Vortex Wave Interaction Research Articles

OAM vortex wave interaction with phase modulated metasurface

Phase modulated metasurface (PMM) can control the transmission state of electromagnetic (EM) wave through phase modulation scheme on the interface. Orbital angular momentum (OAM) vortex wave is denoted by the helical phase distribution of the wave front. The interaction between OAM vortex wave and PMM is investigated in this paper. The mathematical model is firstly established according to the array theory. Whereafter, two typical PMMs of chessboard metasurface (CBM) and phase gradient metasurface (PGM) are exploited as examples to uncover the scattering characteristics under illumination of OAM vortex wave. For CBM, the phase cancellation scheme is found to be broken when the OAM order l equals to ±2 under both normal incidence and oblique incidence. It reveals that the OAM vortex wave is a promising approach for metasurface stealth target detection. For PGM, the scattered wave still keeps the OAM feature but is deflected to the non-specular direction, which reveals that the generalized Snell’s law is also suitable to OAM vortex wave. The discoveries of this paper may find applications in radar detection fields using OAM vortex wave.

Rotating curved spacetime signatures from a giant quantum vortex.

Gravity simulators1 are laboratory systems in which small excitations such as sound2 or surface waves3,4 behave as fields propagating on a curved spacetime geometry. The analogy between gravity and fluids requires vanishing viscosity2-4, a feature naturally realized in superfluids such as liquid helium or cold atomic clouds5-8. Such systems have been successful in verifying key predictions of quantum field theory in curved spacetime7-11. In particular, quantum simulations of rotating curved spacetimes indicative of astrophysical black holes require the realization of an extensive vortex flow12 in superfluid systems. Here we demonstrate that, despite the inherent instability of multiply quantized vortices13,14, a stationary giant quantum vortex can be stabilized in superfluid 4He. Its compact core carries thousands of circulation quanta, prevailing over current limitations in other physical systems such as magnons5, atomic clouds6,7 and polaritons15,16. We introduce a minimally invasive way to characterize the vortex flow17,18 by exploiting the interaction of micrometre-scale waves on the superfluid interface with the background velocity field. Intricate wave-vortex interactions, including the detection of bound states and distinctive analogue black hole ringdown signatures, have been observed. These results open new avenues to explore quantum-to-classical vortex transitions and use superfluid helium as a finite-temperature quantum field theory simulator for rotating curved spacetimes19.

A self-sustaining mechanism for plane Couette flow with a flexible lower boundary

Abstract The stability of plane Couette flow to travelling-wave disturbances is studied asymptotically at high Reynolds numbers Re when the lower boundary possesses a degree of flexibility modelled by a spring-backed plate. First, it is shown that a three-dimensional (3D) linear instability exists, with streamwise and spanwise wavelengths comparable with the channel width. Building on this, nonlinear effects from the self-interaction of the wave are introduced, leading to a self-sustaining interaction between a roll/streak flow and the 3D wave. Governing nonlinear vortex-wave interaction (VWI) equations are derived and a perturbation analysis is carried out to guide a numerical investigation of the equations. The co-existence of two families of finite-amplitude solutions, each with different flow structures, is found. Numerical solutions of the VWI equations in each case show that a small wave amplitude of O(Re−1(log Re)−1/2) is all that is necessary to provoke an O(1) change to the basic Couette flow.

How tidal waves interact with convective vortices in rapidly rotating planets and stars

Context. The dissipation of tidal inertial waves in planetary and stellar convective regions is one of the key mechanisms that drive the evolution of star–planet and planet–moon systems. This dissipation is particularly efficient for young low-mass stars and gaseous giant planets, which are rapid rotators. In this context, the interaction between tidal inertial waves and turbulent convective flows must be modelled in a realistic and robust way. In the state-of-the-art simulations, the friction applied by convection on tidal waves is commonly modeled as an effective eddy viscosity. This approach may be valid when the characteristic length scales of convective eddies are smaller than those of the tidal waves. However, it becomes highly questionable in the case where tidal waves interact with potentially stable large-scale vortices such as those observed at the poles of Jupiter and Saturn. The large-scale vortices are potentially triggered by convection in rapidly-rotating bodies in which the Coriolis acceleration forms the flow in columnar vortical structures along the direction of the rotation axis. Aims. We investigate the complex interactions between a tidal inertial wave and a columnar convective vortex. Methods. We used a quasi-geostrophic semi-analytical model of a convective columnar vortex, which is validated by numerical simulations. First, we carried out linear stability analysis using both numerical and asymptotic Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) methods. We then conducted linear numerical simulations of the interactions between a convective columnar vortex and an incoming tidal inertial wave. Results. The vortex we consider is found to be centrifugally stable in the range –Ωp ≤ Ω0 ≤ 3.62Ωp and unstable outside this range, where Ω0 is the local rotation rate of the vortex at its center and Ωp is the global planetary (stellar) rotation rate. From the linear stability analysis, we find that this vortex is prone to centrifugal instability with perturbations with azimuthal wavenumbers m = {0,1, 2}, which potentially correspond to eccentricity, obliquity, and asynchronous tides, respectively. The modes with m > 2 are found to be neutral or stable. The WKBJ analysis provides analytic expressions of the dispersion relations for neutral and unstable modes when the axial (vertical) wavenumber is sufficiently large. We verify that in the unstable regime, an incoming tidal inertial wave triggers the growth of the most unstable mode of the vortex. This would lead to turbulent dissipation. For stable convective columns, the wave-vortex interaction leads to the mixing of momentum for tidal inertial waves while it creates a low-velocity region around the vortex core and a new wave-like perturbation in the form of a progressive wave radiating in the far field. The emission of this secondary wave is the strongest when the wavelength of the incoming wave is close to the characteristic size (radius) of the vortex. Incoming tidal waves can also experience complex angular momentum exchanges locally at critical layers of stable vortices. Conclusions. The interaction between tidal inertial waves and large-scale coherent convective vortices in rapidly-rotating planets (stars) leads to turbulent dissipation in the unstable regime and complex behaviors such as mixing of momentum and radiation of new waves in the far field or wave-vortex angular momentum exchanges in the stable regime. These phenomena cannot be modeled using a simple effective eddy viscosity.

Three Regimes of Internal Gravity Wave–Stable Vortex Interaction Classified by a Nondimensional Parameter δ: Scattering, Wheel-Trapping, and Spiral-Trapping with Vortex Deformation

Abstract The internal wave–vortex interaction was investigated for a broad parameter range except near inertial waves, by 1) scaling, 2) numerical experiments, and 3) the estimation of possible occurrences. By scaling, we identified a nondimensional parameter, δ = (V/c)[1/(kR)], where V is the vortex flow speed, R is the radius, c is the incident wave phase speed, and k is the horizontal wavenumber. As δ appears in all terms related to the interaction, it is important in the classification of the wave–vortex interaction. Numerical experiments were conducted on internal waves incident on a stable barotropic vortex with a parameter range of δ = [0.001, 1.7], which is much broader than that used in previous studies (δ ≪ 1). We found new phenomena for δ > 0.15, in addition to previously known scattering for δ ≤ 0.15 (scattering regime). For 0.15 < δ ≤ 0.4, part of the incident internal wave is trapped in a vortex, forming a wheel-like shape maintaining a superinertial frequency (wheel-trapping regime). When δ > 0.4, incident waves are trapped, but with a spiral shape (spiral-trapping regime). Spiral-shaped trapped waves release momentum by wave breaking, which deforms the vortex into a zigzag shape in the vertical direction. Vortex deformation produces vertical shear, which rapidly increases the vertical wavenumber of the incident wave. The distribution of δ in the Pacific Ocean was estimated using a high-resolution (1/30°) ocean general circulation model output. We found the occurrences of all three regimes. The scattering and wheel-trapping regimes are distributed broadly and varied seasonally, thus affecting mixing variability. Significance Statement Oceanic internal waves constitute the fundamental forcing of overturning and material circulation, because internal waves eventually break and cause vertical mixing. Interactions between internal waves and vortices affect wave properties and, therefore, mixing. However, as far as we are aware, all previous studies have focused on large weak vortices relative to waves. Here, we investigated such interactions for a much larger parameter space and identified two new regimes, in which vertical mixing is caused by newly found internal wave trapping and vortex deformation processes. We identified a nondimensional parameter that classifies the regimes and estimated their spatiotemporal distribution. These results suggest new energy routes from internal waves to turbulence and are applicable to other types of waves and vortices.

Two-way wave–vortex interactions in a Lagrangian-mean shallow water model

We derive and investigate numerically a reduced model for wave–vortex interactions involving non-dispersive waves, which we study in a two-dimensional shallow water system with an eye towards applications in atmosphere–ocean fluid dynamics. The model consists of a coupled set of nonlinear partial differential equations for the Lagrangian-mean velocity and the wave-related pseudomomentum vector field defined in generalized Lagrangian-mean theory. It allows for two-way interactions between the waves and the balanced flow that is controlled by the distribution of Lagrangian-mean potential vorticity, and for strong solutions it features a desirable exact energy conservation law for the sum of wave energy and mean flow energy. Our model goes beyond standard ray tracing as we can derive weak solutions that contain discontinuities in the pseudomomentum field, using the theory of weakly hyperbolic systems. This allows caustics to form without predicting infinite wave amplitudes, as would be the case in the standard ray-tracing theory. Suitable wave forcing and dissipation terms are added to the model and a numerical scheme for the model is implemented as a coupled set of pseudo-spectral and finite-volume integrators. Idealized examples of interactions between wavepackets and simple vortex structures are presented to illustrate the model dynamics. The unforced and non-dissipative simulations suggest a heuristic rule of ‘greedy’ waves, i.e. in the long run the wave field always extracts energy from the mean flow.

Sound-ring radiation of expanding vortex clusters

We investigate wave-vortex interaction emerging from an expanding compact vortex cluster in a two-dimensional Bose-Einstein condensate. We adapt techniques developed for compact gravitational objects to derive the characteristic modes of the wave-vortex interaction perturbatively around an effective vortex flow field. We demonstrate the existence of orbits or sound-rings, in analogy to gravitational light-rings, and compute the characteristic spectrum for the out-of-equilibrium vortex cluster. The spectrum obtained from numerical simulations of a stochastic Gross-Pitaevskii equation exhibiting an expanding vortex cluster is in excellent agreement with analytical predictions. Our findings are relevant for 2d-quantum turbulence, the semi-classical limit around fluid flows, and rotating compact objects exhibiting discrete circulation.

A vortex–wave interaction theory describing the effect of boundary forcing on shear flows

A strongly nonlinear theory describing the effect of small amplitude boundary forcing in the form of waves on high Reynolds number shear flows is given. The interaction leads to an $O(1)$ change in the unperturbed flow and is relevant to a number of forcing mechanisms. The cases of the shear flow being bounded or unbounded are both considered and the results for the unbounded case apply to quite arbitrary flows. The instability criterion for unbounded flows is expressed in terms of the wall forcing and the friction Reynolds number. As particular examples we investigate wall transpiration or surface undulations as sources of the forcing and both propagating and stationary waves are considered. Results are given for propagating waves with crests perpendicular to the flow direction and for stationary waves with crests no longer perpendicular to the flow direction. In the first of those situations we find the instability induced by transpiration waves is independent of the propagation speed. For wavy walls downstream propagation completely stabilises the flow at a critical speed whereas upstream propagation greatly destabilises the flow. For stationary oblique waves we find that the instability is enhanced and a much wider range of unstable wavenumbers exists. For the bounded case with a wall of fixed wavelength we identify a critical wavelength where the most dangerous mode switches from the aligned to the oblique configuration. For the transpiration problem in the oblique configuration a strong resonance occurs when the vortex wavelength coincides with the spanwise wavelength of the forcing.

Poiseuille flow in rough pipes: linear instability induced by vortex–wave interactions

Abstract

An instability mechanism for channel flows in the presence of wall roughness

Abstract

Wave–vortex interactions and effective mean forces: three basic problems

Three examples of wave–vortex interaction are studied, in analytically tractable weak refraction regimes with attention to the mean recoil forces, local and remote, that are associated with refractive changes in wave pseudomomentum fluxes. Wave-induced mean forces of this kind can be persistent, with cumulative effects, even in the absence of wave dissipation. In each example, a single wavetrain propagates past a single vortex. In the first two examples, in a two-dimensional, non-rotating acoustic or shallow-water setting, the focus is on whether or not the wavetrain overlaps the vortex core. In the overlapping case, the recoil has a local contribution given by the Craik–Leibovich force on the vortex core, the vector product of Stokes drift and mean vorticity. (For a quantum vortex this contribution is called the Iordanskii force arising from the Aharonov–Bohm effect on a phonon current.) However, in all except one special limiting case there are additional “remote” contributions, mediated by Stokes-drift-induced return flows that can intersect the vortex core well away from locations where the waves are refracted. The third example is a non-overlapping, remote-recoil-only example in a rapidly rotating frame, in which the waves are deep-water gravity waves and the mean flow obeys shallow-water quasigeostrophic dynamics. Contrary to what might at first be thought, the Ursell “anti-Stokes flow” induced by the rotation – an Eulerian-mean flow tending to cancel the Stokes drift – fails to suppress remote recoil. There are nontrivial open questions about extending these results to regimes of stronger refraction, especially regarding the scope of the “pseudomomentum rule” for the wave-induced recoil forces.

Wave–vortex interactions, remote recoil, the Aharonov–Bohm effect and the Craik–Leibovich equation

Three examples of non-dissipative yet cumulative interaction between a single wavetrain and a single vortex are analysed, with a focus on effective recoil forces, local and remote. Local recoil occurs when the wavetrain overlaps the vortex core. All three examples comply with the pseudomomentum rule. The first two examples are two-dimensional and non-rotating (shallow water or gas dynamical). The third is rotating, with deep-water gravity waves inducing an Ursell ‘anti-Stokes flow’. The Froude or Mach number, and the Rossby number in the third example, are assumed small. Remote recoil is all or part of the interaction in all three examples, except in one special limiting case. That case is found only within a severely restricted parameter regime and is the only case in which, exceptionally, the effective recoil force can be regarded as purely local and identifiable with the celebrated Craik–Leibovich vortex force – which corresponds, in the quantum fluids literature, to the Iordanskii force due to a phonon current incident on a vortex. Another peculiarity of that exceptional case is that the only significant wave refraction effect is the Aharonov–Bohm topological phase jump.

Centre modes in pipe flow

Abstract In two previous papers, Ozcakir, Tanveer, Hall, & Overman (2016, Travelling waves in pipe flow, J. Fluid Mech., 791, 284–328) and Ozcakir, Hall & Tanveer (2019, Nonlinear exact coherent structures in pipe flow and their instabilities, J. Fluid Mech., 868, 341–368) investigated numerically and asymptotically high Reynolds number exact coherent structures in pipe flow. It was found that, in addition to the structures described by the vortex–wave interaction theory by Hall & Smith (1991, On strongly nonlinear vortex/wave interactions in boundary layer transition. J. Fluid Mech., 227, 641–666), there exists vortical structures localized near the centre of the pipe with a core of size $O(Re^{-1/4})$ convected downstream at a speed that deviates from the pipe centreline speed by $O(Re^{-1/2})$, where $Re$ is the Reynolds number. In the finite Reynolds number calculations by Ozcakir, Tanveer, Hall & Overman (2016, Travelling waves in pipe flow, J. Fluid Mech., 791, 284–328), asymptotic state was referred to as a nonlinear viscous core state (NVC). However the reduced asymptotic equations were not solved and only limited confirmation of the theory was found numerically. Here, in order to conclusively confirm the existence of the NVC state we first describe direct numerical calculations on the asymptotically reduced $Re>>1$ equations for such state states. The results are then compared in detail to the finite $Re$ calculations up-to $Re=10^6$; the latter regime is at much higher values of the Reynolds number than those reported in Ozcakir, Tanveer, Hall & Overman (2016, Travelling waves in pipe flow, J. Fluid Mech., 791, 284–328). The results are found to be in excellent agreement with the finite $Re$ calculations in a region between $Re=10^5$ and $10^6$, thereby confirming that the structure observed by Ozcakir, Tanveer, Hall & Overman (2016, Travelling waves in pipe flow, J. Fluid Mech., 791, 284–328) is indeed a finite Reynolds number realization of an asymptotic NVC state.

Shear stress-driven flow: the state space of near-wall turbulence as

An inner-scaled, shear stress-driven flow is considered as a model of independent near-wall turbulence as the friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}\rightarrow \infty$. In this limit, the model is applicable to the near-wall region and the lower part of the logarithmic layer of various parallel shear flows, including turbulent Couette flow, Poiseuille flow and Hagen–Poiseuille flow. The model is validated against damped Couette flow and there is excellent agreement between the velocity statistics and spectra for the wall-normal height $y^{+}<40$. A near-wall flow domain of similar size to the minimal unit is analysed from a dynamical systems perspective. The edge and fifteen invariant solutions are computed, the first discovered for this flow configuration. Through continuation in the spanwise width $L_{z}^{+}$, the bifurcation behaviour of the solutions over the domain size is investigated. The physical properties of the solutions are explored through phase portraits, including the energy input and dissipation plane, and streak, roll and wave energy space. Finally, a Reynolds number is defined in outer units and the high-$Re$ asymptotic behaviour of the equilibria is studied. Three lower branch solutions are found to scale consistently with vortex–wave interaction (VWI) theory, with wave forcing localising around the critical layer.

Nonlinear exact coherent structures in pipe flow and their instabilities

In this paper, we present computational results of some two-fold azimuthally symmetric travelling waves and their stability. Calculations over a range of Reynolds numbers ($Re$) reveal connections between a class of solutions computed by Wedin & Kerswell (J. Fluid Mech., vol. 508, 2004, pp. 333–371) (henceforth called the WK solution) and the $Re\rightarrow \infty$ vortex–wave interaction theory of Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666) and Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). In particular, the continuation of the WK solutions to larger values of $Re$ shows that the WK solution bifurcates from a shift-and-rotate symmetric solution, which we call the WK2 state. The WK2 solution computed for $Re\leqslant 1.19\times 10^{6}$ shows excellent agreement with the theoretical $Re^{-5/6}$, $Re^{-1}$ and $O(1)$ scalings of the waves, rolls and streaks respectively. Furthermore, these states are found to have only two unstable modes in the large $Re$ regime, with growth rates estimated to be $O(Re^{-0.42})$ and $O(Re^{-0.92})$, close to the theoretical $O(Re^{-1/2})$ and $O(Re^{-1})$ asymptotic results for edge and sinuous instability modes of vortex–wave interaction states (Deguchi & Hall, J. Fluid Mech., vol. 802, 2016, pp. 634–666) in plane Couette flow. For the nonlinear viscous core states (Ozcakir et al., J. Fluid Mech., vol. 791, 2016, pp. 284–328), characterized by spatial a shrinking of the wave and roll structure towards the pipe centre with increasing $Re$, we continued the solution to $Re\leqslant 8\times 10^{6}$ and we find only one unstable mode in the large Reynolds number regime, with growth rate scaling as $Re^{-0.46}$ within the class of symmetry-preserving disturbances.

Vortex–wave interaction arrays: a sustaining mechanism for the log layer?

Vortex–wave interaction theory is used to describe new kinds of localised and distributed exact coherent structures. Starting with a localised vortex–wave interaction state driven by a single inviscid wave, regular arrays of interacting vortex–wave states are investigated. In the first instance the arrays described are operational in an infinite uniform shear flow; we refer to them as ‘uniform shear vortex–wave arrays’. The basic form of the interaction remains identical to the canonical one found by Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666) and subsequently used to describe exact coherent structures by Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). Thus in each cell of a vortex–wave array a roll stress jump is induced across the critical layer of an inviscid wave riding on the streak part of the flow. The theory is extended to arbitrary shear flows using a nonlinear Wentzel–Kramers–Brillouin–Jeffreys or ray theory approach with the wave–roll–streak field operating on a shorter length scale than the mean flow. The evolution equation governing the slow dynamics of the interaction turns out to be a modified form of the well-known mean equation for a turbulent flow, and its particular form can be interpreted as a ‘closure’ between the small and large scales of the flow. If the array structure is taken to be universal, in the sense that it applies to arbitrary shear flows, then the array takes on a form which supports a logarithmic mean velocity profile trapped between what can be identified with the ‘wake region’ and a ‘buffer layer’ well known in the context of wall-bounded turbulent flows. The many similarities between the distributed structures described and wall-bounded turbulence suggest that vortex–wave arrays might be involved in the self-sustaining process supporting the log layer. The modification of the mean profile within each cell of the array leads to ‘staircase’-like streamwise velocity profiles similar to those observed experimentally in turbulent flows. The wave field supporting the ‘staircase’ is concentrated in critical layers which can be associated with the shear layer structures that have been attributed by experimentalists to be the mechanism supporting the uniform-momentum zones of the staircase.

Random forcing of geostrophic motion in rotating stratified turbulence

Random forcing of geostrophic motion is a common approach in idealized simulations of rotating stratified turbulence. Such forcing represents the injection of energy into large-scale balanced motion, and the resulting breakdown of quasi-geostrophic turbulence into inertia–gravity waves and stratified turbulence can shed light on the turbulent cascade processes of the atmospheric mesoscale. White noise forcing is commonly employed, which excites all frequencies equally, including frequencies much higher than the natural frequencies of large-scale vortices. In this paper, the effects of these high frequencies in the forcing are investigated. Geostrophic motion is randomly forced with red noise over a range of decorrelation time scales τ, from a few time steps to twice the large-scale vortex time scale. It is found that short τ (i.e., nearly white noise) results in about 46% more gravity wave energy than longer τ, despite the fact that waves are not directly forced. We argue that this effect is due to wave–vortex interactions, through which the high frequencies in the forcing are able to excite waves at their natural frequencies. It is concluded that white noise forcing should be avoided, even if it is only applied to the geostrophic motion, when a careful investigation of spontaneous wave generation is needed.

Energy production and self-sustained turbulence at the Kolmogorov scale in Couette flow

Several recent studies have reported that there exists a self-similar form of invariant solutions down to the Kolmogorov microscale in the bulk region of turbulent Couette flow. While their role in a fully developed turbulent flow is yet to be identified, here we report that there exists a related mechanism of turbulence production at the Kolmogorov microscale in the bulk region of turbulent Couette flow by performing a set of minimal-span direct numerical simulations up to friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}\simeq 800$. This mechanism is found to essentially originate from the non-zero mean shear in the bulk region of the Couette flow, and involves eddy turn-over dynamics remarkably similar to the so-called self-sustaining process (SSP) and/or vortex–wave interaction (VWI). A numerical experiment that removes all other motions except in the core region is also performed, which demonstrates that the eddies at a given wall-normal location in the bulk region are sustained in the absence of other motions at different wall-normal locations. It is proposed that the self-sustaining eddies at the Kolmogorov microscale correspond to those in uniform shear turbulence at transitional Reynolds numbers, and a quantitative comparison between the eddies in uniform shear and near-wall turbulence is subsequently made. Finally, it is shown that turbulence production by the self-sustaining eddies at the Kolmogorov microscale is much smaller than that of full-scale simulations, and that the difference between the two increases with Reynolds number.

Exact coherent structures in stably stratified plane Couette flow

The existence of exact coherent structures in stably stratified plane Couette flow (gravity perpendicular to the plates) is investigated over Reynolds–Richardson number ($Re$–$Ri_{b}$) space for a fluid of unit Prandtl number $(Pr=1)$ using a combination of numerical and asymptotic techniques. Two states are repeatedly discovered using edge tracking – EQ7 and EQ7-1 in the nomenclature of Gibson & Brand (J. Fluid Mech., vol. 745, 2014, pp. 25–61) – and found to connect with two-dimensional convective roll solutions when tracked to negative $Ri_{b}$ (the Rayleigh–Bénard problem with shear). Both these states and Nagata’s (J. Fluid Mech., vol. 217, 1990, pp. 519–527) original exact solution feel the presence of stable stratification when $Ri_{b}=O(Re^{-2})$ or equivalently when the Rayleigh number $Ra:=-Ri_{b}Re^{2}Pr=O(1)$. This is confirmed via a stratified extension of the vortex wave interaction theory of Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). If the stratification is increased further, EQ7 is found to progressively spanwise and cross-stream localise until a second regime is entered at $Ri_{b}=O(Re^{-2/3})$. This corresponds to a stratified version of the boundary region equations regime of Deguchi, Hall & Walton (J. Fluid Mech., vol. 721, 2013, pp. 58–85). Increasing the stratification further appears to lead to a third, ultimate regime where $Ri_{b}=O(1)$ in which the flow fully localises in all three directions at the minimal Kolmogorov scale which then corresponds to the Osmidov scale. Implications for the laminar–turbulent boundary in the ($Re$–$Ri_{b}$) plane are briefly discussed.

Sustaining processes from recurrent flows in body-forced turbulence

By extracting unstable invariant solutions directly from body-forced three-dimensional turbulence, we study the dynamical processes at play when the forcing is large scale and unidirectional in either the momentum or the vorticity equations. In the former case, the dynamical processes familiar from recent work on linearly stable shear flows – variously called the self-sustaining process (Waleffe, Phys. Fluids, vol. 9 (4), 1997, pp. 883–900) or vortex–wave interaction (Hall & Smith, J. Fluid Mech., vol. 227, 1991, pp. 641–666; Hall & Sherwin, J. Fluid Mech., vol. 661, 2010, pp. 178–205) – are important even when the base flow is linearly unstable. In the latter case, where the forcing drives Taylor–Green vortices, a number of mechanisms are observed from the various types of periodic orbits isolated. In particular, two different transient growth mechanisms are discussed to explain the more complex states found.