List of Figures

View Image Figure 1:
Schematic figure showing the average pressure and mass density stratification of the Sun.
View Image Figure 2:
Image of granulation in the G-continuum, showing hot, bright rising fluid surrounded by cooler, darker intergranular lanes. Granules tile the solar surface and are the dominant feature of solar surface convection. Also shown are a few magnetic concentrations, visible as strings of bright beads along the intergranular lanes (image from the Swedish 1m Solar Telescope and Institute of Theoretical Astrophysics, Oslo).
View Image Figure 3:
Temporal history of typical fluid parcels that reach the surface: height z (Mm), optical depth τ, log(ρ), radiative heating Qrad (103 erg/gm/s), internal energy E (105 erg/gm), specific entropy S (108 erg/gm/K), fraction ionization to total energy, and vorticity ω (10–2 s–1). Time is counted from when the parcel rises through unit optical depth. When fluid reaches optical depth τ ∼ 100, it begins to cool rapidly as the gas starts to recombine. Its entropy and energy drop so quickly that its density increases. As it passes above the surface a small amount of radiative reheating occurs and its entropy increases slightly. When it passes back down through optical depth unity it cools some more with a further drop in entropy. As it heads down into the interior it heats up by adiabatic compression and by diffusive energy exchange. The deeper it gets, the more adiabatic its motion becomes (from Stein and Nordlund, 1998).
View Image Figure 4:
Buoyancy work at a depth of 0.5 Mm as a function of (a) vertical velocity (downflows are positive) and (b) fluid entropy. Most of the convective driving below the surface occurs in the low entropy downflows produced by radiative cooling at the surface (from Stein and Nordlund, 1998).
View Image Figure 5:
Histogram of the entropy (logarithmic color scale with arbitrary units) as a function of depth. Most of the area of a horizontal plane below the surface is occupied by upward moving fluid with close to the maximum entropy. Entropy fluctuations are largest at the surface and decrease with increasing depth due to entrainment of entropy neutral material and, in the case of the simulations, numerical energy diffusion (which however is insignificant in this context – most of the entrainment is due to the overturning forced by mass conservation). Entropy increases above the surface because radiation from the surface heats the gas much above the temperature it would attain if moving adiabatically (from Stein et al., 1997).
View Image Figure 6:
Horizontal slices of vertical velocity (light is downward, dark upward) at depths of 0, 2, 4, 8, 12, and 16 Mm (across then down). Each image is 48 × 48 Mm. Note the gradual and continuously increasing area of the upflows, as outlined by the surrounding downflow lanes, with increasing depth.
Watch/download Movie Figure 7: (mpg-Movie; 37871 KB)
Movie Vertical velocity (red upward, blue downward) and streaklines (seen more clearly in the movie) in a vertical cut through the 24 Mm wide simulation domain. Diverging upflows sweep downflows toward each other at the boundaries of the larger, deeper lying upflows (movie by Chris Henze, NASA Advanced Supercomputing Division, Ames Research Center).
View Image Figure 8:
Velocity and temperature structure of a granule with a corner cutout. A granule resembles a fountain with warm fluid rising up near the center, diverging horizontally, cooling, being pulled back down by gravity in surrounding intergranular lanes (from Stein and Nordlund, 1998).
Watch/download Movie Figure 9: (mov-Movie; 123751 KB)
Movie Entropy fluctuation from a solar convection simulation. Horizontal scale is 6 Mm and the vertical scale range from the temperature minimum to 2.5 Mm below the surface.
View Image Figure 10:
Height of τ = 1 vs vertical velocity (downflows are positive) in the quiet Sun. The rms variation in the τ = 1 surface is ≈ 35 km.
Watch/download Movie Figure 11: (mov-Movie; 48052 KB)
Movie The emergent intensity from a magneto-convection simulation showing changed appearance as one approaches the limb (movie by Mats Carlsson, Oslo).
View Image Figure 12:
Correlation of radiation temperature at 1.6 μm with gas temperature at the depth where ⟨τ ⟩ = 1 1.6μm. We never see the radiation from the high temperature gas because it lies at large local optical depth due to the great temperature sensitivity of the H opacity (10 κ ∝ T) (from Stein and Nordlund, 1998).
View Image Figure 13:
Temperature as a function of geometric depth at several horizontal locations plus the average temperature profile (dashed). Locally the temperature profile is much steeper than the average profile (from Stein and Nordlund, 1998).
View Image Figure 14:
Temperature as a function of optical depth at several horizontal locations plus the average temperature profile (dashed). On an optical depth scale, the temperature profile is similar at all places in the simulation domain, whether in warm upflows or cool downflows. This is because the opacity depends very strongly on temperature, and hence a certain optical depth is reached at nearly the same temperature, whether the temperature rises rapidly (as in upflows) or more slowly (as in downflows) (from Stein and Nordlund, 1998).
View Image Figure 15:
Mean atmosphere structure: T (K), ρ (10–7 g/cm3), P (105 dyne/cm2), S (arbitrary units).
View Image Figure 16:
Mean atmosphere structure: Γ 1, H ii, He ii, He iii.
View Image Figure 17:
Rendering of vorticity around a single granule showing antiparallel vortex tubes (green, opaque surfaces) at the edges of the intergranular lanes (near the right hand side edge) and a ring vortex at the head of a downdraft with two trailing vortex tubes leading up to the surface (center left). The transparent red and blue shows the velocity divergence ∇ ⋅ u red (positive) identifies the diverging flow inside the granule while blue (negative) identifies the converging flow in the intergranular lanes.
Watch/download Movie Figure 18: (mov-Movie; 29987 KB)
Movie Magnitude of the vorticity (entrophy) in a single downdraft. Top of the image is the visible solar surface, bottom is 2.5 Mm below the surface. The vertical scale is stretched. Horizontal tickmarks are 237 km apart.
View Image Figure 19:
Maximum horizontal and vertical flow Mach numbers as a function of depth.
View Image Figure 20:
Mach numbers in horizontal planes (contours at Mach number = 1, 1.2, 1.4, 1.6) for vertical (top) and horizontal flow (bottom) superimposed on images of the vertical velocity at the surface (left) and 1 Mm below the surface (right) (velocity scale on right in  km s–1).
View Image Figure 21:
The average thermal, ionization, acoustic, and kinetic fluxes plus their sum, the total energy flux, as a function of depth. The thermal plus ionization energy fluxes together are the internal energy flux (not plotted), and this plus the acoustic flux constitutes the enthalpy or convective flux. The enthalpy flux plus the kinetic energy flux is the total energy flux transported by fluid motions. (The viscous flux is very small.) Energy is transported upward through the convection zone near the surface mostly as ionization energy (∼ 2/3) and thermal energy (∼ 1/3). The kinetic energy flux is downwards and is 10 – 15% of the total flux near the surface. At larger depths (outside of this plot) both the upward enthalpy flux and the downward kinetic energy flux increase, with the kinetic energy flux reaching about the net solar flux and the enthalpy flux reaching about twice the net solar flux.
View Image Figure 22:
Observed and simulated horizontal velocity amplitudes over a wavenumber range extending from global scales to below granulation scales. Observed velocities are from correlation tracking of TRACE and SOHO white light images (Shine, private communication), and from SOHO/MDI Doppler image modeling (Hathaway et al., 2000, and private communication). Simulation results are from Stein and Nordlund (1998) (granulation scales – orange symbols) and Stein et al. (2006a,b) (supergranulation scales – black symbols).
View Image Figure 23:
Horizontal and vertical components of the convective and oscillatory components (separated by sub-sonic filtering) of the velocity field in a supergranulation scale convection simulation, compared with the convective and oscillatory components of the line-of-sight velocity field, as observed with SOHO/MDI in high resolution mode. The ‘convective’ and ‘oscillatory’ (‘modes’ in the figure) are separated by a line ω = ck, where c = 7 km s–1 (from the same data set as was used in Stein et al., 2006a).
View Image Figure 24:
Subsonic filtered line-of-sight velocities from SOHO, Gaussian filtered to the same effective number of pixel elements, for physical sizes of 400, 200, 100, and 50 Mm. The order of the panels has been scrambled, to make the point that the patterns are very similar, and that it is not obvious how to order the panels in a sequence of increasing size, for example (but see Figure 25).
View Image Figure 25:
Subsonic filtered line-of-sight velocities from SOHO, with the same order of panels as in Figure 24, but without applying the Gaussian filter that was used there.
View Image Figure 26:
Upper panel: A selection of spatially resolved line profiles for a typical Fe i line across the solar granulation pattern. The thick curve denotes the spatially averaged profile. Lower panel: The same as above but instead showing the individual line bisectors. Note that the asymmetry of the spatially resolved lines are not at all representative of the spatially averaged profile (thick line) (from Asplund et al., 2000b).
View Image Figure 27:
The upper left panel shows the predicted continuum intensity across the granulation pattern of one snapshot of a 3D hydrodynamical solar simulation. The other panels illustrate the variation of the predicted (red) and observed (blue) equivalent widths of individual μ = 1 line profiles over the solar surface as a function of the local continuum intensity; each panel is labeled with the species, wavelength, and lower excitation potential of the different transitions. The two crosses denote the values for a typical down- and upflow, with the locations identified in the granulation image. For most lines the 3D LTE and observed behavior agree very well with the exception of low excitation lines of minority ionization stages, such as the 670.8 nm line of Li i, which strongly suggests the presence of non-LTE effects in the line formation (from Asplund, 2005).
View Image Figure 28:
The predicted spatially and temporally averaged 3D LTE solar line profile of a typical Fe i line (solid line) compared with the corresponding calculation when ignoring all Doppler shifts arising from the photospheric velocity field (dashed line), demonstrating the importance of convective line broadening. The latter profile closely resembles 1D line profiles without application of the fudge parameters micro- and macroturbulence.
View Image Figure 29:
The predicted temporally and spatially averaged 3D profile (blue solid line) compared with the observed solar disk-center line (red diamonds). Note the excellent agreement as seen in the residuals (the discrepancies in the far red and blue wings are due to unaccounted for blends). Also shown is the best-fitting 1D line profile after having optimized the micro- and macroturbulence (green solid line), which clearly has the wrong shape, asymmetry, and shift.
View Image Figure 30:
A comparison between the predicted and observed (solid lines with error bars) line bisectors for a few Fe i lines on an absolute wavelength scale. In 1D models all lines are perfectly symmetric with no line shift.
View Image Figure 31:
Upper panel: The observed (dots) feature 630 nm, which corresponds to the main forbidden [O i] line that is blended with a Ni i line. The solid line denote the best fitting 3D-profile consisting of contributions from both [O i] and Ni i (dashed lines). Lower panel: The same as above but ignoring the Ni line. Note that in this case the required O abundance is 0.13 dex higher but that neither the line shape nor the overall line shift of the feature is well explained (from Allende Prieto et al., 2001).
View Image Figure 32:
Center-to-limb variation of the O i 777 nm line strength. The observed behavior is shown as bands with estimated internal errors. The solid lines denote the 3D non-LTE case while the dashed lines correspond to the 3D LTE calculations. Note that in order to have the same disk-integrated equivalent widths, the 3D LTE calculations have been performed with a ≈ 0.2 dex higher O abundance than for the 3D non-LTE profiles (from Asplund et al., 2004).
View Image Figure 33:
Two-dimensional power spectra (ℓ − ν) of the vertical velocity in an 8.5 hour long convection simulation in a domain 48 Mm wide by 20 Mm deep (left) and Doppler velocity from MDI high-resolution observations (right). f- and p1-p4-modes are visible in the simulation data. The dashed curve is the theoretical f-mode ridge. Because of the finite width of the computational box, there are fewer modes which are individually identifiable in the simulation. The low frequency power is convection.
View Image Figure 34:
Comparison the horizontal velocities at a depth of 2 – 3 Mm in the simulation (left) and obtained by inverting travel-times (right) for vertical velocities at a height of 200 km above the surface from the simulation. The small scale flows in the simulation have been removed by filtering.
View Image Figure 35:
A slice through the center of the Born approximation travel-time kernel for a focus depth of 5 Mm in units of s Mm–3. A negative kernel relates the flow in the positive x-direction to a travel-time decrease.
View Image Figure 36:
Comparison of observed and calculated p-mode excitation rates for the entire Sun. Squares are from SOHO GOLF observations for ℓ = 0 − 3 (Roca Cortés et al., 1999) and triangles are the simulation calculations (Equation 45View Equation).
View Image Figure 37:
Exponent n(k) and width w(k) of the convective frequency spectrum as a function of horizontal wavenumber for several depths in the convection zone.
View Image Figure 38:
Mode mass increases with decreasing frequency because the modes extend deeper into the Sun. The simulation modes exist only in a shallow box, so the low frequency modes cannot extend deeply.
View Image Figure 39:
Mode compression decreases with decreasing frequency because the eigenfunctions vary more slowly.
View Image Figure 40:
Convective pressure fluctuations decrease at high frequencies because convection is a low frequency process.
View Image Figure 41:
P-mode excitation as a function of frequency and depth. The image shows the logarithm (base 10) of the absolute value squared of the work integrand, normalized by the factors in front of the integral in expression for the excitation rate, || ∂ξ ||2 ω2|δP∗ω-∂ωr| ∕8Δ νE ω, (in units of erg/cm4/s).
View Image Figure 42:
Pressure as a function of depth for an averaged 3D model (full drawn), for a standard 1D solar model (dashed) and for a 3D model with turbulent pressure removed (dash-dot). Turbulent pressure and the hiding of hot gas contribute about equally to raising the photosphere about one scale height.
View Image Figure 43:
Frequency residuals (observations-calculated) scaled by the ratio Qnℓ of the mode mass to the mode mass of the radial mode with the same frequency. On the left frequencies calculated from the standard 1D solar model S of Christensen-Dalsgaard et al. (1996) and on the right calculated from the horizontal and time averaged 3D simulation extended with a matched mixing length model into the deeper solar layers.
View Image Figure 44:
Granulation image and overlaid magnetogram contours at 30, 50, 70 and 90 G in the Fe i λ6302.5 line (Domínguez Cerdeña et al., 2003). Tickmarks at 1” intervals.
View Image Figure 45:
Emergent intensity (left), and magnetic field (right) at the surface for increasing average vertical field of 0, 200, 800 G (Vögler, 2005). As the magnetic flux increases it fills more of the intergranular lanes, the granules become smaller and eventually have a few large field free granule islands and large field filled lanes with tiny field free granules immersed in them.
View Image Figure 46:
A representative probability density function for the quiet Sun magnetic field (solid line Domínguez Cerdeña et al., 2006b,a), with superimposed results at a fixed height from 3D MHD simulations of Vögler (2003) with mean vertical fields of 10 G (dotted), 50 G (dashed) and 200 G (dash-dot).
View Image Figure 47:
Distribution of magnetic field strengths at unit continuum optical depth for case of 250 G mean vertical field. The distribution is similar to that of Vögler and Schüssler (2003) but extends to larger field strengths because τ = 1 lies deeper in regions of strong field, so the field is more concentrated.
View Image Figure 48:
Distribution of magnetic field strengths at unit continuum optical depth for case of 250 G mean vertical field showing only the weak field portion. Fields less than 1 G occupy only a few percent of the surface area. The most common field strength is of order 10 G.
View Image Figure 49:
Image of magnetic field with superimposed zero velocity contours to outline the granules for the case of a 30 G uniform horizontal seed field. Field magnitudes less than 3, 30, and 300 G respectively are shown in gray. The magnetic field is concentrated into the intergranular lanes. It is highly intermittent, with strong fields occupying a tiny fraction of the total area.
View Image Figure 50:
Temperature, density, and magnetic field strength along a vertical slice through magnetic and non-magnetic regions, with the average formation height for the G-band intensity for a vertical ray (black line) and at μ = 0.6 (white line). Axes are distances in Mm. The bottom panel shows temperature as function of lg τ500.
Watch/download Movie Figure 51: (mov-Movie; 8726 KB)
Movie G-band images calculated from the simulation at disk center and towards the limb at μ = 0.8, 0.6, 0.4. At disk center small magnetic concentrations appear bright, while larger ones appear dark. When looking toward the limb the granulation appears hilly and one sees the bright walls of granules where the line of sight passes through a low density magnetic concentration. Movie shows time evolution. Note the formation of a micropore near the upper center (movie by Mats Carlsson).
View Image Figure 52:
G-band brightness vs. magnetic field strength at continuum optical depth unity for a snapshot of magneto-convection with a unipolar magnetic field. Note that while all bright points correspond to strong magnetic fields there are many locations of strong field that appear dark in the G-band.
View Image Figure 53:
Comparison of G-band intensity at viewing angle μ = 0.63 of observations (left) and at μ = 0.6 simulated (right).
View Image Figure 54:
Schematic sketch of a magnetic flux concentration (region between the thin lines) and adjacent granules (thick lines). The dashed lines enclose the region where 80% of the continuum radiation is formed. Bright facular radiation originates from a thin layer at the hot granule wall behind the limbward side of the optically thin magnetic flux concentration. The line of sight for the dark centerward bands is shown by the dark shaded region (Keller et al., 2004).
View Image Figure 55:
Emergent continuum intensity as a twisted flux tube emerges through the solar surface (Yelles Chaouche et al., 2005).
View Image Figure 56:
Horizontal slices of magnetic field strength. Rows are time (in seconds) increasing downward. Columns are height (in km) above the mean visible surface decreasing toward the right. Red and yellow have opposite polarity to blue and black. Gray is weak field. At the right of each image is an emerging bipole whose legs separated with decreasing height and increasing time. At the left is a submerging bipole whose legs approach with increasing time and which disappears at higher elevations at later times.
View Image Figure 57:
Magnetic field lines in a simulation snapshot viewed from an angle. The red line in the lower left is horizontal field being advected into the domain. In the lower center is a loop like flux concentration rising toward the surface. In the upper right is a vertical flux concentration or “flux tube” through the surface with its field lines connecting chaotically to the outside below the surface.
View Image Figure 58:
Micropore formation sequence. Left panel is images of the magnetic field strength, center panel is emergent intensity, and right panel is mask showing low intensity, strong field locations (Bercik, 2002Bercik et al., 2003).
View Image Figure 59:
Magnetic field (filled contours at 250 G intervals from 0 G to 3500 G) and temperature (1000 K intervals from 4000 K to 16,000 K). The τ = 1 depth is shown as the thick line around z = 0 Mm. The flux concentrations are significantly cooler than their surroundings (Bercik, 2002Bercik et al., 2003).
View Image Figure 60:
Magnetic field (filled contours at 250 G intervals) and ln density (in 0.5 intervals from –2 to 4). The τ = 1 depth is shown as the thick line around z = 0 Mm. The established, strong “flux tube” in the center has been evacuated and is in equilibrium. The smaller flux concentrations on either side are in the process of being evacuated, starting above the surface and piling up plasma below the surface (Bercik, 2002Bercik et al., 2003).
View Image Figure 61:
Image of vertical velocity (red and yellow down, blue and green up in  km s–1) with magnetic field contours at 0.5 kG intervals at the surface (left) and 1.5 Mm below the surface (Bercik, 2002).
View Image Figure 62:
Radiative heating and cooling of flux concentrations. The second from top panel shows temperature contours at 1000 K intervals. The third from top panel shows magnetic field contours at 250 G intervals. Units are 1010 erg g–1 s–1. The top three panels show the net radiative heating/cooling and its relation to the temperature and magnetic fields. The bottom three panels show the contribution of vertical, inclined, and nearly horizontal rays (Bercik, 2002).