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Synchronization is a fundamental phenomenon in nonlinear dynamical systems. In my presentation I will refer to essential features of synchronized systems with focus on a kinematic dynamo model driven by two counter-rotating disks including weak periodic perturbations. Similar concepts can be used to explain the behavior of triadic resonances in a precession driven flow, which may be seen as a precursor for the transition into a turbulent regime found at a critical precession ratio in the DRESDYN water experiment. Finally, I show the connection to the synchronized solar dynamo model presented by Frank Stefani, where the magnetic field frequencies are determined by the tidal forces caused by planets surrounding the sun.

A theoretician of the first water, Axel Brandenburg has always been interested in experimental MHD as well. Anecdotal evidence has it that he is the only person who witnessed in operation the three successful dynamo experiments in Riga, Karlsruhe and Cadarache. Axel’s theoretical work, in turn, has inspired experimentalists more often than not. One case in point is his early work on nonlinear and highly supercritical dynamos which has motivated investigations into the common mechanism underlying the field reversals of the geodynamo and the VKS dynamo. These efforts lead to the complementary explanations of reversals in terms of spectral exceptional points of non-selfadjoint dynamo operators, or via saddle-node bifurcations. In this context, we shortly report the construction progress of the DRESDYN precession driven dynamo, the set-up of which was partly motivated by the signature of Milankovic cycles in the reversal statistics of the geodynamo. The second part of the talk refers to Axel’s work on massively non-linear, “self-creating” dynamos, such as the MRI dynamo or the Tayler-Spruit dynamo. Here we summarize some previous experiments on the magnetorotational instability (MRI) and the Tayler instability (TI), and present some recent results on double-diffusive instabilities for rotating flows with positive shear. The talk will close with a highly speculative idea which connects the oscillatory behaviour of the TI-related alpha-effect to the persistent synchronization of the solar dynamo with the 11.07 years periodic alignments of the tidally dominant planets Venus, Earth and Jupiter.

We present a solar dynamo model of Tayler-Spruit type with an Omega-effect that is conventionally produced by solar-like differential rotation, while its alpha-effect is periodically modulated by planetary tidal forcing. The underlying resonance builds upon the tendency of the Tayler instability to generate helicity oscillations which can be synchronized by periodic tidal forces. We focus on the 11.07 years alignment periodicity of the planets Venus, Earth, and Jupiter and analyze its synchronization with the solar dynamo. The typically emerging dynamo modes are dipolar or quadrupolar fields, oscillating with a 22.14 years period or pulsating with a 11.07 years period. When starting from a conventional alpha-Omega-dynamo, the addition of the periodic alpha-part can lead to entrainment of the entire dynamo via parametric resonance. Phase coherent transitions between dipoles and quadrupoles are discussed, too.

Aiming at a consistent planetary synchronization model of both short-term and long-term solar cycles, we start with an analysis of Schove’s historical data of cycle maxima. Their deviations (residuals) from the average cycle duration of 11.07 years show a high degree of regularity, comprising a dominant 200 year period (Suess-de Vries cycle), and a few periods around 100 years (Gleissberg cycle). Encouraged by their robustness, we support previous forecasts of an upcoming grand minimum in the 21st century. To explain the long-term cycles, we enhance our tidally synchronized solar dynamo model by a modulation of the field storage capacity of the tachocline with the orbital angular momentum of the Sun, which is dominated by the 19.86-year periodicity of the Jupiter–Saturn synodes. This modulation of the 22.14-year Hale cycle leads to a 193-year beat period of dynamo activity which is indeed close to the Suess-de Vries cycle. For stronger dynamo modulation, the model produces additional peaks at typical Gleissberg frequencies, which seem to be explainable by the non-linearities of the basic beat process, leading to a bi-modality of the Schwabe cycle. However, a complementary role of beat periods between the Schwabe cycle and the Jupiter–Uranus/Neptune synodic cycles cannot be completely excluded.

The electrical potential in a battery jumps at each electrode-electrolyte interface. We present a model for computing three-dimensional current and potential distributions, which accounts for such internal voltage jumps. Within the framework of the finite volume method we discretize the Laplace and gradient operators such that they account for internal jump boundary conditions. After implementing a simple battery model in OpenFOAM we validate it using an analytical test case, and show its capabilities by simulating the current distribution and discharge curve of a Li||Bi liquid metal battery.

We present a novel experiment on interfacial wave dynamics in orbitally shaken cylindrical vessels containing two- and three fluid layers. The experiment was designed as a hydrodynamical model for both aluminum reduction cells and liquid metal batteries to gain new insights into the rotational wave motion driven by the metal pad roll instability. Different options are presented to realize stable and measurable multi-layer stratifications. We introduce a new acoustic measurement procedure allowing to reconstruct wave amplitudes also in opaque liquids by tracking ultrasonic pulse echoes reflected on the interfaces. Measurements of resonance curves and phase shifts were conducted for varying interface positions. A strong influence of the top and bottom walls were observed, considerably reducing wave amplitudes and eigenfrequencies, when the interface is getting close. Finally, measured resonance curves were successfully compared with an existing forced wave theory that we extended to two-layer interfacial waves. The comparison stresses the importance to carefully control the boundary condition at the contact line.

Mass and charge transport in liquid metal batteries are closely intertwined because of the fully liquid interior of the cells. We found that charging and discharging cycles may show pronounced asymmetries. They are caused by the presence (charge) or absence (discharge) of solutal convection. While the direction of thermal gradients in the positive electrode of a liquid metal battery depend on boundary conditions and thermodynamics in a non-trivial manner, the solutal gradient predictably changes direction from charge to discharge. The unstable density distribution during charge drives a flow strong enough to prevent any concentration polarization. In contrast, during discharge, the stable density gradient suppresses convection and leads to a substantial mass transport overvoltage. We illustrate this scenario by experimental data, numerical simulations and a physical model.

Liquid metal batteries are promising candidates for low-cost, large-scale stationary electricity storage. Different systems investigated at HZDR are discussed with a focus on fluid dynamic phenomena like interfacial wave interactions, mass transfer, and electro-vortex flows.

Aiming at a simple and consistent planetary synchronization model of both short-term and long-term solar cycles we analyze Schove's data of the cycle minima and maxima. Their residuals from the average cycle duration of 11.07 years show a high degree of regularity, comprising a dominant period of 200 years (Suess-de Vries cycle), and a few periods around 90 years (Gleissberg cycle). Based on Dicke's ratio between the mean square of the residuals to the mean square of the difference of two consecutive residuals, we conjecture that the Schwabe cycle is synchronized, very likely via a helicity-mediated parametric resonance, by the 11.07 years alignment cycle of the tidally dominant planets Venus, Earth and Jupiter. In an attempt to explain the rather regular long-term behaviour, we enhance our Parker-type solar dynamo model by a modulation of the field-storage capacity of the tachocline with the 19.86 years periodicity of the movement of the sun around the barycenter of the solar system, which results basically from the Jupiter-Saturn synodes. Not surprisingly, this modulation leads to a 193-year beat period of dynamo activity which is indeed close to the observed 200 years Suess-de Vries cycle. In cases, the model produces also additional peaks at typical Gleissberg frequencies, which might be interpreted as beat periods of the Schwabe cycle with the Jupiter-Uranus (13.81-year) or the Jupiter-Neptune (12.78-year) synodes.

Materials processing of molten metals often makes use of electrochemical techniques in which large electrical currents pass through the melt. Those currents can cause the melt to flow, both by interacting with Earth’s magnetic field and by interacting with magnetic fields induced by the currents themselves—“electrovortex flow". We present experimental and numerical measurements of the motion of a lead-bismuth melt in which electrovortex flow, flow driven by Earth’s magnetic field, and flow driven by thermal convection all compete and interact. We identify regimes in which the various mechanisms dominate, then consider their implications for materials processing and for liquid metal batteries.

Liquid Metal Batteries (LMBs) are a promising concept for cheap electrical energy storage at grid level. These are built as a stable density stratification of three liquid layers, with two liquid metals separated by a molten salt. In order to ensure a safe and efficient operation, the understanding of transport phenomena in LMBs is essential. With this motivation we study thermal convection induced by internal heat generation. We consider the electrochemical nature of the cell in order to define the heat balance and the operating parameters. Moreover we develop a simple 1D heat conduction model as well as a fully 3D thermo-fluid dynamics model. The latter is implemented in the CFD library OpenFOAM, extending the volume of fluid solver, and validated against a pseudo-spectral code. Both models are used to study a rectangular 10×10 cm Li||Bi LMB cell at three different states of charge.

The solar cycle appears to be remarkably synchronized with the gravitational torques exerted by the tidally dominant planets Venus, Earth and Jupiter. Recently, a possible synchronization mechanism was proposed that relies on the intrinsic helicity oscillation of the current-driven Tayler instability which can be stoked by tidal-like perturbations with a period of 11.07 years. Inserted into a simple alpha-Omega dynamo model these resonantly excited helicity oscillations led to a 22.14 years dynamo cycle. Here, we assess various alternative mechanisms of synchronization. Specifically we study a simple time-delay model of Babcock–Leighton type dynamos and ask whether periodic changes of either the minimal amplitude for rising toroidal flux tubes or the Omega effect could eventually lead to synchronization. In contrast to the easy and robust synchronizability of Tayler–Spruit dynamo models, our answer for those Babcock–Leighton type models is less propitious.

We investigate the interfacial wave coupling dynamics in liquid metal batteries and their effects to the battery's operation safety. Similar to aluminum reduction cells, liquid metal batteries can be highly susceptible to magnetohydrodynamical instabilities that excite undesired interfacial waves capable to provoke short-circuits. However, in liquid metal batteries the wave dynamics is far more complex since two metal-electrolyte interfaces are present that may step into resonance. In the first part of this paper, we present a Potential analysis of coupled gravity-capillary interfacial waves in a three-layer battery model of cylindrical shape. Analytical expressions for the amplitude ratio and the wave frequencies are derived and it is shown that the wave coupling can be completely described by two independent dimensionless parameters. We provide a decoupling criterion clarifying that wave coupling will be present in most future liquid metal batteries. In the second part, the theory is validated by comparing it with multiphase direct numerical simulations. An accompanying parameter study is conducted to analyze the system stability for differently strongly coupled interfaces. Three different coupling regimes are identified involving characteristic coupling dynamics. For strongly coupled interfaces we observe novel instabilities that may have beneficial effects on the operational safety.

The design and performance of liquid metal batteries, a new technology for grid-scale energy storage, depend on fluid mechanics because the battery electrodes and electrolytes are entirely liquid. Here we review prior and current research on the fluid mechanics of liquid metal batteries, pointing out opportunities for future studies. Because the technology in its present form is just a few years old, only a small number of publications have so far considered liquid metal batteries specifically. We hope to encourage collaboration and conversation by referencing as many of those publications as possible here. Much can also be learned by linking to extensive prior literature considering phenomena observed or expected in liquid metal batteries, including thermal convection, magnetoconvection, Marangoni flow, interface instabilities, the Tayler instability, and electro-vortex flow. We focus on phenomena, materials, length scales, and current densities relevant to the liquid metal battery designs currently being commercialized. We try to point out breakthroughs that could lead to design improvements or make new mechanisms important.

Liquid metal batteries (LMBs) were recently proposed as cheap large scale energy storage. Such devices are urgently required for balancing highly fluctuating renewable energies. During discharge, LMBs tend to form intermetallic phases. These do not only limit the up-scalability, but also the efficiency of the cells. Generating a mild fluid flow in the fully liquid cell will smoothen concentration gradients and minimise the formation of intermetallics. In this context we study electro-vortex flow numerically. We simulate a recent LMB related experiment and discuss how the feeding lines to the cell can be optimised to enhance mass transfer. The Lorentz forces have to overcome the stable thermal stratification in the cathode of the cell; we show that thermal effects may reduce electro-vortex flow velocities considerable. Finally, we study the influence of the Earth magnetic field on the flow.

A numerical model for simulating electro-vortical flows in OpenFOAM is developed. Electric potential and current are solved in coupled solid-liquid conductors by a parent-child mesh technique. The magnetic field is computed using a combination of Biot-Savart’s law and induction equation. Further, a PCG solver with special regularisation for the electric potential is derived and implemented. Finally, a performance analysis is presented and the solver is validated against several test cases.

The talk will provide an overview of the liquid metal battery (LMB) related activities at Helmholtz-Zentrum Dresden - Rossendorf (HZDR) with a focus on magnetohydrodynamic aspects of future large scale LMBs. High current densities in the range of 4 up to 130 kA/m-2, as typical for LMBs, together with cells of large cross section will result in substantial currents accompanied by considerable magnetic fields. Thus electromagnetically driven flows and instabilities should be of concern for large enough installations, especially when the thin electrolyte layers necessitated by the limited open circuit voltages are taken into account. Beneficial effects of mild electromagnetically driven flows are to be expected for the cathodes were mixing should improve cell performance.

Liquid metal batteries are introduced as a potential means to future economic large-scale electricity storage indispensable for energy systems with high penetration of fluctuating sources. The talk will concentrate on the role of electrodynamics and fluid mechanics in the design of large single cells, discussing a few instability mechanisms that should be taken into account.

We present a solar dynamo model of the Tayler-Spruit type whose Omega-effect is coventionally produced by differential rotation but whose alpha-effect is assumed as being periodically modulated by planetary tidal forcing. This resonance-like effect has its rationale in the tendency of the current-driven Tayler instability to undergo intrinsic helicity oscillations which can be synchronized by periodic tidal perturbations. Specifically, we focus on the 11.07 years periodicity of the alignment of the tidally dominant planets Venus, Earth, and Jupiter. In the framework of a simple one-dimensional numerical model we prove the subcritical character of this Tayler-Spruit type dynamo. The typical dynamo modes are dipole fields, oscillating with a 22.14 year period, but also quadrupole fields pulsating with an 11.07 years period. Transitions between these field topologies are reminiscent of the observed behavior during the Maunder minimum. Further interesting features of the model are the emergence of mid-term fluctuations, and the intermittent appearance of reversed helicities in both hemispheres. With minor model modifications, the correct direction of the butterfly diagram comes out as a robust feature, too.

The Dresden sodium facility for dynamo and thermohydraulic studies (DRESDYN) is a platform for large-scale liquid sodium experiments devoted to fundamental geo- and astrophysical questions as well as to various applied problems related to the conversion and storage of energy. Its most ambitious part is a precession driven dynamo experiment, comprising 8 tons of liquid sodium supposed to rotate with up to 10 Hz and to precess with up to 1 Hz. Another large-scale set-up is a Tayler-Couette experiment with a gap width of 0.2 m and a height of 2 m, whose inner cylinder rotates with up to 20 Hz. Equipped with a coil system for the generation of an axial field of up to 120 mT and two different axial currents through the center and the liquid sodium, this experiment aims at studying various versions of the magnetorotational instability and their combinations with the Tayler instability. We discuss the physical background of these two experiments and delineate the present status of their technical realization. Other installations, such as a sodium loop and a test stand for In-Service-Inspection experiments, will also be sketched.

Liquid metal batteries (LMBs) are discussed today as a cheap grid scale energy storage, as required for the deployment of fluctuating renewable energies. LMBs incorporate stratified three-layer fluid systems consisting of two liquid metal electrodes separated by a thin molten salt electrolyte (see sketch below). Due to the large electrical conductivities of the liquid metals, LMBs are highly susceptible to become unstable by interactions of induced or external magnetic fields with internal cell currents. Several different types of instabilities have been identified as to be crucial for the LMB operation. Besides the Tayler instability and electrovortex flows, the metal pad roll (MPR) instability, originally known from aluminium reductions cells (ARCs), emerged as a key instability mechanism capable to cause short-circuits by exciting interfacial gravity-capillary waves. The MPR instability can be induced only by the interaction of a homogeneous vertical magnetic field with horizontal compensation currents arising due to small perturbations of the interfaces. While this mechanism is well understood in the case of ARCs, in LMBs an additional interface is present that may strongly influence the global stability depending on several parameters. Both interfaces can be closely coupled for thin salt-layers such that they both may excite each other and may be connected by different oscillating modes. The analysis of the coupling behavior is the main target of this study. As the main part of this talk I will present an analytical analysis using linear wave theory describing coupled gravity-capillary waves enclosed in cylindrical containers. We have derived a fourth-order dispersion relation containing two different coupling modes. Further, we found that the global coupling behavior can be completely described by only two dimensionless parameters. On this basis, we suggest a coupling criterion predicting for which parameter regimes both interfaces can be considered as to be f

Liquid metal batteries are possible candidates for massive and economically feasible large-scale stationary storage and as such could be key components of future energy systems based mainly or exclusively on intermittent renewable electricity sources. The completely liquid interior of liquid metal batteries and the high current densities give rise to a multitude of fluid flow phenomena that will primarily influence the operation of future large cells, but might be important for today’s smaller cells as well. The paper at hand starts with a discussion of the relative merits of using molten salts or ionic liquids as electrolytes for liquid metal cells and touches the choice of electrode materials. This excursus into electrochemistry is followed by an overview of investigations on magnetohydrodynamic instabilities in liquid metal batteries, namely the Tayler instability and electromagnetically excited gravity waves. A section on electro-vortex flows complements the discussion of flow phenomena. Focus of the flow related investigations lies on the integrity of the electrolyte layer and related critical parameters.

The current-driven, kink-type Tayler instability (TI) is a key ingredient of the Tayler-Spruit dynamo model for the generation of stellar magnetic fields, but is also discussed as a mechanism that might limit the up-scaling of liquid metal batteries. Here, we focus on the chiral symmetry breaking and the related alpha-effect that would be needed to close the dynamo loop in the Tayler-Spruit model. For low magnetic Prandtl number, we observe intrinsic oscillations of the alpha-effect. These oscillations serve then as the basis for a synchronized Tayler-Spruit dynamo model, which could possibly link the periodic tidal forces of planets with the oscillation periods of stellar dynamos.

Liquid metal batteries (LMBs) are discussed today as a cheap grid scale energy storage, as required for the deployment of fluctuating renewable energies. Built as a stable density stratification of two liquid metals separated by a thin molten salt layer, LMBs are susceptible to short-circuit by fluid flows. Using direct numerical simulation, we study a sloshing long wave interface instability in cylindrical cells, which is already known from aluminium reduction cells. After characterising the instability mechanism, we investigate the influence of cell current, layer thickness, density, viscosity, conductivity and magnetic background field. Finally we study the shape of the interface and give a dimensionless parameter for the onset of sloshing as well as for the short-circuit.