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We show that connectivity within the high-dimensional recurrent layer of a reservoir network is crucial for its performance. To this end, we systematically investigate the impact of network connectivity on its performance, i.e., we examine the symmetry and structure of the reservoir in relation to its computational power. Reservoirs with random and asymmetric connections are found to perform better for an exemplary Mackey-Glass time series than all structured reservoirs, including biologically inspired connectivities, such as small-world topologies. This result is quantified by the information processing capacity of the different network topologies which becomes highest for asymmetric and randomly connected networks., Comment: 6 pages, 4 figures, supplementary material

Albeit laboratory experiments and numerical simulations have proven themselves successful in enhancing our understanding of long-living large-scale flow structures in horizontally extended Rayleigh-B\'enard convection, some discrepancies with respect to their size and induced heat transfer remain. This study traces these discrepancies back to their origins. We start by generating a digital twin of one standard experimental set-up. This twin is subsequently simplified in steps to understand the effect of non-ideal thermal boundary conditions, and the experimental measurement procedure is mimicked using numerical data. Although this allows explaining the increased observed size of the flow structures in the experiment relative to past numerical simulations, our data suggests that the vertical velocity magnitude has been underestimated in the experiments. A subsequent re-assessment of the latter's original data reveals an incorrect calibration model. The re-processed data show a relative increase in $u_{z}$ of roughly $24 \%$, resolving the previously observed discrepancies. This digital twin of a laboratory experiment for thermal convection at Rayleigh numbers $\Ra = \left\{ 2, 4, 7 \right\} \times 10^{5}$, a Prandtl number $\Pr = 7.1$, and an aspect ratio $\Gamma = 25$ highlights the role of different thermal boundary conditions as well as a reliable calibration and measurement procedure., Comment: 20 pages, 7 figures, 2 tables

We propose a tensor network framework for approximating the evolution of observables of measure-preserving ergodic systems. Our approach is based on a spectrally-convergent approximation of the skew-adjoint Koopman generator by a diagonalizable, skew-adjoint operator $W_\tau$ that acts on a reproducing kernel Hilbert space $\mathcal H_\tau$ with coalgebra structure and Banach algebra structure under the pointwise product of functions. Leveraging this structure, we lift the unitary evolution operators $e^{t W_\tau}$ (which can be thought of as regularized Koopman operators) to a unitary evolution group on the Fock space $F(\mathcal H_\tau)$ generated by $\mathcal H_\tau$ that acts multiplicatively with respect to the tensor product. Our scheme also employs a representation of classical observables ($L^\infty$ functions of the state) by quantum observables (self-adjoint operators) acting on the Fock space, and a representation of probability densities in $L^1$ by quantum states. Combining these constructions leads to an approximation of the Koopman evolution of observables that is representable as evaluation of a tree tensor network built on a tensor product subspace $\mathcal H_\tau^{\otimes n} \subset F(\mathcal H_\tau)$ of arbitrarily high grading $n \in \mathbb N$. A key feature of this quantum-inspired approximation is that it captures information from a tensor product space of dimension $(2d+1)^n$, generated from a collection of $2d + 1$ eigenfunctions of $W_\tau$. Furthermore, the approximation is positivity preserving. The paper contains a theoretical convergence analysis of the method and numerical applications to two dynamical systems on the 2-torus: an ergodic torus rotation as an example with pure point Koopman spectrum and a Stepanoff flow as an example with topological weak mixing., Comment: 52 pages, 10 figures

The time-dependent one-dimensional nonlinear Schr\"odinger equation (NLSE) is solved numerically by a hybrid pseudospectral-variational quantum algorithm that connects a pseudospectral step for the Hamiltonian term with a variational step for the nonlinear term. The Hamiltonian term is treated as an integrating factor by forward and backward Fourier transformations, which are here carried out classically. This split allows us to avoid higher-order time integration schemes, to apply a first-order explicit time stepping for the remaining nonlinear NLSE term in a variational algorithm block, and thus to avoid numerical instabilities. We demonstrate that the analytical solution is reproduced with a small root mean square error for a long time interval over which a nonlinear soliton propagates significantly forward in space while keeping its shape. We analyze the accuracy of the quantum algorithm and compare it with classical approaches. Furthermore, we investigate the influence of algorithm parameters on the accuracy of the results, including the temporal step width and the depth of the quantum circuit.

Anomalous dissipation is a dissipation mechanism of kinetic energy which is established by a sufficiently spatially rough velocity field. It implies that the rescaled mean kinetic energy dissipation rate becomes constant with respect to Reynolds number ${\rm Re}$, the dimensionless parameter that characterizes the strength of turbulence, given that ${\rm Re}\gg 1$. The present study aims at bridging this statistical behavior of high-Reynolds-number turbulence to specific structural building blocks of fluid turbulence -- local vortex stretching configurations which take in the simplest case the form of Burgers' classical vortex stretching model from 1948. We discuss the anomalous dissipation in the framework of Duchon and Robert for this analytical solution of the Navier-Stokes equations, apply the same analysis subsequently to a generalized model of randomly oriented Burgers vortices by Kambe and Hatakeyama, and analyse finally direct numerical simulation data of three-dimensional homogeneous, isotropic box turbulence in this respect. We identify local high-vorticity events in fully developed Navier-Stokes turbulence that approximate the analytical models of strong vortex stretching well. They also correspond to precursors of enhanced anomalous dissipation.

The impact of variable material properties, such as temperature-dependent thermal conductivity and dynamical viscosity, on the dynamics of a fully compressible turbulent convection flow beyond the anelastic limit are studied in the present work by two series of three-dimensional direct numerical simulations in a layer of aspect ratio 4 with periodic boundary conditions in both horizontal directions. One simulation series is for a weakly stratified adiabatic background, one for a strongly stratified one. The Rayleigh number is $10^5$ and the Prandtl number is 0.7 throughout this study. The temperature dependence of material parameters is imposed as a power law with an exponent $\beta$. It generates a superadiabaticity $\varepsilon(z)$ that varies across the convection layer. Central statistical quantities of the flow, such as the mean superadiabatic temperature, temperature and density fluctuations, or turbulent Mach numbers are compared in the form of horizontal plane-time averaged profiles. It is found that the additional material parameter dependence causes systematic quantitative changes of all these quantities, but no qualitative ones. A growing temperature power law exponent $\beta$ also enhances the turbulent momentum transfer in the weak stratification case by 40\%, it reduces the turbulent heat transfer by up to 50\% in the strong stratification case.

We study the dynamics of thermal and momentum boundary regions in three-dimensional direct numerical simulations of Rayleigh-B\'enard convection for the Rayleigh number range $10^5 \le Ra \le 10^{11}$ and $Pr=0.7$. Using a Cartesian slab with horizontal periodic boundary conditions and an aspect ratio of 4, we obtain statistical homogeneity in the horizontal $x$- and $y$-directions, thus approximating best an extended convection layer relevant for most geo- and astrophysical flow applications. We observe upon canonical use of combined long-time and area averages, with averaging periods of at least 100 free-fall times, that a global coherent mean flow is practically absent and that the magnitude of the velocity fluctuations is larger than the mean by up to 2 orders of magnitude. The velocity field close to the wall is a collection of differently oriented local shear-dominated flow patches interspersed by extensive shear-free incoherent regions which can be as large as the whole cross section, unlike for a closed cylindrical convection cell of aspect ratio of the order 1. The incoherent regions occupy a $60\%$ area fraction for all Rayleigh numbers investigated here. Rather than resulting in a pronounced mean with small fluctuations about such a mean, as found in small-aspect-ratio convection, the velocity field is dominated by strong fluctuations of all three components around a non-existent or weak mean. We discuss the consequences of these observations for convection layers with larger aspect ratios, including boundary layer instabilities and the resulting turbulent heat transport., Comment: 18 pages, 10 figures

Two quantum algorithms are presented for the numerical solution of a linear one-dimensional advection-diffusion equation with periodic boundary conditions. Their accuracy and performance with increasing qubit number are compared point-by-point with each other. Specifically, we solve the linear partial differential equation with a Quantum Linear Systems Algorithms (QLSA) based on the Harrow--Hassidim--Lloyd method and a Variational Quantum Algorithm (VQA), for resolutions that can be encoded using up to 6 qubits, which corresponds to $N=64$ grid points on the unit interval. Both algorithms are of hybrid nature, i.e., they involve a combination of classical and quantum computing building blocks. The QLSA and VQA are solved as ideal statevector simulations using the in-house solver QFlowS and open-access Qiskit software, respectively. We discuss several aspects of both algorithms which are crucial for a successful performance in both cases. These are the sizes of an additional quantum register for the quantum phase estimation for the QLSA and the choice of the algorithm of the minimization of the cost function for the VQA. The latter algorithm is also implemented in the noisy Qiskit framework including measurement and decoherence circuit noise. We reflect the current limitations and suggest some possible routes of future research for the numerical simulation of classical fluid flows on a quantum computer., Comment: 45 pages, 20 figures

We employ a linear stability analysis and direct numerical simulations to study the characteristics of wall-modes in thermal convection in a rectangular box under strong and inclined magnetic fields. The walls of the convection cell are electrically insulated. The stability analysis assumes periodicity in the spanwise direction perpendicular to the plane of the homogeneous magnetic field. Our study shows that for a fixed vertical magnetic field, the imposition of horizontal magnetic fields results in an increase of the critical Rayleigh number along with a decrease in the wavelength of the wall modes. The wall modes become tilted along the direction of the resulting magnetic fields and therefore extend further into the bulk as the horizontal magnetic field is increased. Once the modes localized on the opposite walls interact, the critical Rayleigh number decreases again and eventually drops below the value for onset with a purely vertical field. We find that for sufficiently strong horizontal magnetic fields, the steady wall modes occupy the entire bulk and therefore convection is no longer restricted to the sidewalls. The above results are confirmed by direct numerical simulations of the nonlinear evolution of magnetoconvection., Comment: 25 pages, 18 figures

Turbulence parametrizations will remain a necessary building block in kilometer-scale Earth system models. In convective boundary layers, where the mean vertical gradients of conserved properties such as potential temperature and moisture are approximately zero, the standard ansatz which relates turbulent fluxes to mean vertical gradients via an eddy diffusivity has to be extended by mass flux parametrizations for the typically asymmetric up- and downdrafts in the atmospheric boundary layer. In this work, we present a parametrization for a dry convective boundary layer based on a generative adversarial network. The model incorporates the physics of self-similar layer growth following from the classical mixed layer theory by Deardorff. This enhances the training data base of the generative machine learning algorithm and thus significantly improves the predicted statistics of the synthetically generated turbulence fields at different heights inside the boundary layer. The algorithm training is based on fully three-dimensional direct numerical simulation data. Differently to stochastic parametrizations, our model is able to predict the highly non-Gaussian transient statistics of buoyancy fluctuations, vertical velocity, and buoyancy flux at different heights thus also capturing the fastest thermals penetrating into the stabilized top region. The results of our generative algorithm agree with standard two-equation or multi-plume stochastic mass-flux schemes. The present parametrization provides additionally the granule-type horizontal organization of the turbulent convection which cannot be obtained in any of the other model closures. Our work paves the way to efficient data-driven convective parametrizations in other natural flows, such as moist convection, upper ocean mixing, or convection in stellar interiors.

Two hybrid quantum-classical reservoir computing models are presented to reproduce low-order statistical properties of a two-dimensional turbulent Rayleigh-B\'enard convection flow at a Rayleigh number Ra=1e+5 and a Prandtl number Pr=10. These properties comprise the mean vertical profiles of the root mean square velocity and temperature and the turbulent convective heat flux. Both quantum algorithms differ by the arrangement of the circuit layers of the quantum reservoir, in particular the entanglement layers. The second of the two quantum circuit architectures, denoted as H2, enables a complete execution of the reservoir update inside the quantum circuit without the usage of external memory. Their performance is compared with that of a classical reservoir computing model. Therefore, all three models have to learn the nonlinear and chaotic dynamics of the turbulent flow at hand in a lower-dimensional latent data space which is spanned by the time-dependent expansion coefficients of the 16 most energetic Proper Orthogonal Decomposition (POD) modes. These training data are generated by a POD snapshot analysis from direct numerical simulations of the original turbulent flow. All reservoir computing models are operated in the reconstruction mode. We analyse different measures of the reconstruction error in dependence on the hyperparameters which are specific for the quantum cases or shared with the classical counterpart, such as the reservoir size and the leaking rate. We show that both quantum algorithms are able to reconstruct the essential statistical properties of the turbulent convection flow successfully with similar performance compared to the classical reservoir network. Most importantly, the quantum reservoirs are by a factor of 4 to 8 smaller in comparison to the classical case., Comment: 15 pages, 9 figures

Buoyancy-driven turbulent convection leads to a fully compressible flow with a strong top-down asymmetry of first- and second-order statistics when the adiabatic equilibrium profiles of temperature, density and pressure decay very strongly across the convection layer. The growth of this asymmetry and the formation of an increasingly thicker stabilized sublayer with a negative mean convective heat flux at the top of the convection zone is reported here by a series highly resolved three-dimensional direct numerical simulations beyond the Oberbeck-Boussinesq and anelastic limits for dimensionless dissipation numbers $0.1 \le D\le 0.8$ at fixed Rayleigh number $Ra=10^6$ and superadiabaticity. The highly stratified compressible convection regime appears for $D > D_{\rm crit}\approx 0.65$, when density fluctuations collapse to those of pressure; it is characterized by an up to nearly 50\% reduced global turbulent heat transfer and a sparse network of focused thin thermal plumes falling through the top sublayer deep into the bulk.

The present study investigates the non-Oberbeck-Boussinesq (NOB) effects which arise due to the temperature dependence of material properties in cryogenic helium experiments of turbulent Rayleigh-B\'enard convection. They are manifest as a difference of the measured mean temperature at the center of the closed cell, $T_c$, from the arithmetic mean temperature obtained from the prescribed fixed and uniform temperatures at the top and bottom copper plates of the apparatus, $T_m = (T_{bot} +T_{top})=2$. Therefore, the material properties such as specific heat at constant pressure, dynamic viscosity, thermal conductivity, the isobaric expansivity, and the mass density are expanded into power series with respect to temperature up to the quadratic order with coeffcients obtained from the software package HEPAK. A subsequent nonlinear regression that uses deep convolutional networks delivers a dependence of the strength of non-Oberbeck-Boussinesq effects in the pressure-temperature parameter plane. Strength of the NOB effects is evaluated via the deviation of the mean temperature profile $\xi_{NOB} = T_m - T_c$ from the top/bottom-symmetric Oberbeck-Boussinesq case $\xi_{NOB} = 0$. Training data for the regression task are obtained from 236 individual long-term laboratory measurements at different Rayleigh numbers which span 8 orders of magnitude., Comment: 10 figures, 16 pages

We explore the mechanisms of heat transfer in a turbulent constant heat flux-driven Rayleigh-B\'enard convection flow, which exhibits a hierarchy of flow structures from granules to supergranules. Our computational framework makes use of time-dependent flow networks. These are based on trajectories of Lagrangian tracer particles that are advected in the flow. We identify coherent sets in the Lagrangian frame of reference as those sets of trajectories that stay closely together for an extended time span under the action of the turbulent flow. Depending on the choice of the measure of coherence, sets with different characteristics are detected. First, the application of a recently proposed evolutionary spectral clustering scheme allows us to extract granular coherent features that are shown to contribute significantly less to the global heat transfer than their spatial complements. Moreover, splits and mergers of these (leaking) coherent sets leave spectral footprints. Secondly, trajectories which exhibit a small node degree in the corresponding network represent objectively highly coherent flow structures and can be related to supergranules as the other stage of the present flow hierarchy. We demonstrate that the supergranular flow structures play a key role in the vertical heat transport and that they exhibit a greater spatial extension than the granular structures obtained from spectral clustering., Comment: 17 pages, 15 figures

The dynamics in a confined turbulent convection flow is dominated by multiple long-lived macroscopic circulation states, which are visited subsequently by the system in a Markov-type hopping process. In the present work, we analyze the short transition paths between these subsequent macroscopic system states by a data-driven learning algorithm that extracts the low-dimensional transition manifold and the related new coordinates, which we term collective variables, in the state space of the complex turbulent flow. We therefore transfer and extend concepts for conformation transitions in stochastic microscopic systems, such as in the dynamics of macromolecules, to a deterministic macroscopic flow. Our analysis is based on long-term direct numerical simulation trajectories of turbulent convection in a closed cubic cell at a Prandtl number $Pr = 0.7$ and Rayleigh numbers $Ra = 10^6$ and $10^7$ for a time lag of $10^5$ convective free-fall time units. The simulations resolve vortices and plumes of all physically relevant scales resulting in a state space spanned by more than 3.5 million degrees of freedom. The transition dynamics between the large-scale circulation states can be captured by the transition manifold analysis with only two collective variables which implies a reduction of the data dimension by a factor of more than a million. Our method demonstrates that cessations and subsequent reversals of the large-scale flow are unlikely in the present setup and thus paves the way to the development of efficient reduced-order models of the macroscopic complex nonlinear dynamical system., Comment: 24 pages, 12 Figures, 1 table

The impact of turbulent mixing on the droplet size distribution is studied deep inside a warm ice-free cloud. A simplified cloud mixing model was implemented therefore which summarizes the balance equations of water vapor mixing ratio and temperature to an effective advection-diffusion equation for the supersaturation field $s(\textbf{x},t)$. Our three-dimensional direct numerical simulations connect the scalar supersaturation field to the cloud droplet dynamics, in particular to the droplet size distribution for different box sizes. In addition, finite-time Lyapunov exponents are monitored such that we can relate regions of higher compressive strain to those of high local supersaturation amplitudes. We find that the mixing process in terms of the droplet evaporation is always homogeneous in the bulk of the cloud, while being inhomogeneous in view to the relaxation of the supersaturation field. The probability density function of $\lambda_3$ is related to the one of $s$ by a simple one-dimensional aggregation model. The distributions of the compressive finite-time Lyapunov exponents $\lambda_3$, the supersaturation field, and the droplet size are found to be Gaussian.

We study the influence of thermal boundary conditions on large aspect ratio Rayleigh-B\'enard convection by a joint analysis of experimental and numerical data sets for a Prandl number $\mathrm{Pr = 7}$ and Rayleigh numbers $\mathrm{Ra = 10^5 - 10^6}$. The spatio-temporal experimental data are obtained by combined Particle Image Velocimetry and Particle Image Thermometry measurements in a cuboid cell filled with water at an aspect ratio $\Gamma= 25$. In addition, numerical data are generated by Direct Numerical Simulations (DNS) in domains with $\Gamma = 25$ and $\Gamma = 60$ subject to different thermal boundary conditions. Our experimental data show an increased characteristic horizontal extension scale of the flow structures, $\tilde{\lambda}$, for increasing Ra, which is coupled with a raise of the Biot number Bi in particular at the cooling plate. However, we find the experimental flow structure size to range in any case between the ones observed for the idealized thermal conditions captured by the simulations. On the one hand, they are larger than in the numerical case with applied uniform temperatures at the plates, but, on the other hand, smaller than in the case of an applied constant heat flux, the latter of which leads to a structure that grows gradually up to the horizontal domain size. We link this observation qualitatively to theoretical predictions for the onset of convection. Furthermore, we study the effect of the asymmetric boundary conditions on the heat transfer. Contrasting experimental and numerical data reveals an increased probability of far-tail events of reversed heat transfer. The decomposition of the local Nusselt number $\mathrm{Nu_{loc}}$ traces this effect back to the sign of the temperature deviation $\tilde{\Theta}$, revealing asymmetries of the heating and cooling plate on the thermal variance of the generated thermal plumes.

Fully compressible turbulent convection beyond the Oberbeck-Boussinesq limit and anelastic regime is studied in three-dimensional numerical simulations. Superadiabaticity $\epsilon$ and dissipation number $D$, which measures the strength of stratification of adiabatic equilibria, cause two limits of compressible convection -- nearly top-down-symmetric, strongly superadiabatic and highly top-down-asymmetric, strongly stratified convection. Highest turbulent Mach numbers $M_t$ follow for a symmetric blend of these two limits which we term the fully compressible case. Particularly, the strongly stratified convection case leads to a fluctuation-reduced top layer in the convection zone, a strongly reduced global heat transfer, and differing boundary layer dynamics between top and bottom. We detect this asymmetry for growing dissipation number $D$ also in the phase plane which is spanned by the turbulent Mach number $M_t$ and the dilatation parameter $\delta$ which relates the dilatational velocity fluctuations to the solenoidal ones. A detailed analysis of the different transport currents in the fully compressible energy budget relates the low-$D$ convection cases to the standard definition of the dimensionless Nusselt number in the Oberbeck-Boussinesq limit.

We study the influence of fringing magnetic fields on turbulent thermal convection in a horizontally extended rectangular domain. The magnetic field is created in the gap between two semi-infinite planar magnetic poles, with the convection layer located near the edge of the gap. We employ direct numerical simulations in this setup for fixed Rayleigh and small Prandtl numbers, but vary the fringe-width by controlling the gap between the magnetic poles and the convection cell. The magnetic field generated by the magnets is strong enough to cease the flow in high magnetic flux region of the convection cell. We observe that as the local vertical magnetic field strength increases, the large scale structures become thinner and align themselves perpendicular to the longitudinal sidewalls. We determine the local Nusselt and Reynolds numbers as functions of the local Hartmann number (based on the vertical component of the magnetic field) and estimate the global heat and momentum transport. We show that the global heat transport decreases with increasing fringe-width for strong magnetic fields but increases with increasing fringe-width for weak magnetic fields. In the regions of large vertical magnetic fields, the convective motion becomes confined to the vicinity of the sidewalls. The amplitudes of these wall modes show a non-monotonic dependence on the fringe-width., Comment: 29 pages, 18 figures

To understand turbulent convection at very high Rayleigh numbers typical of natural phenomena, computational studies in slender cells are an option if the needed resources have to be optimized within available limits. However, the accompanying horizontal confinement affects some properties of the flow. Here, we explore the characteristics of turbulent fluctuations in the velocity and temperature fields in a cylindrical convection cell of aspect ratio 0.1 by varying the Prandtl number $Pr$ between 0.1 and 200 at a fixed Rayleigh number $Ra = 3 \times 10^{10}$, and find that the fluctuations weaken with increasing $Pr$, quantitatively as in aspect ratio 25. The probability density function (PDF) of temperature fluctuations in the bulk region of the slender cell remains mostly Gaussian, but increasing departures occur as $Pr$ increases beyond unity. We assess the intermittency of the velocity field by computing the PDFs of velocity derivatives and of the kinetic energy dissipation rate, and find increasing intermittency as $Pr$ decreases. In the bulk region of convection, a common result applicable to the slender cell, large aspect ratio cells, as well as in 2D convection, is that the turbulent Prandtl number decreases as $Pr^{-1/3}$., Comment: 13 pages, 16 figures

Turbulent superstructures in horizontally extended three-dimensional Rayleigh-B\'enard convection flows are investigated in controlled laboratory experiments in water at Prandtl number $Pr = 7$. A Rayleigh-B\'enard cell with square cross-section, aspect ratio $\Gamma = l/h = 25$, side length $l$ and height $h$ is used. Three different Rayleigh numbers in the range $10^5 < Ra < 10^6$ are considered. The cell is accessible optically, such that thermochromic liquid crystals can be seeded as tracer particles to monitor simultaneously temperature and velocity fields in a large section of the horizontal mid-plane for long time periods of up to 6 h, corresponding to approximately $10^4$ convective free-fall time units. The joint application of stereoscopic particle image velocimetry and thermometry opens the possibility to assess the local convective heat flux fields in the bulk of the convection cell and thus to analyse the characteristic large-scale transport patterns in the flow. A direct comparison with existing direct numerical simulation data in the same parameter range of $Pr, Ra$ and $\Gamma$ reveals the same superstructure patterns and global turbulent heat transfer scaling $Nu(Ra)$. Slight quantitative differences can be traced back to violations of the isothermal boundary condition at the extended water-cooled glass plate at the top. The characteristic scales of the patterns fall into the same size range, but are systematically larger. It is confirmed experimentally that the superstructure patterns are an important backbone of the heat transfer. The present experiments enable, furthermore, the study of the gradual evolution of the large-scale patterns in time, which is challenging in simulations of large-aspect-ratio turbulent convection., Comment: 25 pages, 11 figures

Inverse cascades of kinetic energy and thermal variance in the subset of vertically homogeneous modes in spectral space are found to cause a slow aggregation to a pair of convective supergranules that eventually fill the whole horizontally extended, three-dimensional, turbulent Rayleigh-B\'{e}nard convection layer when a heat flux is prescribed at the top and bottom. An additional weak rotation of the layer around the vertical axis stops this aggregation at a scale that is smaller than the lateral domain extension and ceases the inverse cascade for the thermal variance. The inverse cascade for the kinetic energy remains intact, even for times at which the root mean square values of temperature and velocity have reached the statistically steady state. This kinetic energy inverse cascade sustains the horizontally extended convection patterns which are best visible in the temperature field. The resulting characteristic length of the aggregated convection patterns depends on the thermal driving and linearly on the strength of rotation. Our study demonstrates the importance of inverse energy cascades beyond the two-dimensional turbulence case in a three-dimensional convection flow that is subject to a multi-scale energy injection by thermal plumes and driven by boundary heat fluxes as typically present in natural geo- and astrophysical systems, such as solar convection., Comment: 15 pages, 10 figures

We simulate the nonlinear chaotic dynamics of Lorenz-type models for a classical two-dimensional thermal convection flow with 3 and 8 degrees of freedom by a hybrid quantum--classical reservoir computing model. The high-dimensional quantum reservoir dynamics are established by universal quantum gates that rotate and entangle the individual qubits of the tensor product quantum state. A comparison of the quantum reservoir computing model with its classical counterpart shows that the same prediction and reconstruction capabilities of classical reservoirs with thousands of perceptrons can be obtained by a few strongly entangled qubits. We demonstrate that the mean squared error between model output and ground truth in the test phase of the quantum reservoir computing algorithm increases when the reservoir is decomposed into separable subsets of qubits. Furthermore, the quantum reservoir computing model is implemented on a real noisy IBM quantum computer for up to 7 qubits. Our work thus opens the door to model the dynamics of classical complex systems in a high-dimensional phase space effectively with an algorithm that requires a small number of qubits., Comment: 15 pages, 7 figures

A combined convolutional autoencoder-recurrent neural network machine learning model is presented to analyse and forecast the dynamics and low-order statistics of the local convective heat flux field in a two-dimensional turbulent Rayleigh-B\'{e}nard convection flow at Prandtl number ${\rm Pr}=7$ and Rayleigh number ${\rm Ra}=10^7$. Two recurrent neural networks are applied for the temporal advancement of flow data in the reduced latent data space, a reservoir computing model in the form of an echo state network and a recurrent gated unit. Thereby, the present work exploits the modular combination of three different machine learning algorithms to build a fully data-driven and reduced model for the dynamics of the turbulent heat transfer in a complex thermally driven flow. The convolutional autoencoder with 12 hidden layers is able to reduce the dimensionality of the turbulence data to about 0.2 \% of their original size. Our results indicate a fairly good accuracy in the first- and second-order statistics of the convective heat flux. The algorithm is also able to reproduce the intermittent plume-mixing dynamics at the upper edges of the thermal boundary layers with some deviations. The same holds for the probability density function of the local convective heat flux with differences in the far tails. Furthermore, we demonstrate the noise resilience of the framework which suggests the present model might be applicable as a reduced dynamical model that delivers transport fluxes and their variations to the coarse grid cells of larger-scale computational models, such as global circulation models for the atmosphere and ocean., Comment: 14 pages, 13 figures

Horizontally extended turbulent convection, termed mesoscale convection in natural systems, remains a challenge to investigate in both experiments and simulations. This is particularly so for very low molecular Prandtl numbers as in stellar convection and in Earth's outer core. The present study reports three-dimensional direct numerical simulations of turbulent Rayleigh-B\'{e}nard convection in square boxes of side length $L$ and height $H$ with the aspect ratio $\Gamma =L/H$ of 25, for Prandtl numbers that span almost 4 orders of magnitude, $10^{-3}\le Pr \le 7$, and Rayleigh numbers $10^5 \le Ra \le 10^7$, obtained by massively parallel computations on grids of up to $5.36\times 10^{11}$ points. The low end of this $Pr$-range cannot be accessed in controlled laboratory measurements. We report the essential properties of the flow and their trends with Rayleigh and Prandtl numbers, in particular the global transport of momentum and heat -- the latter decomposed into convective and diffusive contributions -- across the convection layer, mean vertical profiles of the temperature and temperature fluctuations, and the kinetic energy and thermal dissipation rates. We also explore the degree to which the turbulence in the bulk of the convection layer resembles classical homogeneous and isotropic turbulence in terms of spectra, increment moments, and dissipative anomaly, and find close similarities. Finally, we show that a characteristic scale on the order of the mesoscale seems to saturate to a wavelength of $\lambda\gtrsim 3H$ for $Pr\lesssim 0.005$. We briefly discuss possible implications of these results for the development of subgrid scale parameterization of turbulent convection., Comment: 27 pages, 14 figures, 4 tables

High-amplitude events of the out-of-plane vorticity component $\omega_z$ are analyzed by stereoscopic particle image velocimetry (PIV) in the bulk region of turbulent Rayleigh-B\'{e}nard convection in air. The Rayleigh numbers ${\rm Ra}$ vary from $1.7 \times 10^4$ to $5.1 \times 10^5$. The experimental investigation is connected with a comprehensive statistical analysis of long-term time series of $\omega_z$ and individual velocity derivatives $\partial u_i/\partial x_j$. A statistical convergence for derivative moments up to an order of 6 is demonstrated. Our results are found to agree well with existing high-resolution direct numerical simulation data in the same range of parameters, including the extreme vorticity events which appear in the far exponential tails of the corresponding probability density functions. The transition from a Gaussian to a non-Gaussian velocity derivative statistics in the bulk of a convection flow is confirmed experimentally. The experimental data are used to train a reservoir computing model, one implementation of a recurrent neural network, to reproduce highly intermittent experimental time series of the vorticity and thus reconstruct extreme out-of-plane vorticity events. After training the model with high-resolution PIV data, the machine learning model is run with sparsely seeded, continually available, and unseen measurement data in the reconstruction phase. The dependence of the reconstruction quality on the sparsity of the partial observations is also documented. Our latter result paves the way to machine--learning--assisted experimental analyses of small-scale turbulence for which time series of missing velocity derivatives can be provided by generative algorithms., Comment: 15 pages, 12 figures, journal article

The large-scale flow structure and the turbulent transfer of heat and momentum are directly measured in highly turbulent liquid metal convection experiments for Rayleigh numbers varied between $4 \times 10^5$ and $\leq 5 \times 10^9$ and Prandtl numbers of $0.025~\leq~Pr~\leq ~0.033$. Our measurements are performed in two cylindrical samples of aspect ratios $\Gamma =$ diameter/height $= 0.5$ and 1 filled with the eutectic alloy GaInSn. The reconstruction of the three-dimensional flow pattern by 17 ultrasound Doppler velocimetry sensors detecting the velocity profiles along their beamlines in different planes reveals a clear breakdown of coherence of the large-scale circulation for $\Gamma = 0.5$. As a consequence, the scaling laws for heat and momentum transfer inherit a dependence on the aspect ratio. We show that this breakdown of coherence is accompanied with a reduction of the Reynolds number $Re$. The scaling exponent $\beta$ of the power law $Nu\propto Ra^{\beta}$ crosses \FIN{eventually} over from $\beta=0.221$ to 0.124 when the liquid metal flow at $\Gamma=0.5$ reaches $Ra\gtrsim 2\times 10^8$ and the coherent large-scale flow is completely collapsed., Comment: 4 pages, 5 figures, 1 supplementary with 1 figure and 4 tables, 1 movie

We explore the transport mechanisms of heat in two- and three-dimensional turbulent convection flows by means of the long-term evolution of Lagrangian coherent sets. They are obtained from the spectral clustering of trajectories of massless fluid tracers that are advected in the flow. Coherent sets result from trajectories that stay closely together under the dynamics of the turbulent flow. For longer times, they are always destroyed by the intrinsic turbulent dispersion of material transport. Here, this constraint is overcome by the application of evolutionary clustering algorithms that add a time memory to the coherent set detection and allow individual trajectories to leak in or out of evolving clusters. Evolutionary clustering thus also opens the possibility to monitor the splits and mergers of coherent sets. These rare dynamic events leave clear footprints in the evolving eigenvalue spectrum of the Laplacian matrix of the trajectory network in both convection flows. The Lagrangian trajectories reveal the individual pathways of convective heat transfer across the fluid layer. We identify the long-term coherent sets as those fluid flow regions that contribute least to heat transfer. Thus, our evolutionary framework defines a complementary perspective on the slow dynamics of turbulent superstructure patterns in convection flows that were recently discussed in the Eulerian frame of reference. The presented framework might be well suited for studies in natural flows which are typically based on sparse information from drifters and probes.

We investigate the large-scale circulation (LSC) in a turbulent Rayleigh-B\'enard convection flow in a cubic closed convection cell by means of direct numerical simulations at a Rayleigh number $Ra=10^6$. The numerical studies are conducted for single flow trajectories up to $10^5$ convective free-fall times to obtain a sufficient sampling of the four discrete LSC states, which can be summarized to one macrostate, and the two crossover configurations which are taken by the flow in between for short periods. We find that large-scale dynamics depends strongly on the Prandtl number $Pr$ of the fluid which has values of 0.1, 0.7, and 10. Alternatively, we run an ensemble of 3600 short-term direct numerical simulations to study the transition probabilities between the discrete LSC states. This second approach is also used to probe the Markov property of the dynamics. Our ensemble analysis gave strong indication of Markovianity of the transition process from one LSC state to another, even though the data are still accompanied by considerable noise. It is based on the eigenvalue spectrum of the transition probability matrix, further on the distribution of persistence times and the joint distribution of two successive macrostate persistence times.

We explore the generalization properties of an echo state network applied as a reduced dynamical model to predict flux-driven two-dimensional turbulent convection. To this end, we consider a convection domain at fixed height with a variable ratio of buoyancy fluxes at the top and bottom boundaries, which break the top-down symmetry in comparison to the standard Rayleigh-B\'enard case thus leading to highly asymmetric mean and fluctuation profiles across the layer. Our direct numerical simulation model describes a convective boundary layer in a simple way. The data are used to train and test a recurrent neural network in the form of an echo state network. The input to the echo state networks is obtained in two different ways, either by a proper orthogonal decomposition or by a convolutional autoencoder. In both cases, the echo state network reproduces the turbulence dynamics and the statistical properties of the buoyancy flux, and is able to model unseen data records with different flux ratios.

Convection in the Sun occurs at Rayleigh numbers, $Ra$, as high as $10^{22}$, molecular Prandtl numbers, $Pr$, as low as $10^{-6}$, under conditions that are far from satisfying the Oberbeck-Boussinesq (OB) idealization. The effects of these extreme circumstances on turbulent heat transport are unknown, and no comparable conditions exist on Earth. Our goal is to understand how these effects scale (since we cannot yet replicate the Sun's conditions faithfully). We study thermal convection by using direct numerical simulations, and determine the variation with respect to $Pr$, to values as low as $10^{-4}$, of the turbulent Prandtl number, $Pr_t$, which is the ratio of turbulent viscosity to thermal diffusivity. The simulations are primarily two-dimensional but we draw upon some three-dimensional results as well. We focus on non-Oberbeck-Boussinesq (NOB) conditions of a certain type, but also study OB convection for comparison. The OB simulations are performed in a rectangular box of aspect ratio 2 by varying $Pr$ from $O(10)$ to $10^{-4}$ at fixed Grashof number $Gr \equiv Ra/Pr = 10^9$. The NOB simulations are done in the same box by letting only the thermal diffusivity depend on the temperature. Here, the Rayleigh number is fixed at the top boundary while the mean $Pr$ varies in the bulk from 0.07 to $5 \times 10^{-4}$. The three-dimensional simulations are performed in a box of aspect ratio 25 at a fixed Rayleigh number of $10^5$, and $0.005 \leq Pr \leq 7$. The principal finding is that $Pr_t$ increases with decreasing $Pr$ in both OB and NOB convection: $Pr_t \sim Pr^{-0.3}$ for OB convection and $Pr_t \sim Pr^{-1}$ for the NOB case. The $Pr_t$-dependence for the NOB case especially suggests that convective flows in the astrophysical settings behave effectively as in high-Prandtl-number turbulence., Comment: 17 pages, 11 figures, 3 tables

We study the connection between extreme events of thermal and kinetic energy dissipation rates in the bulk of three-dimensional Rayleigh-B\'{e}nard convection and the wall shear stress patterns at the top and the bottom planes that enclose the layer. Zero points of this two-dimensional vector field stand for detachments of strong thermal plumes. If their position at the opposite planes and a given time is close then they can be considered as precursors for high-amplitude bulk dissipation events triggered by plume collisions or close passings. This scenario requires a breaking of the synchronicity of the boundary layer dynamics at both plates which is found to be in line with a transition of the bulk derivative statistics from Gaussian to intermittent. Our studies are based on three-dimensional high-resolution direct numerical simulations for moderate Rayleigh numbers between $Ra=10^4$ and $5\times 10^5$.

Recurrent neural networks are machine learning algorithms which are suited well to predict time series. Echo state networks are one specific implementation of such neural networks that can describe the evolution of dynamical systems by supervised machine learning without solving the underlying nonlinear mathematical equations. In this work, we apply an echo state network to approximate the evolution of two-dimensional moist Rayleigh-B\'enard convection and the resulting low-order turbulence statistics. We conduct long-term direct numerical simulations in order to obtain training and test data for the algorithm. Both sets are pre-processed by a Proper Orthogonal Decomposition (POD) using the snapshot method to reduce the amount of data. The training data comprise long time series of the first 150 most energetic POD coefficients. The reservoir is subsequently fed by the data and results in predictions of future flow states. The predictions are thoroughly validated by the data of the original simulation. Our results show good agreement of the low-order statistics. This incorporates also derived statistical moments such as the cloud cover close to the top of the convection layer and the flux of liquid water across the domain. We conclude that our model is capable of learning complex dynamics which is introduced here by the tight interaction of turbulence with the nonlinear thermodynamics of phase changes between vapor and liquid water. Our work opens new ways for the dynamic parametrization of subgrid-scale transport in larger-scale circulation models.

We report a linear scaling law for an electrical voltage as a function of the pressure drop in capillary pipes and ducts. This voltage is generated by a process which is termed spin hydrodynamic generation (SHDG), a result of the collective electron spin--coupling to the vorticity field in the laminar flow in combination with an inverse spin-Hall effect. We study this phenomenon in laminar duct flows with different width-to-height aspect ratios ranging from 1 (square ducts) to infinite (two dimensional channels). First, we analytically solve the governing Valet-Fert spin diffusion equations for the SHDG by means of the method of small parameters together with proper boundary conditions for the set of inhomogeneous elliptic partial differential equations. Secondly, the proposed linear scaling law is validated through a series of experiments using capillary tubes with rectangular and square cross-sections. The experimental results show a very good agreement to the analytically found scaling law. A subsequent substitution of the bulk velocity of the laminar wall-bounded flows by the pressure drop reveals a universal scaling law for the electrical voltage that incorporates all pipe and duct geometries which we could study in our experiments. Finally, the efficiency of the system is estimated for circular pipes, rectangular and square ducts. This study shows that the efficiency of a spin hydrodynamic generator is the same for a circular pipe and a square duct with the same diameter and height, respectively. Hence, due to the ease of manufacturing and the possibility to scale the experiments up to parallel settings in a compact form, micro-channels with a square cross-section seem to be the optimum for a spin hydrodynamic generator., Comment: 12 pages, 7 figures

We develop a framework for simulating measure-preserving, ergodic dynamical systems on a quantum computer. Our approach provides a new operator-theoretic representation of classical dynamics by combining ergodic theory with quantum information science. The resulting quantum embedding of classical dynamics (QECD) enables efficient simulation of spaces of classical observables with exponentially large dimension using a quadratic number of quantum gates. The QECD framework is based on a quantum feature map for representing classical states by density operators on a reproducing kernel Hilbert space, $\mathcal H $, and an embedding of classical observables into self-adjoint operators on $\mathcal H$. In this scheme, quantum states and observables evolve unitarily under the lifted action of Koopman evolution operators of the classical system. Moreover, by virtue of the reproducing property of $\mathcal H$, the quantum system is pointwise-consistent with the underlying classical dynamics. To achieve an exponential quantum computational advantage, we project the state of the quantum system to a density matrix on a $2^n$-dimensional tensor product Hilbert space associated with $n$ qubits. By employing discrete Fourier-Walsh transforms, the evolution operator of the finite-dimensional quantum system is factorized into tensor product form, enabling implementation through a quantum circuit of size $O(n)$. Furthermore, the circuit features a state preparation stage, also of size $O(n)$, and a quantum Fourier transform stage of size $O(n^2)$, which makes predictions of observables possible by measurement in the standard computational basis. We prove theoretical convergence results for these predictions as $n\to\infty$. We present simulated quantum circuit experiments in Qiskit Aer, as well as actual experiments on the IBM Quantum System One., Comment: 42 pages, 9 figures

We investigate the free convection processes in the vicinity of a spherical 1 kg mass standard by two- and three-dimensional direct numerical simulations using a spectral element method. Our focus is on the determination and suppression of updraft forces in a high-precision mass comparator which are caused by temperature differences between mass standard and its environment in the millikelvin range - a source of systematic uncertainties in the high-precison mass determination. A two-dimensional model is presented first, which obtains a good agreement with previous laboratory measurements for the smaller temperature differences up to 15 mK. The influence of different boundary conditions and side lengths of the square domain is discussed for the mass standard positioned in the center of the chamber. The complexity is increased subsequently in configurations with additional built-ins for counter heating in form of planar plates or hemispherical shells above the mass standard. The latter ones lead to a full compensation of the updraft force. Three-dimensional simulations in a closed cubic chamber confirm the two-dimensional findings and additionally reveal complex secondary flow pattern in the vicinity of the mass standard. The reduction of the heat transfer due to the built-ins is also demonstrated by a comparison of the Nusselt numbers as a function of the Rayleigh number in the chosen parameter range. Our simulations suggest that such additional constructive measures can enhance the precision of the mass determination by suppression of free convection and related systematic uncertainties., Comment: 29 pages, 20 figures

Spatial regions that do not mix effectively with their surroundings and thus contribute less to the heat transport in fully turbulent three-dimensional Rayleigh-B\'{e}nard flows are identified by Lagrangian trajectories that stay together for a longer time. These trajectories probe Lagrangian coherent sets (CS) which we investigate here in direct numerical simulations in convection cells with square cross section of aspect ratio $\Gamma = 16$, Rayleigh number $Ra = 10^{5}$, and Prandtl numbers $Pr = 0.1, 0.7$ and $7$. The analysis is based on $N=524,288$ Lagrangian tracer particles which are advected in the time-dependent flow. Clusters of trajectories are identified by a graph Laplacian with a diffusion kernel, which quantifies the connectivity of trajectory segments, and a subsequent sparse eigenbasis approximation (SEBA) for cluster detection. The combination of graph Laplacian and SEBA leads to a significantly improved cluster identification that is compared with the large-scale patterns in the Eulerian frame of reference. We show that the detected CS contribute by a third less to the global turbulent heat transport for all investigated $Pr$ compared to the trajectories in the spatial complement. This is realized by monitoring Nusselt numbers along the tracer trajectory ensembles, a dimensionless local measure of heat transfer., Comment: 8 pages, 5 figures

Turbulent convection processes in nature are often found to be organized in a hierarchy of plume structures and flow patterns. The gradual aggregation of convection cells or granules to a supergranule which eventually fills the whole horizontal layer is reported and analysed in spectral element direct numerical simulations of three-dimensional turbulent Rayleigh-B\'{e}nard convection at an aspect ratio of $60$. The formation proceeds over a time span of more than $10^4$ convective time units for the largest accessible Rayleigh number and occurs only when the turbulence is driven by a constant heat flux which is imposed at the bottom and top planes enclosing the convection layer. The resulting gradual inverse cascade process is observed for both temperature variance and turbulent kinetic energy. An additional analysis of the leading Lyapunov vector field for the full turbulent flow trajectory in its high-dimensional phase space demonstrates that turbulent flow modes at a certain scale continue to give rise locally to modes with longer wavelength in the turbulent case. As a consequence successively larger convection patterns grow until the horizontal extension of the layer is reached. This instability mechanism, which is known to exist near the onset of constant heat flux-driven convection, is shown here to persist into the fully developed turbulent flow regime thus connecting weakly nonlinear pattern formation with the one in fully developed turbulence. We discuss possible implications of our study for observed, but not yet consistently numerically reproducible, solar supergranulation which could lead to improved simulation models of surface convection in the Sun., Comment: 15 pages, 11 figures

In an attempt to understand the role of the strong radial dependence of thermal diffusivity on the properties of convection in sun-like stars, we mimic that effect in non-Oberbeck-Boussinesq (NOB) convection in a horizontally-extended rectangular domain (aspect ratio 16), by allowing the thermal diffusivity $\kappa$ to increase with the temperature (as in the case of stars). Direct numerical simulations (i.e., numerical solutions of the governing equations by resolving up to the smallest scales without requiring any modeling) show that, in comparison with Oberbeck-Boussinesq (OB) simulations (two of which we perform for comparison purposes), the symmetry of the temperature field about the mid-horizontal plane is broken, whereas the velocity and heat flux profiles remain essentially symmetric. Our choice of $\kappa(T)$, which resembles the variation in stars, results in the temperature field that loses its fine structures towards the hotter part of the computational domain, but the characteristic large scale of the turbulent thermal `superstructures', which are structures whose size is typically larger than the depth of the convection domain, continue to be largely independent of the depth., Comment: 17 pages, 13 figures

Spin hydrodynamic coupling is a recently discovered method to directly generate electricity from an electrically conducting fluid flow in the absence of Lorentz forces. This method relies on a collective coupling of electron spins - the internal quantum mechanical angular momentum of the electrons - with the local vorticity of a fluid flow. In this work, we experimentally investigate the spin hydrodynamic coupling in circular and non-circular capillary pipe flows and extend a previously obtained range of Reynolds numbers to smaller and larger values, 20

Direct numerical simulations are performed to study turbulent Rayleigh-B\'{e}nard convection in a vertical cylindrical cavity with uniform axial magnetic field. Flows at high Hartmann and Rayleigh numbers are considered. The calculations reveal that, similarly to the behavior observed in Rayleigh-B\'{e}nard convection with strong rotation, flows at strong magnetic field develop a central vortex, while the heat transfer is suppressed.

Direct numerical simulations are carried out to study flow structure and transport properties in turbulent Rayleigh-B\'{e}nard convection in a cylindrical cell of aspect ratio one with an imposed axial magnetic field. Flows at the Prandtl number 0.025 and the Rayleigh and Hartmann numbers up to $10^9$ and 1400 are considered. The results are consistent with those of earlier experimental and numerical data. As anticipated, the heat transfer rate and kinetic energy are suppressed by strong magnetic field. At the same time, their growth with the Rayleigh number is found to be faster in flows at high Hartmann numbers. This behaviour is attributed to the newly discovered flow regime characterized by prominent quasi two-dimensional structures reminiscent of vortex sheets observed earlier in simulations of magnetohydrodynamic turbulence. Rotating wall modes similar to those in the Rayleigh-B\'{e}nard convection with rotation are found in flows near the Chandrasekhar linear stability limit. Detailed analysis of the spatial structure of the flows and its effect of global transport properties is reported.

The effects of a vertical static magnetic field on the flow structure and global transport properties of momentum and heat in liquid metal Rayleigh-B\'enard convection are investigated. Experiments are conducted in a cylindrical convection cell of unity aspect ratio, filled with the alloy GaInSn at a low Prandtl number of $\mathit{Pr}=0.029$. Changes of the large-scale velocity structure with increasing magnetic field strength are probed systematically using multiple ultrasound Doppler velocimetry sensors and thermocouples for a parameter range that is spanned by Rayleigh numbers of $10^6 \le \mathit{Ra} \le 6\times 10^7$ and Hartmann numbers of $\mathit{Ha} \le 1000$. Our simultaneous multi-probe temperature and velocity measurements demonstrate how the large-scale circulation is affected by an increasing magnetic field strength (or Hartmann number). Lorentz forces induced in the liquid metal first suppress the oscillations of the large-scale circulation at low $\mathit{Ha}$, then transform the one-roll structure into a cellular large-scale pattern consisting of multiple up- and downwellings for intermediate $\mathit{Ha}$, before finally expelling any fluid motion out of the bulk at the highest accessible $\mathit{Ha}$ leaving only a near-wall convective flow that persists even below Chandrasekhar's linear instability threshold. Our study thus proves experimentally the existence of wall modes in confined magnetoconvection. The magnitude of the transferred heat remains nearly unaffected by the steady decrease of the fluid momentum over a large range of Hartmann numbers. We extend the experimental global transport analysis to momentum transfer and include the dependence of the Reynolds number on the Hartmann number.

Reservoir computing is applied to model the large-scale evolution and the resulting low-order turbulence statistics of a two-dimensional turbulent Rayleigh-B\'{e}nard convection flow at a Rayleigh number ${\rm Ra}=10^7$ and a Prandtl number ${\rm Pr}=7$ in an extended domain with an aspect ratio of 6. Our data-driven approach which is based on a long-term direct numerical simulation of the convection flow comprises a two-step procedure. (1) Reduction of the original simulation data by a Proper Orthogonal Decomposition (POD) snapshot analysis and subsequent truncation to the first 150 POD modes which are associated with the largest total energy amplitudes. (2) Setup and optimization of a reservoir computing model to describe the dynamical evolution of these 150 degrees of freedom and thus the large-scale evolution of the convection flow. The quality of the prediction of the reservoir computing model is comprehensively tested. At the core of the model is the reservoir, a very large sparse random network charcterized by the spectral radius of the corresponding adjacency matrix and a few further hyperparameters which are varied to investigate the quality of the prediction. Our work demonstrates that the reservoir computing model is capable to model the large-scale structure and low-order statistics of turbulent convection which can open new avenues for modeling mesoscale convection processes in larger circulation models., Comment: 16 pages, 12 figures

We study the fractal scaling of iso-levels sets of a passive scalar mixed by three-dimensional homogeneous and isotropic turbulence at high Reynolds numbers. The Schmidt number is unity. A fractal box-counting dimension $D_F$ can be obtained for iso-levels below about $3$ standard deviations of the scalar fluctuation on either side of its mean value. The dimension varies systematically with the iso-level, with a maximum of about $8/3$ for the iso-level at the mean; this maximum dimension also follows as an upper bound from the geometric measure theory. We interpret this result to mean that mixing in turbulence is always incomplete. A unique box-counting dimension for all iso-levels results when we consider the spatial support of the steep cliffs of the scalar conditioned on local strain; that unique dimension is about $4/3$.

Coherent circulation rolls and their relevance for the turbulent heat transfer in a two-dimensional Rayleigh--B\'{e}nard convection model are analyzed. The flow is in a closed cell of aspect ratio four at a Rayleigh number ${\rm Ra}=10^6$ and at a Prandtl number ${\rm Pr}=10$. Three different Lagrangian analysis techniques based on graph Laplacians -- distance spectral trajectory clustering, time-averaged diffusion maps and finite-element based dynamic Laplacian discretization -- are used to monitor the turbulent fields along trajectories of massless Lagrangian particles in the evolving turbulent convection flow. The three methods are compared to each other and the obtained coherent sets are related to results from an analysis in the Eulerian frame of reference. We show that the results of these methods agree with each other and that Lagrangian and Eulerian coherent sets form basically a disjoint union of the flow domain. Additionally, a windowed time-averaging of variable interval length is performed to study the degree of coherence as a function of this additional coarse graining which removes small-scale fluctuations that cause trajectories to disperse quickly. Finally, the coherent set framework is extended to study heat transport.

Combined measurements of velocity components and temperature in a turbulent Rayleigh-B\'enard convection flow at a low Prandtl number of $\mathit{Pr}= 0.029$ and Rayleigh numbers between $10^6 \le \mathit{Ra} \le 6 \times 10^7$ are conducted in a series of experiments with durations of more than a thousand free-fall time units. Multiple crossing ultrasound beam lines and an array of thermocouples at mid-height allow for a detailed analysis and characterization of the complex three-dimensional dynamics of the single large-scale circulation (LSC) roll in the cylindrical convection cell of unit aspect ratio which is filled with the liquid metal alloy GaInSn. We measure the internal temporal correlations of the complex large-scale flow and distinguish between short-term oscillations associated with a sloshing motion in the mid-plane as well as varying orientation angles of the velocity close to the top/bottom plates and the slow azimuthal drift of the mean orientation of the roll as a whole that proceeds on an up to a hundred times slower time scale. The coherent LSC drives a vigorous turbulence in the whole cell that is quantified by direct Reynolds number measurements at different locations in the cell. The velocity increment statistics in the bulk of the cell displays characteristic properties of intermittent small-scale fluid turbulence. We also show that the impact of the symmetry-breaking large-scale flow persists to small-scale velocity fluctuations thus preventing the establishment of fully isotropic turbulence in the cell centre. Reynolds number amplitudes depend sensitively on beam line position in the cell such that different definitions have to be compared. The global momentum and heat transfer scalings with Rayleigh number are found to agree with those of direct numerical simulations and other laboratory experiments.