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Papers by Sabine Ortiz

Bulletin of the American Physical Society, Nov 19, 2006

Submitted for the DFD06 Meeting of The American Physical Society Three-dimensional instabilities ... more Submitted for the DFD06 Meeting of The American Physical Society Three-dimensional instabilities and transient growth of trailing vortices JEAN-MARC CHOMAZ, CLAIRE DONNADIEU, SABINE ORTIZ, PAUL BILLANT, LadHyX, CNRS-Ecole Polytechnique-An aircraft wake is made of counter-rotating vortices and is known to be affected by a long (Crow) and a short (elliptic) wavelength instabilities. Numerical investigations on the threedimensionnal instabilities and transient growth of such dipole are performed.By means of a three-dimensionnal linear stability analysis, we retrieve the instability bands corresponding to the Crow and elliptic modes but we also observe less unstable oscillatory modes with very broad peaks. The transient growth of this dipole, investigated by computing the optimal perturbations with a direct-adjoint 1 technique, demonstrates the crucial role of the region of maximal strain. Further investigations on the dynamics of trailing vortices in stratified fluids will be performed. Indeed, as such dipoles propagate downwards, they evolve under the influence of the stratification of the atmosphere.

HAL (Le Centre pour la Communication Scientifique Directe), Aug 27, 2007

An aircraft wake is made of counter-rotating vortices and is known to be affected by a long (Crow... more An aircraft wake is made of counter-rotating vortices and is known to be affected by a long (Crow) and a short (elliptic) wavelength instabilities. Numerical investigations on the three-dimensionnal instabilities and transient growth of such dipole are performed. By means of a three-dimensionnal linear stability analysis, we retrieve the instability bands corresponding to the Crow and elliptic modes but we also observe less unstable oscillatory modes with very broad peaks. The transient growth of perturbations on this dipole, investigated by computing the optimal linear perturbations with a direct-adjoint technique, demonstrates the crucial role of the region of maximal strain at short time and of the hyperbolic point at intermediate time. Investigations on the three-dimensionnal dynamics of trailing vortices in stratified fluids are performed. The elliptic instability is almost unaffected by weak and moderate stratifications. Résumé : Le sillage d'un avion est constitué d'un paire de tourbillons contra-rotatifs et est affectée par une instabilité à grande (Crow) et à petite (elliptique) longueur d'onde. On réalise des études numériques sur les instabilités tridimensionnelles d'un tel dipole. Par une étude de stabilité linéaire tridimensionnelle, on retrouve les bandes d'instabilité correspondant aux modes de Crow et elliptiques mais on observe également des modes oscillants moins instables avec des pics très larges. Les croissances transitoires des perturbations sur ce dipole, qui sont étudiées en calculant les perturbations optimales linéaires par une technique direct-adjoint, démontrent le rôle crucial de la région où l'étirement est maximal aux temps courts et du point hyperbolique aux temps intermédiaires. Des études sur la dynamique tridimensionnelle des tourbillons de sillage d'avion en fluide stratifié sont réalisées. L'instabilité elliptique n'est pratiquement pas affectée par des stratifications faible et modérée.

Lecture Notes in Physics

Page 1. An efficient wave interaction mechanism within a turbulent boundary layer Sabine Ortiz 1 ... more Page 1. An efficient wave interaction mechanism within a turbulent boundary layer Sabine Ortiz 1 , Emmanuel Deriat 2 ... Roy. Soc. Lond, A, 406, pp. 1-12 Deriat, E. (1988): Contributions ~ l'd~ude de la structure asymp~otique de la ~urbulence de couehe limite (Phd thesis Univ. ...

APS Division of Fluid Dynamics Meeting Abstracts, Nov 1, 2019

Bulletin of the American Physical Society, 2016

Anexo 19-Fotos ilustrativas das atividades extracurriculares dinamizadas pela professora orientad... more Anexo 19-Fotos ilustrativas das atividades extracurriculares dinamizadas pela professora orientadora, com a colaboração do Núcleo de Estágio .

Submitted for the DFD19 Meeting of The American Physical Society Nonnormal transient growth of tr... more Submitted for the DFD19 Meeting of The American Physical Society Nonnormal transient growth of triadic resonant internal gravity waves KEVIN HA, JEAN MARC CHOMAZ, Ecole Polytechnique, SABINE OR-TIZ, ENSTA-Triadic resonant instability is a key component in the understanding of the dissipation process of inertia gravity waves for geophysical applications like oceanic circulation, which is still incompletely understood. A key process in the oceanic circulation is the vertical mixing, which makes it possible for dense deep water to reach the surface. Global warming regulation by the ocean then depends on mechanisms controlling the vertical mixing of deepwater masses. It was recently proposed by Garrett & Kunze (2007) that the mixing results from the instabilities of internal gravity waves generated by interaction between the barotropic tide and bottom topography (continental shelf, underwater mount). The present work focuses on the energy approach of the triadic resonant instability and demonstrates that due to the nonnormality of the evolution operator, stable triadic resonant interactions result in a transient amplification of perturbation energy. Computations show that they can lead to a longer and more intense transient growth than unstable triads. Instead of being related to the differential growth of a stable and an unstable modes like for unstable triads, the transient growth of stable triads originates from the differential rotation (i.e. phase shift) of two stable eigenmodes.

Internal gravity waves propagate energy from the source and are important to understand the ocean... more Internal gravity waves propagate energy from the source and are important to understand the ocean mixing. We compute the 2D impulse response of a infinite internal wave by direct numerical simulation using an extremely extended computational domain with a resolution up to 16384 by 8192 and integration time (up to 300 Brunt Väisälä periode). Such extended domain and long time are necessary since the base flow is periodic in space and the impulse response has to converge on each ray onto the spatio-temporal Floquet mode. We observe the splitting of the impulse response into 3 different wave packets and show that each of them corresponds to a different branch of the triadic instability. Reanalysis of the triadic instability taking into account the detuning from the exact resonance allows us to show that the group velocity of each leading triads is the average of the group velocity of the two resonant waves. The small-scale wave packet then moves with the fluid where as the large scale mode has a group velocity comparable or larger than the base wave itself. We deduce from this impulse response the absolute and convective nature of each branch of the triadic instability and predict a selection of the instability mode strongly sensitive to the mean advection speed.

Springer Proceedings in Physics, 2009

ABSTRACT Mixing layers (sheared flows in homogeneous or stratified fluid) are present in many geo... more ABSTRACT Mixing layers (sheared flows in homogeneous or stratified fluid) are present in many geophysical contexts and may lead to turbulence and mixing. In several cases, mixing layers are known to exhibit the Kelvin-Helmholtz instability leading to the roll-up of spanwise vortices, the Kelvin-Helmholtz (KH) billows. This is an essentially two-dimensional (2D) process. In fact, in the homogeneous cases the Squire’s theorem implies that the most unstable mode is 2D. However, Squire’s theorem applies only for the exponentially growing perturbations that control the large time dynamics and is not valid for the transient dynamics at short time. Indeed, Iams et al.[1] have shown that, in the non-stratified case, the most amplified optimal perturbations for short times are three-dimensional (3D) and result from a cooperation between the lift-up and Orr mechanisms[2]. This provides a finite time mechanism for spanwise scale selection, scale that may persist at later times if nonlinearities are strong enough.

Physics of Fluids, 2002

In mixing-layers between two parallel streams of different densities, shear and gravity effects i... more In mixing-layers between two parallel streams of different densities, shear and gravity effects interplay; buoyancy acts as a restoring force and the Kelvin-Helmholtz mode is known to be stabilized by the stratification. If the density interface is sharp enough, two new instability modes, known as Holmboe modes, appear, propagating in opposite directions. This mechanism has been studied in the temporal instability framework. The present paper analyzes the associated spatial instability problem. It considers, in the Boussinesq approximation, two immiscible inviscid fluids with a piecewise linear broken-line velocity profile. We show how the classical scenario for transition between absolute and convective instability should be modified due to the presence of propagating waves. In the convective region, the spatial theory is relevant and the slowest propagating wave is shown to be the most spatially amplified, as suggested by intuition. Predictions of spatial linear theory are compared with mixing-layer ͓C. G. Koop and F. K. Browand, J. Fluid Mech. 93, 135 ͑1979͔͒ and exchange flow ͓G. Pawlak and L. Armi, J. Fluid Mech. 376, 1 ͑1999͔͒ experiments. The physical mechanism for Holmboe mode destabilization is analyzed via an asymptotic expansion that predicts the absolute instability domain at large Richardson number.

Springer Proceedings in Physics, 2009

The wake, which forms behind an aircraft due to its lift, is a pair of horizontal counter-rotatin... more The wake, which forms behind an aircraft due to its lift, is a pair of horizontal counter-rotating vortices propagating downwards. Depending on the atmospheric conditions, such dipole can persist over a long time or be rapidely destroyed. This vortex pair, in homogeneous fluids, is unstable with respect to three-dimensional perturbations. Crow [1] has discovered a long-wavelength instability, symmetric with respect

In this talk, we present a stability analysis of the Stuart vortices in the presence of an axial ... more In this talk, we present a stability analysis of the Stuart vortices in the presence of an axial flow by numerically solving the local stability equations derived by Lifschitz & Haimeri (1991). Deriving the criteria for wave vectors to be periodic upon their evolution around flow trajectories that are periodic in a plane perpendicular to the axial direction, we integrate the amplitude equations around periodic trajectories for periodic wave vectors. The elliptic and hyperbolic instabilites, which are present without the axial velocity, disappear beyond a threshold value for the axial velocity strength. Furthermore, a threshold axial velocity strength, above which a new centrifugal instability branch is present, is identified. A heuristic novel criterion, which reduces to the Leibovich & Stewartson (1983) criterion in the limit of an axisymmetric vortex, for centrifugal instability in a non-axisymmetric vortex with an axial flow is then proposed and validated.

Lecture Notes in Physics, 2010

We perform a study on the optimal perturbations developing on mixing layers. The basicly 2-dimens... more We perform a study on the optimal perturbations developing on mixing layers. The basicly 2-dimensional Kelvin-Helmoltz instability that develops in this type of flow is known to become unstable leading to the development of streamwise vortices and eventually turbulence. This process is essential in many geophysical and industrial flows, where it greatly influences mixing and dissipation. We explore different types

Physics of Fluids, 2009

This paper investigates the three-dimensional instabilities and the transient growth of perturbat... more This paper investigates the three-dimensional instabilities and the transient growth of perturbations on a counter-rotating vortex pair. The two dimensional base flow is obtained by a direct numerical simulation initialized by two Lamb-Oseen vortices that quickly adjust to a flow with elliptic vortices. In the present study, the Reynolds number, Re ⌫ = ⌫ / , with ⌫ the circulation of one vortex and the kinematic viscosity, is taken large enough for the quasi steady assumption to be valid. Both the direct linearized Navier-Stokes equation and its adjoint are solved numerically and used to investigate transient and long time dynamics. The transient dynamics is led by different regions of the flow, depending on the optimal time considered. At very short times compared to the advection time of the dipole, the dynamics is concentrated on the points of maximal strain of the base flow, located at the periphery of the vortex core. At intermediate times, depending on the symmetry of the perturbation, one of the hyperbolic stagnation points provides the optimal amplification by stretching of the perturbation vorticity as in the classical hyperbolic instability. The growth of both short time and intermediate time transient perturbations are non-or weakly dependent of the axial wavenumber whereas the long time behavior strongly selects narrow bands of wavenumbers. We show that, for all unstable spanwise wavenumbers, the transient dynamics last until the nondimensional time t = 2, during which the dipole has traveled twice the separation distance between vortices b. During that time, all the wavenumbers exhibit a transient growth of energy by a factor of 50, for the Reynolds number Re ⌫ = 2000. For time larger than t = 2, energy starts growing at a rate given by the standard temporal stability theory. For all wavenumbers and two Reynolds numbers, Re ⌫ = 2000 and Re ⌫ =10 5 , different instability branches have been computed using a high resolution Krylov method. At large Reynolds number, the computed Crow and elliptic instability branches are in excellent agreement with the inviscid theory ͓S.

Linear stability of the Stuart vortices in the presence of an axial flow is studied. 11 The local... more Linear stability of the Stuart vortices in the presence of an axial flow is studied. 11 The local stability equations derived by Lifschitz & Hameiri (Phys. Fluids A, vol. 3 12 (11), 1991, pp. 2644–2651) are rewritten for a three-component (3C) two-dimensional 13 (2D) base flow represented by a 2D streamfunction and an axial velocity that is a 14 function of the streamfunction. We show that the local perturbations that describe 15 an eigenmode of the flow should have wavevectors that are periodic upon their 16 evolution around helical flow trajectories that are themselves periodic once projected 17 on a plane perpendicular to the axial direction. Integrating the amplitude equations 18 around periodic trajectories for wavevectors that are also periodic, it is found that 19 the elliptic and hyperbolic instabilities, which are present without the axial velocity, 20 disappear beyond a threshold value for the axial velocity strength. Furthermore, a 21 threshold axial velocity strength, ab...

We report an investigation of the three-dimensional stability of an horizontal shear flow, the hy... more We report an investigation of the three-dimensional stability of an horizontal shear flow, the hyperbolic tangent velocity profile, in an inviscid, stably stratified fluid. A previous work by Deloncle et al.(2007) shows that the most unstable mode for this flow is two-dimensional. However, for strong stratification, the range of unstable vertical wavenumbers widens proportionally to the inverse of the Froude

Physical review. E, 2021

For the classical problem of the rotation of a solid, we show a somehow surprising behavior invol... more For the classical problem of the rotation of a solid, we show a somehow surprising behavior involving large transient growth of perturbation energy that occurs when the moment of inertia associated to the unstable axis approaches the moment of inertia of one of the two stable axes. In that case, small but finite perturbations around this stable axis may induce a total transfer of energy to the unstable axis, leading to relaxation oscillations where the stable and unstable manifolds of the unstable axis play the role of a separatrix, an edge state. For a fluid in solid-body rotation, a similar linear and nonlinear dynamics apply to the transfer of energy between three inertial waves respecting the triadic resonance condition. We show that the existence of large transient energy growth and of relaxation oscillations may be physically interpreted as in the case of a solid by the existence of two quadratic invariants, the energy and the helicity in the case of a rotating fluid. They occ...

Bulletin of the American Physical Society, Nov 19, 2006

Submitted for the DFD06 Meeting of The American Physical Society Three-dimensional instabilities ... more Submitted for the DFD06 Meeting of The American Physical Society Three-dimensional instabilities and transient growth of trailing vortices JEAN-MARC CHOMAZ, CLAIRE DONNADIEU, SABINE ORTIZ, PAUL BILLANT, LadHyX, CNRS-Ecole Polytechnique-An aircraft wake is made of counter-rotating vortices and is known to be affected by a long (Crow) and a short (elliptic) wavelength instabilities. Numerical investigations on the threedimensionnal instabilities and transient growth of such dipole are performed.By means of a three-dimensionnal linear stability analysis, we retrieve the instability bands corresponding to the Crow and elliptic modes but we also observe less unstable oscillatory modes with very broad peaks. The transient growth of this dipole, investigated by computing the optimal perturbations with a direct-adjoint 1 technique, demonstrates the crucial role of the region of maximal strain. Further investigations on the dynamics of trailing vortices in stratified fluids will be performed. Indeed, as such dipoles propagate downwards, they evolve under the influence of the stratification of the atmosphere.

HAL (Le Centre pour la Communication Scientifique Directe), Aug 27, 2007

An aircraft wake is made of counter-rotating vortices and is known to be affected by a long (Crow... more An aircraft wake is made of counter-rotating vortices and is known to be affected by a long (Crow) and a short (elliptic) wavelength instabilities. Numerical investigations on the three-dimensionnal instabilities and transient growth of such dipole are performed. By means of a three-dimensionnal linear stability analysis, we retrieve the instability bands corresponding to the Crow and elliptic modes but we also observe less unstable oscillatory modes with very broad peaks. The transient growth of perturbations on this dipole, investigated by computing the optimal linear perturbations with a direct-adjoint technique, demonstrates the crucial role of the region of maximal strain at short time and of the hyperbolic point at intermediate time. Investigations on the three-dimensionnal dynamics of trailing vortices in stratified fluids are performed. The elliptic instability is almost unaffected by weak and moderate stratifications. Résumé : Le sillage d'un avion est constitué d'un paire de tourbillons contra-rotatifs et est affectée par une instabilité à grande (Crow) et à petite (elliptique) longueur d'onde. On réalise des études numériques sur les instabilités tridimensionnelles d'un tel dipole. Par une étude de stabilité linéaire tridimensionnelle, on retrouve les bandes d'instabilité correspondant aux modes de Crow et elliptiques mais on observe également des modes oscillants moins instables avec des pics très larges. Les croissances transitoires des perturbations sur ce dipole, qui sont étudiées en calculant les perturbations optimales linéaires par une technique direct-adjoint, démontrent le rôle crucial de la région où l'étirement est maximal aux temps courts et du point hyperbolique aux temps intermédiaires. Des études sur la dynamique tridimensionnelle des tourbillons de sillage d'avion en fluide stratifié sont réalisées. L'instabilité elliptique n'est pratiquement pas affectée par des stratifications faible et modérée.

Lecture Notes in Physics

Page 1. An efficient wave interaction mechanism within a turbulent boundary layer Sabine Ortiz 1 ... more Page 1. An efficient wave interaction mechanism within a turbulent boundary layer Sabine Ortiz 1 , Emmanuel Deriat 2 ... Roy. Soc. Lond, A, 406, pp. 1-12 Deriat, E. (1988): Contributions ~ l'd~ude de la structure asymp~otique de la ~urbulence de couehe limite (Phd thesis Univ. ...

APS Division of Fluid Dynamics Meeting Abstracts, Nov 1, 2019

Bulletin of the American Physical Society, 2016

Anexo 19-Fotos ilustrativas das atividades extracurriculares dinamizadas pela professora orientad... more Anexo 19-Fotos ilustrativas das atividades extracurriculares dinamizadas pela professora orientadora, com a colaboração do Núcleo de Estágio .

Submitted for the DFD19 Meeting of The American Physical Society Nonnormal transient growth of tr... more Submitted for the DFD19 Meeting of The American Physical Society Nonnormal transient growth of triadic resonant internal gravity waves KEVIN HA, JEAN MARC CHOMAZ, Ecole Polytechnique, SABINE OR-TIZ, ENSTA-Triadic resonant instability is a key component in the understanding of the dissipation process of inertia gravity waves for geophysical applications like oceanic circulation, which is still incompletely understood. A key process in the oceanic circulation is the vertical mixing, which makes it possible for dense deep water to reach the surface. Global warming regulation by the ocean then depends on mechanisms controlling the vertical mixing of deepwater masses. It was recently proposed by Garrett & Kunze (2007) that the mixing results from the instabilities of internal gravity waves generated by interaction between the barotropic tide and bottom topography (continental shelf, underwater mount). The present work focuses on the energy approach of the triadic resonant instability and demonstrates that due to the nonnormality of the evolution operator, stable triadic resonant interactions result in a transient amplification of perturbation energy. Computations show that they can lead to a longer and more intense transient growth than unstable triads. Instead of being related to the differential growth of a stable and an unstable modes like for unstable triads, the transient growth of stable triads originates from the differential rotation (i.e. phase shift) of two stable eigenmodes.

Internal gravity waves propagate energy from the source and are important to understand the ocean... more Internal gravity waves propagate energy from the source and are important to understand the ocean mixing. We compute the 2D impulse response of a infinite internal wave by direct numerical simulation using an extremely extended computational domain with a resolution up to 16384 by 8192 and integration time (up to 300 Brunt Väisälä periode). Such extended domain and long time are necessary since the base flow is periodic in space and the impulse response has to converge on each ray onto the spatio-temporal Floquet mode. We observe the splitting of the impulse response into 3 different wave packets and show that each of them corresponds to a different branch of the triadic instability. Reanalysis of the triadic instability taking into account the detuning from the exact resonance allows us to show that the group velocity of each leading triads is the average of the group velocity of the two resonant waves. The small-scale wave packet then moves with the fluid where as the large scale mode has a group velocity comparable or larger than the base wave itself. We deduce from this impulse response the absolute and convective nature of each branch of the triadic instability and predict a selection of the instability mode strongly sensitive to the mean advection speed.

Springer Proceedings in Physics, 2009

ABSTRACT Mixing layers (sheared flows in homogeneous or stratified fluid) are present in many geo... more ABSTRACT Mixing layers (sheared flows in homogeneous or stratified fluid) are present in many geophysical contexts and may lead to turbulence and mixing. In several cases, mixing layers are known to exhibit the Kelvin-Helmholtz instability leading to the roll-up of spanwise vortices, the Kelvin-Helmholtz (KH) billows. This is an essentially two-dimensional (2D) process. In fact, in the homogeneous cases the Squire’s theorem implies that the most unstable mode is 2D. However, Squire’s theorem applies only for the exponentially growing perturbations that control the large time dynamics and is not valid for the transient dynamics at short time. Indeed, Iams et al.[1] have shown that, in the non-stratified case, the most amplified optimal perturbations for short times are three-dimensional (3D) and result from a cooperation between the lift-up and Orr mechanisms[2]. This provides a finite time mechanism for spanwise scale selection, scale that may persist at later times if nonlinearities are strong enough.

Physics of Fluids, 2002

In mixing-layers between two parallel streams of different densities, shear and gravity effects i... more In mixing-layers between two parallel streams of different densities, shear and gravity effects interplay; buoyancy acts as a restoring force and the Kelvin-Helmholtz mode is known to be stabilized by the stratification. If the density interface is sharp enough, two new instability modes, known as Holmboe modes, appear, propagating in opposite directions. This mechanism has been studied in the temporal instability framework. The present paper analyzes the associated spatial instability problem. It considers, in the Boussinesq approximation, two immiscible inviscid fluids with a piecewise linear broken-line velocity profile. We show how the classical scenario for transition between absolute and convective instability should be modified due to the presence of propagating waves. In the convective region, the spatial theory is relevant and the slowest propagating wave is shown to be the most spatially amplified, as suggested by intuition. Predictions of spatial linear theory are compared with mixing-layer ͓C. G. Koop and F. K. Browand, J. Fluid Mech. 93, 135 ͑1979͔͒ and exchange flow ͓G. Pawlak and L. Armi, J. Fluid Mech. 376, 1 ͑1999͔͒ experiments. The physical mechanism for Holmboe mode destabilization is analyzed via an asymptotic expansion that predicts the absolute instability domain at large Richardson number.

Springer Proceedings in Physics, 2009

The wake, which forms behind an aircraft due to its lift, is a pair of horizontal counter-rotatin... more The wake, which forms behind an aircraft due to its lift, is a pair of horizontal counter-rotating vortices propagating downwards. Depending on the atmospheric conditions, such dipole can persist over a long time or be rapidely destroyed. This vortex pair, in homogeneous fluids, is unstable with respect to three-dimensional perturbations. Crow [1] has discovered a long-wavelength instability, symmetric with respect

In this talk, we present a stability analysis of the Stuart vortices in the presence of an axial ... more In this talk, we present a stability analysis of the Stuart vortices in the presence of an axial flow by numerically solving the local stability equations derived by Lifschitz & Haimeri (1991). Deriving the criteria for wave vectors to be periodic upon their evolution around flow trajectories that are periodic in a plane perpendicular to the axial direction, we integrate the amplitude equations around periodic trajectories for periodic wave vectors. The elliptic and hyperbolic instabilites, which are present without the axial velocity, disappear beyond a threshold value for the axial velocity strength. Furthermore, a threshold axial velocity strength, above which a new centrifugal instability branch is present, is identified. A heuristic novel criterion, which reduces to the Leibovich & Stewartson (1983) criterion in the limit of an axisymmetric vortex, for centrifugal instability in a non-axisymmetric vortex with an axial flow is then proposed and validated.

Lecture Notes in Physics, 2010

We perform a study on the optimal perturbations developing on mixing layers. The basicly 2-dimens... more We perform a study on the optimal perturbations developing on mixing layers. The basicly 2-dimensional Kelvin-Helmoltz instability that develops in this type of flow is known to become unstable leading to the development of streamwise vortices and eventually turbulence. This process is essential in many geophysical and industrial flows, where it greatly influences mixing and dissipation. We explore different types

Physics of Fluids, 2009

This paper investigates the three-dimensional instabilities and the transient growth of perturbat... more This paper investigates the three-dimensional instabilities and the transient growth of perturbations on a counter-rotating vortex pair. The two dimensional base flow is obtained by a direct numerical simulation initialized by two Lamb-Oseen vortices that quickly adjust to a flow with elliptic vortices. In the present study, the Reynolds number, Re ⌫ = ⌫ / , with ⌫ the circulation of one vortex and the kinematic viscosity, is taken large enough for the quasi steady assumption to be valid. Both the direct linearized Navier-Stokes equation and its adjoint are solved numerically and used to investigate transient and long time dynamics. The transient dynamics is led by different regions of the flow, depending on the optimal time considered. At very short times compared to the advection time of the dipole, the dynamics is concentrated on the points of maximal strain of the base flow, located at the periphery of the vortex core. At intermediate times, depending on the symmetry of the perturbation, one of the hyperbolic stagnation points provides the optimal amplification by stretching of the perturbation vorticity as in the classical hyperbolic instability. The growth of both short time and intermediate time transient perturbations are non-or weakly dependent of the axial wavenumber whereas the long time behavior strongly selects narrow bands of wavenumbers. We show that, for all unstable spanwise wavenumbers, the transient dynamics last until the nondimensional time t = 2, during which the dipole has traveled twice the separation distance between vortices b. During that time, all the wavenumbers exhibit a transient growth of energy by a factor of 50, for the Reynolds number Re ⌫ = 2000. For time larger than t = 2, energy starts growing at a rate given by the standard temporal stability theory. For all wavenumbers and two Reynolds numbers, Re ⌫ = 2000 and Re ⌫ =10 5 , different instability branches have been computed using a high resolution Krylov method. At large Reynolds number, the computed Crow and elliptic instability branches are in excellent agreement with the inviscid theory ͓S.

Linear stability of the Stuart vortices in the presence of an axial flow is studied. 11 The local... more Linear stability of the Stuart vortices in the presence of an axial flow is studied. 11 The local stability equations derived by Lifschitz & Hameiri (Phys. Fluids A, vol. 3 12 (11), 1991, pp. 2644–2651) are rewritten for a three-component (3C) two-dimensional 13 (2D) base flow represented by a 2D streamfunction and an axial velocity that is a 14 function of the streamfunction. We show that the local perturbations that describe 15 an eigenmode of the flow should have wavevectors that are periodic upon their 16 evolution around helical flow trajectories that are themselves periodic once projected 17 on a plane perpendicular to the axial direction. Integrating the amplitude equations 18 around periodic trajectories for wavevectors that are also periodic, it is found that 19 the elliptic and hyperbolic instabilities, which are present without the axial velocity, 20 disappear beyond a threshold value for the axial velocity strength. Furthermore, a 21 threshold axial velocity strength, ab...

We report an investigation of the three-dimensional stability of an horizontal shear flow, the hy... more We report an investigation of the three-dimensional stability of an horizontal shear flow, the hyperbolic tangent velocity profile, in an inviscid, stably stratified fluid. A previous work by Deloncle et al.(2007) shows that the most unstable mode for this flow is two-dimensional. However, for strong stratification, the range of unstable vertical wavenumbers widens proportionally to the inverse of the Froude

Physical review. E, 2021

For the classical problem of the rotation of a solid, we show a somehow surprising behavior invol... more For the classical problem of the rotation of a solid, we show a somehow surprising behavior involving large transient growth of perturbation energy that occurs when the moment of inertia associated to the unstable axis approaches the moment of inertia of one of the two stable axes. In that case, small but finite perturbations around this stable axis may induce a total transfer of energy to the unstable axis, leading to relaxation oscillations where the stable and unstable manifolds of the unstable axis play the role of a separatrix, an edge state. For a fluid in solid-body rotation, a similar linear and nonlinear dynamics apply to the transfer of energy between three inertial waves respecting the triadic resonance condition. We show that the existence of large transient energy growth and of relaxation oscillations may be physically interpreted as in the case of a solid by the existence of two quadratic invariants, the energy and the helicity in the case of a rotating fluid. They occ...