes the projected wavenumber of the tide on the coast to satisfy a triad relationship with the wavenumbers of the bathymetry and the CTW. In the second set of simulations, CTWs are generated by the interaction of the barotropic tide with topography that has the same scales as the CTW. Under resonant conditions, superinertial coastally trapped waves are a leading order coastal process, with alongshore current magnitudes that can be larger than the barotropic or internal tides from which they are generated.
Abstract
A series of idealized numerical simulations is used to examine the generation of mode-one superinertial coastally trapped waves (CTWs). In the first set of simulations, CTWs are resonantly generated when freely propagating mode-one internal tides are incident on the coast such that the angle of incidence of the internal wave causes the projected wavenumber of the tide on the coast to satisfy a triad relationship with the wavenumbers of the bathymetry and the CTW. In the second set of simulations, CTWs are generated by the interaction of the barotropic tide with topography that has the same scales as the CTW. Under resonant conditions, superinertial coastally trapped waves are a leading order coastal process, with alongshore current magnitudes that can be larger than the barotropic or internal tides from which they are generated.
Abstract
The existence of multiple equilibria (ice-covered and ice-free states) is explored using a set of coupled, nondimensional equations that describe the heat and salt balances in basins, such as the Arctic Ocean, that are subject to atmospheric forcing and two distinct water mass sources. Six nondimensional numbers describe the influences of atmospheric cooling, evaporation minus precipitation, solar radiation, atmospheric temperature, diapycnal mixing, and the temperature contrast between the two water masses. It is shown that multiple equilibria resulting from the dependence of albedo on ice cover exist over a wide range of parameter space, especially so in the weak mixing limit. Multiple equilibria can also occur if diapycnal mixing increases to O(10−4) m2 s−1 or larger under ice-free conditions due to enhanced upward mixing of warm, salty water from below. Sensitivities to various forcing parameters are discussed.
Significance Statement
The purpose of this study is to better understand under what circumstances high-latitude seas, such as the Arctic Ocean, can exist in either an ice-covered or an ice-free state. The temperature and salinity of the ocean, as well as the heat exchange with the atmosphere, are drastically different depending on which state the ocean is in. The theory presented here identifies how forcing from the atmosphere and ocean dynamics determines whether the ocean is ice covered, ice free, or possibly either one.
Abstract
The existence of multiple equilibria (ice-covered and ice-free states) is explored using a set of coupled, nondimensional equations that describe the heat and salt balances in basins, such as the Arctic Ocean, that are subject to atmospheric forcing and two distinct water mass sources. Six nondimensional numbers describe the influences of atmospheric cooling, evaporation minus precipitation, solar radiation, atmospheric temperature, diapycnal mixing, and the temperature contrast between the two water masses. It is shown that multiple equilibria resulting from the dependence of albedo on ice cover exist over a wide range of parameter space, especially so in the weak mixing limit. Multiple equilibria can also occur if diapycnal mixing increases to O(10−4) m2 s−1 or larger under ice-free conditions due to enhanced upward mixing of warm, salty water from below. Sensitivities to various forcing parameters are discussed.
Significance Statement
The purpose of this study is to better understand under what circumstances high-latitude seas, such as the Arctic Ocean, can exist in either an ice-covered or an ice-free state. The temperature and salinity of the ocean, as well as the heat exchange with the atmosphere, are drastically different depending on which state the ocean is in. The theory presented here identifies how forcing from the atmosphere and ocean dynamics determines whether the ocean is ice covered, ice free, or possibly either one.
Abstract
Bubble plumes play a significant role in the air–sea interface by influencing processes such as air–sea gas exchange, aerosol production, modulation of oceanic carbon and nutrient cycles, and the vertical structure of the upper ocean. Using acoustic Doppler current profiler (ADCP) data collected off the west coast of Ireland, we investigate the dynamics of bubble plumes and their relationship with sea state variables. In particular, we describe the patterns of bubble plume vertical extension, duration, and periodicity. We establish a power-law relationship between the average bubble penetration depth and wind speed, consistent with previous findings. Additionally, the study reveals a significant association between whitecapping coverage and observed acoustic volume backscatter intensity, underscoring the role of wave breaking in bubble plume generation. The shape of the probability distribution of bubble plume depths reveals a transition toward stronger and more organized bubble entrainment events during higher wind speeds. Furthermore, we show that deeper bubble plumes are associated with turbulent Langmuir number La t ∼ 0.3, highlighting the potential role of Langmuir circulation on the transport and deepening of bubble plumes. These results contribute to a better understanding of the complex interactions between ocean waves, wind, and bubble plumes, providing valuable insights for improving predictive models and enhancing our understanding of air–sea interactions.
Significance Statement
This research contributes to understanding bubble plume dynamics in the upper ocean and their relationship with sea state variables. The establishment of a power-law relationship between the bubble penetration depth and wind speed, along with the association between whitecapping coverage and acoustic backscatter intensity, contributes to improved predictive capabilities for air–sea interactions and carbon dioxide exchange. The identification of the potential influence of Langmuir circulation on bubble plume dynamics expands our understanding of the role of coherent circulations in transporting bubble plumes. Additionally, this study presents a clear methodology using commercial sensors such as an ADCP, which can be easily replicated by researchers worldwide, leading to potential advancements in our comprehension of bubble plume dynamics.
Abstract
Bubble plumes play a significant role in the air–sea interface by influencing processes such as air–sea gas exchange, aerosol production, modulation of oceanic carbon and nutrient cycles, and the vertical structure of the upper ocean. Using acoustic Doppler current profiler (ADCP) data collected off the west coast of Ireland, we investigate the dynamics of bubble plumes and their relationship with sea state variables. In particular, we describe the patterns of bubble plume vertical extension, duration, and periodicity. We establish a power-law relationship between the average bubble penetration depth and wind speed, consistent with previous findings. Additionally, the study reveals a significant association between whitecapping coverage and observed acoustic volume backscatter intensity, underscoring the role of wave breaking in bubble plume generation. The shape of the probability distribution of bubble plume depths reveals a transition toward stronger and more organized bubble entrainment events during higher wind speeds. Furthermore, we show that deeper bubble plumes are associated with turbulent Langmuir number La t ∼ 0.3, highlighting the potential role of Langmuir circulation on the transport and deepening of bubble plumes. These results contribute to a better understanding of the complex interactions between ocean waves, wind, and bubble plumes, providing valuable insights for improving predictive models and enhancing our understanding of air–sea interactions.
Significance Statement
This research contributes to understanding bubble plume dynamics in the upper ocean and their relationship with sea state variables. The establishment of a power-law relationship between the bubble penetration depth and wind speed, along with the association between whitecapping coverage and acoustic backscatter intensity, contributes to improved predictive capabilities for air–sea interactions and carbon dioxide exchange. The identification of the potential influence of Langmuir circulation on bubble plume dynamics expands our understanding of the role of coherent circulations in transporting bubble plumes. Additionally, this study presents a clear methodology using commercial sensors such as an ADCP, which can be easily replicated by researchers worldwide, leading to potential advancements in our comprehension of bubble plume dynamics.
Abstract
This study adopts a curvature dynamics approach to understand and predict the trajectory of an idealized depth-averaged barotropic outflow onto a slope in shallow water. A novel equation for streamwise curvature dynamics was derived from the barotropic vorticity equation and applied to a momentum jet subject to bottom friction, topographic slope, and planetary rotation. The terms in the curvature dynamics equation have a natural geometric interpretation whereby each physical process can influence the flow direction. It is shown that a weakly spreading jet onto a steep slope admits the formulation of a 1D ordinary differential equation system in a streamline coordinate system, yielding an integrable ordinary differential equation system that predicts the kinematical behavior of the jet. The 1D model was compared with a set of high-resolution idealized depth-averaged circulation model simulations where bottom friction, planetary rotation, and bottom slope were varied. Favorable performance of the 1D reduced physics model was found, especially in the near field of the outflow. The effect of nonlinear processes such as topographic stretching and bottom torque on the fate of the jet outflow is explained using curvature dynamics. Even in the tropics, planetary rotation can have a surprisingly strong influence on the near-field deflection of an intermediate-scale jet, provided that it flows across steep topography.
Abstract
This study adopts a curvature dynamics approach to understand and predict the trajectory of an idealized depth-averaged barotropic outflow onto a slope in shallow water. A novel equation for streamwise curvature dynamics was derived from the barotropic vorticity equation and applied to a momentum jet subject to bottom friction, topographic slope, and planetary rotation. The terms in the curvature dynamics equation have a natural geometric interpretation whereby each physical process can influence the flow direction. It is shown that a weakly spreading jet onto a steep slope admits the formulation of a 1D ordinary differential equation system in a streamline coordinate system, yielding an integrable ordinary differential equation system that predicts the kinematical behavior of the jet. The 1D model was compared with a set of high-resolution idealized depth-averaged circulation model simulations where bottom friction, planetary rotation, and bottom slope were varied. Favorable performance of the 1D reduced physics model was found, especially in the near field of the outflow. The effect of nonlinear processes such as topographic stretching and bottom torque on the fate of the jet outflow is explained using curvature dynamics. Even in the tropics, planetary rotation can have a surprisingly strong influence on the near-field deflection of an intermediate-scale jet, provided that it flows across steep topography.
Abstract
In the global ocean, mesoscale eddies frequently deviate from a circular shape [circular asymmetry (CA)]. On average, anticyclonic eddies display slightly larger asymmetry than that of cyclonic eddies. Both types of eddies exhibit larger asymmetry during the periods of generation and extinction and smaller asymmetry during the intermediate periods. CA’s spatial distribution is dominantly controlled by the eddy’s rotational speed, radius, meridional displacement, and the background oceanic circulation. A decrease in rotational speed, an increase in radius, moving poleward (toward the equator), and/or moving into regions with smaller (larger) mean dynamic topography can enhance the asymmetry of anticyclonic (cyclonic) eddies. A weak advective nonlinearity of the eddy, represented by the ratio of the eddy’s rotational speed to its translation speed, also helps to enhance the asymmetry. Eddy with an asymmetric structure may be important for marine mixing processes.
Significance Statement
Mesoscale eddies are often assumed to be circular and symmetric. However, global observations reveal that eddies frequently diverge from a circular shape, displaying circular asymmetry (CA). The asymmetric structure of eddy is poorly investigated in the global ocean, especially in regions close to the strong oceanic currents. This study reported the evolution of eddy’s asymmetric structure and its spatial distribution. Statistical results show that CA is dominantly controlled by the eddy’s rotational speed, radius, meridional displacement, and the background oceanic circulation. This study provides a new perspective on the evolution of eddy’s structure and may be important in oceanic mixing.
Abstract
In the global ocean, mesoscale eddies frequently deviate from a circular shape [circular asymmetry (CA)]. On average, anticyclonic eddies display slightly larger asymmetry than that of cyclonic eddies. Both types of eddies exhibit larger asymmetry during the periods of generation and extinction and smaller asymmetry during the intermediate periods. CA’s spatial distribution is dominantly controlled by the eddy’s rotational speed, radius, meridional displacement, and the background oceanic circulation. A decrease in rotational speed, an increase in radius, moving poleward (toward the equator), and/or moving into regions with smaller (larger) mean dynamic topography can enhance the asymmetry of anticyclonic (cyclonic) eddies. A weak advective nonlinearity of the eddy, represented by the ratio of the eddy’s rotational speed to its translation speed, also helps to enhance the asymmetry. Eddy with an asymmetric structure may be important for marine mixing processes.
Significance Statement
Mesoscale eddies are often assumed to be circular and symmetric. However, global observations reveal that eddies frequently diverge from a circular shape, displaying circular asymmetry (CA). The asymmetric structure of eddy is poorly investigated in the global ocean, especially in regions close to the strong oceanic currents. This study reported the evolution of eddy’s asymmetric structure and its spatial distribution. Statistical results show that CA is dominantly controlled by the eddy’s rotational speed, radius, meridional displacement, and the background oceanic circulation. This study provides a new perspective on the evolution of eddy’s structure and may be important in oceanic mixing.
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.
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.
