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We use the ASH code to model the convective dynamo of solar-type stars. Based on a series of 15 3-D MHD simulations spanning 4 bins in rotation and mass, we show what mechanisms are at work in these stellar dynamos with and without magnetic cycles and how global stellar parameters affect the outcome. We also derive scaling laws for the differential rotation and magnetic field based on these simulations. We find a weaker trend between differential rotation and stellar rotation rate, ($\Delta \Omega \sim (|\Omega|/\Omega_{\odot})^{0.46}$) in the MHD solutions than in their HD counterpart $(|\Omega|/\Omega_{\odot})^{0.66})$, yielding a better agreement with the observational trends based on power laws. We find that for a fluid Rossby number between $0.15 \lesssim Ro_f \lesssim 0.65$ the solutions possess long magnetic cycle, if $Ro_f \lesssim 0.42$ a short cycle and if $Ro_f \gtrsim 1$ (anti-solar-like differential rotation) a statistically steady state. We show that short-cycle dynamos follow the classical Parker-Yoshimura rule whereas the long-cycle period ones do not. We further demonstrate that the Rossby number dependency of the large-scale surface magnetic field in the simulation ($B_{L,surf} \sim Ro_{f}^{-1.26}$) agrees better with observations ($B_{V} \sim Ro_{s}^{-1.4 \pm 0.1}$) and differs from dynamo scaling based on the global magnetic energy ($B_{bulk} \sim Ro_{f}^{-0.5}$). We also show that up to few percents of the stellar luminosity can be channelled into the star's magnetism, hence providing a large energy reservoir for possible surface eruptive events., Comment: 45 pages, 24 figures, 7 tables, 3 appendices, accepted by ApJ on 12/23/21, published on 02/03/22. (1) DAp/AIM, CEA/IRFU, CNRS/INSU, Univ. Paris-Saclay \& Univ. de Paris, France, (2) CMPA, KU Leuven, Belgium, (3) Universidad Carlos III de Madrid, Spain, (4) Dept. de physique, Univ. de Montr\'eal, Canada, (5) JILA and Dept. of Astrophysical and Planetary Sciences, University of Colorado, USA

The atmosphere of a hot jupiter may be subject to a thermo-resistive instability, in which the increasing electrical conductivity with temperature leads to runaway Ohmic heating. We introduce a simplified model of the local dynamics in the equatorial region of a hot jupiter that incorporates the back reaction on the atmospheric flow as the increasing electrical conductivity leads to flux freezing, which in turn quenches the flow and therefore the Ohmic heating. We demonstrate a new time-dependent solution that emerges for a temperature-dependent electrical conductivity (whereas a temperature-independent conductivity always evolves to a steady-state). The periodic cycle consists of bursts of Alfven oscillations separated by quiescent intervals, with the magnetic Reynolds number alternating between values smaller than and larger than unity, maintaining the oscillation. We investigate the regions of pressure and temperature in which the instability operates. For the typical equatorial accelerations seen in atmospheric models, we find instability at pressures $\sim 0.1$--$1\ {\rm bar}$ and temperatures $\approx 1300$--$1800\ {\rm K}$ for magnetic fields $\sim 10\ {\rm G}$. Unlike previous studies based on a constant wind velocity, we find that the instability is stronger for weaker magnetic fields. Our results add support to the idea that variability should be a feature of magnetized hot jupiter atmospheres, particularly at intermediate temperatures. The temperature-dependence of the electrical conductivity is an important ingredient that should be included in MHD models of hot jupiter atmospheric dynamics., Comment: 10 pages, 5 figures, submitted to ApJ

We present a non-kinematic axisymetric α2Ω mean-field dynamo model in which the complete α-tensor and mean differential rotation profile are both extracted from a global magnetohydrodynamical simulation of solar convection producing cycling large-scale magnetic fields. The nonlinear backreaction of the Lorentz force on differential rotation is the only amplitude-limiting mechanism introduced in the mean-field model. We compare and contrast the amplitude modulation patterns characterizing this mean-field dynamo, to those already well-studied in the context of non-kinematic αΩ models using a scalar α-effect. As in the latter, we find that large quasi-periodic modulation of the primary cycle are produced at low magnetic Prandtl number (Pm), with the ratio of modulation period to the primary cycle period scaling inversely with Pm. The variations of differential rotation remain well within the bounds set by observed solar torsional oscillations. In this low-Pm regime, moderately supercritical solutions can also exhibit aperiodic Maunder Minimum-like periods of strongly reduced cycle amplitude. The inter-event waiting time distribution is approximately exponential, in agreement with solar activity reconstructions based on cosmogenic radioisotopes. Secular variations in low-latitude surface differential rotation during Grand Minima, as compared to epochs of normal cyclic behavior, are commensurate in amplitude with historical inferences based on sunspot drawings. Our modeling results suggest that the low levels of observed variations in the solar differential rotation in the course of the activity cycle may nonetheless contribute to, or perhaps even dominate, the regulation of the magnetic cycle amplitude.

The largest solar flares, of class X and above, are often associated with strong energetic particle acceleration. Based on the self-similar distribution of solar flares, self-organized criticality models such as sandpiles can be used to successfully reproduce their statistics. However, predicting strong (and rare) solar flares turns out to be a significant challenge. We build here on an original idea based on the combination of minimalistic flare models (sandpiles) and modern data assimilation techniques (4DVar) to predict large solar flares. We discuss how to represent a sandpile model over a reduced set of eigenfunctions to improve the efficiency of the data assimilation technique. This improvement is model-independent and continues to pave the way towards efficient near real-time solutions for predicting solar flares.

The solar dipole moment at activity minimum is a good predictor of the strength of the subsequent solar cycle. Through a systematic analysis using a state-of-the-art 2 × 2 D solar dynamo model, we found that bipolar magnetic regions (BMR) with atypical characteristics can modify the strength of the next cycle via their impact on the buildup of the dipole moment as a sunspot cycle unfolds. In addition to summarizing these results, we present further effects of such “rogue” BMRs. These have the ability to generate hemispheric asymmetry in the subsequent sunspot cycle, since they modify the polar cap flux asymmetry of the ongoing cycle. We found strong correlation between the polar cap flux asymmetry of cycle i and the total pseudo sunspot number asymmetry of cycle i + 1 . Good correlation also appears in the case of the time lag of the hemispheres of cycle i + 1 .

Reconstructions of past solar activity based on cosmogenic radioisotopes have reavealed that the Sun spends a significant fraction ( ${\approx}\, 20$ %) of its time in aperiodically recurring states of so-called Grand Minima or Grand Maxima, namely epochs of strongly supressed and markedly above-average levels of magnetic activity, respectively. The physical origin of these episodes is not yet understood. In this article we present a dual-dynamo model of the solar cycle, combining a dominant dynamo based on differential-rotation shear and surface decay of bipolar active regions, and a weak, deep-seated turbulent dynamo. The resulting dynamo simulations are found to exhibit the equivalent of observed Grand Minima and Maxima. By adjusting the magnitude and saturation level of the secondary turbulent dynamo, we can reproduce well the duration and waiting-time distributions of Grand Minima and Maxima inferred from the cosmogenic-isotope record. The exit from Grand Minima episodes is typically characterized by strong hemispheric asymmetries, in agreement with sunspot observations during the 1645 – 1715 Maunder Minimum. In these simulations, Grand Maxima can be unambiguously identified as a distinct dual-dynamo state resulting from constructive interference between the two dynamos mechanisms operating within the simulation. This interaction leads to the autonomous production of long quasi-periodicities in the millennial range, commensurate with the Halstatt cycle. Such a quasi-periodic modulation, readily produced through dynamical backreaction on large-scale flows in non-kinematic dynamo models, is quite uncommon in a purely kinematic solar-cycle model such as the one developed herein. We argue that these long periodicities are set by the long diffusion time of magnetic field accumulating in the stable layers underlying the turbulent convection zone.

Coverage of plasma motions is limited to the line-of-sight component at the Sun’s surface. Multiple tracking and inversion methods were developed to infer the transverse motions from observational data. Recently, the DeepVel neural network was trained with computations performed by numerical simulations of the solar photosphere to recover the missing transverse component at the surface and at two additional optical depths simultaneously from the surface white light intensity in the Quiet Sun. We argue that deep learning could provide additional spatial coverage to existing observations in the form of depth-dependent synthetic observations, i.e. estimates generated through the emulation of numerical simulations. We trained different versions of DeepVel using slices from numerical simulations of both the Quiet Sun and Active Region at various optical and geometrical depths in the solar atmosphere, photosphere and upper convection zone to establish the upper and lower limits at which the neural network can generate reliable synthetic observations of plasma motions from surface intensitygrams. Flow fields inferred in the photosphere and low chromosphere τ ∈ [0.1, 1) are comparable to inversions performed at the surface (τ ≈ 1) and are deemed to be suitable for use as synthetic estimates in data assimilation processes and data-driven simulations. This upper limit extends closer to the transition region (τ ≈ 0.01) in the Quiet Sun, but not for Active Regions. Subsurface flows inferred from surface intensitygrams fail to capture the small-scale features of turbulent convective motions as depth crosses a few hundred kilometers. We suggest that these reconstructions could be used as first estimates of a model’s velocity vector in data assimilation processes to nowcast and forecast short term solar activity and space weather.

We perform a mean-field analysis of the EULAG-MHD millenium simulation of global magnetohydrodynamical convection presented in Passos et al. 2014. The turbulent electromotive force operating in the simulation is assumed to be linearly related to the cyclic axisymmetric mean magnetic field and its first spatial derivatives. At every grid point in the simulation's meridional plane, this assumed relationship involves 27 independent tensorial coefficients. Expanding on Racine et al. 2011, we extract these coefficients from the simulation data through a least-squares minimization procedure based on singular value decomposition. The reconstructed alpha-tensor shows good agreement with that obtained by Racine et al. 2011, who did not include derivatives of the mean-field in their fit, as well as with the alpha-tensor extracted by Augustson et al. 2015 from a distinct ASH MHD simulation. The isotropic part of the turbulent magnetic diffusivity tensor beta is positive definite and reaches values of 5.0x10^7 m2s-1 in the middle of the convecting fluid layers. The spatial variations of both alpha_phiphi and beta_phiphi component are well reproduced by expressions obtained under the SOCA, with a good matching of amplitude requiring a turbulent correlation time about five times smaller than the estimated turnover time of the small-scale turbulent flow. We find the magnetic quenching of the alpha-effect to be driven primarily by a reduction of the small-scale flow's kinetic helicity, with variations of the current helicity playing a lesser role in most locations in the simulation domain. Our measurements of turbulent diffusivity quenching are restricted to the beta_phiphi component, but indicate a weaker quenching, by a factor of 1.36, than of the alpha effect, which in our simulation drops by a factor of three between the minimum and maximum phases of the magnetic cycle., Comment: Accepted in Advances in Space Research

The impressive development of global numerical simulations of turbulent stellar interiors unveiled a variety of possible differential rotation (solar or anti-solar), meridional circulation (single or multi-cellular), and dynamo states (stable large scale toroidal field or periodically reversing magnetic fields). Various numerical schemes, based on the so-called anelastic set of equations, were used to obtain these results. It appears today mandatory to assess their robustness with respect to the details of the numerics, and in particular to the treatment of turbulent sub-grid scales. We report on an ongoing comparison between two global models, the ASH and EULAG codes. In EULAG the sub-grid scales are treated implicitly by the numerical scheme, while in ASH their effect is generally modelled by using enhanced dissipation coefficients. We characterize the sub-grid scales effect in a turbulent convection simulation with EULAG. We assess their effect at each resolved scale with a detailed energy budget. We derive equivalent eddy-diffusion coefficients and use the derived diffusivities in twin ASH numerical simulations. We find a good agreement between the large-scale flows developing in the two codes in the hydrodynamic regime, which encourages further investigation in the magnetohydrodynamic regime for various dynamo solutions., 21 pages, 6 figures, accepted for publication in Advances in Space Research

An algebraic method for the reconstruction and potentially prediction of the solar dipole moment value at sunspot minimum (known to be a good predictor of the amplitude of the next solar cycle) was suggested in the first paper in this series. The method sums up the ultimate dipole moment contributions of individual active regions in a solar cycle: for this, detailed and reliable input data would in principle be needed for thousands of active regions in a solar cycle. To reduce the need for detailed input data, here we propose a new active region descriptor called ARDoR (Active Region Degree of Rogueness). In a detailed statistical analysis of a large number of activity cycles simulated with the 2$\times$2D dynamo model we demonstrate that ranking active regions by decreasing ARDoR, for a good reproduction of the solar dipole moment at the end of the cycle it is sufficient to consider the top $N$ regions on this list explicitly, where $N$ is a relatively low number, while for the other regions the ARDoR value may be set to zero. E.g., with $N=5$ the fraction of cycles where the dipole moment is reproduced with an error exceeding $\pm$30% is only 12%, significantly reduced with respect to the case $N=0$, i.e. ARDoR set to zero for all active regions, where this fraction is 26%. This indicates that stochastic effects on the intercycle variations of solar activity are dominated by the effect of a low number of large ``rogue'' active regions, rather than the combined effect of numerous small ARs. The method has a potential for future use in solar cycle prediction., 15 pages, 2 tables, 4 figs. Accepted for publication in the Journal of Space Weather and Space Climate

We examine the impact of surface inflows into activity belts on the operation of solar cycle models based on the Babcock–Leighton mechanism of poloidal field regeneration. Towards this end we introduce in the solar cycle model of Lemerle & Charbonneau (2017.ApJ834: 133) a magnetic flux-dependent variation of the surface meridional flow based on the axisymmetric inflow parameterization developped by Jiang et al. (2010.ApJ717: 597). The inflow dependence on emerging magnetic flux thus introduces abona fidenonlinear backreaction mechanism in the dynamo loop. For solar-like inflow speeds, our simulation results indicate a decrease of 10–20% in the strength of the global dipole building up at the end of an activity cycle, in agreement with earlier simulations based on linear surface flux transport models. Our simulations also indicate a significant stabilizing effect on cycle characteristics, in that individual cycle amplitudes in simulations including inflows show less scatter about their mean than in the absence of inflows. Our simulations also demonstrate an enhancement of cross-hemispheric coupling, leading to a significant decrease in hemispheric cycle amplitude asymmetries and temporal lag in hemispheric cycle onset. Analysis of temporally extended simulations also indicate that the presence of inflows increases the probability of cycle shutdown following an unfavorable sequence of emergence events. This results ultimately from the lower threshold nonlinearity built into our solar cycle model, and presumably operating in the sun as well.

The periods of magnetic activity cycles in the Sun and solar-type stars do not exhibit a simple or even single trend with respect to rotation rate or luminosity. Dynamo models can be used to interpret this diversity, and can ultimately help us understand why some solar-like stars do not exhibit a magnetic cycle, whereas some do, and for the latter what physical mechanisms set their magnetic cycle period. Three-dimensional non-linear magnetohydrodynamical simulations present the advantage of having only a small number of tunable parameters, and produce in a dynamically self-consistent manner the flows and the dynamo magnetic fields pervading stellar interiors. We conducted a series of such simulations within the EULAG-MHD framework, varying the rotation rate and luminosity of the modeled solar-like convective envelopes. We find decadal magnetic cycles when the Rossby number near the base of the convection zone is moderate (typically between 0.25 and 1). Secondary, shorter cycles located at the top of the convective envelope close to the equator are also observed in our numerical experiments, when the local Rossby number is lower than 1. The deep-seated dynamo sustained in these numerical experiments is fundamentally non-linear, in that it is the feedback of the large-scale magnetic field on the large-scale differential rotation that sets the magnetic cycle period. The cycle period is found to decrease with the Rossby number, which offers an alternative theoretical explanation to the variety of activity cycles observed in solar-like stars., 18 pages, 12 figures, 4 tables, accepted for publication in The Astrophysical Journal

Surface observations indicate that the speed of the solar meridional circulation in the photosphere varies in anti-phase with the solar cycle. The current explanation for the source of this variation is that inflows into active regions alter the global surface pattern of the meridional circulation. When these localized inflows are integrated over a full hemisphere, they contribute to the slow down of the axisymmetric poleward horizontal component. The behavior of this large scale flow deep inside the convection zone remains largely unknown. Present helioseismic techniques are not sensitive enough to capture the dynamics of this weak large scale flow. Moreover, the large time of integration needed to map the meridional circulation inside the convection zone, also masks some of the possible dynamics on shorter timescales. In this work we examine the dynamics of the meridional circulation that emerges from a 3D MHD global simulation of the solar convection zone. Our aim is to assess and quantify the behavior of meridional circulation deep inside the convection zone, where the cyclic large-scale magnetic field can reach considerable strength. Our analyses indicate that the meridional circulation morphology and amplitude are both highly influenced by the magnetic field, via the impact of magnetic torques on the global angular momentum distribution. A dynamic feature induced by these magnetic torques is the development of a prominent upward flow at mid latitudes in the lower convection zone that occurs near the equatorward edge of the toroidal bands and that peaks during cycle maximum. Globally, the dynamo-generated large-scale magnetic field drives variations in the meridional flow, in stark contrast to the conventional kinematic flux transport view of the magnetic field being advected passively by the flow., 20 pages, 13 figures, Accepted in A&A

We present calculations of stratospheric chemical abundances variations between different levels of solar activity using a simple photochemistry model in transient chemistry mode. Different models for the reconstruction of the solar spectrum, as well as observations from the SOLar STellar Irradiance Comparison Experiment (SOLSTICE) and Spectral Irradiance Monitor (SIM) on the SOlar Radiation and Climate Experiment (SORCE) satellite, are used as inputs to the calculations. We put the emphasis on the MOnte CArlo Spectral Solar Irradiance Model (MOCASSIM) reconstructions, which cover the spectral interval from 150 to 400 nm and extend from 1610 to present. We compare our results with those obtained with the Naval Research Laboratory Solar Spectral Irradiance (NRLSSI) model as well as with the Magnesium-Neutron Monitor (MGNM) model over a period of time spanning the ascending phase of Cycle 22. We also perform the calculations using SORCE composite spectra for the descending phase of Cycle 23 and with the reconstructed MOCASSIM, NRLSSI and MGNM spectra for the same period for comparison. Finally, we compare the chemical abundances obtained for the Maunder Minimum with those obtained for the Cycle 23 minimum (in March 2009) and find that stratospheric ozone concentration was slightly higher during the recent minimum, consequent to the small positive variability between the MOCASSIM spectra for both epochs, especially below 260 nm. We find that the response in stratospheric ozone is not only dependent on the variability amplitude in the solar spectrum (especially in the 200–240 nm band), but also significantly on the base level of the minimum solar spectrum.

The magnetic fields of solar-type stars are observed to cycle over decadal periods -11 years in the case of the Sun. The fields originate in the turbulent convective layers of stars and have a complex dependency upon stellar rotation rate. We have performed a set of turbulent global simulations that exhibit magnetic cycles varying systematically with stellar rotation and luminosity. We find that the magnetic cycle period is inversely proportional to the Rossby number, which quantifies the influence of rotation on turbulent convection. The trend relies on a fundamentally non-linear dynamo process and is compatible with the Sun's cycle and those of other solar-type stars., Comment: 24 pages, 12 figures, 2 tables (including supplementary material), accepted for publication in Science

This book investigates complex systems that are idealizations of naturally occurring phenomena characterized by the autonomous generation of structures and patterns at macroscopic scales. It provides material and guidance to allow the reader to learn about complexity through hands-on experimentation with complex systems with the aid of computer programs. Each chapter thus presents a simple computational model of natural complex phenomena ranging from avalanches and earthquakes to solar flares, epidemics, and ant colonies. This introductory chapter explains what complexity is, with emphasis on the fact that defining it is not a simple endeavor, and that it is not the same as randomness or chaos. It also shows that open dissipative systems are complex and clarifies what natural complexity means. Finally, it describes the computer programs listed in this book and suggests materials for further reading about complexity.

This chapter explores how a “natural” process generates dynamically something that is conceptually similar to a percolation cluster by using the case of forest fires. It first provides an overview of the forest-fire model, which is essentially a probabilistic cellular automata, before discussing its numerical implementation using the Python code. It then describes a representative simulation showing the triggering, growth, and decay of a large fire in a representative forest-fire model simulation on a small 100 x 100 lattice. It also considers the behavior of the forest-fire model as well as its self-organized criticality and concludes with an analysis of the advantages and limitations of wildfire management. The chapter includes exercises and further computational explorations, along with a suggested list of materials for further reading.

This chapter examines the complex nature of pattern formation in excitable systems. Many physical, chemical, and biological systems are considered excitable, in which two “components” interact in such a way as to alter each other's state through (nonlinear) processes of inhibition or amplification. Starting from a homogeneous rest state, many systems of this type can spontaneously generate persistent spatiotemporal patterns when subjected to some perturbation. After providing an overview of excitable systems, the chapter discusses the hodgepodge machine and its numerical implementation in the Python code. It then describes four spatial patterns generated by the hodgepodge machine, namely: waves, spirals, spaghettis, and cells. It also explains the spontaneous formation of spatiotemporal patterns. The chapter includes exercises and further computational explorations, along with a suggested list of materials for further reading.

This chapter examines the natural complexity of earthquakes. Earthquakes release energy that is quantified by their magnitude m, a logarithmic measure of seismic wave amplitudes. The number N of earthquakes having a magnitude larger than m in a given area and time interval is expressed by the Gutenberg–Richter law. This power-law reflects scale invariance in the dynamics of earthquakes, and the chapter begins by reproducing this property using a simple mechanical model, the Burridge–Knopoff stick–slip model of seismic faults. The chapter then discusses the Olami–Feder–Christensen lattice-based implementation of the Burridge–Knopoff earthquake model using the Python code. It also describes a representative simulation on a 128 x 128 lattice, the behavior of the Olami–Feder–Christensen model, and earthquake forecasting. The chapter includes exercises and further computational explorations, along with a suggested list of materials for further reading.

This chapter examines the complex nature of flocking. Many living creatures have evolved flocking as a behavioral strategy, including herds, birds, and insects. Models of flocking have been used to understand and control the movement of dense human crowds in socially extreme situations. The chapter begins with a discussion of the flocking model, defined in two spatial dimensions on the periodic unit square in which N agents are moving, under the influence of four forces: repulsion, flocking, self-propulsion, and random forces. It then describes the numerical implementation of the model using the Python code, along with the range of global behaviors generated by the the model. It also considers the spontaneous segregation of active and passive flockers before concluding with an analysis of the perturbing effects of panicked individuals on collective, ordered motion, focusing on two types of agents: strongly flocking “calm” agents and small random force.

This book provides a short, hands-on introduction to the science of complexity using simple computational models of natural complex systems—with models and exercises drawn from physics, chemistry, geology, and biology. By working through the models and engaging in additional computational explorations suggested at the end of each chapter, readers very quickly develop an understanding of how complex structures and behaviors can emerge in natural phenomena as diverse as avalanches, forest fires, earthquakes, chemical reactions, animal flocks, and epidemic diseases. This book provides the necessary topical background, complete source codes in Python, and detailed explanations for all computational models. Ideal for undergraduates, beginning graduate students, and researchers in the physical and natural sciences, this unique handbook requires no advanced mathematical knowledge or programming skills and is suitable for self-learners with a working knowledge of precalculus and high-school physics. The book enables readers to identify and quantify common underlying structural and dynamical patterns shared by the various systems and phenomena it examines, so that they can form their own answers to the questions of what natural complexity is and how it arises.

This chapter describes a simple computational idealization of a sandpile. When sand trickles slowly through your fingers, a small conical pile of sand forms below your hand. Sand avalanches of various sizes intermittently slide down the slope of the pile, which is growing both in width and in height but maintains the same slope angle. The pile of sand is a classic example of self-organized criticality. The chapter first provides an overview of the sandpile model before discussing its numerical implementation and a representative simulation involving a small 100-node lattice. It then examines the invariant power-law behavior of avalanches and the self-organized criticality of a sandpile. The chapter includes exercises and further computational explorations, along with a suggested list of materials for further reading.

This chapter shows how complex shapes and forms encountered in the natural world result from a growth process driven by the repeated action of simple “rules.” To examine this general idea, the chapter focuses on cellular automata, arguably the simplest type of computer programs conceivable yet can sometimes exhibit behaviors that are extremely complex. Cellular automata are also a classic example of simple rules being able to produce complex global “patterns” that cannot be inferred or predicted even if we have complete, a priori knowledge of these rules. The chapter first considers cellular automata in one and two spatial dimensions before discussing a zoo of two-dimensional structures from simple rules. It then describes the role of agents in iterated growth as well as emergent structures and behaviors that can be produced by cellular automata. The chapter includes exercises and further computational explorations, along with suggested materials for further reading.

This chapter explores how naturally occurring inanimate structures grow by accretion of smaller-sized components, focusing on one specific accretion process: diffusion-limited aggregation (DLA). In DLA, particles move about in random fashion, but stick together when they come into contact. Clumps of particles then form and grow further by colliding with other individual particles, or clumps of particles. Over time, one or more aggregates of individual particles will grow. After providing an overview of how DLA works, the chapter describes its numerical implementation and shows a representative simulation of a two-dimensional DLA aggregate. It then considers two peculiar geometrical properties of aggregates resulting from the DLA process, namely self-similarity and scale invariance, and shows that rule based growth through DLA can lead to the buildup of complex structures, sometimes exhibiting fractal geometry. The chapter includes exercises and further computational explorations, along with a suggested list of materials for further reading.

This chapter considers the occurrence of traffic jams in the flow of moving automobiles as an example of complex collective behavior emerging from the interactions of system elements. It begins with a discussion of the basic design principle of an automobile traffic model and the numerical implementation of the model using the Python code. It then describes a representative simulation involving an ensemble of 300 cars initially at rest and distributed randomly, with a mean spacing of 10 units. It also examines how the traffic jam model behaves, traffic jams as avalanches, and the self-organized criticality of car traffic. The chapter includes exercises and further computational explorations, along with a suggested list of materials for further reading.

The surprising thinness of the solar tachocline is still not understood with certainty today. Among the numerous possible scenarios suggested to explain its radial confinement, one hypothesis is based on Maxwell stresses that are exerted by the cyclic dynamo magnetic field of the Sun penetrating over a skin depth below the turbulent convection zone. Our goal is to assess under which conditions (turbulence level in the tachocline, strength of the dynamo-generated field, spreading mechanism) this scenario can be realized in the solar tachocline. We develop a simplified 1D model of the upper tachocline under the influence of an oscillating magnetic field imposed from above. The turbulent transport is parametrized with enhanced turbulent diffusion (or anti-diffusion) coefficients. Two main processes that thicken the tachocline are considered; either turbulent viscous spreading or radiative spreading. An extensive parameter study is carried out to establish the physical parameter regimes under which magnetic confinement of the tachocline that is due to a surface dynamo field can be realized. We have explored a large range of magnetic field amplitudes, viscosities, ohmic diffusivities and thermal diffusivities. We find that, for large but still realistic magnetic field strengths, the differential rotation can be suppressed in the upper radiative zone (and hence the tachocline confined) if weak turbulence is present (with an enhanced ohmic diffusivity of $\eta > 10^{7-8} \, cm^2/s$), even in the presence of radiative spreading. Our results show that a dynamo magnetic field can, in the presence of weak turbulence, prevent the inward burrowing of a tachocline subject to viscous diffusion or radiative spreading., Comment: 10 pages, 7 figures, accepted in A&A

The origin of cycle-to-cycle variations in solar activity is currently the focus of much interest. It has recently been pointed out that large individual active regions with atypical properties can have a significant impact on the long-term behavior of solar activity. We investigate this possibility in more detail using a recently developed $2\times2\mathrm{D}$ dynamo model of the solar magnetic cycle. We find that even a single “rogue” bipolar magnetic region (BMR) in the simulations can have a major effect on the further development of solar activity cycles, boosting or suppressing the amplitude of subsequent cycles. In extreme cases, an individual BMR can completely halt the dynamo, triggering a grand minimum. Rogue BMRs also have the potential to induce significant hemispheric asymmetries in the solar cycle. To study the effect of rogue BMRs in a more systematic manner, a series of dynamo simulations were conducted, in which a large test BMR was manually introduced in the model at various phases of cycles of different amplitudes. BMRs emerging in the rising phase of a cycle can modify the amplitude of the ongoing cycle, while BMRs emerging in later phases will only affect subsequent cycles. In this model, the strongest effect on the subsequent cycle occurs when the rogue BMR emerges around cycle maximum at low latitudes, but the BMR does not need to be strictly cross-equatorial. Active regions emerging as far as $20^{\circ}$ from the equator can still have a significant effect. We demonstrate that the combined effect of the magnetic flux, tilt angle, and polarity separation of the BMR on the dynamo is via their contribution to the dipole moment, $\delta D_{\mathrm{BMR}}$ . Our results indicate that prediction of the amplitude, starting epoch, and duration of a cycle requires an accurate accounting of a broad range of active regions emerging in the previous cycle.

In this paper we complete the presentation of a new hybrid $2\times2$D flux transport dynamo (FTD) model of the solar cycle based on the Babcock-Leighton mechanism of poloidal magnetic field regeneration via the surface decay of bipolar magnetic regions (BMRs). This hybrid model is constructed by allowing the surface flux transport (SFT) simulation described in Lemerle et al. 2015 to provide the poloidal source term to an axisymmetric FTD simulation defined in a meridional plane, which in turn generates the BMRs required by the SFT. A key aspect of this coupling is the definition of an emergence function describing the probability of BMR emergence as a function of the spatial distribution of the internal axisymmetric magnetic field. We use a genetic algorithm to calibrate this function, together with other model parameters, against observed cycle 21 emergence data. We present a reference dynamo solution reproducing many solar cycle characteristics, including good hemispheric coupling, phase relationship between the surface dipole and the BMR-generating internal field, and correlation between dipole strength at cycle maximum and peak amplitude of the next cycle. The saturation of the cycle amplitude takes place through the quenching of the BMR tilt as a function of the internal field. The observed statistical scatter about the mean BMR tilt, built into the model, acts as a source of stochasticity which dominates amplitude fluctuations. The model thus can produce Dalton-like epochs of strongly suppressed cycle amplitude lasting a few cycles and can even shut off entirely following an unfavorable sequence of emergence events.

Requiring that numerical estimates of the flow trajectories comply with the fundamental Euler expansion formula that governs the evolution of a volume of fluid leads to a second-order nonlinear Monge-Ampere partial differential equation (MAE). In Cossette and Smolarkiewicz (2011) [15], a numerical algorithm based on solving the MAE with an inexact Newton-Krylov solver has been developed and used to correct standard estimates of the departure points of flow trajectories provided by a classical semi-Lagrangian scheme in the context of an incompressible fluid. Here we extend the theoretical analysis of the elemental rotational and deformational motions presented in Cossette and Smolarkiewicz (2011) [15]. In particular, closed-form analytic solutions are derived for both cases that serve to illustrate the mechanics of the enhanced trajectory scheme and to address the issues of existence and uniqueness. Scalar advection shows that the MA correction improves mass conservation substantially by suppressing anomalous fluid contraction. The impact of the MA correction on complex flows is studied in the framework of an ideal magneto-fluid in which the formation of current sheets leads to topological changes and reconnection of field lines. The use of the MA correction improves the numerical stability of the solutions and prevents trajectory intersections as well as the spurious growth of the magnetic energy. 2D and 3D examples are presented and the computational performance of the solver is documented.

The prediction of long term solder joint reliability, (SJR), of microelectronic devices and packaging solutions continues to challenge the microelectronic packaging industry, particularly with the introduction of lead-free materials, the push for higher performance (frequency/speed/thermal) and lower unit cost. High performance packages are generally custom designed and therefore have minimal industry data on configuration specific reliability performance. In this application, the package substrate coefficient of thermal expansion, (CTE), was closely matched to the die resulting in a relatively large CTE mismatch between the package and organic PCB. In addition, the package RF and thermal performance requirements required this particular solution to be configured as a “cavity down” perimeter ball array with a large central ground pad to electrically couple the package to the PCB. Given the package's unique design requirements and CTE mismatch, even modest daily temperature swings of 20°C usually found in a controlled or “Central Office” environment could have an adverse impact on the interconnect reliability. This study provides an overview of the solder joint reliability assessment methodologies performed for a custom design lead-free, high performance RF package as part of the requirements to demonstrate compliance to product specifications. SJR life predictions were made for varying package BGA configurations using a multi-tiered approach using constitutive material models, thermo-mechanical finite element simulations, and material specific fatigue models. Empirical accelerated life testing was performed and a life prediction obtained through modeling was validated. Finally, statistical failure distributions were fit to empirical data and discussed in the context of absolute solder life predictions of small fractions unit failures, (100ppm).

The Sun's magnetic field is the engine and energy source driving all phenomena collectively defining solar activity, which in turn structures the whole heliosphere and significantly impacts Earth's atmosphere down at least to the stratosphere. The solar magnetic field is believed to originate through the action of a hydromagnetic dynamo process operating in the Sun's interior, where the strongly turbulent environment of the convection zone leads to flow-field interactions taking place on an extremely wide range of spatial and temporal scales. Following a necessarily brief observational overview of the solar magnetic field and its cycle, this review on solar dynamo theory is structured around three areas in which significant advances have been made in recent years: (a) global magnetohydrodynamical simulations of convection and magnetic cycles, (b) the turbulent electromotive force and the dynamo saturation problem, and (c) flux transport dynamos, and their application to model cycle fluctuations and grand minima and to carry out cycle prediction.

We present a reconstruction of the solar spectrum in the near and mid-ultraviolet spectral range during the Maunder Minimum, a period of strongly suppressed magnetic activity spanning the second half of the 17th century. This spectral reconstruction is based on an extension of the Monte Carlo Solar Spectral Irradiance Model (MOCASSIM). The new version of the model, documented in this paper, extends its spectral range down to 150 nm, its temporal range back to 1610, includes a secular modulation of the quiet-Sun emissivity based on a total solar irradiance reconstruction, and uses the Atmospheric Laboratory for Applications and Science-3 (ATLAS-3) spectrum as a reconstruction baseline. The model is validated against the ATLAS-1 spectrum for 29 March 1992, showing a general agreement varying from ∼ 1 % in the 300 – 400 nm range, up to 3 – 5 % below 200 nm, the largest discrepancies occurring in emission lines formed in the chromosphere and transition region. We also reconstruct ultraviolet spectra for May 2008 and March 2009, spanning the extended phase of low activity separating Cycles 23 and 24. Our results suggest that despite the unusually long temporal extent of this activity minimum, the ultraviolet emission still remained slightly higher than during the Maunder Minimum, due to the lingering presence of decay products from active regions having emerged in the late descending phase of Cycle 23.