Wolf-Christian Müller

The spatial structure of advective transport in two-dimensional homogeneous Rayleigh–Bénard (HRB) convection is investigated by means of direct numerical simulations. The convective driving leads to the emergence of thermal plumes. These create dynamically changing pathways characterised by localised mean flows. The resulting large-scale anisotropy of the system diminishes with increasing nominal Rayleigh number ( Ra ). The key components of advective transport are extracted via a network-based analysis of Lagrangian trajectories. This reveals a coherent structure based on plume-related pathways that governs the transport of heat and matter. A reduced description of the structure is given by the zero isoline of scale-filtered vorticity. While its essential large-scale characteristics display only a weak dependency on Ra , geometric analysis shows that the decrease of large-scale anisotropy with increasing Ra is due to a reduction of the length of vertical transport paths. Mean profiles with respect to the transport paths suggest that this reduction is caused by an enhanced turbulent transfer of temperature fluctuations into adjacent shear layers and vortices. This process leads to a spatial decorrelation of temperature and velocity and, consequently, to a reduced structural impact of the thermal driving on the flow. Spatially resolved nonlinear fluxes indicate that shear layers and vortices next to the transport paths are associated with a spectrally inverse flux of enstrophy. The observed structure of advective transport in HRB convection also displays asymptotically scale-invariant characteristics, contrasting the structural properties of wall-bounded classical Rayleigh–Bénard convection.

Coherent structures in two-dimensional Navier–Stokes turbulence are ubiquitously observed in nature, experiments and numerical simulations. The present study conducts a comparison between several structure detection schemes based on the Okubo–Weiss criterion, the vorticity magnitude and Lagrangian coherent structures (LCSs), focusing on the inverse cascade in two-dimensional hydrodynamic turbulence. A recently introduced vortex scaling phenomenology (Burgess & Scott, J. Fluid Mech., vol. 811, 2017, pp. 742–756) allows the quantification of the respective thresholds required by these methods based on physical properties of the flow. The resulting improved comparability allows us to identify characteristic relative differences in the detection sensitivity between the employed structure detection techniques. With respect to the inverse cascade of energy, coherent structures contribute, as expected, substantially less to the cross-scale flux than the residual incoherent parts of the flow although the energetically dominant coherent structures lead to an important large-scale deformation of the energy spectrum. This cascade inactivity can be understood by an increased misalignment of strain-rate and subgrid stress tensors within coherent structures. At the same time, the structures exhibit strong and localised nonlinear cross-scale interactions that appear to stabilise them. We quantify and interpret the resulting shape preservation of coherent structures in terms of a multi-scale gradient approach (Eyink, J. Fluid Mech., vol. 549, 2006, pp. 191–214) as the depletion of strain rotation and vorticity gradient stretching whereas the dynamics of the residual fluctuations are consistent with the vortex thinning picture.

Turbulence is of fundamental importance for many physical systems on Earth and throughout the universe. A turbulent flow can be described as the superposition of turbulent fluctuations of various length scales, which interact with each other non-linearly, leading to a transfer of energy across scales. We aim at a better understanding of the temporal and spatial properties of this energy transfer process in plasma turbulence, by studying the spatio-temporal correlation between turbulent structures in magnetohydrodynamic (MHD) turbulence.To this end we perform three-dimensional direct numerical simulations with a pseudo-spectral method. We employ the quasi-Lagrangian reference frame, in which tracer particles are followed in the flow each carrying with it a set of probes at fixed distances across which the fluctuations are computed. This avoids the large-scale sweeping effect, which in the case of fixed-grid (Eulerian) measurements would obscure the small-scale temporal dynamics. This approach is based on previous studies in Navier-Stokes turbulence [Physics of Fluids 23.8 (2011): 085107] and has been extended to account for the magnetic field.We investigate systems with different mean magnetic field strength. The spatio-temporal correlation functions yield insight into the nature of the cross-scale transfer of energy in terms of the direction, strength, and time scale of the transfer process. In particular, the scaling of the correlation times perpendicular and parallel to the local magnetic field, the influence of the mean magnetic field and the implications for the current understanding of the cross-scale transfer process are discussed.

Magnetic helicity dynamics are important in the context of the generation of large scale magnetic structures from small scale fluctuations. Up to the present day, these dynamics have remained largely unexplored in compressible plasmas. We present new results from direct numerical simulations of isothermal magnetohydrodynamic turbulence, with Mach numbers ranging from 0.1 to 10 by employing higher-order numerics. A mechanical driving injects kinetic energy at the largest scales, while a small scale electromotive driving injects helical magnetic fluctuations. A large-scale sink of magnetic energy leads to the formation of a turbulent statistically stationary state, which is analyzed, extending the results on nonlinear cross-scale transfer presented in doi:10.1017/jfm.2021.32 and doi:10.1017/jfm.2021.496.

Coherent structures in two-dimensional Navier-Stokes turbulence are ubiquitously observed in nature, experiments and numerical simulations. The present study conducts a comparison between several structure detection schemes based on the Okubo-Weiss criterion, the vorticity magnitude, and Lagrangian coherent structures (LCSs), focusing on the inverse cascade in two-dimensional hydrodynamic turbulence. A recently introduced vortex scaling phenomenology [B. H. Burgess, R. K. Scott, J. Fluid Mech., 811:742--756, 2017] allows the quantification of the respective thresholds required by these methods based on physical properties of the flow. The resulting improved comparability allows to identify characteristic relative differences in the detection sensitivity between the employed structure detection techniques. With respect to the inverse cascade of energy, coherent structures contribute, as expected, substantially less to the cross-scale flux than the residual incoherent parts of the flow although the energetically dominant coherent structures lead to an important large-scale deformation of the energy spectrum. This cascade inactivity can be understood by an increased misalignment of strain-rate and subgrid stress tensors within coherent structures. At the same time, the structures exhibit strong and localised nonlinear cross-scale interactions that appear to stabilize them. We quantify and interpret the resulting shape preservation of coherent structures in terms of a multi-scale gradient approach [G. L. Eyink, J. Fluid Mech., 549:191--214, 2006] as the depletion of strain rotation and vorticity gradient stretching while the dynamics of the residual fluctuations are consistent with the vortex thinning picture.

In this chapter, we aim at presenting the basic techniques necessary to go beyond the widely accepted paradigm of second-order numerics. We specifically focus on finite-volume schemes for hyperbolic conservation laws occuring in fluid approximations such as the equations of ideal magnetohydrodynamics or the Euler equations of gas dynamics. For the sake of clarity, a simple fourth-order ideal magnetohydrodynamic (MHD) solver which allows to simulate strongly shocked systems serves as an instructive example. Issues that only or mainly arise in the world of higher-order numerics are given specific focus. Alternative algorithms as well as refinements and improvements are dicussed and are referenced to in the literature. As an example of application, some results on decaying compressible turbulence are presented.

The application of machine learning (ML) techniques, especially neural networks, has seen tremendous success at processing images and language. This is because we often lack formal models to understand visual and audio input, so here neural networks can unfold their abilities as they can model solely from data. In the field of physics we typically have models that describe natural processes reasonably well on a formal level. Nonetheless, in recent years, ML has also proven useful in these realms, be it by speeding up numerical simulations or by improving accuracy. One important and so far unsolved problem in classical physics is understanding turbulent fluid motion. In this work we construct a strongly simplified representation of turbulence by using the Gledzer-Ohkitani-Yamada (GOY) shell model. With this system we intend to investigate the potential of ML-supported and physics-constrained small-scale turbulence modelling. Instead of standard supervised learning we propose an approach that aims to reconstruct statistical properties of turbulence such as the self-similar inertial-range scaling, where we could achieve encouraging experimental results. Furthermore we discuss pitfalls when combining machine learning with differential equations.

The movement of heat in a convecting system is typically described by the nondimensional Nusselt number, which involves an average over both space and time. In direct numerical simulations of turbulent flows, there is considerable variation in the contributions to the Nusselt number, both because of local spatial variations due to plumes and because of intermittency in time. We develop a statistical approach to more completely describe the structure of heat transfer, using an exit-distance extracted from Lagrangian tracer particles, which we call the Lagrangian heat structure. In a comparison between simulations of homogeneous turbulence driven by Boussinesq convection, the Lagrangian heat structure reveals significant non-Gaussian character, as well as a clear trend with Prandtl number and Rayleigh number. This has encouraging implications for simulations performed with the goal of understanding turbulent convection in natural settings such as Earth’s atmosphere and oceans, as well as planetary and stellar dynamos.