Several readers mentioned that the plot of sea ice anomaly in the Arctic shown in the last post has much greater fluctuation after about 2007. Anomaly values are the difference between a given time’s extent, and the average for that time of year, so to get anomalies you take the original values and subtract the average seasonal cycle. It was correctly concluded that the exaggerated fluctuations after 2007 are because the seasonal cycle itself has changed; subtracting the average seasonal cycle still leaves that difference in place.
We can analyze the actual sea ice extent (not anomalies) to estimate not just the size (i.e. amplitude) and shape of the seasonal cycle, but how that has changed over time. Fourier analysis is an easy way to get at the average seasonal cycle; a straightforward way to get at its changes is windowed Fourier analysis. With it, we can directly estimate how the amplitude has changed over time. I got this:
Clearly, after about 2007 the seasonal cycle is bigger, i.e. the difference between winter-spring and summer-fall sea ice extent has increased.
I’ve often computed what I call an adaptive anomaly, in which anomaly values are computed by subtracting a changing seasonal cycle, one designed to capture the trend in seasonal amplitude but not the fluctuations. One estimate of that is like this:
When we subtract this “adaptive” seasonal cycle from the extent data, we get the adaptive anomaly:
It’s essentially the same as the basic anomaly, and definitely shows the same overall trend, except that the exaggerated fluctuations post-2007 are no longer there because we’ve allowed the average seasonal cycle to “catch up” to its change over time.
Of course the seasonal cycle itself is of interest, and windowed Fourier analysis enables us to estimate more than just its amplitude. In fact the amplitude from windowed Fourier analysis is that of the fundamental harmonic only, so it’s only an approximation to the amplitude of the cycle. The full amplitude depends on the size and timing of the harmonics in a Fourier expansion.
Let’s start by looking at the phase of the fundamental harmonic, the phase of its maximum:
There’s lots of fluctuation of course, but the main result is that the maximum has tended to occur later in the year, by about 3 days, since 2007. This is the maximum of the fundamental harmonic of course.
We can also look at the amplitudes and phases of the higher harmonics, the overtones in a Fourier series expansion. I’ll plot relative amplitude, which is the ratio of an overtone’s amplitude to that of the fundamental (relative phases and amplitudes are used to great advantage in the study of the light curves of variable stars). Here they are for the first three overtones:
The main difference is that the 1st overtone has grown a little bit larger relative to the fundamental since around 2007.
I’ll also plot relative phase, which is the phase of an overtone when the fundamental is at maximum:
These have changed little, except the 3rd overtone. However, the 3rd overtone is weaker so its phase is more uncertain, hence its relative phase is likewise more uncertain.
The fact that the 1st overtone is at relative phase close to 0.5 cycles means that it bottoms out when the fundamental peaks and when the fundamental bottoms out. This makes the cycle more “pointy” at its minimum and more flattened at its maximum. The fact that its relative amplitude has increased means that the degree of “pointiness” has increased.
We can see the same thing if we simply compute the average annual cycle for different times spans of the data. I divided the time span into three sections, from 1979 to 1993, 1993 to 2007, and 2007 to 2017. Here’s how their average annual cycles compare:
The most obvious change is that the cycles have gotten lower — simply because Arctic sea ice is going down overall. But we also see the more “pointy” minima around September and more flattened maxima around March, as well as the fact that since 2007 the cycle has shifted phase to occur a few days later in the year.
What do all these changes signify? I really don’t know; I’m not a sea ice expert and I don’t have a strong intuition about it. Except for one thing: the overall decline is because of global warming. Global warming is because of humans burning fossil fuels. And the changes we’re seeing, so pronounced in the Arctic, are going to mean trouble. We should slow down our carbon emissions, and stop them altogether, as soon as is practical.
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