The evolution of loop structures in flux rings within the solar convection zone

Choudhuri and Gilman (1987) considered certain implications of the hypothesis that the magnetic flux within the Sun is generated at the bottom of the convection zone and then rises through it. Taking flux rings symmetric around the rotation axis and using reasonable values of different parameters, they found that the Coriolis force deflects these flux rings into trajectories parallel to the rotation axis so that they emerge at rather high latitudes. This paper looks into the question of whether the action of the Coriolis force is subdued when the initial configuration of the flux ring has non-axisymmetries in the form of loop structures. The results depend dramatically on whether the flux ring with the loops lies completely within the convection zone or whether the lower parts of it are embedded in the stable layers underneath the convection zone. In the first case, the Coriolis force supresses the non-axisymmetric perturbations so that the flux ring tends to remain symmetric and the trajectories are very similar to those of Choudhuri and Gilman (1987). In the second case, however, the lower parts of the flux ring may remain anchored underneath the bottom of the convection zone, but the upper parts of the loops still tend to move parallel to the rotation axis and emerge at high latitudes. Thus the problem of the magnetic flux not being able to come out at the sunspot latitudes still persists after the non-axisymmetries in the flux rings are taken into account.National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Sunspot velocity correlations: Are they due to Reynolds stresses or to the Coriolis force on rising flux tubes?

Observations have consistently pointed out that the longitudinal and latitudinal motions of sunspots are correlated. The magnitude of the covariance was found to increase with latitude, and its sign was found to be positive in the N-hemisphere and negative in the S-hemisphere. This correlation was believed to be due to the underlying turbulence where the sunspot flux tubes are anchored, and the covariance had the right sign and magnitude needed to explain the transfer of angular momentum toward the equator through Reynolds stresses.Here we present an alternate explanation for these sunspot velocity correlations: It is believed that the dynamo operates in a thin overshoot layer beneath the base of the convection zone, and the flux tubes generated there produce sunspots at the photosphere. By studying the dynamics of flux tubes emerging from the base of the convection zone to the photosphere, we show that these velocity correlations of sunspots could be merely a consequence of the effect of Coriolis force on rising flux tubes. The effect of the Coriolis force, as demonstrated by even a back-of-the-envelope calculation, is to push the faster rotating spots equatorward and the slower rotating spots poleward, giving rise to a correlation in their longitudinal and latitudinal velocities, which is positive in the N-hemisphere and negative in the S-hemisphere. The increase in the correlation with latitude is due to the increase in magnitude of the Coriolis force. Hence we show that these velocity correlations might have nothing to do with the Reynolds stresses of the underlying turbulence.We present analyses of observations, and show that the covariances of plages are an order of magnitude higher than the sunspot covariances. If plages and sunspots share the same origin, and if their horizontal velocity correlations are wholly due to the effect of Coriolis force on rising flux tubes, then the study of their dynamics suggests that the flux tubes that form plages should have diameters of a couple of thousand km at the base of the convection zone and remain intact until they reach the photosphere, whereas sunspots should be formed by a collection of small flux tubes (each measuring about a hundred km in diameter), that rise through the convection zone as individual elements and coalesce when they emerge through the photosphere.Operated by the Association of Universities for Research in Astronomy, Inc., under Cooperative Agreement with the National Science Foundation.     

The Orientational Relaxation of Bipolar Active Regions

In the mean, bipolar active regions are oriented nearly toroidally, according to Hale's polarity law, with a latitude-dependent tilt known as Joy's Law. The tilt angles of individual active regions deviate from this mean behavior and change over time. It has been found that on average the change is toward the mean angle at a rate characteristic of 4.37 days (Howard, 1996). We show that this orientational relaxation is consistent with the standard model of flux tube emergence from a deep dynamo layer. Under this scenario Joy's law results from the Coriolis effect on the rising flux tube (D'Silva and Choudhuri, 1993), and departures from it result from turbulent buffeting of the tubes (Longcope and Fisher, 1996). We show that relaxation toward Joy's angle occurs because the turbulent perturbations relax on shorter time scales than the perturbations from the Coriolis force. The turbulent perturbations relax more rapidly because they are localized to the topmost portion of the convection zone while the Coriolis perturbations are more widely distributed. If a fully-developed active region remains connected to the strong toroidal magnetic field at the base of the convection zone, its tilt will eventually disappear, leaving it aligned perfectly toroidally. On the other hand, if the flux becomes disconnected from the toroidal field the bipole will assume a tilt indicative of the location of disconnection. We compare models which are connected and disconnected from the toroidal field. Only those disconnected at points very deep in the convection zone are consistent with observed time scale of orientational relaxation.     

3-D non-linear evolution of a magnetic flux tube in a spherical shell: The isentropic case

We present recent 3-D MHD numerical simulations of the non-linear dynamical evolution of magnetic flux tubes in an adiabatically stratified convection zone in spherical geometry, using the anelastic spherical harmonic (ASH) code.We seek to understand the mechanism of emergence of strong toroidal fields from the base of the solar convection zone to the solar surface as active regions. We confirm the results obtained in cartesian geometry that flux tubes that are not twisted split into two counter vortices before reaching the top of the convection zone. Moreover, we find that twisted tubes undergo the poleward-slip instability due to an unbalanced magnetic curvature force which gives the tube a poleward motion both in the non-rotating and in the rotating case. This poleward drift is found to be more pronounced on tubes originally located at high latitudes. Finally, rotation is found to decrease the rise velocity of the flux tubes through the convection zone, especially when the tube is introduced at low latitudes. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Connection Between Mean Field Dynamo Theory and Flux Tubes

Mean field dynamo theory deals with various mean quantities and does not directly throw any light on the question of existence of flux tubes. We can, however, draw important conclusions about flux tubes in the interior of the Sun by combining additional arguments with the insights gained from solar dynamo solutions. The polar magnetic field of the Sun is of order 10 G, whereas the toroidal magnetic field at the bottom of the convection zone has been estimated to be 100000 G. Simple order-of-magnitude estimates show that the shear in the tachocline is not sufficient to stretch a 10 G mean radial field into a 100000 G mean toroidal field. We argue that the polar field of the Sun must get concentrated into intermittent flux tubes before it is advected to the tachocline. We estimate the strengths and filling factors of these flux tubes. Stretching by shear in the tachocline is then expected to produce a highly intermittent magnetic configuration at the bottom of the convection zone. The meridional flow at the bottom of the convection zone should be able to carry this intermittent magnetic field equatorward, as suggested recently by Nandy and Choudhuri (2002). When a flux tube from the bottom of the convection zone rises to a region of pre-existing poloidal field at the surface, we point out that it picks up a twist in accordance with the observations of current helicities at the solar surface.     

Comparing Simulations of Rising Flux Tubes Through the Solar Convection Zone with Observations of Solar Active Regions: Constraining the Dynamo Field Strength

We study how active-region-scale flux tubes rise buoyantly from the base of the convection zone to near the solar surface by embedding a thin flux tube model in a rotating spherical shell of solar-like turbulent convection. These toroidal flux tubes that we simulate range in magnetic field strength from 15 kG to 100 kG at initial latitudes of 1^° to 40^° in both hemispheres. This article expands upon Weber, Fan, and Miesch (Astrophys. J. 741, 11, 2011) (Article 1) with the inclusion of tubes with magnetic flux of 10²⁰ Mx and 10²¹ Mx, and more simulations of the previously investigated case of 10²² Mx, sampling more convective flows than the previous article, greatly improving statistics. Observed properties of active regions are compared to properties of the simulated emerging flux tubes, including: the tilt of active regions in accordance with Joy’s Law as in Article 1, and in addition the scatter of tilt angles about the Joy’s Law trend, the most commonly occurring tilt angle, the rotation rate of the emerging loops with respect to the surrounding plasma, and the nature of the magnetic field at the flux tube apex. We discuss how these diagnostic properties constrain the initial field strength of the active-region flux tubes at the bottom of the solar convection zone, and suggest that flux tubes of initial magnetic field strengths of ≥?40 kG are good candidates for the progenitors of large (10²¹ Mx to 10²² Mx) solar active regions, which agrees with the results from Article 1 for flux tubes of 10²² Mx. With the addition of more magnetic flux values and more simulations, we find that for all magnetic field strengths, the emerging tubes show a positive Joy’s Law trend, and that this trend does not show a statistically significant dependence on the magnetic flux.     

A coupled model of magnetic flux generation and transport in stars

We present a combined model for magnetic field generation and transport in cool stars with outer convection zones. The mean toroidal magnetic field, which is generated by a cyclic thin-layer α Ω dynamo at the bottom of the convection zone is taken to determine the emergence probability of magnetic flux tubes in the photosphere. Following the nonlinear rise of the unstable thin flux tubes, emergence latitudes and tilt angles of bipolar magnetic regions are determined. These quantities are put into a surface flux transport model, which simulates the surface evolution of magnetic flux under the effects of large-scale flows and turbulent diffusion. First results are discussed for the case of the Sun and for more rapidly rotating solar-type stars. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Negative Reynolds stress generation by accretion disc convection

The phenomenon of negative viscosity-alpha in convectively unstable Keplerian accretion discs is discussed. The convection is considered as a random flow with an axisymmetric mesoscale pattern. Its correlation tensor is computed with a time-averaging procedure using Kley's 2D hydrocode. There is a distinct anisotropy between the turbulence intensities in the radial and azimuthal directions, i.e. the radial velocity rms dominates the azimuthal one. As a consequence, an extra term in the expression for the turbulent transport of angular momentum appears which does not vanish for rigid rotation ('Λ-effect'). It is negative ('inwards transport') and even seems to dominate the positive contribution of the eddy viscosity representing outwards transport of angular momentum. For a turbulence model close to that of the mixing-length theory, the rotational influence on the anisotropy of the turbulence intensities,     , and the covariance  〈 u ' _R u ' _φ 〉  – representing the angular momentum transport – is computed and compared with the accretion disc simulations. Indeed, the negative angular momentum transport can be explained with the observed dominance of the radial turbulence intensity. If, on the other hand, in turbulence fields the azimuthal intensity would dominate or the turbulence is even isotropic, then we always find a positive transport of the angular momentum.     

Numerical simulations of rotating axisymmetric sunspots

A numerical model of axisymmetric convection in the presence of a vertical magnetic flux bundle and rotation about the axis is presented. The model contains a compressible plasma described by the non-linear MHD equations, with density and temperature gradients simulating the upper layer of the Sun's convection zone. The solutions exhibit a central magnetic flux tube in a cylindrical numerical domain, with convection cells forming collar flows around the tube. When the numerical domain is rotated with a constant angular velocity, the plasma forms a Rankine vortex, with the plasma rotating as a rigid body where the magnetic field is strong, as in the flux tube, while experiencing sheared azimuthal flow in the surrounding convection cells, forming a free vortex. As a result, the azimuthal velocity component has its maximum value close to the outer edge of the flux tube. The azimuthal flow inside the magnetic flux tube and the vortex flow is prograde relative to the rotating cylindrical reference frame. A retrograde flow appears at the outer wall. The most significant convection cell outside the flux tube is the location for the maximum value of the azimuthal magnetic field component. The azimuthal flow and magnetic structure are not generated spontaneously, but decay exponentially in the absence of any imposed rotation of the cylindrical domain.     

Total resonant absorption of acoustic oscillations in sunspots

The question of total resonant absorption of acoustic oscillations in sunspots is studied for cylindrical 1-D flux tubes that are stratified only in the radial direction and surrounded by a uniform, non-magnetic plasma. The numerical investigation of Goossens and Poedts (1992) in linear resistive MHD is taken further by increasing the strength of the azimuthal magnetic field in the equilibrium flux tubes. For relatively strong azimuthal magnetic fields, total absorption is found over a relatively wide range of spot radii.     

Transport of angular momentum and chemical species by anisotropic mixing in stellar radiative interiors

Small levels of turbulence can be present in stellar radiative interiors due to, e.g., the instability of rotational shear. In this paper we estimate turbulent transport coefficients for stably stratified rotating stellar radiation zones. Stable stratification induces strong anisotropy with a very small ratio of radial‐to‐horizontal turbulence intensities. Angular momentum is transported mainly due to the correlation between azimuthal and radial turbulent motions induced by the Coriolis force. This non‐diffusive transport known as the Λ‐effect has outward direction in radius and is much more efficient compared to the effect of radial eddy viscosity. Chemical species are transported by small radial diffusion only. This result is confirmed using direct numerical simulations combined with the test‐scalar method. As a consequence of the non‐diffusive transport of angular momentum, the estimated characteristic time of rotational coupling (≲100 Myr) between radiative core and convective envelope in young solar‐type stars is much shorter compared to the time‐scale of Lithium depletion (∼1 Gyr) (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mechanism of cyclically polarity reversing solar magnetic cycle as a cosmic dynamo

We briefly describe historical development of the concept of solar dynamo mechanism that generates electric current and magnetic field by plasma flows inside the solar convection zone. The dynamo is the driver of the cyclically polarity reversing solar magnetic cycle. The reversal process can easily and visually be understood in terms of magnetic field line stretching and twisting and folding in three-dimensional space by plasma flows of differential rotation and global convection under influence of Coriolis force. This process gives rise to formation of a series of huge magnetic flux tubes that propagate along iso-rotation surfaces inside the convection zone. Each of these flux tubes produces one solar cycle. We discuss general characteristics of any plasma flows that can generate magnetic field and reverse the polarity of the magnetic field in a rotating body in the Universe. We also mention a list of problems which are currently being disputed concerning the solar dynamo mechanism together with observational evidences that are to be constraints as well as verifications of any solar cycle dynamo theories of short and long term behaviors of the Sun, particularly time variations of its magnetic field, plasma flows, and luminosity.     

Axial tilt angles of active regions and their polarity separations

Sunspot group and magnetic (plage) data are examined to search for a relationship between the tilt angles of active regions and the separations of their leading and following portions. A relationship is found in the sense that larger positive tilt angles are associated with larger polarity separations. This is the direction predicted by recent theoretical work (D'Silva and Choudhuri, 1992). The explanation for this appears to be that smaller surface polarity separations lead to larger magnetic tension forces, which diminish the effect of the Coriolis force that acts to twist rising flux tubes.Operated by the Association of Universities for Research in Astronomy, Inc., under Cooperative Agreement with the National Science Foundation.     

The impact of meridional circulation on stellar butterfly diagrams and polar caps

Observations of rapidly rotating solar-like stars show a significant mixture of opposite-polarity magnetic fields within their polar regions. To explain these observations, models describing the surface transport of magnetic flux demand the presence of fast meridional flows. Here, we link subsurface and surface magnetic flux transport simulations to investigate (i) the impact of meridional circulations with peak velocities of  ≤125 m s⁻¹  on the latitudinal eruption pattern of magnetic flux tubes and (ii) the influence of the resulting butterfly diagrams on polar magnetic field properties. Prior to their eruption, magnetic flux tubes with low field strengths and initial cross-sections below  ∼300 km  experience an enhanced poleward deflection through meridional flows (assumed to be polewards at the top of the convection zone and equatorwards at the bottom). In particular, flux tubes which originate between low and intermediate latitudes within the convective overshoot region are strongly affected. This latitude-dependent poleward deflection of erupting magnetic flux renders the wings of stellar butterfly diagrams distinctively convex. The subsequent evolution of the surface magnetic field shows that the increased number of newly emerging bipoles at higher latitudes promotes the intermingling of opposite polarities of polar magnetic fields. The associated magnetic flux densities are about 20 per cent higher than in the case disregarding the pre-eruptive deflection, which eases the necessity for fast meridional flows predicted by previous investigations. In order to reproduce the observed polar field properties, the rate of the meridional circulation has to be of the order of 100 m s⁻¹, and the latitudinal range from which magnetic flux tubes originate at the base of the convective zone (≲50°) must be larger than in the solar case (≲35°).     

Formation of polar starspots through meridional circulation

To explain the observed intermingling of polarities in the magnetic field distributions of rapidly rotating stars, surface magnetic flux transport models demand the presence of fast meridional flows.We combine simulations of the pre-eruptive and post-eruptive magnetic flux transport in cool stars to investigate the influence of a fast meridional circulation on the latitudinal eruption pattern of magnetic flux tubes and on the polar magnetic field properties. Magnetic flux tubes rising through the convection zone experience an enhanced latitude-dependent poleward deflection through meridional flows, which renders the wings of stellar butterfly diagrams convex. The larger amount of magnetic flux emerging at higher latitudes supports the intermingling of opposite polarities of polar magnetic fields and yields magnetic flux densities in the polar regions about 20% higher than in the case disregarding the pre-eruptive deflection. Taking the pre-eruptive evolution of magnetic flux into account therefore eases the need for the fast meridional flows predicted by previous investigations. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new model for flux emergence and the evolution of sunspots and the large-scale fields

Existing models for the evolution of sunspots and sunspot groups, describing the subsurface structure of the magnetic fields and their interactions with the convective motions, are briefly reviewed. It is shown that they are generally unable to account for the most recent data concerning the relationship between the large-scale solar magnetic field structures and the magnetic fields of active regions. In particular, it is shown that the former do not arise directly from the decay of the latter, as required by the Babcock model and all other models based on it. Other observations which are not adequately explained by current models are also cited.A new model is put forward based on the expulsion of toroidal magnetic flux by the dominant (i.e. giant) cells of the convection zone. The flux expelled above these cells forms the large-scale field and thus the configuration of this field provides a clue to the structure of the giant cell patterns. The flux expelled below the cells becomes twisted into a rope as in the Babcock model but a loop or stitch forms only in the region of upflow of the giant cells. The interaction of this loop with intermediate-sized cells as it rises to the surface determines the configuration and extent of the active region which appears at the surface. The compatibility of the model with other observations is discussed and its implications for theories of the solar cycle are noted.     

Hydrodynamic Source of the 11-year Solar Variations

In the traditional axisymmetric models of the 11-year solar cycle, oscillations of the magnetic fields appear in the background of nonoscillating (over time scale considered) turbulent velocity fields and differential rotation. In this paper, an alternative approach is developed: The excitation of magnetic oscillations with the 22-year period is the consequence of hydrodynamic oscillations with the 11-year period. In the excitation of hydrodynamic oscillations, two processes taking place in high latitudes near the interface between the convective and radiative zones play a key role. One is forcing of the westerly zonal flow, the conditions for which are due to deformation of the interfacial surface. The other process is the excitation of a shear instability of zonal flow as a consequence of a strong radial gradient of angular velocity. The development of a shear instability at some stage brings about the disruption of the forcing of differential rotation. In the first (hydrodynamic) part of the paper, the dynamics of axisymmetric flows near the bottom of the convection zone is numerically simulated. Forcing of differential rotation having velocity shear in latitude and the existence of solutions in the form of torsional waves with the 11-year oscillation period are shown. In the second part the dynamics of the magnetic field is studied. The most pronounced peculiarities of the solutions are the existence of forced oscillations with the 22-year period and the drift of the toroidal magnetic field component from the mid latitudes to the equator. In high and low latitudes after cycle maximum, the toroidal component is of opposite sign in accordance with observations. In the third part, the transport of momentum from the bottom of the convection zone to the outer surface by virtue of diffusivity is considered. The existence of some sources of differential rotation in the convection zone is not implied. A qualitative correspondence of the differential rotation profile in the bulk of the convection zone and on its outer surface to experimental data is shown. The time correspondence between torsional and magnetic oscillations is also in accordance with observations.     

Gravitational Instability in the Dust Layer of a Protoplanetary Disk with Interaction between the Layer and the Surrounding Gas

We consider gravitational instability of the dust layer in the midplane of a protoplanetary disk with turbulence and shear stresses between the gas in the disk and that in the dust layer. We solve a linearized system of hydrodynamic equations for perturbations of dust (monodisperse) and gas phases in the incompressible gas approximation. We take into account the gas drag of solid particles (dust aggregates), turbulent diffusion and the velocity dispersion of particles, and the perturbation of the azimuthal velocity of gas in the layer upon the transfer of angular momentum from solid particles to it and from this gas to the surrounding gas in the disk. We obtain and solve the dispersion equation for the layer with the ratio of surface densities of the dust phase and gas being well above unity. The following parameters of gravitational instability in the dust layer are calculated: the critical surface density of solid matter and the Stokes number of particles corresponding to the onset of instability, the wavelength range in which instability occurs, and the rate of its growth as a function of the perturbation wavelength in the circumsolar disk at radial distances of 1 and 10 AU. We show that at 10 AU, the maximum instability growth rate increases due to the transfer of angular momentum of gas in the layer to gas outside it, a new maximum emerges at a longer wavelength, a long-wavelength instability “tail” forms, and the critical surface density initiating instability decreases relative to that determined without the transfer of angular momentum to gas outside the layer. None of these effects are observed at 1 AU, since instability in this region probably develops faster than the transfer of angular momentum to the surrounding gаs of a protoplanetary disk occurs.