Flux Transport Dynamo 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.

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

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.

Nonlinearity, time delay, and Grand Maxima in supercritical Babcock-Leighton dynamos

The physical origin of centennial and millennial-scale variations in solar activity remains ill-understood. Although stochastic fluctuations of the solar dynamo are unavoidable in view of the turbulent nature of the solar convection zone, the quasiperiodic long timescale modulations revealed by the cosmogenic radioisotope records are suggestive of a deterministic process. In this paper, we investigate the nonlinear behavior of two solar cycle models based on the Babcock-Leighton mechanism, with particular emphasis on deterministic amplitude modulation patterns materializing in the moderately to strongly supercritical dynamo regimes. Although formulated quite differently, both models show common long timescale modulation patterns arising from the interaction between the time-delay dynamics inherent to these flux transport dynamos, with the threshold non-linearity characterizing the Babcock-Leighton mechanism of poloidal field regeneration. In particular, we demonstrate the existence of multiple co-existing dynamo branches in the supercritical regime, each retaining a finite-sized basin of attraction over a substantial range in dynamo number. The transition from one branch to another is shown to be possible via the introduction of low-amplitude stochastic noise with short coherence time. On this basis, we propose a novel physical scenario potentially accounting for the occurrence of both Grand Minima and Maxima of solar activity.

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.

A Babcock–Leighton-type Solar Dynamo Operating in the Bulk of the Convection Zone

The toroidal magnetic field is assumed to be generated in the tachocline in most Babcock–Leighton (BL)-type solar dynamo models, in which the poloidal field is produced by the emergence and subsequent dispersal of sunspot groups. However, magnetic activity of fully convective stars and MHD simulations of global stellar convection have recently raised serious doubts regarding the importance of the tachocline in the generation of the toroidal field. In this study, we aim to develop a new BL-type dynamo model, in which the dynamo operates mainly within the bulk of the convection zone. Our 2D model includes the effect of solar-like differential rotation, one-cell meridional flow, near-surface radial pumping, strong turbulent diffusion, BL-type poloidal source, and nonlinear back-reaction of the magnetic field on its source with a vertical outer boundary condition. The model leads to a simple dipolar configuration of the poloidal field that has the dominant latitudinal component, which is wound up by the latitudinal shear within the bulk of the convection zone to generate the toroidal flux. As a result, the tachocline plays a negligible role in the model. The model reproduces the basic properties of the solar cycle, including (a) approximately 11 yr cycle period and 18 yr extended cycle period; (b) equatorward propagation of the antisymmetric toroidal field starting from high latitudes; and (c) polar field evolution that is consistent with observations. Our model opens the possibility for a paradigm shift in understanding the solar cycle to transition from the classical flux transport dynamo.

Impact of anti-solar differential rotation in mean-field solar-type dynamos

Context. Over the course of their lifetimes, the rotation of solar-type stars goes through different phases. Once they reach the zero-age main sequence, their global rotation rate decreases during the main sequence until at least the solar age, approximately following the empirical Skumanich’s law and enabling gyrochronology. Older solar-type stars might then reach a point of transition when they stop braking, according to recent results of asteroseismology. Additionally, recent 3D numerical simulations of solar-type stars show that different regimes of differential rotation can be characterized with the Rossby number. In particular, anti-solar differential rotation (fast poles, slow equator) may exist for high Rossby number (slow rotators). If this regime occurs during the main sequence and, in general, for slow rotators, we may consider how magnetic generation through the dynamo process might be impacted. In particular, we consider whether slowly rotating stars are indeed subject to magnetic cycles. Aims. We aim to understand the magnetic field generation of solar-type stars possessing an anti-solar differential rotation and we focus on the possible existence of magnetic cycles in such stars. Methods. We modeled mean-field kinematic dynamos in solar (fast equator, slow poles) and anti-solar (slow equator, fast poles) differential rotation, using the STELEM code. We consider two types of mean field dynamo mechanisms along with the Ω-effect: the standard α-effect distributed at various locations in the convective envelope and the Babcock-Leighton effect. Results. We find that kinematic αΩ dynamos allow for the presence of magnetic cycles and global polarity reversals for both rotation regimes, but only if the α-effect is saddled on the tachocline. If it is distributed in the convection zone, solar-type cases still possess a cycle and anti-solar cases do not. Conversely, we have not found any possibility for sustaining a magnetic cycle with the traditional Babcock-Leighton flux-transport dynamos in the anti-solar differential rotation regime due to flux addition. Graphic interpretations are proposed in order to illustrate these cases. However, we find that hybrid models containing both prescriptions can still sustain local polarity reversals at some latitudes. Conclusions. We conclude that stars in the anti-solar differential rotation regime can sustain magnetic cycles only for very specific dynamo processes. The detection of a magnetic cycle for such a star would therefore be a particularly interesting constraint in working to decipher what type of dynamo is actually at work in solar-type stars.

Variations of helioseismic parameters due to magnetic field generated by a flux transport model

The change of sound speed has been found at the base of the convection during the solar cycles, which can be used to constrain the solar internal magnetic field. We aim to check whether the magnetic field generated by the solar dynamo can lead to the cyclic variation of the sound speed detected through helioseismology. The basic configuration of magnetic field in the solar interior was obtained by using a Babcock-Leighton (BL) type flux transport dynamo. We reconstructed one-dimensional solar models by assimilating magnetic field generated by an established dynamo and examined their influences on the structural variables. The results show that magnetic field generated by the dynamo is able to cause noticeable change of the sound speed profile at the base of the convective zone during a solar cycle. Detailed features of this theoretical prediction are also similar to those of the helioseismic results in solar cycle 23 by adjusting the free parameters of the dynamo model.

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.

Magnetic Helicity Flux across Solar Active Region Photospheres. I. Hemispheric Sign Preference in Solar Cycle 24

Abstract A hemispheric preference in the dominant sign of magnetic helicity has been observed in numerous features in the solar atmosphere, i.e., left-handed/right-handed helicity in the northern/southern hemisphere. The relative importance of different physical processes that may contribute to the observed hemispheric sign preference (HSP) of magnetic helicity is still under debate. Here, we estimate magnetic helicity flux (dH/dt) across the photospheric surface for 4802 samples of 1105 unique active regions (ARs) that appeared over an 8 yr period from 2010 to 2017 during solar cycle 24, using photospheric vector magnetic field observations by the Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamics Observatory (SDO). The estimates of dH/dt show that 63% and 65% of the investigated AR samples in the northern and southern hemispheres, respectively, follow the HSP. We also find a trend that the HSP of dH/dt increases from ∼50%–60% up to ∼70%–80% as ARs (1) appear at the earlier inclining phase of the solar cycle or higher latitudes and (2) have larger values of , the total unsigned magnetic flux, and the average plasma-flow speed. These observational findings support the enhancement of the HSP mainly by the Coriolis force acting on a buoyantly rising and expanding flux tube through the turbulent convection zone. In addition, the differential rotation on the solar surface as well as the tachocline α-effect of a flux-transport dynamo may reinforce the HSP for ARs at higher latitudes.

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.

Cycle times of early M dwarf stars: mean field models versus observations

AbstractObservations of early-type M stars suggest that there are two characteristic cycle times, one of order one year for fast rotators (Prot < 1 day) and another of order four years for slower rotators. For a sample of fast-rotating stars, the equator-to-pole differences of the rotation rates up to 0.03 rad d−1 are also known from Kepler data. These findings are well-reproduced by mean field models. These models predict amplitudes of the meridional flow, from which the travel time from pole to equator at the base of the convection zone of early-type M stars can be calculated. As these travel times always exceed the observed cycle times, our findings do not support the flux transport dynamo.

Constraining non-linear dynamo models using quasi-biennial oscillations from sunspot area data

Context. Solar magnetic activity exhibits variations with periods between 1.5 and 4 years, the so-called quasi-biennial oscillations (QBOs), in addition to the well-known 11-year Schwabe cycles. Solar dynamo is thought to be the mechanism responsible for the generation of QBOs. Aims. In this work, we analyse sunspot areas to investigate the spatial and temporal behaviour of the QBO signal and study the physical mechanisms responsible using simulations from fully non-linear mean-field flux-transport dynamos. Methods. We investigated the behaviour of the QBOs in the sunspot area data for the full disk, and the northern and southern hemispheres, using wavelet and Fourier analyses. We also ran solar dynamos with two different approaches to generating a poloidal field from an existing toroidal field, namely Babcock–Leighton and turbulent α mechanisms. We then studied the simulated magnetic field strengths as well as meridional circulation and differential rotation rates using the same methods. Results. The results from the sunspot areas show that the QBOs are present in the full disk and hemispheric sunspot areas. These QBOs show slightly different spatial and temporal behaviours, indicating slightly decoupled solar hemispheres. The QBO signal is generally intermittent and in-phase with the sunspot area data, surfacing when the solar activity is at its maximum. The results from the BL-dynamos show that they are neither capable of generating the slightly decoupled behaviour of solar hemispheres nor can they generate QBO-like signals. The turbulent α-dynamos on the other hand generated decoupled hemispheres and some QBO-like shorter cycles. Conclusions. In conclusion, our simulations show that the turbulent α-dynamos with the Lorentz force seem more efficient in generating the observed temporal and spatial behaviour of the QBO signal compared with the BL-dynamos.

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.

Solar activity simulation and forecast with a flux-transport dynamo

We present the assessment of a diffusion-dominated mean field axisymmetric dynamo model in reproducing historical solar activity and forecast for solar cycle 25. Previous studies point to the Sun's polar magnetic field as an important proxy for solar activity prediction. Extended research using this proxy has been impeded by reduced observational data record only available from 1976. However, there is a recognised need for a solar dynamo model with ample verification over various activity scenarios to improve theoretical standards. The present study aims to explore the use of helioseismology data and reconstructed solar polar magnetic field, to foster the development of robust solar activity forecasts. The research is based on observationally inferred differential rotation morphology, as well as observed and reconstructed polar field using artificial neural network methods via the hemispheric sunspot areas record. Results show consistent reproduction of historical solar activity trends with enhanced results by introducing a precursor rise time coefficient. A weak solar cycle 25, with slow rise time and maximum activity $-14.4 \%$ ($\pm 19.5\%$) with respect to the current cycle 24 is predicted.