A model of solar differential rotation

An axisymmetric model for the Sun's differential rotation based upon a mechanism for angular momentum transport by compressible convection is developed. Convective heat transport is also considered. The model is simplified by the neglect of meridional circulation and radiative heat transport but is otherwise a self-consistent one because no adjustable parameters are used and the effective transport coefficients are expressed explicitly in terms of superadiabaticity of stratification rather than assigned externally. The model predictions agree satisfactorily with the observed rotation of the photosphere and with helioseismology data. The dependence of the rotation law, produced by the model, on rotation rate of a Sun-like star is discussed.     

Solar radiation and solar differential rotation

Areas of sunspots and their positions taken from the Greenwich Photoheliographic Results (1874–1976) and typical intensities of the umbrae and penumbrae are used to calculate daily values of the solar flux at a wavelength of about 500 nm. Using overlapping time series of 512 days each solar rotation periods are determined by Fourier transformation. The periods found depend on the phase of the solar activity cycle, as expected from the solar differential rotation. This method may be used for solar type stars to determine relations between activity and rotation too. The problems of errors - e.g. by faculae or the variation of the umbral intensity within the activity cycle - are explained.Mitteilungen aus dem Kiepenheuer-Institut Nr. 284     

The solar differential rotation from the eightteenth solar cycle

The sidereal daily rotation of the Sun, (), depends on the data used. From an appropriate selection of the data — sunspots with regular motion — it is found that ()=14.31–2.70 sin² , where denotes the heliographic latitude. Moreover, it seems that there is a variation, of the order of 3%, with the solar activity.     

Rotating convection and the solar differential rotation

We discuss the implication of a numerical experiment on rotating convection and its relevance to the construction of a model for the solar differential rotation.     

On the differential rotation of the solar photosphere

It is shown that sunspots as tracers can give the same results for the differential rotation of the solar photosphere as the Doppler-shift measurements, if the sunspots used have only insignificant motion relative to their immediate photospheric surroundings.     

22-Year periodicity in the solar differential rotation

Using the data on sunspot groups compiled during 1879–1975, we determined variations in the differential rotation coefficientsA andB during the solar cycle. The variation in the equatorial rotation rateA is found to be significant only in the odd numbered cycles, with an amplitude ∼ 0.01 μ rads^-1. There exists a good anticorrelation between the variations of the differential rotation rateB derived from the odd and even numbered cycles, suggesting existence of a ‘22-year’ periodicity inB. The amplitude of the variation ofB is ∼ 0.05 μ rad s^-1.     

The nature of one mechanism of solar differential rotation

The mechanism of the solar differential rotation usually ascribed to an anisotropic viscosity action is shown to be caused by Coriolis forces which influence anisotropic convective elements in a stratified medium. The estimation of an anisotropy parameters as a function of the convective zone depth is given. The value of (s–1) is positive near the solar surface and negative at the convective zone base, which is in good agreement with observations and the dynamo theory.     

Two types of differential rotation of the solar corona

The differential rotation of the solar corona has been analyzed using as the input data the brightness of the coronal green line Fe xiv 530.3 nm for more than five activity cycles. It is found that the character of rotation of the solar corona changes during the activity cycle. Approximately at the middle of the descending branch the differential rotation is weakly pronounced, while the greatest differential gradient is observed at the ascending branch and, occasionally, at the maximum of the cycle. An explanation of this difference has been suggested. The total rotation rate of the corona can be represented as a superposition of two rotation modes (components) – the fast and slow ones. The synodic period of the fast mode near the equator is about 27 days, increasing slightly with latitude. The synodic period of the slow mode exceeds 30 days. The changing relative fraction of these two modes results in variation of the latitude dependence of the observed rotation rate during the activity cycle. The characteristics of two principal types of differential rotation of the solar corona have been determined. The first type consists of the fast mode alone and is established approximately at the middle of the descending branch of the cycle. The second type is the sum of both modes with the fast mode dominating at low latitudes and the slow mode at high latitudes. The results obtained can be used for in-depth study of interaction of the velocity field and dynamo mechanism in the Sun and stars.     

The solar differential rotation: Present status of observations

The present status of observations regarding the solar differential rotation is reviewed from contributions published in the last two decades. The paper does not deal with the theory; it mentions theoretical aspects only where they are needed to guide and to understand observational efforts and results.Mitteilungen aus dem Kiepenheuer-Institut Nr. 250.     

The maintenance of solar differential rotation by two-dimensional turbulence

A numerical model has been made to test the theory that solar differential rotation is maintained by the Countergradient transport of energy peculiar to two-dimensional turbulence. After a brief discussion of this turbulent process, the numerical methods employed and their application to the sun are reviewed. The results of one problem are presented, indicating that this model can represent the observed large-scale nature of the sun's surface. The reader is referred to the author's dissertation for complete details of the methods and calculations.     

The solar differential rotation and ‘Rossby-type’ waves

It has been suggested that the solar differential rotation might be maintained by nearly horizontal non-spherical convective circulation called the Rossby-type waves (the wave motions characterized by the close balance of the Coriolis force and pressure gradient in horizontal motions). In this paper, such Rossby-type waves which could be excited in the upper solar convection zone are considered, and the possibility of maintenance of the solar differential rotation by such waves is examined. A numerical estimate, in terms of the rate of conversion of the kinetic energy of such wave motions into the mean rotational motion, indicates this possibility. The implications and limitations of the results are also discussed.Visiting Scientist to the High Altitude Observatory on leave of absence from the Department of Astronomy, University of Tokyo, Japan.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

A joined model for solar dynamo and differential rotation

A model for the solar dynamo, consistent in global flow and numerical method employed with the differential rotation model, is developed. The magnetic turbulent diffusivity is expressed in terms of the entropy gradient, which is controlled by the model equations. The magnetic Prandtl number and latitudinal profile of the alpha-effect are specified by fitting the computed period of the activity cycle and the equatorial symmetry of magnetic fields to observations. Then, the instants of polar field reversals and time-latitude diagrams of the fields also come into agreement with observations. The poloidal field has a maximum amplitude of about 10 Gs in the polar regions. The toroidal field of several thousand Gauss concentrates near the base of the convection zone and is transported towards the equator by the meridional flow. The model predicts a value of about 10³⁷ erg for the total magnetic energy of large-scale fields in the solar convection zone.     

The solar surface differential rotation from disk-integrated chromospheric fluxes

Disk-integrated solar chromospheric Caii K-line (3933.68 ) fluxes have been measured almost daily at Sacramento Peak Observatory since 1977. Using observing windows selected to mimic seasonal windows for chromospheric measurements of lower Main-Sequence stars such as those observed by Mount Wilson Observatory's HK Project, we have measured the solar rotation from the modulation of the Caii K-line flux. We track the change of rotation period from the decline of cycle 21 through the maximum of cycle 22. This variation in rotation period is shown to behave as expected from the migration of active regions in latitude according to Maunder's butterfly diagram, including an abrupt change in rotation period at the transition from cycle 21 to cycle 22. These results indicate the successful detection of solar surface differential rotation from disk-integrated observations. We argue that the success of our study compared to previous investigations of the solar surface differential rotation from disk-integrated fluxes lies primarily with the choice of the length of the time-series window. Our selection of 200 days is shorter than in previous studies whose windows are typically on the order of one year. The 200-day window is long enough to permit an accurate determination of the rotation period, yet short enough to avoid complications arising from active region evolution. Thus, measurements of the variation of rotation period in lower Main-Sequence stars, especially those that appear to be correlated with long-term changes in chromospheric activity (i.e., cycles), are probably evidence for stellar surface differential rotation.     

Radial variation of differential rotation in the solar electron corona

Long-lived brightness structures in the solar electron corona persist over many solar rotation periods and permit an observational determination of coronal magnetic tracer rotation as a function of latitude and height in the solar atmosphere. For observations over 1964–1976 spanning solar cycle 20, we compare the latitude dependence of rotation at two heights in the corona. Comparison of rotation rates from East and West limbs and from independent computational procedures is used to estimate uncertainty. Time-averaged rotation rates based on three methods of analysis demonstrate that, on average, coronal differential rotation decreases with height from 1.125 to 1.5 R _S. The observed radial variation of differential rotation implies a scale height of approximately 0.7 R _S for coronal differential rotation.Model calculations for a simple MHD loop show that magnetic connections between high and low latitudes may produce the observed radial variations of magnetic tracer rotation. If the observed tracer rotation represents the rotation of open magnetic field lines as well as that of closed loops, the small scale height for differential rotation suggests that the rotation of solar magnetic fields at the base of the solar wind may be only weakly latitude dependent. If, instead, closed loops account completely for the radial gradients of rotation, outward extrapolation of electron coronal rotation may not describe magnetic field rotation at the solar wind source. Inward extrapolations of observed rotation rates suggest that magnetic field and plasma are coupled a few hundredths of a solar radius beneath the photosphere.     

On the differential rotation with height in the solar atmosphere

Spectroscopic measurements of solar rotation having good height discrimination show no change in angular velocity through the photosphere layers but an increase of 8% for the Hα chromosphere (epoch 1968.9). Spectroscopic results in general are compared with measures made with tracers, i.e. sunspots, filaments, etc., and it is seen that the spectroscopic method always shows increased differential rotation with height, while tracers indicate none. A westward flowing wind is proposed that increases in velocity with height, but produces negligible movement to magnetic regions associated with tracers. Kitt Peak National Observatory Contribution No. 450. Operated by The Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation.     

Observation of solar differential rotation with the aid of magnetic tracers

We tried to search for the manifestation of differential rotation in the distribution of weak remnants of magnetic fields measured with a very low resolution. We found that, during the periods of low solar activity and in parts of the solar photosphere with smaller density of new magnetic flux sources, it was possible to observe the distribution of magnetic tracers in the form of differential rotation parabolas which increase their curvature from one rotation to the next. The obtained differential rotation rates are not far from those given by highly averaged sunspot data or by the daily magnetic fields. The characteristic differential rotation parabolas as well as specific cellular-like features disturbing their smooth patterns are always formed from fields of one main polarity, the sign of which depends on the phase of the activity cycle.Solar Cycle Workshop Paper.     

Large scale circulation in the convection zone and solar differential rotation

In this paper we study the dependence on depth and latitude of the solar angular velocity produced by a meridian circulation in the convection zone, assuming that the main mechanism responsible for setting up and driving the circulation is the interaction of rotation with convection. We solve the first order equations (perturbation of the spherically symmetric state) in the Boussinesq approximation and in the steady state for the axissymmetric case. The interaction of convection with rotation is modelled by a convective transport coefficient k _c = k _co + ?k _c2 P ₂(cos θ) where ? is the expansion parameter, P ₂ is the 2nd Legendre polynomial and k _c2 is taken proportional to the local Taylor number and the ratio of the convective to the total fluxes. We obtain the following results for a Rayleigh number 10³ and for a Prandtl number 1:

1. A single cell circulation extending from poles to the equator and with circulation directed toward the equator at the surface. Radial velocities are of the order of 10 cm s^?1 and meridional ones of the order of 150 cm s^?1.
2. A flux difference between pole and equator at the surface of about 5 percent, the poles being hotter.
3. An angular velocity increasing inwards.
4. Angular velocity constant surfaces of spheroidal shape. The model is consistent with the fact that the interaction of convection with rotation sets up a circulation (driven by the temperature gradient) which carries angular momentum toward the equator against the viscous friction. Unfortunately also a large flux variation at the surface is obtained. Nevertheless it seems that the model has the basic requisites for correct dynamo action.

     

Some controversial issues in theories of the solar differential rotation and dynamo

The following points are discussed:

Simulating the generation of the solar toroidal magnetic field by differential rotation

Within the kinematic dynamo theory, we construct a mathematical model for the evolution of the solar toroidal magnetic field, excited by the differential rotation of the convective zone in the presence of a poloidal field of a relic origin. We use a velocity profile obtained by decoding the data of helioseismological experiments. For the model of ideal magnetic hydrodynamics, we calculate the latitudinal profiles of the increasing-with-time toroidal field at different depths in the solar convection zone. It is found that, in the region of differential rotation, the excited toroidal field shows substantial fluctuations in magnitude with depth. Based on the simulations results, we propose an explanation for the “incorrect polarity” of magnetic bipolar sunspot groups in solar cycles.