**Note:** contributions are sorted in alphabetical order, please refer to the program for sessions and order of appearance

# Multiple-diffusive instabilities in rotating complex fluids

**Propoed by:** Dr. Oleg Kirillov and Prof. Innocent Mutabazi

## Double-diffusive instabilities in rotating hydrodynamic and magnetohydrynamic flows

## Instability windows of Chandrasekhar-Friedman-Schutz instability

## Diffusive and curvature effects on symmetric instability in stratified vortices

We present a local stability analysis of an idealized model of stratified vortices that appear in geophysical settings. The base flow comprises an axisymmetric vortex with background rotation and an out-of-plane stable stratification, and a radial stratification in thermal wind balance with the out-of-plane momentum gradient. Solving the local stability equations along fluid particle trajectories in the base flow, the dependence of short-wavelength instabilities on Schmidt number Sc is studied, in the presence of curvature effects. In the diffusion-free limit, the well-known symmetric instability is recovered. In the viscous, double-diffusive regime, instability characteristics are shown to depend on three non-dimensional parameters (including Sc), and two different instabilities are identified: 1. a monotonic instability (same as symmetric instability at Sc = 1), and 2. an oscillatory instability (absent at Sc = 1). Separating the base flow and perturbation characteristics, two each of base flow and perturbation parameters (apart from Sc) are identified, and the entire parameter space is explored for the aforementioned instabilities. In comparison to Sc = 1, monotonic and oscillatory instabilities are shown to significantly expand the instability region in the space of base flow parameters as Sc moves away from unity. Neutral stability boundaries on the plane of Sc and a modified gradient Richardson number are then identified for both these instabilities. We conclude with a discussion of curvature effects, and the likelihood of monotonic and oscillatory instabilities in typical oceanic settings.

## Effect of the free surface on the stability and energy harvesting efficiency of a tensioned membrane in a uniform current.

Flexible structures have recently been considered as alternative ways to extract energy from ocean waves (Alam 2012, Desmars et al 2018) or tidal currents (Trasch et al 2018), with the objective to find devices with complementary working characteristics compared to non-flexible energy harvesters. We investigate the dynamics of a finite length tensioned membrane with a localized linear damper to mimic energy extraction, which is placed in a uniform current parallel to a free surface. Such configuration resembles the so called Nemtsov’s membrane (Nemtsov 1985), recently studied in details and generalized to finite depth cases by Labarbe & Kirillov (2020, 2021), or to the infinite flag configuration close to a free surface studied recently by Mougel & Michelin (2020). The above studies reveal the importance of the free surface on the stability of the system, due to interactions between surface waves and structural waves when a current is present. In the present study, focus is placed on both forcing by incident waves (as already reported by Achour et al 2020 for weak currents) and stability analysis in order to investigate the role of the current on wave energy extraction by a flexible membrane, and shed additional light on the possible instability mechanism. In this objective, a linear potential flow model coupled to a tensioned beam is considered, and numerical results computed with the finite elements code FreeFEM++ (Hecht 2012) interfaced by StabFEM solver (Fabre et al 2019) are presented for a large range of physical parameters covering both subcritical and supercritical regimes.

## Double-diffusive effects in the local instabilities of an elliptical vortex

## Double-diffusive convection via 2 by 2 matrices

Convection due to competing or cooperating mechanisms, displays a variety of dynamical phenomena. One mechanism is usually a thermal gradient; typical examples of the second are rotation, a magnetic field, or a concentration gradient. The transition from conduction to convection is via a steady or a Hopf bifurcation; the point separating them is the best known codimension-two point. The steady bifurcation may be super or sub-critical, and the amplitude may undergo a qualitative transition from weak to strong.

All these features -- linear and nonlinear -- can be explained as manifestations of the behavior of the eigenvalues of a generic 2 x 2 matrix near the point where the eigenvalue branches intersect. For the stability problem, the eigenvalues have the conventional interpretation as growth rates, while for the nonlinear steady-state problems, they can be interpreted as the energy of steady states. Thus, there is a strong analogy between the stability problem and the bifurcation diagram.