Solar Surface Convection
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We review the properties of solar convection that are directly observable at the solar surface, and discuss the relevant underlying physics, concentrating mostly on a range of depths from the temperature minimum down to about 20 Mm below the visible solar surface.
The properties of convection at the main energy carrying (granular) scales are tightly constrained by observations, in particular by the detailed shapes of photospheric spectral lines and the topology (time- and length-scales, flow velocities, etc.) of the up- and downflows. Current supercomputer models match these constraints very closely, which lends credence to the models, and allows robust conclusions to be drawn from analysis of the model properties.
At larger scales the properties of the convective velocity field at the solar surface are strongly influenced by constraints from mass conservation, with amplitudes of larger scale horizontal motions decreasing roughly in inverse proportion to the scale of the motion. To a large extent, the apparent presence of distinct (meso- and supergranulation) scales is a result of the folding of this spectrum with the effective “filters” corresponding to various observational techniques. Convective motions on successively larger scales advect patterns created by convection on smaller scales; this includes patterns of magnetic field, which thus have an approximately self-similar structure at scales larger than granulation.
Radiative-hydrodynamical simulations of solar surface convection can be used as 2D/3D time-dependent models of the solar atmosphere to predict the emergent spectrum. In general, the resulting detailed spectral line profiles agree spectacularly well with observations without invoking any micro- and macroturbulence parameters due to the presence of convective velocities and atmosphere inhomogeneities. One of the most noteworthy results has been a significant reduction in recent years in the derived solar C, N, and O abundances with far-reaching consequences, not the least for helioseismology.
Convection in the solar surface layers is also of great importance for helioseismology in other ways; excitation of the wave spectrum occurs primarily in these layers, and convection influences the size of global wave cavity and, hence, the mode frequencies. On local scales convection modulates wave propagation, and supercomputer convection simulations may thus be used to test and calibrate local helioseismic methods.
We also discuss the importance of near solar surface convection for the structure and evolution of magnetic patterns: faculae, pores, and sunspots, and briefly address the question of the importance or not of local dynamo action near the solar surface. Finally, we discuss the importance of near solar surface convection as a driver for chromospheric and coronal heating.
KeywordsConvection Zone Solar Surface Local Thermodynamic Equilibrium Solar Convection Zone Poynting Flux
Convection is of central importance to the structure and appearance of the Sun, both with respect to its average internal structure and secular evolution, and with respect to the detailed structure and dynamics of the many diverse phenomena that occur at the solar surface. In this review we discuss convection in those layers of the Sun that directly influence what may be observed at the solar surface. In practice, this means that we consider layers ranging from the temperature minimum down to about 20 Mm below the visible surface. Over this range of depths hydrogen ionization goes from essentially zero, with electrons instead supplied mainly by the most abundant metals, to essentially unity. Helium (2nd) ionization is also nearly complete near the bottom of this range of depths.
In these surface and near-surface layers the equation of state of the solar plasma is strongly influenced by the effects of ionization and molecular dissociation. Deeper down the solar plasma is nearly fully ionized, the equation of state is nearly ideal and, hence, the vertical structure of the lower part of the convection zone (90% by depth) is close to that of a perfect polytrope.
The physics of convection near the surface of the Sun is also greatly influenced by the fact that the solar surface is a radiating surface, where the mode of energy transport all of a sudden changes from convective — with energy being carried by moving fluid — to radiative — with energy carried by essentially free-streaming photons.
Each one of these complications — extreme density stratification, ionization and molecule dissociation, and transition to radiative energy transfer — would by itself be sufficient to render analytical treatments impossible and, hence, major progress in modeling these layers could only be achieved via computer simulations and modeling. The subject area was thus truly opened up for study and revolutionized when computer capacity became large enough for realistic simulations to be performed.
Conversely, because of the proximity and large apparent luminosity of the Sun, it is one of the most ideal objects for quantitative and accurate comparisons between numerical simulations and observations — arguably in five dimensions: one has, to a rather unique extent, access to both the time domain and the spatial domain, over a large range of scales, and the available photon flux is large enough to also essentially fully resolve the wavelength domain, from the extreme ultra-violet to the far infrared.
In addition, the Sun is an ideal object for studying magneto-hydrodynamics in an astrophysical context, in that it displays a variety of magnetically related phenomena, ranging in strength from very weak and kinematic, where the evolution of the magnetic field is almost entirely controlled by the fluid motions to the opposite extreme, where the gas pressure is so feeble that fluid motions are totally dominated by the magnetic field.
The comparison of models and observations is thus both particularly challenging and particularly rewarding in this context, and the evolution of the subject area over the last few decades has amply illustrated this state of affairs.
In this review we try to summarize, in a necessarily incomplete and biased way, our current understanding of solar surface convection.
The review is organized as follows: In Section 2 we discuss the principles of hydrodynamics as they apply to solar convection. In Section 3 we discuss the manifestations and properties of the main energy carrying scales, and the granulation pattern that they give rise to. In Section 4 we discuss how larger scale convective patterns arise and are driven, and how they interact with smaller scale patterns. We introduce the concept of a velocity spectrum, and address the question the extent to which the traditional concepts of meso- and supergranulation represent distinct scales of motion. In Section 5 we discuss how spectral line synthesis and comparisons with observed spectral line profiles may be used to asses the accuracy of numerical simulations, and how the resulting spectral line profiles may be used to accurately determine solar chemical abundances. In Section 6 we discuss applications to global and local helioseismology: wave excitation and damping, the influence of convection on the mean structure, and helioseismic diagnostics related to convective flow patterns. In Section 7 we discuss the interaction of convection and solar magnetic fields and, finally, in Section 8 we discuss open questions and directions for future work. Section 9 ends the review with a summary and concluding remarks.
2 Hydrodynamics of Solar Convection
Although some of the properties of solar convection may appear strange and unexpected at first, especially in comparison to local laboratory and atmospheric hydrodynamics settings, they of course follow rigorously from the laws of physics, which in the fluid approximations are the laws of compressible hydrodynamics and magnetohydrodynamics. These laws form the basis for numerical modeling, but equally importantly also allow a qualitative and semi-quantitative understanding. We therefore start out by writing down these laws, in forms suitable for the discussion (as well as for computational work in some cases).
2.1 Mass conservation
These two forms of the continuity equation may already, before one even considers the rest of the equations of hydrodynamics, be used to understand some very fundamental consequences that conservation of mass has for the structure of solar convection.
For example, the Lagrangian form (Equation 2) shows that a fluid parcel that ascends over a number of density scale heights has to expand correspondingly. The Eulerian form (Equation 1) shows, on the other hand, that if the local density does not change much with time, then a rapid decrease of mean density with height in an ascending flow, can be balanced by rapid expansion sideways.
A rapid vertical expansion could in principle also happen, but as we shall see in the next subsection, the equations of motion dictate that the main expansion/contraction of ascending/descending flows happens sideways.
2.2 Equations of motion
2.3 Kinetic energy equation, buoyancy work, and gas pressure work
But buoyancy forcing of convection is a fundamental and easily understood mechanism; warm gas is lighter and rises, cold gas is denser and sinks. So, how can the total buoyancy work vanish? The solution to this riddle lies in the word “total”. Buoyancy indeed performs positive work on convection cells, maintaining their kinetic energy. It is only when including the ultimate, global scale, the horizontally averaged velocity, that the energy balance score of the buoyancy work is evened out to zero. Because density and velocity are correlated, and because the average mass flux must vanish, the average vertical velocity is upwards, corresponding to negative buoyancy work of a magnitude that exactly cancels the fluctuating buoyancy work!
The root of the somewhat perplexing situation lies in the particular choice of variables that are to be expanded into ‘averages’ and ‘fluctuations’. As a result of choosing density and velocity, rather than for example density and mass flux (which has a vanishing horizontal average) one must include also the (non-vanishing) product of the averages, and not only the average product of the fluctuations.
Perhaps the clearest way of stating the result is to say that there is on the one hand a (generally positive) work done by the buoyancy force (as defined) on the convective motions, and there is on the other hand an equally large but negative work done by the average force of gravity acting on the mean flow. As a result, the total work done by gravity vanishes.
Broadly speaking the positive buoyancy work goes towards maintaining the fluctuating (convective) velocity field against dissipation, but, as illustrated by Equation 17, the most straightforward analysis of work and dissipation is obtained by considering the gas pressure work rather than the buoyancy work.
2.4 Energy transport
2.4.1 Radiative energy transfer
Because of the interplay of the ρκ ν and I ν − S ν factors, the effect is largest in a thin layer near optical depth unity; at larger depths the difference I ν − S ν tends to zero exponentially with optical depth, while at smaller depths the ρκ ν factor diminishes the effect as well.
The discussion above applies frequency by frequency, and because of variations in the opacity κ ν with frequency, in particular across spectral lines, the net heating or cooling can only be found by evaluating the integral over frequency that occurs in Equation 29. An approximate way to estimate that integral using a binning method was developed by Nordlund (1982) and applied in subsequent works by Stein, Nordlund and collaborators (e.g., Stein and Nordlund, 1989, 1998; Asplund et al., 2000b; Carlsson et al., 2004). The method has been tested and used by Ludwig and co-workers (Ludwig et al., 1994; Steffen et al., 1995) and by Vögler and co-workers (Vögler et al., 2004; Vögler, 2004; Vögler et al., 2005). Improvements are being developed by Trampedach and Asplund (2003).
2.4.2 Equation of state
Whenever the constituent atoms or molecules of a gas undergo ionization or dissociation, extra energy is required to heat or cool the gas; i.e., the internal energy e varies more with temperature than for an ideal gas. Near the surface of the Sun, both ionization and dissociation are important; ionization of hydrogen and helium influence the equation of state greatly just below the surface, and dissociation of hydrogen molecules is important in the cooler layers of the photosphere.
2.4.3 Convective and kinetic energy fluxes
At the solar surface there is a very rapid transition between primarily convective and primarily radiative energy transport. In the atmosphere it is more relevant to consider changes of the radiative flux, through Equation 29.
With the previous Section providing the physical background, we now have the tools (and the language) to discuss the various manifestations of convection at the surface of the Sun, treating in this Section the energetically most significant pattern; the solar granulation.
The solar granulation pattern was first observed and described by Herschel (1801), who interpreted the pattern as being due to “hot clouds” floating over a cooler solar surface. Nasmyth (1865) later referred to the pattern as one similar to “willow leaves”. Dawes (1864) was the one who coined the term “granules”, contesting the description of Nasmyth with respect to the shapes — this illustrates that already then spatial resolution of observation was a factor of great importance. The first good photographs, published by Janssen (1896) ended the controversy.
The granulation pattern is in fact, as we now know, associated with heat transport by convection, on horizontal scales of the order of a thousand kilometers, or one megameter (Mm). So, Nasmyth was on the right track concerning the temperature, and must have seen essentially the pattern we know today, but the prolific Dawes (who published 5–6 MNRAS papers per year in that period of time) managed to establish the name for it.
3.1 Observational constraints
Paradoxically, the best observational constraints on solar granulation come from completely unresolved observations; observations of the widths, shapes, and strengths of photospheric spectral lines provide constraints that, when combined with numerical models of convection on granular scales, allow quite robust results to be extracted from the numerical models.
Direct observations (e.g., Title et al., 1989; Wilken et al., 1997; Krieg et al., 2000; Müller et al., 2001; Berrilli et al., 2001; Löfdahl et al., 2001; Hirzberger, 2002; Nesis et al., 2002; Hirzberger, 2002; Roudier et al., 2003b; Del Moro, 2004; Puschmann et al., 2005; Nesis et al., 2006; Stodilka and Malynych, 2006, and references therein), have given a wealth of information about the morphology and evolution of the granulation pattern, and about how it is influenced and advected by larger scale flows. However, direct observations of sub-arcsecond size structures are unavoidably affected by image degradation, caused by limited instrumental resolution, scattering in the instrument and, in the case of Earth-based observations, atmospheric blurring and image distortion. Without the access to a sharp reference image or light source, it is not possible to obtain independent measurements of the amount of image degradation, and hence it is also not possible to perform quantitatively well defined image restorations.
In unresolved observations, on the other hand, key properties such as velocity amplitudes and velocity-intensity correlations are encoded in the shapes of the spectral lines. The many photospheric iron lines that can be observed in the solar spectrum are of particular importance, both because they are numerous, which allows a fair number of practically unblended lines to be found, but also because iron is a relatively heavy atom, so the thermal broadening is small (or at least not completely dominating) compared to the Doppler broadening caused by the velocity field.
The widths of spectral lines are thus heavily influenced by the amplitude of the convective velocity field, which overshoots into the stable layers of the solar photosphere where the iron lines are formed. Similarly, correlations of velocity and temperature cause net blueshifts and characteristic asymmetries of spectral lines (cf. Dravins et al., 1981; Dravins, 1982; Asplund, 2005; Gray, 2005). Together, the widths, shifts, and shapes of spectral lines thus constitute a “fingerprint” of the convective motions, which allows detailed and quantitative comparisons between 3D models and observations to be based on spatially unresolved but spectrally very accurate line profile observations. As shown in Asplund et al. (2000a,b), the agreement between observations and high resolution 3D models is excellent. See Section 5 for additional discussion.
Note that the combination of the excellent agreement of spectral line widths (which constrain the velocity amplitudes) and spectral line shifts and asymmetries (which constrain the product of velocity amplitudes and intensity fluctuations) means that the intensity fluctuations obtained from the simulations are very reliable. If they were too large or too small the spectral line shifts and asymmetries would be correspondingly to large or too small as well (cf. Deubner and Mattig, 1975; Nordlund, 1984). Different numerical models give rms intensity fluctuations that agree to within 1–2 percent of the continuum intensity. Observed rms intensity fluctuations are generally much smaller, presumably due to the combined effects of seeing, limited telescope resolution, and scattered light. A detailed comparison of the rms intensity fluctuations observed with Hinode with the results of forward modeling from numerical simulations (Danilovic et al., 2008) concludes that the results are essentially consistent.
3.2 Convective driving
3.3 Scale selection
Because of the enormous variation of mass density with depth, conservation of mass plays a central role in determining the convective flow patterns; whenever a fluid parcel moves up or down over a density scale height it must expand or contract with a factor of e. This constraint dictates the scales on which the dominant convective motions occur.
The upper layers of the solar convection zone are indeed highly stratified, with density scale heights that are smaller than the typical horizontal dimensions of granules. The rapid decrease in density with height leads to a marked asymmetry in the convective flows. Fluid moving upward into lower density layers cannot carry all its mass higher. Hence, upward moving fluid must diverge and most of it turns over within a scale height. Downward moving fluid on the other hand is moving into denser regions and gets compressed as it descends. As a result, diverging upflows have their fluctuations smoothed out, while converging downflows have their’s increased and become turbulent. Except within 100 km of the surface, upflows occupy about 2/3 and downflows about 1/3 of the area.
The larger scales flow patterns are in a sense also driven by the surface cooling, albeit in a more indirect manner. Rising plasma expands and spreads outwards, and over each density scale height a large fraction turns over and joins the downflowing plasma. Even from a depth of just a few Mm, only a tiny fraction of the ascending flow reaches the surface and is cooled by radiation cooling; the rest turns over and is only cooled by mixing with downflowing material (Stein and Nordlund, 1989). Conversely, descending flows from the surface are being entrained by adiabatic overturning fluid, and turbulent mixing in the downdrafts thus reduces the entropy contrast gradually with depth (cf. Figure 5).
On the average, the plasma at any one level has a slightly superadiabatic stratification, which is a mix of almost perfectly isentropic ascending plasma and cooler descending plasma. Formally, this average structure is convectively unstable according to the Schwarzschild criterion (Schwarzschild, 1958), and one may regard the larger scales as being driven by the corresponding convective instability; a slight perturbation at larger scales is self amplifying, since the diverging upflow sweeps smaller scales downdrafts together, thus amplifying the perturbation (Stein and Nordlund, 1989; Rast, 2003).
3.4 Horizontal patterns and evolution
Mass conservation takes care of itself by acting through the pressure. Large granules require a large pressure perturbation to drive their horizontal flows. If there is insufficient pressure to push enough mass out horizontally, the density builds up over the granule until the pressure is raised sufficiently to expel it. The excess pressure also decelerates the upflow and thus reduces the energy transport to the surface, in particular near the granule center, which then cools (Massaguer and Zahn, 1980). Hence, as granules grow, the upflow velocity near their center decreases and they develop an edge brightened appearance.
The horizontal flow, driven by the excess pressure in the interiors of the granules, must be halted when it reaches an intergranular lane, by which time it has cooled, become denser and is being pulled down by gravity. This requires an excess pressure also in the intergranular lanes (Hurlburt et al., 1984).
3.5 Exploding granules
Particularly undisturbed granules that grow to sufficient size tend to develop dark centers, where the vertical flow reverses. These so called ‘exploding granules’ have been the topic of a number of observational and theoretical papers (e.g., Musman, 1972; Allen and Musman, 1973; Namba and van Rijsbergen, 1977; Namba, 1986; Simon and Weiss, 1991; Simon et al., 1991a,b; Rast, 1995; Hirzberger et al., 1999; Roudier et al., 2001).
As per the discussion in Section 3.3 above, the reason for the phenomenon is the increase with size of the pressure at the center of granules, which is necessary to drive the increasing horizontal flow that results from the requirement of mass conservation. The increase of the density associated with the increasing pressure leads to a negative buoyancy, which eventually reduces the vertical velocity down below the value needed to maintain the surface luminosity. The resulting cooling then accelerates the process, leading to a rapid cooling and reversal of the vertical flow.
3.6 Surface entropy jump
Radiation losses and transport near the solar surface not only produces the low entropy, high density fluid that gravity pulls down to drive the convective motions, but also controls what we observe on the Sun. The surface occurs where photons can escape, so neither the diffusion approximation nor the optically thin approximations are valid. The radiation heating/cooling must be obtained by solving the transfer equation for the radiation. Since the temperature varies both in the vertical and horizontal directions, transfer in three-dimensions is needed. Current computers are still not fast enough to solve the full non-LTE (or even LTE) transfer problem at the large number of frequencies needed to cover the continuum and line spectrum for each of the thousands of time steps needed for a reasonable simulation time sequence, so the binning method mentioned in Section 2.4.1 needs to be used.
Significant absorption and emission occurs in spectral lines above the level from which continuum photons escape. Spectral lines typically cool the plasma where their optical depth is of order unity and heat the plasma where their optical depth is large and they block the escape of radiation; this process is known as “spectral line blanketing” (Mihalas, 1978). Ascending plasma from granule interiors would otherwise adiabatically cool as it expands going from the visible surface to the temperature minimum (∼ 4 scale heights) by a factor of > 3, or from ∼ 6000 K to < 2000 K. This does not happen because the ascending gas is heated by absorbing radiation in optically thick lines and UV continua. Such absorption blocks some of the radiation from escaping producing a greenhouse effect called “back warming”. The photons absorbed above the τ500 = 1 surface reheat the photospheric plasma which then radiates both upward and downward heating the surface above the temperature it would have if all the radiation could escape to space. Realistic horizontal transport in the presence of evacuated magnetic flux concentrations also requires inclusion of optically thick line opacity, because otherwise the flux concentrations become optically thin and do not absorb photons from the hot granule walls and so do not show up as bright points at disk center and faculae toward the limb (Steiner and Stenflo, 1990; Vögler, 2004).
3.7 Temperature fluctuations
3.8 Average structure
In general the average structure of 3D models agrees very closely with mixing-length models of the solar envelope, with the main function of the mixing-length parameter being to calibrate the magnitude of the entropy jump at the surface (Rosenthal et al., 1999; Brandenburg et al., 2005).
A 20 Mm deep simulation also contains the entire hydrogen ionization region, helium first ionization and most of the helium second ionization regions (Figure 16). The adiabatic index becomes very small in the hydrogen ionization zone (reaching a minimum of 1.13). The second helium ionization has only a small effect, producing a plateau at Γ = 1.55.
Supersonic convective flows were first studied by Cattaneo et al. (1990) and Malagoli et al. (1990) in simple models with polytropic stratification and no radiative energy transfer. They predicted that supersonic flows are most likely to occur close to the photosphere in the case of convection in real stars; the horizontal pressure fluctuations need to be large, and the radiative (or boundary) cooling needs to be efficient, so the sound speed can be efficiently reduced while the flow is being accelerated. This prediction agrees well with what has later been found in models with detailed radiative transfer and non-ideal equations of state (Nordlund and Stein, 1991; Stein and Nordlund, 1998; Wedemeyer et al., 2003, 2004; Schaffenberger et al., 2006; Hansteen, 2008).
3.11 Energy fluxes
3.12 Connections with mixing length recipes
The simplest model of turbulent convection is called “Mixing Length Theory” (MLT; see Böhm-Vitense, 1958). It is extensively used in stellar evolution calculations where the convection zone must be modeled many times during the course of a star’s evolution. MLT is based on an unphysical picture of moving fluid parcels in pressure equilibrium with their surroundings. In a convectively unstable region the entropy increases inward. A fluid parcel moving upward adiabatically will have higher entropy than the surrounding mean atmosphere and so has a higher temperature and lower density than the mean stratification, so it is buoyant and is accelerated upward. Conversely, a fluid parcel moving downward adiabatically will have lower entropy than its surroundings and so has a lower temperature and higher density than the mean stratification and is pulled down by gravity. After moving some characteristic distance ℓ (the mixing length) the fluid parcel dissolves back into its surroundings. The mixing length, ℓ, determines the magnitude of the temperature fluctuations, flow velocities, and consequently the energy flux. Typically this mixing length parameter is taken to be some multiple of the pressure scale height or as the distance to the boundary of the convectively unstable region.
In the first place, this is an unphysical picture. Turbulent convection is best pictured as upflows and downdrafts that can extend over many pressure and density scale heights. MLT is a local theory, where the flow velocities and temperature fluctuations are determined by local conditions. In reality, the entropy and temperature fluctuations are determined primarily by radiative cooling in the thin surface thermal boundary layer.
4 Larger Scale Flows and Multi-Scale Convection
Traditionally, larger scale flows at the solar surface have been classified into mesogranulation, with horizontal scales of the order of 5–10 Mm, supergranulation, with horizontal scales of the order of 20–50 Mm, and giant cells, with horizontal scales of the order of 100 Mm or larger. However, as discussed in more detail below, this distinction is largely of historical origin, and current evidence indicates that there is a continuous spectrum of motions, on all scales from global to sub-granular.
The larger flow scales are related to, and unavoidable consequences of, the larger scale convective flows that exist in the subsurface layers of the solar convection zone. Since the scales of these subsurface flows increase continuously with depth it is also hard to imagine a physical reason that would cause distinct flow scales at the solar surface. Already the fact that there is only a factor ∼ 2 gap between the scales traditionally assigned to granulation, mesogranulation and supergranulation makes it very difficult to imagine a true scale separation.
Below, we first summarize the observational evidence categorized by the traditional flow scale labels, ‘mesogranulation’, ‘supergranulation’, and ‘giant cells’. We then consider multi-scale observations and show that unbiased multi-scale observations, as are available, e.g., from SOHO/MDI, show a smoothly increasing spectrum of velocity amplitudes with increasing wave number (decreasing size), incorporating in a continuous manner the scales with the traditional labels.
Mesogranulation was first detected by Larry November and coworkers (November, 1980; November et al., 1981, 1982), and has since been the subject of a large number of observational papers; (e.g., Oda, 1984; Simon et al., 1988; Brandt et al., 1991; Muller et al., 1992; Roudier et al., 1998; Shine et al., 2000; Roudier and Muller, 2004; Leitzinger et al., 2005). There has also been a heated debate as to the nature of mesogranulation, and about whether mesogranulation is a distinct scale of convection or not; (e.g., Wang, 1989; Deubner, 1989; Stein and Nordlund, 1989; Ginet and Simon, 1992; Straus et al., 1992; Ploner et al., 2000; Cattaneo et al., 2001; Rast, 2003). A recent example of the techniques commonly used to observe mesogranulation is Leitzinger et al. (2005), who used correlation tracking and a running (1 h) mean in time to map the horizontal component of the mesogranulation velocity field, and to study the drifts of mesogranules in the supergranular velocity field. Others (e.g., Brandt et al., 1988; Title et al., 1989; DeRosa and Toomre, 1998; Roudier et al., 1999; Shine et al., 2000; DeRosa and Toomre, 2004) have used similar techniques to study flows on scales ranging from meso- to supergranulation. Inherent in these techniques is that, because of the spatial and temporal averaging, smaller spatial scales and shorter time scales cannot be adequately resolved. In a situation where the velocity amplitudes increase with decreasing motion scale, such a cut-off automatically (but incorrectly) brings out the smallest scale that is adequately resolved as a dominant, distinct scale.
Another inherent limitation of the tracking techniques is the assumption that the granule-scale fragments and features used in the tracking process are actually good tracers of the solar surface velocity field. This is not necessarily the case, since granules are active flow features, whose horizontal motions presumably also sample the conditions over a range of depths. Rieutord et al. (2001) compared the velocity field derived from tracking granule motions in numerical simulations to the actual meso-scale velocity field of the plasma in the simulations. They conclude that neither velocity fields at scales less than 2500 km nor time evolution at scales shorter than 30 min can be faithfully measured by granule tracking, but that at larger scales and over longer time scales granule tracking works well.
Supergranulation was first observed by Hart (1956), and was later studied in more detail (and named) by Leighton et al. (1962) and by Simon and Leighton (1964), who emphasized the close correlation between the supergranulation and the chromospheric network.
A large number of works have studied supergranular scale flows observationally and tried to characterize them in terms of, for example, flow velocity as a function of cell size, morphology, clustering, advection of smaller scale flows, divergence and convergence, etc.; some of the most significant ones are, e.g, Deubner (1971); Muller et al. (1992); Hathaway et al. (2000); Zhao and Kosovichev (2003); Del Moro et al. (2004); DeRosa and Toomre (2004) and Meunier et al. (2007).
Numerical hydro- and MHD-simulations, with realistic equations of state and radiative energy transfer, are now reaching the scale of supergranulation (Rieutord et al., 2002; Stein et al., 2006a, 2005, 2006b; Ustyugov, 2006; Stein et al., 2007b; Ustyugov, 2007, 2008), and it is thus becoming possible to make quantitative comparisons between observations and simulations. A great benefit of this approach is that if a good match is obtained between the numerical simulations and observations, one may with some confidence use the numerical simulations as proxies or predictions of the subsurface and small scale structure. This allows, e.g., local helioseismology techniques to be further improved and calibrated (Georgobiani et al., 2004; Zhao et al., 2007a; Georgobiani et al., 2007; Birch et al., 2007).
4.3 Giant cells
Using local helioseismic techniques, flows on scales larger than supergranulation are now routinely detected and mapped (Hindman et al., 2004b; Featherstone et al., 2004; Hindman et al., 2004a; Featherstone et al., 2006; Hindman et al., 2006).
4.4 Multi-scale convection
With access to extended time series of uniform quality data, especially from SOHO/MDI, but also from TRACE and the best Earth-based observatories, it became possible to simultaneously record motions over a range of scales, and to for example study the advection of mesogranulation scale cells by supergranulation scale motions.
Muller et al. (1992) used a 3 h time series from the Pic du Midi observatory to study the evolution and motion of mesogranules. They were able to show that mesogranules are advected by supergranulation scale motions, and also that the evolution of mesogranules is influenced by their location in larger supergranular scale flows. The evolution of supergranular scale flow fields and their influence on mesogranular scale flows were also studied by DeRosa and Toomre (1998), by Shine et al. (2000), and by DeRosa and Toomre (2004). The latter study provides evidence that flows on different scales are to some extent self-similar, and that they have amplitudes that vary smoothly with spatial and temporal scales. Figure 5 of DeRosa and Toomre (2004) illustrates the smooth behavior of the amplitudes characterizing multi-scale solar convection; over a range of time averaging windows extending from 0.1 to 100 hours the velocity amplitude varies smoothly, deviating from a linear fit in the log-log diagram by less than 10%. Figure 14 of DeRosa and Toomre (2004) illustrates the self-similar behavior, in that folding the data with Gaussians of varying size still leaves the size distributions essentially unchanged, after normalization with the filter size.
Other studies also indicate a close similarity and coupling between patterns at different scales. Schrijver et al. (1997a) compared the cellular patterns of the white light granulation and of the chromospheric Ca ii K supergranular network. They matched the patterns to generalized Voronoi foams and concluded that the two patterns are very similar. Roudier et al. (2003a) and Roudier and Muller (2004) demonstrated that a significant fraction of the granules in the photosphere are organized in the form of ‘trees of fragmenting granules’, which consists of families of repeatedly splitting granules, originating from single granules at their beginnings (see also Müller et al., 2001). Trees of fragmenting granules can live much longer than individual granules, with lifetimes typical of mesogranulation; this illustrates that larger scale flows are able to influence and modulate the evolution of smaller scale flows.
A smooth spectrum of motions was anticipated on theoretical grounds from the very beginning, but at the time it was thought that observations showed otherwise; observations were interpreted as though granulation and supergranulation represented distinct scales of motion, with no significant motions at intermediate scales. This lead to theoretical suggestions specifically aimed at explaining such a state of affairs (e.g., Simon and Weiss, 1968).
The numerical experiments by Stein and Nordlund (1989) showed that the topology of convection beneath the solar surface is dominated by effects of stratification, which lead to gentle, expanding and structure-less warm upflows on the one hand, and strong, converging filamentary cool downdrafts, on the other hand. The horizontal flow topology was shown to be cellular, with a hierarchy of cell sizes; granulation being a shallow surface phenomenon associated with the small scale heights near the surface layers, while deeper layers support successively larger cells. The downflows of small cells close to the surface merge into filamentary downdrafts of larger cells at greater depths. It is the radiative cooling at the surface that provides the entropy-deficient material which drives the circulation. The supply of high entropy fluid from the bottom of the convection zone is maintained by diffusive radiative energy transfer from the central, nuclear burning region, and in this sense the heat supply from below maintains the energy flux through the convection zone, but it is the surface that is responsible for generating the entropy contrast patterns that, via the corresponding buoyancy work, maintains the structure of the convective motions.
DeRosa et al. (2002) constructed the first models of solar convection in a spherical geometry that could explicitly resolve both the largest dynamical scales of the system (of order the solar radius) as well as smaller scale convective overturning motions comparable in size to solar supergranulation (20–40 Mm). They found that convection within these simulations spans a large range of horizontal scales, especially near the top of the domain, where convection on supergranular scales is apparent. The smaller cells are advected laterally by the larger scales of convection, which take the form of a connected network of narrow downflow lanes that horizontally divide the domain into regions measuring approximately 100–200 Mm across. Correspondingly, Stein et al. (2006b,a) found a similarly wide range of scales of motion, with a smooth amplitude spectrum consistent with SOHO/MDI observation (Georgobiani et al., 2007), in simulations covering scales from sub-granular to supergranular.
Figure 22 shows a composite of simulated and observed velocities, over a range of scales that extends from global to below granulation scales. Note that the velocities on granular, mesogranular, and supergranular scales are consistent with commonly adopted values (a few km s−1 on granular scales, a few several hundred m s−1 on mesogranular scales, and 100–200 m s−1 on supergranular scales). Note that the velocity spectrum is approximately linear in k, and that it continues to giant cell scales, with no distinct features on scales larger than granulation.
5 Spectral Line Formation
Since the convection zone reaches up to the optical surface in the Sun, convection directly influences the spectrum formation both by modifying the mean stratification and by introducing inhomogeneities and velocity fields in the photosphere. Traditionally, convection is incorporated in classical 1D theoretical model atmospheres through the rudimentary mixing length theory (Böhm-Vitense, 1958) or some close relative thereof (e.g., Canuto and Mazzitelli, 1991). These convection descriptions are local, 1D, time-independent and ignore crucial 3D energy exchange effects between the radiation field and the gas (see Section 3.12). Reality is very far from this simple-minded picture. It should not come as a surprise then that the predicted emergent spectrum based on such 1D model atmospheres may be hampered by significant systematic errors.
As an alternative to 1D theoretical model atmospheres solar physicists often prefer the use of semi-empirical model atmospheres such as the Holweger and Müller (1974), Vernazza et al. (1976) (better known as VAL3C) and Fontenla et al. (2007) models, in which the temperature structure is inferred from observations, notably continuum center-to-limb variation and strengths of various spectral lines; the first of these model atmospheres is based on an LTE inversion while the others account for departures from LTE. While the uncertainty in the temperature structure arising from convective motion is thus hopefully largely bypassed, such semi-empirical models are still 1D and, like all 1D models, predict insufficient line broadening that require the introduction of the fudge parameters micro- and macroturbulence (see Gray, 2005, for a general discussion). Such semi-empirical models are not easily constructed for stars other than the Sun due to the inability to observe center-to-limb variations on stars.
Going somewhat beyond the standard 1D modeling, it is possible to combine several 1D atmosphere models corresponding to, for example, typical up- and down-flows to construct multi-component models of solar/stellar granulation (Voigt, 1956; Schröter, 1957; Nordlund, 1976; Kaisig and Schröter, 1983; Dravins, 1990; Borrero and Bellot Rubio, 2002). Bar a few notable exceptions, little effort has been invested in studying the possible impact of multi-component models on solar/stellar abundance determinations (Lambert, 1978; Hermsen, 1982; Frutiger et al., 2000; Bellot Rubio and Borrero, 2002; Ayres et al., 2006).
A more ambitious approach is to make use of the multi-dimensional, time-dependent radiative-hydrodynamical simulations of solar surface convection that are described in this review. These simulations may then be employed as a 2D or 3D solar model atmosphere for spectral line formation purposes (e.g., Dravins et al., 1981; Nordlund, 1985a; Dravins and Nordlund, 1990a,b; Bruls and Rutten, 1992; Atroshchenko and Gadun, 1994; Gadun and Pavlenko, 1997; Gadun et al., 1997; Uitenbroek, 2000a; Asplund et al., 2000a,b,c; Asplund, 2000; Asplund et al., 2004; Asplund, 2004; Asplund et al., 2005a,b; Asplund, 2005; Allende Prieto et al., 2001, 2002; Steffen and Holweger, 2002; Scott et al., 2006; Ljung et al., 2006; Ludwig and Steffen, 2007; Caffau and Ludwig, 2007; Caffau et al., 2007a,b, 2008b,a; Mucciarelli et al., 2008; Centeno and Socas-Navarro, 2008; Ayres, 2008; Basu and Antia, 2008; Meléndez and Asplund, 2008). For computational reasons, most of the calculations have assumed LTE but some studies have braved going beyond this approximation, either in 1.5D2 (e.g., Shchukina and Trujillo Bueno, 2001) or in 3D (e.g., Nordlund, 1985b; Kiselman and Nordlund, 1995; Kiselman, 1997, 1998, 2001; Uitenbroek, 1998; Barklem et al., 2003; Asplund et al., 2003, 2004; Allende Prieto et al., 2004). A recent review on 3D spectral line formation is given in Asplund (2005).
5.1 Spatially resolved lines
At disk-center, individual line profiles corresponding to different atmospheric columns are largely symmetrical, although the line bisectors in upflows typically have a slight-shape while downflows show slightly more pronounced/-bisectors due to the increasing vertical velocities with depth (Figure 26). Closer to the limb the spatially resolved line profiles become quite distorted. In the line-forming region, the horizontal velocities are typically somewhat larger than the vertical velocities and indeed approach and occasionally exceed the sound speed. The total velocity span of profiles at inclined viewing angles is therefore substantially larger than at disk-center and the lines can develop multiple components as the line-of-sight passes inhomogeneities with distinctly different velocity shifts as well as temperatures. As a consequence the spatially resolved bisectors become very distorted and depend critically on the exact region of the surface that is being observed.
5.2 Spatially averaged lines
5.3 Solar abundance analysis
Due to their realistic nature and successful reproduction of key observational constraints, 3D hydrodynamical model atmospheres can be used for elemental abundance determinations. These will then hopefully be more reliable than corresponding 1D analyses since the traditional free parameters used in stellar/solar spectroscopy (mixing length parameters, micro- and macroturbulence) no longer enter into the line formation calculations and because line profile fitting may be routinely employed instead of relying on often uncertain equivalent width measurements. In a series of papers (Allende Prieto et al., 2001, 2002; Asplund, 2000, 2004; Asplund et al., 2000c, 2004, 2005a,b; Scott et al., 2006), a re-analysis of the solar chemical composition was made, using a carefully constructed 3D solar convection simulation with the correct nominal solar effective temperature (Asplund et al., 2000b). The 3D radiative transfer equation was solved at each time-step of the simulation using a realistic equation-of-state (Mihalas et al., 1988) and opacities (Gustafsson et al., 1975; Kurucz, 1993) with the opacity binning technique (Nordlund, 1982). For most elements, LTE was assumed in the line formation calculations but for Li i and O i full 3D non-LTE calculations have been performed (Asplund et al., 2003, 2004). In these studies special effort has been made to adopt the most reliable and up-to-date atomic and molecular input data, such as transition probabilities, partition functions, line broadening parameters and dissociation energies. In the comparison with observations, both disk-center and disk-integrated flux profiles have been used, which give nearly identical results. Similar solar abundance analyses for selected elements assuming LTE have been carried out using an independent 3D hydrodynamical solar model atmosphere computed with the CO5BOLD code (Freytag et al., 2002) with in general good agreement with those presented by Asplund and collaborators when accounting for differences in the adopted atomic line data (Ludwig and Steffen, 2007; Caffau and Ludwig, 2007; Caffau et al., 2007a,b, 2008a,b; Mucciarelli et al., 2008).
Of all elements carbon has the most diverse set of indicators that can be employed in deriving a solar elemental abundance. The diagnostics include both high-excitation permitted and low-excitation forbidden C I lines as well as a number of different molecular features, including CH vibration-rotation lines in the infrared, CH electronic lines and C2 electronic transitions in the optical (e.g., Lambert, 1978; Asplund et al., 2005b); CO vibration-rotation lines may also be used but it requires prior knowledge of the solar O abundance (Scott et al., 2006; Ayres et al., 2006). These are all formed in very different atmospheric layers and have distinct dependencies to the atmospheric conditions, in particular the temperature. As some lines are quite weak while others are partly saturated they also have different sensitivity to the predicted convective velocities. For these reasons, achieving consistent results from all different abundance indicators is challenging and provides a very stringent test of the appropriateness of the adopted model atmosphere, line formation calculations and input data for the transitions.
The derived solar C, N, and O abundances based on different atomic and molecular indicators using a 3D hydrodynamical model of the solar atmosphere (Asplund et al., 2004, 2005b,a; Scott et al., 2006, Asplund et al., in preparation), together with the corresponding 1D results for the theoretical MARCS model atmosphere (Asplund et al., 1997) and the semi-empirical model of Holweger and Müller (1974). The quoted uncertainty is only the line-to-line scatter.
8.36 ± 0.03
8.39 ± 0.03
8.35 ± 0.03
CH, Δυ = 1
8.38 ± 0.04
8.53 ± 0.04
8.42 ± 0.04
8.45 ± 0.04
8.59 ± 0.04
8.44 ± 0.04
8.44 ± 0.03
8.53 ± 0.03
8.46 ± 0.03
CO, Δυ = 1
8.40 ± 0.01
8.60 ± 0.01
8.55 ± 0.02
CO, Δυ = 2
8.37 ± 0.01
8.69 ± 0.02
8.58 ± 0.02
7.79 ± 0.08
7.90 ± 0.08
7.86 ± 0.08
NH Δυ = 1
7.79 ± 0.05
8.01 ± 0.05
7.88 ± 0.05
7.81 ± 0.04
8.03 ± 0.04
7.88 ± 0.04
8.68 ± 0.01
8.76 ± 0.02
8.72 ± 0.01
8.64 ± 0.02
8.64 ± 0.08
8.72 ± 0.03
OH, Δυ = 0
8.65 ± 0.02
8.82 ± 0.01
8.83 ± 0.03
OH, Δυ = 1
8.61 ± 0.03
8.87 ± 0.03
8.74 ± 0.03
OH, Δυ = 2
8.57 ± 0.06
8.80 ± 0.06
8.61 ± 0.06
The observed strong vibrational-rotational lines of CO in the Sun, in particular towards the limb, have for a long time been a challenge to model and has been argued to be inconsistent with the canonical temperature rise in the lower chromosphere (e.g., Ayres et al., 2006, and references therein). Departures from LTE in the line formation have been demonstrated to be unimportant and thus can not be invoked to explain the strong CO lines (Uitenbroek, 2000b). The presence of temperature atmosphere inhomogeneities in 3D hydrodynamical surface convection simulations on the other hand strengthens the predicted CO lines into better agreement with observations (Uitenbroek, 2000a; Scott et al., 2006); the typically cooler horizontally mean temperature stratification in 3D modeling than in the Holweger and Müller (1974) semi-empirical model is also conducive for molecular formation. Scott et al. (2006) have shown that the 3D-based C abundance from weak CO lines from the fundamental and first overtone bands are in excellent agreement with the other atomic and molecular C abundance indicators (Table 1). Although the situation is markedly better than with the Holweger and Müller (1974) model, the stronger low-excitation CO lines, however, are still predicted to be too weak with the 3D model, suggesting even lower temperatures around the heights of the nominal temperature minimum than the current generation of 3D models implies. In the solar photosphere up to heights of about 700 km above the optical surface, the time-dependent molecule formation of CO has been shown to proceed according to LTE expectations (Asensio Ramos et al., 2003; Wedemeyer-Böhm et al., 2005), which will therefore not affect the derived abundances. The dynamical time-scale in the higher atmospheric layers are much shorter than the time-scale for radiative relaxation, which prevents a CO-driven self-amplified thermal instability to be operative (Wedemeyer-Böhm and Steffen, 2007).
Since previously most trust has been put in the molecular lines, the study by Asplund et al. (2005b) has resulted in a significant lowering of the recommended solar photospheric C abundance to log ∊C = 8.39±0.05. Compared with the compilations of for example Anders and Grevesse (1989) and Grevesse and Sauval (1998) the new value is lower by 0.17 and 0.13 dex, respectively. While for the molecular lines the dominant factor is the use of a realistic 3D model instead of 1D model atmospheres, it should be emphasized that it is not the only factor causing this downward revision. Indeed, for the [C I] 872.7 nm line an improved transition probability is of equal importance while for the C I lines allowing for non-LTE effects is even more important than the 3D effects. The fact that after these significant improvements (3D model atmospheres, non-LTE line formation and better atomic/molecular line data) the derived abundances are in such excellent agreement is very encouraging. Finally, the derived carbon isotopic ratio (12C/13C = 87 ± 4) from CO lines using the same 3D model as above (Scott et al., 2006) is in excellent agreement with the terrestrial value (89 ± 1) while with 1D models the implied ratio is significantly lower (Scott et al., 2006; Ayres et al., 2006).
Also for N there are both atomic and molecular lines that may be utilized in an abundance analysis, albeit not as many different indicators as for C or O (Asplund et al., 2005a). The forbidden [N I] lines cannot be employed as they are all too weak in the solar spectrum but there are some 20 weak high-excitation N I lines that are suitable for the purpose. In addition there are both vibration-rotation and pure rotation lines of NH in the infrared as well as a multitude of CN transitions from various bands; in the latter case, however, one has to have prior knowledge of the C abundance from other diagnostics. The (still preliminary) results (Asplund, et al., in preparation) in terms of the solar N abundance are given in Table 1. Again, there is very satisfactory agreement between the different indicators in 3D although similar consistency is in fact achieved also with the 1D MARCS model while the Holweger-Müller model returns somewhat larger discrepancies. The mean 3D-based abundance is significantly lower than in previous studies: log ϵN = 7.80 ± 0.05. This is 0.25 dex lower than the value recommended by Anders and Grevesse (1989) and is mainly due to the use of the 3D model with minor changes coming from non-LTE effects for N I, refined atomic/molecular line data and exact choice of lines.
Over the years a large number of studies have been devoted to estimate the solar O abundance, because of its importance in astrophysics stemming from its relatively large abundance, its significant contribution to the opacity in the stellar/solar interior and its diagnostics value for stellar nucleosynthesis and cosmic chemical evolution. In the solar case both atomic and molecular transitions are available for abundance analysis (e.g., Lambert, 1978; Asplund et al., 2004). The high-excitation O I lines (including the well-known 777 nm triplet) are easily measured but are susceptible to significant departures from LTE (e.g., Altrock, 1968; Kiselman, 1993; Kiselman and Nordlund, 1995; Asplund et al., 2004). The forbidden [O I] lines from the ground state at 630 and 636 nm do not suffer from non-LTE effects but are very weak and blended by other lines (Allende Prieto et al., 2001; Asplund et al., 2004; Caffau et al., 2008a; Ayres, 2008). In the infrared there are numerous vibration-rotation as well as pure rotation lines of OH that can be employed (e.g., Sauval et al., 1984; Grevesse et al., 1984; Asplund et al., 2004; Meléndez, 2004); the electronic OH lines in the UV are less suitable for abundance determinations due to the possibility of missing UV opacity (Asplund, 2004). Table 1 summarizes the results of Asplund et al. (2004) based on an analysis using a 3D hydrodynamical solar model atmosphere.
Caffau et al. (2008a) have recently performed an independent analysis of the 630 nm line using an alternative 3D hydrodynamical solar model atmosphere computed with the CO5BOLD code. The two 3D models are based on essentially the same approximations, assumptions and input data but with different and completely independent numerical implementations. The resulting temperature structures are similar, although the CO5BOLD model is slightly warmer in the line-forming region than the 3D model by Asplund et al. (2000b). Caffau et al. (2008a) predict the Ni contribution to the feature by adopting the Johansson et al. (2003) gf-value and the solar Ni abundance of Grevesse and Sauval (1998), which is 0.06 dex higher than the revised value of Scott et al. (in preparation). They find an essentially identical O abundance from this line as Allende Prieto et al. (2001) and Asplund et al. (2004): log ϵO = 8.68±0.15. Ayres (2008) has carried out a similar study but based only on one snapshot of the CO5BOLD 3D solar model. More importantly, he follows the procedure of Allende Prieto et al. (2001) to leave the Ni contribution as a free parameter when obtaining the best fitting line profile, in spite of the new accurate experimental transition probability. He concludes that the Ni line is only 70% as strong as would be expected based on the new gf-value and the Ni abundance of Grevesse and Sauval (1998), a difference which greatly exceeds the quoted uncertainties of these two values. As a result he arrives at a correspondingly higher O abundance: log ϵO = 8.81 ± 0.04.
Centeno and Socas-Navarro (2008) have used a novel alternative method, namely to analyze spectro-polarimetric observations of sunspots and the asymmetry of the Stokes V profile of the [O i] + Ni i blend. They derive an atomic ratio of O/Ni = 210 ± 24, which according to their analysis corresponds to log ϵO = 8.86 ± 0.07 after applying the Ni abundance of Grevesse and Sauval (1998) and correcting for the ∼ 50% of O locked up in CO in the cool environments of sunspots. The asymmetry of the Stokes V profile is relatively model-independent, but the derived oxygen abundance is of course still sensitive to the choice of (gf-values and the C and Ni abundances. Correcting for the use of an outdated gf-value for the [O I] line and employing the new Ni abundance of Scott et al. (in preparation) revises their estimated atomic O abundance from log ϵO = 8.56 to log ϵOI = 8.43. If other molecules can be ignored, as supported by their calculations, the maximum possible total O abundance is obtained by assuming that all of C is tied up in CO. When adopting the solar C abundance of Grevesse and Sauval (1998) this leads to log ϵO = 8.78 while using the value from Asplund et al. (2005b) leads to log ϵO = 8.71, which is quite close to the values found by Allende Prieto et al. (2001) and Asplund et al. (2004). According to the analysis of Asplund et al. (2004), the even weaker [O I] 636 nm line gives results consistent with the 630 nm line, after allowance for two blending CN lines. This is not the case with the analysis of Caffau et al. (2008a), in which the 636 nm line leads to a 0.10 dex higher abundance than the 630 nm line: log ϵO = 8.78 instead of log ϵO = 8.68. The exact reasons for this have not yet been established but it may be partly related to their use of a relatively high solar Ni abundance (Grevesse and Sauval, 1998), which under-estimates the O contribution to the 630 nm feature.
Meléndez and Asplund (2008) has performed a 3D-based analysis of the [O I] line at 557.7 nm not previously considered for abundance purposes. They argue that the significant blending due to two C2 lines can be rather accurately constrained by other nearby C2 lines from the same molecular band with essentially identical excitation potential, line strengths, and thus line formation properties. Using the same 3D solar atmosphere model as employed in the series of papers by Asplund and collaborators, Meléndez and Asplund (2008) find log ϵO = 8.70 ± 0.08. A range of 1D models give however very similar results (e.g., 8.73 with the Holweger and Müller, 1974 model) with the mean value of the tested 3D and 1D models being log ϵO = 8.71 ± 0.02.
An independent 3D-based analysis of the permitted O I lines has recently been performed by Caffau et al. (2008a) using a CO5BOLD solar model. In addition to the lines considered by Asplund et al. (2004) they also include the 1130.2 and 1316.4 nm transitions. They include non-LTE abundance corrections but only computed using 1D models, i.e., not full 3D non-LTE computations as in Asplund et al. (2004). Their weighted mean of the eight O I and two [O I] lines is log ϵO = 8.76 ± 0.07, i.e., a value in between the recommended values of Asplund et al. (2005a) and Grevesse and Sauval (1998). Most of the differences with the atomic-based results of Asplund et al. (2004) can be traced to Caffau et al. (2008a) use of rather efficient H collisions for the 1D non-LTE calculations (Δlog ϵO = 0.03 dex) and large adopted equivalent widths.3 Their measured equivalent widths are particularly large for the O I 777 nm triplet, indeed deviating by > 3σ compared with the eight most recent influential published solar abundance investigations starting from Lambert (1978). The reasons for these differences have not yet been established. Encouragingly, the particular choice of 3D model does not influence the results significantly. Somewhat earlier, Holweger (2001) obtained a similar O abundance based on an analysis taking multi-dimensional effects into account from a 2D solar granulation simulation and 1D non-LTE line formation: log ϵO = 8.74 ± 0.08.
Asplund et al. (2004) selected the best unblended 70 fundamental vibration-rotation (located around 3 μm) and 127 pure rotation lines (> 9 μm) of OH for abundance analysis. The resulting O abundances are much lower than previously thought from 1D studies: log ϵO = 8.61 and 8.65, respectively. This is a direct reflection of the great temperature sensitivity of the OH lines. There are no apparent abundance trends with wavelength, excitation potential or line strength for the vibration-rotation lines but there is a clear correlation with line strength for the strongest pure rotation lines in the 3D analysis. Since these lines are formed in very high atmospheric layers, this is probably a reflection of shortcomings of the 3D model atmospheres at those heights, such as too high temperatures. The derived low O abundance presented by Asplund et al. (2004) has been supported by a study of the extremely weak first overtone vibration-rotation lines of OH (Meléndez, 2004), which are also included in Table 1.
Asplund et al. (2004) argued that the mean 3D-based oxygen abundance from all the different abundance indicators is log ϵO = 8.66 ± 0.05. The agreement between the different diagnostics is excellent in the 3D case (Δϵ ≤ 0.1 dex) while the corresponding 1D results show quite discrepant results (Δϵ ≥ 0.2 dex) with typically the molecular lines implying a higher abundance than the atomic due to the neglect of atmospheric inhomogeneities and the, in general warmer, mean temperature structure. The estimated abundance is 0.27 dex lower than recommended by Anders and Grevesse (1989), i.e., almost a factor of two downward revision. Given the recent flurry of studies devoted to the topic and the somewhat disparate results obtained, it is clear though that the solar O abundance issue has not yet been resolved once and for all. There are discouraging differences in terms of the observational data used, the collisional data required for the O I non-LTE calculations, the treatment of blending lines and the more reliable 3D atmospheric stratification, which urgently need to be addressed.
The estimated isotopic ratio (16O/18O = 479 ± 29) (Scott et al., 2006) from CO lines using a 3D solar model is in very good agreement with the terrestrial value (499 ± 1). In contrast, with 1D models the implied ratio is significantly lower than the terrestrial value (Ayres et al., 2006), which would be rather surprising.
Iron is a standard reference element for the overall metallicity of stars and galaxies and thus the solar Fe abundance is of fundamental importance for astronomy. In the 1980s and 1990s a debate raged whether the solar value was low (log ϵFe ≈ 7.50, e.g., Holweger et al., 1995) or high (log ϵFe ≈ 7.65, e.g., Blackwell et al., 1995) which centered around the appropriate choices for the equivalent widths, microturbulence, pressure broadening constants, and transition probabilities. With the application of a 3D hydrodynamical solar model atmosphere the microturbulence parameter becomes obsolete (Asplund et al., 2000b) while the excellent agreement between predicted and observed line shapes allows the use of profile fitting instead of relying on equivalent widths measurements. Asplund et al. (2000c) performed a 3D LTE abundance analysis with the most up-to-date atomic data of both Fe i and Fe ii lines, in both cases finding a low abundance: log ϵFeI = 7.44 and log ϵFeI = 7.45, respectively. There is, however, some evidence that departures from LTE are not insignificant for Fe i, mainly driven by over-ionization due to hot radiation field from below in the upflows (see discussion in Asplund, 2005, and references therein). As for O I lines, the main uncertainty in the available non-LTE calculations appears to be the treatment of inelastic collisions with H. Some calculations (e.g., Shchukina and Trujillo Bueno, 2001) which ignore these collisions altogether imply 1.5D non-LTE abundance corrections of ≈ +0.05 dex for Fe i for the Sun. On the other hand, Korn et al. (2003) find that the H collisions are indeed efficient in thermalizing the Fe i level populations and, hence, that the resulting 1D non-LTE abundance corrections are very small. The jury is still out regarding the H collisions but it is clear that the solar Fe abundance is low, log ϵFe ≤ 7.55, in particular since the solar Fe ii lines now have quite reliable gf-values and are little affected by the uncertainties surrounding the H collisions.
5.3.5 Other elements
To date at least a preliminary 3D-based abundance analysis of all elements up through Ni in the period table plus Zr, Eu, Hf, and Th has been performed; more elements are continually added to this inventory. For the vast majority of elements, solar spectoscopists have to settle for employing only various types of atomic transitions when attempting to estimate the solar abundances. Furthermore, for most elements only atomic lines from one ionization stage are available while often very few lines are suitable for abundance purposes (indeed, in several cases only a handful or fewer lines can be used). Finally, most of the time the appropriate atomic lines for a given element have quite similar sensitivity to the atmospheric conditions. It is therefore difficult to use the results as an indication of the appropriateness of the adopted model atmosphere and line formation calculations, as can be done for C, N, and O as described above, and the resulting 3D-based solar abundances must therefore be taken at face value.
Asplund (2000) found that the solar photospheric Si abundance only needed a 0.04 dex downward revision compared with previous studies. This is quite a significant finding nevertheless as it sets the absolute abundance scale of meteorites since those are depleted in all volatile elements, including hydrogen. In meteorites elemental abundances are therefore measured relative Si. As a consequence knowledge of the photospheric Si abundance is necessary to anchor the photospheric and meteoritic abundances to the same absolute scale. Of the other elements, Asplund et al. (2005a) reported new abundances of the intermediate elements (Na-Ca) while a study of the Fe-peak elements (Sc-Ni) is currently ongoing (Scott et al., in preparation). Asplund (2004) estimated the amount of missing UV opacity from a comparison of the OH lines in the infrared and UV and from this could conclude that the solar photospheric Be abundance is not significantly depleted compared with the meteoritic value. Equipped with new transition probabilities Ljung et al. (2006) re-derived the solar Zr abundance with a 3D-analysis. The here mentioned works have all been based on the 3D solar model of Asplund et al. (2000c), which was also employed in the solar C, N, and O abundance analyses described in previous sections (Asplund et al., 2004, 2005b,a). For most atoms, the impact of 3D models compared with various often-used 1D models is relatively modest in general in contrast to the very model atmosphere-sensitive molecular lines. Typically the derived solar abundances are 0.02–0.1 dex lower than in a corresponding analysis with the Holweger and Müller (1974) model atmosphere when adopting the same input physics otherwise (Asplund et al., 2005a); lines from a minority ionization stage are the most affected in this regard due to their greater temperature dependence (Steffen and Holweger, 2002; Asplund, 2005). This is mainly a reflection of the steeper mean temperature gradient in the line forming region in the 3D model. For the elements for which two ionization stages can be employed, in general more consistent results are obtained with the 3D solar model compared with 1D model atmospheres, theoretical or semi-empirical. Furthermore, the agreement with the revised meteoritic abundance scale (Asplund, 2000) is typically improved, lending further support to the results.
Similar 3D-based determinations of the solar abundances using a CO5BOLD 3D hydrodynamical solar model atmosphere have recently been carried out. Caffau and Ludwig (2007) and Caffau et al. (2007a) have revisited the solar S abundance and find a value in perfect agreement with that of Asplund et al. (2005a). On the other hand, Caffau et al. (2007b) find a solar P abundance 0.1 dex higher than the preliminary analysis of Asplund et al. (2005a); the exact reasons for this difference have not yet been established. In addition, Caffau et al. (2008b) and Mucciarelli et al. (2008) have determined the solar abundances of Eu, Hf, and Th, typically finding ≈ 0.02 dex lower values than the corresponding results with the Holweger and Müller (1974) 1D semi-empirical model; these elements have not yet been studied by Asplund and collaborators.
5.3.6 Implications and reliability of the new solar abundances
The recent revisions of the solar chemical composition based on 3D hydrodynamical model atmosphere, non-LTE line formation when necessary and improved atomic/molecular input data (e.g., Asplund et al., 2004, 2005b,a) has some major implications, if true. First and foremost, due to the large changes of the solar C, N, and O (and by extension also to Ne and Ar4 abundances in particular, the overall photospheric metallicity is greatly reduced, from Z = 0.0194 (Anders and Grevesse, 1989) or Z = 0.0170 (Grevesse and Sauval, 1998) to Z = 0.0122 (Asplund et al., 2005a). This relatively dramatic downward adjustment brings with it both good and bad news.
The new low solar abundances bring the Sun into agreement with the solar neighborhood as measured by nearby OB-type stars and the local interstellar medium (e.g., Turck-Chièze et al., 2004; Esteban et al., 2005; Przybilla et al., 2008), in particular when considering that the proto-solar abundances must have been 15% higher than the present-day photospheric values due to elemental diffusion (Turcotte et al., 1998). Furthermore, these abundances are consistent with existing models of the Galactic chemical evolution over the past 4.5 Gyr, which predicts that the oxygen abundance in the solar neighborhood has increased with ≈ 0.05 dex since the birth of the Sun. With the old C, N, and O abundances of Anders and Grevesse (1989) and Grevesse and Sauval (1998) there was a sharp conflict between the Sun and its surroundings.
However, the new abundances wreck havoc with the previous impressive agreement between the predicted solar interior sound speed and that measured with helioseismology (e.g., Delahaye and Pinsonneault, 2006; Basu and Antia, 2008). Since in particular C, O, and Ne are significant opacity sources in the solar interior, the reduced abundances change the predicted temperature and density structure of the Sun based on standard solar interior models (e.g., Bahcall et al., 2006). The most noticeable disagreement occurs immediately below the convection zone, which also means that the predicted depth of the convection zone with the new abundances is inconsistent with the value inferred from helioseismology. A great number of possible explanations have been put forward in attempts to resolve the discrepancy, including missing opacity (Badnell et al., 2005), underestimated diffusion (Guzik et al., 2005), accretion of low-metallicity gas (Guzik et al., 2006), internal gravity waves (Young and Arnett, 2005; Charbonnel and Talon, 2005), and underestimated solar Ne abundance (Bahcall et al., 2005; Drake and Testa, 2005) but to date no satisfactory solution has been found to this dilemma.
Another possibility is of course that the new solar abundances are not fully trustworthy and that the real solar C, N, and/or O abundances are significantly higher. Indeed with the values recommended by Grevesse and Sauval (1998) based on an analysis using the modified5 Holweger and Müller (1974) semi-empirical 1D model atmosphere this solar model problem largely disappears. The reasons for the lowering of the C, N, and O abundances are different for the various abundance indictors. As already mentioned, molecular transitions are particularly sensitive to the model atmosphere temperature structure and presence of inhomogeneities. The atomic lines are also dependent on the adopted model atmosphere but to a lesser extent. Instead the permitted lines such as O I are particularly vulnerable to non-LTE effects and thus the treatment of the poorly known inelastic H collisions but also the measurement of equivalent widths/fitting of line profiles. The forbidden lines are mostly dependent on how blends are accounted for. In all cases, improved atomic/molecular data and selection of lines also have played a role.
One of the key ingredients is the application of 3D, hydrodynamical models of the solar atmosphere instead of 1D hydrostatic models, either theoretical or semi-empirical ones. Such 3D models are still relatively new and thus the possibility of remaining problems with the predicted atmospheric structure in the line-forming region cannot yet be ruled out. As shown in this Section, theoretical 3D line profiles, including their shifts and asymmetries, agree almost perfectly with observations in spite of having no tunable free parameters in the modeling (Asplund et al., 2000b). No 1D model comes close to achieving this. Furthermore, the predicted properties of the 3D simulations resemble very closely the observed solar granulation such as geometry, brightness contrast, velocities, both in terms of statistics and the temporal evolution. However, further confrontations with observational constraints are needed to remove lingering doubts about the 3D models’ appropriateness for abundance analysis purposes. For example, Ayres et al. (2006) argued on the basis of continuum center-to-limb variations and CO line strengths that the 3D model employed by As-plund and co-workers has an erroneous temperature structure. Their conclusion is based on their own calculations using the temporally and spatially averaged mean 3D model; full 3D calculations using the same opacities and equation-of-state used for the hydrodynamical convection simulation instead reveal smaller differences between the predicted and observed limb-darkening behavior by about a factor of two (see also discussion in Koesterke et al. (2008)). It is true though that the 3D model by Asplund et al. (2000b) is not perfect in this respect: in terms of the center-to-limb variation it performs better than theoretical 1D MARCS and Kurucz models but not quite as well as the Holweger and Müller (1974) model, as shown by Koesterke et al. (2008) and Pereira et al. (in preparation). However, in the case of the solar hydrogen lines, which also trace the temperature structure in deep atmospheric layers, the 3D predictions agree better with observations than the Holweger-Müller model, which results in too shallow line wings (Pereira et al., in preparation), in particular when considering the possibility of non-LTE effects in the H line formation (Barklem, 2007b). The CO5BOLD 3D solar model employed by for example Caffau et al. (2008a) performs better in terms of continuum center-to-limb variation than the corresponding 3D model of Asplund et al. and, at least, as well as the Holweger-Müller model, yet it implies essentially the same solar oxygen abundance for atomic lines as advocated by Asplund et al. (2004) when allowing for differences in non-LTE corrections and adopted line strengths. The CO5BOLD model has not yet been applied to molecular transitions but it is expected that it will yield somewhat higher abundances than estimated by Asplund et al. (2004), in which the molecular-based O abundances anyway are slightly lower than the atomic-based values. New 3D solar simulations are currently being carried out with an improved treatment of radiative transfer, including a selective (or sparse) opacity sampling technique rather than opacity binning and proper allowance of scattering (Trampedach et al., in preparation; Hayek et al., in preparation).
Finally, it must be emphasized that the application of a 3D solar model atmosphere is only one of the reasons for the recent downward revision of the solar C, N, and O abundances. Important in this respect are allowances of departures from LTE in the line formation for permitted atomic transitions, recognition of significant blends perturbing some weak lines and improved atomic and molecular input data. Even with the Holweger and Müller (1974) model with its shallow temperature gradient and high temperatures in the line-forming region quite low abundances are derived, as seen in Table 1; note that the molecular abundances are over-estimated in all 1D models on account of the neglect of temperature inhomogeneities.
As described above, there is today no clear consensus what the solar oxygen abundance is (unfortunately carbon and nitrogen have not received nearly as much attention). Asplund and collaborators are currently performing a re-analysis of the solar chemical composition for most accessible elements using a new improved 3D solar atmosphere simulation compared to the old 3D model applied in their previous works on the topic. Preliminary calculations reveal that the value recommended by Asplund et al. (2005a) may need to be revised upwards slightly to log ϵO ≈ 8.70–8.75. This would be in line with the recent estimates of Caffau et al. (2008a), Meléndez and Asplund (2008), and Centeno and Socas-Navarro (2008), when taking into account the recommended adjustments to the input data and analyses discussed in detail above. Such an O abundance would ease but not remove the discrepancy between solar interior models and helioseismology.
6 Applications to Helioseismology
The Sun oscillates. Think of a drum. A drum vibrates when it is hit, which excites its resonant modes. Which modes are excited depends on the skin head and where it is hit. The Sun is similar. It has a resonant cavity in which acoustic waves are trapped, like an organ pipe. Resonance occurs when an integral number of half wavelengths fit into the cavity between the turning points where the waves are reflected or refracted. In the Sun acoustic waves are trapped by reflections near the surface where the temperature drops rapidly and the wavelength of the waves becomes larger than the scale height of the density stratification. The waves are trapped in the solar interior where the increasing temperature and sound speed refracts that waves back up toward the surface.
The Sun’s resonant modes are excited by turbulence in the convection zone. In realistic solar convection zone simulations, acoustic oscillations are also naturally excited. Such simulations have been used to study wave propagation properties, resonant mode excitation, effects on mode frequencies, and as a known data set to test, validate and refine local helioseismic inversion methods. Local helioseismology is a powerful tool for probing the subsurface layers of the Sun. For a review of its methods and results see Gizon and Birch (2005) and Zhao (2008).
6.1 Wave propagation in the convection zone
Recently, numerical calculations of linearized wave propagation have been used to study properties for the more complex conditions of the solar atmosphere which is both inhomogeneous and dispersive. There are two important issues: first, because the solar atmosphere is dispersive, waves of different frequencies travel at different speeds (Jefferies et al., 1994), and second, the solar atmosphere’s inhomogeneities in temperature, magnetic field, and density are especially large close to the surface, so they are not small perturbations on a mean background state.
Numerical solutions for linear wave propagation in a mean background solar model with its structure modified so as to be convectively stable (sometimes with a given perturbation) have been used to study various helioseismic issues: How does the dispersive nature of acoustic waves affect their propagation (Tong et al., 2003c; D’Silva, 1998). What is the interaction and coupling between different MHD wave modes (Cally and Goossens, 2007; Rosenthal et al., 2002; Bogdan et al., 2003). How do various types of perturbations and distribution of acoustic sources affect acoustic wave propagation (Tong et al., 2003b,a; Shelyag et al., 2006; Cameron et al., 2007a; Parchevsky and Kosovichev, 2007; Parchevsky et al., 2008). What is the accuracy of time-distance inversion methods — ray tracing and the Born approximation in determining perturbation properties (Hung et al., 2001; Birch et al., 2001; Birch and Felder, 2004; Shelyag et al., 2007). Are there techniques for improving the signal/noise in the observations (Hanasoge et al., 2007). Recently, it has become possible to use realistic models of the near surface layers of the Sun from numerical simulations of solar surface convection to explore the properties of wave propagation and test the accuracy of local helioseismic methods. In such models, p-modes are naturally excited by the convective motions and entropy fluctuations. The mean atmospheric structure is different from the one-dimensional models of the solar atmosphere, which alters the resonant cavity. A complex hierarchy of fluctuations in temperature and sound speed, flow velocities and magnetic flux are present. Such simulations provide the needed test bed of known data against which to test and validate the methods of local helioseismology.
6.2 Excitation of p-modes
Solar oscillations are excited by the convection, both by the entropy fluctuations produced by radiative cooling near the surface and by the Reynolds stresses below the surface where the convective velocities are large (Goldreich et al., 1994). The same processes excite oscillations in convection simulations. In simulations, the spectrum of excited oscillations depends on the size of the computational domain, the wider and deeper it is the richer the spectrum of resonant modes that exist (Figure 33).
Several people have derived equations for p-mode excitation using analytic expressions for the convective kinetic energy and entropy spectra, e.g., (Balmforth, 1992; Goldreich et al., 1994; Samadi and Goupil, 2001). These authors’ formulas and Equation (50) are similar. In all, the excitation rate is proportional to the turbulent pressure and entropy induced pressure fluctuations times the gradient of the displacement eigenfunction divided by the mode mass. The main difference is that Equation (50) contains the absolute value squared of the product of the pressure fluctuations with the mode compression, while the others contain the the absolute value squared of the mode compression times the sum of the absolute values squared of the turbulent pressure and entropy fluctuation contributions. This difference results from the assumption (explicit or implicit) that there are no correlations between the turbulence at different scales and that the mode compression, ∂ξ/∂r, does not change on the length scale of the turbulence and so can be removed from the integral over the local turbulence. Neither of these assumptions is correct and is not necessary. Chaplin et al. (2005) have derived an analytic expression for the mode amplitude where the pressure-mode compression interaction is retained.
Samadi et al. (2003a,b) investigated in detail the effects of different assumptions about the turbulence properties on the p-mode excitation rate. They found that the mode driving is particularly sensitive to the turbulence anisotropy factor and to the dynamic properties of the turbulence as represented by the temporal part of the turbulence spectrum. Using the simulation results, fit by analytic expressions for these factors, increases the excitation rate and the contribution of turbulent pressure relative to that of entropy fluctuations.
In the Sun and cool stars the contributions from the Reynolds stresses and entropy fluctuations are comparable at the peak driving frequencies, while the Reynolds stress excitation dominates at lower frequencies where the excitation occurs over a larger range of depths. In hotter stars and giants the Reynolds stress excitation dominates at all frequencies.
Jacoutot et al. (2008b) and Jacoutot et al. (2008a) have investigated the influence of the presence of magnetic fields on the excitation of p-modes. They find that regions with magnetic field strengths typical of the peripheries of active regions have enhanced excitation of p-modes, particularly at high frequencies. This is consistent with the observations of so-called “acoustic halos” (Braun et al., 1992; Brown et al., 1992; Hindman and Brown, 1998; Jain and Haber, 2002) around active regions.
6.3 p-mode frequencies
6.4 p-mode line profiles
The interaction between the oscillations, convection and radiation is also responsible for the asymmetry of the mode’s spectra and the reversal of asymmetry between velocity and intensity. There is disagreement as to which is the dominant mechanism producing the asymmetry and its reversal. The Green’s function for the response to excitation has an asymmetry that depends on the depth of the source. The superposition of correlated noise to the oscillation signal will produce a reversal of asymmetry between velocity and intensity (Nigam et al., 1998; Kumar and Basu, 1999). Such a reversal can also be produced by radiative transfer effects (Georgobiani et al., 2003). In addition, the vertical oscillation motions move the height of the steep background temperature gradient in the low photosphere up and down, which tends to cancel the effect caused by oscillation induced opacity fluctuations (Straus et al., 2006). These effects dominate over the intrinsic wave properties in controlling the phase relation between temperature and velocity.
7 Interaction with Magnetic Fields
Note that the transportation and concentration of magnetic flux in the solar photosphere — and more generally at a free surface — is a different process than the “flux expulsion” mechanism (Weiss, 1966; Maheswaran, 1969; Peckover and Weiss, 1978; Galloway and Weiss, 1981). Convective motions inside a box with closed boundaries tends to — over a period of time and in the presence of magnetic diffusion — expel magnetic fields from the interior of convection cells. Magnetic field lines penetrating a free surface boundary are efficiently concentrated by purely advective transport, and can subsequently be further concentrated due the suppression of convection associated with a strong magnetic field (Spruit, 1976, 1977a, 1979; Spruit and Zweibel, 1979; Bushby et al., 2008).
7.1 Effects of magnetic fields on convection
7.2 Center-to-limb behavior
As one observes towards the limb several differences in the surface appearance occur (Figure 51 and Figure 11). First, the granules develop a three-dimensional pillow appearance with the granules higher than the intergranular lanes (see Figure 10), with their near sides brighter than their far sides. This is partly due to the true three-dimensional structure of granulation, the granules are observed higher than the intergranular lanes. Bright granules have their continuum optical depth unity 80 km above the mean surface. Partly it is due to the near side being seen more normal to the line of sight and the far side at a more glancing angle.
High resolution observations of solar faculae show that they have an asymmetric contrast profile, with some brightness extending up to one arcsecond in the limbward direction from their peak in brightness (Hirzberger and Wiehr, 2005). The wide contrast profile cannot be explained solely by the “hot wall” effect, as was noted by Lites et al. (2004). The works by Keller et al. (2004) and Steiner (2005) address this issues, with somewhat conflicting but broadly consistent explanations. One conclusion is that the limbward extension of brightness comes from seeing the granule behind the facular magnetic field through the rarefied facular magnetic flux concentration; a circumstance that observers suspected decades ago. The explanation is corroborated by the direct comparisons of observations and simulations by De Pontieu et al. (2006).
7.3 Magnetic flux emergence
Magnetic flux emergence through the solar surface is driven by two processes: buoyancy and advection. Magnetic concentrations with strong fields rise toward the surface first, because they are buoyant (to maintain approximate pressure equilibrium with their surroundings the density inside the concentration must be smaller than in its surroundings), and second, because they are advected by convective upflows. As it rises, a magnetic concentration will encounter convective downflows piercing the upflows on smaller and smaller scales, with downflow speeds significantly larger than the upflow speeds. The portions of the magnetic concentration in the downflows will be dragged down or at least have their upward motion slowed, while the portions still in the upflows continue to ascend rapidly. This leads to the formation of Ω-loops of successively smaller dimensions as the surface is approached. In general, the asymmetry of upflow and downflows (amplitudes and topology) leads to a tendency for downward transport of magnetic flux; a process known as “magnetic flux pumping” (Petrovay and Szakaly, 1993; Tobias et al., 1998; Dorch and Nordlund, 2000, 2001; Tobias et al., 2001). This process is likely to be of central importance for the overall flux balance of the solar convection zone, and allows storage of considerable magnetic flux inside the convection zone proper.
The fact that magnetic fields that are concentrated close to the surface tend to tangle and spread out in many directions below the surface has been demonstrated earlier by Grossmann-Doerth et al. (1998) — see also Vögler et al. (2005) and Schaffenberger et al. (2005).
The ubiquitous occurrence of downflows in the close vicinity but outside magnetic flux concentrations (see for example also Steiner et al., 1998) has been explained in terms of baroclinic flows by Deinzer et al. (1984). The effect has been observationally verified by Langangen et al. (2007).
Emonet and Cattaneo (2001)have shown that dynamo action will occur in a turbulent medium even in the absence of rotation. This led to the suggestion that in addition to the global solar dynamo there is a local, surface dynamo acting in the Sun, a proposition that has recently been confirmed by Vögler and Schüssler (2007). It must be remembered, however, that even though dynamo action occurs in the surface layers of the Sun, these layers are not isolated from the rest of the convection zone. The plasma and magnetic field are mixed throughout the convection zone.
7.4 Convection as a driver of chromospheric and coronal heating
In a broad sense it is of course generally accepted that the solar convection zone is the ultimate driver of the activity that transpires in the solar chromosphere and corona, since the convection zone is the only available source of mechanical energy. Moreover, since upper atmosphere activity and heating is empirically so intimately connected with the presence of magnetic fields, one must also conclude that it is the Poynting flux (the flux of electromagnetic energy) rather than the acoustic or advective/convective kinetic energy flux that is the main transport agent.
From that point of view the question of chromospheric and coronal driving can in principle be reduced to a question of being able to understand and estimate the Poynting flux passing through the solar surface. In practice, this is not an easy task, and one can furthermore convincingly argue that the magnitude of the Poynting flux indeed depends as much on the sink (the chromosphere and corona) as it depends on the source (the subsurface layers of the convection zone).
If one assumes, for example, that one aspect of the subsurface driving is to twist the magnetic field that passes through the surface, then if there is only a weak resistance to the twist, so the twisting motion can proceed with almost no counteracting force, then very little work is performed, and consequently the Poynting flux through the solar surface must be small. Conversely, if there is a large resistance to the twisting motion, and strong counteracting forces develop, that must correspond to a large Poynting flux through the solar surface.
The key factor that regulates the magnitude of the Poynting flux is the angle that the magnetic field lines make relative to the surface, or relative to the motions that attempt to twist the field lines. If there is a strong resistance to the twisting motion the angle is changed — the field becomes twisted and provides a counteracting force, while if there is almost no resistance the field lines remain straight, and the magnetic field hardly transmits any Poynting flux.
Apart from the twist angle, the Poynting flux is also proportional to the energy density of the magnetic field and to the velocity with which the field is being transported. A knowledge of the these three factors on a boundary is in principle enough to compute the Poynting flux through the boundary. Scaling formulae that express these relations were presented by Parker (1983) and van Ballegooijen (1986). The main uncertainty that one encounters when trying to estimate actual heating rates is the angle factor. Galsgaard and Nordlund (1996) found that simple experiments with boundary driving adhere closely to a scaling law that may be interpreted to mean that the twist angle is always of a size that allows the magnetic field lines to twist around their neighbors about once, from end to end. This rule of thumb is consistent with the stability properties of twisted flux tube, which tend to become unstable (to kinking and similar phenomena) when twisted more than about once.
An important consequence of the scaling is that increased resistivity leads to a decrease of the energy dissipation, in that a larger resistivity allows magnetic field lines to diffuse more quickly, straighten out, and hence become less tilted at the driving boundary (Parker, 1988). Conversely, the work and dissipation obtained for a given resistivity is a lower bound on the work and dissipation obtained at very low resistivities. It is this property that gives hope that numerical experiments can provide accurate predictions of chromospheric and coronal heating.
Indeed, Gudiksen and Nordlund (2005a,b) were able to construct a 3D corona model where sufficient heat was provided exclusively by this mechanism. Synthetic diagnostics from the model (Peter et al., 2005, 2006) fits observed values very well.
The models by Gudiksen and Nordlund (2005a,b) had an artificial velocity field, but with statistical properties consistent with the observed solar surface velocity field, from granular to supergranular scales. Using a fully self-consistent subsurface velocity field, evolved simultaneously with the chromospheric and coronal dynamics requires higher numerical resolution, since one must ideally simultaneously resolve scales from sub-granular sizes up to supergranular or active region sizes. The first steps towards this ultimate goal have been taken (Hansteen and Gudiksen, 2005; Hansteen et al., 2007; Abbett, 2007). Similar studies concentrated on the chromosphere have been initiated by Wedemeyer et al. (2003); see also Schaffenberger et al. (2005) and Schaffenberger et al. (2006).
8 Current Status and Future Directions
As discussed in the previous sections of this review, numerical models of the hydrodynamics and magneto-hydrodynamics of the solar surface layers are becoming increasingly realistic and accurate, both because of the steady growth of computing power, and because of the incorporation of increasingly complex physical mechanisms in the models.
The hydrodynamic aspects of the near-surface regions are now largely understood, both qualitatively and quantitatively, but this cannot at all be said about even the most well-known and and well-studied aspect of solar magneto-hydrodynamics; sunspots and the solar activity cycle.
The study of the magneto-hydrodynamics of the solar surface layers, and even more so the study of the overlying chromosphere and corona, which also provides the opportunity to study phenomena that go beyond classical magneto-hydrodynamics, is now the main focus area in solar physics research.
The results and experiences that will come out of these future efforts will be extremely valuable, also for other branches of astrophysics, where the “five dimensions” that are accessible in solar physics (space, time, and wavelength) are projected down to just a few — essentially one dimension when dealing with stars, and essentially only 2D projections of temporal snapshots when dealing with extended objects such as galaxies or molecular clouds.
To continue the progress the realism of the numerical simulations still needs to be improved: radiation, resolution, diffusion, non-LTE excitation, and ionization are all issues that need more work. Especially the chromospheric and coronal heating problems are also going to benefit from advances in the raw computing power, and from improved and and more efficient approximate methods to model radiation and non-equilibrium processes.
For studies that go beyond classical hydrodynamics and magneto-hydrodynamics a range of new tools need to be developed and deployed: particle-in-cell codes that can describe the near-collisionless conditions in the corona with billions to trillions of particles, codes that use hybrid fluid and particle pictures, diagnostic modules that show what would be observed with the range of new instruments and satellites that will be deployed, etc.
9 Summary and Concluding Remarks
Solar convection is the driver that ultimately controls the Sun’s magnetic field, its explosive events, the interplanetary medium and Earth’s weather and space weather. In this review we have discussed the principles of hydrodynamics as they apply to convection near the solar surface.
In Section 3 we discussed the manifestations and properties of the main energy carrying scales, and the granulation pattern that they give rise to. We emphasized the crucial importance of the density stratification in determining the size and evolution of the granulation pattern, and the importance of the radiative cooling at the solar surface as the provider of the entropy deficient plasma that then drives convective motions over a range of scales.
In Section 4 we discussed how larger scale convective patterns arise and are driven, and how they interact with smaller scale patterns. We introduced the concept of a velocity spectrum, and addressed the question of to what extent or not the traditional concepts of meso- and supergranulation represent distinct scales of motion. We showed that both observations and theoretical models give a smooth velocity spectrum, with velocity amplitudes decreasing approximately linearly with horizontal wave number on scales larger than granulation.
In Section 5 we discussed how spectral line synthesis and comparisons with observed spectral line profiles may be used to assess the accuracy of numerical simulations and how the resulting spectral line profiles may be used to accurately determine solar chemical abundances. We conclude that 3D models with numerical resolutions of the order of 2003 reproduce the widths, shifts, and shapes of photospheric spectral lines with high accuracy, and that the chemical abundances derived from such models have a higher degree of internal consistency with less scatter and fewer systematic trends than the ones obtains from traditional abundance analysis, based on empirical or theoretical one-dimensional model atmospheres.
In Section 6 we discussed applications to global and local helioseismology; wave excitation and damping, the influence of convection on the mean structure, and helioseismic diagnostics related to convective flow patterns. We showed that solar oscillations with realistic overall amplitudes are spontaneously excited in numerical simulations of the solar surface layers, and that solar surface convection directly influences the frequency of the oscillations by changing the size of the resonant cavity.
In Section 7 we discussed the interaction of solar surface convection and solar magnetic fields. In active regions, the presence of magnetic fields causes the brightening of plage regions towards the limb, and increased possibilities for convection to heat the chromosphere and corona.
Finally, in Section 8, we briefly discussed open questions and directions for future work, and emphasized the importance of the Sun as a test bed for astrophysical hydrodynamics, magneto-hydrodynamics, and beyond.
In conclusion, we submit that the Sun is a superb laboratory for investigating dynamical as-trophysical processes and especially plasma physics. Combinations of observations and numerical simulations can provide detailed insights into these physical processes, and can also provide data sets that may be used to validate observational analysis methods and to design future observing programs.
Erratum 2009-04-15: corrected +P(∇ · u) to −P(∇ · u)
1.5D denotes the case when considering each column in a 2D or 3D model as a separate entity and, thus, ignoring all non-vertical radiative transfer.
We note that the equivalent widths quoted in Asplund et al. (2004) are not measured from the observed spectrum but theoretical values from the best-fitting 3D non-LTE line profiles.
The solar Ne and Ar contents are not inferable from the photospheric spectrum. Instead these abundances are deduced from the coronal Ne/O and Ar/O ratios coupled with the photospheric O abundance, assuming that these ratios are the same in the photosphere and the corona.
In their analysis of the C, N, and O, Grevesse and Sauval (1998) allowed for modifications of the temperature structure of the Holweger and Müller (1974) model in order to remove otherwise unavoidable trends with excitation potential and line strengths for certain species, see also Grevesse and Sauval (1999) for further details.
RFS was supported by NASA grants NNG04GB92G and NAG 5-12450 and by NSF grants AST0205500 and AST0605738. ÅN is grateful to JILA and to the Danish Natural Science Research Council (FNU) for support during a one year sabbatical at the University of Colorado, Boulder. The authors’ simulations were performed on NASA’s project Columbia computer system, at the National Center for Supercomputing Applications, which is supported by the National Science Foundation, at Michigan State University and at the Danish Center for Scientific Computing, Copenhagen.
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