Instability of Lenticular Vortices: Results from Laboratory Experiments, Linear Stability Analysis and Numerical Simulations
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
2.2. Methods
2.2.1. Setup of the Laboratory Experiments
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 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 Dominant
4.2. Nonlinear Saturation with Wavenumber(s)
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 .
- Some unstable modes may be found in the zone of stability in the lower-left corner of the parameter space.
- For very large aspect ratio, we found a very unstable ageostrophic unstable mode with high wavenumber (), 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 , as shown by both the linear stability analysis and the nonlinear simulations. The growth rate is slightly lower for large depth ratio. For , 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 ).
- for , two baroclinic dipoles form and propagate away from the initial eddy, leaving eventually a weak monopole at the center;
- for , 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
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 |
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0.1 | 0.14 | 0.2 | 0.28 | 0.4 | 0.56 | 0.8 | |
3 | 10 | 16 | 25 | 40 | 63 | 90 |
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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
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 StyleLahaye, 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
APA StyleLahaye, N., Paci, A., & Llewellyn Smith, S. G. (2021). Instability of Lenticular Vortices: Results from Laboratory Experiments, Linear Stability Analysis and Numerical Simulations. Fluids, 6(11), 380. https://doi.org/10.3390/fluids6110380