A resent post by Roger and comments thereon led to my realising there are misunderstandings on the intepretation of the polar field relationships.
Wilcox Observatory measure and publish a time series of the solar polar magnetic field, a difficult measurement. Started 31st May 1976, data point every 10 days.
Figure 1, straight plot of f10.7 radio noise as a proxy for solar activity and mean solar polar magnetic field.
Firstly here are some clarification notes.
The polar field is not the interplanetary field indirectly associated with terrestrial cosmic ray flux. This field at earth roughly follows the F10.7 / sunspot shape, is very noisy.
Neither is it the Livingstone & Penn finding about the change in sunspot magnetic field possibly reducing with time.
Particularly the polar field passes through zero field at about solar cycle maximum. It is believed the solar maximum, which is a matter of opinion published after the event occurs 6 months to 3 years after the zero crossing. The maximum polar fields occur when the solar activity from a sunspot or radio noise perspective is at a minimum. This phase difference might be counter-intuitive.
Also remember the solar magnetic field has half the frequency (twice the period) of the sunspot cycle.
In addition alternate sunspot cycles have opposite polarity magnetically but the human eye cannot see the difference, all look the same. In reality there are a mixture a lot of the time. 
Polar field relationships
Figure 2. Differentiating the low pass filtered mean polar field to extract a shape / change trace, probably a novel result. Filter uses end correction.
This is where things get interesting. The polar field as a rule of thumb transits zero field a little before the sunspot maximum which we can see at years about 1980, 1990 and 2000. We can also see the greatest magnetic field is around sunspot cycle minimum.
My opinion is the magnetic field tends to stabilise the sun hence a lower field when there sun is active, stronger field when it is passive. There are however contradictions, unsurprising in a mysterious object.
In differentiating the curvature and rate of change is brought out. This shows maximum values about 1980, 1990 and 2000, about the same as zero crossing points.
The current sunspot cycle is peculiar. The polar field zero crossing is past and there appears to be a peak in the differentiated so we are probably past or at sunspot cycle maximum. I’ve not seen official confirmation, generally a late announcement. However the Talkshop yields up, I posted on this during 2012, a report from the Japanese Hinode satellite team.
This was taken as jumping the gun but neither has the effect been seen before, moreover the north and south pole do not seem to change concurrently. Put together this means that signs of a zero field forming appear earlier than previously thought. More than three years after then also points to currently past sunspot cycle maximum.
The compromise filter and differentiator
This was done to try and gain a useful result, there is no right or wrong.
Applying the differentiator to the polar field data as-is produces massive noise, the process is very sensitive to data noise.
Figure 3. Effect of filter cutoff.
Unfortunately the WSO Polar magnetic field data is compromised, a matter I’ve written on in the past. Briefly it is a difficult measurement done by telescope. The geometric view of the solar pole varies by time of earth year and in consequence there is a modulation effect on the data. A spectral analysis of the data
Figure 4. A spectral analysis (windowed chirp z-transform) reveals a tell-tale of amplitude modulation of some kind, a doublet. This explains the wobble visible on north and south individual data. To a degree summation cancels the wobble but this is incomplete and cannot undo related higher order effects.
The modulation is primarily between the large signal excursion and the varying field of view which changes with an annual period. (technically a product of terms (literal multiplication)). The usual sum and difference computations confirm this, details not part of this work.
Work on this a few years ago as a remedy only achieved a partially successful synchronous demodulation. The data collection has varied over time so this probably in my opinion irrecoverable. I hope someone succeeds.
An example of similar modulation in a natural system can be seen in Spectrum of 100-kyr glacial cycle: Orbital inclination, not eccentricity
A closer look at the published polar data
Figure 5. Probability plot of normalised 300 day low pass filtered mean data.
Looking at the data produced this extraordinary data shape where a straight line or simple curvature would be reasonable confirms the data is unsafe for detail work. Filtering noise reduction has clarified what is not obvious in straight data.
Figure 6. Probability plot of various data
Solar cycle 24 is past maximum. Attempting extrapolation based on the polar magnetic data is too uncertain.
Putting the data together is a little wrinkly so here is a portable spreadsheet of polar data and pre-decimated F10.7 data sampled concurrently with magnetic data. (2MB XLS ’97 format)
[UPDATE] Added a demonstration of modulation
Figure 7. How an approximation to the North pole data can be mimiced via amplitude modulation. One year sine trace is shown magnified 100X for clarity. DSB + C is double sideband + carrier, which is the slow signal. Plotted as points so that the granularity of the signal can be seen.
- Long-term Evolution of Sunspot Magnetic Fields
- F10.7 NOAA FTP site
F10.7[cm] is 2800MHz measured by pointing a very directional aerial (antenna) at the sun. This is a large interesting subject.
- Hathaway at NASA site, Backwards sunspot
- Slide 19 from presentation, “Hale’s sunspot polarity law”
- Interplanetary Magnetic Field
- Moderately technical presentation on F10.7 solar noise measurement at 2800 MHz
- Smooth noise-robust differentiators (N=5)
- Spectrum of 100-kyr glacial cycle: Orbital inclination, not eccentricity, Richard A. Muller and Gordon J. MacDonald, NAS 1997, open access at PNAS http://www.ncbi.nlm.nih.gov/pmc/articles/PMC33747/
Perhaps particularly Figure 2 therein.
Post by Tim