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Paul Billant - One of the best experts on this subject based on the ideXlab platform.
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stability of an isolated pancake vortex in continuously stratified Rotating Fluids
Journal of Fluid Mechanics, 2016Co-Authors: Eunok Yim, Paul Billant, Claire MenesguenAbstract:This paper investigates the stability of an axisymmetric pancake vortex with Gaussian angular velocity in radial and vertical directions in a continuously stratified-Rotating fluid. The different instabilities are determined as a function of the Rossby number Ro, Froude number Fh, Reynolds number Re and aspect ratio α. Centrifugal instability is not significantly different from the case of a columnar vortex due to its short-wavelength nature: it is dominant when the absolute Rossby number |Ro| is large and is stabilized for small and moderate |Ro| when the generalized Rayleigh discriminant is positive everywhere. The Gent–McWilliams instability, also known as internal instability, is then dominant for the azimuthal wavenumber m=1 when the Burger number Bu=α2Ro2/(4F2h) is larger than unity. When Bu≲0.7Ro+0.1, the Gent–McWilliams instability changes into a mixed baroclinic–Gent–McWilliams instability. Shear instability for m=2 exists when Fh/α is below a threshold depending on Ro. This condition is shown to come from confinement effects along the vertical. Shear instability transforms into a mixed baroclinic–shear instability for small Bu. The main energy source for both baroclinic–shear and baroclinic–Gent–McWilliams instabilities is the potential energy of the base flow instead of the kinetic energy for shear and Gent–McWilliams instabilities. The growth rates of these four instabilities depend mostly on Fh/α and Ro. Baroclinic instability develops when Fh/α|1+1/Ro|≳1.46 in qualitative agreement with the analytical predictions for a bounded vortex with angular velocity slowly varying along the vertical.
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On the mechanism of the Gent–McWilliams instability of a columnar vortex in stratified Rotating Fluids
Journal of Fluid Mechanics, 2015Co-Authors: Eunok Yim, Paul BillantAbstract:In stably stratified and Rotating Fluids, an axisymmetric columnar vortex can be unstable to a special instability with an azimuthal wavenumber m = 1 which bends and slices the vortex into pancake vortices (Gent & McWilliams Geophys. Astrophys. Fluid Dyn., vol. 35 (1-4), 1986, pp. 209-233). This bending instability, called the 'Gent-McWilliams instability' herein, is distinct from the shear, centrifugal or radiative instabilities. The goals of the paper are to better understand the origin and properties of this instability and to explain why it operates only in stratified Rotating Fluids. Both numerical and asymptotic stability analyses of several velocity profiles have been performed for wide ranges of Froude number Fr-h = Omega(0)/N and Rossby number R-0 = 2 Omega(0)/f, where Omega(0) is the angular velocity on the vortex axis, N the Brunt-Vaisala frequency and f the Coriolis parameter. Numerical analyses restricted to the centrifugally stable range show that the maximum growth rate of the Gent-McWilliams instability increases with R-0 and is independent of Frh for Fr-h 1, the maximum growth rate decreases dramatically with Fr-h. Long axial wavelength asymptotic analyses for isolated vortices prove that the Gent-McWilliams instability is due to the destabilization of the long-wavelength bending mode by a critical layer at the radius r(c) where the angular velocity Omega is equal to the frequency omega : Omega(r(c)) = omega. A necessary and sufficient instability condition valid for long wavelengths, finite Rossby number and Fr-h 0 is necessary but not sufficient because the destabilizing effect of the critical layer r(c) is strongly reduced by a second stabilizing critical layer r(c2) existing at the radius where the angular velocity is equal to the Brunt-Vaisala frequency. For non-isolated vortices, numerical results show that only finite axial wavenumbers are unstable to the Gent-McWilliams instability.
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Towards a theory for vortex filaments in stratified-Rotating Fluids
Fluid Dynamics Research, 2014Co-Authors: Paul Billant, Axel Deloncle, Jean-marc Chomaz, Pantxika OtheguyAbstract:In inviscid Fluids with uniform density, it is common to idealize three-dimensional vortex tubes by filaments (i.e., single lines of an infinitesimal cross section). Thanks to the Kelvin and Helmholtz theorems, it is known that these vortex filaments are transported with the fluid and their circulation is conserved. The induced motions can be computed by the Biot–Savart law, with an appropriate cut off in the integral to avoid singularity. Hence, this approach allows one to model the linear or nonlinear dynamics of vortex flows. A priori, vortex filaments cannot be used in density-stratified and Rotating Fluids since the circulation is not conserved and the vortex lines are not material lines. However, in this paper we review a theory that is equivalent to vortex filaments. It is based on matched asymptotic expansions for small vortex-core size, weak curvature, and small vortex displacements. The resulting stability equations are formally identical to those of vortex filaments in homogeneous Fluids. However, striking differences between homogeneous and stratified-Rotating Fluids exist, such as the reversal of the self-induced motion for strong stratification or complex self-induction for moderate stratification due to the presence of critical points. The three-dimensional linear stability of vertical vortex pairs and vortex arrays (Karman street, double symmetric row) in stratified and Rotating Fluids has been investigated using this analytical approach. The results are in very good agreement with the results of direct numerical stability analyses of smooth vortex configurations. Possible extensions to include nonlinear and baroclinic effects are briefly discussed.
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zigzag instability of vortex pairs in stratified and Rotating Fluids part 1 general stability equations
Journal of Fluid Mechanics, 2010Co-Authors: Paul BillantAbstract:In stratified and Rotating Fluids, pairs of columnar vertical vortices are subjected to three-dimensional bending instabilities known as the zigzag instability or as the tall-column instability in the quasi-geostrophic limit. This paper presents a general asymptotic theory for these instabilities. The equations governing the interactions between the strain and the slow bending waves of each vortex column in stratified and Rotating Fluids are derived for long vertical wavelength and when the two vortices are well separated, i.e. when the radii R of the vortex cores are small compared to the vortex separation distance b . These equations have the same form as those obtained for vortex filaments in homogeneous Fluids except that the expressions of the mutual-induction and self-induction functions are different. A key difference is that the sign of the self-induction function is reversed compared to homogeneous Fluids when the fluid is strongly stratified: | max | N (where N is the Brunt–Vaisala frequency and max the maximum angular velocity of the vortex) for any vortex profile and magnitude of the planetary rotation. Physically, this means that slow bending waves of a vortex rotate in the same direction as the flow inside the vortex when the fluid is stratified-Rotating in contrast to homogeneous Fluids. When the stratification is weaker, i.e. | max | > N , the self-induction function is complex because the bending waves are damped by a viscous critical layer at the radial location where the angular velocity of the vortex is equal to the Brunt–Vaisala frequency. In contrast to previous theories, which apply only to strongly stratified non-Rotating Fluids, the present theory is valid for any planetary rotation rate and when the strain is smaller than the Brunt–Vaisala frequency: Γ/(2π b 2 ) ≪ N , where Γ is the vortex circulation. Since the strain is small, this condition is met across a wide range of stratification: from weakly to strongly stratified Fluids. The theory is further generalized formally to any basic flow made of an arbitrary number of vortices in stratified and Rotating Fluids. Viscous and diffusive effects are also taken into account at leading order in Reynolds number when there is no critical layer. In Part 2 (Billant et al ., J. Fluid Mech ., 2010, doi:10.1017/S002211201000282X), the stability of vortex pairs will be investigated using the present theory and the predictions will be shown to be in very good agreement with the results of direct numerical stability analyses. The existence of the zigzag instability and the distinctive stability properties of vortex pairs in stratified and Rotating Fluids compared to homogeneous Fluids will be demonstrated to originate from the sign reversal of the self-induction function.
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zigzag instability of vortex pairs in stratified and Rotating Fluids part 2 analytical and numerical analyses
Journal of Fluid Mechanics, 2010Co-Authors: Paul Billant, Axel Deloncle, Jean-marc Chomaz, Pantxika OtheguyAbstract:The three-dimensional stability of vertical vortex pairs in stratified and Rotating Fluids is investigated using the analytical approach established in Part 1 and the predictions are compared to the results of previous direct numerical stability analyses for pairs of co-Rotating equal-strength Lamb–Oseen vortices and to new numerical analyses for equal-strength counter-Rotating vortex pairs. A very good agreement between theoretical and numerical results is generally found, thereby providing a comprehensive description of the zigzag instability. Co-Rotating and counter-Rotating vortex pairs are most unstable to the zigzag instability when the Froude number F h = Γ/(2π R 2 N ) (where Γ is the vortex circulation, R the vortex radius and N the Brunt–Vaisala frequency) is lower than unity independently of the Rossby number Ro = Γ/(4π R 2 Ω b ) (Ω b is the planetary rotation rate). In this range, the maximum growth rate is proportional to the strain Γ/(2π b 2 ) ( b is the separation distance between the vortices) and is almost independent of F h and Ro . The most amplified wavelength scales like F h b when the Rossby number is large and like F h b /| Ro | when | Ro | ≪ 1, in agreement with previous results. While the zigzag instability always bends equal-strength co-Rotating vortex pairs in a symmetric way, the instability is only quasi-antisymmetric for finite Ro for equal-strength counter-Rotating vortex pairs because the cyclonic vortex is less bent than the anticyclonic vortex. The theory is less accurate for co-Rotating vortex pairs around Ro ≈ −2 because the bending waves rotate very slowly for long wavelength. The discrepancy can be fully resolved by taking into account higher-order three-dimensional effects. When F h is increased above unity, the growth rate of the zigzag instability is strongly reduced because the bending waves of each vortex are damped by a critical layer at the radius where the angular velocity of the vortex is equal to the Brunt–Vaisala frequency. The zigzag instability, however, continues to exist and is dominant up to a critical Froude number, which mostly depends on the Rossby number. Above this threshold, equal-strength co-Rotating vortex pairs are stable with respect to long-wavelength bending disturbances whereas equal-strength counter-Rotating vortex pairs become unstable to a quasi-symmetric instability resembling the Crow instability in homogeneous Fluids. However, its growth rate is lower than in homogeneous Fluids because of the damping by the critical layer. The structure of the critical layer obtained in the computations is in excellent agreement with the theoretical solution. Physically, the different stability properties of vortex pairs in stratified and Rotating Fluids compared to homogeneous Fluids are shown to come from the reversal of the direction of the self-induced motion of bent vortices.
Pantxika Otheguy - One of the best experts on this subject based on the ideXlab platform.
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Towards a theory for vortex filaments in stratified-Rotating Fluids
Fluid Dynamics Research, 2014Co-Authors: Paul Billant, Axel Deloncle, Jean-marc Chomaz, Pantxika OtheguyAbstract:In inviscid Fluids with uniform density, it is common to idealize three-dimensional vortex tubes by filaments (i.e., single lines of an infinitesimal cross section). Thanks to the Kelvin and Helmholtz theorems, it is known that these vortex filaments are transported with the fluid and their circulation is conserved. The induced motions can be computed by the Biot–Savart law, with an appropriate cut off in the integral to avoid singularity. Hence, this approach allows one to model the linear or nonlinear dynamics of vortex flows. A priori, vortex filaments cannot be used in density-stratified and Rotating Fluids since the circulation is not conserved and the vortex lines are not material lines. However, in this paper we review a theory that is equivalent to vortex filaments. It is based on matched asymptotic expansions for small vortex-core size, weak curvature, and small vortex displacements. The resulting stability equations are formally identical to those of vortex filaments in homogeneous Fluids. However, striking differences between homogeneous and stratified-Rotating Fluids exist, such as the reversal of the self-induced motion for strong stratification or complex self-induction for moderate stratification due to the presence of critical points. The three-dimensional linear stability of vertical vortex pairs and vortex arrays (Karman street, double symmetric row) in stratified and Rotating Fluids has been investigated using this analytical approach. The results are in very good agreement with the results of direct numerical stability analyses of smooth vortex configurations. Possible extensions to include nonlinear and baroclinic effects are briefly discussed.
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zigzag instability of vortex pairs in stratified and Rotating Fluids part 2 analytical and numerical analyses
Journal of Fluid Mechanics, 2010Co-Authors: Paul Billant, Axel Deloncle, Jean-marc Chomaz, Pantxika OtheguyAbstract:The three-dimensional stability of vertical vortex pairs in stratified and Rotating Fluids is investigated using the analytical approach established in Part 1 and the predictions are compared to the results of previous direct numerical stability analyses for pairs of co-Rotating equal-strength Lamb–Oseen vortices and to new numerical analyses for equal-strength counter-Rotating vortex pairs. A very good agreement between theoretical and numerical results is generally found, thereby providing a comprehensive description of the zigzag instability. Co-Rotating and counter-Rotating vortex pairs are most unstable to the zigzag instability when the Froude number F h = Γ/(2π R 2 N ) (where Γ is the vortex circulation, R the vortex radius and N the Brunt–Vaisala frequency) is lower than unity independently of the Rossby number Ro = Γ/(4π R 2 Ω b ) (Ω b is the planetary rotation rate). In this range, the maximum growth rate is proportional to the strain Γ/(2π b 2 ) ( b is the separation distance between the vortices) and is almost independent of F h and Ro . The most amplified wavelength scales like F h b when the Rossby number is large and like F h b /| Ro | when | Ro | ≪ 1, in agreement with previous results. While the zigzag instability always bends equal-strength co-Rotating vortex pairs in a symmetric way, the instability is only quasi-antisymmetric for finite Ro for equal-strength counter-Rotating vortex pairs because the cyclonic vortex is less bent than the anticyclonic vortex. The theory is less accurate for co-Rotating vortex pairs around Ro ≈ −2 because the bending waves rotate very slowly for long wavelength. The discrepancy can be fully resolved by taking into account higher-order three-dimensional effects. When F h is increased above unity, the growth rate of the zigzag instability is strongly reduced because the bending waves of each vortex are damped by a critical layer at the radius where the angular velocity of the vortex is equal to the Brunt–Vaisala frequency. The zigzag instability, however, continues to exist and is dominant up to a critical Froude number, which mostly depends on the Rossby number. Above this threshold, equal-strength co-Rotating vortex pairs are stable with respect to long-wavelength bending disturbances whereas equal-strength counter-Rotating vortex pairs become unstable to a quasi-symmetric instability resembling the Crow instability in homogeneous Fluids. However, its growth rate is lower than in homogeneous Fluids because of the damping by the critical layer. The structure of the critical layer obtained in the computations is in excellent agreement with the theoretical solution. Physically, the different stability properties of vortex pairs in stratified and Rotating Fluids compared to homogeneous Fluids are shown to come from the reversal of the direction of the self-induced motion of bent vortices.
Jean-marc Chomaz - One of the best experts on this subject based on the ideXlab platform.
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Towards a theory for vortex filaments in stratified-Rotating Fluids
Fluid Dynamics Research, 2014Co-Authors: Paul Billant, Axel Deloncle, Jean-marc Chomaz, Pantxika OtheguyAbstract:In inviscid Fluids with uniform density, it is common to idealize three-dimensional vortex tubes by filaments (i.e., single lines of an infinitesimal cross section). Thanks to the Kelvin and Helmholtz theorems, it is known that these vortex filaments are transported with the fluid and their circulation is conserved. The induced motions can be computed by the Biot–Savart law, with an appropriate cut off in the integral to avoid singularity. Hence, this approach allows one to model the linear or nonlinear dynamics of vortex flows. A priori, vortex filaments cannot be used in density-stratified and Rotating Fluids since the circulation is not conserved and the vortex lines are not material lines. However, in this paper we review a theory that is equivalent to vortex filaments. It is based on matched asymptotic expansions for small vortex-core size, weak curvature, and small vortex displacements. The resulting stability equations are formally identical to those of vortex filaments in homogeneous Fluids. However, striking differences between homogeneous and stratified-Rotating Fluids exist, such as the reversal of the self-induced motion for strong stratification or complex self-induction for moderate stratification due to the presence of critical points. The three-dimensional linear stability of vertical vortex pairs and vortex arrays (Karman street, double symmetric row) in stratified and Rotating Fluids has been investigated using this analytical approach. The results are in very good agreement with the results of direct numerical stability analyses of smooth vortex configurations. Possible extensions to include nonlinear and baroclinic effects are briefly discussed.
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zigzag instability of vortex pairs in stratified and Rotating Fluids part 2 analytical and numerical analyses
Journal of Fluid Mechanics, 2010Co-Authors: Paul Billant, Axel Deloncle, Jean-marc Chomaz, Pantxika OtheguyAbstract:The three-dimensional stability of vertical vortex pairs in stratified and Rotating Fluids is investigated using the analytical approach established in Part 1 and the predictions are compared to the results of previous direct numerical stability analyses for pairs of co-Rotating equal-strength Lamb–Oseen vortices and to new numerical analyses for equal-strength counter-Rotating vortex pairs. A very good agreement between theoretical and numerical results is generally found, thereby providing a comprehensive description of the zigzag instability. Co-Rotating and counter-Rotating vortex pairs are most unstable to the zigzag instability when the Froude number F h = Γ/(2π R 2 N ) (where Γ is the vortex circulation, R the vortex radius and N the Brunt–Vaisala frequency) is lower than unity independently of the Rossby number Ro = Γ/(4π R 2 Ω b ) (Ω b is the planetary rotation rate). In this range, the maximum growth rate is proportional to the strain Γ/(2π b 2 ) ( b is the separation distance between the vortices) and is almost independent of F h and Ro . The most amplified wavelength scales like F h b when the Rossby number is large and like F h b /| Ro | when | Ro | ≪ 1, in agreement with previous results. While the zigzag instability always bends equal-strength co-Rotating vortex pairs in a symmetric way, the instability is only quasi-antisymmetric for finite Ro for equal-strength counter-Rotating vortex pairs because the cyclonic vortex is less bent than the anticyclonic vortex. The theory is less accurate for co-Rotating vortex pairs around Ro ≈ −2 because the bending waves rotate very slowly for long wavelength. The discrepancy can be fully resolved by taking into account higher-order three-dimensional effects. When F h is increased above unity, the growth rate of the zigzag instability is strongly reduced because the bending waves of each vortex are damped by a critical layer at the radius where the angular velocity of the vortex is equal to the Brunt–Vaisala frequency. The zigzag instability, however, continues to exist and is dominant up to a critical Froude number, which mostly depends on the Rossby number. Above this threshold, equal-strength co-Rotating vortex pairs are stable with respect to long-wavelength bending disturbances whereas equal-strength counter-Rotating vortex pairs become unstable to a quasi-symmetric instability resembling the Crow instability in homogeneous Fluids. However, its growth rate is lower than in homogeneous Fluids because of the damping by the critical layer. The structure of the critical layer obtained in the computations is in excellent agreement with the theoretical solution. Physically, the different stability properties of vortex pairs in stratified and Rotating Fluids compared to homogeneous Fluids are shown to come from the reversal of the direction of the self-induced motion of bent vortices.
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Zigzag instability of vortex pairs in stratified and Rotating Fluids. Part 2. Analytical and numerical analyses
Journal of Fluid Mechanics, 2010Co-Authors: P. Billant, Jean-marc Chomaz, A. Deloncle, P. OtheguyAbstract:The three-dimensional stability of vertical vortex pairs in stratified and Rotating Fluids is investigated using the analytical approach established in Part 1 and the predictions are compared to the results of previous direct numerical stability analyses for pairs of co-Rotating equal-strength Lamb-Oseen vortices and to new numerical analyses for equal-strength counter-Rotating vortex pairs. A very good agreement between theoretical and numerical results is generally found, thereby providing a comprehensive description of the zigzag instability. Co-Rotating and counter-Rotating vortex pairs are most unstable to the zigzag instability when the Froude number Fh =/(2pR 2N) (where is the vortex circulation, R the vortex radius and N the Brunt-Väiäsälä frequency) is lower than unity independently of the Rossby number Ro =/(4pR2?b) (?b is the planetary rotation rate). In this range, the maximum growth rate is proportional to the strain/(2pb2) (b is the separation distance between the vortices) and is almost independent of F h and Ro. The most amplified wavelength scales like Fhb when the Rossby number is large and like Fhb/|Ro| when |Ro| 1, in agreement with previous results. While the zigzag instability always bends equal-strength co-Rotating vortex pairs in a symmetric way, the instability is only quasi-antisymmetric for finite Ro for equal-strength counter-Rotating vortex pairs because the cyclonic vortex is less bent than the anticyclonic vortex. The theory is less accurate for co-Rotating vortex pairs around Ro 2 because the bending waves rotate very slowly for long wavelength. The discrepancy can be fully resolved by taking into account higher-order three-dimensional effects. When Fh is increased above unity, the growth rate of the zigzag instability is strongly reduced because the bending waves of each vortex are damped by a critical layer at the radius where the angular velocity of the vortex is equal to the Brunt-Visl frequency. The zigzag instability, however, continues to exist and is dominant up to a critical Froude number, which mostly depends on the Rossby number. Above this threshold, equal-strength co-Rotating vortex pairs are stable with respect to long-wavelength bending disturbances whereas equal-strength counter-Rotating vortex pairs become unstable to a quasi-symmetric instability resembling the Crow instability in homogeneous Fluids. However, its growth rate is lower than in homogeneous Fluids because of the damping by the critical layer. The structure of the critical layer obtained in the computations is in excellent agreement with the theoretical solution. Physically, the different stability properties of vortex pairs in stratified and Rotating Fluids compared to homogeneous Fluids are shown to come from the reversal of the direction of the self-induced motion of bent vortices. © 2010 Cambridge University Press.
P. Otheguy - One of the best experts on this subject based on the ideXlab platform.
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Zigzag instability of vortex pairs in stratified and Rotating Fluids. Part 2. Analytical and numerical analyses
Journal of Fluid Mechanics, 2010Co-Authors: P. Billant, Jean-marc Chomaz, A. Deloncle, P. OtheguyAbstract:The three-dimensional stability of vertical vortex pairs in stratified and Rotating Fluids is investigated using the analytical approach established in Part 1 and the predictions are compared to the results of previous direct numerical stability analyses for pairs of co-Rotating equal-strength Lamb-Oseen vortices and to new numerical analyses for equal-strength counter-Rotating vortex pairs. A very good agreement between theoretical and numerical results is generally found, thereby providing a comprehensive description of the zigzag instability. Co-Rotating and counter-Rotating vortex pairs are most unstable to the zigzag instability when the Froude number Fh =/(2pR 2N) (where is the vortex circulation, R the vortex radius and N the Brunt-Väiäsälä frequency) is lower than unity independently of the Rossby number Ro =/(4pR2?b) (?b is the planetary rotation rate). In this range, the maximum growth rate is proportional to the strain/(2pb2) (b is the separation distance between the vortices) and is almost independent of F h and Ro. The most amplified wavelength scales like Fhb when the Rossby number is large and like Fhb/|Ro| when |Ro| 1, in agreement with previous results. While the zigzag instability always bends equal-strength co-Rotating vortex pairs in a symmetric way, the instability is only quasi-antisymmetric for finite Ro for equal-strength counter-Rotating vortex pairs because the cyclonic vortex is less bent than the anticyclonic vortex. The theory is less accurate for co-Rotating vortex pairs around Ro 2 because the bending waves rotate very slowly for long wavelength. The discrepancy can be fully resolved by taking into account higher-order three-dimensional effects. When Fh is increased above unity, the growth rate of the zigzag instability is strongly reduced because the bending waves of each vortex are damped by a critical layer at the radius where the angular velocity of the vortex is equal to the Brunt-Visl frequency. The zigzag instability, however, continues to exist and is dominant up to a critical Froude number, which mostly depends on the Rossby number. Above this threshold, equal-strength co-Rotating vortex pairs are stable with respect to long-wavelength bending disturbances whereas equal-strength counter-Rotating vortex pairs become unstable to a quasi-symmetric instability resembling the Crow instability in homogeneous Fluids. However, its growth rate is lower than in homogeneous Fluids because of the damping by the critical layer. The structure of the critical layer obtained in the computations is in excellent agreement with the theoretical solution. Physically, the different stability properties of vortex pairs in stratified and Rotating Fluids compared to homogeneous Fluids are shown to come from the reversal of the direction of the self-induced motion of bent vortices. © 2010 Cambridge University Press.
Nader Masmoudi - One of the best experts on this subject based on the ideXlab platform.
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Stability of oscillating boundary layers in Rotating Fluids
Annales Scientifiques de l'École Normale Supérieure, 2008Co-Authors: Nader Masmoudi, Frédéric RoussetAbstract:We prove the linear and non-linear stability of oscillating Ekman boundary layers for Rotating Fluids in the so-called ill-prepared case under a spectral hypothesis. Here, we deal with the case where the viscosity and the Rossby number are both equal to epsilon. This study generalizes the study of [23] where a smallness condition was imposed and the study of [26] where the well-prepared case was treated.
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ekman layers of Rotating Fluids the case of general initial data
Communications on Pure and Applied Mathematics, 2000Co-Authors: Nader MasmoudiAbstract:In this paper we describe the weak solutions of the Navier-Stokes equations with a large Coriolis term as the Rossby and the Ekman numbers go to zero in a special domain with various boundary conditions. This work extends the results (in the case of well-prepared initial data) in [19, 30]. © 2000 John Wiley & Sons, Inc.
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the euler limit of the navier stokes equations and Rotating Fluids with boundary
Archive for Rational Mechanics and Analysis, 1998Co-Authors: Nader MasmoudiAbstract:In this paper we study the convergence of weak solutions of the Navier-Stokes equations in some particular domains, with different horizontal and vertical viscosities, when they go to zero with different speeds. The difficulty here comes from the Dirichlet boundary conditions. Precisely we show that if the ratio of the vertical viscosity to the horizontal viscosity also goes to zero, then the solutions converge to the solution of the Euler system. We study the same limit for Rotating Fluids with Rossby number also going to zero.
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ekman layers of Rotating Fluids the case of well prepared initial data
Communications in Partial Differential Equations, 1997Co-Authors: E Grenier, Nader MasmoudiAbstract:In this paper we study the convergence of weak solutions of the Navier Stokes equations with a large Coriolis term as the Rossby and Ekman numbers go to zero, and in particular the so called Ekman boundary layers, and justify some classical expansions in geophysical fluid dynamics (see [14], chapter 4).
