tokens: of toroidal field, attributable to a magnetic buoyancy instability (see for e.g., Vasil Brummell 2008 ; Guerrero Käpylä 2011 ). If this were to be the case, we might then expect multiple flux structures to be formed at the same time in a turbulent environment, and such structures would most likely possess random strengths, twists, and separations drawn from some distribution. We therefore choose each of our simulations in the MC series for a particular background field strength to start with five flux concentrations, with random initial twists (−5 ≤ q ≤ 5), and random distance, d , between them (with the constraint, d ≥ 2 R , so that they do not overlap), embedded initially at the same (canonical) height in the prescribed overlying large-scale background field. The random values are drawn from a uniform distribution over the range given. Our setup certainly has some significant degree of arbitrariness, owing to the limitations in our quantitative understanding of the amount of twist in flux tubes, the distance between them, and their strengths at their origination in the deep solar interior. For example, our choice of drawing the random values of twist from a uniform distribution might not be very realistic, and a Gaussian with a larger range might be better (or perhaps even a skewed distribution, to account for the effects in the formation of tubes not included in our model, e.g., rotation: see Chatterjee et al. 2011 ). However, as mentioned above, there are some constraints on the value of ∣ q ∣. Firstly, ∣ q ∣ ≳ 12 (the dashed line in Figure 8 , which is actually a very weakly decreasing function of B s ) is unphysical, since in this case the azimuthal field is sufficiently strong to induce a negative density in the structure. Secondly, structures where ∣ q ∣ ≳ 8 are unstable to kink instabilities (according to Linton et al. ( 1996 ), which does not fit our model exactly, but is probably a reasonable guide). Therefore, our chosen distribution (uniform, ∣ q ∣ ≤ 5) seems reasonable, in that it makes it likely that the initial simulation setup has a mixture of weak and reasonably strong concentrations of both helicities, and, if anything, overemphasizes the stronger twists (i.e., those more likely to violate the SHHR). An MC series of this type (fifty simulations, five random tubes each) is reported for four different background field strengths, ranging from relatively weak to relatively influential (0.01 ≤ B s ≤ 0.10). We identify the weak background field with the beginning and end of the cycle, and the stronger background with the peak of the cycle, although we have no direct observational evidence from the deep solar interior of the relative strengths of these fields from which to set their values. Figure 13 shows the results of the MC simulations for each of the four different background field strengths. Each circle in the plot represents an accumulation over a specified time interval [ t 0 , t 1 ] in the simulation of the absolute value
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