Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-24T12:34:08.541Z Has data issue: false hasContentIssue false
  • English
  • Français

Modal stability analysis of arrays of stably stratified vortices

Published online by Cambridge University Press:  17 December 2020

Yuji Hattori Open the ORCID record for Yuji Hattori [Opens in a new window] ,
Shota Suzuki ,
Makoto Hirota  and
Manish Khandelwal
Yuji Hattori*
Affiliation:
Institute of Fluid Science, Tohoku University, Sendai980-8577, Japan
Shota Suzuki
Affiliation:
Institute of Fluid Science, Tohoku University, Sendai980-8577, Japan Graduate School of Information Sciences, Tohoku University, Sendai980-8579, Japan
Makoto Hirota
Affiliation:
Institute of Fluid Science, Tohoku University, Sendai980-8577, Japan
Manish Khandelwal
Affiliation:
Department of Mathematics, Indira Gandhi National Tribal University, Amarkantak484887, India
*
†Email address for correspondence: hattori@ifs.tohoku.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

The linear stability of a periodic array of vortices in stratified fluid is studied by modal stability analysis. Two base flows are considered: the two-dimensional Taylor–Green vortices and the Stuart vortices. In the case of the two-dimensional Taylor–Green vortices, four types of instability are identified: the elliptic instability, the pure hyperbolic instability, the strato-hyperbolic instability and the mixed hyperbolic instability, which is a mixture of the pure hyperbolic and the strato-hyperbolic instabilities. Although the pure hyperbolic instability is most unstable for the non-stratified case, it is surpassed by the strato-hyperbolic instability and the mixed hyperbolic instability for the stratified case. The strato-hyperbolic instability is dominant at large wavenumbers. Its growth rate tends to a constant along each branch in the large-wavenumber and inviscid limit, implying that the strato-hyperbolic instability is not stabilized by strong stratification. Good agreement between the structure of the strato-hyperbolic instability mode and the corresponding local solution is observed. In the case of the Stuart vortices, the unstable modes are classified into three types: the pure hyperbolic instability, the elliptic instability and the mixed-type instability, which is a mixture of the pure hyperbolic and the elliptic instabilities. Stratification decreases the growth rate of the elliptic instability, which is expected to be stabilized by stronger stratification, although it is not completely stabilized within the range of Froude numbers considered. The present results imply that both the pure hyperbolic instability and the strato-hyperbolic instability are important in stably stratified flows of geophysical or planetary scale.

JFM classification

Geophysical and Geological Flows: Stratified flows Vortex Flows: Vortex instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 909 , 25 February 2021 , A4
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Arratia, C., Caulfield, C. P. & Chomaz, J.-M. 2013 Transient perturbation growth in time-dependent mixing layers. J. Fluid Mech. 717, 90–133.CrossRefGoogle Scholar
Aspden, J. M. & Vanneste, J. 2009 Elliptical instability of a rapidly rotating, strongly stratified fluid. Phys. Fluids 21, 074104.CrossRefGoogle Scholar
Bayly, B. J., Holm, D. D. & Lifschitz, A. 1996 Three-dimensional stability of elliptical vortex columns in external strain flows. Phil. Trans. R. Soc. Lond. A 354, 895–926.Google Scholar
Billant, P. 2000 Zigzag instability of vortex pairs in stratified and rotating fluids. Part 1. General stability equations. J. Fluid Mech. 660, 354–395.CrossRefGoogle Scholar
Billant, P. & Chomaz, J.-M. 2000 a Experimental evidence for a new instability of a vertical columnar vortex pair in a strongly stratified fluid. J. Fluid Mech. 418, 167–188.CrossRefGoogle Scholar
Billant, P. & Chomaz, J.-M. 2000 b Theoretical analysis of the zigzag instability of a vertical columnar vortex pair in a strongly stratified fluid. J. Fluid Mech. 419, 29–63.CrossRefGoogle Scholar
Billant, P. & Chomaz, J.-M. 2000 c Three-dimensional stability of a vertical columnar vortex pair in a stratified fluid. J. Fluid Mech. 419, 65–91.CrossRefGoogle Scholar
Billant, P. & Chomaz, J.-M. 2001 Self-similarity of strongly stratified inviscid flows. Phys. Fluids 13, 1645–1651.CrossRefGoogle Scholar
Billant, P., Deloncle, A., Chomaz, J.-M. & Otheguy, P. 2010 Zigzag instability of vortex pairs in stratified and rotating fluids. Part 2. Analytical and numerical analyses. J. Fluid Mech. 660, 396–429.CrossRefGoogle Scholar
Caulfield, C. P. & Peltier, W. R. 2000 The anatomy of the mixing transition in homogeneous and stratified free shear layers. J. Fluid Mech. 413, 1–47.CrossRefGoogle Scholar
Deloncle, A., Billant, P. & Chomaz, J.-M. 2008 Nonlinear evolution of the zigzag instability in stratified fluids: a shortcut on the route to dissipation. J. Fluid Mech. 599, 229–239.CrossRefGoogle Scholar
Donnadieu, C., Ortiz, S., Chomaz, J.-M. & Billant, P. 2009 Three-dimensional instabilities and transient growth of a counter-rotating vortex pair. Phys. Fluids 21, 094102.CrossRefGoogle Scholar
Edwards, W. S., Tuckerman, L. S., Friesner, R. A. & Sorensen, D. C. 1994 Krylov methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 110, 82–102.CrossRefGoogle Scholar
Friedlander, S. & Vishik, M. M. 1991 Instability criteria for the flow of an inviscid incompressible fluid. Phys. Rev. Lett. 66, 2204–2206.CrossRefGoogle ScholarPubMed
Gau, T. & Hattori, Y. 2014 Modal and non-modal stability of two-dimensional Taylor–Green vortices. Fluid Dyn. Res. 46, 031410.CrossRefGoogle Scholar
Guimbard, D., Le Dizès, S., Le Bars, M., Le Gal, P. & Leblanc, S. 2000 Elliptic instability of a stratified fluid in a rotating cylinder. J. Fluid Mech. 660, 240–257.CrossRefGoogle Scholar
Hattori, Y. 2016 Concentration of vorticity in a destabilized vortex due to selective decay. J. Fluid Mech. 797, 630–643.CrossRefGoogle Scholar
Hattori, Y. 2018 Concentration of vorticity due to selective decay in doubly periodic vortices and a vortex pair. Fluid Dyn. Res. 50, 011405.CrossRefGoogle Scholar
Julien, S., Ortiz, S. & Chomaz, J.-M. 2004 Secondary instability mechanisms in the wake of a flat plate. Eur. J. Mech. B/Fluids 23, 157–165.CrossRefGoogle Scholar
Leblanc, S. & Cambon, C. 1998 Effects of the Coriolis force on the stability of Stuart vortices. J. Fluid Mech. 356, 353–379.CrossRefGoogle Scholar
Leblanc, S. & Godeferd, F. S. 1999 An illustration of the link between ribs and hyperbolic instability. Phys. Fluids 11, 497–499.CrossRefGoogle Scholar
Le Dizès, S. & Billant, P. 2009 Radiative instability in stratified vortices. Phys. Fluids 21, 096602.CrossRefGoogle Scholar
Leweke, T. & Williamson, C. H. K. 1998 Three-dimensional instabilities in wake transition. Eur. J. Mech. B/Fluids 17, 571–586.CrossRefGoogle Scholar
Lifschitz, A. & Hameiri, E. 1991 Local stability conditions in fluid dynamics. Phys. Fluids A 3, 2644–2651.CrossRefGoogle Scholar
Morales-Juberías, R., Sayanagi, K. M., Dowling, T. E. & Ingersoll, A. P. 2011 Emergence of polar-jet polygons from jet instabilities in a Saturn model. Icarus 211, 1284–1293.CrossRefGoogle Scholar
Miyazaki, T. & Adachi, K. 1998 Short-wavelength instabilities of waves in rotating stratified fluids. Phys. Fluids 10, 3168–3177.CrossRefGoogle Scholar
Miyazaki, T. & Fukumoto, Y. 1992 Three-dimensional instability of strained vortices in a stably stratified flow. Phys. Fluids A 4, 2515–2522.CrossRefGoogle Scholar
Otheguy, P., Billant, P. & Chomaz, J.-M. 2006 a Elliptic and zigzag instabilities on co-rotating vertical vortices in a stratified fluid. J. Fluid Mech. 553, 253–272.CrossRefGoogle Scholar
Otheguy, P., Billant, P. & Chomaz, J.-M. 2006 b The effect of planetary rotation on the zigzag instability of co-rotating vortices in a stratified fluid. J. Fluid Mech. 553, 273–281.CrossRefGoogle Scholar
Peyret, R. 2010 Spectral Methods for Incompressible Viscous Flow. Springer.Google Scholar
Potylitsin, P. G. & Peltier, W. R. 1998 Stratification effects on the stability of columnar vortices on the $f$-plane. J. Fluid Mech. 355, 45–79.CrossRefGoogle Scholar
Potylitsin, P. G. & Peltier, W. R. 1999 Three-dimensional destabilization of Stuart vortices: the influence of rotation and ellipticity. J. Fluid Mech. 387, 205–226.CrossRefGoogle Scholar
Pralits, J. O., Giannetti, F. & Brandt, L. 2013 Three-dimensional instability of the flow around a rotating circular cylinder. J. Fluid Mech. 730, 5–18.CrossRefGoogle Scholar
Sayanagi, K. M., Dyudina, U. A., Ewald, S. P., Fischer, G., Ingersoll, A. P., Kurth, W. S., Muro, G. D., Porco, C. C. & West, R. A. 2013 Dynamics of Saturn's great storm of 2010–2011 from Cassini ISS and RPWS. Icarus 223, 460–478.CrossRefGoogle Scholar
Sipp, D. & Jacquin, L. 1998 Elliptic instability in two-dimensional flattened Taylor–Green vortices. Phys. Fluids 10, 839–849.CrossRefGoogle Scholar
Suzuki, S., Hirota, M. & Hattori, Y. 2018 Strato-hyperbolic instability: a new mechanism of instability in stably stratified vortices. J. Fluid Mech. 854, 293–323.CrossRefGoogle Scholar
Waite, M. L. & Smolarkiewicz, P. K. 2008 Instability and breakdown of a vertical vortex pair in a strongly stratified fluid. J. Fluid Mech. 606, 239–273.CrossRefGoogle Scholar
Waleffe, F. 1990 On the three-dimensional instability of strained vortices. Phys. Fluids A 2, 76–80.CrossRefGoogle Scholar
Youssef, A. & Marcus, P. S. 2003 The dynamics of jovian white ovals from formation to merger. Icarus 162, 74–93.CrossRefGoogle Scholar