scholarly journals Stability of a Gaussian pancake vortex in a stratified fluid

2013 ◽  
Vol 718 ◽  
pp. 457-480 ◽  
Author(s):  
M. Eletta Negretti ◽  
Paul Billant
Keyword(s):  
Aspect Ratio ◽  
Froude Number ◽  
Gravitational Instability ◽  
Shear Instability ◽  
Vertical Shear ◽  
Minimum Thickness ◽  
Horizontal Size ◽  
Vertical Thickness ◽  
3 Omega ◽  
Pancake Vortex

AbstractVortices in stably stratified fluids generally have a pancake shape with a small vertical thickness compared with their horizontal size. In order to understand what mechanism determines their minimum thickness, the linear stability of an axisymmetric pancake vortex is investigated as a function of its aspect ratio $\alpha $, the horizontal Froude number ${F}_{h} $, the Reynolds number $\mathit{Re}$ and the Schmidt number $\mathit{Sc}$. The vertical vorticity profile of the base state is chosen to be Gaussian in both radial and vertical directions. The vortex is unstable when the aspect ratio is below a critical value, which scales with the Froude number: ${\alpha }_{c} \sim 1. 1{F}_{h} $ for sufficiently large Reynolds numbers. The most unstable perturbation has an azimuthal wavenumber either $m= 0$, $\vert m\vert = 1$ or $\vert m\vert = 2$ depending on the control parameters. We show that the threshold corresponds to the appearance of gravitationally unstable regions in the vortex core due to the thermal wind balance. The Richardson criterion for shear instability based on the vertical shear is never satisfied alone. The dominance of the gravitational instability over the shear instability is shown to hold for a general class of pancake vortices with angular velocity of the form $\tilde {\Omega } (r, z)= \Omega (r)f(z)$ provided that $r\partial \Omega / \partial r\lt 3\Omega $ everywhere. Finally, the growth rate and azimuthal wavenumber selection of the gravitational instability are accounted well by considering an unstably stratified viscous and diffusive layer in solid body rotation with a parabolic density gradient.

Analogies and differences between the stability of an isolated pancake vortex and a columnar vortex in stratified fluid

2016 ◽  
Vol 796 ◽  
pp. 732-766 ◽  
Author(s):  
Eunok Yim ◽  
Paul Billant
Keyword(s):  
Reynolds Number ◽  
Aspect Ratio ◽  
Froude Number ◽  
Gravitational Instability ◽  
Baroclinic Instability ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Shear Instability ◽  
Axisymmetric Mode ◽  
Columnar Vortex

In order to understand the dynamics of pancake shaped vortices in stably stratified fluids, we perform a linear stability analysis of an axisymmetric vortex with Gaussian angular velocity in both the radial and axial directions with an aspect ratio of ${\it\alpha}$. The results are compared to those for a columnar vortex (${\it\alpha}=\infty$) in order to identify the instabilities. Centrifugal instability occurs when $\mathscr{R}>c(m)$ where $\mathscr{R}=ReF_{h}^{2}$ is the buoyancy Reynolds number, $F_{h}$ the Froude number, $Re$ the Reynolds number and $c(m)$ a constant which differs for the three unstable azimuthal wavenumbers $m=0,1,2$. The maximum growth rate depends mostly on $\mathscr{R}$ and is almost independent of the aspect ratio ${\it\alpha}$. For sufficiently large buoyancy Reynolds number, the axisymmetric mode is the most unstable centrifugal mode whereas for moderate $\mathscr{R}$, the mode $m=1$ is the most unstable. Shear instability for $m=2$ develops only when $F_{h}\leqslant 0.5{\it\alpha}$. By considering the characteristics of shear instability for a columnar vortex with the same parameters, this condition is shown to be such that the vortex is taller than the minimum wavelength of shear instability in the columnar case. For larger Froude number $F_{h}\geqslant 1.5{\it\alpha}$, the isopycnals overturn and gravitational instability can operate. Just below this threshold, the azimuthal wavenumbers $m=1,2,3$ are unstable to baroclinic instability. A simple model shows that baroclinic instability develops only above a critical vertical Froude number $F_{h}/{\it\alpha}$ because of confinement effects.

scholarly journals High resolution parameter study of the vertical shear instability

2020 ◽  
Vol 499 (2) ◽  
pp. 1841-1853
Author(s):  
Natascha Manger ◽  
Hubert Klahr ◽  
Wilhelm Kley ◽  
Mario Flock
Keyword(s):  
Aspect Ratio ◽  
Large Scale ◽  
Dead Zone ◽  
Pressure Ratio ◽  
Theoretical Models ◽  
Shear Instability ◽  
Vertical Shear ◽  
Parameter Study ◽  
Aspect Ratios ◽  
Strong Scaling

ABSTRACT Theoretical models of protoplanetary discs have shown the vertical shear instability (VSI) to be a prime candidate to explain turbulence in the dead zone of the disc. However, simulations of the VSI have yet to show consistent levels of key disc turbulence parameters like the stress-to-pressure ratio α. We aim to reconcile these different values by performing a parameter study on the VSI with focus on the disc density gradient p and aspect ratio h = H/R. We use full 2π 3D simulations of the disc for chosen set of both parameters. All simulations are evolved for 1000 reference orbits, at a resolution of 18 cells per h. We find that the saturated stress-to-pressure ratio in our simulations is dependent on the disc aspect ratio with a strong scaling of α∝h2.6, in contrast to the traditional α model, where viscosity scales as ν∝αh2 with a constant α. We also observe consistent formation of large scale vortices across all investigated parameters. The vortices show uniformly aspect ratios of χ ≈ 10 and radial widths of approximately 1.5H. With our findings we can reconcile the different values reported for the stress-to-pressure ratio from both isothermal and full radiation hydrodynamics models, and show long-term evolution effects of the VSI that could aide in the formation of planetesimals.

scholarly journals Stability of an isolated pancake vortex in continuously stratified-rotating fluids

2016 ◽  
Vol 801 ◽  
pp. 508-553 ◽  
Author(s):  
Eunok Yim ◽  
Paul Billant ◽  
Claire Ménesguen
Keyword(s):  
Angular Velocity ◽  
Baroclinic Instability ◽  
Base Flow ◽  
Shear Instability ◽  
Vertical Shear ◽  
Rossby Number ◽  
Centrifugal Instability ◽  
Pancake Vortex ◽  
To Come ◽  
The Stability

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 $F_{h}$, Reynolds number $Re$ and aspect ratio ${\it\alpha}$. 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 $Bu={\it\alpha}^{2}Ro^{2}/(4F_{h}^{2})$ is larger than unity. When $Bu\lesssim 0.7Ro+0.1$, the Gent–McWilliams instability changes into a mixed baroclinic–Gent–McWilliams instability. Shear instability for $m=2$ exists when $F_{h}/{\it\alpha}$ 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 $F_{h}/{\it\alpha}$ and $Ro$. Baroclinic instability develops when $F_{h}/{\it\alpha}|1+1/Ro|\gtrsim 1.46$ in qualitative agreement with the analytical predictions for a bounded vortex with angular velocity slowly varying along the vertical.

scholarly journals Gravity Wave Emission by Shear Instability

2019 ◽  
Vol 49 (9) ◽  
pp. 2393-2406 ◽  
Author(s):  
Carsten Eden ◽  
Manita Chouksey ◽  
Dirk Olbers
Keyword(s):  
Gravity Wave ◽  
Single Layer ◽  
Internal Gravity Waves ◽  
Shear Instability ◽  
Vertical Shear ◽  
Wave Signal ◽  
Wave Emission ◽  
Balanced Flow ◽  
Single Layer Model

AbstractGravity wave emission by geostrophically balanced flow is diagnosed in numerical simulations of lateral and vertical shear instabilities. The diagnostic method in use allows for a separation of balanced flow and residual wave signal up to fourth order in the Rossby number (Ro). While evidence is found for a small but finite gravity wave emission from balanced flow in a single-layer model with large lateral shear and large Ro, a vertically resolved model with moderate velocity amplitudes appropriate to the interior ocean hardly shows any wave emission. Only when static instabilities generated by the shear instability of the balanced flow are allowed can a gravity wave signal similar to the ones reported in earlier studies be detected in the vertically resolved case. This result suggests a relatively small role of spontaneous wave emission in the classical sense of Lighthill radiation, and emphasizes the role of convective or symmetric instabilities during frontogenesis for the generation of internal gravity waves in the ocean and atmosphere.

scholarly journals Kelvin–Helmholtz Billows in the Eyewall of Hurricane Erin

2006 ◽  
Vol 134 (3) ◽  
pp. 1036-1038 ◽  
Author(s):  
Sim D. Aberson ◽  
Jeffrey B. Halverson
Keyword(s):  
Tropical Cyclone ◽  
High Altitude ◽  
Shear Instability ◽  
Vertical Shear ◽  
Vertically Aligned ◽  
Aircraft Pilots

Abstract A photograph of vertically aligned Kelvin–Helmholtz billows in the eastern eyewall of Hurricane Erin on 10 September 2001 is presented. The vertical shear instability in the horizontal winds necessary to produce the billows is confirmed with a high-altitude dropwindsonde observation. This shear instability is not known to be common in tropical cyclone eyewalls and is likely only in cases with a very large eyewall tilt. However, research and reconnaissance aircraft pilots need to be aware of the possibility of their existence, along with other types of hazardous conditions, in such rare circumstances.

Vertical shear-induced resonant triads in Keplerian discs

2019 ◽  
Vol 488 (3) ◽  
pp. 4207-4219 ◽  
Author(s):  
Yuri Shtemler ◽  
Michael Mond
Keyword(s):  
Temporal Evolution ◽  
Shear Instability ◽  
Vertical Shear ◽  
Class Iii ◽  
Radial Temperature ◽  
Linear Interaction ◽  
Linear Mode ◽  
Proposed Model ◽  
Non Linear ◽  
Resonant Triads

ABSTRACT The vertical-shear instability (VSI) is studied through weakly non-linear analysis of unmagnetized vertically isothermal thin Keplerian discs under small radial temperature gradients. Vertically global and radially local axisymmetric compressible perturbations are considered. The VSI excites three classes of quasi-resonant triads of non-linearly interacting modes characterized by distinct temporal evolution. Most of the triads belong to the two-mode regime of non-linear interaction. Such triads are comprised of one unstable non-linear mode that grows quasi-exponentially, and two other modes that practically decoupled from the former. The latter two modes perform non-linear oscillations around their either linear prototypes (class I) or respective initial values (class II). The rest of the resonant triads belong to class III where all three modes exhibit non-linear oscillations. The proposed model describes an intermediate non-linear stage of the VSI prior to its saturation.

Clustering of the resonant triads induced by vertical-shear instability in astrophysical discs

2020 ◽  
Vol 499 (3) ◽  
pp. 3222-3232
Author(s):  
Yuri Shtemler ◽  
Michael Mond
Keyword(s):  
Characteristic Frequency ◽  
Characteristic Time ◽  
Temperature Gradients ◽  
Shear Instability ◽  
Vertical Shear ◽  
Higher Dimension ◽  
Small Temperature ◽  
Common Mode ◽  
Vertical Speed ◽  
Resonant Triads

ABSTRACT Clustering of resonant triads that are induced by vertical-shear instability (VSI), driven by the combined effect of the vertical speed shear and small temperature gradients, is studied for vertically isothermal thin unmagnetized Keplerian discs. The authors’ recent study of isolated VSI resonant triads is extended to illustrate their clustering. The coupling conditions for two VSI resonant triads with one common mode are derived and generalized to higher dimension clustering. The clustering of two, three, and four triads connected via one common mode is numerically simulated. The numerical simulations demonstrate the chaotization of non-linear oscillations about the prototypes of the linearly stable modes with a growing cluster’s dimension that is accompanied by a decrease of the characteristic time of chaotization and an increase of the characteristic frequency of perturbations. The chaos associated with the VSI resonant clustering is believed to precede transition to sustainable turbulence in astrophysical discs.

Pebble accretion onto planets in turbulent discs

2018 ◽  
Vol 14 (S345) ◽  
pp. 237-238
Author(s):  
Wilhelm Kley ◽  
Giovanni Picogna ◽  
Moritz H. R. Stoll
Keyword(s):  
Gas Flow ◽  
Mass Range ◽  
Shear Instability ◽  
Accretion Discs ◽  
Vertical Shear ◽  
Turbulent Motion ◽  
Hydrodynamic Drag ◽  
Embedded Particles ◽  
Hydrodynamical Simulations

AbstractPlanets form in protoplanetary accretion discs around young protostars. These discs are driven by internal turbulence and the gas flow is not laminar but has stochastic components. For weakly ionised discs the turbulence can be generated purely hydrodynamically through the vertical shear instability (VSI). Embedded particles (dust/pebbles) experience a hydrodynamic drag and drift inward radially and are stirred up vertically by the turbulent motion of the disc. We study the accretion of particles onto a forming planet embedded in a VSI turbulent protoplanetary disc through a series of 3D hydrodynamical simulations for locally isothermal discs with embedded planets in the mass range from 5 to 100 Earth masses (M2295).

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