Numerical Validation of the Inverse Cascade of Surface Gravity Wave Action.

We report numerical simulations of surface gravity waves forced at small scale and the subsequent inverse cascade of wave action. We combine the spectral approach to simulating weakly nonlinear waves with the capabilities of modern graphics processing units to reach unprecedented scale separation between the forcing and domain scales. The resulting broad inertial range allows for an unambiguous confirmation of the theoretical prediction for the spectrum in the inverse cascade regime, both in terms of spectral index and dependence of the spectral level on the action flux.

Velocity-informed upper bounds on the convective heat transport induced by internal heat sources and sinks.

Three-dimensional convection driven by internal heat sources and sinks (CISS) leads to experimental and numerical scaling laws compatible with a mixing-length-or 'ultimate'-scaling regime [Formula: see text]. However, asymptotic analytic solutions and idealized two-dimensional simulations have shown that laminar flow solutions can transport heat even more efficiently, with [Formula: see text]. The turbulent nature of the flow thus has a profound impact on its transport properties. In the present contribution, we give this statement a precise mathematical sense. We show that the Nusselt number maximized over all solutions is bounded from above by [Formula: see text], before restricting attention to 'fully turbulent branches of solutions', defined as families of solutions characterized by a finite non-zero limit of the dissipation coefficient at large driving amplitude. Maximization of [Formula: see text] over such branches of solutions yields the better upper-bound [Formula: see text]. We then provide three-dimensional numerical and experimental data of CISS compatible with a finite limiting value of the dissipation coefficient at large driving amplitude. It thus seems that CISS achieves the maximal heat transport scaling over fully turbulent solutions. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.

Experimental observation of the geostrophic turbulence regime of rapidly rotating convection.

The competition between turbulent convection and global rotation in planetary and stellar interiors governs the transport of heat and tracers, as well as magnetic field generation. These objects operate in dynamical regimes ranging from weakly rotating convection to the "geostrophic turbulence" regime of rapidly rotating convection. However, the latter regime has remained elusive in the laboratory, despite a worldwide effort to design ever-taller rotating convection cells over the last decade. Building on a recent experimental approach where convection is driven radiatively, we report heat transport measurements in quantitative agreement with this scaling regime, the experimental scaling law being validated against direct numerical simulations (DNS) of the idealized setup. The scaling exponent from both experiments and DNS agrees well with the geostrophic turbulence prediction. The prefactor of the scaling law is greater than the one diagnosed in previous idealized numerical studies, pointing to an unexpected sensitivity of the heat transport efficiency to the precise distribution of heat sources and sinks, which greatly varies from planets to stars.

Enhanced dynamo growth in nonhomogeneous conducting fluids.

We address magnetic-field generation by dynamo action in systems with inhomogeneous electrical conductivity and magnetic permeability. More specifically, we first show that the Taylor-Couette kinematic dynamo undergoes a drastic reduction of its stability threshold when a (zero-mean) modulation of the fluid's electrical conductivity or magnetic permeability is introduced. These results are obtained outside the mean-field regime, for which this effect was initially proposed. Beyond this illustrative example, we extend a duality argument put forward by Favier and Proctor (2013) to show that swapping the distributions of conductivity and permeability and changing u→-u leaves the dynamo threshold unchanged. This allows one to make connections between a priori unrelated dynamo studies. Finally, we discuss the possibility of observing such an effect both in laboratory and astrophysical settings.

Shortcut to Geostrophy in Wave-Driven Rotating Turbulence: The Quartetic Instability.

We report on laboratory experiments of wave-driven rotating turbulence. A set of wave makers produces inertial-wave beams that interact nonlinearly in the central region of a water tank mounted on a rotating platform. The forcing thus injects energy into inertial waves only. For moderate forcing amplitude, part of the energy of the forced inertial waves is transferred to subharmonic waves, through a standard triadic resonance instability. This first step is broadly in line with the theory of weak turbulence. Surprisingly however, stronger forcing does not lead to an inertial-wave turbulence regime. Instead, most of the kinetic energy condenses into a vertically invariant geostrophic flow, even though the latter is unforced. We show that resonant quartets of inertial waves can trigger an instability-the "quartetic instability"-that leads to such spontaneous emergence of geostrophy. In the present experiment, this instability sets in as a secondary instability of the classical triadic instability.

The vortex gas scaling regime of baroclinic turbulence.

The mean state of the atmosphere and ocean is set through a balance between external forcing (radiation, winds, heat and freshwater fluxes) and the emergent turbulence, which transfers energy to dissipative structures. The forcing gives rise to jets in the atmosphere and currents in the ocean, which spontaneously develop turbulent eddies through the baroclinic instability. A critical step in the development of a theory of climate is to properly include the eddy-induced turbulent transport of properties like heat, moisture, and carbon. In the linear stages, baroclinic instability generates flow structures at the Rossby deformation radius, a length scale of order 1,000 km in the atmosphere and 100 km in the ocean, smaller than the planetary scale and the typical extent of ocean basins, respectively. There is, therefore, a separation of scales between the large-scale gradient of properties like temperature and the smaller eddies that advect it randomly, inducing effective diffusion. Numerical solutions show that such scale separation remains in the strongly nonlinear turbulent regime, provided there is sufficient drag at the bottom of the atmosphere and ocean. We compute the scaling laws governing the eddy-driven transport associated with baroclinic turbulence. First, we provide a theoretical underpinning for empirical scaling laws reported in previous studies, for different formulations of the bottom drag law. Second, these scaling laws are shown to provide an important first step toward an accurate local closure to predict the impact of baroclinic turbulence in setting the large-scale temperature profiles in the atmosphere and ocean.

Quantitative Experimental Observation of Weak Inertial-Wave Turbulence.

We report the quantitative experimental observation of the weak inertial-wave turbulence regime of rotating turbulence. We produce a statistically steady homogeneous turbulent flow that consists of nonlinearly interacting inertial waves, using rough top and bottom boundaries to prevent the emergence of a geostrophic flow. As the forcing amplitude increases, the temporal spectrum evolves from a discrete set of peaks to a continuous spectrum. Maps of the bicoherence of the velocity field confirm such a gradual transition between discrete wave interactions at weak forcing amplitude and the regime described by weak turbulence theory (WTT) for stronger forcing. In the former regime, the bicoherence maps display a near-zero background level, together with sharp localized peaks associated with discrete resonances. By contrast, in the latter regime, the bicoherence is a smooth function that takes values of the order of the Rossby number in line with the infinite-domain and random-phase assumptions of WTT. The spatial spectra then display a power-law behavior, both the spectral exponent and the spectral level being accurately predicted by WTT at high Reynolds number and low Rossby number.

Radiative heating achieves the ultimate regime of thermal convection.

The absorption of light or radiation drives turbulent convection inside stars, supernovae, frozen lakes, and Earth's mantle. In these contexts, the goal of laboratory and numerical studies is to determine the relation between the internal temperature gradients and the heat flux transported by the turbulent flow. This is the constitutive law of turbulent convection, to be input into large-scale models of such natural flows. However, in contrast with the radiative heating of natural flows, laboratory experiments have focused on convection driven by heating and cooling plates; the heat transport is then severely restricted by boundary layers near the plates, which prevents the realization of the mixing length scaling law used in evolution models of geophysical and astrophysical flows. There is therefore an important discrepancy between the scaling laws measured in laboratory experiments and those used, e.g., in stellar evolution models. Here we provide experimental and numerical evidence that radiatively driven convection spontaneously achieves the mixing length scaling regime, also known as the "ultimate" regime of thermal convection. This constitutes a clear observation of this regime of turbulent convection. Our study therefore bridges the gap between models of natural flows and laboratory experiments. It opens an experimental avenue for a priori determinations of the constitutive laws to be implemented into models of geophysical and astrophysical flows, as opposed to empirical fits of these constitutive laws to the scarce observational data.

Transition to Turbulent Dynamo Saturation.

While the saturated magnetic energy is independent of viscosity in dynamo experiments, it remains viscosity dependent in state-of-the-art 3D direct numerical simulations (DNS). Extrapolating such viscous scaling laws to realistic parameter values leads to an underestimation of the magnetic energy by several orders of magnitude. The origin of this discrepancy is that fully 3D DNS cannot reach low enough values of the magnetic Prandtl number Pm. To bypass this limitation and investigate dynamo saturation at very low Pm, we focus on the vicinity of the dynamo threshold in a rapidly rotating flow: the velocity field then depends on two spatial coordinates only, while the magnetic field consists of a single Fourier mode in the third direction. We perform numerical simulations of the resulting set of reduced equations for Pm down to 2×10^{-5}. This parameter regime is currently out of reach to fully 3D DNS. We show that the magnetic energy transitions from a high-Pm viscous scaling regime to a low-Pm turbulent scaling regime, the latter being independent of viscosity. The transition to the turbulent saturation regime occurs at a low value of the magnetic Prandtl number, Pm≃10^{-3}, which explains why it has been overlooked by numerical studies so far.

Energy-dissipation anomaly in systems of localized waves.

We study the statistics of the power P dissipated by waves propagating in a one-dimensional disordered medium with damping coefficient ν. An operator imposes the wave amplitude at one end, therefore injecting a power P that balances dissipation. The typical realization of P vanishes for ν→0: Disorder leads to localization and total reflection of the wave energy back to the emitter, with negligible losses. More surprisingly, the mean dissipated power 〈P〉 averaged over the disorder reaches a finite limit for ν→0. We show that this "anomalous dissipation" lim_{ν→0}〈P〉 is directly given by the integrated density of states of the undamped system. In some cases, this allows us to compute the anomalous dissipation exactly, using properties of the undamped system only. As an example, we compute the anomalous dissipation for weak correlated disorder and for Gaussian white noise of arbitrary strength. Although the focus is on the singular limit ν→0, we finally show that this approach is easily extended to arbitrary ν.

Wave-turbulence description of interacting particles: Klein-Gordon model with a Mexican-hat potential.

In field theory, particles are waves or excitations that propagate on the fundamental state. In experiments or cosmological models, one typically wants to compute the out-of-equilibrium evolution of a given initial distribution of such waves. Wave turbulence deals with out-of-equilibrium ensembles of weakly nonlinear waves, and is therefore well suited to address this problem. As an example, we consider the complex Klein-Gordon equation with a Mexican-hat potential. This simple equation displays two kinds of excitations around the fundamental state: massive particles and massless Goldstone bosons. The former are waves with a nonzero frequency for vanishing wave number, whereas the latter obey an acoustic dispersion relation. Using wave-turbulence theory, we derive wave kinetic equations that govern the coupled evolution of the spectra of massive and massless waves. We first consider the thermodynamic solutions to these equations and study the wave condensation transition, which is the classical equivalent of Bose-Einstein condensation. We then focus on nonlocal interactions in wave-number space: we study the decay of an ensemble of massive particles into massless ones. Under rather general conditions, these massless particles accumulate at low wave number. We study the dynamics of waves coexisting with such a strong condensate, and we compute rigorously a nonlocal Kolmogorov-Zakharov solution, where particles are transferred nonlocally to the condensate, while energy cascades towards large wave numbers through local interactions. This nonlocal cascading state constitutes the intermediate asymptotics between the initial distribution of waves and the thermodynamic state reached in the long-time limit.

Disentangling inertial waves from eddy turbulence in a forced rotating-turbulence experiment.

We present a spatiotemporal analysis of a statistically stationary rotating-turbulence experiment, aiming to extract a signature of inertial waves and to determine the scales and frequencies at which they can be detected. The analysis uses two-point spatial correlations of the temporal Fourier transform of velocity fields obtained from time-resolved stereoscopic particle image velocimetry measurements in the rotating frame. We quantify the degree of anisotropy of turbulence as a function of frequency and spatial scale. We show that this space-time-dependent anisotropy is well described by the dispersion relation of linear inertial waves at large scale, while smaller scales are dominated by the sweeping of the waves by fluid motion at larger scales. This sweeping effect is mostly due to the low-frequency quasi-two-dimensional component of the turbulent flow, a prominent feature of our experiment that is not accounted for by wave-turbulence theory. These results question the relevance of this theory for rotating turbulence at the moderate Rossby numbers accessible in laboratory experiments, which are relevant to most geophysical and astrophysical flows.

Sudden chain energy transfer events in vibrated granular media.

In a mixture of two species of grains of equal size but different mass, placed in a vertically vibrated shallow box, there is spontaneous segregation. Once the system is at least partly segregated and clusters of the heavy particles have formed, there are sudden peaks of the horizontal kinetic energy of the heavy particles, that is otherwise small. Together with the energy peaks the clusters rapidly expand and the segregation is partially lost. The process repeats once segregation has taken place again, either randomly or with some regularity in time depending on the experimental or numerical parameters. An explanation for these events is provided based on the existence of a fixed point for an isolated particle bouncing with only vertical motion. The horizontal energy peaks occur when the energy stored in the vertical motion is partly transferred into horizontal energy through a chain reaction of collisions between heavy particles.

From reversing to hemispherical dynamos.

We show that hemispherical dynamos can result from weak equatorial symmetry breaking of the flow in the interior of planets and stars. Using a model of spherical dynamo, we observe that the interaction between a dipolar and a quadrupolar mode can localize the magnetic field in only one hemisphere when the equatorial symmetry is broken. This process is shown to be related to the one that is responsible for reversals of the magnetic field. These seemingly very different behaviors are thus understood in a unified framework.

Reduction of velocity fluctuations in a turbulent flow of gallium by an external magnetic field.

The magnetic field of planets or stars is generated by the motion of a conducting fluid through a dynamo instability. The saturation of the magnetic field occurs through the reaction of the Lorentz force on the flow. In relation to this phenomenon, we study the effect of a magnetic field on a turbulent flow of liquid gallium. The measurement of electric potential differences provides a signal related to the local velocity fluctuations. We observe a reduction of velocity fluctuations at all frequencies in the spectrum when the magnetic field is increased.