Flux Transport Dynamo Model Research Articles

Evaluating solar-like behavior in turbulent alpha and Babcock-Leighton mechanisms using non-kinematic nonlinear flux-transport solar dynamos.

In this study, we simulate 30,000years of solar activity using turbulent-alpha (TA) and Babcock-Leighton (BL) mechanisms in a non-kinematic, nonlinear mean field flux-transport solar dynamo model. We evaluate their performances against observational data from proxies, like C, and direct solar observations. Both the TA and BL dynamos generate Schwabe-like variations, with the TA dynamo also generating periods that can be related to the QBOs and the Gleissberg cycle. The TA dynamo spends 13.3% (12.2%) of its time in a grand minimum (maximum) state, closely match the historical solar activity reconstructions from proxy records, while the BL dynamo underperforms. For the TA dynamo, the Schwabe cycle length variations during grand minima and the latitudinal and radial dependencies of the amplitude of variations both in Schwabe and QBO timescales align well with C data and solar observations, whereas the BL dynamo fails to reproduce these features.

Algebraic quantification of the contribution of active regions to the Sun’s dipole moment: applications to century-scale polar field estimates and solar cycle forecasting

ABSTRACT The solar cycle is generated by a magnetohydrodynamic dynamo mechanism which involves the induction and recycling of the toroidal and poloidal components of the Sun’s magnetic field. Recent observations indicate that the Babcock–Leighton (BL) mechanism – mediated via the emergence and evolution of tilted bipolar active regions – is the primary contributor to the Sun’s large-scale dipolar field. Surface flux transport models and dynamo models have been employed to simulate this mechanism, which also allows for physics-based solar cycle forecasts. Recently, an alternative analytic method has been proposed to quantify the contribution of individual active regions to the Sun’s dipole moment (DM). Utilizing solar cycle observations spanning a century, here, we test the efficacy of this algebraic approach. Our results demonstrate that the algebraic quantification approach is reasonably successful in estimating DMs at solar minima over the past century – providing a verification of the BL mechanism as the primary contributor to the Sun’s dipole field variations. We highlight that this algebraic methodology serves as an independent approach for estimating DMs at the minima of solar cycles, relying on characteristics of bipolar solar active regions. We also show how this method may be utilized for solar cycle predictions; our estimate of the Sun’s dipole field at the end of cycle 24 using this approach indicates that solar cycle 25 would be a moderately weak cycle, ranging between solar cycle 20 and cycle 24.

A Babcock-Leighton dynamo model of the Sun incorporating toroidal flux loss and the helioseismically inferred meridional flow

Context. Key elements of the Babcock-Leighton model for the solar dynamo are increasingly constrained by observations. Aims. We investigate whether the Babcock-Leighton flux-transport dynamo model remains in agreement with observations if the meridional flow profile is taken from helioseismic inversions. Additionally, we investigate the effect of the loss of toroidal flux through the solar surface. Methods. We employ the two-dimensional flux-transport Babcock-Leighton dynamo framework. We use the helioseismically inferred meridional flow profile, and include toroidal flux loss in a way that is consistent with the amount of poloidal flux generated by Joy’s law. Our model does not impose a preference for emergences at low latitudes; however, we do require that the model produces such a preference. Results. We can find solutions that are in general agreement with observations, including the latitudinal migration of the butterfly wings and the 11 year period of the cycle. The most important free parameters in the model are the depth to which the radial turbulent pumping extends and the turbulent diffusivity in the lower half of the convection zone. We find that the pumping needs to extend to depths of about 0.80 R⊙ and that the bulk turbulent diffusivity needs to be around 10 km2 s−1 or less. We find that the emergences are restricted to low latitudes without the need to impose such a preference. Conclusions. The flux-transport Babcock-Leighton model, incorporating the helioseismically inferred meridional flow and toroidal field loss term, is compatible with the properties of the observed butterfly diagram and with the observed toroidal loss rate. Reasonably tight constraints are placed on the remaining free parameters. The pumping needs to be just below the depth corresponding to the location where the meridional flow changes direction, and where numerical simulations suggest the convection zone becomes marginally subadiabatic. However, our linear model does not reproduce the observed ‘rush to the poles’ of the diffuse surface radial field resulting from the decay of sunspots; reproducing this might require the imposition of a preference for flux to emerge near the equator.

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.

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

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.

Evidence of persistence of weak magnetic cycles driven by meridional plasma flows during solar grand minima phases

ABSTRACT Long-term sunspot observations and solar activity reconstructions reveal that the Sun occasionally slips into quiescent phases known as solar grand minima, the dynamics during which is not well understood. We use a flux transport dynamo model with stochastic fluctuations in the mean-field and Babcock–Leighton poloidal field source terms to simulate solar cycle variability. Our long-term simulations detect a gradual decay of the polar field during solar grand minima episodes. Although regular active region emergence stops, compromising the Babcock–Leighton mechanism, weak magnetic activity continues during minima phases sustained by a mean-field α-effect; surprisingly, periodic polar field amplitude modulation persists during these phases. A spectral analysis of the simulated polar flux time series shows that the 11-yr cycle becomes less prominent while high frequency periods and periods around 22 yr manifest during grand minima episodes. Analysis of long-term solar open flux observations appears to be consistent with this finding. Through numerical experimentation, we demonstrate that the persistence of periodic amplitude modulation in the polar field and the dominant frequencies during grand minima episodes are governed by the speed of the meridional plasma flow – which appears to act as a clock.

Bayesian approach for modeling global magnetic parameters for the solar active region

Context. Active regions (ARs) appear in the solar atmosphere as a consequence of the emergence of magnetic flux tubes. The presence of elongated magnetic polarities in line-of-sight (LOS) magnetograms indicates the existence of twist in the flux tubes that form them. These polarity elongations, referred to as magnetic tongues, bias the measurement of AR characteristics obtained during their emergence phase (e.g., their tilt angle and magnetic flux). In particular, obtaining a good estimation of the tilt angle evolution plays a key role in constraining flux-transport dynamo models. Aims. In this work, we aim to estimate the intrinsic properties of the twisted flux tubes, or flux ropes, that form ARs by quantitatively comparing observed LOS magnetograms with synthetic ones derived from a toroidal magnetic flux tube model. Methods. We developed a Bayesian inference method to obtain the statistical distributions of the inferred model parameters. As an example, we applied the method to NOAA AR 10268. Next, we tested the results using a synthetic-AR generator to quantify the effect of small-scale perturbations over the inferred parameter distributions. Results. We conclude that this method can significantly remove the effects of magnetic tongues on the derived AR global characteristics, providing a better understanding of the intrinsic properties of the emerging flux rope. Conclusions. These results provide a framework for future analyses of the physical properties of emerging ARs using Bayesian statistics.

Supercriticality of the Dynamo Limits the Memory of the Polar Field to One Cycle

Abstract The polar magnetic field precursor is considered to be the most robust and physics-based method for the prediction of the next solar cycle strength. However, to make a reliable prediction of a cycle, is the polar field at the solar minimum of the previous cycle enough or do we need the polar field of many previous cycles? To answer this question, we performed several simulations using Babcock–Leighton-type flux-transport dynamo models with a stochastically forced source for the poloidal field (α term). We show that when the dynamo is operating near the critical dynamo transition or only weakly supercritical, the polar field of cycle n determines the amplitude of the next several cycles (at least three). However, when the dynamo is substantially supercritical, this correlation of the polar field is reduced to one cycle. This change in the memory of the polar field from multiple to one cycle with the increase of the supercriticality of the dynamo is independent of the importance of various turbulent transport processes in the model. Our this conclusion contradicts the existing idea. We further show that when the dynamo operates near the critical transition, it produces frequent extended episodes of weaker activity, resembling the solar grand minima. The occurrence of grand minima is accompanied by the multicycle correlation of the polar field. The frequency of grand minima decreases with the increase of supercriticality of the dynamo.

The Polar Precursor Method for Solar Cycle Prediction: Comparison of Predictors and Their Temporal Range

Abstract The polar precursor method is widely considered to be the most robust physically motivated method to predict the amplitude of an upcoming solar cycle. It uses indicators of the magnetic field concentrated near the poles around the sunspot minimum. Here, we present an extensive analysis of the performance of various such predictors, based on both observational data (Wilcox Solar Observatory (WSO) magnetograms, Mount Wilson Observatory polar faculae counts, and Pulkovo A(t) index) and outputs (polar cap magnetic flux and global dipole moment) of various existing flux transport dynamo models. We calculate Pearson correlation coefficients (r) of the predictors with the next cycle amplitude as a function of time measured from several solar cycle landmarks: setting r = 0.8 as a lower limit for acceptable predictions, we find that observations and models alike indicate that the earliest time when the polar predictor can be safely used is 4 yr after the polar field reversal. This is typically 2–3 yr before the solar minimum and about 7 yr before the predicted maximum, considerably extending the usual temporal scope of the polar precursor method. Reevaluating the predictors another 3 yr later, at the time of the solar minimum, further increases the correlation level to r ≳ 0.9. As an illustration of the result, we determine the predicted amplitude of Cycle 25 based on the value of the WSO polar field at the now official minimum date of 2019 December as 126 ± 3. A forecast based on the value in early 2017, 4 yr after the polar reversal would have only differed from this final prediction by 3.1 ± 14.7%.

The meridional circulation of the Sun: Observations, theory and connections with the solar dynamo

The meridional circulation of the Sun, which is observed to be poleward at the surface, should have a return flow at some depth. Since large-scale flows like the differential rotation and the meridional circulation are driven by turbulent stresses in the convection zone, these flows are expected to remain confined within this zone. Current observational (based on helioseismology) and theoretical (based on dynamo theory) evidences point towards an equatorward return flow of the meridional circulation at the bottom of the convection zone. Assuming the mean values of various quantities averaged over turbulence to be axisymmetric, we study the large-scale flows in solar-like stars on the basis of a 2D mean field theory. Turbulent stresses in a rotating star can transport angular momentum, setting up a differential rotation. The meridional circulation arises from a slight imbalance between two terms which try to drive it in opposite directions: a thermal wind term (arising out of the higher efficiency of convective heat transport in the polar regions) and a centrifugal term (arising out of the differential rotation). To make these terms comparable, the poles of the Sun should be slightly hotter than the equator. We discuss the important role played by the meridional circulation in the flux transport dynamo model. The poloidal field generated by the Babcock-Leighton process at the surface is advected poleward, whereas the toroidal field produced at the bottom of the convection zone is advected equatorward. The fluctuations in the meridional circulation (with coherence time of about 30–40 yr) help in explaining many aspects of the irregularities in the solar cycle. Finally, we discuss how the Lorentz force of the dynamo-generated magnetic field can cause periodic variations in the large-scale flows with the solar cycle.

The flow of plasma inside the Sun

Solar Physics The activity of the Sun, including the occurrence of sunspots, is driven by magnetic fields that originate from the motion of charged plasma beneath the surface. Helioseismology uses acoustic oscillations to probe the Sun's interior, analogous to seismology's use of earthquakes to investigate Earth's interior. Gizon et al. analyzed helioseismology data from 1996 to 2019, covering two 11-year solar cycles. They measured the latitudinal and radial flow of plasma as a function of depth within the Sun and how it varies with time. The results support magnetic flux-transport dynamo models, which can explain the distribution of sunspots over each solar cycle. Science , this issue p. [1469][1] [1]: /lookup/doi/10.1126/science.aaz7119

Meridional flow in the Sun’s convection zone is a single cell in each hemisphere

The Sun's magnetic field is generated by subsurface motions of the convecting plasma. The latitude at which the magnetic field emerges through the solar surface (as sunspots) drifts toward the equator over the course of the 11-year solar cycle. We use helioseismology to infer the meridional flow (in the latitudinal and radial directions) over two solar cycles covering 1996-2019. Two data sources are used, which agree during their overlap period of 2001-2011. The time-averaged meridional flow is shown to be a single cell in each hemisphere, carrying plasma toward the equator at the base of the convection zone with a speed of ~4 meters per second at 45° latitude. Our results support the flux-transport dynamo model, which explains the drift of sunspot-emergence latitudes through the meridional flow.

Exploring the Cycle Period and Parity of Stellar Magnetic Activity with Dynamo Modeling

Abstract Observations of chromospheric and coronal emissions from various solar-type stars show that the stellar magnetic activity varies with the rotation rates of the stars. The faster the star rotates, the stronger its magnetic activity becomes, but the activity cycle period does not show a straightforward variation with the rotation rate. For slowly rotating stars, the cycle period decreases with the increase in rotation rate, while for the fast rotators, the dependency of cycle period on rotation is presently quite complicated. We aim to provide an explanation of these observational trends of stellar magnetic activity using a dynamo model. We construct a theoretical dynamo model for stars of mass 1 M ⊙ based on the kinematic flux transport dynamo model including radial pumping near the surface of the stars. The inclusion of this near-surface downward radial pumping is found to be necessary to match the observed surface magnetic field in the Sun. The main ingredients of our dynamo model, meridional circulation and differential rotation for stars, are obtained from a mean-field hydrodynamic model. Our model shows a decrease in cycle period with increasing rotation rate in the slowly rotating regime and a slight increase in cycle period with rotation rate for the rapid rotators. The strength of the magnetic field is found to increase as the rotation rate of the star increases. We also find that the parity of the stellar magnetic field changes with rotation. According to our model, the parity flips to quadrupolar from dipolar if the rotation period of the star is less than 17 days.

A New Formula for Predicting Solar Cycles

A new formula for predicting solar cycles based on the current theoretical understanding of the solar cycle from flux transport dynamo is presented. Two important processes---fluctuations in the Babcock-Leighton mechanism and variations in the meridional circulation, which are believed to be responsible for irregularities of the solar cycle---are constrained by using observational data. We take the polar field near minima of the cycle as a measure of the randomness in the Babcock-Leighton process, and the decay rate near the minima as a consequence of the change in meridional circulation. We couple these two observationally derived quantities into a single formula to predict the amplitude of the future solar cycle. Our new formula suggests that the cycle 25 would be a moderate cycle. Whether this formula for predicting the future solar cycle can be justified theoretically is also discussed using simulations from the flux transport dynamo model.

Solar meridional circulation from twenty-one years of SOHO/MDI and SDO/HMI observations

Context. The solar meridional flow is an essential ingredient in flux-transport dynamo models. However, no consensus on its subsurface structure has been reached.Aims. We merge the data sets from SOHO/MDI and SDO/HMI with the aim of achieving a greater precision on helioseismic measurements of the subsurface meridional flow.Methods. The south-north travel-time differences are measured by applying time-distance helioseismology to the MDI and HMI medium-degree Dopplergrams covering May 1996–April 2017. Our data analysis corrects for several sources of systematic effects:P-angle error, surface magnetic field effects, and center-to-limb variations. For HMI data, we used theP-angle correction provided by the HMI team based on the Venus and Mercury transits. For MDI data, we used aP-angle correction estimated from the correlation of MDI and HMI data during the period of overlap. The center-to-limb effect is estimated from the east-west travel-time differences and is different for MDI and HMI observations. An interpretation of the travel-time measurements is obtained using a forward-modeling approach in the ray approximation.Results. In the latitude range 20°–35°, the travel-time differences are similar in the southern hemisphere for cycles 23 and 24. However, they differ in the northern hemisphere between cycles 23 and 24. Except for cycle 24’s northern hemisphere, the measurements favor a single-cell meridional circulation model where the poleward flows persist down to ∼0.8R⊙, accompanied by local inflows toward the activity belts in the near-surface layers. Cycle 24’s northern hemisphere is anomalous: travel-time differences are significantly smaller when travel distances are greater than 20°. This asymmetry between northern and southern hemispheres during cycle 24 was not present in previous measurements, which assumed a differentP-angle error correction where south-north travel-time differences are shifted to zero at the equator for all travel distances. In our measurements, the travel-time differences at the equator are zero for travel distances less than ∼30°, but they do not vanish for larger travel distances. This equatorial offset for large travel distances need not be interpreted as a deep cross-equator flow; it could be due to the presence of asymmetrical local flows at the surface near the end points of the acoustic ray paths.Conclusions. The combined MDI and HMI helioseismic measurements presented here contain a wealth of information about the subsurface structure and the temporal evolution of the meridional circulation over 21 years. To infer the deep meridional flow, it will be necessary to model the contribution from the complex time-varying flows in the near-surface layers.

Incorporating Surface Convection into a 3D Babcock–Leighton Solar Dynamo Model

Abstract The convective flows observed on the photosphere (e.g., supergranulation, granulation) play a key role in the Babcock–Leighton (BL) process to generate large-scale polar fields from sunspot fields. In most surface flux transport (SFT) and BL dynamo models, the dispersal and migration of surface fields are modeled as an effective turbulent diffusion. Recent SFT models have incorporated explicit, realistic convective flows in order to improve the fidelity of convective transport but, to our knowledge, this has not yet been implemented in previous BL models. We present the first kinematic 3D Flux-Transport/BL model to explicitly incorporate realistic convective flows based on solar observations. Though we describe a means to generalize these flows to 3D, we find that the kinematic small-scale dynamo action they produce disrupts the operation of the cyclic dynamo. The cyclic solution is found by limiting the convective flow to act only on the vertical radial component of the magnetic field. The results obtained are generally in good agreement with the observed surface flux evolution and with non-convective models that have a turbulent diffusivity of the order of 3 × 1012 cm2 s−1 (300 km2 s−1). However, we find that the use of a turbulent diffusivity underestimates the dynamo efficiency, producing weaker mean fields and a shorter cycle than in the convective models. Also, the convective models exhibit bands of mixed polarity in the polar regions that have no counterpart in solar observations, and the poleward migration speed of poloidal flux is determined mainly by the meridional flow and the vertical diffusion.

Transport of Internetwork Magnetic Flux Elements in the Solar Photosphere

Abstract The motions of small-scale magnetic flux elements in the solar photosphere can provide some measure of the Lagrangian properties of the convective flow. Measurements of these motions have been critical in estimating the turbulent diffusion coefficient in flux-transport dynamo models and in determining the Alfvén wave excitation spectrum for coronal heating models. We examine the motions of internetwork flux elements in Hinode/Narrowband Filter Imager magnetograms and study the scaling of their mean squared displacement and the shape of their displacement probability distribution as a function of time. We find that the mean squared displacement scales super-diffusively with a slope of about 1.48. Super-diffusive scaling has been observed in other studies for temporal increments as small as 5 s, increments over which ballistic scaling would be expected. Using high-cadence MURaM simulations, we show that the observed super-diffusive scaling at short increments is a consequence of random changes in barycenter positions due to flux evolution. We also find that for long temporal increments, beyond granular lifetimes, the observed displacement distribution deviates from that expected for a diffusive process, evolving from Rayleigh to Gaussian. This change in distribution can be modeled analytically by accounting for supergranular advection along with granular motions. These results complicate the interpretation of magnetic element motions as strictly advective or diffusive on short and long timescales and suggest that measurements of magnetic element motions must be used with caution in turbulent diffusion or wave excitation models. We propose that passive tracer motions in measured photospheric flows may yield more robust transport statistics.

A Three dimensional Babcock-Leighton Solar Dynamo Model with Non-axisymmetric Convective Flows

AbstractThe observed convective flows on the photosphere (e.g., supergranulation, granulation) play a key role in the Babcock-Leighton (BL) process to generate large scale polar fields from sunspots fields. In most surface flux transport (SFT) and BL dynamo models, the dispersal and migration of surface fields is modeled as an effective turbulent diffusion. We present the first kinematic 3D FT/BL model to explicitly incorporate realistic convective flows based on solar observations. The results obtained are generally in good agreement with the observed surface flux evolution and with non-convective models that have a turbulent diffusivity on the order of 3 × 1012 cm2 s−1 (300 km2 s−1). However, we find that the use of a turbulent diffusivity underestimates the dynamo efficiency, producing weaker mean fields and shorter cycle.

Explaining the variation of the meridional circulation with the solar cycle

AbstractThe meridional circulation of the Sun is observationally found to vary with the solar cycle, becoming slower during the solar maxima. We explain this by constructing a theoretical model in which the equation of the meridional circulation (the φ component of the vorticity equation) is coupled with the equations of the flux transport dynamo model. We find that the Lorentz force of the dynamo-generated magnetic fields can slow down the meridional circulation during the solar maxima in broad conformity with the observations.