The rhythms of the sea: a primer on tides

West Sands Beach near St. Andrews, Scotland, where the tidal range (the difference between high tide and low tide) can be up to 4 meters. 

 

Few natural phenomena capture the imagination and, at the same time, affect daily experience as much as the oceanic tides. With their daily rhythm, tides may have even shaped the evolution of life itself: it has been suggested that tides played a role in the transition from marine to land animals at the end of the Devonian period [1].  Understanding tides allows us to fully appreciate a hidden clockwork that connects celestial mechanics with the ebb and flow of the sea, and its impact on life on Earth. 

Tides follow highly regular patterns, as is immediately obvious to anyone who lives near a coastal area with a tidal range of the order of meters (as opposed to the puny tens of cm in my Mediterranean hometown!). In this post I will discuss how the most readily observable periodicities in the rise and fall of the sea level can be traced to specific features of the motion of the Moon.

 

The equilibrium tide model

Tides are caused by the gravitational pull of the Moon (and, to a lesser extent, the Sun) on the Earth’s oceans. About two-thirds of the total effect comes from the Moon, and one-third from the Sun. As the relative positions of the Moon and the Sun with respect to the Earth change continuously, their contributions sometimes reinforce each other, and sometimes partially cancel, producing regularly repeating patterns. The simplest way to understand the tidal rhythms is in terms of the equilibrium tide model. 

In this highly idealized picture, the Earth is imagined to be completely covered by a uniform ocean that instantly responds to the tide-generating force. At any point on the surface of the Earth, the tide-generating force due to the Moon (the main contributor) is the difference between the lunar gravitational attraction on that point, and that acting on the center of the Earth. On the side of the Earth facing the Moon, the pull is slightly stronger than at the center, drawing the ocean outward into a bulge. On the far side, the pull is slightly weaker than at the center, so the solid Earth is pulled away more strongly than the distant ocean, leaving another bulge behind. The rotation of the Earth under these two opposite bulges explains qualitatively why in most locations there are two high tides and two low tides each day.

Although highly simplified, the equilibrium tide model provides a baseline for understanding the most apparent tidal patterns, which have been known for a very long time (for a masterfully written historical account of the scientific study of tides, see [2]):

  • Adapted from Figure 1 of [3].
    Most locations on Earth experience a semi-diurnal tidal pattern, with two consecutive high tides observed in a "tidal day", which lasts 24 h 50 min.
  • Typically, of two consecutive high tides, one is higher than the other ("dominant" vs. "subordinate" cf. panel A), a phenomenon known as "diurnal inequality". 
  • High tides are higher ("spring tides") when the Earth, Moon, and Sun are aligned (full or new Moon phases), and smaller ("neap tides") when the Sun and Moon are at right angles from each other with respect to the Earth (first and last quarter Moon phases); forming a cycle with a period of 29.53 days  (synodic month; panel B);
  • The diurnal inequality is largest when the Moon is at its maximum angular distance from the equator, and vanishes when the Moon is on the equator, resulting in a cycle with a period of 27.32 days (tropical month; see panel C);
  • The distance between the Earth and the Moon varies with the anomalistic period of 27.55 days (panel D); this additional frequency further modulates the amplitude of the spring-neap cycle, with 3-4 unusually high "perigean spring tides" occurring during a year.

Due to its highly idealized formulation, the equilibrium tide model fails to provide an accurate description of the real-world tides for most locations on Earth. Specifically, it cannot explain the diurnal tidal regimes commonly observed at low latitudes (i.e., one high-tide and one low-tide in a tidal day), or the existence of spring-neap cycles synchronized with the tropical, instead of the synodic, month. 

In general, these shortcomings stem from ignoring the presence of the continents, which hinder the progress of the tide, slowing it down through friction, and can induce resonant phenomena which amplify or suppress some specific periodicities present in the purely astronomical forcing (for more details, see [3] and references therein). 


Tides at Kwajalein Atoll

Intuitively, the predictions of the equilibrium tide model can be expected to be less wrong in the case of small islands in the middle of the ocean. Even in this case, the agreement is only qualitative; nevertheless the comparison with real data can be instructive.

In the following figure I compare the equilibrium tide prediction with actual measurement for the Kwajalein Atoll, part of the Marshall Islands, in the Pacific Ocean. Note that, being an atoll, Kwajalein lacks a significant continental shelf, and thus the idealization inherent in the equilibrium tide model is less inaccurate. 

The data are from the GESLA-4 dataset, retrieved from https://gesla787883612.wordpress.com/downloads/; in the equilibrium tide calculation, I have used the geocentric positions of the Sun and the Moon obtained from the DE440 JPL Planetary and Lunar Ephemerides. For clarity, I have not removed the arbitrary offset between the two curves, which is due to different zero point definitions.

 

Observed vs. predicted sea level at Kwajalein, Marshall Islands, from March 15 to June 4, 1991. The blue, green, and red labels at the top refer to the anomalistic, synodic, and tropical cycles, respectively ("Prg." = Perigee, "Apg." = Apogee; "New" = New Moon, "Qtr." = First/Last quarter, "Full" = Full Moon; "C": Moon crossing over the equator, "No" = Moon at maximum northern declination, "So" = Moon at maximum southern declination; cf. panel E of Figure 1 of [3]).

 

It is immediately apparent that the equilibrium tide model reproduces the overall pattern observed in the sea level measurements.  Similarly, we note that the simple rules described above hold; for instance, April 16 provides an example of a nearly perigean spring tide; the diurnal inequality is largest at maximum lunar declination (e.g., May 16), and nearly zero at equator crossover (e.g., May 10). These qualitative comparisons suggest that, in this case, the tidal signal is largely governed by the purely astronomical effects, and thus successfully captured by the equilibrium tide model.

A quantitative comparison, on the other hand, reveals discrepancies in both amplitude and phase between the two measured and predicted curves, mostly reflecting the neglect of friction and resonances in the equilibrium tide model.

 

References

[1] Balbus, S. A. 2014, Proceedings of the Royal Society of London Series A, 470, 2168.

[2] Cartwright, D. E. 2000, Tides: a scientific history. Cambridge University Press. 

[3] Kvale, E. P. 2006, Marine Geology, 235, 1-4, 5. 

 

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