Dynamo Model Research Articles

Development method of American heavy vehicle low-load test cycle

Heavy vehicles have higher driving time and NOx emission level under low-load scenarios. Developing and inducting corresponding test cycle into emission valuation and certification standards is an effective method to improve translation from test emission results to real-world emission control benefits. At present, China is short of systematic research and achievements in this field. Present certification cycle failed to evaluate low-load emission. In America, CARB firstly applied least-square method to fit the multi-modal normal distribution with moving window average load, thus defined low-load data set. Secondly, applied K-means clustering method to classify and identity typical low-load driving scenarios. Thirdly, applied model to translate vehicle profile to construct low-load profile library. Finally, selected and combined representative profiles to construct the Low-load test cycle (LLC). Further more, by engine dynamo meter test and model simulation, LLC was used to evaluate the after treatment system and improve emission control technology under low-load scenarios. This paper comprehensively studied the overall process of cycle construction, verification and application, explained in detail the technical details of cycle construction routine, thus, provided practical technical support for development of China’s low-load test cycle.

Quantum molecular dynamics simulations of methane-water mixture at extreme conditions in Uranus and Neptune

Based on first-principles molecular dynamics simulations, the equation of state and Hugoniot curve of a methane-water mixture are investigated at relevant conditions in Uranus and Neptune. Under a strong shock wave, the compression ratio of the methane-water mixture is significantly higher than that of pure methane. Calculations of the pair-correlation function show that methane molecules are dissociated by ionized water above 4000 K. Above 6000 K, the methane and water molecules are completely dissociated, and many short-lived chemical species such as ${\mathrm{H}}_{3}\mathrm{O}$ and $\mathrm{OH}$ ions, C--O bonds, and free H atoms are formed. These results may provide a reasonable material basis for the unusual magnetic fields and the thin-shell dynamo models in Uranus and Neptune. The temperature and density ranges of our simulations are also relevant to dynamic shock experiments, and the results may provide reasonable models for future laboratory studies.

Effects of Emerging Bipolar Magnetic Regions in Mean-field Dynamo Model of Solar Cycles 23 and 24

We model the physical parameters of Solar Cycles 23 and 24 using a nonlinear dynamical mean-field dynamo model that includes the formation and evolution of bipolar magnetic regions (BMRs). The Parker-type dynamo model consists of a complete MHD system in the mean-field formulation: the 3D magnetic induction equation, and 2D momentum and energy equations in the anelastic approximation. The initialization of BMRs is modeled in the framework of Parker’s magnetic buoyancy instability. It defines the depths of BMR injections, which are typically located at the edge of the global dynamo waves. The distribution with longitude and latitude and the size of the initial BMR perturbations are taken from the NOAA database of active regions. The modeling results are compared with various observed characteristics of the solar cycles. Only the BMR perturbations located in the upper half of the convection zone lead to magnetic active regions on the solar surface. While the BMRs initialized in the lower part of the convection zone do not emerge on the surface, they still affect the global dynamo process. Our results show that BMRs can play a substantial role in the dynamo processes and affect the strength of the solar cycles. However, the data driven model shows that the BMR’s effect alone cannot explain the weak Cycle 24. This weak cycle and the prolonged preceding minimum of magnetic activity were probably caused by a decrease of the turbulent helicity in the bulk of the convection zone during the decaying phase of Cycle 23.

Spatio-temporal non-localities in a solar-like mean-field dynamo

ABSTRACT The scale separation approximation, which is in the base of the solar mean-field dynamo models, can be hardly justified both by observations and theoretical applications to astrophysical dynamos. The general expression for the mean turbulent electromotive force can be written in integral form with convolution of the turbulent effects and mean magnetic field variations over scales of the turbulent flows and global scales of the mean-field dynamo. Following results of direct numerical simulations (DNS), which had been reported earlier, we take the Lorentzian form of the integral convolution kernels as an experimental fact. It allows us to approximate the governing equation for the mean electromotive force by the reaction–diffusion type equation. Solution of the eigenvalue problem reveals a few curious properties of the dynamo model with the non-local mean electromotive force. We find a decrease of the critical dynamo instability threshold, and an increase the dynamo periods of the unstable modes, as reported in earlier studies. Simultaneously, the non-local model shows substantially lower growth rate of the unstable dynamo modes in proximity of the critical threshold than the model which employs the scale separation approximation. We verify these findings using the non-linear solar dynamo model. For the supercritical regime, when the α-effect magnitude is about twice of the instability threshold, the model shows the Parker’s dynamo wave solutions with the wave propagating from the mid-latitude at the bottom of the convection zone towards the solar equator at the surface.

Dynamo modelling for cycle variability and occurrence of grand minima in Sun-like stars: rotation rate dependence

ABSTRACT Like the solar cycle, stellar activity cycles are also irregular. Observations reveal that rapidly rotating (young) Sun-like stars exhibit a high level of activity with no Maunder-like grand minima and rarely display smooth regular activity cycles. On the other hand, slowly rotating old stars like the Sun have low activity levels and smooth cycles with occasional grand minima. We, for the first time, try to model these observational trends using flux transport dynamo models. Following previous works, we build kinematic dynamo models of one solar mass star with different rotation rates. Differential rotation and meridional circulation are specified with a mean-field hydrodynamic model. We include stochastic fluctuations in the Babcock–Leighton source of the poloidal field to capture the inherent fluctuations in the stellar convection. Based on extensive simulations, we find that rapidly rotating stars produce highly irregular cycles with strong magnetic fields and rarely produce Maunder-like grand minima, whereas the slowly rotating stars (with a rotation period of 10 d and longer) produce smooth cycles of weaker strength, long-term modulation in the amplitude, and occasional extended grand minima. The average duration and the frequency of grand minima increase with decreasing rotation rate. These results can be understood as the tendency of less supercritical dynamo in slower rotating stars to be more prone to produce extended grand minima.

Исключение интегрального члена в уравнениях одной эредитарной системы, связанной с задачей гидромагнитного динамо

В работе изучается двумерная система интегро-дифференциальных уравнений, которая является простейшей эредитарной моделью двумодового гидромагнитного динамо. Учет пространственной и временной нелокальности взаимодействий в динамо-системах сейчас активно исследуется. В маломодовых приближениях уравнений динамо можно рассматривать только временную нелокальность, т.е. эредитарность (память). Память в исследуемой системе реализована в виде обратной связи, распределенной по всем прошлым состояниям системы. Обратная связь представлена с помощью интегрального члена типа свертки от квадратичной комбинации фазовых переменных с ядром достаточно общего вида. Этот член моделирует подавление турбулентного генератора поля (α-эффекта) квадратичной формой от фазовых переменных. В реальных динамо-системах такое подавление обеспечивается силой Лоренца. Основной результат работы – доказательство возможности исключения интегрального члена для одного класса ядер. Такие ядра являются решениями однородного линейного дифференциального уравнения с постоянными коэффициентами. Доказано, что исследуемую интегро-дифференциальную систему можно заменить дифференциальной системой большей размерности с подходящими начальными условиями на дополнительные фазовые переменные. Если ядро является решением уравнения n-го порядка, то размерность системы может достигать 3n−2 и зависит от начальных условий, которым удовлетворяет ядро. В работе используются классические методы теории дифференциальных уравнений. Приводятся примеры динамических систем, возникающих при некоторых ядрах в результате исключения интегрального члена. Результаты работы можно использовать для верификации вычислительных алгоритмов и программных кодов, разработанных для решения интегро-дифференциальных уравнений.We study a two-dimensional system of integro-differential equations, which is the simplest hereditary model of a two-mode hydromagnetic dynamo. Accounting for the spatial and temporal nonlocality of interactions in dynamo systems is currently being actively studied. In the low-mode approximations of the dynamo equations, one can consider only temporal nonlocality, i.e. heredity (memory). Memory in the system under study is implemented in the form of feedback distributed over all past states of the system. The feedback is represented by a convolution-type integral term of a quadratic combination of phase variables with a fairly general kernel. This term models the quenching of the turbulent field generator (α-effect) by a quadratic form in phase variables. In real dynamo systems, such quenchingn is provided by the Lorentz force. The main result of the work is a proof of the possibility of eliminating the integral term for one class of kernels. Such kernels are solutions of a homogeneous linear differential equation with constant coefficients. It is proved that the studed integro-differential system can be replaced by a higher-dimensional differential system with suitable initial conditions for additional phase variables. If the kernel is a solution to an n-order equation, then the dimension of the system can reach 3n−2 and depends on the initial conditions that the kernel satisfies. The work uses classical methods of the theory of differential equations. Examples are given of dynamical systems that arise for some kernels as a result of the elimination of the integral term. The results of the work can be used to verify computational algorithms and program codes developed for solving integro-differential equations.

Nonlinear Parametric Resonance in the Simplest Model of a Solar Dynamo

The properties of nonlinear parametric resonance are investigated using the example of the low-mode Parker dynamo model. This model is a system of four ordinary differential equations and in the simplest approximation describes the processes of generation and oscillation of large-scale magnetic fields in stellar systems. In the absence of nonlinear effects, the problem under consideration, by analogy with a system of harmonic oscillations, admits an asymptotic division of multiple resonant frequencies. However, despite the fact that at first glance at these frequencies it is reasonable to expect an amplification of the amplitude in the nonlinear case, it is demonstrated that in the presence of nonlinear terms, the behavior of the system is significantly more complex. In particular, generation suppression can be observed at resonant or low frequencies, while amplification occurs in the immediate vicinity of the resonance or at sufficiently high frequencies. The reasons are discussed for this behavior, as well as the possibility of the influence of parametric resonance on the establishment of planetary dynamo cycles.

Kinematic Dynamo Modeling and Its Peculiarities in Polar Latitudes

Kinematic Dynamo Modeling and Its Peculiarities in Polar Latitudes

The emergence and growth of the flux transport dynamo model of the sunspot cycle

The emergence and growth of the flux transport dynamo model of the sunspot cycle

Correlation between activity indicators: Hα and Ca II lines in M-dwarf stars

Context. Different approaches have been adopted to study both short- and long-term stellar magnetic activity, and although the mechanisms by which low-mass stars generate large-scale magnetic fields are not well understood, it is known that stellar rotation plays a key role. Aims. There are stars that show a cyclical behaviour in their activity studied on the blue side of the visible spectrum, which can be explained by solar dynamo or αΩ dynamo models. However, when studying late-type dwarf stars, they become redder and it is necessary to implement other indicators to analyse their magnetic activity. In the present work, we perform a comparative study between the best-known activity indicators so far defined from the Ca II and Hα lines to analyse M-dwarf stars. Methods. We studied a sample of 29 M stars with different chromospheric activity levels and spectral classes ranging from dM0 to dM6. To do so, we employed 1796 spectra from different instruments with a median time span of observations of 21 yr. The spectra have a wide spectral range that allowed us to compute the chrosmospheric activity indicators based on Ca II and Hα. In addition, we complemented our data with photometric observations from the TESS space mission for better stellar characterisation and short-term analysis. Results. We obtained a good and significant correlation (ρ = 0.91) between the indexes defined from the two lines for the whole set of stars in the sample. However, we found that there is a deviation for faster rotators (with Prot < 4 days) and higher flare activity (at least one flare per day). For the individual analysis, we found that the indexes computed individually for each star correlate independently of the level of chromospheric emission and the rotation period. Conclusions. There is an overall positive correlation between Ca II and Hα emission in dM stars, except during flare events. In particular, we found that low-energy high-frequency flares could be responsible for the deviation in the linear trend in fast-rotator M dwarfs. This implies that the rotation period could be a fundamental parameter to study the stellar activity and that the rotation could drive the magnetic dynamo in low-mass active stars.

Mantle-driven north–south dichotomy in geomagnetic polar minima

The geomagnetic field on the core-mantle boundary (CMB) is characterized by weak or even reversed field in the polar regions and intense flux patches at the edges of the intersection of the inner-core tangent cylinder (TC) with the CMB. This high-latitude field morphology is in agreement with thermal wind theory inside the TC in which polar upwellings disperse magnetic field lines. Furthermore, inferences from the geomagnetic secular variation hint to the presence of a westward jet at high latitudes of the northern hemisphere, also in agreement with the TC dynamical theory, but not in the southern hemisphere. Here we study polar minima in an ensemble of geomagnetic field models that span the historical era and in a set of numerical dynamo simulations with a heterogeneous outer boundary heat flux inferred from a tomographic model of lowermost mantle seismic anomalies. We quantified the polar minima using a previously-proposed expression as well as a new measure which may better capture this phenomenon. We found that throughout the historical era the geomagnetic field is characterized by stronger polar minima and more reversed flux inside the northern TC than inside the southern TC. Likewise, almost all dynamo models exhibit on average stronger polar minima in the northern hemisphere. This north/south dichotomy is explained in terms of the pattern of lowermost mantle seismic anomalies, in particular the southern centers of the two Large Low Shear-wave Velocity Provinces below Africa and the Pacific. We also investigated polar minima in planets where magnetic field models at the top of the dynamo region are available. We speculate that the absence of polar minima in Mercury’s field is likely due to the thick stratified layer at the top of its core, while the strong polar minima in Jupiter’s field might have a different dynamical origin than the geomagnetic polar minima.

Kinematic Dynamo Modeling and Its Peculiarities in Polar Latitudes

The results of calculations of the magnetic field structure in the kinematic approximation are presented, testifying to the consistency of the constructed numerical model with the structure of the flows of a conducting liquid: the instability of the convective process in a rotating spherical shell is reflected in the evolution of the magnetic field, manifesting itself in the pulsation regime of the field. It is shown that the features of the field evolution in the calculations are most clearly pronounced at high latitudes and have analogs in the behavior of the real geomagnetic field.

No evidence for synchronization of the solar cycle by a “clock”

The length of the solar activity cycle fluctuates considerably. The temporal evolution of the corresponding cycle phase, that is, the deviation of the epochs of activity minima or maxima from strict periodicity, provides relevant information concerning the physical mechanism underlying the cyclic magnetic activity. An underlying strictly periodic process (akin to a perfect “clock”), with the observer seeing a superposition of the perfect clock and a small random phase perturbation, leads to long-term phase stability in the observations. Such behavior would be expected if cycles were synchronized by tides caused by orbiting planets or by a hypothetical torsional oscillation in the solar radiative interior. Alternatively, in the absence of such synchronization, phase fluctuations accumulate and a random walk of the phase ensues, which is a typical property of randomly perturbed dynamo models. Based on the sunspot record and the reconstruction of solar cycles from cosmogenic 14C, we carried out rigorous statistical tests in order to decipher whether there exists phase synchronization or random walk. Synchronization is rejected at significance levels of between 95% (28 cycles from sunspot data) and beyond 99% (84 cycles reconstructed from 14C), while the existence of random walk in the phases is consistent with all data sets. This result strongly supports randomly perturbed dynamo models with little inter-cycle memory.

Realization of Bullard’s disc dynamo

We report experimental results from three successful runs of a Bullard-type homopolar disc dynamo. The set-up consisted of a copper disc with a radius of 30 cm and thickness of 3 cm which was placed co-axially beneath a flat, multi-arm spiral coil of the same size and connected to it electrically at the centre and along the circumference by sliding liquid-metal contacts. The magnetic field was measured using Hall probes which were fixed on the top face of the coil. We measured also the radial voltage drop across the coil. When the disc rotation rate reached Ω ≈ 7 Hz , the magnetic field increased steeply approaching B 0 ≈ 40 mT in the central part of the coil. This field was more than two orders of magnitude stronger than the background magnetic field. In the first two runs, the electromagnetic torque braking the disc in the dynamo regime exceeded the breakdown torque of the electric motor driving the disc. As a result, the motor stalled and the dynamo was interrupted. Stalling did not occur in the third run when the driving frequency was set higher and increased faster. We also propose an extended disc dynamo model which qualitatively reproduces the experimental results.

Topologically protected quantum dynamo effect in a driven spin-boson model

We describe a quantum dynamo effect in a driven system coupled to a harmonic oscillator describing a cavity mode or to a collection of modes forming an Ohmic bosonic bath. When the system Hamiltonian changes in time, this induces a dynamical field in the bosonic modes having resonant frequencies with the driving velocity. This field opposes the change of the external driving field in a way reminiscent of Faraday's law of induction, justifying the term 'quantum dynamo effect'. For the specific situation of a periodically driven spin-$\frac{1}{2}$ on the Bloch sphere, we show that the work done by rolling the spin from north to south pole can efficiently be converted into a coherent displacement of the resonant bosonic modes, the effect thus corresponds to a work-to-work conversion and allows to interpret this transmitted energy into the bath as work. We study this effect, its performance and limitations in detail for a driven spin-$\frac{1}{2}$ in the presence of a radial magnetic field addressing a relation with topological systems through the formation of an effective charge in the core of the sphere. We show that the dynamo effect is directly related to the dynamically measured topology of this spin-$\frac{1}{2}$ and thus in the adiabatic limit provides a topologically protected method to convert driving work into a coherent field in the reservoir. The quantum dynamo model is realizable in mesoscopic and atomic systems.

Alpha tensor and dynamo excitation in turbulent fluids with anisotropic conductivity fluctuations

AbstractA mean‐field theory of the electrodynamics of a turbulent fluid is formulated under the assumption that the molecular electric conductivity is correlated with the turbulent velocity fluctuation in the (radial) direction, . It is shown that for such homogeneous fluids a strong turbulence‐induced field advection anti‐parallel to arises almost independently of rotation. For rotating fluids, an extra effect appears with the known symmetries and with the expected maximum at the poles. Fast rotation, however, with Coriolis number exceeding unity suppresses this term. Numerical simulations of forced turbulence using the nirvana code demonstrate that the radial advection velocity, , always dominates the term. We show finally with simplified models that dynamos are strongly influenced by the radial pumping: for the solutions become oscillatory, while for they become highly exotic if they exist at all. In conclusion, dynamo models for slow and fast solid‐body rotation on the basis of finite conductivity–velocity correlations are unlikely to work, at least for dynamos without strong shear.

Three-dimensional non-kinematic simulation of the post-emergence evolution of bipolar magnetic regions and the Babcock-Leighton dynamo of the Sun

Context. The Babcock-Leighton flux-transport model is a widely accepted dynamo model of the Sun that can explain many observational aspects of solar magnetic activity. This dynamo model has been extensively studied in a two-dimensional (2D) mean-field framework in both kinematic and non-kinematic regimes. Recent three-dimensional (3D) models have been restricted to the kinematic regime. In these models, the surface poloidal flux is produced by the emergence of bipolar magnetic regions (BMRs) that are tilted according to Joy’s law. Aims. We investigate the prescription for emergence of a BMR in 3D non-kinematic simulations. In particular, we examine the effect of the radial extent of the BMR. We also report our initial results based on a cyclic Babcock-Leighton dynamo simulation. Methods. We extended a conventional 2D mean-field model of the Babcock-Leighton flux-transport dynamo into 3D non-kinematic regime, in which a full set of magnetohydrodynamic (MHD) equations are solved in a spherical shell using a Yin-Yang grid. The large-scale mean flows, such as differential rotation and meridional circulation, are not driven by rotationally constrained convection, but rather by the parameterized Λ-effect in this model. For the induction equation, we used a Babcock-Leighton α-effect source term by which the surface BMRs are produced in response to the dynamo-generated toroidal field inside the convection zone. Results. We find that in the 3D non-kinematic regime, the tilt angle of a newly-emerged BMR is very sensitive to the prescription for the subsurface structure of the BMR (particularly, its radial extent). Anti-Joy tilt angles are found unless the BMR is deeply embedded in the convection zone. We also find that the leading spot tends to become stronger (higher field strengths) than the following spot. The anti-Joy’s law trend and the morphological asymmetry of the BMRs can be explained by the Coriolis force acting on the Lorentz-force-driven flows. Furthermore, we demonstrate that the solar-like magnetic cycles can be successfully obtained if Joy’s law is explicitly given in the Babcock-Leighton α-effect. In these cyclic dynamo simulations, a strong Lorentz force feedback leads to cycle modulations in the differential rotation (torsional oscillation) and meridional circulation. The simulations, however, do not include radiative effects (e.g., enhanced cooling by faculae) that are required to properly model the torsional oscillations. The non-axisymmetric components of the flows are found to exist as inertial modes such as the equatorial Rossby modes.

Comparison of physics-based prediction models of solar cycle 25

Physics-based solar cycle predictions provide an effective way to verify our understanding of the solar cycle. Before the start of cycle 25, several physics-based solar cycle predictions were developed. These predictions use flux transport dynamo (FTD) models, surface flux transport (SFT) models, or a combination of the two kinds of models. The common physics behind these predictions is that the surface poloidal fields around cycle minimum dominate the subsequent cycle strength. In the review, we first give short introductions to SFT and FTD models. Then we compare 7 physics-based prediction models from 4 aspects, which are what the predictor is, how to get the predictor, how to use the predictor, and what to predict. Finally, we demonstrate the large effect of assimilated magnetograms on predictions by two SFT numerical tests. We suggest that uncertainties in both initial magnetograms and sunspot emergence should be included in such physics-based predictions because of their large effects on the results. In addition, in the review we put emphasis on what we can learn from different predictions, rather than an assessment of prediction results.

Kinematic Dynamo Model of a Solar Magnetic Cycle

The paper deals with the problem of explaining the origin and nature of the space-time variations in the magnetic activity of the Sun. It presents a new hydrodynamic model of the solar magnetic cycle, which uses helioseismological data on the differential rotation of the solar convective zone. The model is based on the hypothesis of the emergence of global flows as a result of the loss of stability of a differentially rotating plasma layer in the convective zone. First, the hydrodynamic global plasma flows are calculated without accounting for the effect of a magnetic field on them. Under this condition, it is shown that the solutions found describe all global flows observed on the surface of the Sun: permanent meridional circulation from the equator to the poles, torsional oscillations and space-time variations of the meridional flow. We conclude that the last two flows are azimuthal and meridional components of a single three-dimensional global hydrodynamic flow. Second, to simulate the dynamics of the magnetic field, the found velocities of global migrating flows and the spatial profile of the angular velocity of the internal differential rotation of the solar convective zone obtained from helioseismic measurements were used. Good coincidences have been obtained between the characteristics of the calculated dynamics of global migrating flows and the variable global magnetic fields generated by them with the observed values on the solar surface. An explanation is given for some phenomena on the surface of the Sun, which could not be explained within the framework of the available models.

Shear-driven magnetic buoyancy in the solar tachocline: the mean electromotive force due to rotation

ABSTRACT The leading theoretical paradigm for the Sun’s magnetic cycle is an αω-dynamo process, in which a combination of differential rotation and turbulent, helical flows produces a large-scale magnetic field that reverses every 11 yr. Most αω solar dynamo models rely on differential rotation in the solar tachocline to generate a strong toroidal field. The most problematic part of such models is then the production of the large-scale poloidal field, via a process known as the α-effect. Whilst this is usually attributed to small-scale convective motions under the influence of rotation, the efficiency of this regenerative process has been called into question by some numerical simulations. Motivated by likely conditions within the tachocline, the aim of this paper is to investigate an alternative mechanism for the poloidal field regeneration, namely the magnetic buoyancy instability in a shear-generated, rotating magnetic layer. We use a local, fully compressible model in which an imposed vertical shear winds up an initially vertical magnetic field. The field ultimately becomes buoyantly unstable, and we measure the resulting mean electromotive force (EMF). For sufficiently rapid rotation, we find that a significant component of the mean EMF is aligned with the direction of the mean magnetic field, which is the characteristic feature of the classical αω-dynamo model. Our results therefore suggest that magnetic buoyancy could contribute directly to the generation of large-scale poloidal field in the Sun.