Observations of the solar magnetic field show that the magnetic flux is indeed concentrated in
the intergranular lanes, as seen in Figure 44 (Domínguez Cerdeña et al., 2003) – see also
(Bellot Rubio and Collados, 2003; Martínez González et al., 2006; Khomenko 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., 2003; Domínguez Cerdeña et al., 2006c). 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 (2 × 10^{19} 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, , cannot be
uniquely determined by Zeeman splitting and Hanle depolarization observations. However,
Domínguez Cerdeña et al. (2006c) used numerical magneto-convection simulations to constrain
(Figure 46) 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 46). 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 47). 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 48). 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 49).

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