Dynamo Model Research Articles

Effects of anisotropic turbulent thermal diffusion on spherical magnetoconvection in the Earth’s core

We re-consider the plate-like model of turbulence in the Earth’s core, proposed by Braginsky and Meytlis (1990), and show that it is plausible for core parameters not only in polar regions but extends to mid- and low-latitudes where rotation and gravity are not parallel, except in a very thin equatorial layer. In this model the turbulence is highly anisotropic with preferred directions imposed by the Earth’s rotation and the magnetic field. Current geodynamo computations effectively model sub-grid scale turbulence by using isotropic viscous and thermal diffusion values significantly greater than the molecular values of the Earth’s core. We consider a local turbulent dynamo model for the Earth’s core in which the mean magnetic field, velocity and temperature satisfy the Boussinesq induction, momentum and heat equations with an isotropic turbulent Ekman number and Roberts number. The anisotropy is modelled only in the thermal diffusion tensor with the Earth’s rotation and magnetic field as preferred directions. Nonlocal organising effects of gravity and rotation (but not aspect ratio in the Earth’s core) such as an inverse cascade and nonlocal transport are assumed to occur at longer length scales, which computations may accurately capture with sufficient resolution. To investigate the implications of this anisotropy for the proposed turbulent dynamo model we investigate the linear instability of turbulent magnetoconvection on length scales longer than the background turbulence in a rotating sphere with electrically insulating exterior for no-slip and isothermal boundary conditions. The equations are linearised about an axisymmetric basic state with a conductive temperature, azimuthal magnetic field and differential rotation. The basic state temperature is a function of the anisotropy and the spherical radius. Elsasser numbers in the range 1–20 and turbulent Roberts numbers 0.01–1 are considered for both equatorial symmetries of the magnetic basic state. It is found that anisotropic turbulent thermal diffusivity has a strong destabilising effect on magneto-convective instabilities, which may relax the tight energy budget constraining geodynamo models. The enhanced instability is not due to a reduction of the total diffusivity. The anisotropy also strengthens instabilities which break the symmetry of the underlying state, which may facilitate magnetic field reversal. Geostrophic flow appears to suppress the symmetry breaking modes and magnetic instabilities. Through symmetry breaking and the geostrophic flow the anisotropy may provide a mechanism of magnetic field reversal and its suppression in computational dynamo models.

Analytic mean-fieldα2-dynamo with a force-free corona

Stellar dynamos are affected by boundary conditions imposed by stellar coronae. Under some approximations it is possible to find analytical solutions. Interior dynamo models often consider a current-free coronae without taking into account the constraints imposed by the presence of currents in the corona. We aim to analytically evaluate the effect of coronal currents and of an outer boundary condition on the efficiency of an $\alpha^2$ dynamo. We intend to estimate the change in geometry and dinamo excitation numbers with respect to the current-free case. We analytically solve the turbulent dynamo induction equation for a homogeneous, non-mirror symmetric turbulence, in a spherical domain surrounded by a linear force-free corona with magnetic field B satisfying curl B = \beta B. The main result is that the dynamo number is a decreasing function of \beta. Moreover, if the current is parallel to the field (\beta > 0) the dynamo number is smaller than in the force-free case. On the contrary (\beta < 0) the dynamo number is greater than the in force-free case. We conclude that the presence of currents in the corona needs to be taken into account because it affects the condition for excitation of a dynamo.

A theoretical model of the variation of the meridional circulation with the solar cycle

Observations of the meridional circulation of the Sun, which plays a key role in the operation of the solar dynamo, indicate that its speed varies with the solar cycle, becoming faster during the solar minima and slower during the solar maxima. To explain this variation of the meridional circulation with the solar cycle, we construct a theoretical model by coupling the equation of the meridional circulation (the $\phi$ component of the vorticity equation within the solar convection zone) with the equations of the flux transport dynamo model. We consider the back reaction due to the Lorentz force of the dynamo-generated magnetic fields and study the perturbations produced in the meridional circulation due to it. This enables us to model the variations of the meridional circulation without developing a full theory of the meridional circulation itself. We obtain results which reproduce the observational data of solar cycle variations of the meridional circulation reasonably well. We get the best results on assuming the turbulent viscosity acting on the velocity field to be comparable to the magnetic diffusivity (i.e. on assuming the magnetic Prandtl number to be close to unity). We have to assume an appropriate bottom boundary condition to ensure that the Lorentz force cannot drive a flow in the subadiabatic layers below the bottom of the tachocline. Our results are sensitive to this bottom boundary condition. We also suggest a hypothesis how the observed inward flow towards the active regions may be produced.

Dynamo transition in a five-mode helical model

We construct a five-mode helical dynamo model containing three velocity and two magnetic modes and solve it analytically. This model exhibits dynamo transition via supercritical pitchfork bifurcation. We show that the critical magnetic Reynolds number for dynamo transition (Rmc) asymptotes to constant values for very low and very high magnetic Prandtl numbers (Pm). Beyond dynamo transition, secondary bifurcations lead to periodic, quasi-periodic, and chaotic dynamo states as the forcing amplitude is increased and chaos appears through a quasi-periodic route.

North–South Asymmetry in Rieger-type Periodicity during Solar Cycles 19–23

Abstract Rieger-type periodicity has been detected in different activity indices over many solar cycles. It was recently shown that the periodicity correlates with solar activity having a shorter period during stronger cycles. Solar activity level is generally asymmetric between northern and southern hemispheres, which could suggest the presence of a similar behavior in the Rieger-type periodicity. We analyze the sunspot area/number and the total magnetic flux data for northern and southern hemispheres during solar cycles 19–23, which had remarkable north–south asymmetry. Using wavelet analysis of sunspot area and number during the north-dominated cycles (19–20), we obtained the periodicity of 160–165 days in the stronger northern hemisphere and 180–190 days in the weaker southern hemisphere. On the other hand, south-dominated cycles (21–23) display the periodicity of 155–160 days in the stronger southern hemisphere and 175–188 days in the weaker northern hemisphere. Therefore, the Rieger-type periodicity has the north–south asymmetry in sunspot area/number data during solar cycles with strong hemispheric asymmetry. We suggest that the periodicity is caused by magnetic Rossby waves in the internal dynamo layer. Using the dispersion relation of magnetic Rossby waves and observed Rieger periodicity, we estimated the magnetic field strength in the layer as 45–49 kG in more active hemispheres (north during cycles 19–20 and south during cycles 21–23) and 33–40 kG in weaker hemispheres. The estimated difference in the hemispheric field strength is around 10 kG, which provides a challenge for dynamo models. Total magnetic flux data during cycles 20–23 reveals no clear north–south asymmetry, which needs to be explained in the future.

Solar cycle variations of rotation and asphericity in the near-surface shear layer

The precise shape of the Sun is sensitive to the influence of gravity, differential rotation, local turbulence and magnetic fields. So its precise measurement is a long-standing astrometric objective. It has been previously shown by different methods that the solar shape exhibits asphericity that evolves with the solar cycle. Thanks to the Michelson Doppler Imager (MDI) on Solar and Heliospheric Observatory (SoHO) and the Helioseismic and Magnetic Imager (HMI) aboard NASA's Solar Dynamics Observatory (SDO), and their capability to observe with an unprecedented accuracy the surface gravity oscillation (f) modes, it is possible to extract information concerning the coefficients of rotational frequency splitting, a1, a3 and a5, that measure the latitudinal differential rotation, together with the a2, a4 and a6 asphericity coefficients. Analysis of these helioseismology data with time for almost two solar cycles, from 1996 to 2017, reveals a close correlation of the a1 and a5 coefficients with the solar activity, whilst a3 exhibits a long-term trend and a weak correlation with the solar activity in the current solar cycle indicating a substantial change of the global solar rotation, potentially associated with a long-term evolution of the solar cycles. Looking in more details, the asphericity coefficients, a2, a4 and a6 are more strongly associated with the solar cycle when applying a time lag of respectively 0.1, 1.6 and −1.6 years. The magnitude of a6-coefficient varies in phase with the sunspot number (SN), but its amplitude is ahead of the SN variation. The latest measurements made in mid 2017 indicate that the magnitude of the a6-coefficient has probably reached its minimum; therefore, the next solar minimum can be expected by the end of 2018 or in the beginning of 2019. The so-called “seismic radius” in the range of f-mode angular degree: ℓ=137−299 exhibits a temporal variability in anti-phase with the solar activity; its relative value decreased by ∼2.3×10−5 in Solar Cycle 23 and ∼1.7×10−5 in Cycle 24. Such results will be useful for better understanding the physical mechanisms which act inside the Sun, and so, better constrain dynamo models for forecasting the solar cycles.

Flux transport dynamo: From modelling irregularities to making predictions

The flux transport dynamo, in which the poloidal magnetic field is generated by the Babcock–Leighton mechanism and the meridional circulation plays a crucial role, has emerged as an attractive model for the solar cycle. Based on theoretical calculations done with this model, we argue that the fluctuations in the Babcock–Leighton mechanism and the fluctuations in the meridional circulation are the most likely causes of the irregularities of the solar cycle. With our increased theoretical understanding of how these irregularities arise, it can be possible to predict a future solar cycle by feeding the appropriate observational data in a theoretical dynamo model.

On the effect of laterally varying boundary heat flux on rapidly rotating spherical shell convection

The onset of convection in a rotating spherical shell subject to laterally varying heat flux at the outer boundary is considered in this paper. The focus is on the geophysically relevant regime of rapid rotation (low Ekman number) where the natural length scale of convection is significantly smaller than the length scale imposed by the boundary heat flux pattern. Contrary to earlier studies at a higher Ekman number, we find a substantial reduction in the onset Rayleigh number Rac with increasing lateral variation. The decrease in Rac is shown to be closely correlated to the equatorial heat flux surplus in the steady, basic state solution. The consistency of such a correlation makes the estimation of Rac possible without solving the full stability problem. The steady baroclinic flow has a strong cyclone–anticyclone asymmetry in the kinetic helicity only for equatorially symmetric lateral variations, with possible implications for dynamo action. Equatorially antisymmetric variations, on the other hand, break the symmetry of the mean flow, in turn negating its helicity. Analysis of the perturbation solution reveals strongly localized clusters through which convection rolls drift in and out at a frequency higher than that for the reference case with homogeneous boundary heat flux. Large lateral variations produce a marked decrease in the azimuthal length scale of columns, which indicates that small-scale motions are essential to the transport of heat in rapidly rotating, localized convection. With an equatorially antisymmetric heat flux pattern, convection in individual clusters goes through an asynchronous wax-wane cycle whose frequency is much lower than the drift rate of the columns. These continual variations in convection intensity may in turn result in fluctuations in the magnetic field intensity, an effect that needs to be considered in dynamo models. Finally, there is a notable analogy between the role of a laterally varying boundary heat flux and the role of a laterally varying magnetic field in confining small-scale convection.

Physical conditions for Jupiter-like dynamo models

The Juno mission will measure Jupiter’s magnetic field with unprecedented precision and provide a wealth of additional data that will allow us to constrain the planet’s interior structure and dynamics. Here we analyse 66 different numerical simulations in order to explore the sensitivity of the dynamo-generated magnetic field to the planets interior properties. Jupiter field models based on pre-Juno data and up-to-date interior models based on ab initio simulations serve as benchmarks. Our results suggest that Jupiter-like magnetic fields can be found for a number of different models. These complement the steep density gradients in the outer part of the simulated shell with an electrical conductivity profile that mimics the low conductivity in the molecular hydrogen layer and thus renders the dynamo action in this region largely unimportant. We find that whether we assume an ideal gas or use the more realistic interior model based on ab initio simulations makes no difference. However, two other factors are important. A low Rayleigh number leads to a too strong axial dipole contribution while the axial dipole dominance is lost altogether when the convective driving is too strong. The required intermediate range that yields Jupiter-like magnetic fields depends on the other system properties. The second important factor is the convective magnetic Reynolds number radial profile Rmc(r), basically a product of the non-axisymmetric flow velocity and electrical conductivity. We find that the depth where Rmc exceeds about 50 is a good proxy for the top of the dynamo region. When the dynamo region sits too deep, the axial dipole is once more too dominant due to geometric reasons. Extrapolating our results to Jupiter and the result suggests that the Jovian dynamo extends to 95% of the planetary radius.The zonal flow system in our simulations is dominated by an equatorial jet which remains largely confined to the molecular layer. Where the jet reaches down to higher electrical conductivities, however, it gives rise to a secondary αΩ dynamo that modifies the dipole-dominated field produced deeper in the planet. This secondary dynamo can lead to strong magnetic field patches at lower latitudes that seem compatible with the pre-Juno field models.

The relation between stellar magnetic field geometry and chromospheric activity cycles – I. The highly variable field of ε Eridani at activity minimum

Abstract The young and magnetically active K dwarf ε Eridani exhibits a chromospheric activity cycle of about 3 yr. Previous reconstructions of its large-scale magnetic field show strong variations at yearly epochs. To understand how ε Eridani’s large-scale magnetic field geometry evolves over its activity cycle, we focus on high-cadence observations spanning 5 months at its activity minimum. Over this time-span, we reconstruct three maps of ε Eridani’s large-scale magnetic field using the tomographic technique of Zeeman–Doppler imaging. The results show that at the minimum of its cycle, ε Eridani’s large-scale field is more complex than the simple dipolar structure of the Sun and 61 Cyg A at minimum. Additionally, we observe a surprisingly rapid regeneration of a strong axisymmetric toroidal field as ε Eridani emerges from its S-index activity minimum. Our results show that all stars do not exhibit the same field geometry as the Sun, and this will be an important constraint for the dynamo models of active solar-type stars.

Strong dipole magnetic fields in fast rotating fully convective stars

M dwarfs are the most numerous stars in our Galaxy with masses between approximately 0.5 and 0.1 solar mass. Many of them show surface activity qualitatively similar to our Sun and generate flares, high X-ray fluxes, and large-scale magnetic fields. Such activity is driven by a dynamo powered by the convective motions in their interiors. Understanding properties of stellar magnetic fields in these stars finds a broad application in astrophysics, including, e.g., theory of stellar dynamos and environment conditions around planets that may be orbiting these stars. Most stars with convective envelopes follow a rotation-activity relationship where various activity indicators saturate in stars with rotation periods shorter than a few days. The activity gradually declines with rotation rate in stars rotating more slowly. It is thought that due to a tight empirical correlation between X-ray and magnetic flux, the stellar magnetic fields will also saturate, to values around ~4kG. Here we report the detection of magnetic fields above the presumed saturation limit in four fully convective M-dwarfs. By combining results from spectroscopic and polarimetric studies we explain our findings in terms of bistable dynamo models: stars with the strongest magnetic fields are those in a dipole dynamo state, while stars in a multipole state cannot generate fields stronger than about four kilogauss. Our study provides observational evidence that dynamo in fully convective M dwarfs generates magnetic fields that can differ not only in the geometry of their large scale component, but also in the total magnetic energy.

Understanding Solar Cycle Variability

Abstract The level of solar magnetic activity, as exemplified by the number of sunspots and by energetic events in the corona, varies on a wide range of timescales. Most prominent is the 11-year solar cycle, which is significantly modulated on longer timescales. Drawing from dynamo theory, together with the empirical results of past solar activity and similar phenomena for solar-like stars, we show that the variability of the solar cycle can be essentially understood in terms of a weakly nonlinear limit cycle affected by random noise. In contrast to ad hoc “toy models” for the solar cycle, this leads to a generic normal-form model, whose parameters are all constrained by observations. The model reproduces the characteristics of the variable solar activity on timescales between decades and millennia, including the occurrence and statistics of extended periods of very low activity (grand minima). Comparison with results obtained with a Babcock–Leighton-type dynamo model confirm the validity of the normal-mode approach.

Tuning of the mean-field geodynamo model

Parker’s two-dimensional (2D) dynamo model with an algebraic form of nonlinearity for the α-effect is considered. The model uses geostrophic distributions for the α-effect and differential rotation, which are derived from the three-dimensional (3D) convection models. The resulting configurations of the magnetic field in the liquid core are close to the solutions in Braginsky’s Z-model. The implications of the degree of geostrophy observed in the 3D dynamo models for the behavior of the mean magnetic field are explored. It is shown that the reduction in geostrophy leads to magnetic field reversals accompanied by the relative growth of the nondipole component of the field on the surface of the liquid core. The simulations with a random α-effect which causes turbulent pulsations are carried out. The approach is capable of producing realistic sequences of magnetic reversals.

Evolution of the Sun’s non-axisymmetric toroidal field

Aims. We aim to infer the sub-surface distribution of the Sun's non-axisymmetric azimuthal magnetic flux from observable quantities, such as the surface magnetic field and the large scale plasma flows. Methods. We have built a kinematic flux transport model of the solar dynamo based on the Babcock-Leighton framework. We constructed the source term for the poloidal field using SOLIS magnetograms spanning three solar cycles. Based on this source we calculated the azimuthal flux below the surface. The flux transport model has two free parameters which we constrain using sunspot observations from cycle 22. We compared the model results with observations from cycle 23. Results. The structure of the azimuthal field is mainly axisymmetric. The departures from axisymmetry represent, on average, ~3% of the total azimuthal flux. Owing to its relative weakness, the non-axisymmetric structure of the azimuthal field does not have a significant impact on the location in which the emergences appear or on the amount of flux contained in them. We find that the probability of emergence is a function of the ratio between the flux content of an active region and the underlying azimuthal flux.

Spatial Inhomogeneity in Solar Faculae

AbstractIn this paper, we investigate the inhomogeneous spatial distribution of solar faculae. The focus is on the latitudinal and longitudinal distributions of these highly localised features covering ubiquitously the solar surface. The statistical analysis is based on white light observations of the Solar and Heliospheric Observatory (SOHO) and Solar Dynamics Observatory (SDO) between 1996 and 2014. We found that the fine structure of the latitudinal distribution of faculae displays a quasi-biennial oscillatory pattern. Furthermore, the longitudinal distribution of photospheric solar faculae does not show homogeneous behaviour either. In particular, the non-axisymmetric behaviour of these events show similar properties as that of the active longitude (AL) found in the distribution of sunspots. Our results, preliminary though, may provide a valuable observational constrain for developing the next-generation solar dynamo model.

The equatorial asymmetry of a magnetic field

Solution of the inverse problem for Parker’s one-dimensional mean-field dynamo model in a thin spherical layer is considered. The method allows the spatial distribution of energy sources, the α- and Ω-effects, to be found provided specified constraints occur on the solution. The highest ratio of the magnetic energies for the Northern and Southern hemispheres is discussed as such a constraint. The method is a modification of the Monte-Carlo technique; it is convenient for parallel computations and based on minimization of the cost function that characterizes the deviation of the model solution properties from the desired ones. The calculations show that the ratio of the energies in the hemispheres may exceed an order of magnitude for both poloidal and toroidal components of the magnetic energy. The ratio depends on the distance of the effective zone of the generation of the magnetic field from the equator and the number of harmonics in the spectrum. The greater this distance is and the higher the number of harmonics is, the stronger the magnetic field asymmetry can be.

Towards Estimating the Solar Meridional Flow and Predicting the 11-yr Cycle Using Advanced Variational Data Assimilation Techniques

AbstractWe present in this work the development of a solar data assimilation method based on an axisymmetric mean field dynamo model and magnetic surface data. Our mid-term goal is to predict the solar quasi cyclic activity. We focus on the ability of our variational data assimilation algorithm to constrain the deep meridional circulation of the Sun based on solar magnetic observations. Within a given assimilation window, the assimilation procedure minimizes the differences between data and the forecast from the model, by finding an optimal meridional circulation in the convection zone, and an optimal initial magnetic field, via a quasi-Newton algorithm. We demonstrate the capability of the technique to estimate the meridional flow by a closed-loop experiment involving 40 years of synthetic, solar-like data. We show that the method is robust in estimating a (stochastic) time-varying flow fluctuating 30% about the average, and that the horizon of predictability of the method is ~ 1 cycle length.

Predicting a Solar Cycle Before its Onset Using a Flux Transport Dynamo Model

AbstractWe begin with a review of the predictions for cycle 24 before its onset. After summarizing the basics of the flux transport dynamo model, we discuss how this model had been used to make a successful prediction of cycle 24, on the assumption that the irregularities of the solar cycle arise due to the fluctuations in the Babcock–Leighton mechanism. We point out that fluctuations in the meridional circulation can be another cause of irregularities in the cycle.

Reinforcing a Double Dynamo Model with Solar-Terrestrial Activity in the Past Three Millennia

AbstractBy applying Principal Components Analysis (PCA) to solar magnetic synoptic maps in cycle 21-23 obtained with Wilcox Solar Observatory we derived analytical expressions for two principal components and their summary curve of solar magnetic field oscillations defined by dipole magnetic sources. In this paper we extrapolate backwards three millennia the summary curve describing solar activity and compare it with the relevant historic data. The extrapolated summary curve shows a remarkable resemblance to the sunspot and terrestrial activity reported in the past millennia: the Maunder Minimum (1645-1715), Wolf minimum (1200), Oort minimum (1010-1050), Homer minimum (800-900 BC), the medieval warm period (900-1200), the Roman warm period (400-10BC). We note that Sporer minimum (1460-1550) derived from the increased abundance of isotope Δ14C is likely produced by a strong increase of galactic cosmic rays caused by a supernova Vela Junior occurred in the Southern hemisphere.

Limits on radial differential rotation in Sun-like stars from parametric fits to oscillation power spectra

Rotational shear in Sun-like stars is thought to be an important ingredient in models of stellar dynamos. Thanks to helioseismology, rotation in the Sun is characterized well, but the interior rotation profiles of other Sun-like stars are not so well constrained. Until recently, measurements of rotation in Sun-like stars have focused on the mean rotation, but little progress has been made on measuring or even placing limits on differential rotation. Using asteroseismic measurements of rotation we aim to constrain the radial shear in five Sun-like stars observed by the NASA Kepler mission: KIC004914923, KIC005184732, KIC006116048, KIC006933899, and KIC010963065. We used stellar structure models for these five stars from previous works. These models provide the mass density, mode eigenfunctions, and the convection zone depth, which we used to compute the sensitivity kernels for the rotational frequency splitting of the modes. We used these kernels as weights in a parametric model of the stellar rotation profile of each star, where we allowed different rotation rates for the radiative interior and the convective envelope. This parametric model was incorporated into a fit to the oscillation power spectrum of each of the five Kepler stars. This fit included a prior on the rotation of the envelope, estimated from the rotation of surface magnetic activity measured from the photometric variability. The asteroseismic measurements without the application of priors are unable to place meaningful limits on the radial shear. Using a prior on the envelope rotation enables us to constrain the interior rotation rate and thus the radial shear. In the five cases that we studied, the interior rotation rate does not differ from the envelope by more than approximately +/-30%. Uncertainties in the rotational splittings are too large to unambiguously determine the sign of the radial shear.