7.1 Effects of magnetic fields on convection

As the magnetic flux through a surface patch increases, it has a profound effect on the convective motions and the granulation pattern (Cattaneo et al., 2003Vögler, 2005Jump To The Next Citation Point). Their simulations, starting with a uniform vertical magnetic field of varying strength, show that as the mean field strength increases first thin sheets of strong (kilogauss) field fill up more and more of the intergranular lanes (Figure 45View Image). Occasionally, micropores form at the downflow vertices. The average size of granules is reduced. When all the intergranular lanes are filled with kilogauss flux, the lanes widen. A few widely separated granule sized upflows persists, with numerous small, weakly magnetized upflow plumes inside the broad lanes of kilogauss field. At average field strengths characteristic of sunspots (2.5 kG) convection occurs as narrow plumes (Weiss et al., 19901996Schüssler and Vögler, 2006). A feeble initial upflow radiatively cools when it reaches the surface. The reduced pressure induces a strong upflow plume. A narrow return flow surrounding the plume develops. Expansion of the rising plume reduces the magnetic field inside it to a few hundred Gauss. Plumes tend to be elongated in random directions. The upflow is eventually strongly braked by the radiative losses at the surface. The sudden halting of the upflow produces a density buildup at the surface which increases the opacity, so that the τ = 1 surface lies at cooler temperatures and leads to the appearance of a dark lane down the center of the plume.
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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.

Observations of the solar magnetic field show that the magnetic flux is indeed concentrated in the intergranular lanes, as seen in Figure 44View Image (Domínguez Cerdeña et al., 2003) – see also (Bellot Rubio and Collados, 2003Martínez González et al., 2006Khomenko et al., 2008). More magnetic flux emerges as small bipoles in the quiet Sun than in active regions. The number of magnetic concentrations decreases exponentially with increasing magnetic flux (Hagenaar et al., 2003Jump To The Next Citation PointDomínguez Cerdeña et al., 2006cJump To The Next Citation Point). The distribution function for the number of magnetic flux concentrations as a function of the flux is a sum of two exponentials: one for small and the other for large fluxes. The distribution of smaller fluxes (<× 1019 Mx) does not vary with the solar cycle, while the number of large flux concentrations (and their size) increases from cycle minimum to maximum. This occurs because the larger concentrations are dominated by unipolar regions fed by the dispersal of active regions. The rate of emergence of small bipoles is anti-correlated with the number of sunspots in the magnetic cycle (Hagenaar et al., 2003). The probability density function for magnetic field strength, P (B ), cannot be uniquely determined by Zeeman splitting and Hanle depolarization observations. However, Domínguez Cerdeña et al. (2006c) used numerical magneto-convection simulations to constrain P(B ) (Figure 46View Image) Magnetic fields with strengths from 0 to 2.5 kG occur at the quiet Sun surface. The distribution function for weak fields (< 500 G) has a log-normal form. The strong fields, observable by Zeeman splitting, occupy only a small fraction (1 – 10%) of the solar surface, however, they contribute half or more of the magnetic energy and up to half of the magnetic flux. Weak fields cover most of the quiet Sun surface. The magnetic energy density is a significant fraction of the kinetic energy density of granular motions. The most probable magnetic field value is not zero, but of order 100 G. There is a local maximum near the maximum field strength (Figure 46View Image). Simulations with a mean vertical field of 250 G (strong plage) and a horizontal grid size of 25 km have a similar magnetic field distribution but with more area covered by significant field strengths and a larger maximum field strength (Figure 47View Image). Here too only a few percent of the surface has fields below 1 G and the most likely field strength is about 10 G (Figure 48View Image). Steiner (2003) provides a flux based integrated probability density distribution which may be a more robust way to compare observations and models. The main difference between the observations and the simulations is a pronounced maximum in the observed but not the simulated probability density function near the maximum field strength (note that the simulation results may be sensitive to numerical resolution, boundary and initial conditions, and may be influenced by limited statistics). Simulations with no net vertical flux have a stretched exponential distribution of field strengths. A stretched exponential distribution means that the stronger the field the tinier the fraction of the area it occupies. Fields of 3 G, fill all the intergranular lanes and exist even inside some of the granules. Fields stronger than 30 G have been swept out of the granules into the intergranular lanes and even some the intergranular lanes have no field stronger than 30 G. Fields stronger than 300 G are highly intermittent (Figure 49View Image).

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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).
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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.
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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.
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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.

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