Sea Ice Seasonal Cycle

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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17 responses to “Sea Ice Seasonal Cycle

  1. Steep, steady volume loss is likely to blame for the behavior at the minimum. Extent, being a two dimensional measurement, is behaving how you would expect (in a non-linear fashion) as volume gradually approaches zero. 75% of September volume has already been lost (with respect to 1979 values).

    The slower, flatter March maximum extent loss is more complicated. March volume loss is less – on the order of 30%. Also, due to geographical factors, getting extent to 12M km^2 is fairly easy during the refreeze season, as this represents the rough area contained by the Arctic Ocean (minus some of the peripheral seas). Getting low-thickness, first-year ice is fairly easy to do. In fact, even in this very warm winter, we achieved 12M extent on just 40% of the historical average Freezing Degree Day (FDD) totals (volume is another matter, of course). The flattening at the maximum is probably related to the decreasing propensity of the outer peripheral seas to freeze in winter. The Barents, Kara and Bering Seas are holding back extent numbers this year, for instance.

    The slowness of the winter extent decreases is deceiving, though. Volume loss continues apace. (It is especially bad this winter with the extraordinary warmth the Arctic has been subjected to.) In the future, once winter maximum extent crosses below the 12M mark, the losses should speed up as this means the main Arctic Ocean basin is unable to completely freeze up and volume will be very low by that time.

    • “The Barents, Kara and Bering Seas are holding back extent numbers this year”

      Well, sort of:

      Personally I reckon the timing/value of the annual maximum extent is currently more to do with “weather” than “climate”. Volume is undoubtedly a better metric at this time of year, if only it were easier to measure!

      “Getting low-thickness, first-year ice is fairly easy to do”

      Playing the Devil’s Advocate for a moment, what do you make of the “Slow Transition” argument?,933.0.html

      • Ah yes, I’ve read it. It’s a valuable argument. Getting MYI to melt out was considerably easier than substantially thinning FYI. It requires a much warmer winter to do so (to overcome the negative feedback of increased seasonal volume ice growth). That was the thrust of his argument. Well, it appears we’re getting a much warmer winter. If that continues, then it will likely have a significant impact on FYI thickness for this upcoming season (20-30cm averaged across the pack, more towards the edges and less in the middle). Not extreme, but noticeable.

        The interesting thing is that it doesn’t take much further warming from this winter’s pattern to start seriously reducing FYI thickness, especially around the edges. Note how long it took to finally freeze the Chukchi — the first week of January. In that region, substantial thickening stops in March, so there’s really not much margin left.

  2. Seems pretty clear to me: it’s a natural consequence of the sea ice being thinner. During the dark cold of Arctic winter the sea still develops a thin layer of ice of nearly the same extent as before – but being thinner, it melts away much more easily during the summer. So you get an only slightly changed and flattened maximum, but progressively deeper and sharper minima. I guess the maxima will only start to show very large changes when/if large parts of the Arctic stay above the ocean freezing point for much of the winter.

  3. Very interesting, tamino. Personally, this is an area of stats I know exceedingly little about. What little I do know is that the “cyclists” have routinely misused it to posit multi-decadal/century-sized cycles that “explain” the present warming over the years.

    It’s nice to see an example of how such an analysis _ought_ to be done. Especially that you’re working with cycles at many wavelengths relative to your range.

  4. Later cycle and flatter maximum and pointier minimum seem reasonably easy to explain: In summer albedo feedback dominates so if extent/area is low heat builds in the ocean and keeps extent loss declining at a steeper rate but this heat runs out and being later in the season the extent then rises more rapidly. At the maximum, if the extent is lower then the edge is further north so has to wait that little bit longer before the sun reaches an angle that can hold the extent constant. Perhaps this leads me to expect that the lead up to the maximum should be flatter but then the descent might be similar or even steeper due to less ice volume rather than flatter.

  5. Excellent work. I would be curious to see what give a simple extrapolation.

  6. A very instructive plot again, thank you.
    Generally, with thinner ice, higher thickness amplitudes are to be expected, because the growth rate of an ice sheet, with a given air temperature, behaves inversely to its thickness.
    This can heuristically be translated into extent by visualizing, that the ice sheet is thicker in the middle and zero at the fringes, so that a stronger thickness growth rate at one fringe point is resulting in a faster movement of the fringe point away from the center.
    So the extent oscillation is somehow the shadow of the volume oscillation – with one restriction: the arctic basis has an upper bound for the extent given by the surrounding shores of Eurasia, Canada and Greenland. Without those, we wouldn’t see the flattened top, but a more sinusoidal curve in wintertime, and we would probably see a stronger decrease of the winter maximum.
    As the surrounding of the polar basin is not closed, the upper bound is not absolute and there is some outgrowth of ice through the holes in it, especially east of greenland.
    What’s intriguing is the V shape of the summer minimum, which is becoming more pronounced. Again we can heuristically explain that with the form of the ice cap. If it has a low thickness gradient at the fringes, which comes with a generally thinner ice cap, the melt- or buildup rate translates into a high edge speed.
    I f you look at the annual cycle of the ice volume ( , it has no such V shaped summer minimum. So in the summer, a moderate volume loss / growth is compared to a more distinct extent loss / growth.
    IOW the V shape indicates a very flat summer ice sheet.

  7. Correction: “arctic basin” instead of “arctic basis”.

  8. I think the contrasting shape change of the maximum and minimum is probably due mainly to the geography of the continents. You wrote a post some time ago regarding the rate of decline in area being affected by how much of earth’s surface at a given northern latitudes was covered by land vs ocean. Isn’t that right?

  9. I’m no sea ice expert either, but another thing you’d expect is that as the extent of sea ice decreases, if the errors in measurement or spatial coverage in the absolute sense remain the same, the relative size of the error will get bigger. I don’t know the sizes of those absolute errors, but, eventually, observational systems which have built in assumptions about large extents of sea ice as a baseline will need to have those assumptions revisited when these are no longer true.

  10. If the various graphs of ice behavior are something like the operational history of an important piece of machinery then fairly clearly something’s broken or failing and is in need of intervention. I wonder if the average denier would be so dismissive if this were a HUMS graph for a helicopter carrying loved ones?

  11. The PIOMAS (modelled!) Arctic sea ice volume numbers for January have just been released. As anticipated they reveal a new record low for the date by a considerable margin:

  12. Larry Edwards

    Tamino, can you please explain what the secondary and tertiary harmonics and overtones are, regarding what is happening to the ice during the annual period? Physically, what do the harmonics represent?

    [Response: A complete answer is beyond the scope of this blog post. But in a nutshell, the fundamental maps out a “sine wave” approximation to the annual cycle. However, the waveform of the cycle isn’t precisely a sinusoid. By adding higher harmonics we can make the waveform mimic its real shape. An infinite number of harmonics is required to mimic a waveform generally, but the lowest few will usually get you close enough.

    The harmonics don’t really have a *physical* meaning, but a mathematical one.]

  13. Harry Twinotter

    One of the people presenting the Arctic Report Card 2016 at the AGU mentioned a regime shift around 2007?

  14. Thanks Tamino; now I better understand why what I obtained by monthly averaging and baselining out of
    looks like this:

    I really had the impression of having made a bad evaluation of their climatology data!

    Do the Antarctic sea ice extent anomalies not follow a similar scheme?

  15. The assembled throng may be interested to hear this “Shock News!”?

    The NSIDC daily Antarctic sea ice extent metric is at the lowest level EVER in their records going back to 1979.

    JAXA concur

    Following the recent heatwave the Arctic equivalent is of course still at the lowest level for the date in NSIDC’s records.