Why are there no ocean tides at the equator?


Tides are a very complex phenomenon. For any particular location, their height and fluctuation in time depends to varying degrees on the location of the Sun and the Moon, and to the details of the shape of the beach, coastline, coastline depth and prevailing ocean currents. Here is a figure that shows the difference between high tide and low tide around the world.

Newton's explanation is that, when you calculate the difference in gravity between Earth and moon at each point on the surface of Earth, you get the customary graph shown here:

This is also the shape of the 'equipotential surface' where mass would be in equilibrium and instantaneously 'at rest'. There are two gravitational tides: The Body Tide and the Water Tide. The Body Tide is the response of the solid Earth to this gravitational distortion in the solid rock of Earth. The lunar body tide has a height of 0.3 meters relative to the unstressed shape of Earth while the solar body tide is about half this high. The water tides are far higher because water is lower density than rock and is free to flow around Earth's surface with lower inertia than rock. Water tide heights can exceed 10 meters!

You would think that the solid body tide would flex the ground so severely that pipes,railroad tracks and other systems would flex and break over time. The good news is that the scale of this distortion is continent-spanning as the figure below shows.

Lunar tidal force: these images depict the Moon directly over 30 N (or 30 S) viewed from above the Northern Hemisphere, showing both sides of the planet. Red up, blue down.(Wikipedia 'Earth Tide') Red denotes a 0.3-meter upward shift due to the body tide. Each square tile is about 600 kilometers on a side, so if you calculated the distortion as you walked from the top of the bulge (red) to the minimum distortion 'blue' region, you would get a vertical shift of about 0.027 millimeters per kilometer! If you had a 1 kilometer-long pipe, its ends would flex by 0.027 millimeters or 27 microns...which is not enough to cause structural damage. Large scientific instruments such as the Large Hadron Collider in CERN have to include this distortion in the mechanical design of the circular ring, which must be precisely designed so that the path of the circulating protons is not distorted by this effect.

So what does this all have to do with whether tides are found at the Equator?

Although Newton gave us the basic gravitational theory for solid body tides, his application of this theory to the behavior of water was not correct in detail. The French mathematician Laplace used Newton's gravitational theory, but realized that its application to water tides had much more to do with the gravitational forcing of various water oscillations. Water oscillations, treated as a harmonic system with many different resonant frequencies is a much more powerful description of the details of water tides on Earth. When you combine the main lunar water tide and the solar tides acting on a complex shaped layer of water along Earth's surface, what you get is a very different pattern of high and low water tides shown in this figure.

This figure created by Dr. Richard Ray/Space Geodesy branch, NASA/GSFC, shows the M2 lunar tidal constituent. Amplitude is indicated by color, and the white lines are cotidal differing by 1 hr. The curved arcs around the amphidromic points show the direction of the tides, each indicating a synchronized 6 hour period. Note that this response of ocean water has virtually nothing to do with the simple two-bulges, gravitational stress pattern expected from Newton's calculation above.

So are there water tides at the Equator? Yes there are, and in fact the only locations that have very weak tides are near the poles!

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