Moon Over Miami Boston

Most people aren’t surprised that Miami is suffering through more flooding because of sea level rise. After all, the surrounding land lies very low (there really aren’t any hills in Miami, in fact there are precious few in the whole state of Florida). The sea level at any given moment doesn’t have to rise far above MHHW (Mean Higher High Water) to flood the streets.

Other areas aren’t so vulnerable. Boston, for instance, is on the coast but the region isn’t so low-elevation; there are plenty of hills. And according to Sweet & Marra, for local flooding to occur the sea needs to rise 0.68 m (a bit over two feet) above MHHW. Of course that can happen; when storms hit the storm surge is sometimes considerably greater than that and they sometimes hit at or near high tide. But the threshhold at Boston in higher than in most locations, in fact it’s the 3rd-highest threshhold level in the list of 27 eastern U.S. cities studied by Sweet & Marra.

So — is high tide alone enough to cause flooding in Boston, now that the sea has risen to higher heights?

I retrieved hourly sea level data for Boston from the Joint Archive for Sea Level (JASL) at the Univeristy of Hawaii Sea Level Center (UHSLC). Then I tallied, for each day, the highest value to estimate HHW (Higher High Water). I removed days that didn’t have data for all 24 hours (there were only a few dozen in the nearly 92 years of data). I then computed the mean value during the time span 1960 through 1979, the standard choice for calculating MHHW. Subtracting MHHW from the daily high water values yields a record of daily high water level in Boston above MHHW, which looks like this:


The horizontal red line is the 0.68-meter value, the threshhold for flooding due to high water (flooding can also happen due to torrential rainfall).

Clearly, that has happened more often recently. In fact we can count the number of days on which HHW exceeds 0.68 m above MHHW:


The red line is a lowess smooth, the blue line a fit using Poisson regression. Both suggest, and the data make clear, that flooding is on the increase in Boston. In fact, it’s happening a lot more frequently recently. But that’s not the question at hand: is it possible for such flooding to occur in Boston due to tide alone, without requiring storm surge or precipitation?

To answer that question, one needs to know what part of the ocean height (at HHW) is due to the tides. The tides are multiperiodic, i.e. they are a combination of periodic factors, so first I Fourier-analyzed the data, giving this amplitude spectrum (I’m fond of amplitude spectra rather than power spectra):


There are plenty of spectral peaks, but not all of them are due to independent periods. However, the two strongest ones are, at 13.26 and 24.74 cycles/yr. Their associated periods are 27.55 and 14.77 days.

That first period is the anomalistic month. It’s the time required for the moon to orbit the earth once, then move a little further yet so it returns to perigee, it’s closest approach to Earth. This takes longer than its orbital period because the moon’s orbit itself changes, with perigee advancing a wee bit each orbit. Perigee affects tides because when the moon is closer the tides are stronger, so once each anomalistic month, when we reach perigee, that effect is at its peak.

The second period is one half of the synodic month. It’s the time required for the moon to orbit the earth once, than move a bit further so it returns to alignment with the sun. The sun’s gravity affects the tides just as the moon’s does, and when the two are aligned (either in the same, or opposite, directions) their influences join forces to make very strong tides. When they are out of alignment, they oppose each other and the tides are weak. A strong alignment tide is called a spring tide while a weak out-of-alignment tide is a neap tide.

That period is half the synodic month because we have spring tide twice each cycle, when the sun and moon are in the same direction, and when they’re in opposite directions.

Yet another cyclic component of daily high water is at a frequency of precisely one year, with a period of precisely one year. That’s because the sun also has a stronger influence when we’re closer, and it takes just about 1 year for Earth (in its orbit) to return to perihelion, the point of closest approach to the sun.

I also identified a cycle component with period 18.32 years. Honestly, I haven’t figured out the cause of that.

In addition to the cyclic components, there’s also spectral power at very low frequencies, not due to periodic behavior, but because of the long-term trend. For Boston, it’s well approximated by a cubic polynomial.

In addition to the fundamental frequencies of oscillation, there are harmonics (multiples of the fundamental frequencies) and combinations (sums of multiples of more than one fundamental frequency). In all, I’ve identified three harmonics and seven combinations in the spectrum of daily high water at Boston.

With four fundamental, three harmonic, and seven combination frequencies, that gives us a 14-frequency Fourier fit to model daily high water. Plus a cubic polynomial for the long term trend. The model tells us, with excellent accuracy, what the tidal part of high water is, apart from the impacts of storm surge and precipitation. And it looks like this:


That answers the question. Yes, tide alone is sufficient to cause flooding in Boston, even without storm surge or precipitation.

It didn’t used to be. In fact, it wasn’t this way until 2011. These data end with 2012, but it has continued to be that way, and will continue to be so. In fact it will get worse because sea level rise continues. It may get especially worse for Boston, because at the moment, sea level rise there is happening faster than the global average, by quite a bit.


Interesting story.


20 responses to “Moon Over Miami Boston

  1. Yes, I have pictures (one from last week, across a parking lot from our front door. Lots of flooding around, particularly in northeasters. Here’s a local one (Steve H is a friend):

    A lot of Boston is about 3 feet above current king tide level.
    Send me an email and I will pass on some pictures (and a donation, I won’t use paypal and besides don’t like giving credit card companies their weregild).

  2. The Very Reverend Jebediah Hypotenuse

    I also identified a cycle component with period 18.32 years. Honestly, I haven’t figured out the cause of that.

    Period doesn’t quite match – but this may be relevant:

  3. The 18.32 year period is probably an interaction of the Lunar node with the shorter period species (I can ask Zygmunt later). Chapter 1.7

    Click to access tide_book.pdf

  4. An obvious conclusion drawn from your first plot is that flooding of low-lying areas will progressively become more common, more widespread and more severe.

  5. Interesting and unsettling.

  6. Excellent posting. Thanks. I particularly like the “tides only” analysis.

  7. Tamino, is the data for the reconstructed 14-frequency Fourier fit available online some place? Could I have a copy? I’m interested in seeing how much of that is compatible with an RTS fit to the global trends at I’m interested in the recent secular acceleration, noting the deceleration of the same around 2011 on the other side. They are not symmetric, but I want to get quantitative about that.

    I’m also interested in prediction intervals going forward.

    Also, and this may be another little project, I’m interested in knowing to what degree (quantitative again) could this be explained by the Gulf Stream high water shifting northwards.

    [Response: You can get the hourly sea level data here. You might be able to reproduce the analysis yourself, I hope the description is adequate. Bear in mind, this particular approach really only identifies the tidal component because it uses a simple cubic polynomial for secular stuff (decent approximation for Boston, maybe not so good for other locations). I doubt it’ll be very informative about secular trends.

    I’ll put together some comprehensible data about the reconstruction, and see if I can post it here.]

  8. Yeah, our Florida republicans still have their heads buried in the sand, while sea level rise covers their asses.

  9. OT: In a post in September, Tamino offered an opinion (which I agree with) that something went wrong with the satellite data used in the UAH and RSS data sets, after 2000. I’m not sure which satellites are used for those but a recent paper suggests that degraded MODIS satellites are responsible for much of the assumed darkening of Arctic ice sheets. Could this be related to the degraded data in satellite data sets?

  10. “I also identified a cycle component with period 18.32 years. Honestly, I haven’t figured out the cause of that.”
    According to Wikipedia the longitude of ascending node of the Moon is regressing by one revolution in 18.6 years.
    So it maybe also a lunar cause.

  11. Given that polynomial fits tend to go crazy at the extremes of the data on which they are based, is there a different fit you could use for the trend?

    [Response: Yes, but this was partly a convenience choice (my Fourier fit program already includes a simultaneous poly fit if desired), mostly because the Boston data (*not* other stations) matches a cubic quite well. I intend to try other choices for other stations, and yes, I have plans to apply this methodology to other locations, and to extrapolate into the future.]

  12. Everett F Sargent


    The research quality data at UHSLC has been updated through 2014 …

    Click to access JASL2015DataReport.pdf

    I typically just download the from the ftp site (see above documentation).

    Not sure if you are at all interested in ‘nailing’ the 18.6 year nodal period (via FFT or whatever) …

    (I usually use an FFT to guide me in harmonic analysis)

    But this NOAA page will get you another 10-months of 2015 verified data for Boston (datum/units/timezone are the same) …
    (The NOAA interface is limited to 366 days for hourly data and 32 days for 6-minute data)

    Also see this 6-minute series for Boston which confirms the high tide at the end of October (it does appear that there was a little bit of an onshore wind) …

    Also, a while back I used the JPL Horizons interface to determine (as best as I could) the date of the nodal minor standstill and obtained October 6th 2015.

  13. On the left coast, we have El Nino which results in sea level rise events of a few months every few years. Maximum intensity (total rise above 1915 sea level) of El Nino events tends to increase faster than average sea level rise as in

    USCOE understands, but NOAA tends to average the El Nino events out of existence as in

    Thus, dikes and levees around the SF Bay and Delta must be designed and built to withstand sea level rise events that are increasing faster than sea level is rising.

    And, some recent evidence suggests that the 2015 El Nino may be an event of some interest.

  14. It’s a sad statement on what twenty-plus years of motivated denialism has done to public discourse that I read Rahmstorf’s article and immediately thought “sensible, well-written, persuasive… I wonder what idiotic means will be used to tear it down?”

    Maybe the general readership of the SMH isn’t quite as jaded.

  15. Besides beeing interesting with regard to climate change and it’s consequences, this is a very beautiful analysis in itself. Thank you.

  16. Couple of points, as Susan said, Boston has a lot of low lying areas. Yeah there are a few hills, but also a lot of reclaimed marshland (like Fenway Park and Logan Airport). There is lots of flooding.

    Second, Boston lies in the North American Hotspot where sea level rise has been unusually fast