Tides and Amphidromic Nodes
Lunar Gravitational Tractive Forces, Kelvin Waves, and Resonant Bays
The Sun's gravitational pull on Earth is 178 times stronger than the Moon's, yet the Moon dominates ocean tides by more than two to one. Tides are driven by the gradient of gravity across Earth's 12,742-kilometer diameter. Because that differential force falls off with the inverse cube of distance ($1/r^3$), proximity beats mass.
When the Moon sits directly overhead, its upward pull reduces a water drop's weight by only one ten-millionth of Earth's gravity. Such a tiny vertical tug cannot lift the sea. Instead, tides are piled up by horizontal tractive forces at 45 degrees from zenith, sliding water sideways where gravity offers zero resistance.
Most coastlines experience two high tides every 24 hours and 50 minutes because water bulges on opposite sides of the planet simultaneously. On the near side, lunar gravity exceeds the centrifugal force of the Earth-Moon orbital wobble. On the far side, orbital centrifugal force exceeds lunar gravity, piling up an equal antipodal bulge.
Because continents block tidal bulges from circling the globe, the Coriolis force deflects the sloshing water into rotating Kelvin waves. Each wave pinwheels around a stationary hub called an amphidromic point. At roughly a dozen open-ocean nodes, the tide never rises or falls while high-water crests sweep around like clock hands.
In the open Pacific and Atlantic oceans, the surface tide rises and falls by less than one meter. Yet Canada's Bay of Fundy experiences a 16-meter tidal range. Its 270-kilometer rock funnel has a natural sloshing period of 12.5 hours, matching the 12.42-hour lunar tide in quarter-wave acoustic resonance.
Isaac Newton's static model assumed the ocean molds itself instantaneously to the Moon, predicting high tide whenever the Moon crosses the meridian. Pierre-Simon Laplace proved in 1775 that the real tide is a shallow-water wave. In a 4,000-meter ocean, wave speed ($\sqrt{gh}$) is 200 meters per second—trailing the Moon's 460-meter-per-second equatorial track.
Solving fluid equations across jagged coasts was impossible in the nineteenth century, so William Thomson (Lord Kelvin) decomposed harbor tides into astronomical sine waves. In 1872, he built a brass analog computer of pulleys and steel wire. Cranked by hand, it summed lunar and solar frequencies to trace a port's annual tides in four hours.
When a rising spring tide pushes into a shallow river estuary whose depth drops faster than the tide rises, the crest overtakes its trough. The leading edge steepens into a breaking hydraulic jump called a tidal bore. On China's Qiantang River, the bore forms a 4-meter wall charging upstream at 40 kilometers per hour.
The tallest tidal waves in the sea are invisible at the surface. As tidal currents slosh across submerged mountain chains like the Hawaiian Ridge, they heave dense deep water over the peaks. That push launches subsurface internal tides along the density jump that oscillate 150 meters vertically while ruffling the surface by centimeters.
Bottom friction and internal waves dissipate 3.7 terawatts of tidal power daily, transferring angular momentum from Earth's spin to the Moon's orbit. As the Moon recedes at 3.8 centimeters per year, Earth's day lengthens. John Wells confirmed this in 1963 by counting 400 daily growth bands per annual ring in 380-million-year-old Devonian corals.
Cold polar water sinking into the abyss would fill the ocean into a stagnant ice bath within millennia unless turbulence mixed warm surface heat downward. Walter Munk calculated in 1998 that maintaining global stratification requires 2 terawatts of mixing power. Roughly half comes from internal tides breaking against deep-sea ridges.
Along California beaches, silvery grunion fish (Leuresthes tenuis) time their spawning to the 14-day spring-neap cycle. Just after the highest spring tide following a full or new moon, grunion ride waves onto wet sand to bury their eggs. The embryos wait ten days in damp sand until the next spring tide's surf triggers hatching.