Taylor State Research Articles

Self-propulsion of a Quincke droplet on a superhydrophobic wall under low electric Reynolds number ReE≤ 1

A rotating object placed on a wall can generate an additional translating motion. Inspired by this phenomenon, we unfold a novel approach to the self-propulsion of a Quincke rotating drop in the current two-dimensional numerical simulation based on the resting wall effect. Accordingly, the impact of two controlling variables, the electric field strength E0* and viscosity ratio λ, is examined in detail for a Quincke drop resting on a superhydrophobic wall. We consider a fixed conductivity ratio and permittivity ratio to (i) explore the dynamic activities of the droplet to verify the proposed self-propulsion scheme and (ii) reveal the physical propelling mechanism. Our results show that the Quincke drop displays three distinct states. (I) Taylor state (where the symmetry in dynamic behaviors is the primary indicator). (II) Transition stage from a Taylor regime to the Quincke regime, when the symmetry is broken and the created asymmetric flow causes the droplet to detach from the wall. At this stage, the tuned controlling parameters led to diverse droplet detachment processes, significantly influencing the subsequent self-propulsion. Additionally, based on the droplet behaviors in the transition stage for 6.78 < E0* ≤ 57.63 at fixed λ = 50, three distinct propulsion patterns are discovered: one-way propulsion for 6.78 < E0* < 9.5, round trip propulsion for 9.5 ≤ E0* < 33.9, and liquid film-breakup propulsion for 33.9 ≤ E0* ≤ 57.63. (III) Self-propulsion stage. Here, the levitated droplet entrains the bulk fluid into the bottom, preventing its re-depositing on the wall by creating a liquid cushion between the Quincke rotating drop and the wall. This thin liquid cushion generates a higher viscous stress at the droplet's bottom, causing a significant velocity difference between its upper and lower halves. This velocity difference produces the crucial horizontal translation for the rotating droplet, i.e., the self-propulsion. Moreover, the liquid cushion's thickness (h*) affects the translation velocity. A higher E0* or λ leads to a smaller h* and expedites the droplet translation.

State and evolution of the geodynamo from numerical models reaching the physical conditions of Earth’s core

SUMMARY Our understanding of the geodynamo has recently progressed thanks to geomagnetic data of improved quality, and analyses resting on numerical simulations of increasing realism. Here, these two advances are combined in order to diagnose the state and present dynamics of Earth’s core in physically realistic conditions. A sequential, ensemble-based framework assimilates the output of geomagnetic field models covering the past 180 yr into a numerical geodynamo simulation, the physical realism of which is also advanced as data is assimilated. The internal dynamical structure estimated for the geodynamo at present reproduces previously widely documented features such as a planetary-scale, eccentric westwards gyre and localization of buoyancy release beneath the Eastern (0°E−180°E) hemisphere. Relating the typical magnetic variation timescale of the assimilated states to the power at which they operate, the present convective power of the geodynamo is estimated at 2.95 ± 0.2 TW, corresponding to an adiabatic heat flow out of the core of 14.8 ± 1 TW if the top of the core is convectively neutrally stratified at present. For the first time, morphologically and dynamically relevant trajectories are obtained by integrating the estimated states forward for a few decades of physical time using a model reaching the physical conditions of Earth’s core. Such simulations accurately account for the spatio-temporal content of high-resolution satellite geomagnetic field models and confirm earlier interpretations in terms of rapid core dynamics. The enforcement of a realistic force balance approaching a Taylor state allows for propagation of weak (velocity perturbation of about 0.6 $\mathrm{km\,yr^{-1}}$) axisymmetric torsional waves with period about 5 yr, supported by a magnetic field of root-mean-squared amplitude of 5.6 mT inside the core. Quasi-geostrophic magneto-Coriolis waves of interannual periods and significantly stronger velocity perturbation (about $7 \mathrm{km\,yr^{-1}}$) are also reproduced, with properties that converge towards those recently retrieved from the analysis of geomagnetic variations before fully achieving Earth’s core conditions. The power spectral density of magnetic variations falls off rapidly at frequencies exceeding the inverse Alfvén time (about $0.6\mathrm{yr^{-1}}$), which indicates that the excitation of hydromagnetic waves occurs preferentially at large spatial scales. The possibility to account for geomagnetic variations from years to centuries in physically realistic models opens the perspective of better constraining properties of the deep Earth through geomagnetic data assimilation.

On the relationship between the multi-region relaxed variational principle and resistive inner-layer theory

We show that the variational energy principle of the multi-region relaxed magnetohydrodynamic (MRxMHD) model can be used to predict finite-pressure linear tearing instabilities. In this model, the plasma volume is sliced into sub-volumes separated by ‘ideal interfaces’, and in each volume the magnetic field relaxes to a Taylor state, where the pressure gradient . The MRxMHD model is implemented in the Stepped-Pressure Equilibrium Code (SPEC) so that the equilibrium solution in each region is computed while preserving the force balance across the interfaces. As SPEC computes the Hessian matrix (a discretized stability matrix), the stability of an MRxMHD equilibrium can also be computed with SPEC. In this article, using SPEC, we investigate the effect of local pressure gradients and the in the vicinity of the resonant surface of a tearing mode. For low-beta plasma, we have been able to illustrate a relationship between the resistive singular-layer theory (Coppi et al 1966 Nucl. Fusion 6 101; Glasser et al 1975 Phys. Fluids 18 875–88) and the MRxMHD model. Within the singular layer, the volume-averaged magnetic helicity and the flux-averaged toroidal flux are shown to be the invariants for the linear tearing modes in SPEC simulations. Our technique to compute MRxMHD stability is first tested numerically in a cylindrical tokamak and its application in toroidal geometry is demonstrated. We demonstrate an agreement between the stability boundary obtained with SPEC simulation and the resistive inner-layer theories.

Unsteady electrorotation of a viscous drop in a uniform electric field

A dielectric drop suspended in an immiscible dielectric fluid of higher conductivity can spontaneously generate the so-called Quincke rotation (a rotating activity that a weakly conducting drop/solid particle displays in an electric field) subjected to sufficiently strong electric field strength. The steady tilt has been extensively studied and is well elucidated now. However, the unsteady electrorotation of drop remains a largely unclear, complex issue. Motivated by this, we examine the unsteady drop electrorotation in this work with the required integrated convective bulk charge transport effect. First, for the steady rotation, the transient evolution to a steady droplet tilt from the symmetric Taylor state is analyzed in-depth. Here we discover several new phenomena, including the evolving equatorial charge jets. For unsteady rotation, based on a drop's interfacial charge variation, deformation, and tilt angle, the study reports the growth of three distinct rotating patterns in the viscosity ratio range 0.2≤λ=μi/μo≤20.0 and electric field strength E0≤25 kv/cm at a fixed conductivity ratio Q ( = σi/σo) = 0.026 and permittivity ratio S (= ϵi/ϵo) = 0.566. A low-viscosity drop ( λ≤2.5) exhibits only the periodic rotation. For the viscosity ratio 2.5<λ<7.0, the increased electric intensity creates two new unsteady rotation modes: the pseudo-periodic tumbling and the irregular one. For λ≥7, the periodic mode remains absent; instead, the drop displays the electric intensity-dependent tumbling and irregular rotation patterns. Our study shows that the rotation reduces a drop's transitory interfacial charge. At this stage, the drop rotation behavior is controlled by competing charge convection due to fluid flow and charge supply by conduction. The resulting varying electric Reynolds number ReE (the time ratio of charge relaxation and charge convection) explains the created different rotation mechanisms. For ReE>1, owing to lacking enough interfacial charge to sustain rotation, the drop's transition to a temporary non-rotating Taylor state occurs until the interface recharges. The resultant mechanism supports the periodic batch-type rotation for a low-viscosity drop and the irregular rotation for a high-viscosity drop in a higher electric field. In contrast, for ReE<1, the drop timely acquires sufficient charge to support continuous tumbling electrorotation.

Editor’s Column: Summoning the “Honest Spirit” of John Steinbeck: Back to Ireland

Editor’s Column: Summoning the “Honest Spirit” of John Steinbeck: Back to Ireland

Chandrasekhar–Kendall–Woltjer–Taylor state in a resistive plasma

We give a criterion for the Chandrasekhar–Kendall–Woltjer–Taylor (CKWT) state in a resistive plasma. We find that the lowest momentum (longest wavelength) of the initial helicity amplitudes of magnetic fields are the key to the CKWT state, which can be reached if one helicity is favored over the other. This indicates that the imbalance between two helicities at the lowest momentum or longest wavelength in the initial conditions is essential to the CKWT state. A few examples of initial conditions for helicity amplitudes are taken to support the above statement both analytically and numerically.

Effect of injected flux and current temporal phasing on self-organization in the HIT-SI3 experiment

The HIT-SI3 device at the University of Washington uses three oscillating inductive helicity injectors to form and sustain spheromak plasma equilibria. By adjusting the temporal phase of the injector waveforms with respect to each other, the toroidal spectrum of the imposed perturbations can be controlled. Using a recently implemented GPU-based control system, the available mode spectra were explored experimentally by scanning the space of relative injector phasing. In this space, significant variation in the toroidal mode spectrum (n = 1, 2, 3) of the perturbations was observed. Additionally, variation in characteristics of driven equilibria was also observed, including a ≈30% range in toroidal current gain (Iϕ/IInj). Experimental results are compared with both a composite-equilibria and nonlinear dynamic model, including extended MHD simulations using the NIMROD code and composite Taylor state equilibria computed using the PSI-Tet code. Qualitative agreement is seen with the nonlinear models, but not with composite-equilibria models, suggesting the use of nonlinear models to better capture observed plasma dynamics and provide predictive use for future experiments.

Numerical study of δ-function current sheets arising from resonant magnetic perturbations

General three-dimensional toroidal ideal magnetohydrodynamic equilibria with a continuum of nested flux surfaces are susceptible to forming singular current sheets when resonant perturbations are applied. The presence of singular current sheets indicates that, in the presence of non-zero resistivity, magnetic reconnection will ensue, leading to the formation of magnetic islands and potentially regions of stochastic field lines when islands overlap. Numerically resolving singular current sheets in the ideal magnetohydrodynamics (MHD) limit has been a significant challenge. This work presents numerical solutions of the Hahm–Kulsrud–Taylor (HKT) problem, which is a prototype for resonant singular current sheet formation. The HKT problem is solved by two codes: a Grad–Shafranov (GS) solver and the Stepped Pressure Equilibrium Code (SPEC) code. The GS solver has built-in nested flux surfaces with prescribed magnetic fluxes. The SPEC code implements multi-region relaxed magnetohydrodynamics (MRxMHD), whereby the solution relaxes to a Taylor state in each region while maintaining force balance across the interfaces between regions. As the number of regions increases, the MRxMHD solution appears to approach the ideal MHD solution assuming a continuum of nested flux surfaces. We demonstrate agreement between the numerical solutions obtained from the two codes through a convergence study.

Fugitive Freedom: The Improbable Lives of Two Impostors in Late Colonial Mexico

Fugitive Freedom: The Improbable Lives of Two Impostors in Late Colonial Mexico

On Woltjer's force‐free minimizers and Moffatt's magnetic relaxation

In this note, we exhibit a situation where a stationary state of Moffatt's ideal magnetic relaxation problem is different than the corresponding force-free L 2 $L^2$ energy minimizer of Woltjer's variational principle. Such examples have been envisioned in Moffatt's seminal work on the subject and involve divergence free vector fields supported on collections of essentially linked magnetic tubes. Justification of Moffatt's examples requires the strong convergence of a minimizing sequence. What is proven in the current note is that there is a gap between the global minimum (Woltjer's minimizer) and the minimum over the weak L 2 $L^2$ closure of the class of vector fields obtained from a topologically nontrivial field by energy-decreasing diffeomorphisms. In the context of Woltjer-Taylor relaxation, our result shows that the Taylor state cannot be reached during the viscous relaxation in the perfectly conducting magneto-fluid, if the initial field has a nontrivial topology. The result also applies to any other relaxation process which evolves a divergence free field by means of energy-decreasing diffeomorphisms, such processes were proposed by Vallis et al. and more recently by Nishiyama.

Numerical integration of particle orbits in discontinuous fields using VENUS-LEVIS and SPEC

The orbit code VENUS-LEVIS is adapted to follow particles in plasma equilibria with discontinuous fields generated by the Stepped Pressure Equilibrium Code (SPEC). The latter is an implementation of the Multi-Region relaxed MHD model, which efficiently computes Taylor states in a series of nested toroidal volumes and supports the formation of magnetic islands and chaotic field regions. To adapt VENUS-LEVIS, an event location procedure is implemented in the existing numerical integrator, which ensures the particle sees the correct field along its trajectory, regardless of the discontinuities present in the Stepped Pressure Equilibrium model. The algorithm is tested in the case where the magnetic field is uniform in the upper and lower half-spaces but has a discontinuity in its direction (shear) on the plane z=0. Particle drifts due to the discontinuity are studied. The convergence properties of the numerical scheme are highlighted by the numerical accuracy, and conservation of the system's invariants, such as energy and momentum. Simulations and convergence studies using the SPEC-LEVIS interface in axisymmetric geometry are then presented. Finally, illustrative particle drifts due to the discontinuity are studied and explained: we examine drifts associated with the change in Larmor radius of passing particles with small excursion from flux surfaces.

Internal Defense in Containment Theory in the Context of Teenagers Engagement in Indonesia Terrorism

The history of acts of terrorism in Indonesia, which began in 2003-2019, shows several shifting trends in terrorism crimes, terms of action strategies, targets, and characteristics of perpetrators. Based on the action strategy and the features of the perpetrators, the authors found the involvement of children and adolescents. Horgan, Taylor, Bloom, and Winter (2017) state that the involvement of children in acts of terrorism is part of the regeneration of terrorist organizations to continue to exist and survive for a long time. This research aims to illustrate how the internal defense system in the containment theory contributes to the terrorism context. This research is qualitative which uses in-depth interviews for gathering data. Results showed that internal defense could not support teenagers to avoid their engagement with radical or terrorist groups.

Observations of a plectonemic configuration in a stable magnetized plasma jet

Astrophysical jets are collimated high-speed outflows emerging from spinning and accreting matter around celestial objects and may spontaneously result from self-organized processes. Magnetic self-organization is commonly observed in laboratory plasma physics experiments; however, they require close-fitting flux conservers to constrain and stabilize the toroidal or cylindrical structures. Here we report the first observations of a long, stable, free-boundary plasma jet far from chamber walls, embedding a double-helix magnetic structure resembling a force-free plectonemic Taylor state. The jets arise from an experimental setup that mimics an accretion disk and has no close-fitting solid flux conserver. The results support the hypothesis that self-organization could be a universal, intrinsic explanation for jet formation, collimation, and stability and may help explain double-helix features in celestial observations.

Scaling of magnetic dissipation and particle acceleration in ABC fields

Using particle-in-cell numerical simulations with electron–positron pair plasma, we study how the efficiencies of magnetic dissipation and particle acceleration scale with the initial coherence length $\lambda _0$ in relation to the system size $L$ of the two-dimensional ‘Arnold–Beltrami–Childress’ (ABC) magnetic field configurations. Topological constraints on the distribution of magnetic helicity in two-dimensional systems, identified earlier in relativistic force-free simulations, that prevent the high- $(L/\lambda _0)$ configurations from reaching the Taylor state, limit the magnetic dissipation efficiency to about $\epsilon _{\textrm {diss}} \simeq 60\,\%$ . We find that the peak growth time scale of the electric energy $\tau _{E,{\textrm {peak}}}$ scales with the characteristic value of initial Alfvén velocity $\beta _{A,{\textrm {ini}}}$ like $\tau _{E,\textrm {peak}} \propto (\lambda _0/L)\beta _{A,{\textrm {ini}}}^{-3}$ . The particle energy change is decomposed into non-thermal and thermal parts, with non-thermal energy gain dominant only for high initial magnetisation. The most robust description of the non-thermal high-energy part of the particle distribution is that the power-law index is a linear function of the initial magnetic energy fraction.

Effect of geometric and magnetic boundary conditions on magnetic islands in 3D force-free ideal MHD equilibria

Three-dimensional boundary features and magnetic perturbations affect the magnetic topology of the force-free ideal MHD equilibria formed within a given volume. The use of the PSI-Tet 3D finite-element equilibrium code reveals the existence of magnetic islands inside such equilibria within boundaries with three-dimensional features in configurations that directly depend on the natures of those features. The nature and limits of the influence of such features are explored, as is the influence of flux boundary conditions to qualitatively identify the structure of magnetic islands in Taylor state equilibria with three-dimensional boundary conditions.

Coordinate parameterisation and spectral method optimisation for Beltrami field solver in stellarator geometry

The numerical solution of the stepped pressure equilibrium (Hudson et al 2012 Phys. Plasmas 19 112502) requires a fast and robust solver to obtain the Beltrami field in three-dimensional geometry such as stellarators. The spectral method implemented in the stepped pressure equilibrium code (SPEC) is efficient when the domain is a hollow torus, but ill-conditioning of the discretised linear equations occurs in the (solid) toroid due to the artificially singular coordinate parameterisation near the axis. In this work, we propose an improved choice for the reference axis to prevent coordinates surfaces from overlapping. Then, we examine the parity and asymptotics of the magnetic vector potential near the axis and suggest the use of recombined and rescaled Zernike radial basis functions. The maximum relative error in the magnetic field of the Wendelstein 7-X geometry is shown to reach 10−9 at high resolution in a series of convergence tests and benchmarks against the boundary integral equation solver for Taylor states. The new method is also reported to significantly improve the accuracy of multi-volume SPEC calculations. A comparison between free-boundary SPEC and the analytical Dommaschk potential is presented with higher-than-usual Fourier resolution. It is illustrated that we are able to resolve low amplitude current sheets when an interface is placed where there is no flux surface in the analytic solution. This was previously concealed because of insufficient numerical resolution.

Taylor State Merging at SSX: Experiment and Simulation

We describe experiments and simulations of dynamical merging with two Taylor state plasmas in a Swarthmore Spheromak Experiment (SSX) device. Taylor states are formed by magnetized plasma guns at opposite ends of the device. We performed experiments with Taylor states of both senses of magnetic helicity (right-handed twist or left-handed twist). We present results of both counter-helicity merging (one side left-handed, the other right-handed) and co-helicity merging (both sides left-handed). Experiments show significant ion heating, consistent with magnetic reconnection. We suggest that the merged, warm state could be a suitable target for future magneto-inertial fusion experiments. Magnetohydrodynamic simulations of these experiments reveal the structure of the final relaxed, merged state.

Kink Instability: Evolution and Energy Dissipation in Relativistic Force-free Nonrotating Jets

Abstract We study the evolution of kink instability in a force-free, nonrotating plasma column of high magnetization. The main dissipation mechanism is identified as reconnection of magnetic field lines with various intersection angles, driven by the compression of the growing kink lobes. We measure dissipation rates dU Bϕ /dt ≈ −0.1U Bϕ /τ, where τ is the linear growth time of the kink instability. This value is consistent with the expansion velocity of the kink mode, which drives the reconnection. The relaxed state is close to a force-free Taylor state. We constrain the energy of that state using considerations from linear stability analysis. Our results are important for understanding magnetic field dissipation in various extreme astrophysical objects, most notably in relativistic jets. We outline the evolution of the kink instability in such jets and derive constraints on the conditions that allow for the kink instability to grow in these systems.

Force balance in convectively driven dynamos with no inertia

We study dynamo action in rotating, plane layer Boussinesq convection in the absence of inertia. This allows a decomposition of the velocity into a thermal part driven by buoyancy, and a magnetic part driven by the Lorentz force. We have identified three families of solutions, defined in terms of what is the dominant contribution to the velocity. In weak field dynamos the dominant contribution is the thermal component, in super strong field dynamos the dominant contribution is magnetic and in strong field dynamos the two components are comparable. For each of these solutions we investigate the force balance in the momentum equation to determine the relative importance of the viscous, buoyancy, Coriolis and magnetic forces. We do this by extracting the solenoidal part of the individual terms in the momentum equation, thereby removing their pressure contributions. This is numerically preferable to the more common practice of taking the curl of the momentum equation, which introduces an extra derivative. We find that, irrespective of the type of dynamo solution, the dynamics is controlled by the horizontal forces (in projection). Furthermore, in the progression from weak to strong to super strong dynamos, we find that the viscous forces in the thermal equation become negligible, thereby leading to a balance between buoyancy and Coriolis forces. On the other hand, no corresponding trend is observed in the magnetic part of the momentum equation: the viscous stresses always remain significant. This can be attributed to the different degrees of smoothness of the Coriolis and Lorentz forces, the latter having contributions from strong, filamentary structures. We discuss how our findings relate to dynamo solutions in which viscosity plays no role whatsoever – so-called Taylor states.

Taylor states in stellarators: A fast high-order boundary integral solver

We present a boundary integral equation solver for computing Taylor relaxed states in non-axisymmetric solid and shell-like toroidal geometries. The computation of Taylor states in these geometries is a key element for the calculation of stepped pressure stellarator equilibria. The integral representation of the magnetic field in this work is based on the generalized Debye source formulation, and results in a well-conditioned second-kind boundary integral equation. The integral equation solver is based on a spectral discretization of the geometry and unknowns, and the computation of the associated weakly-singular integrals is performed with high-order quadrature based on a partition of unity. The resulting scheme for applying the integral operator is then coupled with an iterative solver and suitable preconditioners. Several numerical examples are provided to demonstrate the accuracy and efficiency of our method, and a direct comparison with the leading code in the field is reported.