Elliptical Instability Research Articles

Dynamics of an unsteady stagnation vortical flow via dynamic mode decomposition analysis

The dynamics of a large-scale stagnation vortex pair in an axisymmetric stagnation flow subject to a laminar wake disturbance is measured by time-resolved two-dimensional particle image velocimetry, and then quantitatively characterized by both the Eulerian velocity/vorticity fields and the Lagrangian finite-time Lyapunov exponents fields. This vortex pair is found to be the result of the forced response of the stagnation flow to the upstream shearing disturbances, and presents a dynamical evolution of quasi-periodic shedding due to short-wave elliptical instability. Dynamic mode decomposition analysis of both the Eulerian measure and the Lagrangian measure is taken for a quantitative description of this process. The sparsity-promoting scheme (Jovanovic et al. Phys Fluids 26(2):024,103, 2014), which integrates the mode identification and truncation as a whole, is used to distinguish those modes with dynamical significance from irrelevant ones with transient behavior. The superiority of this scheme is evidenced by the facts that it avoids the eigenvalue contamination problem, and credits higher priority to the sub-dominant modes directly associated with the system dynamics. It is found that the energetic mode with a frequency of 0.177 Hz, or about 10% of the maximum shear rate of the upstream wake, determines the quasi-periodical vortex formation process. Its half-order harmonic represents the vortex shedding event along one fixed direction. High-order even-quarter harmonics jointly contribute to the circular pattern of the vortex tube. In addition, a set of low-frequency odd-quarter harmonics are highlighted as the elliptical instability and the following vortex deformation process. Based on this finding, a reduce-order representation with 8 Eulerian modes or 56 Lagrangian modes is proposed to characterize the dominant dynamics of this unsteady vortical stagnation flow. In addition, the Eulerian measure seems to be more efficient than the Lagrangian measure in terms of reduced-order representation.

Curvature instability of a curved Batchelor vortex

In this paper, we analyse the curvature instability of a curved Batchelor vortex. We consider this short-wavelength instability when the radius of curvature of the vortex centreline is large compared with the vortex core size. In this limit, the curvature instability can be interpreted as a resonant phenomenon. It results from the resonant coupling of two Kelvin modes of the underlying Batchelor vortex with the dipolar correction induced by curvature. The condition of resonance of the two modes is analysed in detail as a function of the axial jet strength of the Batchelor vortex. In contrast to the Rankine vortex, only a few configurations involving $m=0$ and $m=1$ modes are found to become the most unstable. The growth rate of the resonant configurations is systematically computed and used to determine the characteristics of the most unstable mode as a function of the curvature ratio, the Reynolds number and the axial flow parameter. The competition of the curvature instability with another short-wavelength instability, which was considered in a companion paper (Blanco-Rodríguez & Le Dizès, J. Fluid Mech., vol. 804, 2016, pp. 224–247), is analysed for a vortex ring. A numerical error found in this paper, which affects the relative strength of the elliptic instability, is also corrected. We show that the curvature instability becomes the dominant instability in large rings as soon as axial flow is present (vortex ring with swirl).

Parametric resonance in the dynamics of an elliptic vortex in a periodically strained environment

Abstract. The model of an elliptic vortex evolving in a periodically strained background flow is studied in order to establish the possible unbounded regimes. Depending on the parameters of the exterior flow, there are three classical regimes of the elliptic vortex motion under constant linear deformation: (i) rotation, (ii) nutation, and (iii) infinite elongation. The phase portrait for the vortex dynamics features critical points which correspond to the stationary vortex not changing its form and orientation. We demonstrate that, given superimposed periodic oscillations to the exterior deformation, the phase space region corresponding to the elliptic critical point experiences parametric instability leading to locally unbounded dynamics of the vortex. This dynamics manifests itself as the vortex nutates along the strain axis while continuously elongating. This motion continues until nonlinear effects intervene near the region associated with the steady-state separatrix. Next, we show that, for specific values of the perturbation parameters, the parametric instability is effectively suppressed by nonlinearity in the primal parametric instability zone. The secondary zone of the parametric instability, on the contrary, produces an effective growth of the vortex's aspect ratio.

Instability of cylinder wake under open-loop active control

Instability of a wake controlled by a streamwise Lorentz force is investigated through a Floquet stability analysis. The streamwise Lorentz force, which is a two-dimens- ional control input created by an electromagnetic actuator located on the cylinder surface, adjusts the base flow to affect the three-dimensional wake instability and achieve wake stabilization and transition delay. The instability mode at a Reynolds number Re = 300 can be transformed from B to A with N = 1.0, where N is an interaction number repre- senting the strength of the Lorentz force relative to the inertial force in the fluid. The wake flow is Floquet stable when N increases to 1.3. The spanwise perturbation wavelengths are 3.926D and 0.822D in the modes A and B, respectively, where D is the cylinder diameter. In addition, the oscillating amplitudes of drag and lift are reduced with the increase in the interaction number. Particle tracing is used to explore the essential physical mechanism for mode transformation. The path lines show that suppression of flow separation hinders the fluid deformation and rotation, leading to the decrease in elliptic and hyperbolic instability regions, which is the material cause of mode transformation. All of the results indicate that wake stabilization and transition delay can be achieved under open-loop active control via the streamwise Lorentz force.

Tidally-forced turbulence in planetary interiors

The turbulence generated in the liquid metal cores and subsurface oceans of planetary bodies may be due to the role of mechanical forcing through precession/nutation, libration, tidal forcing and collisions. Here, we model the response of an enclosed constant density fluid to tidal forcing by combining laboratory equatorial velocity measurements with selected high-resolution numerical simulations to show, for the first time, the generation of bulk filling turbulence. The transition to saturated turbulence is characterized by an elliptical instability that first excites primary inertial modes of the system, then secondary inertial modes forced by the primary inertial modes, and then bulk filling turbulence. The amplitude of this saturated turbulence scales with the body's elliptical distortion, U ∼ β, while a time- and radially averaged azimuthal zonal flow scales with β2. The results of the current tidal experiments are compared with recent studies of the libration-driven turbulent flows studied by Grannan et al. and Favier et al. Tides and libration correspond to two end-member types of geophysical mechanical forcings. For satellites dominated by tidal forcing, the ellipsoidal boundary enclosing the internal fluid layers is elastically deformed while, for librational forcing, the core-mantle boundary possesses an inherently rigid, frozen-in ellipsoidal shape. We find striking similarities between tidally and librationally driven flow transitions to bulk turbulence and zonal flows. This suggests a generic fluid response independent of the style of mechanical forcing. Since β ≲ 10−4 in planetary bodies, it is often argued that mechanically driven zonal velocities will be small. In contrast, our linear scaling for mechanically driven bulk turbulence, U ∼ β, suggests geophysically significant velocities that can play a significant role in planetary processes including tidal dissipation and magnetic field generation.

Elliptic instability of a curved Batchelor vortex

The occurrence of the elliptic instability in rings and helical vortices is analysed theoretically. The framework developed by Moore & Saffman (Proc. R. Soc. Lond. A, vol. 346, 1975, pp. 413–425), where the elliptic instability is interpreted as a resonance of two Kelvin modes with a strained induced correction, is used to obtain the general stability properties of a curved and strained Batchelor vortex. Explicit expressions for the characteristics of the three main unstable modes are obtained as a function of the axial flow parameter of the Batchelor vortex. We show that vortex curvature adds a contribution to the elliptic instability growth rate. The results are applied to a single vortex ring, an array of alternate vortex rings and a double helical vortex.

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.

Vortices in stratified protoplanetary disks

Large scale vortices could play a key role in the evolution of protoplanetary disks, particularly in the dead-zone where no turbulence associated with magnetic field is expected. Their possible formation by the subcritical baroclinic instability is a complex issue due to the vertical structure of the disk and to the elliptical instability.} {In two-dimensional disks the baroclinic instability is studied as a function of the thermal transfer efficiency. In three-dimensional disks we explore the importance of radial and vertical stratification on the processes of vortex formation and amplification.} {Numerical simulations are performed using a fully compressible hydrodynamical code based on a second order finite volume method. We assume a perfect gas law in inviscid disk models in which heat transfer is due to either relaxation or diffusion.} {In 2D, the baroclinic instability with thermal relaxation leads to the formation of large-scale vortices, which are unstable with respect to the elliptic instability. In the presence of heat diffusion, hollow vortices are formed which evolve into vortical structures with a turbulent core. In 3D, the disk stratification is found to be unstable in a finite layer which can include the mid-plane or not. When the unstable layer contains the mid-plane, the 3D baroclinic instability with thermal relaxation is found to develop first in the unstable layer as in 2D, producing large-scale vortices. These vortices are then stretched out in the stable layer, creating long-lived columnar vortical structures extending through the height of the disk. They are also found to be the source of internal vortex layers that develop across the whole disk along baroclinic critical layer surfaces, and form new vortices in the upper region of the disk.} {In three-dimensional disks, vortices can survive for a very long time if the production of vorticity by...

Concentration of vorticity in a destabilized vortex due to selective decay

The formation of concentrated vortices like tornadoes and tropical cyclones in rotating fluids is of much interest in atmospheric flows. It is shown by direct numerical simulation that the selective decay of inviscid invariants leads to concentration of vorticity in a destabilized vortex. By selective decay we mean here that the circulation of the mean flow decays faster than the angular momentum or energy. Initially localized disturbances are superimposed onto the two-dimensional flattened Taylor–Green vortices to trigger the elliptic instability. In the later stage of nonlinear evolution of the disturbance circulation decays faster than angular momentum and energy, giving rise to a sharp peak in the vorticity distribution of the mean flow. During the selective decay vortex pairs reconnect and eventually annihilate at the cell boundaries of the Taylor–Green vortices. By evaluating the weight function of the inviscid invariants it is shown that the loss of angular momentum is much smaller than that of circulation when vorticity is lost at the cell boundary by reconnection or annihilation. Thus the reconnection and subsequent annihilation of vortex pairs is responsible for the selective decay and concentration of vorticity. The relevance of the mechanism to previous experiments and general cases is also discussed.

Non-linear tides in a homogeneous rotating planet or star: global modes and elliptical instability

We revisit the global modes and instabilities of homogeneous rotating ellipsoidal fluid masses, which are the simplest global models of rotationally and tidally deformed gaseous planets or stars. The tidal flow in a short-period planet may be unstable to the elliptical instability, a hydrodynamic instability that can drive tidal evolution. We perform a global (and local WKB) analysis to study this instability using the elegant formalism of Lebovitz & Lifschitz. We survey the parameter space of global instabilities with harmonic orders $\ell\leq 5$, for planets with spins that are purely aligned (prograde) or anti-aligned (retrograde) with their orbits. In general, the instability has a much larger growth rate if the planetary spin and orbit are anti-aligned rather than aligned. We have identified a violent instability for anti-aligned spins outside of the usual frequency range for the elliptical instability (when $\frac{n}{\Omega}\lesssim -1$, where $n$ and $\Omega$ are the orbital and spin angular frequencies, respectively) if the tidal amplitude is sufficiently large. We also explore the instability in a rigid ellipsoidal container, which is found to be quantitatively similar to that with a realistic free surface. Finally, we study the effect of rotation and tidal deformation on mode frequencies. We find that larger rotation rates and larger tidal deformations both decrease the frequencies of the prograde sectoral surface gravity modes. This increases the prospect of their tidal excitation, potentially enhancing the tidal response over expectations from linear theory. In a companion paper, we use our results to interpret global simulations of the elliptical instability.

Non-linear tides in a homogeneous rotating planet or star: global simulations of the elliptical instability

I present results from the first global hydrodynamical simulations of the elliptical instability in a tidally deformed gaseous planet (or star) with a free surface. The elliptical instability is potentially important for tidal evolution of the shortest-period hot Jupiters. I model the planet as a spin-orbit aligned or anti-aligned, and non-synchronously rotating, tidally deformed, homogeneous fluid body. A companion paper presented an analysis of the global modes and instabilities of such a planet. Here I focus on the nonlinear evolution of the elliptical instability. This is observed to produce bursts of turbulence that drive the planet towards synchronism with its orbit in an erratic manner. If the planetary spin is initially anti-aligned, the elliptical instability also drives spin-orbit alignment on a similar timescale as the spin synchronisation. The instability generates differential rotation inside the planet in the form of zonal flows, which play an important role in the saturation of the instability, and in producing the observed burstiness. These results are broadly consistent with the picture obtained using a local Cartesian model (where columnar vortices played the role of zonal flows). I also simulate the instability in a container that is rigid (but stress-free) rather than free, finding broad quantitative agreement. The dissipation resulting from the elliptical instability could explain why the shortest-period hot Jupiters tend to have circular orbits inside about 2-3 days, and predicts spin-synchronisation (and spin-orbit alignment) out to about 10-15 days. However, other mechanisms must be invoked to explain tidal circularisation for longer orbital periods.

Ransom test results from various two-fluid schemes: Is enforcing hyperbolicity a thermodynamically consistent option?

The basic elliptic ill-posedness of physical models and numerical schemes for two-fluid flows is a recurring issue that has motivated the introduction of numerous possible correction strategies. In practical applications physical terms are generally present and regularize the models (viscosity, drag, surface tension, etc.). Yet, many numerical schemes were developed with the stringent and self-imposed constraint that the convective part of the models to be solved had to be hyperbolic, regardless of the type and magnitude of the particular physical regularizing terms. This leads to consider the simplest possible two-fluid “backbone” models corrected with the simplest “universal” terms to ensure hyperbolicity.Among the proposed corrections is the introduction of an interfacial pressure, either closed by algebraic relations or by supplementary evolution equations. Concurrently with the shift to hyperbolic behavior, these techniques also affect other features of systems: Kelvin–Helmholtz type instabilities are notably quenched at all scales, a highly undesirable effect in many practical situations. Less commonly recognized are also distortions in the transfers between kinetic, reversible, and irreversible energies, sometimes up to thermodynamic inconsistency.The present work aims at comparing on the standard Ransom-faucet test the results from various available hyperbolic and elliptic schemes and models against an explicit double Lagrange-plus-remap discretization of the basic elliptic, one-pressure, compressible, six-equations system (i.e. with energy equations). Four features are examined on this test: the entropy preservation, the stretched stream profile, the volume fraction discontinuity, and the unstable character of the analytical solution for the simplest backbone model.The paper highlights the fact that the convective part of two-fluid models might not be necessarily hyperbolic provided that it is physically consistent and numerically robust. Observation of published results for Ransom’s test shows that by enforcing hyperbolicity regardless of thermodynamical consistency, numerical models remove instabilities at the volume fraction discontinuity, but at the expense of distorted profiles of the stretched stream due to excessive numerical diffusion and to spurious forces in the momentum equation. The present approach provides a form of neutral starting point before including dissipative terms: robust but not excessively diffusive, with accurate capture of the stretched stream and volume fraction discontinuity for any practical mesh refinement. Moreover, and consistently with the chosen elliptic model, this numerical scheme eventually generates the elliptic instabilities for late times or fine meshes (but remains robust under the appropriate time step restrictions). It can be supplemented by any kind of small-scale regularization term in order to introduce a cut-off under which physical or numerical stability may be necessary.

Vortex formation in protoplanetary discs induced by the vertical shear instability

We present the results of 2D and 3D hydrodynamic simulations of idealized protoplanetary discs that examine the formation and evolution of vortices by the vertical shear instability (VSI). In agreement with recent work, we find that discs with radially decreasing temperature profiles and short thermal relaxation time-scales, are subject to the axisymmetric VSI. In three dimensions, the resulting velocity perturbations give rise to quasi-axisymmetric potential vorticity perturbations that break-up into discrete vortices, in a manner that is reminiscent of the Rossby wave instability. Discs with very short thermal evolution time-scales (i.e. {\tau}<0.1 local orbit periods) develop strong vorticity perturbations that roll up into vortices that have small aspect ratios ({\chi}<2) and short lifetimes (~ a few orbits). Longer thermal time-scales give rise to vortices with larger aspect ratios (6<{\chi}<10), and lifetimes that depend on the entropy gradient. A steeply decreasing entropy profile leads to vortex lifetimes that exceed the simulation run times of hundreds of orbital periods. Vortex lifetimes in discs with positive or weakly decreasing entropy profiles are much shorter, being 10s of orbits at most, suggesting that the subcritical baroclinic instability plays an important role in sustaining vortices against destruction through the elliptical instability. Applied to the outer regions of protoplanetary discs, where the VSI is most likely to occur, our results suggest that vortices formed by the VSI are likely to be short lived structures.

Internal structure of vortex rings and helical vortices

The internal structure of vortex rings and helical vortices is studied using asymptotic analysis and numerical simulations in cases where the core size of the vortex is small compared to its radius of curvature, or to the distance to other vortices. Several configurations are considered: a single vortex ring, an array of equally-spaced rings, a single helix and a regular array of helices. For such cases, the internal structure is assumed to be at leading order an axisymmetric concentrated vortex with an internal jet. A dipolar correction arises at first order and is shown to be the same for all cases, depending only on the local vortex curvature. A quadrupolar correction arises at second order. It is composed of two contributions, one associated with local curvature and another one arising from a non-local external 2-D strain field. This strain field itself is obtained by performing an asymptotic matching of the local internal solution with the external solution obtained from the Biot–Savart law. Only the amplitude of this strain field varies from one case to another. These asymptotic results are thereafter confronted with flow solutions obtained by direct numerical simulation (DNS) of the Navier–Stokes equations. Two different codes are used: for vortex rings, the simulations are performed in the axisymmetric framework; for helices, simulations are run using a dedicated code with built-in helical symmetry. Quantitative agreement is obtained. How these results can be used to theoretically predict the occurrence of both the elliptic instability and the curvature instability is finally addressed.

Three-dimensional instabilities and optimal perturbations of a counter-rotating vortex pair in stratified flows

This paper investigates the three-dimensional instabilities and the optimal perturbations on a pair of horizontal counter-rotating Lamb-Oseen vortices in a vertically stably stratified flow. Two-dimensional (2D) simulations are first performed, showing that while the dipole moves vertically against the stratification the vortex parameters: the radius a∗, the separation distance b∗, and the circulation Γ∗ are solely function of the time rescaled by the Brunt-Väisälä frequency N, independently of the Froude number, when the Reynolds number is large enough. Here, the Froude number is Fr = W0/Nb0 with W0 the initial advection velocity of the dipole and b0 the initial separation distance between the two vortices. For weak and moderate stratifications (large Fr), the stratification acts on a long time scale compared to the advection time of the dipole implying that the 2D flow can be considered as quasi-steady. In that case, when three dimensional instabilities are added, a linear stability analysis of this 2D flow at different instants retrieves the instability peaks corresponding to the Crow instability for the long wavelengths and to the elliptic instability for the short wavelengths showing that the dynamics is almost unaffected by buoyancy effects. The Crow and elliptic instabilities scale with the instantaneous dipole parameters showing in particular that stratification promotes instability by reducing the distance b∗ between vortices [K. K. Nomura et al., “Short-wavelength instability and decay of a vortex pair in stratified fluid,” J. Fluid Mech. 553, 283-322 (2006)]. For strong stratifications (Froude numbers of order unity or smaller), the quasi-steady approximation is not valid, and the question of stability should be formulated in a different way, by, for example, searching for the transient growth of the energy of perturbation that may be computed for steady or unsteady base flow. Then, for each time horizon τ, we should determine the critical perturbation leading to the largest energy growth by the time τ. Presently, we compute the optimal perturbations at two time horizons τ = 4 and τ = 10 dimensionalized by 2πb02/Γ0 with a direct-adjoint technique which takes into account the evolution of the base flow. In the homogeneous case, this technique allows to investigate the effect of the weak unsteadiness of the flow due to viscous diffusion which induces a growth of the vortex core radius a∗. Both Crow and elliptic instabilities are retrieved in the optimal response and in the energy gain curves. Even if very slow, the viscous diffusion is found to increase the gain of the antisymmetric elliptic perturbation compared to the symmetric one. When the fluid is stratified, peaks at small wavenumber and at wavenumber of the order of the vortex core size are found for all Froude numbers with optimal responses strongly resembling, respectively, the Crow and the elliptic modes with optimal gains corresponding to mean growth rates larger than in the homogeneous case for both modes. However, as the strength of stratification increases (Froude numbers smaller than 2), optimal perturbations start departing from their homogeneous counterpart with large perturbation in the wake of the dipole associated with density effects.

Convection in deformed bodies: The effect of equatorial ellipticity on convective behavior

Tidal interactions between bodies such as hot jupiter planets and their host stars are likely to result in non-spherical geometries. These elliptical instabilities may have interesting effects on interior fluid convective patterns, which in turn could influence the nature of the magnetic dynamo within these planets. Simulations of thermal convection in the 2D rotating equatorial plane are conducted to determine to first order the effect of equatorial eccentricity on convection for varying density contrasts with differing convective vigor and rotation rates. This survey is conducted in two dimensions in order to simulate a broad range of eccentricities and to maximize the parameter space explored. The location of the three regimes documented in previous work (Evonuk and Samuel, 2012), dipolar flow, transitional flow, and differential flow, are found to be offset by the introduction of equatorial eccentricity to the system. The introduction of equatorial eccentricity changes the fluid behavior such that bodies with high amounts of deformation are likely to have weaker differential flows shifting their behavior towards transitional and dipolar flow structures. A scaling law based on the convective Rossby number, density contrast, and the eccentricity of the equatorial plane can therefore provide a way to estimate which regime a given body lies in.

Generation and maintenance of bulk turbulence by libration-driven elliptical instability

Longitudinal libration corresponds to the periodic oscillation of a body’s rotation rate and is, along with precessional and tidal forcings, a possible source of mechanically-driven turbulence in the fluid interior of satellites and planets. In this study, we present a combination of direct numerical simulations and laboratory experiments, modeling this geophysically relevant mechanical forcing. We investigate the fluid motions inside a longitudinally librating ellipsoidal container filled with an incompressible fluid. The elliptical instability, which is a triadic resonance between two inertial modes and the oscillating base flow with elliptical streamlines, is observed both numerically and experimentally. The large-scale inertial modes eventually lead to small-scale turbulence, provided that the Ekman number is small enough. We characterize this transition to turbulence as additional triadic resonances develop while also investigating the properties of the turbulent flow that displays both intermittent and sustained regimes. These turbulent flows may play an important role in the thermal and magnetic evolution of bodies subject to mechanical forcing, which is not considered in standard models of convectively driven magnetic field generation.

A study of the turbulence within a spiralling vortex filament using proper orthogonal decomposition

The stability and turbulence characteristics of a vortex filament emanating from a single-bladed rotor in hover are investigated using proper orthogonal decomposition (POD). The rotor is operated at a tip chord Reynolds number and tip Mach number of 218 000 and 0.23, respectively, and with a blade loading of $C_{T}/{\it\sigma}=0.066$. In-plane components of the velocity field (normal to the axis of the vortex filament) are captured by way of two-dimensional particle image velocimetry with corrections for vortex wander being performed using the ${\it\Gamma}_{1}$ method. The first POD mode alone is found to encompass nearly 75 % of the energy for all vortex ages studied and is determined using a grid of sufficient resolution to avoid numerical integration errors in the decomposition. The findings reveal an equal balance between the axisymmetric and helical modes during vortex roll-up, which immediately transitions to helical mode dominance at all other vortex ages. This helical mode is one of the modes of the elliptic instability. The spatial eigenfunctions of the first few Fourier-azimuthal modes associated with the most energetic POD mode is shown to be sensitive to the choice of the wander correction technique used. Higher Fourier-azimuthal modes are observed in the outer portions of the vortex and appeared not to be affected by the choice of the wander correction technique used.

Experimental study of global-scale turbulence in a librating ellipsoid

We present laboratory experimental results demonstrating that librational forcing of an ellipsoidal container of water can produce intense motions through the mechanism of a libration driven elliptical instability (LDEI). These libration studies are conducted using an ellipsoidal acrylic container filled with water. A particle image velocimetry method is used to measure the 2D velocity field in the equatorial plane over hundreds libration cycles for a fixed Ekman number, E = 2 × 10−5. In doing so, we recover the libration induced base flow and a time averaged zonal flow. Further, we show that LDEI in non-axisymmetric container geometries is capable of driving both intermittent and saturated turbulent motions in the bulk fluid. Additionally, we measure the growth rate and amplitude of the LDEI induced excited flow in a fully ellipsoidal container at more extreme parameters than previously studied [Noir et al., “Experimental study of libration-driven flows in nonaxisymmetric containers,” Phys. Earth Planet. Inter. 204-205, 1 (2012); Cébron et al., Phys. Fluids 24, 061703, “Libration driven elliptical instability,” (2012)]. Excitation of bulk filling turbulence by librational forcing provides a mechanism for transferring rotational energy into turbulent fluid motion and thus can play an important role in the thermal evolution, interior dynamics, and magneto-hydrodynamics of librating bodies, as appear to be common in solar system settings [e.g., Comstock and Bills, “A solar system survey of forced librations in longitude,” J. Geophys. Res. Planets 108, 1 (2003)].

Effects of an axial flow on the centrifugal, elliptic and hyperbolic instabilities in Stuart vortices

AbstractLinear stability of the Stuart vortices in the presence of an axial flow is studied. The local stability equations derived by Lifschitz &amp; Hameiri (Phys. Fluids A, vol. 3 (11), 1991, pp. 2644–2651) are rewritten for a three-component (3C) two-dimensional (2D) base flow represented by a 2D streamfunction and an axial velocity that is a function of the streamfunction. We show that the local perturbations that describe an eigenmode of the flow should have wavevectors that are periodic upon their evolution around helical flow trajectories that are themselves periodic once projected on a plane perpendicular to the axial direction. Integrating the amplitude equations around periodic trajectories for wavevectors that are also periodic, it is found that the elliptic and hyperbolic instabilities, which are present without the axial velocity, disappear beyond a threshold value for the axial velocity strength. Furthermore, a threshold axial velocity strength, above which a new centrifugal instability branch is present, is identified. A heuristic criterion, which reduces to the Leibovich &amp; Stewartson criterion in the limit of an axisymmetric vortex, for centrifugal instability in a non-axisymmetric vortex with an axial flow is then proposed. The new criterion, upon comparison with the numerical solutions of the local stability equations, is shown to describe the onset of centrifugal instability (and the corresponding growth rate) very accurately.