tides

I have always been fascinated with the tides. For me, the place at the edge of the ocean, where the earth, air and water converge, is where the magic and bioluminescence are found. When I was in college, a few of my courses contained lessons on what drives the tides, and while most of the learning from the courses of that time period has ebbed away from the shores of my mind, I tucked those particular sets of notes away in a folder to one day revisit, so that I could stitch together the fragments I had learned into a more cohesive understanding.

Twenty years after college graduation, the time has arrived! I saturated my brain with this topic over the first weekend of 2020, autodidacting my way through a weekend graduate course in tides. The second chapter of my book (now that one chapter is written I’m daring myself to say out loud that I’m writing one) contains a tidepooling scene, and in my quest to represent factually both my level of understanding of our local tides at the time, as well as to avoid putting any dubious information out into the world, it felt like an auspicious moment to take this deep dive. A big part of my quest is to take on the challenge of writing about such subjects without wringing all the life out of them, but instead retaining as much of the magic and bioluminescence they naturally contain as I possibly can, to write descriptively and with metaphor about topics that often get handled, instead, with stuffy textbook sterility.

It took immersing myself in this coastal reality with a tide chart in my pocket to be able to holistically appreciate the factoids I had memorized in school. With theory and experience tucked under each arm, I waded in.

In disentangling the enigmatic cycle of the tides and putting it into words, I’ve noticed that my notes from each of my undergraduate courses contain only incomplete segments of the information, and that online resources, even from my beloved NOAA dot gov, are likewise incomplete and occasionally provide some misleading or conflicting information. I gleaned useful clues where I could and sorted through inconsistencies to integrate the theory with my lived experience to support the scene in chapter two.

Then I ended up with 700% more brainiac thoughts on the matter than the chapter requires, so the overflow is resulting in this blog post. This way when I forget what I learned, I can come refer back to this synthesis.

In my notes from Dr. Serafy’s Field Bio taken in the Spring of 1997 was a list of three types of tides that came closest to being understandable and seeming accurate, but it was missing one level, and some further nuance. His hierarchy of tides spanned daily, monthly and what he called yearly categories. I would compile a new list with four levels including daily, monthly, I would change his third category to monthly-yearly, and add a fourth level yearly:

Daily aka Diurnal tides: twice a day high, twice a day low, at least where I live. We can visualize the earth rotating through a bulge of water created by the gravitational forces on the earth’s oceans of the moon (primarily) and the sun (secondarily). When our geographic location on earth rotates through this bulge, we experience high tide, and as it rotates through the skinny part of the bulge, we experience low tide. On the far side of the earth away from the moon is a similar bulge, the result of centrifugal force – so we get one more high and one more low before we return to the point of origin on our rotation. Though the earth takes only 24 hours to rotate once on its axis with respect to the sun, it actually takes 24 hours and 50 minutes for the earth to return to the same relationship to the moon because the moon has also been moving during the 24 hour period and it takes 50 extra minutes for the earth to catch up.

Monthly aka Lunar tides: The pulling of sun and moon in synchrony or at odds with one another result in spring and neap tides. Full moon and new moon are created under conditions of syzygy, you’re welcome for this awesome new word in your life with its three sometimes vowels, which means the tidal forces of moon and sun act to reinforce each other, and cause the diurnal tides to be spring tides (higher highs, lower lows). Closer to half moons, the lunar and solar forces run perpendicular to and therefore counteract each other, causing more moderate neap tides which see the smallest change between high and low tide (higher lows and lower highs).

Monthly-yearly aka Perigee tides (Dr. Serafy called perigee tides simply “yearly”): Each trip of the earth around the sun (we call this a year) contains about 13 lunar cycles of 27 1/3 days. During each of these thirteen trips of the moon and earth around each other, the moon is closer to the earth at one point each month. This state is called perigee and is a consequence of the moon’s orbit being elliptical; apogee is when the moon is farthest from earth. Perigee produces even higher/lower spring tides. Twice-ish per year, the timing of this close approach of the moon to the earth coincides with the lunar spring tide on the new moon or full moon, and the effects are additive, resulting in spring tides of even greater magnitude. We can call this perigee syzygy, okay?

The internet will tell you that perigean spring tides occur “about three or four times per year, in spring and fall” but I took some time and dissected what this meant, and what it means is that in spring and fall, the syzygy-perigee phenomena are most likely to overlap each other in time, so the best bets are around March and October. Like blue moons, sometimes we can get an extra one or two of these perigee-syzygies happening in, say, April, or September. This year, in March, a full moon (syzygy) will take place on the 9th, while perigee will occur on the 10^th! A nice block of negative low tides (6^th through 11^th) coincides with this lunar magic. In October, we’ll have new moon perigee syzygy, both occurring on the 16^th, with several days of negative tides once again.

A September 17^th-18^th new moon perigee, and a full moon perigee on April 7^th seem like they bring us to four perigee syzygies for this year! It seems that we probably have perigee-syzygy a minimum of twice per year, and could have it up to four times, with two of them having possibly slightly decreased magnitude based on how well synchronized the perigees and syzygies are.

Finally, Yearly aka Perihelion tides would be level four on my list: The sun is closest to earth (a condition called perihelion) on January 2, and farthest (aphelion) on July 2. (Aside: I know that sounds backwards, trust me it’s not, we have summer in the northern hemisphere because of the tilt of the earth on its axis, and that tilt being oriented sunwards in summer; summer is not a condition of the earth’s distance from the sun.) When perihelion and perigee coincide, the result is what in the past decade I started hearing referred to as “king” tides. If we picked ONE king tide of the year, it would be the high tide associated with the perigee-syzygy situated closest to that January 2^nd perihelion: in 2020, the closest perigee is January 13, and the closest full moon is January 10 (hey, that’s today! First full moon of 2020!). The absolute highest high tide for the year, then, is predicted to occur on January 11, according to my tide table. Oh hey, that’s tomorrow!

last full moon of 2019

Sea level matters… so, the January king tide will have a nice negative low tide corresponding to it, but the lowest of low tides for the year will not occur until June. Sea level will be at its lowest in summer, which is one factor contributing to this being the case (but not the only one).

The folks in charge of king tides have named quite a few dates as king tides, not just ONE regal tide reigning for the year. Still, these dates do cluster around the perihelion, corresponding to when the perigee-syzygies occur.

Winter 2019-20 king tide Dates:

November 25-27, 2019

December 24-26, 2019

January 10-12, 2020

February 8-10, 2020

I looked at them closely, and I believe they can be summarized as occurring around perihelion, around perigee UPON (some of) the new/full moons between November and February. Let’s call them perihelion perigee syzygy tides instead of king tides, alrighty? The syzygies of interest in November and December happen to be new moons, while the ones in January and February happen to be full moons. This isn’t coincidence, it has to do with the stuff I said before about “more nuance” that I was missing from my notes from college (keep reading).

After I had tackled this much of the tide cycle picture, I still didn’t understand why the “clamming” negative tides always take place in the morning during summer time, and in the evening during winter time. I took it as a clue that these big tides are close to solstices, and another clue that the “switch” from the high magnitude tidal variations from morning to evening occurs very close to spring and fall equinoxes  – you can see this in very abrupt jumps from the AM column to the PM column of the shaded “negative tide” segments of the months if you thumb through your pocket tide table. Okay, so this is something seasonal, which means, it relates at least in part to the sun.

Seasons happen, as I mentioned earlier in an aside, because the earth is tilted on its axis, and as it revolves around the sun, the angle of the relationship between our equator and the sun changes. Let’s be geocentric for a second, and we’ll talk in terms of declination, which is the angle of a celestial body above our earth’s equator. Declination is a term I learned during celestial navigation, because if I knew the angle of the sun above the horizon wherever I was in time and space on a schooner, leaning against the rail measuring said angle with my sextant, right at exactly noon, this translated very accurately (through some complicated mathematical gymnastics) into my latitude. Which is good to know! At dawn or dusk, when a few stars and the horizon could be seen through the sextant, triangulating the latitudes indicated by the declinations of several of these navigational stars could be used for another accurate latitude estimate. We can talk about the declination of any celestial body – stars, sun, planets, moon, even though we realize they’re not all revolving around us.

From our earth’s equator, the declination of the sun can range from 0 degrees (at both equinoxes) to 23.5 degrees north latitude, (on ~June 21 or Northern hemisphere summer solstice) or to 23.5 degrees south latitude (on ~Dec 22 or Northern hemisphere winter solstice). When the sun’s declination is 23.5° North, it is the closest it is going to get to our latitude of roughly 45° north, giving us our longest, warmest days we fondly refer to as summer.

Guess what! The moon also has a declination! The moon’s orbit is at about a 5 degree angle to the earth’s orbit around the sun (confusing, I know, but we’re juggling three balls here) and so when considering all three orbs, we arrive at maximum north and south declinations for the moon of 28.6 degrees. The moon travels between its maximum extremes of declination north and south (and through zero declination relative to our equator) during each lunar cycle (aka month). Here in the north, it pulls on us hardest at its maximum northern declination. Any phase of the moon can correspond to any declination, but!!! There are times of year when the moon’s maximum northern declination (when it pulls on us most and makes our higher tides higher) coincides with syzygy (which is when our high tides are already higher because of full or new moon). These effects are additive as well, so when we have both conditions coinciding, we get extra big tides. As much as I read about this, it took drawing myself a diagram to fully grasp how this interaction of the moon’s phase and its declination indicated June low tides in the morning and December low tides in the evening.

When we are at the part of our trip around the sun where our axis is tilted towards the sun (and it’s summer solstice) and we are at the part of the moon’s cycle where it reaches its most extreme northern declination (it’s at 28.6 degrees just because that’s where it is in its orbit), AND we are in a state of syzygy (new moon in the case of summer solstice), we get the most extreme “diurnal inequality” of tides, in other words, our extreme low low razor clamming tides. In the case of summer, the highs will not be quite as high as the winter king tides, because sea level is lowest; the corresponding highs are maybe 9.5 ft instead of the 10 ft levels of king tides.) This June, we will have a very low tide on June 23 of -1.7 ft (perigee will occur June 30, syzygy/new moon will take place on June 20, and maximum moon declination will take place on June 22, with, of course, maximum sun declination on June 21). We will also have a -2.3 ft low tide (the most negative low tide predicted for the coming year) on June 6, with a perigee on June 3 and full moon on June 5. I think that the close correspondence of perigee and full moons in May, June and July are the reason the tides closest to full moon on these months will be the lower lows for those months.

In winter, the maximum northern declination of the moon (Dec 30) will instead correspond to a full moon (Dec 29) near winter solstice (Dec 22). Still with me?

Finally, to really get why the summer lows are in morning and the winter lows are in evening, we need to consider that the moon, when it is new, crosses overhead at noon in our geocentric paradigm. When the moon is full, it crosses overhead at midnight! (I have always felt that it is magic that the moon rises right at 6 pm whenever it is full, so we get to see it, and not be asleep for it.) The big low tides following syzygy occur about 18 hours after this passage of the moon overhead, at either time of year. Without getting too technical, the actions of gravity are not instantaneous because we are subject to laws of physics like friction; there is lag time. Summer solstice low tide associated with new moon, therefore, takes place 18 hours after noon, or the next morning around 6:00 a.m. Winter solstice low tide associated with full moon happens 18 hours after midnight, the following evening around 6:00 p.m.

Okay! That’s all there is!

Just kidding, that’s not it… there’s more! (But you might need to go make more tea.) Please…. Consider the following:

One of the most memorable and mind-blowing days of my undergraduate career was the day my physical oceanography professor had us perform a harmonic analysis and compile a tide chart from 11 different harmonic constituents of the local (Long Island) tide. Harmonic constituents are the things I was talking about above – the various forces acting in concert to produce the tidal cycle, mainly exerted by the moon and the sun, except summarized into coefficients, numbers that tell exactly the magnitude and frequency at which these forces are exerted.

Harmonic analysis is based on Fourier theorem which says that any repeating disturbance can be written as the sum of a series of sinusoidal waves. You can superimpose a collection of sine/cosine curves of different frequencies and amplitudes upon each other, adding and subtracting their contributing magnitudes at any point along the x axis of time to result in a function representing any “repeating disturbance,” say, a tidal cycle. I like that it is called harmonic, because that makes me think of music, and the harmony of all the different instruments and voices working together to produce one cohesive piece of music.

Here are two cosine curves with different magnitudes and frequencies.

Here I have marked them up to show how you can add them together. At any value on the x axis, you come up with a new y value that is the sum of the y values of the two separate curves. At certain points along the x axis, the effects of the two curves will reinforce each other, while at other points, they will counteract each other.

All you need to do is connect the dots and you have found the sum!

Here the sum of the two blue curves is shown superimposed in orange. You can see that in some places, the two original curves worked together to produce larger peaks, while in other places, they offset each other and resulted in a more moderate value.

The various cosine curves we add together each represent one component of the moon’s or sun’s influence on the tides, that when combined, will either reinforce one another and add up to bigger tides, or will cancel or counteract each other and neutralize the curve to result in smaller tides. If you visit the NOAA tidal prediction page, you can download a set of 37 harmonic constituents for each of a whole bunch of different geographic locations where we might care about the tides, and then generate the tide tables for that area. My professor had us do this with just 11 of the harmonic constituents, of course he chose 11 of the ones that play the largest role in formulating the tides. First, we plotted 11 different cosine curve time series as separate functions. Then we added them up.

Tidal predictions will not always equal tidal observations in real time, because the predictions are based only on the gravitational forces acting on the tides, while other factors such as onshore winds and storm surges can play an unpredictable role as well.

OK! Let’s do the same fun exercise for right here on the Oregon coast!

For South Beach I found the harmonic constituents, starting with M2:  amplitude = 2.91, phase = 359.3, speed = 28.984104

For each constituent, we need to graph the function:

h(t) = R cos (at-φ)

Substituting in amplitude for R, speed for a, and phase for φ (that’s lowercase phi for the Greek fans in the live studio audience).

For our first constituent in South Beach, M2, we plug in our values and graph this equation:

h(t) = 2.91 cos (28.984104t-359.3)

And it looks like this:

M2 is the Principal lunar semidiurnal constituent, one of the many semidiurnal factors acting on the tides, semidiurnal meaning twice a day. Some constituents do their thing once a day, and others with more frequent (terdiurnal like meals!) or less frequent (fortnightly like fried chicken cravings! Annual like birthdays!)

Here is K1, the Lunar diurnal constituent. Compared to the twice a day constituent, you can see the difference in the wavelength, even if you don’t know the Greek symbol for that. This one does its thing just once a day.

Wait till you see the next one! Hold onto your teacup!

Looking at the same time frame for the SA Solar annual constituent, it looks like it’s not a repeating function at all… because we’re only looking at one day! It will take a whole year for this one to repeat itself.

See it now? I changed the scale of the x axis to show a whole year of the solar annual constituent, instead of just one day.

A whole bunch of constituents graphed as separate equations on the same grid

You can see how much each variable of the constituent matters as you graph each one, and what they do to the graph as they change. Large amplitudes will add more height to the tide… They will add it at greater frequency based on the speed, so that a new peak happens, say, every 27.3 days instead of every 365 days. The amount the whole curve is shifted along the x axis away from the origin is determined by the phase.

You can see how each of the constituents doing their own thing is a pretty graph, but not a tide table. Each curve repeats… well, repetitively, without the variation we’re used to in our tides, for example, here is what our January 2020 looks like:

We get to this by adding our separate curves together! Let’s start by adding just two components together. This is M2 + K1 or

h(t) = 2.91 cos (28.984104t-359.3)+ 1.42 cos (15.041069t-117.7)

showing this graph within the desmos interface where I made them all

Wow! It already looks more like a tide table! Each time we add in another constituent, the influences reinforce or counteract each other in such a way that the curve starts to approach what we are used to in our tide tables – now there are some higher highs and lower lows aka spring tides, and some more moderate neap tides, and they come and go in a wave of their own.

I made this!

For 730 hours (aka one month), behold an approximation of the tides in South Beach based on 8 tidal constituents namely:

M2 Principal lunar semidiurnal constituent

K1 Lunar diurnal constituent

O1 Lunar diurnal constituent (they’re fraternal twins, OK?)

S2 Principal solar semidiurnal constituent

N2 Larger lunar elliptic semidiurnal constituent

P1 Solar diurnal constituent

SA Solar annual constituent

K2 Lunisolar semidiurnal constituent

M2+K1+O1+S2+N2+P1+SA+K2

Which really looks like this in equation format:

h(t) = 2.91 cos (28.984104t-359.3) + 1.42 cos (15.041069t-117.7) + 0.86 cos (13.943035t-109.8) + 0.79 cos (30.0t-19.3) + 0.6 cos (28.43973t-339.9) + 0.44 cos (14.958931t-114.2) + 0.4 cos (0.0410686t-285.5) + 0.21 cos (30.082138t-9.9)        

My understanding is that there are hundreds of harmonic constituents if not an infinite supply beyond the 37 reported by NOAA, if one wanted to measure the way each star in the galaxy exerts a (nearly negligible) gravitational tug at one’s shoreline. But we can achieve a functional tidal prediction based on just a handful of the ones that have the greatest influence. We don’t always have to include the Lunisolar Synodic Fortnightly Constituent to be able to predict our tides reasonably well. Nor the Lunar Terdiurnal Constituent. Not even the Shallow Water Overtides of Principal Lunar Constituent (aren’t you glad these names exist though?). But you can, if you choose, or if those constituents play a larger role in determining the tides in your local area.

Back on Long Island during undergrad, our tides were also semidiurnal, and we had two highs and two lows per day, but they were much more equal in height, unlike our Pacific Northwest unequal semidiurnal tide pattern. Plugging in the harmonic constituents for Montauk, NY (I picked this station of the available ones for Long Island because of the purple sand) paints a very different wavy line on our graph!

Some constituents exert a greater influence on tides in different geographic locations! There is still a lot more to learn! Yippee!

I could keep doing this for other locations! Maybe I will! We can try the Bay of Fundy! Or an exotic island someplace! If you want to check out this fun graphing interface I used called Desmos, click on the embedded graph below. I think it could be fun for youngsters taking algebra to play with and explore their equations in a more interactive, engaging, and artful way (hint hint camp boss I’m talking about Panda). Also, isn’t it fun to know those cosines are useful for something you care about in your real life?

Ok, thanks for sticking with me through the peaks and troughs of this learning adventure! I’m off to write more of chapter two!

one year of tides; math oceanography art fun