In this memorable photograph (courtesy of NASA), we see astronaut Buzz Aldrin holding in his right hand a sophisticated mirror: the Laser Ranging Retro-Reflector (LR3). This mirror has now been standing on the Moon for 50 years.

By sending a laser beam from the Earth to the mirror, and measuring the time it takes for the light to return, the distance to the Moon can be accurately calculated because we know the speed of light. This sounds simple in principle, but the execution and interpretation of this kind of measurement is far from straightforward.

First of all, the beam has to be pointed accurately to a tiny mirror over a vast distance. Moreover, there is in fact no such thing as “the” distance to the Moon: it varies a lot due to the lunar orbital characteristics (notably the ellipticity). On average, it is about 385000 km (i.e., 60 times the Earth’s radius), but it varies by as much as 50000 km. Notwithstanding these enormous variations, it has been possible to deduce with remarkable accuracy that the Moon recedes from the Earth at a net rate of 3.8 cm per year.

Something of this order of magnitude was already expected on theoretical grounds even before the Apollo mission took place – based, in part, on Babylonian clay tablets! It is not that the Babylonians were measuring the distance to the Moon – accomplished astronomers though they were – but they documented diligently where and when total solar eclipses took place.

Modern astronomical calculations can reconstruct these events, but assuming the present length of day would place them westward of the actual location. From this discrepancy, the change in the length of day can be inferred as an increase of about 2 milliseconds per century.

Classical mechanics provides the link between the slowing of the Earth’s spin and the recession of the Moon (via the principle of conservation of angular momentum): if one is known, the other can be calculated. The slowing of the Earth’s spin also implies a loss of kinetic energy of the Earth.The beneficiary of this loss is the tide: every second, 3.2×10^{12 }J is put into the (lunar) tides. This number, too, follows from the lunar recession rate. Thus, the Apollo mission to the Moon helped us a step forward in understanding the tides on Earth.

From a geological perspective, the present recession rate seems unusually large. Numerical modelling on paleotides – the tides during the geological past when the continents were shaped differently and located elsewhere – indicates that tides were mostly weaker, implying a smaller energy input and a smaller lunar recession rate.

Theo Gerkema, Department of Estuarine and Delta Systems, NIOZ Royal Netherlands Institute for Sea Research, and Utrecht University, Yerseke, The Netherlands

]]>Some decades later, this notion of a hidden form of attraction was placed in a new light by Newton’s universal theory of gravity. For the understanding of tides, this was a fundamental step forward in two ways. First, the theory at once put the Moon and Sun on an analogous footing as tide-generating bodies; second, the mathematical formulation of the force of gravity shifted the focus from the unanswerable why to the more productive how.

With that, the force behind the tides was well explained. However, what we are ultimately interested in is the response of the oceans and seas to this force. This response is complex because the tides propagate as waves, as was elucidated by Laplace in the late 18th century. The complexity of the oceans, with their irregular shapes due to the presence of continents, precludes a purely mathematical treatment of the problem.

From a practical point of view, a correct prediction of high and low tides in harbors and adjacent coastal seas was the most pressing issue. Knowing the fundamental frequencies involved the tides from astronomy and hydrodynamics, the art of prediction reached a high level of reliability in the 19th century, again with some of the greatest physicists involved (notably Lord Kelvin).

By the early 20th century, the main gaps in understanding concerned the pattern of tidal propagation in the open ocean, where measurements are scarce, and the dissipation of tides. Meanwhile, another kind of tide had been discovered: the internal tides that propagate in the interior of the ocean.

The pieces finally came together with the advancement of satellite altimetry, combined with numerical modelling and increasing computer power. For the first time, an accurate picture arose of the propagation of tides in the open ocean. Moreover, it could now be estimated where the losses of energy occur. Deep ocean ridges turned out to be key areas, where energy leaks into internal tides, which in turn induce abyssal mixing. Remarkably, this small-scale mixing was found to play an essential role in maintaining the large-scale meridional overturning circulation in the ocean – revealing yet another dimension of tides. Over half a century, key contributions to these ideas came from Walter Munk, who died this year at the age of 101. Numerical models also opened up a way to study tides in the geological past, the subject of paleotides.

Writing a textbook on tides feels like a journey through time. Some parts of the book could have been written by ancient Greek astronomers, others by 17th century students of Newton’s theory, still others by a physical oceanographer in the 19th century. But the advances of recent decades enable us to bring the elements together in a newly unified way.

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