Solar meridional motions derived from sunspot observations

Sunspot drawings obtained at the National Astronomical Observatory of Japan during the years 1954–1986 were used to determine meridional motions of the Sun. A meridional flow of a few ms^–1 was found, which is equatorward in the latitude range from -20° to +15° and is poleward at higher latitudes in both hemispheres. A northward flow of 0.01° day^–1 or 1.4 ms^–1 at mid-latitudes (between 10° and 20°) was also detected. From our limited data-set of three solar cycles, an indication of solar-cycle dependence of meridional motions was found.     

Dependence of the meridional motions of sunspot groups on life spans and age of the groups,and on the phase of the solar cycle

We have analyzed data on sunspot groups compiled during 1874–1981 and investigated the following: (i) dependence of the `initial' meridional motion (v <sub>ini</sub>()) of sunspot groups on the life span () of the groups in the range 2–12 days, (ii) dependence of the meridional motion (v(t)) of sunspot groups of life spans 10–12 days on the age (t) of the spot groups, and (iii) variations in the mean meridional motion of spot groups of life span 2–12 days during the solar cycle. In each of the latitude intervals 0°–10°, 10°–20° and 20°–30°, the values of both v <sub>ini</sub>() and v(t) often differ significantly from zero. In the latitude interval 20°–30°, the forms of v <sub>ini</sub>() and v(t) are largely systematic and mutually similar in both the north and south hemispheres. The form of v(t) suggests existence of periodic variation in the solar meridional motion with period of 4 days and amplitude 10–20 m s<sup>–1</sup>. Using the anchoring depths of magnetic structures for spot groups of different and testimated earlier, (Javaraiah and Gokhale, 1997), we suggest that the forms of v <sub>ini</sub>() and v(t) may represent radial variation of meridional flow in the Sun's convection zone, rather than temporal variation of the flow. The meridional flows (v <sub>e</sub>(t)) determined from the data during the last few days (i.e., age t: 10–12 days) of spot groups of life spans of 10–12 days are found to have magnitudes (10–20 m s<sup>–1</sup>) and directions (poleward) similar to the those of the surface meridional plasma flows determined from the Dopplergrams and magnetograms. The mean meridional velocity of sunspot groups living 2–12 days seems to vary during the solar cycle. The velocity is not significantly different from zero during the rising phase of the cycle and there is a suggestion of equatorward motion (a few m s<sup>–1</sup>at lower latitudes and 10 m s<sup>–1</sup>at higher latitudes) during the declining phase (last few years) of the cycle. The variation during the odd numbered cycles seems to anticorrelate with the variation during the even numbered cycles, suggesting existence of 22-year periodicity in the solar meridional flow. The amplitude of the anticorrelation seems to be depending on latitude and the cycle phase. In the latitude interval 20°–30° the `surface plasma meridional motion', v _e(t), is found to be poleward during maximum years (v _e(t) 20 m s^–1at 4th year) and equatorward during ending years of the cycle (v _e(t) –17 m s^–1at 10th year).     

Implications of a strongly peaked polar magnetic field

Using the flux-transport equation in the absence of sources, we study the relation between a highly peaked polar magnetic field and the poleward meridional flow that concentrates it. If the maximum flow speed _m greatly exceeds the effective diffusion speed /R, then the field has a quasi-equilibrium configuration in which the poleward convection of flux via meridional flow approximately balances the equatorward spreading via supergranular diffusion. In this case, the flow speed () and the magnetic field B() are related by the steady-state approximation () (/R)B()/B() over a wide range of colatitudes from the poles to midlatitudes. In particular, a general flow profile of the form sin ^p cos ^q which peaks near the equator (q p) will correspond to a cos ⁿ magnetic field at high latitudes only if p = 1 and _m = n /R. Recent measurements of n 8 and 600 km² s^–1 would then give _m 7 m s^–1.     

Evolution and nature of north-south asymmetry in the heliospheric current sheet

The nature and evolution of north-south asymmetry in the heliospheric current sheet (HCS) has been investigated using solar and interplanetary magnetic field (IMF) observations for the past few solar cycles. The mean heliographic latitude of the HCS (averaged over the solar longitude) a ₀ is found to be non-zero during many solar rotations indicating that the large-scale solar magnetic field is more ordered in a system where the origin is shifted away from the centre of the Sun. We have shown that the asymmetry in HCS manifests in different forms depending on the transition heliographic latitude of the reversal of dominant polarity of the IMF ( _T) and the difference in the maximum latitudinal extension of the HCS in the two solar hemispheres (). The classification of the observed asymmetry during 1971–1985 and its effect on IMF observations near Earth has been studied. We have also inferred the sign of _T during 1947–1971 using inferred IMF polarity data. The observed sign reversals of _T suggest the importance of periodicities less than the solar cycle period to be associated with the evolution of asymmetry in HCS. Asymmetry in sunspot activity about the solar equator does not seem to relate consistently well with the asymmetry in HCS about the heliographic equator.     

LONG-TERM VARIATIONS OF THE TORSIONAL OSCILLATIONS OF THE SUN

We investigated long-term variations of the differential rotation of the solar large-scale magnetic field on 1024 H charts in the latitude zones from +45° to -45° in the period 1915–1990. We used the expansion in terms of Walsh functions. It turns out that the rotation of the Sun becomes more rigid than average during the cycle maximum and the rotation is more differential during minimum. From 1915 to 1990, 7 bands of faster- and 7 bands of slower-than-average rotation are revealed showing an 11-year period. These bands drift towards the equator: 45° in 2.5 to 8 years. The time span of the bands varies from 4 to 6.8 years and is in anti-phase with long-term solar activity. The latitude span of the bands of torsional oscillations varies from 0.5 R to 1.3 R and shows a long-term variation of about 55 years. The poloidal component of velocity, V varies from 2 ms ^-1 to 6 ms ^-1. The maximum rate of the equatorial drift occurs in the period between 1935 and 1955 and it develops prior to the highest maximum activity. At the modern epoch from 1965 to 1985, V does not exceed 3 ms ^-1, but now it has a tendency to increase. The bands of slower-than-average rotation correspond to the evolution of the magnetic activity towards the equator in the butterfly diagram.     

Latitudinal Distribution of Polar Faculae

The distribution of polar faculae with respect to latitude is investigated, using data obtained at the Ussuriysk Observatory during the years 1963–1994. To correct the data for the effect of visibility, a visibility function of polar faculae is derived. Corrected surface density of polar faculae is calculated as a function of latitude and time. During most part of each solar cycle, polar faculae exhibit pronounced concentrations at high latitudes with maxima of the surface density located near the poles. Such concentrations of polar faculae (below referred to as `polar condensations') are formed after a lapse of 1–2 years from the polar magnetic field reversals, and then they persist for 7–9 years, until the high-latitude magnetic fields again start to reverse. During several years after the sunspot minima, the polar condensations co-exist with the new latitudinal belts of polar faculae which appear at middle latitudes and then migrate toward the poles. To describe the evolution of the polar condensations quantitatively, the polar faculae density n at latitudes above 60° has been approximated by means of the power law nn ₀ cos^m where is polar angle. The parameters n ₀ and m both are found to vary during the course of the solar cycle, reaching maximum values near or shortly after the minimum of sunspot activity. At the minimum phase of the solar cycle, on average, the surface density of polar faculae varies as cos¹⁴. In addition to the 11-yr variation, the latitude–time distribution of polar faculae exhibits short-term variations occurring on the time scale of 2–3 years.     

On mass flow in a complex sunspot

The velocity field in a large complex sunspot is investigated in Fe i 6302.5 Å and in H with a spatial resolution of about 2.5. The Evershed flow is almost parallel to the solar surface. For the inclination angle between the velocity and the horizontal = 4.4°±1.3° is estimated; = 11° is the definite upper limit.     

Some comments on the limb shift of solar lines

We searched for a variation with heliographic latitude of the solar limb effect by comparing the relative wavelengths of weak and strong Fraunhofer lines. The blue shifts associated with the limb effect appear 9%±5% larger in the polar radius vector than in an equatorial radius vector at cos = 0.5. This should perhaps be interpreted as an increase with latitude of either solar convection or of convective overshoot. Recent observations of poleward meridional flows of 30m s^–1 should be corrected for this limb effect variation. This correction increases this flow velocity to 70 m s^–1. A search for a similar variation in plages and in network boundaries had negative results, the variation being +1%±5% and -1%±6% respectively.Now at the Multiple Mirror Telescope Observatory, University of Arizona, Tucson, Ariz. 85721, U.S.A. The MMTO is jointly operated by the University of Arizona and the Smithsonian Institution.The Sacramento Peak Observatory is operated by the Association of Universities for Research in Astronomy, Inc. under contract AST 78-17292 with the National Science Foundation.     

Study of the nonradial directional property of the rays of the streamer belt and chains in the solar corona

Based on analyzing corona images taken by the LASCO C1, C2, and C3 instruments, a study is made of the behavior of the streamer belt spanning one half of the 1996–2001 cycle of solar activity, from minimum to maximum activity, in the absence of coronal mass ejections. It is shown that: (1) The position of the streamer belt relative to the solar equator is generally characterized by two angles: _o and _E, where _o is the latitudinal position (near the solar surface) of the middle of the base of the helmet, the top of which gradually transforms to a ray of the streamer belt with a further distance from the Sun, and _E is the latitude of this ray for R>5–6 R from the Sun's center where the ray becomes radial. (2) Only rays lying at some of the selected latitudes _o retain their radial orientation (_oE) throughout their extent. Namely: _o0° (equator), _o±90° (north and south poles), and the angle _o lying in the range ±(65°–75°) in the N- and S-hemispheres. (3) A deviation of rays from their radial orientation in the direction normal to the surface of the streamer belt occurs: for latitudes _o<|65°–75°| toward the equator (>0°) reaching a maximum in the N and S hemispheres, respectively, when _OM40°, and _OM–42° for latitudes _o>|65°–75°| toward the pole (<0°). The regularities obtained here are a numerical test which can be used to assess of the validity of the theory for describing the behavior of the Sun's quasi-stationary corona over a cycle of solar activity.     

Continuum photometry of solar white-light faculae

We have determined absolute continuum intensities and brightness temperatures of individual facular grains at a spatial resolution limited by the =50 cm aperture of the SVST on La Palma. A facular region at 57° was observed simultaneously in three narrow continuum windows at 450.5, 658.7, and 863.5 nm. We corrected for image degradation by the Earth's atmosphere using the speckle masking method. The brightness temperatures do not exactly follow the Planck law. The differences of T _blue–T _red=220 K and T _ir–T _red=–42 K reflect the wavelength dependence of the continuum formation depth. The (red) temperatures of 250 facular grains show excesses between 250 and 450 K above their undisturbed neighborhood. The wavelength dependence of the relative intensity ratios C= [I ^fac/I ^phot] show a large scatter around mean values of C _blue/C _red=1.075 and C _ir/C _red=0.98. We determined the center-to-limb variation of the 863.5 nm continuum contrast for 0.17>cos>0.39 by measuring 270 grains in reconstructed facular images. The upper envelope of the data points increases linearly to 1.5 at cos=0.17. Application of the mean color dependence yields green contrasts up to C ₅₅₀=1.7, which is far higher than previously observed values. The behaviour for cos>0.17 is estimated from (unreconstructed) frame-selected best images taken over a time interval of 7 hours. Six distinct facular regions clearly discernible during the whole time interval indicate a slight contrast decrease towards the extreme limb. The observed quantities are useful for an adjustment of model calculations and for a discrimination of competing models.     

Meridional motions of magnetic features in the solar photosphere

We cross-correlate pairs of Mt. Wilson magnetograms spaced at intervals of 24–38 days to investigate the meridional motions of small magnetic features in the photosphere. Our study spans the 26-yr period July 1967–August 1993, and the correlations determine longitude averages of these motions, as functions of latitude and time. The time-average of our results over the entire 26-yr period is, as expected, antisymmetric about the equator. It is poleward between 10° and 60°, with a maximum rate of 13 m s^–1, but for latitudes below ±10° it is markedly equatorward, and it is weakly equatorward for latitudes above 60°. A running 1-yr average shows that this complex latitude dependence of the long-term time average comes from a pattern of motions that changes dramatically during the course of the activity cycle. At low latitudes the motion is equatorward during the active phase of the cycle. It tends to increase as the zones of activity move toward the equator, but it reverses briefly to become poleward at solar minimum. On the poleward sides of the activity zones the motion is most strongly poleward when the activity is greatest. At high latitudes, where the results are more uncertain, the motion seems to be equatorward except around the times of polar field reversal. The difference-from-average meridional motions pattern is remarkably similar to the pattern of the magnetic rotation torsional oscillations. The correspondence is such that the zones in which the difference-from-average motion is poleward are the zones where the magnetic rotation is slower than average, and the zones in which it is equatorward are the zones where the rotation is faster.Our results suggest the following characterization: there is a constant and generally prevailing motion which is perhaps everywhere poleward and varies smoothly with latitude. On this is superimposed a cycle-dependent pattern of similar amplitude in which the meridional motions of the small magnetic features are directed away from regions of magnetic flux concentration. This is suggestive of simple diffusion, and of the models of Leighton (1964) and Sheeley, Nash, and Wang (1987). The correspondence between the meridional motions pattern and the torsional oscillations pattern in the magnetic rotation suggests that the latter may be an artifact of the combination of meridional motion and differential rotation.     

High-Latitude Solar Rotation Traced by Magnetic Elements

Sequential observations at Huairou Solar Observation Station, China, and Kitt Peak, U.S.A., show that polar magnetic elements can live from several to more than 58 hours. This enables measurement of the solar rotation rate near the polar region by tracing magnetic element motions. With observations carried out on 8–15 July 1997, we identify and trace more than 1300 elements at north heliographic latitudes between 55°–85° using two methods, and fit the mean sidereal rotation rate as =14.0±0.54–(2.24±1.22)sin²–(1.78±0.79)sin⁴ deg per day.     

General-relativistic hierarchical cosmology: An exact model

An exact solution of Einstein's equation is stated in which the density (), pressure (p), scale factorS and metric coefficients are functions of only one dimensionless self-similar variable,ct/R, wheret is cosmic time andR is a co-moving radial coordinate. The solution represents a cosmology that begins as a static sphere having R ^–2 and evolves into an expanding model which is pressure-free and has a hierarchical type of density law ( R ^–, approximately, with =a number, 02). It is suggested that this model should supersede the previous models of Wesson and other workers, since it appears to be the simplest cosmology for a hierarchy.     

On symmetrical planetary corotations

Exact corotations are equilibrium points in the phase space of the asteroidal elliptic restricted problem of three bodies averaged over the synodic period, at a mean-motions resonance. If the resonant critical angle is =(p+q) _jup –p–q, exact corotations are double resonant motions defined by the conditionsd/dt=0 andd(– _jup )/dt=0. The first condition is characteristic of the periods resonance(p + q) : p and the second one is a secular resonance equivalent to that usually known as thev ₅-resonance. This paper presents the symmetric solutions =0 (mod ), = _jup (mod ). Corotations have a coherence property which is unique in non-collisional Celestial Mechanics: An elementary calculation shows that, in the neighbourhood of these solutions, the motions cluster aroundp independent longitude values and are, in each cluster, as close together as and are close to the equilibrium values.     

The Energy Equation in the Lower Solar Convection Zone

As a consequence of the Taylor–Proudman balance, a balance between the pressure, Coriolis and buoyancy forces in the radial and latitudinal momentum equations (that is expected to be amply satisfied in the lower solar convection zone), the superadiabatic gradient is determined by the rotation law and by an unspecified function of r, say, S(r), where r is the radial coordinate. If the rotation law and S(r) are known, then the solution of the energy equation, performed in this paper in the framework of the ML formalism, leads to a knowledge of the Reynolds stresses, convective fluxes, and meridional motions. The ML-formalism is an extension of the mixing length theory to rotating convection zones, and the calculations also involve the azimuthal momentum equation, from which an expression for the meridional motions in terms of the Reynolds stresses can be derived. The meridional motions are expanded as U _r(r,)=P ₂(cos)₂(r)/r ²+P ₄(cos)₄(r)/r ² +..., and a corresponding equation for U (r,). Here is the polar angle, is the density, and P ₂(cos), P ₄(cos) are Legendre polynomials. A good approximation to the meridional motion is obtained by setting ₄(r)=–H₂(r) with H–1.6, a constant. The value of ₂(r) is negative, i.e., the P ₂ flow rises at the equator and sinks at the poles. For the value of H obtained in the numerical calculations, the meridional motions have a narrow countercell at the poles, and the convective flux has a relative maximum at the poles, a minimum at mid latitudes and a larger maximum at the equator. Both results are in agreement with the observations.     

Subsurface Meridional Flow from HMI Using the Ring-Diagram Pipeline

We have determined the meridional flows in subsurface layers for 18 Carrington rotations (CR 2097 to 2114) analyzing high-resolution Dopplergrams obtained with the Helioseismic and Magnetic Imager (HMI) instrument onboard the Solar Dynamics Observatory (SDO). We are especially interested in flows at high latitudes up to 75^° in order to address the question whether the meridional flow remains poleward or reverses direction (so-called counter cells). The flows have been determined in depth from near-surface layers to about 16 Mm using the HMI ring-diagram pipeline. The measured meridional flows show systematic effects, such as a variation with the B ₀-angle and a variation with central meridian distance (CMD). These variations have been taken into account to lead to more reliable flow estimates at high latitudes. The corrected average meridional flow is poleward at most depths and latitudes with a maximum amplitude of about $20~\mathrm{m\,s}^{-1}$ near 37.5^° latitude. The flows are more poleward on the equatorward side of the mean latitude of magnetic activity at 22^° and less poleward on the poleward side, which can be interpreted as convergent flows near the mean latitude of activity. The corrected meridional flow is poleward at all depths within ±?67.5^° latitude. The corrected flow is equatorward only at 75^° latitude in the southern hemisphere at depths between about 4 and 8 Mm and at 75^° latitude in the northern hemisphere only when the B ₀ angle is barely large enough to measure flows at this latitude. These counter cells are most likely the remains of an insufficiently corrected B ₀-angle variation and not of solar origin. Flow measurements and B ₀-angle corrections are difficult at the highest latitude because these flows are only determined during limited periods when the B ₀ angle is sufficiently large.     

A model of photospheric faculae deduced from white light high resolution pictures

High resolution pictures (about 0.3) of photospheric faculae near the solar limb have been obtained with the Pic du Midi 50 cm refractor; their granular structure then clearly appears. The microphotometric study of these facular granules shows that the ratio of their intensity to the photospheric intensity, I ^f/I ^ph (cos) reaches a maximum near cos = 0.3 and then decreases towards the limb. The values of this ratio have been corrected with a most likely spread function. Then a temperature model of a facular granule is obtained: with respect to the neighbouring photosphere, this granule appears as a photospheric hot cloud which does not extend high in the solar atmosphere (thickness 100 km above ₅₀₀₀ = 1). The temperature excess is 750K at maximum. This hot region is located over a layer which is cooler than the normal photosphere at the same level. Another hot region might extend above the photospheric hot cloud, possibly up to the chromosphere. This photospheric facula model which is confined to the lower photosphere seems to indicate that this phenomenon is different from the photospheric network which is visible up to the lower chromosphere.     

On the influence of gradients in the angular velocity on the solar meridional motions

If fluctuations in the density are neglected, the large-scale, axisymmetric azimuthal momentum equation for the solar convection zone (SCZ) contains only the velocity correlations and where u are the turbulent convective velocities and the brackets denote a large-scale average. The angular velocity, , and meridional motions are expanded in Legendre polynomials and in these expansions only the two leading terms are retained (for example, where is the polar angle). Per hemisphere, the meridional circulation is, in consequence, the superposition of two flows, characterized by one, and two cells in latitude respectively. Two equations can be derived from the azimuthal momentum equation. The first one expresses the conservation of angular momentum and essentially determines the stream function of the one-cell flow in terms of : the convective motions feed angular momentum to the inner regions of the SCZ and in the steady state a meridional flow must be present to remove this angular momentum. The second equation contains also the integral indicative of a transport of angular momentum towards the equator.With the help of a formalism developed earlier we evaluate, for solid body rotation, the velocity correlations and for several values of an arbitrary parameter, D, left unspecified by the theory. The most striking result of these calculations is the increase of with D. Next we calculate the turbulent viscosity coefficients defined by whereC _ro ⁰ and C _o ⁰ are the velocity correlations for solid body rotation. In these calculations it was assumed that ₂ was a linear function of r. The arbitrary parameter D was chosen so that the meridional flow vanishes at the surface for the rotation laws specified below. The coefficients v _ro ⁱ and v _0o ⁱ that allow for the calculation of C _ro and C _0o for any specified rotation law (with the proviso that ₂ be linear) are the turbulent viscosity coefficients. These coefficients comply well with intuitive expectations: v _ro ¹ and –v _0o ³ are the largest in each group, and v _0o ³ is negative.The equations for the meridional flow were first solved with ₀ and ₂ two linear functions of r ( ₀ ¹ = – 2 × 10 ^–12 cm ^–1) and ( ₂ ¹ = – 6 × 10 ¹² cm ^–1). The corresponding angular velocity increases slightly inwards at the poles and decreases at the equator in broad agreement with heliosismic observations. The computed meridional motions are far too large ( 150m s^–1). Reasonable values for the meridional motions can only be obtained if _o (and in consequence ), increase sharply with depth below the surface. The calculated meridional motion at the surface consists of a weak equatorward flow for gq < 29° and of a stronger poleward flow for > 29°.In the Sun, the Taylor-Proudman balance (the Coriolis force is balanced by the pressure gradient), must be altered to include the buoyancy force. The consequences of this modification are far reaching: is not required, now, to be constant along cylinders. Instead, the latitudinal dependence of the superadiabatic gradient is determined by the rotation law. For the above rotation laws, the corresponding latitudinal variations of the convective flux are of the order of 7% in the lower SCZ.     

The peculiarities of rotation of the solar polar regions

The sidereal rotation rate of the high-latitude solar regions is examined using long-lived photospheric polar faculae. The observations were carried out with the photoheliograph of Kislovodsk Mountain Station of the Pulkovo Observatory from 1982 to 1986. The following facts have been established: (a) There is a differential rotation of the polar faculae close to the maximum of solar activity, while the amount of latitude gradient of solar rotation decreases towards the sunspot minimum; (b) small differences of rotation in the northern and southern hemispheres of the Sun are observed; (c) some deviations of differential rotation curves constructed for each Carrington rotation from the mean curve of differential rotation are revealed. The total amplitude of the maximum positive and negative excesses is about 40–50 m s^–1. The positive surplus velocities of solar rotation (the amplitude of which is about 20–25 m s^–1) move in the form of a wave from heliographic latitudes 40° with a velocity of 1.6 m s^–1. The latitude width of this flow is B 15°. This wave of abnormally high velocity starts in the year of minimum solar activity and reaches the pole 11 years later. The picture is symmetrical relative to the equator.