fluids Article Instability of Lenticular Vortices: Results from Laboratory Experiments, Linear Stability Analysis and Numerical Simulations Noé Lahaye 1, * , Alexandre Paci 2 and Stefan G. Llewellyn Smith 3   Citation: 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 Academic Editors: Xavier Carton and Sabrina Speich Received: 31 August 2021 Accepted: 14 October 2021 Published: 23 October 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Inria & IRMAR, Campus Universitaire de Beaulieu, 35042 Rennes, France 2 CNRM, Université de Toulouse, METEO-FRANCE, CNRS, 31100 Toulouse, France; alexandre.paci@meteo.fr 3 Department of Mechanical and Aerospace Engineering and Scripps Institution of Oceanography, University of California San Diego, La Jolla, CA 92093-0411, USA; sgls@ucsd.edu * Correspondence: noe.lahaye@inria.fr Abstract: The instability of surface lenticular vortices is investigated using a comprehensive suite of laboratory experiments combined with numerical linear stability analysis as well as nonlinear numerical simulations in a two-layer Rotating Shallow Water model. The development of instabilities is discussed and compared between the different methods. The linear stability analysis allows for a clear description of the origin of the instability observed in both the laboratory experiments and numerical simulations. While global qualitative agreement is found, some discrepancies are observed and discussed. Our study highlights that the sensitivity of the instability outcome is related to the initial condition and the lower-layer flow. The inhibition or even suppression of some unstable modes may be explained in terms of the lower-layer potential vorticity profile. Keywords: Vortex; instability; ocean eddies; numerical simulations; laboratory experiments 1. Introduction Mesoscale vortices are a major component of the global oceanic circulation. They are particularly common in oceanic regions of high mesoscale activity and can have long lifetimes, up to the order of a year (e.g., [1–4]). These coherent structures play an active role in the energetics of the ocean general circulation, in the transport of biological species, heat and salt anomalies, and in air–sea interaction (e.g., [5,6]). For example, eddies that detach from currents separating oceanic regions with different properties may propagate from one side to the other, transporting into a new region water with anomalous properties. These warm- and cold-core vortices are associated with the intersection of the isopycnals with the surface of the ocean, forming a front (e.g., [7]), but can also be sub-surface intensified [8] or form lenticular vortices at depth, a well known example of such vortices being the meddies [3]. Typical eddy size in the ocean are a few Rossby deformation radii (between 1 and 5 for eddies described in details in the literature), with Rossby numbers smaller than unity but finite (typically below 0.3 in absolute value) (e.g., [4,9–11]). Smaller eddies can also be found, down to the submesoscale and with higher Rossby numbers (e.g., [12,13]). Warm-core surface anticyclonic vortices are the subject of this study and will be referred as lenticular vortices. Early modelling studies of the dynamics of such vortices date back to the 1970s, start- ing with experimental studies [14,15]. Subsequently Griffiths and Linden [16] performed laboratory experiments in a rotating tank, using two different methods for generating the vortex. The first method (referred to as constant-flux) consists of slowly injecting a lighter fluid at the surface of the tank filled with heavier fluid until the desired volume for the vortex is obtained (this technique was already used by Gill et al. [15]). The second tech- nique, the so-called constant-volume method previously used by Saunders [14], consists of releasing a given volume of lighter fluid initially contained in a suspended cylinder. Fluids 2021, 6, 380. https://doi.org/10.3390/fluids6110380 https://www.mdpi.com/journal/fluids