Stretching solar dynamos from differential rotation in a Riemannian manifold setting is presented. The spherical model follows closely a twisted magnetic flux tube Riemannian geometrical model or flux rope in solar physics, presented previously by Ricca [Solar Physics (1997)]. The spherical model presented here present new and interesting feature concerning its connection with spherical steady solar dynamos. One of this new feature is represented by the fact that the by considering poloidal magnetic field component much weaker than its toroidal counterpart, one obtains a stretch dynamo action where the Riemannian solar spherical line element is proportional to differential rotation. This result is obtained also by using the Vainshtein-Zeldovich stretch, twist and fold (STF) method to generate dynamos. One notes that for high magnetic Reynolds of $R{m}=O({10^{7}})$ the dynamo action is present for a corresponding small stretching factor of $K^{2}=1.6$ where $K^{2}=1$ represents the unstretched dynamo. The constant stretched dynamos considered here are shown to be Riemann-flat, where the Riemann curvature tensor vanishes. Solar cycle dynamos are therefore compatible with the Riemann-flat stretching dynamo model from solar differential rotation presented here.
A new antidynamo theorem for non-stretched twisted magnetic flux tube dynamo is obtained. Though Riemannian curvature cannot be neglected since one considers curved magnetic flux tube axis, the stretch can be neglect since one only considers the limit of thin magnetic flux tubes. The theorem states that marginal or slow dynamos along curved (folded), torsioned (twisted) and non-stretched flux tubes endowed with diffusionless plasma flows, if a constraint is imposed on the relation between poloidal and toroidal magnetic fields in the helical dynamo case. A formula for the stretch of flux tubes …
Differential rotation plays a crucial role in the alpha-omega dynamo, and thus also in creation of magnetic fields in stars with convective outer envelopes. Still, measuring the radial differential rotation on stars is impossible with the current techniques, and even the measurement of surface differential rotation is difficult. In this work we investigate the surface differential rotation obtained from dynamo models using similar techniques as are used on observations, and compare the results with the known radial differential rotation used when creating the Dynamo model.
Magnetic fields can be created in stably stratified (non-convective) layers in a differentially rotating star. A magnetic instability in the toroidal field (wound up by differential rotation) replaces the role of convection in closing the field amplification loop. Tayler instability is likely to be the most relevant magnetic instability. A dynamo model is developed from these ingredients, and applied to the problem of angular momentum transport in stellar interiors. It produces a prodominantly horizontal field. This dynamo process might account for the observed pattern of rotation in the solar…
We present the first results from three-dimensional spherical shell simulations of magnetic dynamo action realized by turbulent convection penetrating downward into a tachocline of rotational shear. This permits us to assess several dynamical elements believed to be crucial to the operation of the solar global dynamo, variously involving differential rotation resulting from convection, magnetic pumping, and amplification of fields by stretching within the tachocline. The simulations reveal that strong axisymmetric toroidal magnetic fields (about 3000 G in strength) are realized within the lowe…
We present numerical simulations of a self-sustaining magnetic field in a differentially rotating non-convective stellar interior. A weak initial field is wound up by the differential rotation; the resulting azimuthal field becomes unstable and produces a new meridional field component, which is then wound up anew, thus completing the `dynamo loop'. This effect is observed both with and without a stable stratification. A self-sustained field is actually obtained more easily in the presence of a stable stratification. The results confirm the analytical expectations of the role of Tayler instabi…
The topological mapping between a torus of big radius and a sphere is applied to the Riemannian geometry of a stretched and twisted very thick magnetic flux tube, to obtain spherical dynamos solving the magnetohydrodynamics (MHD) self-induction equation for the magnetic flux tubes undergoing differential (non-uniform) rotation along the tube magnetic axis. Constraints on the shear is also computed. It is shown that when the hypothesis of the convective cyclonic dynamo is used the rotation is constant and a solid rotational body is obtained. As usual toroidal fields are obtained from poloidal m…
In this paper we discuss a dynamic flux-transport dynamo model that includes the feedback of the induced magnetic field on differential rotation and meridional flow. We consider two different approaches for the feedback: meanfield Lorentz force and quenching of transport coefficients such as turbulent viscosity and heat conductivity. We find that even strong feedback on the meridional flow does not change the character of the flux-transport dynamo significantly; however it leads to a significant reduction of differential rotation. To a large degree independent from the dynamo parameters, the s…
Increasing evidence is becoming available about not only the surface differential rotation of rapidly rotating cool stars but, in a small number of cases, also about temporal variations, which possibly are analogous to the solar torsional oscillations. Given the present difficulties in resolving the precise nature of such variations, due to both the short length and poor resolution of the available data, theoretical input is vital to help assess the modes of behaviour that might be expected, and will facilitate interpretation of the observations. Here we take a first step in this direction by …
Many stars exhibit strong magnetic fields, some of which are thought to be of primordial origin and others a sign of magnetic dynamo processes. We briefly review the results of observational studies of solar-type stars seeking to evaluate the linkage between rotation rate and possible magnetic cycles of activity. Clearly turbulent convection and rotation within spherical shell geometries provide ingredients essential for dynamo action. However, intensive efforts over several decades in solar research have demonstrated that it is no easy matter to achieve cyclic magnetic activity that is in acc…
We study axisymmetric mean-field dynamo models containing differential rotation, the $α$ effect and the additional turbulent induction effects. The additional effects result from the combined action of rotation and an inhomogeneity of the large-scale magnetic field. The best known of them is the $\vecΩ\times\vec{J}$ effect. We also include anisotropic diffusion and a new dynamo term which is of third order in the rotation vector $\vecΩ$ The model calculations are carried out using the rotation profile of the Sun as obtained from helioseismic measurements and radial profiles of other quantities…
The generation of solar non-axisymmetric magnetic fields is studied based on a linear alpha2-Omega dynamo model in a rotating spherical frame. The model consists of a solar-like differential rotation, a magnetic diffusivity varied with depth, and three types of alpha-effects with different locations, i.e. the tachocline, the whole convective zone and the sub-surface. Some comparisons of the critical alpha-values of axisymmetric m=0 and longitude-dependent modes m=1,2,3 are presented to show the roles of the magnetic diffusivity in the problem of modes selection. With the changing of diffusivit…
In this paper we investigate the idea of Hanasz & Lesch 1993 that the galactic dynamo effect is due to the Parker instability of magnetic flux tubes. In addition to the former approach, we take into account more general physical conditions in this paper, by incorporating cosmic rays and differential forces due to the axisymmetric differential rotation and the density waves as well. We present the theory of slender magnetic flux tube dynamics in the thin flux tube approximation and the Lagrange description. This is the application of the formalism obtained for solar magnetic flux tubes by Sprui…
The consequences of the shear-induced alpha effect to the long-term modulation of magnetic activity are examined with the help of the axisymmetric numerical dynamo model that includes the self-consistent description of the angular momentum balance, heat transport and magnetic field generation in the spherical shell. We find that the shear contributions to alpha effect can complicate the long-term behaviour of the large-scale magnetic activity and differential rotation in nonlinear dynamo. Additionally we consider the impact secular magnetic activity variations to the secular modulations of the…
An essential ingredient in kinematic dynamo models is the velocity field within the solar convection zone. In particular, the differential rotation is now well constrained by helioseismic observations. Helioseismology also gives us information about the depth-dependence of the meridional circulation in the near-surface layers. The typical velocity inputs used in solar dynamo models, however, continue to be an analytic fit to the observed differential rotation and a theoretically constructed meridional flow profile that matches only the peak flow speed at the surface. Here we take the first ste…
Flux-dominated solar dynamo models have demonstrated to reproduce the main features of the large scale solar magnetic cycle, however the use of a solar like differential rotation profile implies in the the formation of strong toroidal magnetic fields at high latitudes where they are not observed. In this work, we invoke the hypothesis of a thin-width tachocline in order to confine the high-latitude toroidal magnetic fields to a small area below the overshoot layer, thus avoiding its influence on a Babcock-Leighton type dynamo process. Our results favor a dynamo operating inside the convection …
We present the results of 3--D nonlinear simulations of magnetic dynamo action by core convection within A-type stars of 2 solar masses, at a range of rotation rates. We consider the inner 30% by radius of such stars, with the spherical domain thereby encompassing the convective core and a portion of the surrounding radiative envelope. The compressible Navier-Stokes equations, subject to the anelastic approximation, are solved to examine highly nonlinear flows that span multiple scale heights, exhibit intricate time dependence, and admit magnetic dynamo action. Small initial seed magnetic fiel…
Core convection and dynamo activity deep within rotating A-type stars of 2 solar masses are studied with 3--D nonlinear simulations. Our modeling considers the inner 30% by radius of such stars, thus capturing within a spherical domain the convective core and a modest portion of the surrounding radiative envelope. The MHD equations are solved using the ASH code to examine turbulent flows and magnetic fields, both of which exhibit intricate time dependence. By introducing small seed magnetic fields into our progenitor hydrodynamic models rotating at one and four times the solar rate, we assess …
Observations show that solar activity is distributed non-axisymmetrically, concentrating at "preferred longitudes". This indicates the important role of non-axisymmetric magnetic fields in the origin of solar activity. We investigate the generation of the non-axisymmetric fields and their coupling with axisymmetric solar magnetic field. Our kinematic generation (dynamo) model operating in a sphere includes solar differential rotation, which approximates the differential rotation obtained by inversion of helioseismic data, modelled distributions of the turbulent resistivity, non-axisymmetric me…
The theory of the solar/stellar activity cycles is presented, based on the mean-field concept in magnetohydrodynamics. A new approach to the formulation of the electromotive force and the theory of differential rotation and meridional circulation is described. Dynamo cycles in the overshoot layer and distributed dynamos are compared, with the latter including the influence of meridional flow. The overshoot layer dynamo reproduces the solar cycle periods and the butterfly diagram only if alpha=0 in the convection zone (CZ). The distributed dynamo including meridional flows shows the observed bu…
We present three-dimensional nonlinear magnetohydrodynamic simulations of the interiors of fully convective M-dwarfs. Our models consider 0.3 solar-mass stars using the Anelastic Spherical Harmonic code, with the spherical computational domain extending from 0.08-0.96 times the overall stellar radius. Like previous authors, we find that fully convective stars can generate kG-strength magnetic fields (in rough equipartition with the convective flows) without the aid of a tachocline of shear. Although our model stars are everywhere unstably stratified, the amplitudes and typical pattern sizes of…
Rapidly rotating spherical kinematic dynamos are computed using the combination of a quasi geostrophic (QG) model for the velocity field and a classical spectral 3D code for the magnetic field. On one hand, the QG flow is computed in the equatorial plane of a sphere and corresponds to Rossby wave instabilities of a geostrophic internal shear layer produced by differential rotation. On the other hand, the induction equation is computed in the full sphere after a continuation of the QG flow along the rotation axis. Differential rotation and Rossby-wave propagation are the key ingredients of the …
The isospectrality problem is studied for the operator of the spherical hydromagnetic alpha^2-dynamo. It is shown that this operator is formally pseudo-Hermitian (J-symmetric) and lives in a Krein space. Based on the J-symmetry, an operator intertwining Ansatz with first-order differential intertwining operators is tested for its compatibility with the structure of the alpha^2-dynamo operator matrix. An intrinsic structural inconsistency is obtained in the set of associated matrix Riccati equations. This inconsistency is interpreted as a no-go theorem which forbids the construction of isospect…
Differential rotation is widely supposed to be essential for the dynamo effects that sustain solar and planetary magnetic fields, but dynamo effects tend to oppose the flows that drive them, and it is uncertain what drives differential rotation. The relative sign of the differential rotation and meridional circulation is not consistent with simple convection modified by Coriolis forces. We investigate dynamo mechanisms consistent with the observed solar circulation, and discuss how reactive JxB forces would affect such flows. We formulate scaling rules that relate the magnetic field strength t…
Observations of the Sun and solar-type stars continue to reveal phenomena whose understanding is very likely to require a nonlinear framework. Here we shall concentrate on two such phenomena, namely the grand minima type behaviour observed in the Sun and solar-type stars and the recent dynamical variations of the differential rotation in the solar convection zone, deduced from the helioseismic observations, and discuss how their explanations have recently motivated the development/employment of novel ideas from nonlinear dynamics.
