We review the possible role of spontaneous emission and subsequent capture of internal gravity wa... more We review the possible role of spontaneous emission and subsequent capture of internal gravity waves (IGWs) for dissipation in oceanic flows under conditions characteristic for the ocean circulation. Dissipation is necessary for the transfer of energy from the essentially balanced large-scale ocean circulation and mesoscale eddy fields down to smaller scales where instabilities and subsequent small-scale turbulence complete the route to dissipation. Spontaneous wave emission by flows is a viable route to dissipation. For quasi-balanced flows, characterized by a small Rossby number, the amplitudes of emitted waves are expected to be small. However, once being emitted into a three-dimensional eddying flow field, waves can undergo refraction and may be “captured.” During wave capture, the wavenumber grows exponentially, ultimately leading to breakup and dissipation. For flows with not too small Rossby number, e.g., for flows in the vicinity of strong fronts, dissipation occurs in a more complex manner. It can occur via spontaneous wave emission and subsequent wave capture, with the amplitudes of waves emitted in frontal systems being expected to be larger than amplitudes of waves emitted by quasi-balanced flows. It can also occur through turbulence and filamentation emerging from frontogenesis. So far, quantitative importance of this energy pathway—crucial for determining correct eddy viscosities in general circulation models—is not known. Toward an answer to this question, we discuss IGWs diagnostics, review spontaneous emission of both quasi-balanced and less-balanced frontal flows, and discuss recent numerical results based on a high-resolution ocean general circulation model.
Geophysical flows comprise a broad range of spatial and temporal scales, from planetary- to meso-... more Geophysical flows comprise a broad range of spatial and temporal scales, from planetary- to meso-scale and microscopic turbulence regimes. The relation of scales and flow phenomena is essential in order to validate and improve cur- rent numerical weather and climate prediction models. While regime separation is often possible on a formal level via multi-scale analysis, the systematic exploration, structure preservation, and mathematical details remain challenging. This chapter provides an entry to the literature and reviews fundamental notions as background for the later chapters in this collection and as a departure point for original research in the field.
This book describes the derivation of the equations of motion of fluids as well as the dynamics o... more This book describes the derivation of the equations of motion of fluids as well as the dynamics of ocean and atmospheric currents on both large and small scales through the use of variational methods. In this way the equations of Fluid and Geophysical Fluid Dynamics are re-derived making use of a unifying principle, that is Hamilton’s Principle of Least Action. The equations are analyzed within the framework of Lagrangian and Hamiltonian mechanics for continuous systems. The analysis of the equations’ symmetries and the resulting conservation laws, from Noether’s Theorem, represent the core of the description. Central to this work is the analysis of particle relabeling symmetry, which is unique for fluid dynamics and results in the conservation of potential vorticity. Different special approximations and relations, ranging from the semi-geostrophic approximation to the conservation of wave activity, are derived and analyzed. Thanks to a complete derivation of all relationships, this book is accessible for students at both undergraduate and graduate levels, as well for researchers. Students of theoretical physics and applied mathematics will recognize the existence of theoretical challenges behind the applied field of Geophysical Fluid Dynamics, while students of applied physics, meteorology and oceanography will be able to find and appreciate the fundamental relationships behind equations in this field.
In this work we investigate the statistical mechanics of a family of two dimensional (2D) fluid f... more In this work we investigate the statistical mechanics of a family of two dimensional (2D) fluid flows, described by the generalized Euler equations, or α-models. These models describe both nonlocal and local dynamics, with one example of the latter given by the surface quasi geostrophy (SQG) model for which the existence of singularities is still under discussion. Furthermore, SQG is relevant both for atmosphere and ocean dynamics, and in particular, it is proposed to understand the oceanic submesoscale structures, front and filaments, associated with horizontal gradients of buoyancy. We discuss under which conditions the statistical theory suggests a principle of selective decay for the whole family of turbulent models, and then we explore the selective decay principle numerically, the transition to equilibrium and the formation of singularities, starting from initial conditions (i.c.s) corresponding to a hyperbolic saddle. We study the topological transitions in the flow configuration induced by filaments breaking. Furthermore we compare the theoretical equilibrium states, and the functional relation between the generalized vorticity q and its correspondent stream function ψ, with the results of the simulations. For the particular i.c.s investigated, and domain used for the simulations, we have not noticed transition from tanh-like to sinh-like ψ −q functional relation, which would be expected by the emergence of coherent structures that are, however, filtered by the time averages used to compute the mean fields.
Geophysical and Astrophysical Fluid Dynamics, 2019
The velocity fluctuations for point vortex models are studied for the $\alpha$-turbulence equatio... more The velocity fluctuations for point vortex models are studied for the $\alpha$-turbulence equations, which are characterized by a fractional Laplacian relation between active scalar and the streamfunction. In particular, we focus on the local dynamics regime. The local dynamics differ from the well-studied case of $2$D turbulence as it allows to consider the true thermodynamic limit, that is, to consider an infinite set of point vortices on an infinite plane keeping the density of the vortices constant. The consequence of this limit is that the results obtained are independent on the number of point vortices in the system. This limit is not defined for $2$D turbulence. We show an analytical form of the probability density distribution of the velocity fluctuations for different degrees of locality. The central region of the distribution is not Gaussian, in contrast to the case of $2D$ turbulence, but can be approximated with a Gaussian function in the small velocity limit. The tails of the distribution exhibit a power law behavior and self similarity in terms of the density variable. Due to the thermodynamic limit, both the Gaussian approximation for the core and the steepness of the tails are independent on the number of point vortices, but depend on the $\alpha$-model. We also show the connection between the velocity statistics for point vortices uniformly distributed, in the context of the $\alpha$-model in classical turbulence, with the velocity statistics for point vortices non-uniformly distributed. Since the exponent of the power law depends just on $\alpha$, we test the power law approximation obtained with the point vortex approximation, by simulating full turbulent fields for different values of $\alpha$ and we compute the correspondent probability density distribution for the absolute value of the velocity field. These results suggest that the local nature of the turbulent fluctuations in the ocean or in the atmosphere might be deduced from the shape of the tails of the probability density functions.
In this article we derive the equations for a rotating stratified fluid governed by inviscid Eule... more In this article we derive the equations for a rotating stratified fluid governed by inviscid Euler–Boussinesq and primitive equations that account for the effects of the perturbations upon the mean. Our method is based on the concept of the geometric generalized Lagrangian mean recently introduced by Gilbert and Vanneste, combined with generalized Taylor and horizontal isotropy of fluctuations as turbulent closure hypotheses. The models we obtain arise as Euler–Poincaré equations and inherit from their parent systems conservation laws for energy and potential vorticity. They are structurally and geometrically similar to Euler–Boussinesq-α and primitive equations-α models, however feature a different regularizing second order operator.
We examine the relative dispersion of surface drifters deployed in groups of triplets at the boun... more We examine the relative dispersion of surface drifters deployed in groups of triplets at the boundaries of a filament in the upwelling region off Namibia for both the entire ensemble and the two main subgroups. For the drifters in the group released at the northern boundary of the filament, close to the upwelling front, we find that the mean-square pair separation hs 2 (t)i shows the characteristic distinct dispersion regimes [nonlocal, local (Richardson), and diffusive] of an ocean surface mixed layer. We confirm the different dispersion regimes by a rescaled presentation of the moments hs n (t)i and thereby also explain the anomalous slow decay of the kurtosis in the transient regime. For the drifter group released at the southern boundary, hs 2 (t)i remains constant for a short period, followed by a steep ''Richardson like'' increase and an asymptotic diffusive increase. In contrast to the northern release, the corresponding moments reveal a narrow distribution of pair separations for all regimes. The analysis of finite-size Lyapunov exponents (FSLEs) reveals consistent results when applied to the two releases separately. When applied to the entire drifter ensemble, the two measures yield inconsistent results. We relate the breakdown of consistency to the impact of the different dynamics on the respective averages: whereas, because of separation in scale, hs 2 (t)i is dominated by the northern release, the decay of the FSLEs for small distances reflects the drifter dynamics within the filament.
The scale-dependent stability of surface quasi-geostrophic (SQG) vortices is studied both analyti... more The scale-dependent stability of surface quasi-geostrophic (SQG) vortices is studied both analytically and numerically. In particular, we study the sensitivity of the stability of SQG vortices on a nondimensional number σ, namely the square root of the Burger number, which sets the transition scale between different dynamical regimes corresponding to local and nonlocal dynamics. We analyse the stability of two different examples. The first example is given by a Rankine vortex, characterised by constant buoyancy. For this case, asymptotic analysis suggests that the frequencies of the perturbations at scales smaller than the transition scale show a σ −1 dependence. At scales larger than the transition scale, the frequencies scale instead like σ−2. The second example consists of a Rankine vortex shielded by a filament characterised by a different value of constant buoyancy. For this example we study the dispersion relation for the perturbations for the cases in which the inner vortex and the outer filament have different asymptotic properties behaviour.
Point-vortex models are presented for the generalized Euler equations, which are characterized by... more Point-vortex models are presented for the generalized Euler equations, which are characterized by a fractional Laplacian relation between the active scalar and the stream function. Special focus is given to the case of the surface quasigeostrophic (SQG) equations, for which the existence of finite-time singularities is still a matter of debate. Point-vortex trajectories are expressed using Nambu dynamics. The formulation is based on a noncanonical bracket and allows for a geometrical interpretation of trajectories as intersections of level sets of the Hamiltonian and Casimir. Within this setting, we focus on the collapse of solutions for the three-point-vortex model. In particular, we show that for SQG the collapse can be either self-similar or non-self-similar. Self-similarity occurs only when the Hamiltonian is zero, while non-self-similarity appears for nonzero values of the same. For both cases, collapse is allowed for any choice of circulations within a permitted interval. These results differ strikingly from the classical point-vortex model, where collapse is self-similar for any value of the Hamiltonian, but the vortex circulations must satisfy a strict relationship. Results may also shed a light on the formation of singularities in the SQG partial differential equations, where the singularity is thought to be reached only in a self-similar way.
The North Atlantic Oscillation (NAO) and the Arctic Oscillation (AO) describe the dominant part o... more The North Atlantic Oscillation (NAO) and the Arctic Oscillation (AO) describe the dominant part of the variability in the Northern Hemisphere extratropical troposphere. Due to the strong connection of these patterns with surface climate, recent years have shown an increased interest and an increasing skill in forecasting them. However, it is unclear what the intrinsic limits of short-term predictability for the NAO and AO patterns are. This study compares the variability and predictability of both patterns, using a range of data and index computation methods for the daily NAO/AO indices. Small deviations from Gaussianity are found and characteristic decorrelation time scales of around one week. In the analysis of the Lyapunov spectrum it is found that predictability is not significantly different between the AO and NAO or between reanalysis products. Differences exist however between the indices based on EOF analysis, which exhibit predictability time scales around 12 - 16 days, and the station-based indices, exhibiting a longer predictability of 18 - 20 days. Both of these time scales indicate predictability beyond that currently obtained in ensemble prediction models for short-term predictability. Additional longer-term predictability for these patterns may be gained through local feedbacks and remote forcing mechanisms.
A new method to describe hyperbolic patterns in two-dimensional flows is proposed. The method is ... more A new method to describe hyperbolic patterns in two-dimensional flows is proposed. The method is based on the Covariant Lyapunov Vectors (CLVs), which have the properties of being covariant with the dynamics, and thus, being mapped by the tangent linear operator into another CLVs basis, they are norm independent, invariant under time reversal and cannot be orthonormal. CLVs can thus give more detailed information about the expansion and contraction directions of the flow than the Lyapunov vector bases, which are instead always orthogonal. We suggest a definition of Hyperbolic Covariant Coherent Structures (HCCSs), which can be defined on the scalar field representing the angle between the CLVs. HCCSs can be defined for every time instant and could be useful to understand the long-term behavior of particle tracers. We consider three examples: a simple autonomous Hamiltonian system, as well as the non-autonomous " double gyre " and Bickley jet, to see how well the angle is able to describe particular patterns and barriers. We compare the results from the HCCSs with other coherent patterns defined on finite time by the Finite Time Lyapunov Exponents (FTLEs), to see how the behaviors of these structures change asymptotically.
A systematic method to derive the Hamiltonian and Nambu form for the shallow water equations usin... more A systematic method to derive the Hamiltonian and Nambu form for the shallow water equations using the conservation for energy and potential enstrophy is presented. Different mechanisms, such as vortical flows and emission of gravity waves, emerge from different conservation laws for total energy and potential enstrophy. The equations are constructed using exterior differential forms and self-adjoint operators, and result in the sum of two Nambu brackets—one for the vortical flow and one for the wave-mean flow interaction—and a Poisson bracket representing the interaction between divergence and geostrophic imbalance. The advantage of this approach is that the Hamiltonian and Nambu forms can here be written in a coordinate-independent form.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2016
Characterizing the stratosphere as a turbulent system, temporal fluctuations often show different... more Characterizing the stratosphere as a turbulent system, temporal fluctuations often show different correlations for different time scales as well as intermittent behaviour that cannot be captured by a single scaling exponent. In this study, the different scaling laws in the long-term stratospheric variability are studied using multifractal de-trended fluctuation analysis (MF-DFA). The analysis is performed comparing four re-analysis products and different realizations of an idealized numerical model, isolating the role of topographic forcing and seasonal variability, as well as the absence of climate teleconnections and small-scale forcing. The Northern Hemisphere (NH) shows a transition of scaling exponents for time scales shorter than about 1 year, for which the variability is multifractal and scales in time with a power law corresponding to a red spectrum, to longer time scales, for which the variability is monofractal and scales in time with a power law corresponding to white noise. Southern Hemisphere (SH) variability also shows a transition at annual scales. The SH also shows a narrower dynamical range in multifractality than the NH, as seen in the generalized Hurst exponent and in the singularity spectra. The numerical integrations show that the models are able to reproduce the low- frequency variability but are not able to fully capture the shorter term variability of the stratosphere.
In this study we give a characterization of semi-geostrophic turbulence by performing freely deca... more In this study we give a characterization of semi-geostrophic turbulence by performing freely decaying simulations for the case of constant uniform potential vorticity, a set of equations known as the surface semi-geostrophic approximation. The equations are formulated as conservation laws for potential temperature and potential vorticity, with a nonlinear Monge–Ampère type inversion equation for the streamfunction, expressed in a transformed coordinate system that follows the geostrophic flow. We perform model studies of turbulent surface semi-geostrophic flows in a domain doubly periodic in the horizontal and limited in the vertical by two rigid lids, allowing for variations of potential temperature at one of the boundaries, and we compare the results with those obtained in the corresponding surface quasi-geostrophic case. The results show that, while the surface quasi-geostrophic dynamics is dominated by a symmetric population of cyclones and anticyclones, the surface semi-geostrophic dynamics features a more prominent role of fronts and filaments. The resulting distribution of potential temperature is strongly skewed and peaked at non-zero values at and close to the active boundary, while symmetry is restored in the interior of the domain, where small-scale frontal structures do not penetrate. In surface semi-geostrophic turbulence, energy spectra are less steep than in the surface quasi-geostrophic case, with more energy concentrated at small scales for increasing Rossby number. The energy related to frontal structures, the lateral strain rate and the vertical velocities are largest close to the active boundary. These results show that the semi-geostrophic model could be of interest for studying the lateral mixing of properties in geophysical fows.
We review the possible role of spontaneous emission and subsequent capture of internal gravity wa... more We review the possible role of spontaneous emission and subsequent capture of internal gravity waves (IGWs) for dissipation in oceanic flows under conditions characteristic for the ocean circulation. Dissipation is necessary for the transfer of energy from the essentially balanced large-scale ocean circulation and mesoscale eddy fields down to smaller scales where instabilities and subsequent small-scale turbulence complete the route to dissipation. Spontaneous wave emission by flows is a viable route to dissipation. For quasi-balanced flows, characterized by a small Rossby number, the amplitudes of emitted waves are expected to be small. However, once being emitted into a three-dimensional eddying flow field, waves can undergo refraction and may be “captured.” During wave capture, the wavenumber grows exponentially, ultimately leading to breakup and dissipation. For flows with not too small Rossby number, e.g., for flows in the vicinity of strong fronts, dissipation occurs in a more complex manner. It can occur via spontaneous wave emission and subsequent wave capture, with the amplitudes of waves emitted in frontal systems being expected to be larger than amplitudes of waves emitted by quasi-balanced flows. It can also occur through turbulence and filamentation emerging from frontogenesis. So far, quantitative importance of this energy pathway—crucial for determining correct eddy viscosities in general circulation models—is not known. Toward an answer to this question, we discuss IGWs diagnostics, review spontaneous emission of both quasi-balanced and less-balanced frontal flows, and discuss recent numerical results based on a high-resolution ocean general circulation model.
Geophysical flows comprise a broad range of spatial and temporal scales, from planetary- to meso-... more Geophysical flows comprise a broad range of spatial and temporal scales, from planetary- to meso-scale and microscopic turbulence regimes. The relation of scales and flow phenomena is essential in order to validate and improve cur- rent numerical weather and climate prediction models. While regime separation is often possible on a formal level via multi-scale analysis, the systematic exploration, structure preservation, and mathematical details remain challenging. This chapter provides an entry to the literature and reviews fundamental notions as background for the later chapters in this collection and as a departure point for original research in the field.
This book describes the derivation of the equations of motion of fluids as well as the dynamics o... more This book describes the derivation of the equations of motion of fluids as well as the dynamics of ocean and atmospheric currents on both large and small scales through the use of variational methods. In this way the equations of Fluid and Geophysical Fluid Dynamics are re-derived making use of a unifying principle, that is Hamilton’s Principle of Least Action. The equations are analyzed within the framework of Lagrangian and Hamiltonian mechanics for continuous systems. The analysis of the equations’ symmetries and the resulting conservation laws, from Noether’s Theorem, represent the core of the description. Central to this work is the analysis of particle relabeling symmetry, which is unique for fluid dynamics and results in the conservation of potential vorticity. Different special approximations and relations, ranging from the semi-geostrophic approximation to the conservation of wave activity, are derived and analyzed. Thanks to a complete derivation of all relationships, this book is accessible for students at both undergraduate and graduate levels, as well for researchers. Students of theoretical physics and applied mathematics will recognize the existence of theoretical challenges behind the applied field of Geophysical Fluid Dynamics, while students of applied physics, meteorology and oceanography will be able to find and appreciate the fundamental relationships behind equations in this field.
In this work we investigate the statistical mechanics of a family of two dimensional (2D) fluid f... more In this work we investigate the statistical mechanics of a family of two dimensional (2D) fluid flows, described by the generalized Euler equations, or α-models. These models describe both nonlocal and local dynamics, with one example of the latter given by the surface quasi geostrophy (SQG) model for which the existence of singularities is still under discussion. Furthermore, SQG is relevant both for atmosphere and ocean dynamics, and in particular, it is proposed to understand the oceanic submesoscale structures, front and filaments, associated with horizontal gradients of buoyancy. We discuss under which conditions the statistical theory suggests a principle of selective decay for the whole family of turbulent models, and then we explore the selective decay principle numerically, the transition to equilibrium and the formation of singularities, starting from initial conditions (i.c.s) corresponding to a hyperbolic saddle. We study the topological transitions in the flow configuration induced by filaments breaking. Furthermore we compare the theoretical equilibrium states, and the functional relation between the generalized vorticity q and its correspondent stream function ψ, with the results of the simulations. For the particular i.c.s investigated, and domain used for the simulations, we have not noticed transition from tanh-like to sinh-like ψ −q functional relation, which would be expected by the emergence of coherent structures that are, however, filtered by the time averages used to compute the mean fields.
Geophysical and Astrophysical Fluid Dynamics, 2019
The velocity fluctuations for point vortex models are studied for the $\alpha$-turbulence equatio... more The velocity fluctuations for point vortex models are studied for the $\alpha$-turbulence equations, which are characterized by a fractional Laplacian relation between active scalar and the streamfunction. In particular, we focus on the local dynamics regime. The local dynamics differ from the well-studied case of $2$D turbulence as it allows to consider the true thermodynamic limit, that is, to consider an infinite set of point vortices on an infinite plane keeping the density of the vortices constant. The consequence of this limit is that the results obtained are independent on the number of point vortices in the system. This limit is not defined for $2$D turbulence. We show an analytical form of the probability density distribution of the velocity fluctuations for different degrees of locality. The central region of the distribution is not Gaussian, in contrast to the case of $2D$ turbulence, but can be approximated with a Gaussian function in the small velocity limit. The tails of the distribution exhibit a power law behavior and self similarity in terms of the density variable. Due to the thermodynamic limit, both the Gaussian approximation for the core and the steepness of the tails are independent on the number of point vortices, but depend on the $\alpha$-model. We also show the connection between the velocity statistics for point vortices uniformly distributed, in the context of the $\alpha$-model in classical turbulence, with the velocity statistics for point vortices non-uniformly distributed. Since the exponent of the power law depends just on $\alpha$, we test the power law approximation obtained with the point vortex approximation, by simulating full turbulent fields for different values of $\alpha$ and we compute the correspondent probability density distribution for the absolute value of the velocity field. These results suggest that the local nature of the turbulent fluctuations in the ocean or in the atmosphere might be deduced from the shape of the tails of the probability density functions.
In this article we derive the equations for a rotating stratified fluid governed by inviscid Eule... more In this article we derive the equations for a rotating stratified fluid governed by inviscid Euler–Boussinesq and primitive equations that account for the effects of the perturbations upon the mean. Our method is based on the concept of the geometric generalized Lagrangian mean recently introduced by Gilbert and Vanneste, combined with generalized Taylor and horizontal isotropy of fluctuations as turbulent closure hypotheses. The models we obtain arise as Euler–Poincaré equations and inherit from their parent systems conservation laws for energy and potential vorticity. They are structurally and geometrically similar to Euler–Boussinesq-α and primitive equations-α models, however feature a different regularizing second order operator.
We examine the relative dispersion of surface drifters deployed in groups of triplets at the boun... more We examine the relative dispersion of surface drifters deployed in groups of triplets at the boundaries of a filament in the upwelling region off Namibia for both the entire ensemble and the two main subgroups. For the drifters in the group released at the northern boundary of the filament, close to the upwelling front, we find that the mean-square pair separation hs 2 (t)i shows the characteristic distinct dispersion regimes [nonlocal, local (Richardson), and diffusive] of an ocean surface mixed layer. We confirm the different dispersion regimes by a rescaled presentation of the moments hs n (t)i and thereby also explain the anomalous slow decay of the kurtosis in the transient regime. For the drifter group released at the southern boundary, hs 2 (t)i remains constant for a short period, followed by a steep ''Richardson like'' increase and an asymptotic diffusive increase. In contrast to the northern release, the corresponding moments reveal a narrow distribution of pair separations for all regimes. The analysis of finite-size Lyapunov exponents (FSLEs) reveals consistent results when applied to the two releases separately. When applied to the entire drifter ensemble, the two measures yield inconsistent results. We relate the breakdown of consistency to the impact of the different dynamics on the respective averages: whereas, because of separation in scale, hs 2 (t)i is dominated by the northern release, the decay of the FSLEs for small distances reflects the drifter dynamics within the filament.
The scale-dependent stability of surface quasi-geostrophic (SQG) vortices is studied both analyti... more The scale-dependent stability of surface quasi-geostrophic (SQG) vortices is studied both analytically and numerically. In particular, we study the sensitivity of the stability of SQG vortices on a nondimensional number σ, namely the square root of the Burger number, which sets the transition scale between different dynamical regimes corresponding to local and nonlocal dynamics. We analyse the stability of two different examples. The first example is given by a Rankine vortex, characterised by constant buoyancy. For this case, asymptotic analysis suggests that the frequencies of the perturbations at scales smaller than the transition scale show a σ −1 dependence. At scales larger than the transition scale, the frequencies scale instead like σ−2. The second example consists of a Rankine vortex shielded by a filament characterised by a different value of constant buoyancy. For this example we study the dispersion relation for the perturbations for the cases in which the inner vortex and the outer filament have different asymptotic properties behaviour.
Point-vortex models are presented for the generalized Euler equations, which are characterized by... more Point-vortex models are presented for the generalized Euler equations, which are characterized by a fractional Laplacian relation between the active scalar and the stream function. Special focus is given to the case of the surface quasigeostrophic (SQG) equations, for which the existence of finite-time singularities is still a matter of debate. Point-vortex trajectories are expressed using Nambu dynamics. The formulation is based on a noncanonical bracket and allows for a geometrical interpretation of trajectories as intersections of level sets of the Hamiltonian and Casimir. Within this setting, we focus on the collapse of solutions for the three-point-vortex model. In particular, we show that for SQG the collapse can be either self-similar or non-self-similar. Self-similarity occurs only when the Hamiltonian is zero, while non-self-similarity appears for nonzero values of the same. For both cases, collapse is allowed for any choice of circulations within a permitted interval. These results differ strikingly from the classical point-vortex model, where collapse is self-similar for any value of the Hamiltonian, but the vortex circulations must satisfy a strict relationship. Results may also shed a light on the formation of singularities in the SQG partial differential equations, where the singularity is thought to be reached only in a self-similar way.
The North Atlantic Oscillation (NAO) and the Arctic Oscillation (AO) describe the dominant part o... more The North Atlantic Oscillation (NAO) and the Arctic Oscillation (AO) describe the dominant part of the variability in the Northern Hemisphere extratropical troposphere. Due to the strong connection of these patterns with surface climate, recent years have shown an increased interest and an increasing skill in forecasting them. However, it is unclear what the intrinsic limits of short-term predictability for the NAO and AO patterns are. This study compares the variability and predictability of both patterns, using a range of data and index computation methods for the daily NAO/AO indices. Small deviations from Gaussianity are found and characteristic decorrelation time scales of around one week. In the analysis of the Lyapunov spectrum it is found that predictability is not significantly different between the AO and NAO or between reanalysis products. Differences exist however between the indices based on EOF analysis, which exhibit predictability time scales around 12 - 16 days, and the station-based indices, exhibiting a longer predictability of 18 - 20 days. Both of these time scales indicate predictability beyond that currently obtained in ensemble prediction models for short-term predictability. Additional longer-term predictability for these patterns may be gained through local feedbacks and remote forcing mechanisms.
A new method to describe hyperbolic patterns in two-dimensional flows is proposed. The method is ... more A new method to describe hyperbolic patterns in two-dimensional flows is proposed. The method is based on the Covariant Lyapunov Vectors (CLVs), which have the properties of being covariant with the dynamics, and thus, being mapped by the tangent linear operator into another CLVs basis, they are norm independent, invariant under time reversal and cannot be orthonormal. CLVs can thus give more detailed information about the expansion and contraction directions of the flow than the Lyapunov vector bases, which are instead always orthogonal. We suggest a definition of Hyperbolic Covariant Coherent Structures (HCCSs), which can be defined on the scalar field representing the angle between the CLVs. HCCSs can be defined for every time instant and could be useful to understand the long-term behavior of particle tracers. We consider three examples: a simple autonomous Hamiltonian system, as well as the non-autonomous " double gyre " and Bickley jet, to see how well the angle is able to describe particular patterns and barriers. We compare the results from the HCCSs with other coherent patterns defined on finite time by the Finite Time Lyapunov Exponents (FTLEs), to see how the behaviors of these structures change asymptotically.
A systematic method to derive the Hamiltonian and Nambu form for the shallow water equations usin... more A systematic method to derive the Hamiltonian and Nambu form for the shallow water equations using the conservation for energy and potential enstrophy is presented. Different mechanisms, such as vortical flows and emission of gravity waves, emerge from different conservation laws for total energy and potential enstrophy. The equations are constructed using exterior differential forms and self-adjoint operators, and result in the sum of two Nambu brackets—one for the vortical flow and one for the wave-mean flow interaction—and a Poisson bracket representing the interaction between divergence and geostrophic imbalance. The advantage of this approach is that the Hamiltonian and Nambu forms can here be written in a coordinate-independent form.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2016
Characterizing the stratosphere as a turbulent system, temporal fluctuations often show different... more Characterizing the stratosphere as a turbulent system, temporal fluctuations often show different correlations for different time scales as well as intermittent behaviour that cannot be captured by a single scaling exponent. In this study, the different scaling laws in the long-term stratospheric variability are studied using multifractal de-trended fluctuation analysis (MF-DFA). The analysis is performed comparing four re-analysis products and different realizations of an idealized numerical model, isolating the role of topographic forcing and seasonal variability, as well as the absence of climate teleconnections and small-scale forcing. The Northern Hemisphere (NH) shows a transition of scaling exponents for time scales shorter than about 1 year, for which the variability is multifractal and scales in time with a power law corresponding to a red spectrum, to longer time scales, for which the variability is monofractal and scales in time with a power law corresponding to white noise. Southern Hemisphere (SH) variability also shows a transition at annual scales. The SH also shows a narrower dynamical range in multifractality than the NH, as seen in the generalized Hurst exponent and in the singularity spectra. The numerical integrations show that the models are able to reproduce the low- frequency variability but are not able to fully capture the shorter term variability of the stratosphere.
In this study we give a characterization of semi-geostrophic turbulence by performing freely deca... more In this study we give a characterization of semi-geostrophic turbulence by performing freely decaying simulations for the case of constant uniform potential vorticity, a set of equations known as the surface semi-geostrophic approximation. The equations are formulated as conservation laws for potential temperature and potential vorticity, with a nonlinear Monge–Ampère type inversion equation for the streamfunction, expressed in a transformed coordinate system that follows the geostrophic flow. We perform model studies of turbulent surface semi-geostrophic flows in a domain doubly periodic in the horizontal and limited in the vertical by two rigid lids, allowing for variations of potential temperature at one of the boundaries, and we compare the results with those obtained in the corresponding surface quasi-geostrophic case. The results show that, while the surface quasi-geostrophic dynamics is dominated by a symmetric population of cyclones and anticyclones, the surface semi-geostrophic dynamics features a more prominent role of fronts and filaments. The resulting distribution of potential temperature is strongly skewed and peaked at non-zero values at and close to the active boundary, while symmetry is restored in the interior of the domain, where small-scale frontal structures do not penetrate. In surface semi-geostrophic turbulence, energy spectra are less steep than in the surface quasi-geostrophic case, with more energy concentrated at small scales for increasing Rossby number. The energy related to frontal structures, the lateral strain rate and the vertical velocities are largest close to the active boundary. These results show that the semi-geostrophic model could be of interest for studying the lateral mixing of properties in geophysical fows.
Three dimensional (3D) Finite Time Lyapunov Exponents (FTLEs) are computed from numerical simulat... more Three dimensional (3D) Finite Time Lyapunov Exponents (FTLEs) are computed from numerical simulations of a freely evolving mixed layer (ML) front in a zonal channel undergoing baroclinic instability. The 3D FTLEs show a complex structure, with features that are less defined than the two-dimensional (2D) FTLEs, suggesting that stirring is not confined to the edges of vortices and along filaments and posing significant consequences on mixing. The magnitude of the FTLEs is observed to be strongly determined by the vertical shear. A scaling law relating the local FTLEs and the nonlocal density contrast used to initialize the ML front is derived assuming thermal wind balance. The scaling law only converges to the values found from the simulations within the pycnocline, while it displays differences within the ML, where the instabilities show a large ageostrophic component. The probability distribution functions of 2D and 3D FTLEs are found to be non Gaussian at all depths. In the pycnocl...
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Books by Gualtiero Badin
Papers by Gualtiero Badin
The local dynamics differ from the well-studied case of $2$D turbulence as it allows to consider the true thermodynamic limit, that is, to consider an infinite set of point vortices on an infinite plane keeping the density of the vortices constant. The consequence of this limit is that the results obtained are independent on the number of point vortices in the system. This limit is not defined for $2$D turbulence.
We show an analytical form of the probability density distribution of the velocity fluctuations for different degrees of locality.
The central region of the distribution is not Gaussian, in contrast to the case of $2D$ turbulence, but can be approximated with a Gaussian function in the small velocity limit.
The tails of the distribution exhibit a power law behavior and self similarity in terms of the density variable.
Due to the thermodynamic limit, both the Gaussian approximation for the core and the steepness of the tails are independent on the number of point vortices, but depend on the $\alpha$-model.
We also show the connection between the velocity statistics for point vortices uniformly distributed, in the context of the $\alpha$-model in classical turbulence, with the velocity statistics for point vortices non-uniformly distributed.
Since the exponent of the power law depends just on $\alpha$, we test the power law approximation obtained with the point vortex approximation, by simulating full turbulent fields for different values of $\alpha$ and we compute the correspondent probability density distribution for the absolute value of the velocity field.
These results suggest that the local nature of the turbulent fluctuations in the ocean or in the atmosphere might be deduced from the shape of the tails of the probability density functions.
The local dynamics differ from the well-studied case of $2$D turbulence as it allows to consider the true thermodynamic limit, that is, to consider an infinite set of point vortices on an infinite plane keeping the density of the vortices constant. The consequence of this limit is that the results obtained are independent on the number of point vortices in the system. This limit is not defined for $2$D turbulence.
We show an analytical form of the probability density distribution of the velocity fluctuations for different degrees of locality.
The central region of the distribution is not Gaussian, in contrast to the case of $2D$ turbulence, but can be approximated with a Gaussian function in the small velocity limit.
The tails of the distribution exhibit a power law behavior and self similarity in terms of the density variable.
Due to the thermodynamic limit, both the Gaussian approximation for the core and the steepness of the tails are independent on the number of point vortices, but depend on the $\alpha$-model.
We also show the connection between the velocity statistics for point vortices uniformly distributed, in the context of the $\alpha$-model in classical turbulence, with the velocity statistics for point vortices non-uniformly distributed.
Since the exponent of the power law depends just on $\alpha$, we test the power law approximation obtained with the point vortex approximation, by simulating full turbulent fields for different values of $\alpha$ and we compute the correspondent probability density distribution for the absolute value of the velocity field.
These results suggest that the local nature of the turbulent fluctuations in the ocean or in the atmosphere might be deduced from the shape of the tails of the probability density functions.