Systematically Asymmetric Heliospheric Magnetic Field: Evidence for a Quadrupole Mode and Non-Axisymmetry with Polarity Flip-Flops

Recent studies of the heliospheric magnetic field (HMF) have detected interesting, systematic hemispherical and longitudinal asymmetries which have a profound significance for the understanding of solar magnetic fields. The in situ HMF measurements since the 1960s show that the heliospheric current sheet (HCS) is systematically shifted (coned) southward during solar minimum times, leading to the concept of a bashful ballerina. While temporary shifts can be considerably larger, the average HCS shift (coning) angle is a few degrees, less than the 7.2^∘ tilt of the solar rotation axis. Recent solar observations during the last two solar cycles verify these results and show that the magnetic areas in the northern solar hemisphere are larger and their intensity weaker than in the south during long intervals in the late declining to minimum phase. The multipole expansion reveals a strong quadrupole term which is oppositely directed to the dipole term. These results imply that the Sun has a symmetric quadrupole S0 dynamo mode that oscillates in phase with the dominant dipole A0 mode. Moreover, the heliospheric magnetic field has a strong tendency to produce solar tilts that are roughly opposite in longitudinal phase. This implies is a systematic longitudinal asymmetry and leads to a “flip-flop” type behaviour in the dominant HMF sector whose period is about 3.2 years. This agrees very well with the similar flip-flop period found recently in sunspots, as well as with the observed ratio of three between the activity cycle period and the flip-flop period of sun-like stars. Accordingly, these results require that the solar dynamo includes three modes, A0, S0 and a non-axisymmetric mode. Obviously, these results have a great impact on solar modelling.     

Can we look inside a dynamo?

For a simple spherically symmetric mean‐field dynamo model we investigate the possibility of determining the radial dependence of the coefficient α. Growth rates for different magnetic field modes are assumed to be known by measurement. An evolutionary strategy (ES) is used for the solution of the inverse problem. Numerically, we find quite different α‐profiles giving nearly the same eigenvalues. The ES is also applied to find functions α(r) yielding zero growth rates for the lowest four magnetic field modes. Additionally, a slight modification of the ES is utilized for an “energetic” optimization of α²‐dynamos. The consequences of our findings for inverse dynamo theory and for the design of future dynamo experiments are discussed.     

Flip-Flops as Observational Signatures of Different Dynamo Modes in Cool Stars

Cool, rapidly rotating stars exhibit enhanced magnetic activity with cyclic behavior on various time scales. In particular, the longitude of the dominant activity region switches quasi-periodically by 180^∘, which is known as the “flip-flop” phenomenon. In the present paper we introduce a new approach for the interpretation of stellar cycles based on light curve modeling with dipole and quadrupole dynamo modes. We discuss the observational signatures of different combinations of the dynamo modes. The proposed simple model is able to reproduce the basic properties of long-term photometric behavior of active stars and allows us to study different mechanisms resulting in flip-flops.     

A search for the age-dependency of Ap star parameters

Some observational data of the sample of the magnetic chemically peculiar stars (MCP stars) are investigated statistically. For the MCP stars of spectral types later than A2 both the frequency distribution and the R ⋅ sin i-values suggest the existence of a linear relation between stellar diameter and rotation period. The MCP stars of spectral types earlier than B9 show an overpopulation of small R ⋅ sin i which may indicate the existence of a second group with smaller radius in this sample. The equatorially symmetric rotator is used as the magnetic model. With respect to its temporal behaviour the effective magnetic field is separated into dipolar and quadrupolar contribution. Both signs of the axisymmetric quadrupole moment appear with equal frequency. The dipole moment which produces the amplitude of the B_eff(t) curve forms for longer periods two groups which are separated by a distinct gap. Both of the groups exhibit magnetic fields which are the stronger the greater the stellar radius is, contrary to what is expected for frozen-in fields. The dominance of magnetic curves without polarity reversal for longer-period stars is in accordance with predictions of the dynamo theory.     

Non‐holonomic filamentary dynamos as generalized Arnold's maps in Riemannian space

A filamentary non‐holonomic dynamo solution of self‐induction magnetic field equation is found by considering highly conducting filaments. It is shown that planar filaments cannot support dynamo action since the flow along the filament vanishes for torsion‐free filaments. This is a generalization of the Zeldovich theorem for linear magnetic dynamo filaments. The flow of filament is proportionally to the product between Frenet torsion and curvature. This shows that filamentary dynamos must possess Frenet torsion. A well‐known example of this result is the α ‐dynamo in solar physics. Magnetic helicity and magnetic energy for this filamentary dynamo are computed. Magnetic helicity vanishes by construction and the magnetic field decays with torsion energy in helicoidal dynamos. The approach considered here is useful for the investigation of anisotropic turbulent cascades. As a particular simple example it is shown that under certain constraints the solution can be reduced to the Arnold cat dynamo map solution where the non‐holonomic directional mixed derivative, would play the role of the Lyapunov exponent which appears on stretching the magnetic field in Riemannian space. The solution seems to describe marginal slow dynamos when the velocities involved in the dynamo flows are constants. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The disk dynamo

The simplest dynamo in a thin disk is analysed. It the antisymmetric helicity function (z) (wherez is a coordinate perpendicular to the disk plane) is smooth and limited, then the conditions for generating a magnetic field and the symmetry of the resulting solutions depend only on the form of at the segment (0,h) — whereh is the half-thickness of the disk — and the value of the dimensionless dynamo numberD. When (z) does not change its sign at this segment andD>D _c (the critical dynamo number), the excitation of non-oscillating even (quadrupole) and oscillating odd (dipole) fields are possible. When (z) changes its sign at the segment indicated, non-oscillating odd magnetic fields can also be excited.The old exact solutions of the disk dynamo are studied and new ones are found. The results can be of importance when applied to the problem of the generation of a magnetic field in galactic and turbulent disks appearing around some X-ray sources.     

Models of magnetic field generation in partly stable planetary cores: Applications to Mercury and Saturn

A substantial part of Mercury's iron core may be stably stratified because the temperature gradient is subadiabatic. A dynamo would operate only in a deep sublayer. We show that such a situation arises for a wide range of values for the heat flow and the sulfur content in the core. In Saturn the upper part of the metallic hydrogen core could be stably stratified because of helium depletion. The magnetic field is unusually weak in the case of Mercury and unusually axisymmetric at Saturn. We study numerical dynamo models in rotating spherical shells with a stable outer region. The control parameters are chosen such that the magnetic Reynolds number is in the range of expected Mercury values. Because of its slow rotation, Mercury may be in a regime where the dipole contribution to the internal magnetic field is weak. Most of our models are in this regime, where the dynamo field consists mainly of rapidly varying higher multipole components. They can hardly pass the stable conducting layer because of the skin effect. The weak low-degree components vary more slowly and control the structure of the field outside the core, whose strength matches the observed field strength at Mercury. In some models the axial dipole dominates at the planet's surface and in others the axial quadrupole is dominant. Differential rotation in the stable layer, representing a thermal wind, is important for attenuating non-axisymmetric components in the exterior field. In some models that we relate to Saturn the axial dipole is intrinsically strong inside the dynamo. The surface field strength is much larger than in the other cases, but the stable layer eliminates non-axisymmetric modes. The Messenger and Bepi Colombo space missions can test our predictions that Mercury's field is large-scaled, fairly axisymmetric, and shows no secular variations on the decadal time scale.     

The solar dynamo

The basic features of the solar activity mechanism are explained in terms of the dynamo theory of mean magnetic fields. The field generation sources are the differential rotation and the mean helicity of turbulent motions in the convective zone. A nonlinear effect of the magnetic field upon the mean helicity results in stabilizing the amplitude of the 22-year oscillations and forming a basic limiting cycle. When two magnetic modes (with dipole and quadrupole symmetry) are excited nonlinear beats appear, which may be related to the secular cycle modulation.The torsional waves observed may be explained as a result of the magnetic field effect upon rotation. The magnetic field evokes also meriodional flows.Adctual variations of the solar activity are nonperiodic since there are recurrent random periods of low activity of the Maunder minimum type. A regime of such a magnetic hydrodynamic chaos may be revealed even in rather simple nonlinear solar dynamo models.The solar dynamo gives rise also to three-dimensional, non-axisymmetric magnetic fields which may be related to a sector structure of the solar field.     

The dipole-quadrupole cycle of the background solar magnetic field

The evolution of the background magnetic field with the solar cycle has been studied using the dipole-quadrupole magnetic energy behaviour in a cycle. The combined energy of the axisymmetric dipole, non-axisymmetric quadrupole, and equatorial dipole is relatively lowly variable over the solar cycle. The dipole field changed sign when the quadrupole field was near a maximum, andvice versa. A conceptual picture involving four meridional magnetic polarity sectors proposed to explain these features may be in agreement with equatorial coronal hole observations. The rate of sector rotation is estimated to be 8 heliographic degrees per year faster than the Carrington rotation (P = 27.23^d synodic). Polarity boundaries of sectors located 180° apart show meridional migrations in one direction, while the boundaries of the other two sectors move in the opposite direction. A simple model of how the magnetic field energy varies, subject to specifying reasonable initial photospheric magnetic and velocity field patterns, follows the observed evolution of the dipole and quadrupole field energies quite nicely.     

On the non-symmetric solar dynamo

The external field of the solar magnetohydrodynamic dynamo is expressed in terms of spherical harmonics and in powers of 1/r. The non-symmetric dynamo is stabilized by a -dependent rotational oscillation which interacts with the magnetic field, thus compensating for Ohmic loss. As a consequence, the axis of a dipole wave is found to move on a great circle, with revolution time equal to the magnetic cycle.Proceedings of the 14th ESLAB Symposium on Physics of Solar Variations, 16–19 September 1980, Scheveningen, The Netherlands.     

A non-adiabatic analysis for axisymmetric pulsations of magnetic stars

This paper presents the results of a non-adiabatic analysis for axisymmetric non-radial pulsations including the effect of a dipole magnetic field. Convection is assumed to be suppressed in the stellar envelope, and the diffusion approximation is used to radiative transport. As in a previous adiabatic analysis, the eigenfunctions are expanded in a series of spherical harmonics. The analysis is applied to a  1.9-M_⊙  , main-sequence model  (log  T _eff= 3.913)  . The presence of a magnetic field always stabilizes low-order acoustic modes. All the low-order modes of the model that are excited by the κ-mechanism in the He  ii ionization zone in the absence of a magnetic field are found to be stabilized if the polar strength of the dipole magnetic field is larger than about 1 kG. For high-order p modes, on the other hand, distorted dipole and quadrupole modes excited by the κ-mechanism in the H ionization zone remain overstable, even in the presence of a strong magnetic field. It is found, however, that all the distorted radial high-order modes are stabilized by the effect of the magnetic field. Thus, our non-adiabatic analysis suggests that distorted dipole modes and distorted quadrupole modes are most likely excited in rapidly oscillating Ap stars. The latitudinal amplitude dependence is found to be in reasonable agreement with the observationally determined one for HR 3831. Finally, the expected amplitude of magnetic perturbations at the surface is found to be very small.     

Modeling the total and polarized emission in evolving galaxies: “Spotty” magnetic structures

Future radio observations with the Square Kilometre Array (SKA) and its precursors will be sensitive to trace spiral galaxies and their magnetic field configurations up to redshift z ≈ 3. We suggest an evolutionary model for the magnetic configuration in star‐forming disk galaxies and simulate the magnetic field distribution, the total and polarized synchrotron emission, and the Faraday rotation measures for disk galaxies at z ≲ 3. Since details of dynamo action in young galaxies are quite uncertain, we model the dynamo action heuristically relying only on well‐established ideas of the form and evolution of magnetic fields produced by the mean‐field dynamo in a thin disk. We assume a small‐scale seed field which is then amplified by the small‐scale turbulent dynamo up to energy equipartition with kinetic energy of turbulence. The large‐scale galactic dynamo starts from seed fields of 100 pc and an averaged regular field strength of 0.02 μG, which then evolves to a “spotty” magnetic field configuration in about 0.8 Gyr with scales of about one kpc and an averaged regular field strength of 0.6 μG. The evolution of these magnetic spots is simulated under the influence of star formation, dynamo action, stretching by differential rotation of the disk, and turbulent diffusion. The evolution of the regular magnetic field in a disk of a spiral galaxy, as well as the expected total intensity, linear polarization and Faraday rotation are simulated in the rest frame of a galaxy at 5GHz and 150 MHz and in the rest frame of the observer at 150 MHz. We present the corresponding maps for several epochs after disk formation. Dynamo theory predicts the generation of large‐scale coherent field patterns (“modes”). The timescale of this process is comparable to that of the galaxy age. Many galaxies are expected not to host fully coherent fields at the present epoch, especially those which suffered from major mergers or interactions with other galaxies. A comparison of our predictions with existing observations of spiral galaxies is given and discussed (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bimodal mass and angular momentum losses by magnetically controlled winds during stellar evolution

In recent times evidence for bimodal distributions of stars in the H–R diagram has reached a striking evidence. These bimodal distributions seem to be correlated with a bimodal distribution of masses and angular velocities. The approach we propose to explain the observed bimodality suggests that this latter is due to a bimodal mass loss by magnetically controlled stellar winds during stellar evolution, owing to different magnetic field configurations. It is assumed a mechanism analogous to that which produces solar wind, with magnetic field generated by dynamo working in the convection zone. Different field geometries (dipole cr quadrupole), which depend on the mode the dynamo operates, can produce different but discrete mass losses during stellar evolution, thus producing bimodal distributions of masses and angular velocities.     

An unconventional accretion disc dynamo

In the light of recent results from numerical simulations of accretion disc MHD turbulence, we revisit the problem of the configuration of large-scale magnetic fields resulting from an α Ω dynamo operating in a thin accretion disc. In particular, we analyse the consequences of the peculiar sign of the α -effect suggested by numerical simulations . We determine the symmetry of the fastest-growing modes in the kinematic dynamo approximation and, in the framework of an ' α -quenched' dynamo model, study the evolution of the magnetic field. We find that the resulting field for this negative polarity of the α -effect generally has dipole symmetry with respect to the disc midplane, although the existence of an equilibrium configuration depends on the properties of the turbulence. The role of magnetic field dragging is discussed and, finally, the presence of an external uniform magnetic field is included to address the issue of magneto centrifugal wind launching from accretion discs.     

A dynamo driven by zonal jets at the upper surface: Applications to giant planets

We present a dynamo mechanism arising from the presence of barotropically unstable zonal jet currents in a rotating spherical shell. The shear instability of the zonal flow develops in the form of a global Rossby mode, whose azimuthal wavenumber depends on the width of the zonal jets. We obtain self-sustained magnetic fields at magnetic Reynolds numbers greater than 10³. We show that the propagation of the Rossby waves is crucial for dynamo action. The amplitude of the axisymmetric poloidal magnetic field depends on the wavenumber of the Rossby mode, and hence on the width of the zonal jets. We discuss the plausibility of this dynamo mechanism for generating the magnetic field of the giant planets. Our results suggest a possible link between the topology of the magnetic field and the profile of the zonal winds observed at the surface of the giant planets. For narrow Jupiter-like jets, the poloidal magnetic field is dominated by an axial dipole whereas for wide Neptune-like jets, the axisymmetric poloidal field is weak.     

The residual permanent magnetic dipole moment of the moon

The residual dipole moment of the outer spherical shell of the Moon, magnetized in the field of an internal dipole is calculated for the case when the permeability of the shell differs from unity. It is shown that, using an average value of surface magnetization from returned lunar crystalline rock samples and a global figure for the lunar permeability of 1.012, that a residual moment of the order of 10¹⁵ to 10¹⁶ Am² is expected. This value is some two or three orders of magnitude lower than the moment for a shell magnetized in an external uniform field and is of the same order as the upper limit of the residual moment detected by Russellet al. (1974). At present the magnetic data and the thermal state of the Moon are not known with sufficient accuracy to distinguish between a crust magnetized in an internal dipole field of constant polarity and a crust magnetized in the dipole field of a self-reversing core dynamo. Refined measurements of the relevant parameters together with the theory presented in this paper could enable these two possibilities to be distinguished.     

Polarization of continuum radiation in magnetic atmospheres

A numerical solution is presented for the problem of continuum radiative transfer in a magnetoactive medium. The continuum opacities are calculated in the presence of a strong magnetic field (H=10⁷ G) typical of magnetic white dwarfs. The L.T.E. pure absorption model is assumed for calculating the polarized radiation field emitted by a realistic model atmosphere in the plane parallel approximation. The wavelength dependence of the linear and circular polarizations are calculated for both uniform and dipole field configurations.     

Solar cycle according to mean magnetic field data

To investigate the shape of the solar cycle, we have performed a wavelet analysis of the large–scale magnetic field data for 1960–2000 for several latitudinal belts and have isolated the following quasi-periodic components: ∼22, 7 and 2 yr. The main 22-yr oscillation dominates all latitudinal belts except the latitudes of ±30° from the equator. The butterfly diagram for the nominal 22-yr oscillation shows a standing dipole wave in the low-latitude domain  (∣θ∣≤ 30°)  and another wave in the sub-polar domain  (∣θ∣≥ 35°)  , which migrates slowly polewards. The phase shift between these waves is about π. The nominal 7-yr oscillation yields a butterfly diagram with two domains. In the low-latitude domain  (∣θ∣≤ 35°)  , the dipole wave propagates equatorwards and in the sub-polar region, polewards. The nominal 2-yr oscillation is much more chaotic than the other two modes; however the waves propagate polewards whenever they can be isolated.
We conclude that the shape of the solar cycle inferred from the large-scale magnetic field data differs significantly from that inferred from sunspot data. Obviously, the dynamo models for a solar cycle must be generalized to include large-scale magnetic field data. We believe that sunspot data give adequate information concerning the magnetic field configuration deep inside the convection zone (say, in overshoot later), while the large-scale magnetic field is strongly affected by meridional circulation in its upper layer. This interpretation suggests that the poloidal magnetic field is affected by the polewards meridional circulation, whose velocity is comparable with that of the dynamo wave in the overshoot layer. The 7- and 2-yr oscillations could be explained as a contribution of two sub-critical dynamo modes with the corresponding frequencies.     

Model of the magnetic field in the inner heliosphere with regard to radial field strength leveling in the solar corona

On the basis of experimental results from the Ulysses spacecraft, a model is proposed for calculating the magnetic field in the corona and the heliosphere in the potential approximation, which is a modification of the potential-field source-surface model. In addition to the photospheric surface and the source surface, a new demarcated spherical surface (leveling surface) is introduced in this model. The magnitude of the radial component of the magnetic field on this surface is assumed to be constant, and its sign abruptly changes from one hemisphere to another. General analytical formulas are given to calculate the potential and field for this model. Calculations are described in detail using the dipole and quadrupole harmonics as examples. Expressions are obtained for the surface currents. The results of visualization of the magnetic field for an axial dipole are discussed.     

North-south asymmetry in the heliospheric current sheet and the IMF sector structure

It is shown that in a heliomagnetic field the presence of a magnetic quadrupole in addition to a magnetic dipole introduces a north-south asymmetry in the heliospheric current sheet (HCS) about the heliographic equator. The dominant polarity of the interplanetary magnetic field (IMF) for the above type of current sheet reverses sign at a transition latitude _T, which lies in a heliohemisphere opposite to the one in which the HCS has more heliolatitudinal extension. The position of _T in the heliosphere and the north-south asymmetry introduced in the HCS change with the relative phase of the dipole and quadrupole components present in the solar magnetic field. The effect of the above type of asymmetric HCS in the IMF mean sector width is evaluated and the results are in agreement with the observations during the minima of solar cycle 21.