More frequent hot days do not come in a three year cycle, but in a 1.5 year cycle related to ENSO.

The Hot Day data set

The graph of number of hot days per year

The graph on the left is one I posted earlier. The height of each data point represents the number of hot days in a year, plotted near January. The pattern of points led me to join them by a smooth curve. This curve swings up and down rather regularly, with five peaks and five dips in the fifteen years. That is, more frequent hot days seem to come in a three-year cycle.
Is this cycle “real”? Should we look for a cause? Will the cycle continue?
Probably not! The points of measurement are one year apart. Cycles that are only three years long may be “aliases” of different and shorter undetectable cycles. (See Note below on Nyquist frequency.)

More detailed hot day data

Other graphs already shown include further data: the number of hot days in each month, and the 13-year average number of hot days in each calendar month. From these I have calculated a relative frequency. That is, the ratio of the actual number to the average number for that month.
Only the months of November, December, January and February have enough hot days to calculate a relative frequency, but these can show changes within the hotter months of each year.

The daily maximum temperature data set

A graph that I posted in “El Niño and my climate” shows a curve of smoothed monthly means of daily maximum temperature anomalies. The yearly cycle of summer-to winter temperature has been removed. I have also applied a smoothing function, which makes the monthly points of measurement effectively two or three months apart. As a result, cycles longer than about six months can be detected.
There are about 10 peaks and 10 dips in the 15.5 year curve. They define a cycle of about 1.5 years wavelength. That cycle is so much longer than the minimum-detectable six month cycle that “aliasing” is not likely.
The reality of this temperature curve is supported by its close similarity to the recognised curve of the El Niño – Southern Oscillation (ENSO), as read from NINO3.4 Pacific Ocean sea surface temperature anomalies.

A combined graph of hot day and temperature data

The graph at the top of the page presents the monthly smoothed maximum temperature anomaly again, using the scale at the left. To this I have added data on the number and frequency of hot days.
The annual number of hot days is shown in blue, in blue boxes. The boxes are placed higher or lower according to the number, but the height is adjusted to match other data better.
A “Hot Day Index” is shown by blue diamonds. This index is based on the relative frequency of hot days in each month that has data.  I have re-scaled the values to improve the match. (See Note on Re-scaling below.)

Matching hot days with temperature

While the match is not perfect, the frequency of hot days varies with the temperature curve in great detail.
* Seasons with more frequent hot days are those with higher temperatures, with exceptions only in 2011 and 2014.
* In most years, the frequency of hot days within a single season rises or falls (or peaks) with the rising or falling (or peaking) in temperature. Exceptions occur in 2001, 2007, 2008, 2012 and 2014.

Conclusion

The frequency of hot days in these fifteen years at this site has varied in close agreement with cycles in the anomaly of daily maximum temperature. These irregular, unpredictable cycles are about 1.5 years long, and relate to ENSO to some degree.
The total number of hot days in each year is a very crude measure of the incidence of hot days. It gives the misleading impression that there is a three year cycle in hot days. This apparent cycle is an alias of the 1.5 year cycle.

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Notes

Nyquist frequency

When something is sampled at regular intervals (e.g. one year in this case), it is not possible to detect cycles with wavelength shorter than twice the sampling interval (i.e. shorter than two years in this case). This is the Nyquist sampling theorem. When cycles appear that are not much longer than the minimum detectable wavelength, it is likely that they are spurious aliases for shorter, undetectable cycles.

Re-scaling the relative hot day data

Monthly hot day data used here are relative frequencies of hot days in a given month. To reduce noise, I have smoothed these values (1:2:1)/4. Then, to improve the match  of the curve with that of temperature, I have taken a fourth root. I chose a suitable scale (not shown) on the graph, and aligned the relative frequency “1.0” with the zero temperature anomaly.