Abstract
Under the assumption that the photospheric quiet Sun magnetic field is turbulent, the cancellation function has previously been used to estimate the true, resolution-independent mean, unsigned vertical flux 〈|B z |〉true. We show that the presence of network elements, noise, and seeing complicate the measurement of accurate cancellation functions and their power law exponents κ. Failure to exclude network elements previously led to estimates that were too low for both the cancellation exponent κ and 〈|B z |〉true. However, both κ and 〈|B z |〉true are overestimated due to noise in magnetograms. While no conclusive value can be derived with data from current instruments, our Hinode/SP results of κ⪅0.38 and 〈|B z |〉true⪅270 gauss can be taken as upper bounds.
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Acknowledgement
JPG gratefully acknowledges the support of the U.S. Department of Energy through the LANL/LDRD Program for this work. Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in cooperation with ESA and NSC (Norway) data provided by the SOHO/MDI consortium. SOHO is a mission of international cooperation between ESA and NASA. SDO is a mission for NASA’s Living With a Star program.
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Appendix
Appendix
For a completely self-similar field, knowledge of net flux at any scale and of the power law scaling exponent implies knowledge of net flux at all scales and, hence, the unsigned flux. This is most easily seen in the example of a vertical random field f z . For a random field, the integral over an area is proportional to the square root of the area (random walk),
while the integral over a strictly positive field is proportional to its area,
Therefore,
and we identify κ=1 for pure noise. For scales smaller than the correlation length of the noise, the net flux is equal to the unsigned flux,
and for larger scales,
The constant C is determined by equating Equations (11) and (12) at the correlation length, l=l 0:
This power law dependence, Equation (13), is shown in Figure 13. This means that the unsigned flux can be exactly calculated given the net flux at any scale and the scale at the end of the power law, l 0,
for a random field. For any other purely self-similar field with cancellation exponent κ,
For the extreme case of a unipolar field, κ=0 and Flux l =Flux0 for all scales l. In general, 0≤κ≤1.
Ratio of net flux to unsigned flux χ(l) vs. scale (l) for noise with correlation length l 0. The ratio is given by \(\frac{l_{0}}{l}\) (dashed line).
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Pietarila, A., Pietarila Graham, J. Instrumental and Observational Artifacts in Quiet Sun Magnetic Flux Cancellation Functions. Sol Phys 282, 389–404 (2013). https://doi.org/10.1007/s11207-012-0140-4
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DOI: https://doi.org/10.1007/s11207-012-0140-4