# Instability of Lenticular Vortices: Results from Laboratory Experiments, Linear Stability Analysis and Numerical Simulations

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation and Methods

#### 2.1. RSW Equations and Basic State

#### 2.1.1. The Two-Layer RSW Model

#### 2.1.2. Vortex Profiles

#### 2.1.3. Range of Parameters Investigated and Estimates of Realistic Values

^{2}as reported by the authors, gives the same value). The resulting Burger number ranges between $0.05$ and $0.07$, while the Rossby number is between around $0.04$ and $0.07$. These typical dynamical parameters were confirmed by observations from satellites in ths region [11], who reported typical eddy size ranging from 50 km–170 km.

#### 2.2. Methods

#### 2.2.1. Setup of the Laboratory Experiments

^{−1}and $1.35$ rad s

^{−1}within an accuracy of $0.01\%$. We used two different cylinders with radius ${R}_{c}$ $5.5$ cm and 20 cm. Dyed fresh water was used to visualize the evolution of the vortex using a CCD camera placed above the tank. The lower layer was salt water with concentration in the range 3.6–80 $\mathrm{g}\text{}{\mathrm{L}}^{-1}$, thus setting the value of the reduced gravity. The total fluid depth was varied from 8–40 $\mathrm{cm}$ while the thickness of lighter fluid in the cylinder was $3.5\le {h}_{0}\le 7$ cm, covering a range of initial depth ratio $0.088\le {\delta}_{0}\le 0.67$. In total, 42 different experiments were performed, with initial Burger numbers ${\theta}_{0}={g}^{\prime}{h}_{0}/\left(f{R}_{c}^{2}\right)$ varying between $0.015$ and $0.88$. A table summarizing the parameters of the laboratory experiments is given in Appendix D.

#### 2.2.2. Linear Stability Analysis

#### 2.2.3. Nonlinear Simulations: Numerics and Initial Conditions

## 3. Early Stage of the Instability

#### 3.1. Preliminary Considerations from Linear Stability Analysis

#### 3.2. Observations in Laboratory Experiments

- The parameters $\delta ,{Q}_{1}$ in the laboratory experiments are deduced from qualitative considerations and semi-empirical relations, neglecting the details of the initial adjustment;
- Friction effects decreases the azimuthal velocity, especially during the initial adjustment and near the edge of the eddy [37].
- The initial adjustment leaves out high-wavenumber perturbations to the vortex before the instability starts developing (see Figure 4a). The vortex may be unstable with respect to several wavenumbers, which are unevenly excited by this perturbation, so that a mode that is not the most unstable may initially gain more energy than the most unstable one and develop faster.

#### 3.3. Results from Nonlinear Numerical Simulations

#### 3.4. Sensitivity to the Initial Conditions

## 4. Nonlinear Saturation of the Instability

#### 4.1. Nonlinear Saturation for $m=2$ Dominant

#### 4.2. Nonlinear Saturation with Wavenumber(s) $m>2$

## 5. Discussion: Impact of a Lower-Layer Flow

#### 5.1. Piecewise-Constant Lower-Layer PV

#### 5.2. Constant Lower-Layer PV

## 6. Conclusions

- The instability domain is dominated by the hybrid instability, associated with resonance between a frontal mode and a lower-layer Rossby wave.
- The growth rate of the most unstable mode increases with the depth ratio and slightly decreases (increases) with the value of the PV of the eddy for small (large) values of the depth ratio. The wavenumber of the most unstable mode increases with both depth ratio and the eddy PV. Several unstable modes with close growth rates and wavenumbers co-exist for $m>2$.

- Some unstable modes may be found in the zone of stability in the lower-left corner of the $\delta ,{Q}_{1}$ parameter space.
- For very large aspect ratio, we found a very unstable ageostrophic unstable mode with high wavenumber ($m\approx 10$), associated with the resonance between Poincare-like waves.

- If the lower-layer PV is constant (relevant for oceanic application and “constant-flux” experiments, and associated with a weak co-rotating lower-layer circulation), the hybrid instability vanishes as no lower-layer PV gradient supporting the Rossby wave propagation is present. As a result, the vortex is found to be stable, as was previously suggested (e.g., [24,25]) and confirmed by a linear stability analysis in the same model by Cohen et al. [26] with a different numerical method. We further confirmed this inhibition of the instability by means of nonlinear numerical simulations. The ageostrophic instability for very high depth ratio persists (from the linear stability analysis), which was not previously reported.
- If the lower layer is piecewise-constant (corresponding to the inviscid adjusted state in “constant-volume” laboratory experiments and associated with a weak counter-rotating lower-layer circulation), the stability properties are hardly changed for ${Q}_{1}>5$, as shown by both the linear stability analysis and the nonlinear simulations. The growth rate is slightly lower for large depth ratio. For ${Q}_{1}<5$, no significant unstable mode has been found, but weakly unstable mode may exist for high values of the depth ratio.

- The wavenumber observed in laboratory experiments was sometimes different than the linear prediction, especially for large values of the depth ratio;
- The growth rate in the numerical experiments is lower that expected from the linear stability analysis (between 10% and 40%, increasing with ${Q}_{1}$).

- for $m=2$, two baroclinic dipoles form and propagate away from the initial eddy, leaving eventually a weak monopole at the center;
- for $m>2$, several dipoles propagate with a strongly curved direction. As a result, they sometimes stay in the neighbourhood of the initial eddy, interacting in a complex manner with each other. Again, a weak monopole can be left at the location of the initial location of the eddy, depending on the parameters.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PV | Potential Vorticity |

RSW | Rotating Shallow Water |

QG | Quasi-Geostrophic |

## Appendix A. Linear Problem in the Exterior Domain (r > 1) and Boundary Conditions

## Appendix B. Initial Conditions for the Nonlinear Numerical Simulations

## Appendix C. Linear Stability Analysis with a Free Surface

## Appendix D. Laboratory Experiments Parameters

**Table A1.**Parameters of the laboratory experiments: experimental configuration are in the first 7 columns, estimated nondimensional parameters of the adjusted vortex in the next 4 columns, and observed most unstable mode number in last column. Symbols are explained in the text. Bold values are for the experiment shown in Figure 4.

${\mathit{\theta}}_{0}$ | ${\mathit{\delta}}_{0}$ | ${\mathit{g}}^{\prime}$ | ${\mathit{R}}_{\mathit{c}}$ | ${\mathit{H}}_{0}$ | ${\mathit{h}}_{0}$ | $\mathsf{\Omega}$ | ${\mathit{L}}_{\mathit{e}}$ | $\mathit{\delta}$ | ${\mathit{Q}}_{1}$ | $\mathit{Bu}$ | m |
---|---|---|---|---|---|---|---|---|---|---|---|

cm/s | cm | cm | cm | rad/s | |||||||

0.330 | 0.110 | 5.0 | 5.5 | 40.0 | 4.5 | 0.75 | 8.66 | 0.10 | 7.5 | 0.12 | 2 |

0.160 | 0.095 | 5.0 | 5.5 | 40.0 | 3.8 | 1.00 | 7.70 | 0.09 | 12.3 | 0.08 | 2 |

0.880 | 0.175 | 3.8 | 5.5 | 40.0 | 7.0 | 0.50 | 10.66 | 0.14 | 4.3 | 0.18 | 2 |

0.280 | 0.150 | 3.8 | 5.5 | 40.0 | 6.0 | 0.83 | 8.41 | 0.14 | 8.4 | 0.11 | 2 |

0.190 | 0.150 | 3.8 | 5.5 | 40.0 | 6.0 | 1.00 | 7.90 | 0.14 | 10.9 | 0.09 | 2 |

0.120 | 0.150 | 3.8 | 5.5 | 40.0 | 6.0 | 1.25 | 7.41 | 0.15 | 15.1 | 0.07 | 2 |

0.230 | 0.100 | 6.9 | 5.5 | 40.0 | 4.0 | 1.00 | 8.14 | 0.10 | 9.5 | 0.10 | 2 |

0.144 | 0.100 | 6.8 | 5.5 | 40.0 | 4.0 | 1.25 | 7.59 | 0.10 | 13.2 | 0.07 | 2 |

0.260 | 0.195 | 4.4 | 5.5 | 20.5 | 4.0 | 0.75 | 8.30 | 0.18 | 8.8 | 0.10 | 2 |

0.093 | 0.195 | 4.4 | 5.5 | 20.5 | 4.0 | 1.25 | 7.18 | 0.19 | 18.3 | 0.05 | 3 |

0.146 | 0.195 | 4.4 | 5.5 | 20.5 | 4.0 | 1.00 | 7.60 | 0.19 | 13.1 | 0.07 | 2 |

0.420 | 0.400 | 3.2 | 5.5 | 10.0 | 4.0 | 0.50 | 9.06 | 0.35 | 6.5 | 0.13 | 2 |

0.185 | 0.400 | 3.2 | 5.5 | 10.0 | 4.0 | 0.75 | 7.87 | 0.38 | 11.1 | 0.09 | 2 |

0.067 | 0.400 | 3.2 | 5.5 | 10.0 | 4.0 | 1.25 | 6.92 | 0.39 | 23.7 | 0.04 | 4 |

0.086 | 0.400 | 3.2 | 5.5 | 10.0 | 4.0 | 1.10 | 7.11 | 0.39 | 19.4 | 0.05 | 4 |

0.066 | 0.630 | 2.5 | 5.5 | 8.0 | 5.0 | 1.25 | 6.91 | 0.62 | 23.9 | 0.04 | 4 |

0.460 | 0.560 | 6.3 | 5.5 | 9.0 | 5.0 | 0.75 | 9.23 | 0.49 | 6.1 | 0.14 | 2 |

0.170 | 0.630 | 6.3 | 5.5 | 8.0 | 5.0 | 1.25 | 7.77 | 0.60 | 11.7 | 0.08 | 4 |

0.100 | 0.088 | 6.3 | 5.5 | 40.0 | 3.5 | 1.35 | 7.24 | 0.09 | 17.3 | 0.06 | 2 |

0.055 | 0.175 | 17.6 | 20.0 | 28.5 | 28.5 | 1.00 | 24.69 | 0.17 | 27.7 | 0.04 | 4 |

0.037 | 0.175 | 11.9 | 20.0 | 28.5 | 28.5 | 1.00 | 23.85 | 0.17 | 38.4 | 0.03 | 5 |

0.023 | 0.175 | 7.4 | 20.0 | 28.5 | 28.5 | 1.00 | 23.03 | 0.17 | 57.7 | 0.02 | 6 |

0.088 | 0.193 | 25.8 | 20.0 | 28.5 | 28.5 | 1.00 | 25.93 | 0.19 | 19.1 | 0.05 | 3 |

0.135 | 0.175 | 43.4 | 20.0 | 28.5 | 28.5 | 1.00 | 27.35 | 0.17 | 13.9 | 0.07 | 3 |

0.015 | 0.175 | 6.8 | 20.0 | 28.5 | 28.5 | 1.00 | 22.45 | 0.17 | 84.0 | 0.02 | 7 |

0.196 | 0.190 | 36.5 | 20.0 | 28.7 | 28.7 | 1.20 | 28.85 | 0.18 | 10.6 | 0.04 | 3 |

0.046 | 0.100 | 18.5 | 20.0 | 39.5 | 39.5 | 0.80 | 24.29 | 0.10 | 32.1 | 0.05 | 3 |

0.030 | 0.100 | 17.1 | 20.0 | 39.5 | 39.5 | 1.00 | 23.46 | 0.10 | 45.9 | 0.03 | 4 |

0.022 | 0.100 | 11.0 | 20.0 | 39.5 | 39.5 | 1.20 | 22.97 | 0.10 | 59.9 | 0.01 | 5 |

0.056 | 0.100 | 28.7 | 20.0 | 39.0 | 39.0 | 1.20 | 24.73 | 0.10 | 27.3 | 0.03 | 3 |

0.018 | 0.100 | 10.5 | 20.0 | 39.0 | 39.0 | 1.20 | 22.68 | 0.10 | 71.5 | 0.01 | 5 |

0.200 | 0.600 | 53.7 | 20.0 | 10.0 | 6.0 | 1.00 | 28.94 | 0.56 | 10.5 | 0.09 | 3 |

0.100 | 0.670 | 25.9 | 20.0 | 10.5 | 7.0 | 1.00 | 26.32 | 0.65 | 17.3 | 0.06 | 4 |

0.062 | 0.600 | 16.4 | 20.0 | 10.0 | 6.0 | 1.00 | 24.98 | 0.59 | 25.2 | 0.04 | 5 |

0.041 | 0.600 | 11.0 | 20.0 | 10.0 | 6.0 | 1.00 | 24.05 | 0.58 | 35.3 | 0.03 | 6 |

0.055 | 0.600 | 14.8 | 20.0 | 10.0 | 6.0 | 1.00 | 24.69 | 0.59 | 27.7 | 0.04 | 5 |

0.052 | 0.600 | 13.9 | 20.0 | 10.0 | 6.0 | 1.00 | 24.56 | 0.59 | 29.0 | 0.03 | 5 |

0.025 | 0.400 | 9.8 | 20.0 | 10.0 | 4.0 | 1.00 | 23.16 | 0.40 | 53.6 | 0.02 | 7 |

0.030 | 0.400 | 12.0 | 20.0 | 10.0 | 4.0 | 1.00 | 23.46 | 0.40 | 45.9 | 0.02 | 5 |

0.031 | 0.194 | 8.4 | 20.0 | 31.0 | 6.0 | 1.00 | 23.52 | 0.19 | 44.6 | 0.02 | 5 |

0.037 | 0.194 | 9.8 | 20.0 | 31.0 | 6.0 | 1.00 | 23.85 | 0.19 | 38.4 | 0.03 | 5 |

0.026 | 0.194 | 6.9 | 20.0 | 31.0 | 6.0 | 1.00 | 23.22 | 0.19 | 51.9 | 0.02 | 5 |

## References

- McWilliams, J.C. Submesoscale, coherent vortices in the ocean. Rev. Geophys.
**1985**, 23, 165–182. [Google Scholar] [CrossRef] - Olson, D.B. Rings in the Ocean. Annu. Rev. Earth Planet. Sci.
**1991**, 19, 283–311. [Google Scholar] [CrossRef] - Carton, X. Oceanic vortices. In Fronts, Waves and Vortices in Geophysical Flows; Flòr, J., Ed.; Springer: Berlin/Heidelberg, Germany, 2010; Volume 805, pp. 61–108. [Google Scholar]
- Chelton, D.; Schlax, M.; Samelson, R. Global observations of nonlinear mesoscale eddies. Progr. Oceanogr.
**2011**, 91, 167–216. [Google Scholar] [CrossRef] - Paci, A.; Caniaux, G.; Gavart, M.; Giordani, H.; Lévy, M.; Prieur, L.; Reverdin, G. A High-Resolution Simulation of the Ocean during the POMME Experiment: Simulation Results and Comparison with Observations. J. Geophys. Res. Ocean.
**2005**, 110. [Google Scholar] [CrossRef] [Green Version] - Paci, A.; Caniaux, G.; Giordani, H.; Lévy, M.; Prieur, L.; Reverdin, G. A High-Resolution Simulation of the Ocean during the POMME Experiment: Mesoscale Variability and near Surface Processes. J. Geophys. Res. Ocean.
**2007**, 112. [Google Scholar] [CrossRef] [Green Version] - Flierl, G.R. A simple model for the structure of warm and cold core rings. J. Geophys. Res. Ocean.
**1979**, 84, 781–785. [Google Scholar] [CrossRef] - de Marez, C.; L’Hégaret, P.; Morvan, M.; Carton, X. On the 3D Structure of Eddies in the Arabian Sea. Deep Sea Res. Part I Oceanogr. Res. Pap.
**2019**, 150, 103057. [Google Scholar] [CrossRef] - Olson, D.B. The Physical Oceanography of Two Rings Observed by the Cyclonic Ring Experiment. Part II: Dynamics. J. Phys. Oceanogr.
**1980**, 10, 514–528. [Google Scholar] [CrossRef] [Green Version] - Olson, D.B.; Evans, R.H. Rings of the Agulhas Current. Deep Sea Res. Part A Oceanogr. Res. Pap.
**1986**, 33, 27–42. [Google Scholar] [CrossRef] - Goni, G.J.; Garzoli, S.L.; Roubicek, A.J.; Olson, D.B.; Brown, O.B. Agulhas Ring Dynamics from TOPEX/POSEIDON Satellite Altimeter Data. J. Mar. Res.
**1997**, 55, 861–883. [Google Scholar] [CrossRef] [Green Version] - Petersen, M.R.; Williams, S.J.; Maltrud, M.E.; Hecht, M.W.; Hamann, B. A Three-Dimensional Eddy Census of a High-Resolution Global Ocean Simulation. J. Geophys. Res. Ocean.
**2013**, 118, 1759–1774. [Google Scholar] [CrossRef] - McWilliams, J.C. A Survey of Submesoscale Currents. Geosci. Lett.
**2019**, 6, 3. [Google Scholar] [CrossRef] [Green Version] - Saunders, P.M. The Instability of a Baroclinic Vortex. J. Phys. Oceanogr.
**1973**, 3, 61–65. [Google Scholar] [CrossRef] [Green Version] - Gill, A.; Smith, J.; Cleaver, R.; Hide, R.; Jonas, P. The vortex created by mass transfer between layers of a rotating fluid. Geophys. Astrophys. Fluid Dyn.
**1979**, 12, 195–220. [Google Scholar] [CrossRef] - Griffiths, R.W.; Linden, P.F. The stability of vortices in a rotating, stratified fluid. J. Fluid Mech.
**1981**, 105, 283–316. [Google Scholar] [CrossRef] - Flierl, G.R. Rossby Wave Radiation from a Strongly Nonlinear Warm Eddy. J. Phys. Oceanogr.
**1984**, 14, 47–58. [Google Scholar] [CrossRef] - Gula, J.; Zeitlin, V.; Bouchut, F. Instabilities of Buoyancy-Driven Coastal Currents and Their Nonlinear Evolution in the Two-Layer Rotating Shallow Water Model. Part 2. Active Lower Layer. J. Fluid Mech.
**2010**, 665, 209–237. [Google Scholar] [CrossRef] [Green Version] - Ribstein, B.; Zeitlin, V. Instabilities of Coupled Density Fronts and Their Nonlinear Evolution in the Two-Layer Rotating Shallow-Water Model: Influence of the Lower Layer and of the Topography. J. Fluid Mech.
**2013**, 716, 528–565. [Google Scholar] [CrossRef] - Boss, E.; Paldor, N.; Thompson, L. Stability of a Potential Vorticity Front: From Quasi-Geostrophy to Shallow Water. J. Fluid Mech.
**1996**, 315, 65–84. [Google Scholar] [CrossRef] [Green Version] - Ripa, P. Instability of a solid-body rotating vortex in a two-layer model. J. Fluid Mech.
**1992**, 242, 395–417. [Google Scholar] [CrossRef] - Paldor, N.; Nof, D. Linear instability of an anticyclonic vortex in a two-layer ocean. J. Geophys. Res.
**1990**, 95, 18075–18079. [Google Scholar] [CrossRef] [Green Version] - Cohen, Y.; Dvorkin, Y.; Paldor, N. Linear instability of warm core, constant potential vorticity, eddies in a two-layer ocean. Q. J. R. Meteorol. Soc.
**2015**, 141, 1884–1893. [Google Scholar] [CrossRef] - Dewar, W.K.; Killworth, P.D. On the stability of oceanic rings. J. Phys. Oceanogr.
**1995**, 25, 1467–1487. [Google Scholar] [CrossRef] [Green Version] - Benilov, E.S. Stability of vortices in a two-layer ocean with uniform potential vorticity in the lower layer. J. Fluid Mech.
**2004**, 502, 207–232. [Google Scholar] [CrossRef] [Green Version] - Cohen, Y.; Dvorkin, Y.; Paldor, N. On the stability of outcropping eddies in a constant PV ocean. Q. J. R. Meteorol. Soc.
**2016**, 142, 1920–1928. [Google Scholar] [CrossRef] - Ikeda, M. Instability and splitting of mesoscale rings using a two-layer quasi-geostrophic model on a f-plane. J. Phys. Oceanogr.
**1981**, 11, 987–998. [Google Scholar] [CrossRef] - Gent, P.R.; McWilliams, J.C. The instability of barotropic circular vortices. Geophys. Astrophys. Fluid Dyn.
**1986**, 24, 209–233. [Google Scholar] [CrossRef] - Baey, J.M.; Carton, X. Vortex multipoles in two-layer rotating shallow-water flows. J. Fluid Mech.
**2002**, 460, 151–1753. [Google Scholar] [CrossRef] [Green Version] - Lahaye, N.; Zeitlin, V. Centrifugal, barotropic and baroclinic instabilities of isolated ageostrophic anticyclones in the two-layer rotating shallow water model and their nonlinear saturation. J. Fluid Mech.
**2015**, 762, 5–34. [Google Scholar] [CrossRef] [Green Version] - Katsman, C.A. Stability of Multilayer Ocean Vortices: A Parameter Study Including Realistic Gulf Stream and Agulhas Rings. J. Phys. Oceanogr.
**2003**, 33, 22. [Google Scholar] [CrossRef] - Verzicco, R.; Lalli, F.; Campana, E. Dynamics of baroclinic vortices in a rotating, stratified fluid: A numerical study. Phys. Fluids
**1997**, 9, 419–432. [Google Scholar] [CrossRef] - Thivolle-Cazat, E.; Sommeria, J.; Galmiche, M. Baroclinic instability of two-layer vortices in laboratory experiments. J. Fluid Mech.
**2005**, 544, 69–97. [Google Scholar] [CrossRef] - Zeitlin, V. Geophysical Fluid Dynamics: Understanding (Almost) Everything with Rotating Shallow Water Models; Oxford University Press: Oxford, UK, 2018. [Google Scholar]
- Olson, D.B.; Schmitt, R.W.; Kennelly, M.; Joyce, T.M. A Two-Layer Diagnostic Model of the Long-Term Physical Evolution of Warm-Core Ring 82B. J. Geophys. Res. Ocean.
**1985**, 90, 8813–8822. [Google Scholar] [CrossRef] - Chelton, D.B.; deSzoeke, R.A.; Schlax, M.G.; El Naggar, K.; Siwertz, N. Geographical Variability of the First Baroclinic Rossby Radius of Deformation. J. Phys. Oceanogr.
**1998**, 28, 433–460. [Google Scholar] [CrossRef] - Stegner, A.; Bouruet-Aubertot, P.; Pichon, T. Nonlinear adjustment of density fronts. Part 1. The Rossby scenario and the experimental reality. J. Fluid Mech.
**2004**, 502, 335–360. [Google Scholar] [CrossRef] [Green Version] - Bouchut, F.; Zeitlin, V. A robust well-balanced scheme for multi-layer shallow water equations. Disc. Cont. Dyn. Syst.
**2010**, 13, 739–758. [Google Scholar] [CrossRef] [Green Version] - Gula, J.; Zeitlin, V. Instabilities of shallow-water flows with vertical shear in the rotating annulus. In Modeling Atmospheric and Oceanic Fluid Flows: Insights From Laboratory Experiments; AGU Book Series: Washington, DC, USA, 2015. [Google Scholar]
- Sokolovskiy, M.A.; Verron, J. Finite-Core Hetons: Stability and Interactions. J. Fluid Mech.
**2000**, 423, 127–154. [Google Scholar] [CrossRef] [Green Version] - Perrot, X.; Carton, X. Instability of a two-step Rankine vortex in a reduced gravity QG model. Fluid Dyn. Res.
**2014**, 46, 031417. [Google Scholar] [CrossRef] - Gryanik, V.M.; Sokolovskiy, M.A.; Verron, J. Dynamics of heton-like vortices. Regul. Chaotic Dyn.
**2006**, 11, 383–434. [Google Scholar] [CrossRef] - Benilov, E.S.; Flanagan, J.D. The effect of ageostrophy on the stability of vortices in a two-layer ocean. Ocean Model.
**2008**, 23, 49–58. [Google Scholar] [CrossRef] - Boyd, J.P. Orthogonal rational functions on a semi-infinite interval. J. Comput. Phys.
**1987**, 70, 63–88. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Basic vortex profile: nondimensional upper-layer thickness ${H}_{1}$ (upper left panel), lower-layer PV ${Q}_{2}$ (lower left panel) and upper and lower layer azimuthal velocity ${V}_{1}$ and ${V}_{2}$ (upper and lower right panels, respectively). The vortex parameters are ${Q}_{1}=10$, $\delta =0.2$, and the solution with a quiescent lower layer (black) is compared to the solution with a piecewise-constant lower-layer PV (blue). The analytical solution for ${Q}_{1}=0$, ${V}_{2}=0$ is plotted as a grey dashed line for reference.

**Figure 2.**Schematics of the laboratory experiment setup, showing the rotating tank and the inner cylinder with lighter fluid inside, before removal.

**Figure 3.**Growth rate (colours) and wavenumber (black contours) of the most unstable mode, as found from the linear stability analysis, in the $(\delta ,{Q}_{1})$ parameter space, for lenticular vortices with a quiescent lower layer (${V}_{2}=0$). Following the discussion of realistic parameters (see Section 2.1.3), typical values for large ocean rings are $log\left({Q}_{1}\right)\in [0,1.5]$ and $log\left(\delta \right)<-0.5$.

**Figure 4.**Snapshots of the evolution of the flow during a developing instability with wavenumber $m=6$ observed in the laboratory experiment. Small-scale dissipative processes during the initial adjustment are visible in the first snapshot. Estimated eddy parameters are $\delta \approx 0.17$, ${Q}_{1}\approx 59$.

**Figure 5.**

**Left panel**: wavenumbers of the most unstable modes from the linear stability analysis (contours) and observed in laboratory experiments (text). Green labels are reproduced from (Griffiths and Linden [16], their Figure 5), and blue labels are from our set of experiments. Contours in the lower left part of the panel has been removed for visibility, using the condition $\sigma >2\times {10}^{-2}$ for modes higher than $m=2$ in this region of the $(\delta ,{Q}_{1})$ space.

**Right panel**: comparison of the growth rate predicted by linear theory (colours) and measured in the numerical experiments (colored bullets) in the $(\delta ,{Q}_{1})$ parameter space (zoomed over the range of $\delta $ values investigated in the nonlinear simulations).

**Figure 6.**Upper (

**left**) and lower (

**right**) non-axisymmetric parts of the pressure (lines) and velocity (arrows) at $t=40\phantom{\rule{3.33333pt}{0ex}}{f}^{-1}$ for the simulation initialized with a $m=3$ perturbation in the upper layer. Positive values of the pressure perturbation are in blue and negative values are in red.

**Figure 7.**(

**Left**): Evolution of the upper layer thickness perturbation (logarithm of norm) for different wavenumbers for the simulation with initial conditions randomly perturbed (several wavenumbers). Dashed lines indicate the estimated growth rate ($\sigma =3.79\times {10}^{-2}$ for $m=2$ and $\sigma =2.91\times {10}^{-2}$ for $m=3$). (

**Right**): Evolution of the energy anomaly (total: black, potential: blue, kinetic: red) normalized by the initial total energy anomaly. Thick continuous lines: perturbation containing a range of wavenumbers, thick dashed lines: perturbation with $m=2$, thin continuous lines: perturbation with $m=3$. Vortex parameters: $\delta =0.2$, ${Q}_{1}=12$.

**Figure 8.**Snapshots of thickness (grey shading, black for ${h}_{1}=\delta =0.2$) and pressure in the upper layer (black contour at intervals of $4\times {10}^{-4}$ starting from 0) and lower layer (positive in blue, negative in red, at intervals of ${10}^{-4}$) for the simulations initialized with $m=2$ perturbation (first row), $m=3$ perturbation (second row) and several wavenumbers (last row) at $t=130\phantom{\rule{3.33333pt}{0ex}}{f}^{-1}$ (left column) ant $260{f}^{-1}$ (right panel), corresponding roughly to 5 and 10 inverse growth rate. Initial vortex profile parameters: ${Q}_{1}=12$, $\delta =0.2$.

**Figure 9.**Evolution of the lower-layer PV at $t=130{f}^{-1}$ (

**upper row**) and $t=260{f}^{-1}$ (

**lower row**), for the eddy with ${Q}_{1}=12$ and $\delta =0.20$ initialised with azimuthal mode number $m=2$ (

**left column**), $m=3$ (

**middle column**) or several wavenumbers (

**right column**); corresponding to the pressure and layer thickness shown in Figure 8 (panels are transposed). The nondimensional lower-layer PV anomaly is contoured every $0.15$ from black to purple, starting at $0.05$. The green contour indicates upper layer thickness at value $0.05=\delta {H}_{0}/4$.

**Figure 10.**Vortex profiles (same fields as Figure 1) with different lower-layer conditions: quiescent (${V}_{2}=0$; black), piecewise-constant PV (blue) and constant PV (orange). Vortex parameters: ${Q}_{1}=10$, $\delta =0.25$.

**Figure 11.**Evolution of the growth rate of the most unstable mode, $\sigma $, as the lower-layer flow evolves from the rest-state to the zero PV anomaly solution, for a lenticular vortex with zero PV in the upper layer. Vortex depth ratios $\delta =0.49$, $0.35$ and azimuthal wave numbers $m=2$, 7. No other unstable mode was found.

**Table 1.**Values of the parameters $\delta $ and ${Q}_{1}$ used in the initial conditions of the numerical simulations (every pairs).

$\delta $ | 0.1 | 0.14 | 0.2 | 0.28 | 0.4 | 0.56 | 0.8 |

${Q}_{1}$ | 3 | 10 | 16 | 25 | 40 | 63 | 90 |

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**MDPI and ACS Style**

Lahaye, N.; Paci, A.; Llewellyn Smith, S.G.
Instability of Lenticular Vortices: Results from Laboratory Experiments, Linear Stability Analysis and Numerical Simulations. *Fluids* **2021**, *6*, 380.
https://doi.org/10.3390/fluids6110380

**AMA Style**

Lahaye N, Paci A, Llewellyn Smith SG.
Instability of Lenticular Vortices: Results from Laboratory Experiments, Linear Stability Analysis and Numerical Simulations. *Fluids*. 2021; 6(11):380.
https://doi.org/10.3390/fluids6110380

**Chicago/Turabian Style**

Lahaye, Noé, Alexandre Paci, and Stefan G. Llewellyn Smith.
2021. "Instability of Lenticular Vortices: Results from Laboratory Experiments, Linear Stability Analysis and Numerical Simulations" *Fluids* 6, no. 11: 380.
https://doi.org/10.3390/fluids6110380