Turbulence generation through an iterative cascade of the elliptical instability

The essence of turbulent flow is the conveyance of energy through the formation, interaction, and destruction of eddies over a wide range of spatial scales—from the largest scales where energy is injected down to the smallest scales where it is dissipated through viscosity. Currently, there is no mechanistic framework that captures how the interactions of vortices drive this cascade. We show that iterations of the elliptical instability, arising from the interactions between counter-rotating vortices, lead to the emergence of turbulence. We demonstrate how the nonlinear development of the elliptical instability generates an ordered array of antiparallel secondary filaments. The secondary filaments mutually interact, leading to the formation of even smaller tertiary filaments. In experiments and simulations, we observe two and three iterations of this cascade, respectively. Our observations indicate that the elliptical instability could be one of the fundamental mechanisms by which the turbulent cascade develops.

Experimental study of the nonlinear saturation of the elliptical instability: inertial wave turbulence versus geostrophic turbulence

In this paper, we present an experimental investigation of the turbulent saturation of the flow driven by the parametric resonance of inertial waves in a rotating fluid. In our set-up, a half-metre wide ellipsoid filled with water is brought to solid-body rotation, and then undergoes sustained harmonic modulation of its rotation rate. This triggers the exponential growth of a pair of inertial waves via a mechanism called the libration-driven elliptical instability. Once the saturation of this instability is reached, we observe a turbulent state for which energy is injected into the resonant inertial waves only. Depending on the amplitude of the rotation rate modulation, two different saturation states are observed. At large forcing amplitudes, the saturation flow mainly consists of a steady, geostrophic anticyclone. Its amplitude vanishes as the forcing amplitude is decreased while remaining above the threshold of the elliptical instability. Below this secondary transition, the saturation flow is a superposition of inertial waves which are in weakly nonlinear resonant interaction, a state that could asymptotically lead to inertial wave turbulence. In addition to being a first experimental observation of a wave-dominated saturation in unstable rotating flows, the present study is also an experimental confirmation of the model of Le Reun et al. (Phys. Rev. Lett., vol. 119 (3), 2017, 034502) who introduced the possibility of these two turbulent regimes. The transition between these two regimes and their relevance to geophysical applications are finally discussed.

The turbulent response to tidal and libration forcing

In conjunction with thermo-solutal convection, the turbulence generated in planetary liquid cores may be due to the role of boundary forcing through geophysically relevant mechanisms such as precession, libration and tidal forcing (Le Bars et al. 2015). In this paper, we discuss laboratory equatorial velocity measurements and selected high-resolution numerical simulations to show the generation of developed turbulence driven by longitudinal libration or tidal forcing. In both cases, the transition to saturated turbulence is driven by an elliptical instability that excites inertial modes of the system. We find striking similarities in both the transition to bulk turbulence and the enhanced zonal flow hinting at a generic fluid response independent of the forcing mechanism. We finally discuss the relevance of this work to the planetary regime and possible directions for future investigations.

Nonlinear tidal flows in short-period planets

I discuss two related nonlinear mechanisms of tidal dissipation that require finite tidal deformations for their operation: the elliptical instability and the precessional instability. Both are likely to be important for the tidal evolution of short-period extrasolar planets. The elliptical instability is a fluid instability of elliptical streamlines, such as in tidally deformed non-synchronously rotating or non-circularly orbiting planets. I summarise the results of local and global simulations that indicate this mechanism to be important for tidal spin synchronisation, planetary spin-orbit alignment and orbital circularisation for the shortest period hot Jupiters. The precessional instability is a fluid instability that occurs in planets undergoing axial precession, such as those with spin-orbit misalignments (non-zero obliquities). I summarise the outcome of local MHD simulations designed to study the turbulent damping of axial precession, which suggest this mechanism to be important in driving tidal evolution of the spin-orbit angle for hot Jupiters. Avenues for future work are also discussed.

Elliptical instability of the Moore–Saffman model for a trailing wingtip vortex

In this paper, we investigate the elliptical instability exhibited by two counter-rotating trailing vortices. This type of instability can be viewed as a resonance between two normal modes of a vortex and an external strain field. Recent numerical investigations have extended earlier results that ignored axial flow to include models with a simple wake-like axial flow such as the similarity solution found by Batchelor (J. Fluid Mech., vol. 20, 1964, pp. 645–658). We present herein growth rates of elliptical instability for a family of velocity profiles found by Moore & Saffman (Proc. R. Soc. Lond. A, vol. 333, 1973, pp. 491–508). These profiles have a parameter $n$ that depends on the wing loading. As a result, unlike the Batchelor vortex, they are capable of modelling both the jet-like and the wake-like axial flow present in a trailing vortex at short and intermediate distances behind a wingtip. Direct numerical simulations of the linearized Navier–Stokes equations are performed using an efficient spectral method in cylindrical coordinates developed by Matsushima & Marcus (J. Comput. Phys., vol. 53, 1997, pp. 321–345). We compare our results with those for the Batchelor vortex, whose velocity profiles are closely approximated as the wing loading parameter $n$ approaches 1. An important conclusion of our investigation is that the stability characteristics vary considerably with $n$ and $W_{0}$, a parameter measuring the strength of the mean axial velocity component. In the case of an elliptically loaded wing ($n=0.50$), we find that the instability growth rates are up to 50 % greater than those for the Batchelor vortex. Our results demonstrate the significant effect of the distribution and intensity of the axial flow on the elliptical instability of a trailing vortex.