The Photospheric Convection Spectrum

Spectra of the cellular photospheric flows are determined from observations acquired by the MDI instrument on the SOHO spacecraft. Spherical harmonic spectra are obtained from the full-disk observations. Fourier spectra are obtained from the high-resolution observations. The p-mode oscillation signal and instrumental artifacts are reduced by temporal filtering of the Doppler data. The resulting spectra give power (kinetic energy) per wave number for effective spherical harmonic degrees from 1 to over 3000. Significant power is found at all wavenumbers, including the small wavenumbers representative of giant cells. The time evolution of the spectral coefficients indicates that these small wavenumber components rotate at the solar rotation rate and thus represent a component of the photospheric cellular flows. The spectra show distinct peaks representing granules and supergranules but no distinct features at wavenumbers representative of mesogranules or giant cells. The observed cellular patterns and spectra are well represented by a model that includes two distinct modes – granules and supergranules.     

Spherical harmonic analysis of steady photospheric flows,II

The use of the spherical harmonic functions to analyse the nearly steady flows in the solar photosphere is extended to situations in which B ₀, the latitude at disk center, is nonzero and spurious velocities are present. The procedures for extracting the rotation profile and meridional circulation are altered to account for the seasonal tilt of the Sun's rotation axis toward and away from the observer. A more robust and accurate method for separating the limb shift and meridional circulation signals is described. The analysis procedures include the ability to mask out areas containing spurious velocities (velocity-like signals that do not represent true flow velocities in the photosphere). The procedures are shown to work well in extracting the various flow components from realistic artificial data with a broad, continuous spectrum for the supergranulation. The presence of this supergranulation signal introduces errors of a few m s ^-1 in the measurements of the rotation profile, meridional circulation, and limb shift from a single Doppler image. While averaging the results of 24 hourly measurements has little effect in reducing these errors, an average of 27 daily measurements reduces the errors to well under 1 m s ^-1.     

Simulating photospheric Doppler velocity fields

A method is described for constructing artificial data that realistically simulate photospheric velocity fields. The velocity fields include rotation, differential rotation, meridional circulation, giant cell convection, supergranulation, convective limb shift, p-mode oscillations, and observer motion. Data constructed by this method can be used for testing algorithms designed to extract and analyze these velocity fields in real Doppler velocity data.     

Time-Distance Helioseismology with f Modes as a Method for Measurement of Near-Surface Flows

Travel times measured for the f mode have been used to study flows near the solar surface in conjunction with simultaneous measurements of the magnetic field. Previous flow measurements of Doppler surface rotation, small magnetic feature rotation, supergranular pattern rotation, and surface meridional circulation have been confirmed. In addition, the flow in supergranules due to Coriolis forces has been measured. The spatial and temporal power spectra for a six-day observing sequence have been measured.     

Nonlinear simulations of solar rotation effects in supergranules

Nonlinear calculations for the three-dimensional and time dependent convective flow in a plane parallel layer of fluid are carried out with parameter values appropriate for supergranules on the Sun. A rotation vector is used which is tilted from the vertical to represent various latitudes. For the incompressible fluid used in this model the solar rotation produces turning motions sufficient to completely twist a fluid column in about one day. It is suggested that this effect will be greatly enhanced in a compressible fluid. The tilted rotation vector produces anisotropies and systematic Reynolds stresses which drive mean flows. The resulting flows produce a rotation rate which increases inward and a meridional circulation with poleward flow along the outer surface.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

On axisymmetric hydromagnetic equilibria

The physical characteristics of possible axisymmetric equilibria are examined on the basis of the integrals of hydromagnetic equations. It is shown for nearly spherical configurations that a surface differential rotation is possible only in the absence of a meridional circulation with either purely toroidal or purely poloidal magnetic field. In the presence of a meridional circulation, it is shown that no surface rotation or constant rotation is possible if the magnetic field is purely toroidal, and that no rotation is possible if the magnetic field is purely poloidal. A brief discussion is given on the possible solutions including the case of stellar winds with force-free magnetic fields.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Peering over the Sun's pole: behavior of its rotational vortex

A new method for measuring spectroscopically the rotation at the Sun's poles is described. Using solar CO lines at 4.666 µm, infrared spectra are recorded at a fixed limb distance of 4.8 arc sec while progressing along an arc ±5.7 deg from the Sun's rotational pole. Since the poles dip twice a year to about 7 arc sec from the limb, our observations can range either side of and through the vortex axis. Advantages to this technique are: (1) a low disturbing signal from supergranules owing to their superposition at the limb, (2) no ‘limb shift’ error since limb distance is constant and the CO lines have no known limb shift, (3) emphasis is on the quiet Sun since the CO molecule is confined there, (4) negligible scattered light in the IR (<1%), and (5) the improved seeing afforded by the IR. Although any definitive determination of solar rotation requires observations over an extended time span, our preliminary results suggest two features peculiar to the extreme pole: (1) the occasional apparent cessation of rotation, (2) some sort of singularity, again occasional, producing a sharp velocity signal (a vortex?) within 1 deg of the pole.     

Mass loss from the inner regions of accretion discs due to centrifugally driven magnetic wind flows

Wind flows and collimated jets are believed to be a feature of a range of disc accreting systems. These include active galactic nuclei, T Tauri stars, X-ray binaries and cataclysmic variables. The observed collimation implies large-scale magnetic fields and it is known that dipole-symmetry fields of sufficient strength can channel wind flows emanating from the surfaces of a disc. The disc inflow leads to the bending of the poloidal magnetic field lines, and centrifugally driven magnetic winds can be launched when the bending exceeds a critical value. Such winds can result in angular momentum transport at least as effective as turbulent viscosity, and hence they can play a major part in driving the disc inflow.
It is shown here that if the standard boundary condition of vanishing viscous stress close to the stellar surface is applied, together with the standard connection between viscosity and magnetic diffusivity, then poloidal magnetic field bending increases as the star is approached with a corresponding increase in the wind mass loss rate. A significant amount of material can be lost from the system via the enhanced wind from a narrow region close to the stellar surface. This occurs for a Keplerian angular velocity distribution and for a modified form of angular velocity, which allows for matching of the disc and stellar rotation rates through a boundary layer above the stellar surface. The enhanced mass loss is significantly affected by the behaviour of the disc angular velocity as the stellar surface is approached, and hence by the stellar rotation rate. Such a mechanism may be related to the production of jets from the inner regions of disc accreting systems.     

Supergranulation rotation

Simple convection models estimate the depth of supergranulation at approximately 7500 km which suggests that supergranules would rotate at the rate of the plasma in the outer 1% of the solar radius. The supergranulation rotation obtained from MDI dopplergrams shows that supergranules rotate faster than the outer 5% of the convection zone and show zonal flows matching results from inversions of f-mode splittings. Additionally, the rotation rate depends on the size scale of the features.     

Temporal filters for isolating steady photospheric flows

A variety of temporal filters are tested on artificial data with 60 and 75 s sampling intervals to determine their accuracy in separating the nearly-steady photospheric flows from the p-mode oscillations in Doppler velocity data. Longer temporal averages are better at reducing the residual signal due to p-modes but they introduce additional errors from the rotation of the supergranule pattern across the solar disk. Unweighted filters (boxcar averages) leave residual r.m.s. errors of about 6 m s^–1 from the p-modes after 60 min of averaging. Weighted filters, with nearly Gaussian shapes, leave similar residual errors after only 20 min of averaging and introduce smaller errors from the rotation of the supergranule pattern. The best filters found are weighted filters that use data separated by 150 or 120 s so that the p-modes are sampled at opposite phases. These filters achieve an optimum error level after about 20 min, with the r.m.s. errors due to the p-mode oscillations and the rotation of the supergranules both at a level of only 1.5 m s^–1.     

Solar-Cycle-Related Variations in the Solar Differential Rotation and Meridional Flow: A Comparison

We have analysed a large set of sunspot group data (1874 – 2004) and find that the meridional flow strongly varies with the phase of the solar cycle, and the variation is quite different in the northern and the southern hemispheres. We also find the existence of considerable cycle-to-cycle variation in the meridional velocity, and about a 11-year difference between the phases of the corresponding variations in the northern and the southern hemispheres. In addition, our analysis also indicates the following: (i) the existence of a considerable difference (about 180^°) between the phases of the solar-cycle variations in the latitude-gradient terms of the northern and the southern hemispheres’ rotations; (ii) the existence of correlation (good in the northern hemisphere and weak in the southern hemisphere) between the mean solar-cycle variations of meridional flow and the latitude-gradient term of solar rotation; (iii) in the northern hemisphere, the cycle-to-cycle variation of the mean meridional velocity leads that of the equatorial rotation rate by about 11 years, and the corresponding variations have approximately the same phase in the southern hemisphere; and (iv) the directions of the mean meridional velocity is largely toward the pole in the longer sunspot cycles and largely toward the equator in the shorter cycles.     

Solar rotation measurements at Mount Wilson

This paper describes a thorough reevaluation of the procedures for reducing the data acquired at the Mt. Wilson Observatory synoptic program of solar observations at the 150-foot tower. We also describe a new program of acquiring as many scans per day as possible of the solar magnetic and velocity fields. We give a new fitting formula which removes the background velocity field from each scan. An important new feature of our reduction algorithm is our treatment of the limb shift which permits time variation in this function. We identify the difference between the limb shift along the north-south axis and the east-west axis as potentially being a result of meridional circulation. Our analysis interprets the time variation in the east-west limb shift as being the result of changes in a vertical component of the meridional circulation.The performance of the system improved in 1982 as a result of the installation of a new exit slit assembly. The amplitude of the limb shift variations found prior to 1982 is larger than is easily explained with simple ideas of meridional circulation. However, we have not been able to firmly identify instrumental causes for the variations although small changes in the band-pass of the exit slit assembly could have contributed.We have established a correlation between the observed stray light in the system and a component of the velocity field which is antisymmetric with respect to the solar central meridian. We remove this stray light effect by adding an additional term to the fitting function.Finally, we show that the inclusion of the above improvements allows us to study the torsional oscillations at high latitude using a procedure which can retain the longitude dependent information about the velocity pattern.     

Differential rotation of giant stars

From a set of high-resolution spectral observations of late type giant stars we used Doppler imaging to derive time-series temperature maps of the stellar surfaces. Using these temperature maps, it is possible to track the temporal changes of the spot features and derive estimates of the strength and sign of the differential surface rotation of these stars. Looking into the latitudinal changes of the surface maps, it is also possible to derive meridional flows on these stars. But due to the lower accuracy of the latitudes of the reconstructed spot features, the data requirements are higher than for the detection of differential rotation. Nevertheless, a correlation between the differential rotation and meridional flow estimates is suggested. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Subsurface Velocity of Emerging and Decaying Active Regions

We study the temporal variation of subsurface flows of 828 active regions and 977 quiet regions. The horizontal flows cover a range of depths from the surface to about 16 Mm and are determined by analyzing Global Oscillation Network Group high-resolution Doppler data with ring-diagram analyses. The vertical velocity component is derived from the divergence of the measured horizontal flows using mass conservation. For comparison, we analyze Michelson Doppler Imager (MDI) Dynamics Run data covering 68 active regions common to both data sets. We determine the change in unsigned magnetic flux during the disk passage of each active region using MDI magnetograms binned to the ring-diagram grid. We then sort the data by their flux change from decaying to emerging flux and divide the data into five subsets of equal size. We find that emerging flux has a faster rotation than the ambient fluid and pushes it up, as indicated by enhanced vertical velocity and faster-than-average zonal flow. After active regions are formed, downflows are established within two days of emergence in shallow layers between about 4 and 10 Mm. Emerging flux in existing active regions shows a similar scenario, where the upflows at depths greater than about 10 Mm are enhanced and the already established downflows at shallower depths are weakened. When active regions decay, the corresponding flow pattern disappears as well; the zonal flow slows down to values comparable to that of quiet regions and the upflows become weaker at deeper layers. The residual meridional velocity is mainly poleward and shows no obvious variation. The magnitude of the residual velocity, defined as the sum of the squares of the residual velocity components, increases with increasing magnetic flux and decreases with decreasing flux.     

Power spectral analysis of Jupiter’s clouds and kinetic energy from Cassini

We present suggestive evidence for an inverse energy cascade within Jupiter’s atmosphere through a calculation of the power spectrum of its kinetic energy and its cloud patterns. Using Cassini observations, we composed full-longitudinal mosaics of Jupiter’s atmosphere at several wavelengths. We also utilized image pairs derived from these observations to generate full-longitudinal maps of wind vectors and atmospheric kinetic energy within Jupiter’s troposphere. We computed power spectra of the image mosaics and kinetic energy maps using spherical harmonic analysis. Power spectra of Jupiter’s cloud patterns imaged at certain wavelengths resemble theoretical spectra of two-dimensional turbulence, with power-law slopes near −5/3 and −3 at low and high wavenumbers, respectively. The slopes of the kinetic energy power spectrum are also near −5/3 at low wavenumbers. At high wavenumbers, however, the spectral slopes are relatively flatter than the theoretical prediction of −3. In addition, the image mosaic and kinetic energy power spectra differ with respect to the location of the transition in slopes. The transition in slope is near planetary wavenumber 70 for the kinetic energy spectra, but is typically above 200 for the image mosaic spectra. Our results also show the importance of calculating spectral slopes from full 2D velocity maps rather than 1D zonal mean velocity profiles, since at large wavenumbers the spectra differ significantly, though at low wavenumbers, the 1D zonal and full 2D kinetic energy spectra are practically indistinguishable. Furthermore, the difference between the image and kinetic energy spectra suggests some caution in the interpretation of power spectrum results solely from image mosaics and its significance for the underlying dynamics. Finally, we also report prominent variations in kinetic energy within the equatorial jet stream that appear to be associated with the 5 μm hotspots. Other eddies are present within the flow collar of the Great Red Spot, suggesting caution when interpreting snapshots of the flow inside these features as representative of a time-averaged state.     

Meridional Motion and Reynolds Stress from Debrecen Photoheliographic Data

The Debrecen Photoheliographic Data catalogue is a continuation of the Greenwich Photoheliographic Results providing daily positions of sunspots and sunspot groups. We analyse the data for sunspot groups focussing on meridional motions and transfer of angular momentum towards the solar equator. Velocities are calculated with a daily shift method including an automatic iterative process of removing the outliers. Apart from the standard differential rotation profile, we find meridional motion directed towards the zone of solar activity. The difference in measured meridional flow in comparison to Doppler measurements and some other tracer measurements is interpreted as a consequence of different flow patterns inside and outside of active regions. We also find a statistically significant dependence of meridional motion on rotation velocity residuals confirming the transfer of angular momentum towards the equator. Analysis of horizontal Reynolds stress reveals that the transfer of angular momentum is stronger with increasing latitude up to about \(40^{\circ}\), where there is a possible maximum in absolute value.     

A large-eddy simulation of turbulent compressible convection: differential rotation in the solar convection zone

We present the results of two simulations of the convection zone, obtained by solving the full hydrodynamic equations in a section of a spherical shell. The first simulation has cylindrical rotation contours (parallel to the rotation axis) and a strong meridional circulation, which traverses the entire depth. The second simulation has isorotation contours about mid-way between cylinders and cones, and a weak meridional circulation, concentrated in the uppermost part of the shell.
We show that the solar differential rotation is directly related to a latitudinal entropy gradient, which pervades into the deep layers of the convection zone. We also offer an explanation of the angular velocity shear found at low latitudes near the top. A non-zero correlation between radial and zonal velocity fluctuations produces a significant Reynolds stress in that region. This constitutes a net transport of angular momentum inwards, which causes a slight modification of the overall structure of the differential rotation near the top. In essence, the thermodynamics controls the dynamics through the Taylor–Proudman momentum balance . The Reynolds stresses only become significant in the surface layers, where they generate a weak meridional circulation and an angular velocity 'bump'.     

Waves and instabilities in a differentially rotating disc containing a poloidal magnetic field

The theory of waves and instabilities in a differentially rotating disc containing a poloidal magnetic field is developed within the framework of ideal magnetohydrodynamics. A continuous spectrum, for which the eigenfunctions are localized on individual magnetic surfaces, is identified but is found not to contain any instabilities associated with differential rotation. The normal modes of a weakly magnetized thin disc are studied by extending the asymptotic methods used previously to describe the equilibria. Waves propagate radially in the disc according to a dispersion relation which is determined by solving an eigenvalue problem at each radius. The dispersion relation for a hydrodynamic disc is re-examined and the modes are classified according to their behaviour in the limit of large wavenumber. The addition of a magnetic field introduces new, potentially unstable, modes and also breaks up the dispersion diagram by causing avoided crossings. The stability boundary to the magnetorotational instability in the parameter space of polytropic equilibria is located by solving directly for marginally stable equilibria. For a given vertical magnetic field in the disc, bending of the field lines has a stabilizing effect and it is shown that stable equilibria exist which are capable of launching a predominantly centrifugally driven wind.     

P – ω Dynamo in Spherical Geometry

Based on the fundamental P – ω dynamo equation, using spherical polar coordinates, we carry out a study of turbulent plasma wave dynamo effect. For various rotation laws, different analytical solutions are derived. In the cases of no rotation and rigid rotation, the dynamo generates poloidal field only, while with differential rotation, regardless the differential rotation is radial or latitudinal, poloidal and toroidal fields are all generated. We may think that the solutions are the analytical forms of the magnetic field in a turbulent source region of celestial bodies. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.