List of Figures

View Image Figure 1:
A full disk magnetogram from the Kitt Peak Solar Observatory showing the line of sight magnetic flux density on the photosphere of the Sun on May 11, 2000. White (Black) color indicates a field of positive (negative) polarity.
View Image Figure 2:
A continuum intensity image of the Sun taken by the MDI instrument on board the SOHO satellite on the same day as Figure 1. It shows the sunspots that are in some of the active regions in Figure 1.
View Image Figure 3:
A full disk soft X-ray image of the solar coronal taken on the same day as Figure 1 from the soft X-ray telescope on board the Yohkoh satellite. Active regions appear as sites of bright X-ray emitting loops.
View Image Figure 4:
A soft X-ray image of the solar coronal on May 27, 1999, taken by the Yohkoh soft X-ray telescope. The arrows point to two “sigmoids” at similar longitudes north and south of the equator showing an inverse-S and a forward-S shape respectively.
View Image Figure 5:
Schematic illustrations based on Schüssler and Rempel (2002) of the various forces involved with the mechanical equilibria of an isolated toroidal flux ring (a) and a magnetic layer (b) at the base of the solar convection zone. In the case of an isolated toroidal ring (see the black dot in (a) indicating the location of the tube cross-section), the buoyancy force has a component parallel to the rotation axis, which cannot be balanced by any other forces. Thus mechanical equilibrium requires that the buoyancy force vanishes and the magnetic curvature force is balanced by the Coriolis force resulting from a prograde toroidal flow in the flux ring. For a magnetic layer (as indicated by the shaded region in (b)), on the other hand, a latitudinal pressure gradient can be built up, so that an equilibrium may also exist where a non-vanishing buoyancy force, the magnetic curvature force and the pressure gradient are in balance with vanishing Coriolis force (vanishing longitudinal flow).
View Image Figure 6:
Upper panel: Regions of unstable toroidal flux tubes in the (B0, λ0)-plane (with B0 being the magnetic field strength of the flux tubes and λ0 being the equilibrium latitude). The subadiabaticity at the location of the toroidal flux tubes is assumed to be δ ≡ ∇ − ∇ad = − 2.6 × 10− 6. The white area corresponds to a stable region while the shaded regions indicate instability. The degree of shading signifies the azimuthal wavenumber of the most unstable mode. The contours correspond to lines of constant growth time of the instability. Thicker lines are drawn for growth times of 100 days and 300 days. Lower panel: Same as the upper panel except that the subadiabaticity at the location of the toroidal tubes is δ ≡ ∇ − ∇ad = − 1.9 × 10−7. From Caligari et al. (1995).
Watch/download Movie Figure 7: (mpg-Movie; 250 KB)
Movie: The formation of arched flux tubes as a result of the non-linear growth of the undulatory buoyancy instability of a neutrally buoyant equilibrium magnetic layer perturbed by a localized velocity field. From Fan (2001a). The images show the volume rendering of the absolute magnetic field strength |B |. Only one half of the wave length of the undulating flux tubes is shown, and the left and right columns of images show, respectively, the 3D evolution as viewed from two different angles.
View Image Figure 8:
Horizontal average of the magnetic field B x as a function depth for the initial state (dotted line), at a later time when the instability saturates (dashed line), and in the final steady state (solid line). The magnetic pressure gradient is maintained at the top and bottom boundaries during the non-linear evolution of the magnetic buoyancy instability. From Kersalé et al. (2007). Figure reproduced by permission of the AAS.
View Image Figure 9:
Evolution of the kinetic energy density. The system eventually establishes a modulated periodic state with two disparate time scales. From Kersalé et al. (2007). Figure reproduced by permission of the AAS.
View Image Figure 10:
Space-time plots for fixed values of x and z of the magnetic energy density (left), the transverse horizontal (y) velocity (middle), and the vertical velocity (right), with the horizontal axis being the y-axis and vertical axis denoting the time. From Kersalé et al. (2007). Figure reproduced by permission of the AAS.
View Image Figure 11:
From Vasil and Brummell (2008). A 3D MHD simulation of the build up and subsequent buoyancy break up of a layer of horizontal magnetic field forced by a vertical shear on an initially weak vertical field in a subadiabatically stratified atmosphere. The sequence of images show the volume renderings of the magnetic field strength. Figure reproduced by permission of the AAS.
View Image Figure 12:
Latitude of loop emergence as a function of the initial latitude at the base of the solar convection zone, for tubes with initial field strengths B = 30, 60, and 100 kG and fluxes Φ = 1021 and 1022 Mx. From Fan and Fisher (1996).
View Image Figure 13:
Tilt angles at the apex of the emerging flux loops as a function of the emergence latitudes. The squares and the asterisks denote loops originating from initial toroidal tubes located at different depths with different local subadiabaticity (squares: δ ≡ ∇ − ∇ad = − 2.6 × 10−6 and field strength ranges between 105 G and 1.5 × 105 G; asterisks: δ ≡ ∇ − ∇ad = − 1.9 × 10−7 and field strength ranges between 4 × 104 G and 6 × 104 G). The shaded region indicates the range of the observed tilt angles of sunspot groups measured by Howard (1991b). From Caligari et al. (1995).
View Image Figure 14:
Tilt angles at the apex of the emerging loops versus emerging latitudes resulting from thin flux tube simulations of Fan and Fisher (1996). The numbers used as data points indicate the corresponding initial field strength values in units of 10 kG (‘X’s represent 100 kG). Also plotted (solid line) is the least squares fit: tilt angle = 15.7° × sin (latitude), obtained in Fisher et al. (1995) by fitting to the measured tilt angles of 24701 sunspot groups observed at Mt. Wilson, same data set studied in Howard (1991b).
View Image Figure 15:
The distribution of the tilt angle as a function of sine latitude: (a) at the beginning of flux emergence, (b) at the middle of the emergence period, and (c) at the end of emergence. From Kosovichev and Stenflo (2008). Figure reproduced with permission of the AAS.
View Image Figure 16:
Plots of the magnetic field strength as a function of depth along the emerging loops calculated from the thin flux tube model of Fan and Fisher (1996) showing the asymmetry in field strength between the leading leg (solid curve) and the following leg (dash-dotted curve) of each loop. Panels (a), (b), and (c) correspond to the cases with initial toroidal field strengths of 3 × 104 G, 6 × 104 G, and 105 G respectively. The flux Φ = 1022 Mx and the initial latitude 𝜃 = 5° are the same for the three cases shown.
View Image Figure 17:
A view from the north pole of the configuration of an emerging loop obtained from a thin flux tube simulation of a buoyantly unstable initial toroidal flux tube. The initial field strength is 1.2 × 105 G, and the initial latitude is 15°. Note the strong asymmetry in the east-west inclination of the two sides of the emerging loop. From Caligari et al. (1995).
View Image Figure 18:
The evolution at the tube apex of (top-left panel) the Alfvén velocity va, rise velocity vr, convective velocity vconv from a model solar convection zone of Christensen-Dalsgaard (Christensen-Dalsgaard et al., 1993), the azimuthal velocity v ϕ, (top-right panel) the magnetic field strength B, (bottom-left panel) time elapsed since the onset of the Parker instability, and (bottom-right panel) the tube radius a and the local pressure scale height Hp, as a function of depth, resulting from a thin flux tube simulation of an emerging Ω-shaped tube described in Fan and Gong (2000, corresponding to the case shown in the top panel of Figure 1 in that paper). From Fan (2009a).
View Image Figure 19:
The figure shows the latitudinal profile of αbest (see Pevtsov et al., 1995, for the exact way of determining αbest) for (a) 203 active regions in cycle 22 (Longcope et al., 1998), and (b) 263 active regions in cycle 23. Error bars (when present) correspond to 1 standard deviation of the mean αbest from multiple magnetograms of the same active region. Points without error bars correspond to active regions represented by a single magnetogram. The solid line shows a least-squares best-fit linear function. From Pevtsov et al. (2001).
View Image Figure 20:
This figure illustrates that in the northern hemisphere, when a toroidal flux tube (whose cross-section is the hashed area with a magnetic field going into the paper) rising into a region of poloidal magnetic field (in the clockwise direction) generated by the Babcock–Leighton type α-effect of earlier emerging flux tubes of the same type, the poloidal field gets wrapped around the cross-section of the toroidal tube and reconnects behind it, creating an emerging flux tube with left-handed twist. In this figure, the north-pole is to the left, equator to the right, and the dashed line indicating the solar surface. Note the α-effect for the Babcock–Leighton type solar dynamo model mentioned above is not to be confused with the α value measured in solar active region discussed in this section. From Choudhuri (2003).
View Image Figure 21:
Simulated butterfly diagram of active region emergence based on a circulation-dominated mean-field dynamo model with Babcock–Leighton α-effect. The sign of the twist of the emerging active region flux tube is determined by considering poloidal flux accretion during its rise through the convection zone. Right handed twist (left handed twist) is indicated by plus signs (circles). From Choudhuri et al. (2004).
View Image Figure 22:
Upper panel: Evolution of a buoyant horizontal flux tube with purely longitudinal magnetic field. Lower panel: Buoyant rise of a twisted horizontal flux tube with twist that is just above the minimum value given by Equation (26View Equation). The color indicates the longitudinal field strength and the arrows describe the velocity field. From Fan et al. (1998a). (For a corresponding movie showing the evolution of the tube for the untwisted and the twisted cases refer to Figure 23.)
Watch/download Movie Figure 23: (mpg-Movie; 196 KB)
Movie: The evolution of a rising flux tube. From Fan et al. (1998a). For a detailed description see Figure 22.
View Image Figure 24:
The rise of a buoyant Ω-loop with an initial field strength B = 105 G in a rotating model solar convection zone at a local latitude of 15° (from Abbett et al. (2001)). The Ω-loop rises cohesively even though it is untwisted. The loop develops an asymmetric shape with the leading side (leading in the direction of rotation) having a shallower angle relative to the horizontal direction compared to the following side.
Watch/download Movie Figure 25: (mpg-Movie; 432 KB)
Movie: The evolution of a weakly twisted, buoyantly rising Ω-tube, resulting from a simulation described in Fan (2008, see the LNT run in that paper). From Fan (2008). Figure and movie reproduced with permission of the AAS.
View Image Figure 26:
(a) 3D volume rendering of the magnetic field strength of a weakly twisted, rising Ω-tube, whose apex is approaching the top boundary, resulting from a simulation described in Fan (2008, see the LNT run in that paper). (For a corresponding movie see Figure 25.) (b) A cross section of B near the top boundary at r = 0.937R ⊙; (c) selected field lines threading through the coherent apex cross-section of the Ω-tube.
View Image Figure 27:
Dots show values of α ≡ J ⋅ B ∕B2 computed along each of the selected field lines of the final Ω-tube shown in Figure 26(c) as a function of depth for the following side (left panel) and the leading side (right panel). The field-line averaged mean α is shown as the solid curve. From Fan (2009a).
Watch/download Movie Figure 28: (mpg-Movie; 314 KB)
Movie: The rise of a kink unstable magnetic flux tube through an adiabatically stratified model solar convection zone (result from a simulation in Fan et al. (1999) with an initial right-handed twist that is 4 times the critical level for the onset of the kink instability). In this case, the initial twist of the tube is significantly supercritical so that the e-folding growth time of the most unstable kink mode is smaller than the rise time scale. The flux tube is perturbed with multiple unstable modes. The flux tube becomes kinked and arches upward at the center where the kink concentrates, with a rotation of the tube orientation at the apex that exceeds 90°.
View Image Figure 29:
A horizontal cross-section near the top of the upward arching kinked loop shown in the last panel of Figure 28. The contours denote the vertical magnetic field Bz with solid (dotted) contours representing positive (negative) B z. The arrows show the horizontal magnetic field. One finds a compact bipolar region with sheared transverse field at the polarity inversion line. The apparent polarity orientation (i.e. the direction of the line drawn from the peak of the positive pole to the peak of the negative pole) is rotated clockwise by about 145° from the +x direction (the east-west direction) of the initial horizontal flux tube.
View Image Figure 30:
The evolution of a uniformly buoyant magnetic flux tube in a stratified convective velocity field from the simulations of Fan et al. (2003). Top-left image: A snapshot of the vertical velocity of the 3D convective velocity field in a superadiabatically stratified fluid. The density ratio between the bottom and the top of the domain is 20. Top-right image: The velocity field (arrows) and the tube axial field strength (color image) in the vertical plane that contains the axis of the uniformly buoyant horizontal flux tube inserted into the convecting box. Lower panel: The evolution of the buoyant flux tube with B = Beq (left column) and with B = 10Beq (right column). The color indicates the absolute field strength of the flux tube scaled to the initial tube field strength at the axis. (For a corresponding movie see Figure 31.)
Watch/download Movie Figure 31: (mpg-Movie; 328 KB)
Movie: The evolution of a uniformly buoyant magnetic flux tube. From Fan et al. (2003). For a detailed description see Figure 30.
View Image Figure 32:
Cut at the latitude of 30°of the radial velocity (color) and of the magnetic energy (line contours) for three different simulations of the rise of a buoyant toroidal flux ring with different initial field strengths: 2.5Beq (top panel), 5Beq (middle panel), and 10Beq (bottom panel). From Jouve and Brun (2009). Figure reproduced with permission of the AAS.
View Image Figure 33:
Depth from the surface where the apex of an emerging flux loop with varying initial field strength rising from the bottom of the convection zone looses pressure confinement or “explodes” as a result of tube plasma establishing hydrostatic equilibrium along the tube. The explosion height is computed by considering an isentropic thin flux loop with hydrostatic equilibrium along the field lines (see Moreno-Insertis et al., 1995) in a model solar convection zone of Christensen-Dalsgaard (Christensen-Dalsgaard et al., 1993).
View Image Figure 34:
Evolution of magnetic field strength (gray scale: darker gray denotes stronger field) and velocity field (arrows) during the flux loop explosion. The horizontal part of the field is amplified by a factor of 3. From Rempel and Schüssler (2001).
Watch/download Movie Figure 35: (mpg-Movie; 7921 KB)
Movie: Continuum intensity images and synthetic magnetograms of the simulated emerging flux region resulting from a simulation of flux emergence from the top layer of the solar convection zone into the solar photosphere. From (Cheung et al., 2008). Figure and movie reproduced with permission of the AAS.
View Image Figure 36:
The left panel shows the 3D coronal magnetic field produced by the emergence of a twisted magnetic flux tube from the solar interior into the solar atmosphere, resulting from a simulation of Fan (2009b). The right panel shows the z component of the vorticity ωz on the photosphere overlaid with contours of Bz with solid (dotted) contours representing positive (negative) Bz. It shows counter-clockwise vortical motion (i.e. positive ω z) centered on the peaks of the vertical flux concentrations of the two polarities of the emerging region. Figures reproduced with permission of the AAS.
Watch/download Movie Figure 37: (mpg-Movie; 2984 KB)
Movie: 3D evolution of a set of tracked field lines as they are being twisted up and rotate in the atmosphere due to the shear and rotational motions at their footpoints on the photosphere. In these images and the movie, a field line of a particular color corresponds to the same field line carrying the same plasma. The black field line corresponds to the original tube axis, and all the other field lines have their mid points (at x = 0, y = 0) above the mid point of the black field line (reddish field lines have mid points above bluish field lines). From (Fan, 2009b). Figure and movie reproduced with permission of the AAS.
View Image Figure 38:
The variation of 2 α ≡ (∇ × B) ⋅ B ∕B as a function of z along three field lines: the black field line shown in Figure 37, which is the original tube axis, and its two neighboring blue field lines at time t = 118. The α values are plotted along these three field lines (using the same colors for the data points as those of the corresponding field Figures 37) as a function of z, from their left ends to the left apices in the atmosphere. From Fan (2009b). Figure reproduced with permission of the AAS.
View Image Figure 39:
Emerging magnetic field in the solar atmosphere resulting from the 3D simulation of the emergence of a left-hand-twisted magnetic flux tube by Magara (2004). The colors of the field lines represent the square value of the current density at their footpoints on a chromospheric plane located at z = 5. Top left: Top view of the magnetic field lines. Note the inverse-S shape of the brighter field lines, which is consistent with the X-ray sigmoid morphology preferentially seen in the northern hemisphere. Top right: The square of the current density (color image) and vertical magnetic flux (contours) at the chromospheric plane. Bottom left: Side view of the magnetic field lines. Bottom right: Another perspective view of the magnetic field.
View Image Figure 40:
Comparison between the results from a simulation of emerging flux tube (left and middle columns) and the XRT/Hinode observations (right column). The left column shows the evolution of the constant current surfaces, the middle one shows the result computed from a heating proxy, and the right column shows XRT images at three different times during the evolution of the sigmoid structure. From Archontis et al. (2009). Figure reproduced by permission of the AAS.