Solar Interior and Variability (SIV)

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Radiative Processes and Small-scale MHD in the Convection Zone and Photosphere

Synthesis of radiative outputs

-Synthetic Models of the Solar Spectrum- P. Fox (HAO), with O. R. White (HAO), J. Fontenla, J. Harder (both University of Colorado, Laboratory for Atmospheric and Space Physics [LASP]), E. Avrett and R. Kurucz (both Harvard Smithsonian Center for Astrophysics) continue to develop comprehensive, semi-empirical, synthetic models of the solar spectrum. Fox worked with Thuillier (CNRS, France) to make detailed comparisons between the SOLSPEC solar spectrum observed on shuttle flights and the SunRISE synthesis for solar activity levels early in October, 1992. Extremely good agreement was found over the wavelength range 900 nm to 2.4 microns, especially for the slope of the spectrum. Fox investigated the natural redundancy and coherency in the solar spectrum, identifying wavelength regions where both continuum and line variations due to solar activity have similar magnitude and sign response, and where they do not. Fox presented the latest status on the compliance of the SunRISE spectral synthesis to the ISO-DIS 21348 (for solar irradiances) which will be published in Advances in Space Research as part of the standards compliance process. Fontenla, with colleagues at LASP and HAO, made comparisons between spectra from the Spectral Irradiance Monitor (SIM) instrument on board the Solar Radiation and Climate Experiment (SORCE) spacecraft and the synthesis calculations, initially between 300 nm and 1 micron (Fontenla et al. 2004)

-Solar Irradiance Variability During Cycle 23- Giuliana de Toma and collaborators have conducted a detailed analysis of the variation in the Sun's radiative output during solar cycle 23. At the onset of solar cycle 23, the Total Solar Irradiance (TSI) appeared to be increasing faster than expected relative to estimates derived from San Fernando Observatory (SFO) solar image analysis (de Toma et al., 2001). This early observation forced comparison with similar results from cycle 22. Careful study of the TSI record by the Solar and Heliospheric Observatory (SOHO)/VIRGO team removed this discrepancy with ground-based estimates and gives a solid basis for study of cycle 23.

As cycle 23 evolved from the minimum of sunspot activity in 1996 to the maximum in 2000-2002, it was obvious that cycle 23 was quite different from its immediate predecessors, with significantly lower magnetic activity. Cycle 23 not only had fewer sunspots, but they were on average smaller and had weaker magnetic flux than in cycle 22 (Livingston, 2002). Average sunspot number, sunspot area, and facular area decreased by approximately 33%, 37%, and 45%, respectively, between the maxima of cycles 22 and 23. In contrast to this decrease in magnetic activity, observations showed that the strength of the TSI cycle did not change significantly in cycle 23. TSI values during the maximum phase of cycle 23 are comparable to those measured in the more active cycles 22 and 21. This is because TSI is more sensitive to the balance between dark sunspots and bright faculae/plages than to their individual values. It also indicates that reconstructions of TSI back in time, when only the sunspot record was available, are likely to have large uncertainties.

de Toma and co-workers extended (de Toma et al., 2004) their empirical models to the maximum phase of cycle 23 using the new measurements of TSI and activity indices. An important finding was the consistency between the TSI record and their regression fits, due to the improvement in both TSI and the activity indices time series in the last two years. Analysis of these new observational data resolved the questions about differences in TSI measurements and empirical estimates between solar cycles 22 and 23 raised in our earlier study. de Toma and co-workers now find that the TSI increase from solar minimum to maximum in cycle 23 agrees with estimates from irradiance indices. They are able to fit the TSI record from 1986 to the present to rms accuracy of 130 ppm, which is comparable to that reported for precision of the TSI measurements, provided indices containing all information from the solar disk are used. The best fits come from the new full-disk indices developed at SFO and the Mg II 280 nm index. They also find that a quasi-periodic TSI variation with a period very close to 1 year occurs between 2000 and 2003. It cannot be accounted for by errors in orbital determinations for either the SOHO or ACRIMSat satellites. This periodicity also remains with a lower amplitude in the residual between TSI and the best surrogate. Although this annual variation is then not completely accounted for in the analysis, de Toma and co-workers suggest that it may originate from the timing of sunspot emergence in the maximum of cycle 23 and the differences in lifetimes of dark and bright structures on the Sun.

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Flux Tube Models

-Simulations of Rising Flux Tubes with a New 3D Anelastic MHD Code- Yuhong Fan (HAO) has made significant progress in developing a code that solves the 3D anelastic MHD equations in a spherical shell. In this code, the equations are discretized spatially using a staggered finite-difference scheme, and are advanced in time with an explicit two-step predictor-corrector time stepping. An upwind, monotonicity-preserving interpolation scheme is used for evaluating the fluxes of all the advection terms. The constrained transport algorithm is used to guarantee (to machine round-off errors) that the magnetic field satisfies the divergence-free condition, and a method of characteristics that is upwind in the Alfven waves is used in evaluating both the v times B field in the induction equation and the Lorentz force in the momentum equation. The elliptic pressure equation is solved at every sub-timestep using a preconditioned conjugate-residual scheme with an implicit Richardson preconditioner (Skamarock, Smolarkiewicz, and Klemp 1996). Fan is now testing the code by performing 2D axisymmetric MHD simulations of the buoyant rise of twisted, toroidal magnetic flux rings in the solar convective envelope, and comparing the result with the previous studies by Choudhuri and Gilman (1987) based on a highly simplified thin flux tube model.

The three movies below show three different simulations where a weakly twisted toroidal flux ring, initially in thermal equilibrium with the surroundings (and hence buoyant), starts its ascent from the base of the solar convective envelope at 15° latitude. The three different simulations correspond to cases with the magnetic Rossby number R_b = v_A0 / 2ΩH_p (where v_A0 is the Alfven speed at the initial tube center, Ω is the angular speed of solar rotation, and H_p is the pressure scale height at the base of the convection zone) equal to, respectively, infinity (i.e. ignoring solar rotation), 2.26 (corresponding to an initial tube field strength of 105 G), and 3.85 (corresponding to an initial tube field strength of 1.7 times 105 G). When solar rotation is ignored, the magnetic flux tube simply rises radially (see the movie for Case 1, which shows the evolution of the tube cross-section in the meridional plane). Here, the flux tube is sufficiently twisted that most of the magnetic flux rises cohesively in the head of the tube, with some flux pulled into to wake behind. With the solar rotation included, the Coriolis force acting on the flux ring becomes important compared to the magnetic buoyancy force when the initial field strength B₀ of the toroidal flux ring goes below about 105 G. In Case 2, where B₀ = 105 G (see the movie for Case 2), the rising trajectory of the cohesive head of the tube cross-section is nearly radial during its rise in the lower half of the convection zone but is deflected to become nearly parallel to the rotation axis in the outer half of the solar convection zone. The tube emerges at 31° latitude, compared to its initial latitude of 15°. When the initial field strength is increased to 1.7 times 105 G (see the movie for Case 3), the rising trajectory for the cohesive head of the tube cross-section becomes radial with little poleward deflection. At this field strength, the Coriolis force remains small compared to the magnetic buoyancy of the flux ring during the entire rise through the solar convective envelope. More simulations are being carried out to explore the parameter space. After studying the axisymmetric rise of toroidal flux rings, Fan will then perform 3D simulations of the non- axisymmetric emergence of initially toroidal magnetic flux tubes through the solar convective envelope.

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Supergranulation

-The Spectrum of the Solar Supergranulation- Mark Rast (HAO), together with Jason Lisle and Juri Toomre (both of the Joint Institute for Laboratory Astrophysics [JILA], University of Colorado), demonstrated that the spectrum of the solar supergranulation is strongly influenced by the existence of multiple scales of motion within the solar photosphere, each of which exhibits a different rotation rate with respect to a stationary observer. Supergranules were shown to be asymmetrically distributed in space, exhibiting a weak north-south alignment. This large scale organization of the supergranular pattern is persistent in time, spanning many supergranular lifetimes, and rotates in the prograde direction at a rate that is faster than the supergranular pattern itself.

Figure a shows a horizontal flow divergence map of a 15x15 degree equatorial region of the Sun after tracking it at a rate 100 m/s faster than the Carrington rate and averaging 192 images over an eight day time series. The weak north-south alignment of the supergranular cells is apparent even to the eye. Figure b plots the y and x averages of the image in Fig. a plotted as a function of longitude (dark curves) and latitude (light curves), respectively. The temporally averaged divergence field shows variation with longitude that is noticeably greater than its variation with latitude. As seen in Figure c the ratio of the standard deviation in longitude to that in latitude (σ_x/σ_y) depends on the averaging time interval, with that ratio approaching three for long averaging times. It also depends on the rate at which the solar photosphere is tracked (Figure d) with the vertical alignment of the supergranular flow exhibiting a maximum when the region is tracked at a rate 110 m/s greater than the Carrington rate, about 20 m/s faster than the rate at which the equatorial supergranulation itself superrotates. These results are discussed in detail in Lisle et al, 2004.

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Global Hydrodynamics (HD) and Magnetohydrodynamics (MHD) of the Solar Interior

Differential Rotation, Meridional Circulation and Global Convection

Even though many details of the solar differential rotation are known from helioseismology to a large extent, theoretical approaches still have problems in explaining the observed pattern. Differential rotation has been primarily addressed using two approaches in the past: 3D full-sphere simulations of compressible convection, and axisymmetric mean-field models that parametrize processes (turbulent diffusivities, turbulent angular momentum transport) on the convective scale. While 3D simulations have been successful in predicting the correct amplitude of the differential rotation, the computed profile is still closer to the Taylor-Proudman state (with cylindrical differential rotation) than the observed pattern (contours of constant Ω show an inclination of about 25° to the axis of rotation). Axisymmetric mean-field models have recently been used to show that solar-like differential rotation is possible if the rotation-induced anisotropy of the convective energy flux is taken into account, an effect that leads to a temperature difference of a few degrees K between pole and equator (Küker & Stix 2001, and references therein). Matthias Rempel (HAO) has developed a mean-field model for differential rotation in order to address the specific question of whether a sub-adiabatic tachocline, in conjunction with turbulent heat conductivity within the convection zone and overshoot region, can break the Taylor-Proudman constraint requiring the differential rotation to be constant on cylinders in the case of isentropic stratification. Rempel found that the entropy perturbation generated in the sub-adiabatic tachocline is sufficient to explain the observed deviations from the Taylor-Proudman state, if the contribution of the convection zone is more or less neutral. If a non-adiabatic convection zone is considered, then roughly the lower 40-50% of the convection zone is required to be sub-adiabatic (due to non-local convection effects), otherwise additional effects like anisotropic heat transport are needed to explain the observed differential rotation. The accompnaying Figure shows a solution in which the convection zone is sub-adiabatic up to 0.8375, R_⊙. The observed 25° inclination of the Ω contours to the axis of rotation is reproduced very well. The meridional flow predicted by this approach shows a counter-clockwise flow (if the radial angular momentum flux is negative), which is observed through helioseismology in the upper-half of the convection zone. Such a flow is favorable for flux-transport dynamo models, where the equatorward meridional flow at the base of the convection zone ensures the equatorward propagation of magnetic activity throughout the solar cycle.

-Simulations of Global-Scale Solar Convection- High-resolution numerical simulations of turbulent solar convection which incorporate the full spherical geometry of the convective envelope are continually being improved and extended by Mark Miesch (HAO) and collaborators (principally A. S. Brun of Saclay, CEA, France, J. Toomre, M. Browning, and B. Brown, all of JILA and the University of Colorado, and N. N. Mansour, M. Rogers, and Y.-N. Young, all of the Center for Turbulence Research at NASA Ames Research Center and Stanford University). As Miesch and his co-workers achieve ever higher resolution and consequently more turbulent parameter regimes, new dynamics are emerging, and the flows are becoming increasingly dominated by isolated, intermittent downflow plumes, as can be seen in the accompanying Figure. Recent hydromagnetic dynamo simulations by Brun, Miesch & Toomre (2004) in particular have achieved unprecedented spatial resolution and are providing new insight into the generation and transport of magnetic fields in the solar convection zone and their feedback on large-scale flows such as differential rotation. Other areas of recent emphasis include the development of improved sub-grid-scale models for unresolved motions and more realistic treatments of the complex boundary regions near the top and bottom of the convection zone. We have also initiated an investigation into the propagation of acoustic waves in the solar interior and their interaction with flow fields, thermal inhomogeneities, and magnetic structures (Mansour et al. 2004; Miesch, Mansour & Rogers 2004). This project promises to elucidate the (forward problem) of helioseismology, providing essential theoretical support to ongoing observational efforts.

Radial velocity (a), temperature (b), and enstrophy (c) patterns are shown in a high-resolution simulation of turbulent solar convection. Images are shown as orthographic projections for a layer near the middle of the convection zone (simulation by Brun, Miesch & Toomre).

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Global MHD of the Tachocline

-Nonlinear HD and MHD Instabilities in the Tachocline- Mausumi Dikpati (HAO) has been developing a spectral code to study the nonlinear evolution of HD and MHD shallow-water instabilities in the solar tachocline. Starting from the so-called primitive equations, Dikpati has completed the purely HD part of the nonlinear shallow-water code which is now producing results showing that the solar tachocline latitudinal differential rotation stabilizes by forming high-latitude prograde jets. The tendency for jet formation was previously predicted in the linear calculations of Dikpati and Gilman (2001). By representing the scalar variables (the tachocline thickness with deformable top surface) in terms of the scalar spherical harmonics and the vector variables (velocities and magnetic fields) in terms of vector spherical harmonics (as described in Morse and Feshbach 1953), a fully explicit spectral code has been developed. Currently, Dikpati is using this code to explore the amplitude of the jet that can be produced with a given initial energy in the system, by employing the third-order Adams-Bashforth predictor corrector method for the time evolution. Dikpati is making further progress in developing the semi-implicit time-evolution scheme in which the linear terms, such as gravity wave-type terms in the pure HD case, are being treated implicitly and, the other non-linear terms explicitly. This semi-implicit scheme will allow time steps that are considerably larger than the limit set by the CFL criteria. The code has several computation-intensive modules which are parallelized using OpenMP directives for faster throughput in shared memory machines, such as dual-Xeons and in a single bluesky node.

-Limits on the Penetration Depth of the Solar Meridional Flow- An issue that has recently emerged in the dynamo community concerns the characteristics of the meridional flow in flux-transport dynamo models. It has been suggested by a number of authors (e.g. Nandy & Choudhuri 2002; Guerrero & Munoz 2004) that in order for the flux-transport models to produce magnetic cycles in agreement with solar observations, the meridional flow must penetrate deep within the convection zone, even through and beneath the region of strong toroidal field known as the tachocline. In response to this suggestion, Gilman and Miesch (2004) investigated the ability of a meridional flow to penetrate the convection zone in a hydrodynamic, rotating, thin shell model. They found that the two boundary layers present (the well-known Ekman layer, and less well-studied buoyancy-diffusion layer) restricted the penetration of the circulation to depths much smaller than that required by the above-mentioned authors. In a continuation of this project, Joanne Mason (HAO) and Gilman have recently started to investigate how the inclusion of a magnetic field would affect the above results. Preliminary calculations reveal the existence of an additional boundary layer, the Hartmann layer, which is expected to limit the penetration depth of the meridional flow even more.

-Thin-Shell Modeling of the Solar Tachocline- The solar tachocline is a thin layer of strong rotational shear located near the base of the solar convection zone. By exploiting its thin radial extent, much progress can be made in understanding its dynamics through both linear analysis and nonlinear simulation. Recently, Miesch and Gilman (2004) have developed a tachocline model based on the thin-shell limit of the 3D MHD equations. The resulting system can be regarded as the MHD generalization of the HYdrostatic Primitive Equations (HYPE) often used in meteorology. Gilman, Dikpati, and Miesch (2004) have used the HYPE system to study joint instabilities of the differential rotation and toroidal magnetic fields in the solar tachocline. For strongly stable stratification and relatively weak field strengths, the HYPE results coincide with previous instabilities found in 2D and shallow-water systems, with peak growth rates of several months. However, if the stratification is nearly adiabatic or if the toroidal fields are strongly super-equipartition, we find a distinct mode of instability characterized by nonzero vertical wavenumbers and high growth rates (of order days; see Figure). Work is proceeding to understand the nature of these high- growth-rate modes and their implications for the Sun.

Growth rates are shown for tachocline instabilities as a function of reduced gravity, G (bottom axis), for vertical wavenumbers n=1 (red), n=3 (blue) and n=10 (green). The growth rate for shallow-water modes is also shown for comparison (dashed line). The left and right vertical axes indicate, respectively, the non-dimensional growth rate and the corresponding dimensional growth time. Ranges of G characteristic of the solar overshoot region and radiative zone are indicated (Gilman, Dikpati & Miesch 2004).

-Global MHD Instabilities in a Diffusive Tachocline- Mausumi Dikpati, Paul Cally (NCAR Affiliate Scientist, Monash University, Australia), and Peter Gilman have developed a more realistic 2D model for global MHD instabilities in the solar tachocline, by including diffusion in the form of kinetic and magnetic drag (following a Newton's cooling law formulation). This instability has previously been studied by Dikpati, Cally, Gilman and others for the case of an idealized tachocline with no kinematic viscosity and magnetic diffusivity. Since radial diffusion is more important than latitudinal diffusion in the thin solar tachocline, diffusive decay of flow and magnetic fields can be considered as proportional to those variables. Dikpati, Cally, and Gilman found that, for solar-like toroidal magnetic fields of strength about 100 kG, instability exists for a wide range of kinetic and magnetic drag parameters, providing a mechanism for enhanced angular momentum transport in latitude, which could explain why the solar tachocline is so thin. From a detailed parameter space survey, they set upper limits of 5x10¹¹ cm² / s and 3x10¹⁰ cm² / s for kinematic viscosity and magnetic diffusivity, respectively, such that this instability occurs in the solar tachocline on a timescale shorter than a sunspot cycle. They found that magnetic drag has much more influence than kinetic drag in damping this instability. This happens because the sink due to magnetic drag dissipates perturbation magnetic energy faster than the vorticity-sink from kinetic drag dissipates perturbation kinetic energy. Consequently, in presence of a small magnetic drag, the non-solar-like clam-shell pattern found by Cally to be the inevitable final state of a broad magnetic profile undergoing an ideal MHD tachocline instability, does not occur in a diffusive tachocline; a banded profile, however, still tips with no reduction in tip-angle. Dikpati, Cally, and Gilman have also examined how tipping may affect various surface manifestations of magnetic features, such as the latitudes and orientations of bipolar active regions.

Tip angle (degrees) as a function of (dimensionless) time for 10° Gaussian bands of peak field strength 105 G with various drag coefficients, for bands initially placed at 40° latitude.

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Solar Dynamos: Physics and Predictions

-A Non-Axisymmetric Flux-Transport Dynamo Model- Dikpati (HA0) is leading a multi-year effort to develop a non-axisymmetric flux-transport dynamo model, in collaboration with Eric McDonald (HAO), Peter Gilman (HAO) and Ad Van Ballegooijen (Harvard/Smithsonian Center for Astrophysics). Building on the 2D kinematic flux-transport dynamo models and their successes in reproducing many large-scale longitude-averaged solar cycle features, Dikpati and collaborators will now be theoretically investigating the mechanism for producing the longitude-dependent solar cycle features, such as "active longitudes" and sector boundary structures. Formulating the large-scale, non-axisymmetric magnetic field components in terms of scalar potentials, Dikpati has derived the evolution equations for the axisymmetric as well as non-axisymmetric magnetic fields. Since the large-scale longitude dependence has been Fourier analyzed in this model, the resulting equations are coupled partial differential equations in radius, latitude and time. Dikpati is leading the development of a numerical scheme based on the Peaceman-Rachford Alternating-Direction-Implicit method to solve this coupled system of complex PDEs. The model will first be run by switching off the large-scale source of non-axisymmetry, in order to investigate which are the dominant non-axisymmetric features produced in this case. Then the large-scale non-axisymmetry will be introduced into the model, using the knowledge gained from the theory of global MHD instabilities in the tachocline to guide the development of the model.

-Predicting the Onset of the Upcoming Cycle 24- Understanding solar cycle mechanisms and predicting the features of an upcoming cycle have become an increasingly necessary and challenging task for our technological society. In the past, the so-called "precursor method" predicted some cycles well, but not the current cycle 23, which has behaved anomalously (de Toma et al. 2004). Following the postulate of previous authors (Schatten et al. 1978) that there is "magnetic persistence" or a memory of past magnetic fields in the Sun, and demonstrating the physical origins of such a memory in a flux-transport dynamo model of the solar cycle, Dikpati and collaborators (2004) recently built the first physical model for large-scale solar cycle prediction. Dikpati et al. (2004) have been able to show why solar cycle 23 behaved anomalously, and therefore why its features were not accurately predicted. By incorporating observed dynamical variations of some of the dynamo ingredients, namely, the surface poloidal field source and the meridional circulation, Dikpati and co-workers showed that a 10-20% weakening of the large-scale, surface poloidal field source in cycle 23 relative to the previous cycle 22 was the primary reason for the substantial delay in the polar reversal of cycle 23. Helioseismic observations indicate that the meridional flow speed decreased systematically during 1996-2002, and that it remained slow until March 2004. Dikpati et al. have shown that this systematic decrease in the meridional flow speed caused the unusually slow rise of cycle 23. They have also made a preliminary prediction (Dikpati et al 2004) that the onset of the upcoming cycle 24 should be delayed, not starting until late in 2007 or early in 2008.

Left frame shows the dynamo-generated magnetic flux (computed from the fields that exceed about 40 kG), as a function of time, in the shear layer (computed from the peak field above 40 kG). Right frame shows the extension of the model beyond the present, by assuming three different variations for the meridional flow noted in the curves. The yellow patch denotes the time-span for which observed surface flow data is available. If the flow continues to be slow during next few years, the dynamo-generated magnetic flux would follow the blue curve, whereas it will follow green curve if flow accelerates. In both cases, preliminary calculations indicate that the onset of cycle 24 will be late.

-Solar Torsional Oscillations: Theory Versus Observation- The feedback of the Lorentz force on the meridional flow and differential rotation in the convection zone leads to variations in the rotation rate with the solar cycle, which are known as torsional oscillations. Although this oscillation pattern has been detected in helioseismic observations down to the base of the convection zone, it is uncertain how reliable the inversions at the convection zone bottom are. The dynamo model with Lorentz force feedback developed by Rempel yields torsional oscillations as part of the dynamo solution. In order to test the accuracy of the helioseismic inversions, Rempel used his model to provide Rachel Howe (National Solar Observatory [NSO]) with simulated torsional oscillations data from which an artifical helioseismic data set could be constructed. Howe, Rempel, and collaborators (Howe et al. 2004) found that most features present in the artificial data could be recovered through the inversion process, even when realistic noise was added to the data.

-Depth Dependence of Turbulent Diffusivity in the Sun- Night Song, E. J. Zita (both Evergreen State College), together with Dikpati and McDonald, have studied the influence of various diffusivity profiles on the evolution of the large-scale, diffuse magnetic fields of the Sun. The only estimate available for the value of the diffusivity in the solar surface layers comes from the mixing-length model for convection, and not much is known about how it should vary as a function of depth down to the base of the convection zone or below it. Dikpati and collaborators constructed various theoretical profiles for the depth-dependent diffusivity, and used an an advective-diffusive flux-transport model to study how each profile affected the production of certain magnetic features at the surface. They compared the model output with observed solar magnetic features, and evaluated the successes and drawbacks of each profile, in order to determine which among them was most relevant to the Sun.

-Dynamo Models with Spatially Separated Generation Layers- Both the interface dynamo model and the traditional Babcock-Leighton dynamo are characterized by the two generation mechanisms operating in spatially disjoint regions. Due to this spatial separation, the models admit many interesting properties in addition to their ability to generate magnetic fields. In the case in which the α and Ω-effects are assumed to operate in sufficiently thin parallel layers that they can be described mathematically by δ-functions, the interface model has been shown to admit both long wave modes with small wavenumber, and short wave modes with wavenumber comparable to the depth of the region responsible for the α-effect (Mason et al. 2002). Although dynamo theorists have primarily concentrated on the short wave mode and analyzed its capability to produce magnetic cycles in analogy with the solar cycle, it is necessary to investigate the long wave mode as well. Using weakly nonlinear theory and a multiple scales technique, Mason (HAO) and Edgar Knobloch (University of Leeds, UK, and University of California, Berkeley) have derived an equation that governs the slow evolution of the amplitude of the long wave mode. The evolution equation takes the form of a modified Korteweg-de Vries equation, the solutions of which are described by snoidal waves: nonlinear waves of magnetic activity that propagate towards the equator as observed in the Sun. The leading order contributions to both the toroidal and poloidal fields are shown together in accompanying Figure.

The leading order contributions to the toroidal field (solid line) and poloidal field (|B_P| for Ξ ≤ π and |B_P| for Ξ ≥ π, dashed line) taken at the interface between the tachocline and convection zone (Ξ denotes a reference frame traveling with speed v northwards with respect to an already equatorwards moving frame ξ).

A further consequence of the two generation mechanisms operating in spatially separated regions is that the dynamo is only efficient in so far as the magnetic flux may be transported between the two regions. It is believed that the magnetic buoyancy instability is responsible for the transport of newly generated toroidal flux from the tachocline to the convection zone (Parker 1955), and that the magnetic pumping mechanism returns the poloidal field to the tachocline for the cycle to repeat. The turbulent compressible penetrative convection studies of Tobias et al. (2001) illustrate the ability of the strong vortical downflows to wrap up the magnetic field and drag it downwards with them as they penetrated the stable overshoot region. Mason, David Hughes, and Steve Tobias (both of the University of Leeds, UK) have investigated the effects of incorporating the pumping mechanism into a kinematic, axisymmetric, mean-field model, and have shown that there exists a preferred magnitude of pumping at which the dynamo is at its most efficient (see the second accompanying Figure). Current studies of a nonlinear extension to the above described model are aimed at investigating the effect that the pumping mechanism has on the phase difference between the poloidal and toroidal fields. Preliminary results show that the pumping mechanism changes this phase relation and also has an effect on the period of the magnetic cycle.

The minimum of the critical dynamo number governing the onset of the dynamo instability as a function of the magnitude of the pumping. The diffusion is constant throughout the convection zone and tachocline, but the pumping operates only within the convection zone. The dynamo is most easily excited with a pumping of magnitude approximately 1.1 (the negative sign illustrates the direction of pumping from the convection zone to the tachocline).

-Flux-transport Dynamos with JxB Feedback- Flux-transport dynamos have proven to be successful in modeling the evolution of the large-scale solar magnetic field. However, these studies addressed the transport of magnetic field by the meridional circulation in a purely kinematic regime. The toroidal field strength at the base of the solar convection zone, as inferred from studies of rising magnetic flux tubes, is around 100 kG, and thus is orders of magnitude larger than the equipartition field strength estimated from a meridional flow speed of a few m/s. Therefore, it is crucial for flux-transport dynamos to include the feedback of the jxB force on the meridional flow. Rempel, Dikpati, and Keith MacGregor (HAO) addressed this problem using two approaches: (1) a kinematic model in which the feedback is parametrized in terms of a non-linear quenching of the meridional flow in regions exceeding a certain field strength; and (2), an MHD approach in which the full set of hydrodynamic equations together with the dynamo equations is solved. In this latter approach, the mean-field differential rotation model developed by Rempel is used to compute both the differential rotation and the meridional flow, which are then used to evolve the magnetic field through the dynamo equations. The Lorentz force is allowed to feedback on the differential rotation and meridional flow, leading to a dynamo that operates in the "dynamic" regime. A typical solution for the magnetic field (butterfly diagram) and meridional flow is shown in the Figure.

Butterfly diagrams of the toroidal field strength at r=0.735 R_⊙ (the range is around -13 kG to +13 kG), the equator-ward meridional flow velocity (the range is around -0.1 m/s to +1.2 m/s), and the poleward meridional flow at r=0.95 R_⊙ (the range is around -4.5 m/s to +0.4 m/s). The Lorentz force feedback changes the meridional flow significantly throughout the convection zone. In regions of strong field (around 10 kG) at the base of the convection zone the flow speed is significantly reduced (by around 50%). Near to the surface the Lorentz force has the tendency to significantly weaken the poleward flow above 50° latitude. Above 60° latitude a second cell forms with a reversed flow direction.

In this combined investigation of kinematic and dynamic dynamo models, Rempel, Dikpati, and MacGregor found that flux transport dynamos work even with significant feedback of the Lorentz force on the meridional flow and differential rotation for two main reasons: (1) the meridional flow avoids regions of strong toroidal field, but still transports the weaker poloidal field that is the source for the toroidal field via the Ω-effect; and (2),the transport capacity of the meridional flow is much larger than estimates based on its energy density suggest, since it is driven by the small difference of several large forces, namely, the Coriolis force, viscous force, and buoyancy/pressure force. The dynamical model sets an upper limit of about 30 kG to the strength of the toroidal field that can be transported. Stronger fields will either move towards the pole or reach an equilibrium, depending on how well angular momentum is conserved within the toroidal band. In this case, a dynamo with equator-ward propagating toroidal field would still be possible, since the meridional flow can transport the weaker poloidal field, which provides the source for the toroidal field. The strong toroidal field would reach an equilibrium state in which the magnetic tension is compensated by the Coriolis force of an prograde jet. Rempel, Dikpati, and MacGregor found as a further result that the feedback of the Lorentz force can produce in conjunction with the hotter pole (required for balance of differential rotation) a reverse polar cell in the meridional flow that varies with the dynamo phase as found by Haber et al. (2002).

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Inferences about Interior Global MHD from Surface Observations

-Surface Signatures of Internal Magnetic Fields- Aimee Norton (HAO) and Gilman used solar surface magnetism data to search for signatures of the toroidal magnetic field and its solar-cycle dynamics. They were interested in what can be learned about the dynamical behavior of the interior toroidal magnetic field from the statistical study of solar surface magnetism. Through analysis of sunspot data over significant time periods, Norton and Gilman partially recovered certain properties of the interior toroidal field. They analyzed both Kitt Peak magnetogram data and continuum intensity sunspot data from the Michaelson Doppler Imager (MDI) on board the Solar and Heliospheric Observatory (SOHO) spacecraft to search for the following solar toroidal band properties: width in latitude and the existence of a tipping instability (longitudinal m=1 mode) for any time during the solar cycle. This tipping has been predicted by recent theoretical work using a nonlinear model of a 2D MHD tachocline. In order to determine the extent to which toroidal field dynamics can be recovered, Norton and Gilman modeled artificially generated sunspot distributions from sub-surface toroidal fields that were assigned certain properties. They found that a relatively wide toroidal band, 15-25° in latitude over which sunspots emerge, best fit the data.

A tipping of 5° early in the solar cycle, gradually decreasing to 0° as the band moves equatorward is compatible with both the MDI and Kitt Peak data and modeling efforts. Norton and Gilman also analyzed and modeled the MDI data in two time periods when the toroidal band was at high and low latitudes, since the tipping amplitude is predicted to be greater at high latitudes. The analysis indicated a tip was more likely to exist for high latitude data, but it was on average <5°, which was consistent with modeling results. They found that the band widens from 15-20°; early in the solar cycle to 20-25° late in the solar cycle. This could be explained by magnetic drag spreading the toroidal band due to altered flow along the tipped field lines. The tipping instability is difficult to recover with the analysis techniques used by Norton and Gilman when the toroidal band width is much greater than the tipping amplitude. In addition, the existence of a prograde jet that acts to stabilize the band against tipping, or the presence of modes with m > 1 if the toroidal magnetic field is of order 20 kG would either suppress the instability or make it difficult to measure.

-LOWL/ECHO-Extraction of Mode Parameters and Science Ojectives- David Salabert (HAO) is currently working on the analysis of the helioseismic data acquired by LOWL/ECHO (Experiment for Coordinated Helioseismic Observations) network, built and operated by HAO since 1994. Such data can be analyzed to constrain the structure and rotation of the solar interior. The LOWL/ECHO instruments observe the solar oscillations as perturbations of the radial velocity over the surface of the Sun. This is accomplished by taking images in narrow bandpass filters displaced slightly redward and blueward of a solar absorption line. A velocity image is obtained from the difference between the two intensity images. Modes are separated in the data by projecting the velocity images onto spherical harmonic functions. The time series of each mode coefficient is then Fourier transformed in time to yield a power spectrum which has discrete peaks at the mode eigenfrequencies. Accurately determined eigenfrequencies are the basic data product of this instrument. The internal structure and rotational characteristics of the Sun can be determined from these frequencies. In a first step, the estimation of the solar radius from the images was computed with a better accuracy; thus, the decomposition of the velocity images in spherical harmonics is more accurate, thereby improving the quality of the time series of each mode. However, the spherical harmonics are not orthogonal over the observed area, so the observed Fourier spectra in the case of resolved observations, are a linear combination of different modes. The correlation between the Fourier spectra of each individual mode is represented by the leakage matrix. To extract correct mode parameters for a particular mode, it is necessary to have a good knowledge of the other modes which leak and interfere with the studied mode in the Fourier domain. Before undertaking any extensive analysis, Salabert is working on the leakage matrix problem to obtain good estimates of leaked modes, and thus extract reliable mode parameters. Then the so-called a-coefficents for each degree, representing the shift in frequency induced mainly by the internal rotation, can be estimated.

Salabert, in collaboration with Sebastien Jimenez-Reyes (Insítuto de Astrofísica de Canarias, Spain) and Michael Thompson (NCAR Affiliate Scientist, University of Sheffield, UK) is using LOWL/ECHO data to study the Sun's internal rotation. Of particular interest is the rotation rate near the tachocline, the transition zone between the solid rotation of the radiative interior and the differential rotation of the convection zone. The solar magnetic field is believed to be generated in this shear layer, therefore observations of the dynamics of this region are extremely important. In both ground-based GONG (Global Oscillations Network Group) data and space-based MDI data, an oscillation of period about 1.3 years of the rotation rate near the tachocline has been observed. The existence of this periodic change in the tachocline needs to be confirmed, and Salabert will utilize the long time series of LOWL/ECHO data as an independent check, to search for these tachocline variations. Time series of 108 d over 6 years of LOWL data have been computed and decomposed onto spherical harmonics. The resulting Fourier spectra are currently being analyzed in order to study the dynamics of the tachocline. In additon, Salabert is utilizing the LOWL/ECHO data to place better constraints on the structure and rotation of the solar core. The study of the solar core requires the measurement of p-modes of low degree, which penetrate deeply into the Sun. The LOWL/ECHO instruments have the advantage that they are optimized for observing the low degree modes, unlike GONG and MDI. The capability for observing low degree oscillations, together with the longer time series of LOWL/ECHO data compared to GONG and MDI, make these observations very valuable for placing constraints on the structure and rotation of the deep solar interior. The existence of a single-instrument helioseismic data base extending back over a decade in time also makes it possible to study manifestations of solar activity in the interior. Mode parameters obtained from the LOWL/ECHO data over 10 years of operation will be used to follow the evolution of the near surface dynamics, and more importantly, to search for signatures of variations of the dynamo with the solar cycle in the dynamics and structure of the tachocline.

Example for the mode l=9, m=9 at 2543.4 μHz. The p-mode parameters are extracted by fitting a Lorenztian profile (green solid line - the dashed line corresponds to the initial parameters). The first sidebands are included in the profile. The m-leakage and the l-leakage for this mode l=9, m=9 are represented by the dot-dashed line and the dashed line, respectively.

Fitted frequencies for the mode of angular degree l=76 at 2869.9 μHz. The a-coefficients, representing the shift in frequency induced mainly by the internal solar rotation, are computed as Clebsch-Gordan coefficients (solid line).

-Detecting Tachocline Jets- Tachocline toroidal fields are likely to exist in the form of narrow bands, at least during some phases of the solar cycle. Recent theoretical studies show that a narrow toroidal band in the solar tachocline can be held in equilibrium against its poleward slip due to the curvature stress by the Coriolis force produced by a prograde jet inside the band. Previous attempts to detect convection-zone jets using 7-month GONG data have been made, but to date, no clear evidence of jet-like flows has been detected. Recently, Dikpati, Gilman, Thierry Corbard (Observatoire de la Cote d'Azur, France), Jorgen Christensen-Dalsgaard (NCAR Affiliate Scientist, Aarhus University, Denmark), and Thompson (University of Sheffield) have explored the use of long-term GONG data, in order to detect the amplitude, width and the latitude-location of jets that could exist in addition to background zonal flows. Using the latitudinal force-balance equation for toroidal bands in a typical solar tachocline, they generated artificial helioseismic data containing jets. They then inverted these synthetic observations in an effort to recover the jets. Following validation of the technique through experiments of this sort (see the accompanying Figure), they are applying it to the inversion of real data from different epochs.

Jet and total internal rotation including background for a 20° band with 60 kG peak field, assuming that balance is entirely between magnetic curvature stress and Coriolis force.

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Extra-Solar Stars and Planets

-SCF Models of Differentially Rotating Stars- The computation of stellar models that consistently include effects arising from rapid, non-uniform rotation is a formidable task, requiring the solution of Poisson's equation in addition to the equations governing the internal structure of a star. To avoid this difficulty, researchers have often utilized methods based on approximate expressions for the gravitational potential that are strictly valid only in the limit of vanishing rotation. In an effort to develop an approach to this problem that is both more accurate and more widely applicable, Stephen Jackson, Keith MacGregor, and Andrew Skumanich (all of HAO) have devised a new version of the self-consistent-field (SCF) method for calculating the structure of a rotating star. As originally implemented in the 1970's, the SCF method was capable of producing consistent, converged models only for stars more massive than 9 M_⊙ and, as a result, received limited use as a tool to investigate how rotation affects stellar properties.

The SCF method as reformulated by Jackson, MacGregor, and Skumanich is an iterative scheme, initialized by specifying one-dimensional trial distributions of the temperature and pressure (each normalized by their central values), and a two-dimensional trial function describing the shape of constant-density surfaces. The normalized trial density distribution itself follows from the equation of state, and is used as the source term in Poisson's equation to determine the gravitational potential. For a rotation law in which the angular velocity depends only on the perpendicular distance from the axis of rotation, the total potential and its equipotential surfaces can then be evaluated and used in solving the stellar structure equations. This step yields updated temperature and pressure distributions, and allows the process to be restarted and continued to convergence. When self-consistency of the normalized temperature and pressure profiles has been attained, the corresponding central values are adjusted in order to bring them closer to the physical conditions at the center of the final equilibrium configuration. The entire procedure is then begun again and repeated until an acceptable level of agreement between the current and previous central values has been obtained.

Jackson, MacGregor, and Skumanich have carried out an extensive series of tests, including comparisons with models computed using other techniques, to validate the revised SCF method. Their results indicate that the method yields converged models for chemically homogeneous, main sequence stars of all masses, and is capable of treating cases in which rapid, differential rotation causes the photospheric shape of a star to deviate significantly from sphericity. They have shown that the models can be characterized by two quantities measuring the degree and rate of differential rotation, and have delineated the regions in the corresponding two-dimensional parameter space wherein equilibrium stellar models can be obtained (see first accompanying figure). They have also surveyed how basic model properties such as photospheric size and shape, central thermodynamic conditions, and luminosity depend on the presumed internal rotational state of the star (see second accompanying figure). To interpret the behavior of many of the attributes of computed upper main sequence models, they have derived an approximate, semi-analytic model in which the outer, radiative envelopes of such stars are treated polytropically. To compare the model properties with the observed characteristics of rotating stars, Jackson, MacGregor, and Skumanich have begun work on synthesizing the shapes of spectral lines formed in a differentially rotating stellar photosphere. Although the effects of gravity darkening have yet to be incorporated, preliminary results suggest that differential rotation of the kind used in their recent models for the Be star Achernar (Jackson, MacGregor, & Skumanich 2004) is difficult to detect through analysis of photospheric absorption line profiles. After updating some of the input physics and adding a mixing-length treatment of convection, Jackson, MacGregor, and Skumanich anticipate using the modified SCF method to address a variety of problems; these include an examination of the structure and properties of a rapidly and differentially rotating young Sun, an investigation of the circumstances under which stars of intermediate mass can have rotationally induced convective envelopes, and a study of extremely distended, highly flattened configurations as representations of star/disk systems.

Map of rotational parameter space for 6 M_⊙ stars showing regions where converged models can and cannot be obtained. The quantity α is a measure of the degree of differential rotation, while η is the ratio of the axial rotation rate to the critical rotation rate at the equator on the surface of the star. The four green lines are contours on which the ratio of the rotational kinetic energy to the gravitational potential energy, t, assumes values of 3, 6, 9, and 12%, respectively. Region I, the domain over which converged models can be readily obtained, is separated from Region II by the red line, from Region III by the purple line, and from Region IV by the blue line. Regions II and III are domains in which equilibrium models presumably exist but cannot be obtained with the present code. Some of the models in these two regions apparently have level surfaces with toroidal topology, and some or all of the models in these regions may be unstable. The blue line marking the boundary of Region IV is the parabola η = 1 + α², the locus of points for which Ω_e, the equatorial angular velocity, is equal to Ω_cr, the critical angular velocity. No equilibrium model exists in Region IV because the net force at the equator would be directed outward.

The dependence on the kinetic-energy parameter t of six basic physical properties for 3 M_⊙ models. Each variable is plotted in units of the value of the same variable for the corresponding nonrotating reference model. The solid curves are cubic-spline fits to data for models computed by Clement (1979) using a two-dimensional relaxation technique. The dashed lines are cubic-spline fits to data for SCF 3 M_⊙ models, with the filled circles indicating the actual data points for those models. Panel (a) displays the variation of the equatorial radius R_e and the polar radius R_p, Panel (b) the variation of the mean radius R_a and central temperature T_c, Panel (c) the variation of the central density ρ_c and the luminosity L.

-Results in Stellar and Planetary Astrophysics- HAO's program in stellar and planetary astrophysics reached significant milestones in FY2003, driven by 3 noteworthy observations.

For the last 3 years, Tim Brown (HAO) has supervised operation of the STellar Astrophysics and Research on Exoplanets (STARE) telescope, now sited on the island of Tenerife, Spain. This work is carried out with extensive assistance from the Astrophysical Institute of the Canaries (IAC), including collaboration with J. Belmonte (IAC), and Brown's co-supervision of two graduate students (R. Alonso and O. Creevey, both of the IAC and the University of La Laguna). For about 18 months, STARE has observed in conjunction with two similar telescopes: one at Flagstaff, Arizona (operated by E. Dunham and G. Mandushev, both of Lowell Observatory), and the other at Mt. Palomar, California (operated by D. Charbonneau, Harvard University, and F. T. O'Donovan, CalTech). Together, these three telescopes make up the Transatlantic Exoplanet Survey (TrES) network. Recent improvements in analysis techniques, combined with the higher duty cycle possible with the 3-site network, have enabled a flurry of recent discoveries of interesting eclipsing objects.

In May 2004, during analysis of a data set obtained about a year earlier, R. Alonso noticed a star in the constellation Lyra that showed periodic eclipses that were consistent with a transiting planet. Extensive follow-up observations showed that this object is indeed a planet, now dubbed TrES-1, the first transiting planet detected by the TrES network. This was the fifth transiting planet known, and only the second that is near enough to the Sun that it can be thoroughly characterized using presently available telescopes. TrES-1 has a mass about 0.75 times that of Jupiter, a radius of about 1.04 Jupiter radii, and it circles its parent star (which is slightly smaller and cooler than the Sun) once every 3.02 days. At present, the most puzzling point concerning TrES-1 is that its mass and equilibrium surface temperature are quite similar to those of the planet HD 209458b (the first known transiting planet, also co-discovered by the STARE telescope), but its radius is 20% smaller (see the first accompanying Figure). Extensive studies of both planets to understand the origin of this difference are underway.

Graduate student O. Creevey investigated a newer TrES data set and found a relatively faint and unusually red eclipsing binary star that showed an apparent orbital period of 0.6 day. Further examination proved this to be a nearly-symmetrical pair of M dwarf stars, each with a mass only about 0.4 times the solar mass. Such binary systems present an opportunity for precise estimates of the radii and masses of both components; they are important because the theoretical mass/radius relation is ill-determined for stars with such cool temperatures, and because only 3 similar systems are known. In collaboration with F. Benedict and W. Cochran (both University of Texas), Creevey obtained radial velocity data confirming the characterization of the system, and giving mass and radius values accurate within a few percent. Future observations will refine these values further.

On June 8 2004, the planet Venus transited the Sun for the first time in more than 120 years (left). T. Brown and M. Knölker (HAO) traveled to Tenerife to observe the transit using the Vacuum Tower Telescope of the Kiepenheuer Institue for Solar Physics (KIS). Working with a team of German and Spanish scientists (including W. Schmidt and H. Schleicher from KIS, H. Rauer from the University of Berlin, and R. Alonso and M. Collados Vera from the IAC), they observed the spectrum of sunlight that had traveled through the outer reaches of the Venusian atmosphere. By measuring the strengths and Doppler-shifted wavelengths of near-infrared lines of carbon dioxide, they aim to measure the composition, excitation temperature, and wind speed as a function of Venusian latitude and height above Venus's cloud deck. Initial analysis of the data shows promising results and more detailed study is in progress.