4.2 Subsurface rise and emergence of magnetic flux

Magnetic fields are produced by dynamo action throughout the solar convection zone. Their emergence through the visible surface is driven by two processes: advection by convective upflows and buoyancy (to maintain approximate pressure equilibrium with their surroundings the density inside the concentration is smaller than in its surroundings). Fan (2009) has reviewed the rise of magnetic flux through the deep convection zone. Simulations of magnetic flux emerging from the surface layers of the convection zone have been initiated in three ways: either from coherent twisted flux tubes forced into the computational domain through the bottom boundary or made buoyant by lowering their density (Yelles Chaouche et al., 2005Jump To The Next Citation Point; Cheung et al., 2007Jump To The Next Citation Point, 2008Jump To The Next Citation Point; Martínez-Sykora et al., 2008, 2009; Cheung et al., 2010Jump To The Next Citation Point), or by inflows at the bottom advecting minimally structured, uniform, untwisted, horizontal field advected by inflows into the domain through the bottom boundary (Stein et al., 2010a,bJump To The Next Citation Point), or produced locally by dynamo action (Abbett, 2007; Abbett and Fisher, 2010). These very different approaches, using several different computer codes, show several, robust, common features.

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Figure 10: mpg-Movie (37039 KB) Log B showing multiple loops and several vertical flux concentrations, one of which has become a pore (see Section 4.4). The movie shows that most of the magnetic features are being pushed down by convective downflows, but some of the loops are rising toward the surface and in places loops have opened out through the top boundary leaving vertical “flux tubes” behind. Movie produced by Sandstrom, CSC, NASA Ames Res. Ctr.

Convective flows produce a hierarchy of loop structures in rising magnetic flux (Figure 10Watch/download Movie). Magnetic flux rises through the convection zone because it is advected by broad upflows and because it is buoyant. Along the way, it encounters 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 or that have large density deficits and so large buoyancy continue to ascend rapidly (Figures 10Watch/download Movie, 11Watch/download Movie). The different scales of motions produce a hierarchy of magnetic Ω- and U-loops with small loops riding piggy-back on larger loops in a serpentine structure (Cheung et al., 2007Jump To The Next Citation Point, 2008; Kitiashvili et al., 2010Jump To The Next Citation Point; Stein et al., 2010bJump To The Next Citation Point) (Figure 12View Image). As a result, emergence of large, undulated Ω-loops occurs as a collection of small-scale, mixed polarity, emergence events. 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 “turbulent pumping” (Drobyshevskii et al., 1980; Nordlund et al., 1992; Petrovay and Szakály, 1993; Tobias et al., 1998; Dorch and Nordlund, 2000, 2001; Tobias et al., 2001) (Figure 10Watch/download Movie).

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Figure 11: mov-Movie (270352 KB) Rise and emergence of initially uniform, untwisted horizontal magnetic field continuously being advected into the computational domain by inflows in the centers of supergranule cells at the bottom. Left: |B | image and velocity vectors. Right: |B| image and magnetic field vectors both in a vertical plane. The boundary field strength was gradually increased from 0.2 kG with an e-folding time of 5 hrs until it reached 5 kG and thereafter held constant.

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Figure 11: mov-Movie (190181 KB) Rise and emergence of initially uniform, untwisted horizontal magnetic field continuously being advected into the computational domain by inflows in the centers of supergranule cells at the bottom. Left: |B | image and velocity vectors. Right: |B| image and magnetic field vectors both in a vertical plane. The boundary field strength was gradually increased from 0.2 kG with an e-folding time of 5 hrs until it reached 5 kG and thereafter held constant.
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Figure 12: Several magnetic field lines showing large scale loops with smaller serpentine loops riding piggy-back on them. Shading shows downflows.

As the magnetic flux rises it expands (Figure 13View Image). For horizontal flux tubes, the horizontal expansion is much larger than the vertical expansion. Consider a purely horizontal field B = Bxˆ in the x-direction. Suppose the rates of expansion in the horizontal and vertical directions are α = ∂vx∕∂x = ∂vy∕∂y and ∂v ∕∂z = 𝜖α z, respectively. The rate of change of the magnetic field following the fluid motion is given by the Lagrangian derivative

DB ---- = − (∇ ⋅ u )B + (B ⋅ ∇)u , (13 ) Dt
which becomes,
D ln B -------= − (1 + 𝜖)α . (14 ) Dt
The continuity equation (1View Equation) becomes
D--ln-ρ Dt = − ∇ ⋅ u = − (2 + 𝜖)α . (15 )
So for emerging horizontal fields, (1+𝜖)∕(2+𝜖) B ∝ ρ (Cheung et al., 2010Jump To The Next Citation Point). For isotropic expansion (𝜖 = 1), B ∝ ρ2∕3. For vertical expansion small compared to horizontal expansion (𝜖 ≪ 1) B ∝ ρ1∕2.
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Figure 13: Time sequence of vertical cross-sections perpendicular to an initial coherent twisted flux tube. The grey scale is temperature and the color coding is magnetic field strength |B |. The purple line is the τ500 = 1. Image reproduced by permission from Cheung et al. (2007), copyright by ESO.

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Figure 14: mov-Movie (106111 KB) Time sequence of emergent continuum intensity (left), vertical magnetic field at τRoss = 0.1 (right), for an initial twisted flux half torus driven into the computational domain at 7.5 Mm depth. The initial central field strength was 21 kG and total flux 7.6 × 1021 Mx. The domain is 92 × 49 Mm wide. Image reproduced by permission from Cheung et al. (2010Jump To The Next Citation Point), copyright by AAS.
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Figure 15: Sweeping of magnetic field into intergranular lanes. Initially the entire surface is covered with 30 G horizontal field. Surface magnetic field strength is shown at 0.5 min (left), 10 min (center) and 30 min (right). Black contours are zero velocity contours which outline the granules. Fields stronger than 0.5 kG appear as white and black. Field magnitudes less than 30 G are shown in grey. Diverging upflows first sweep the granules clear of strong fields and on a longer time scale sweep the interiors of mesogranules free of strong fields. Image reproduced by permission from Stein and Nordlund (2006Jump To The Next Citation Point), copyright by AAS.

The fields first appear at the surface in localized regions as small bipoles with a small-scale, mixed pepper and salt polarity. The emergence region spreads in time (Figure 14Watch/download Movie). As the bipoles begin to emerge, horizontal and vertical fields have similar strengths, but horizontal fields are more common (cover more area) than vertical fields, except for the strongest fields. The mixed polarity fields collect into separated unipolar regions due to the underlying large scale magnetic structures (Figure 14Watch/download Movie).

Diverging, overturning convective motions quickly sweep magnetic fields (on granular times of minutes) from the granules into the intergranular lanes (Figure 15View Image) (Weiss, 1966; Hurlburt and Toomre, 1988; Tao et al., 1998a; Emonet and Cattaneo, 2001; Weiss et al., 2002; Steiner et al., 1998Jump To The Next Citation Point; Stein and Nordlund, 2004; Vögler et al., 2005Jump To The Next Citation Point; Stein and Nordlund, 2006Jump To The Next Citation Point). In hours (mesogranular times) the field tends to collect on a mesogranule scale. Observations averaged over several hours reveal this magnetic pattern (de Wijn et al., 2005; Ishikawa and Tsuneta, 2010), even though there is no distinct convective meso-granule scale. In days (supergranule times) the slower, large scale supergranule motions sweep the fields to the supergranule boundaries. Eventually a balance is reached where the rate of emergence of new flux balances the rate at which flux is swept to larger horizontal scales. This balance empirically occurs at supergranulation scales and produces the magnetic network at the supergranule boundaries. Here the new flux encounters existing magnetic flux, which it either cancels or augments (Simon et al., 2001; Krijger and Roudier, 2003; Isobe et al., 2008).

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Figure 16: Schematic scenario of how granular motions pinch off U-tubes to produce O-loops in 2D and plasmoides in 3D. Image reproduced by permission from Cheung et al. (2010Jump To The Next Citation Point), copyright by AAS.

The rising magnetic loops are not coherent, but rather have a filamentary structure (Figure 10Watch/download Movie). Some individual filaments rise more rapidly than others. The small-scale crenulation of the loops produces the “pepper and salt” pattern as the flux emerges through the visible surface. As the bulk of the magnetic loop reaches the surface, the different polarities concentrate in unipolar regions accompanied by flux cancellation where opposite polarities come in contact (Figures 14Watch/download Movie and 17Watch/download Movie). This happens because the mixed polarity emergence is due to the undulating magnetic field lines produced by convective upflows and downdrafts distorting the large loops rising from below. The underlying large-scale magnetic structures organize the mixed polarity fields when they approach the surface. In order for the like polarity branches to collect, the mass trapped in the U-loops between the peaks of the small Ω-loops must be removed. This occurs by the U-loop getting pinched off and forming plasmoids which may either sink below the surface or get ejected into space (Figure 16View Image) (Lites, 2009; Cheung et al., 2010Jump To The Next Citation Point).

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Figure 17: mov-Movie (165960 KB) Emergent continuum intensity (left) and vertical magnetic field at τcont = 0.01 (right) from simulation with initial/boundary condition of convective inflows advecting 1 kG uniform, untwisted, horizontal field into the computational domain at 20 Mm depth. The intensity range is I∕ ⟨I ⟩ = [0.13,2.5] and the magnetic field range is ± 3.5 kG. The pores may form spontaneously in vertical flux tubes from magnetic loops that have reached the surface and opened out through top boundary. Compare this with Figure 14Watch/download Movie for the rise of a coherent twisted flux tube. (Movie shows the initial “pepper and salt” emergence, the horizontal advection of the field, its concentration into unipolar regions with cancellation where opposite polarities meet and merging of like polarities to form pores. Resolution was increased from 48 km to 24 km horizontally at time 51.7 hrs.)

Magnetic flux emergence simulations starting with horizontal, uniform, untwisted field at 20 Mm depth is shown in Figures 10Watch/download Movie, 11Watch/download Movie and 18View Image. Figure 18View Image shows magnetic field lines in the simulation box viewed from the side and slightly above. 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 (Stein and Nordlund, 2006Jump To The Next Citation Point). While the field lines form a coherent bundle near the surface, below the surface they become tangled and spread out in many different directions (Vögler et al., 2005Jump To The Next Citation Point; Schaffenberger et al., 2005Jump To The Next Citation Point). A “flux tube” at the surface forms by a loop-like flux concentration rising up through the surface and opening up through the upper boundary where the condition is that the field tends toward a potential field. This leaves behind the legs of the loop. Typically one leg is more compact and coherent than the other and persists for a longer time as a coherent entity while the other is quickly dispersed by the convective motions. Cattaneo et al. (2006) have studied the existence of flux tubes using an idealized simulation of a stably stratified atmosphere with shear in both the vertical and one horizontal direction driven by a forcing term in the momentum equation. They find that in the absence of symmetries, even in this laminar flow case, there are no flux surfaces separating the inside of a flux concentration from the outside, so that the magnetic field lines in the concentration connect chaotically to the outside and the “flux tube” is leaky (Figure 19View Image). 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. (1998Jump To The Next Citation Point) – see also Vögler et al. (2005Jump To The Next Citation Point), Schaffenberger et al. (2005Jump To The Next Citation Point), and Stein and Nordlund (2006Jump To The Next Citation Point).

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Figure 18: 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. Image reproduced by permission from Stein and Nordlund (2006Jump To The Next Citation Point), copyright by AAS.
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Figure 19: Magnetic flux concentration at the solar surface and magnetic field lines showing the complex field line connections below the surface. The “flux tube” is a local surface phenomena. Image reproduced by permission from Schaffenberger et al. (2005Jump To The Next Citation Point).

Magnetic fields alter the granule properties – producing smaller, lower intensity contrast, “abnormal” granules (Bercik et al., 1998Jump To The Next Citation Point; Vögler, 2005). Strong magnetic flux concentrations typically form in convective downflow lanes, especially at the vertices of several such lanes, due to the sweeping of flux by the diverging convective upflows (Vögler et al., 2005Jump To The Next Citation Point; Stein and Nordlund, 2006Jump To The Next Citation Point). They are surrounded by downflows which sometimes become supersonic. The normal convective downflows are enhanced surrounding the flux concentrations by baroclinic driving due to the influx of radiation into the concentration (Deinzer et al., 1984Jump To The Next Citation Point; Knölker et al., 1991; Bercik, 2002Jump To The Next Citation Point; Vögler et al., 2005Jump To The Next Citation Point).

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Figure 20: Emergent continuum intensity as a twisted flux tube emerges through the solar surface. Image reproduced by permission from Yelles Chaouche et al. (2005).

Granules become larger and darker as the field first emerges (due to the suppression of vertical motions by the horizontal section of the bipoles and adiabatic cooling due to their expansion) and elongate in the direction of the horizontal component of the field (Figure 20View Image).

The main differences between these two approaches are that a coherent initial flux tube leads to a more coherent symmetrical structure when it emerges through the surface and field line connections below the surface are more localized. In the minimally structured approach organized magnetic field concentrations develop spontaneously when sufficient flux is present, instead of being imposed as initial and boundary conditions, so the emergent structures are less coherent.


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