Tide

The tide is the regular rising and falling of the ocean's surface caused by changes in gravitational forces external to the Earth. The main changing gravitational field is due to the Moon while a lesser field is caused by the Sun.

Tidal physics

Ignoring external forces, the ocean's surface defines a geopotential surface or geoid, where the gravitational force is directly towards the centre of the Earth and there is no net lateral force and hence no flow of water.

Related Topics:
Geopotential surface - Geoid

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Now consider the effect of added external, massive bodies such as the Moon and Sun. These massive bodies have strong gravitational fields that diminish with distance in space. It is the spatial differences in these fields that deform the geoid shape. This deformation has a fixed orientation relative to the influencing body and the rotation of the Earth relative to this shape drives the tides around. Gravitational forces follow the inverse-square law (force is inversely proportional to the square of the distance), but tidal forces are inversely proportional to the cube of the distance. The Sun's gravitational pull on Earth is 179 times bigger than the Moon's, but because of its much greater distance, the Sun's tidal effect is smaller than the Moon's (about 46% as strong). For simplicity, the next few sections use the word "Moon" where also "Sun" can be understood.

Related Topics:
Inverse-square law - Square - Cube

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Since the Earth's crust is solid, it moves with everything inside as one whole, as defined by the average force on it. For a geoid shape this average force is equal to the force on its centre. The water at the surface is free to move following forces on its particles. It is the difference between the forces at the Earth's centre and surface which determine the effective tidal force.

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At the point right "under" the Moon (the sub-lunar point), the water is closer than the solid Earth; so it is pulled more and rises. On the opposite side of the Earth, facing away from the Moon (the antipodal point), the water is farther than the solid earth, so it is pulled less and moves away from Earth, rising as well. On the lateral sides, the water is pulled in a slightly different direction than at the centre. The vectorial difference with the force at the centre points almost straight inwards to Earth. It can be shown that the forces at the sub-lunar and antipodal points are approximately equal and that the inward forces at the sides are about half that size. Somewhere in between there is a point where the tidal force is parallel to the Earth's surface. Those parallel components actually contribute most to the formation of tides, since the water particles are free to follow. The actual force on a particle is only about a ten millionth of the force caused by the Earth's gravity.

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These minute forces all work together:

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• pull up under and away from the Moon
• pull down at the sides
• pull towards the sub-lunar and antipodal points at intermediate points

So two bulges are formed pointing towards the Moon just under it and away from it on Earth's far side.

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Tidal amplitude and cycle time

Since the Earth rotates relative to the Moon in one lunar day (24 hours, 48 minutes), each of the two bulges travels around at that speed, leading to one high tide every 12 hours and 24 minutes. The theoretical amplitude of oceanic tides due to the Moon is about 54 cm at the highest point. This is the amplitude that would be reached if the ocean were uniform with no landmasses and Earth not rotating.

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The Sun similarly causes tides, of which the theoretical amplitude is about 25 cm (46 % of that of the Moon) and the cycle time is 12 hours.

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At spring tide the two effects add to each other to a theoretical level of 79 cm, while at neap tide the theoretical level is reduced to 29 cm.

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Real amplitudes differ considerably, not only because of global topography as explained above, but also because the natural period of the oceans is in the same order of magnitude as the rotation period: about 30 hours (by comparison, the natural period of the Earth's crust is about 57 minutes). This means that, if the Moon suddenly vanished, the level of the oceans would oscillate with a period of 30 hours with a slowly decreasing amplitude while dissipating the stored energy. This 30 h value is a simple function of terrestrial gravity and the average depth of the oceans.

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The distances of Earth from the Moon or the Sun vary, because the orbits are not circular, but elliptical. This causes a variation in the tidal force and theoretical amplitude of about ±18% for the Moon and ±5% for the Sun. So if both are in closest position and aligned, the theoretical amplitude would reach 93 cm.

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Tidal lag

Because the Moon's tidal forces drive the oceans with a period of about 12.42 hours (half of the Earth's synodic period of rotation), which is considerably less than the natural period of the oceans, complex resonance phenomena take place. The lag between the Moon's passage and the tidal response varies between 2 hours in the southern oceans, to two days in the North Sea. The global average tidal lag is six hours (which means low tide occurs when the Moon is at its zenith or its nadir, a result that goes against common intuition). Tidal lag and the transfer of momentum between sea and land causes the Earth's rotation to slow down and the Moon to be moved further away in a process known as tidal acceleration.

Related Topics:
North Sea - Tidal lag - Tidal acceleration

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Alternative explanation

Some other explanations in articles on the physics of tides include the (apparent) centrifugal force on the Earth in its orbit around the common centre of mass (the barycentre) with the Moon. The barycentre is located at about ¾ of the radius from the Earth's centre. It is important to note that the Earth has no "rotation" around this point. It just "displaces" around this point in a circular way. Every point on Earth has the same angular velocity and the same radius of orbit, but with a displaced centre. So the centrifugal force is uniform and does not contribute to the tides. However, this uniform centrifugal force is just equal (but with opposite sign) to the gravitational force acting on the centre of mass of Earth. So subtracting the gravitational force at the centre of Earth from the local gravitational forces at the surface, has the same effect as adding the (uniform) centrifugal forces. Although these two explanations seem very different, they yield the same results.

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