The Sea Level Cycles Get More Elusive

Guest Post by Willis Eschenbach

In my last post on the purported existence of the elusive ~60-year cycle in sea levels as claimed in the recent paper “Is there a 60-year oscillation in global mean sea level?”, I used a tool called “periodicity analysis” (discussed here) to investigate cycles in the sea level. However, some people said I wasn’t using the right tool for the job. And since I didn’t find the elusive 60-year cycle, I figured they might be right about periodicity analysis. In the process, however I found a more sensitive tool, which is to just fit a sine wave to the tidal data at each cycle length and measure the peak-to-peak amplitude of the best-fit sine wave. I call this procedure “sinusoidal periodicity”, for a simple reason—I’m a self-taught mathematician, so I don’t know the right name for the procedure. I’m sure this analysis method is known, but since I made it up I don’t know what it’s actually called.

I like to start with a look at the rawest view of the data. In this case, here’s the long-term Stockholm tide gauge record itself, before any further analysis. This is the longest complete monthly tidal gauge record I know of, at 200 years.

Figure 1. Stockholm monthly average sea level. This is a relative sea level, measured against an arbitrary zero point.

As you can see, Stockholm is (geologically speaking) rapidly leaping upwards after the removal of the huge burden of ice and glaciers about 12,000 years ago. As a result, the relative sea level (ocean relative to the land) has been falling steadily for the last 200 years, at a surprisingly stable rate of about 4 mm per year.

In any case, here’s what the sinusoidal periodicity analysis looks like for the Stockholm tide data, both with and without the annual cycle:

Figure 1a. “Sinusoidal Periodicity” of the Stockholm tide gauge data, showing the peak-to-peak amplitude (in millimetres) of the best-fit sine wave at each period length. Upper panel shows the data including the annual variations. In all cases, the underlying dataset is linearly detrended before sinusoidal periodicity analysis. Note the different scales of the two panels.

Now, I could get fond of this kind of sinusoidal analysis. To begin with, it shares one advantage of periodicity analysis, which is that the result is linear in period, rather than linear with frequency as is the case with Fourier transforms and spectral analysis. This means that from monthly data you get results in monthly increments of cycle length. Next, it outperforms periodicity analysis in respect of the removal of the short-period signals. As you can see above, unlike with periodicity analysis, removing the annual signal does not affect the results for the longer-term cycles. The longer cycles are totally unchanged by the removal of the annual cycle. Finally, I very much like the fact that the results are in the same units as the input data, which in this case is millimetres. I can intuitively get a sense of a 150-mm (6 inch) annual swing in the Stockholm sea level as shown above, or a 40 mm (1.5 inch) swing at both ~5.5 and ~31 years.

Let me start with a few comments on the Stockholm results above. The first one is that there is no significant power in the ~ 11-year period of the sunspot cycle, or the 22-year Hale solar cycle, as many people have claimed. There is a small peak at 21 years, but it is weak. After removal of the annual cycle, the next strongest cycles peak at ~5.5, 31.75, and 15 years.

Next, there are clearly cycle lengths which have very little power, such as 19.5, 26.5, and 35 years.

Finally, in this record I don’t see much sign of the proverbial ~60 cycle. In this record, at least, there isn’t much power in any of the longer cycles.

My tentative conclusion from the sinusoidal analysis of the Stockholm tide record is that we are looking at the resonant frequencies (and non-resonant frequencies) of the horizontal movement of the ocean within its surrounding basin.

So let me go through all of the datasets that are 120 years long or longer, using this tool, to see what we find.

So lets move on to the other 22 long-term tidal datasets that I linked to in my last post. I chose 120 years because I’m forced to use shorter datasets than I like. Normally, I wouldn’t consider results from a period less than three times the length of the cycle in question to be significant. However, there’s very few datasets that long, so the next step down is to require at least 120 years of data to look for a 60-year cycle. Less than that and you’re just fooling yourself. So without further ado, here are the strengths of the sinusoidal cycles for the first eight of the 22 datasets …

Figure 2. Sinusoidal amplitude, first eight of the 22 long-term (>120 year) datasets in the PSMSL database. Note that the units are different in different panels.

The first thing that strikes me about these results? The incredible variety. A few examples. Brest has lots of power in the longer-term cycles, with a clear peak at ~65 years. Wismar 2, on the other hand, has very little power in the long-term cycles, but a clear cycle at ~ 28 years. San Francisco has a 55-year peak, but the strongest peak there is at 13 years. In New York, on the other hand, the ~51 year peak is the strongest cycle after the annual cycle. Cuxhaven 2 has a low spot between 55 and 65 years, as does Warnemunde 2, which goes to zero at about 56 years … go figure.

Confused yet? Here’s another eight …

Figure 3. Sinusoidal periodicity, second eight of the 22 long-term (>120 year) datasets in the PSMSL database. Note that the units are different in different panels.

Again the unifying theme is the lack of a unifying theme. Vlissingen and Ijmuiden bottom out around 50 years. Helsinki has almost no power in the longer cycles, but the shorter cycles are up to 60 mm in amplitude.. Vlissingen is the reverse. The shorter cycles are down around 15-20 mm, and the longer cycles are up to 60 mm in amplitude. And so on … here’s the final group of six:

Figure 4. Sinusoidal periodicity, final six of the 22 long-term (>120 year) datasets in the PSMSL database. Note that the units are different in different panels.

Still loads of differences. As I noted in my previous post, the only one of the datasets that showed a clear peak at ~55-years was Poti, and I find the same here. Marseilles, on the other hand, has power in the longer term, but without a clear peak. And the other four all bottom out somewhere between 50 and 70 years, no joy there.

In short, although I do think this method of analysis gives a better view, I still cannot find the elusive 60-year cycle. Here’s an overview of all 22 of the datasets, you tell me what you see:

Figure 5. Sinusoidal periodicity, all twenty-two of the long-term tide gauge datasets.

Now, I got started on this quest because of the statement in Abstract of the underlying study, viz:

We find that there is a significant oscillation with a period around 60-years in the majority of the tide gauges examined during the 20th Century …

(As an aside, waffle-words like “a period around 60-years” drive me spare. The period that they actually tested for was 55-years … so why not state that in the abstract? Whenever one of these good cycle-folk says “a period around” I know they are investigating the upper end of the stress-strain curve of veracity … but I digress.)

So they claim a 55-year cycle in “the majority of the tide gauges” … sorry, I’m still not seeing it. The Poti record in violet in Figure 5 is about the only tide gauge to show a significant 55-year peak.

On average (black line), for these tide gauge records, the strongest cycle is 6 years 4 months. There is another peak at 18 years 1 month. All of them have low spots at 12-14 years and at 24 years … and other than that, they have very little in common. In particular, there seems to be no common cycles longer than about thirty years or so.

So once again, I have to throw this out as an opportunity for those of you who think the authors were right and who believe that there IS a 55-year cycle “in the majority of the tide gauges”. Here’s your chance to prove me wrong, that’s the game of science. Note again that I’m not saying there is no 55-year signal in the tide data. I’m saying I’ve looked for it in a couple of different ways now, and gotten the same negative result.

I threw out this same opportunity in my last post on the subject … to date, nobody has shown such a cycle exists in the tide data. Oh, there are the usual number of people who also can’t find the signal, but who insist on telling me how smart they are and how stupid I am for not finding it. Despite that, so far, nobody has demonstrated the 55-year signal exists in a majority of the tide gauges.

So please, folks. Yes, I’m a self-taught scientist. And yes, I’ve never taken a class in signal analysis. I’ve only taken two college science classes in my life, Introductory Physics 101 and Introductory Chemistry 101. I freely admit I have little formal education.

But if you can’t find the 55-year signal either, then please don’t bother telling me how smart you are or listing all the mistakes you think I’m making. If you’re so smart, find the signal first. Then you can explain to me where I went wrong.

What’s next for me? Calculating the 95% CIs for the sinusoidal periodicity, including autocorrelation. And finding a way to calculate it faster, as usual optimization is slow, double optimization (phase and amplitude) is slower, and each analysis requires about a thousand such optimizations. It takes about 20 seconds on my machine, doable, but I’d like some faster method.

Best regards to each of you,

w.

As Always: Please quote the exact words that you disagree with, it avoids endless misunderstandings.

Also: Claims without substantiation get little traction here. Please provide links, citations, locations, observations and the like, it’s science after all. I’m tired of people popping up all breathless to tell us about something they read somewhere about what happened some unknown amount of time ago in some unspecified location … links and facts are your friend.

Data: All PSMSL stations in one large Excel file, All Tide Data.xlsx

Just the 22 longest stations as shown in Figs. 2-4 as a CSV text file, Tide Data 22 Longest.csv .

Stockholm data as an excel worksheet, eckman_2003_stockholm.xls 

Code: The function I wrote to do the analysis is called “sinepower”, available here. If that link doesn’t work for you, try here. The function doesn’t call any external functions or packages … but it’s slow. There’s a worked example at the end of the file, after the function definition, that imports the 22-station CSV file. Suggestions welcome.