Abstract
The goal of this paper is to describe the formation of Kelvin-Helmholtz instabilities at the interface of two fluids of different densities and the ability of various shallow water models to reproduce correctly the formation of these instabilities.
Working first in the so called rigid lid case, we derive by a simple linear analysis an explicit condition for the stability of the low frequency modes of the interface perturbation, an expression for the critical wave number above which Kelvin-Helmholtz instabilities appear, and a condition for the stability of all modes when surface tension is present. Similar conditions are derived for several shallow water asymptotic models and compared with the values obtained for the full Euler equations. Noting the inability of these models to reproduce correctly the scenario of formation of Kelvin-Helmholtz instabilities, we derive new models that provide a perfect matching. A comparisons with experimental data is also provided.
Moreover, we briefly discuss the more complex case where the rigid lid is replaced by a free surface. In this configuration, it appears that some frequency modes are stable when the velocity jump at the interface is large enough; we explain why such stable modes do not appear in the rigid lid case.
To Walter Craig, for his 60th birthday, with friendship and admiration.
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Notes
- 1.
- 2.
The endpoint case \(g^{{\prime}}- e(\mathbf{k}) = 0\) actually corresponds to a linear amplification.
- 3.
If \(\sigma\) is very large, \(\alpha _{GN_{\sigma }}(\cdot )\) is no longer decreasing over \(\mathbb{R}^{+}\). However, if \(\sigma\) satisfies
$$\displaystyle{\sigma \leq \frac{1} {3}g^{{\prime}}(\rho ^{+} +\rho ^{-})\min \{(H^{+})^{2}, (H^{-})^{2}\},}$$which is always satisfied in realistic physical configurations, then \(\alpha _{GN_{\sigma }}(\cdot )\) is indeed a decreasing function. We always assume that we are in such a regime.
- 4.
When r ≥ 1∕6, there is no obvious physical meaning for \(v_{r}^{\pm }\). For 0 ≤ r ≤ 1∕6, and for small amplitude waves \(v_{r}^{\pm }\) is the horizontal velocity evaluated on the level line \(\{z =\hat{ z}_{r}^{\pm }(1 \pm \frac{\zeta } {H^{\pm }})+\zeta \}\) (see [33], Sect. 5.6.2).
- 5.
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Acknowledgements
The authors want to address their warmest thanks to the referee for his/her careful reading and valuable suggestions. D. L. acknowledges support from the ANR-13-BS01-0003-01 DYFICOLTI and the ANR BOND. M.. This work was supported by Fondation Sciences Mathématiques de Paris (FSMP) when M. Ming was a postdoc with D. L. at DMA, l’École Normale Supérieure in 2012.
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Lannes, D., Ming, M. (2015). The Kelvin-Helmholtz Instabilities in Two-Fluids Shallow Water Models. In: Guyenne, P., Nicholls, D., Sulem, C. (eds) Hamiltonian Partial Differential Equations and Applications. Fields Institute Communications, vol 75. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2950-4_7
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