The Sea Does Not Look Up

Nothing below is modelled. A lunar ephemeris gives the moment the Moon crossed each station's meridian, to better than a second. The station's own record gives the moment the water peaked. The dial shows every one of those gaps, for a whole year, laid on top of each other. Noon on the dial is the Moon overhead.

The gaps are not scattered at random and they are not near zero. Each station has its own, tight and repeatable, and the stations do not agree with each other. That is the first thing the bulge picture cannot survive: a bulge held under the Moon by the Moon's own gravity has nowhere to put an eleven-hour delay.

Every dot is a tide station that publishes harmonic constants, 1,362 of them. No coastline is drawn underneath; the gauges trace their own. Run the clock and each one lights when its high water arrives. The pale vertical line is the meridian the Moon is over.

Watch the Gulf of Maine and the Gulf of Alaska. They light nearly half a cycle apart, and neither of them lights when the Moon is over it. What is travelling along the coast is a wave, moving at the speed the water and the depth allow, refracting into bays and running up estuaries. The Moon is only the thing that keeps pushing it.

Laplace worked this out in 1775, and the field has called it the dynamic theory of tides ever since. The reason the bulge picture survives anyway is that it is easy to draw and the correction is a map, not a sentence. So: the map.

Take the stations along one open coast in order and plot when each one crests. If the tide is a travelling wave, the crest times should march; the slope is its speed. Every station in the region is drawn, including the ones excluded from the fit, so you can see exactly what was left out and why.

This one needs no ephemeris, no constants, no model. Count the high waters in the record and measure how far apart they are.

The northern Gulf of Mexico is where the principal lunar tide nearly disappears. Run the map above on how big the lunar tide is and watch the Louisiana and Florida panhandle coast go dark. At Pensacola the M2 amplitude is 1.7 cm; at Boston it is 137. The daily constituents are not similarly suppressed (K1 is 12.5 cm at Pensacola and 14.3 cm at Boston, near enough the same), so what is left is a once-a-day tide.

A basin can do that. Semidiurnal forcing that is badly matched to the basin's own response gets cancelled; the region sits near an amphidromic point, a place the co-tidal lines rotate around and where that constituent's range goes to nothing. The map below shows the collapse and the phase reversal that goes with it, walked along the coast from Texas to the Florida Keys.

A tide prediction is computed from the Moon's and the Sun's mean motions and a table of local constants. It does not know there is a storm coming, and it never will. So observed minus predicted is, by construction, everything the astronomy does not contain.

The residual is not noise, and here is the test that says so. If it were the gauge misbehaving it would be independent between stations. If it were an error in the harmonic model it would repeat with the tide. It does neither: on 9–10 January 2024 it arrived at Grand Isle, Louisiana, then Pensacola, then Charleston, then Atlantic City, then New York, then Boston, then Eastport, in that order, over 3,844 km and 30.7 hours. Rank correlation between distance along the coast and arrival time: ρ = 0.976, exact permutation p = 0.0002 over all 40,320 orderings.

The same test run on the quietest week of the year, chosen by measurement rather than by eye, gives ρ = 0.467, p = 0.11: nothing. And run on a twelve-day window in late September, when the eastern seaboard saw more than one system, it gives ρ = −0.867: strongly ordered the wrong way, which is what you get when the largest departure at Florida and the largest departure at New York belong to different storms. A test that finds a front where there is one, no front where there is none, and nonsense when handed two fronts at once is worth more than a test that always agrees with you.

That September window had to be widened to get an honest answer. The first version of this analysis used four days, and the peak search cannot return an arrival inside the first 1.5 hours of a window, that being its smoothing half-width, so two stations came back with arrivals sitting on the very first admissible sample. Those were not measurements, they were the edge of the window. Every window now refuses to report an arrival within two hours of its own boundary, and says so.

One more thing has to be said before that table can be read as "the astronomy". Every harmonic table, NOAA's included, carries two constituents called Sa and Ssa: an annual and a semiannual term, 5 to 12 cm at these stations. The Sun's actual annual gravitational tide is under a millimetre. What Sa and Ssa are fitting is seasonal warming, steric expansion and river discharge: weather with an astronomical name, sitting inside the prediction.

So the split is worth doing twice. Refit each station with the seasonal and long-period terms removed, leaving only what the Moon and Sun actually pull, and the numbers move: hardly at all in some places, enormously in others.