solar tides

To calculate the amplitude of the solar tides, two imaginary wells are built from the surface to the center of the Earth. One is parallel to the line that joins the Earth and the Sun and the other is perpendicular.

The force and acceleration felt by the water in the perpendicular well are almost parallel to the Earth-Sun axis, but not exactly. The reason is that the Sun is a finite distance away and the forces are directed towards the center of the Sun and are not completely parallel. Let us calculate the component of the acceleration of gravity perpendicular to the Earth-Sun axis, , experienced by water located at a distance from the center of the Earth. Without further ado than projecting the acceleration vector, we arrive at:

Here, it is the acceleration due to the Sun's attraction:

In this last formula, it is the mass of the Sun and it is the distance from the Earth to the Sun. For its part, the component perpendicular to the axis is:.

This acceleration varies linearly between the center of the Earth and the surface. The mean value is obtained by replacing by , where is the radius of the Earth. This acceleration adds additional "weight" to the water column of the well and causes the pressure at the bottom to increase by an amount , where is the density of the water. This increase in pressure, transmitted to the ocean surface, corresponds to a variation in ocean level given by the formula (where is the acceleration of Earth's gravity):

The numerical calculation gives a variation of 8.14 cm.

We will now proceed to calculate the decrease in the acceleration of gravity caused by the Sun at a point located at a distance from the center of the Earth. Adding this additional distance into the gravitational acceleration formula:

The first addend corresponds to the acceleration for a body located at a distance . Therefore, the decrease in acceleration is:.

In turn, the average acceleration is:.

The pressure variation is, as in the previous case, , so:.

This acceleration gives an increase in the height of the ocean of 16.28 cm.

With the sum of the two effects, the semimajor axis of the ellipsoid is 24.4 cm greater than the semiminor axis. As the Earth rotates, a point on the equator sees the height of the sea reach a maximum (high tide) twice a day: each time that point passes through the semimajor axis. In the same way, every time the point passes through a semi-minor axis, the height of the sea passes through a minimum (low tide). Although the difference obtained between high tide and low tide is 24.4 cm, we must not forget that this is only the part due to the Sun and that the continents or the inclination of the Earth's axis of rotation have not been taken into account. The variation in sea height can be approximated by a sinusoid with a period of 12 hours.

lunar tides

The Moon revolves around the Earth, but the latter is not stationary. In reality, both the Moon and the Earth rotate around the center of mass of the two stars. This point is located approximately 4670 km from the center of the Earth, that is, about 1700 km deep, under the place on the Earth's surface that moves from west to east with the lunar translation movement, where the attraction of our satellite is greatest at any given time. The Moon has a mass kg and is at an average distance from the Earth of m.

The calculation of lunar tides is similar to the calculation of solar tides. Simply replace the mass and distance of the Sun with those of the Moon. The difference in height of the ocean due to the non-parallelism of the forces is:.

The numerical calculation gives us a variation of 17.9 cm.

The difference in height of the ocean caused by the difference in attraction due to different distances from the Moon is:.

The numerical calculation gives us a variation of 35.6 cm.

The difference in length between the semimajor axis and the semiminor axis of the ellipsoid due to lunar tides of 35.6 cm. Therefore, the amplitude of lunar tides is approximately twice that of solar tides. As for solar tides, the variation in sea height at a point on the Earth's surface can be approximated by a sinusoid. This time, the period is 12 hours, 25 minutes and 10 s.

Spring tides and neap tides

The ellipsoid due to solar tides has the long axis directed towards the Sun. The ellipsoid due to lunar tides has the long axis directed towards the Moon. As the Moon rotates around the Earth, the major axes of the ellipsoids do not rotate at the same speed. With respect to stars, the rotation period of the solar ellipsoid is one year. The ellipsoid of the Moon is 27.32 days. The result is that the axes of the two ellipsoids approach each other every 14.7652943 days. When the major axes of the two ellipsoids are aligned, the tidal amplitude is maximum and they are called spring tides or syzygy tides. This happens on new moons and full moons. On the other hand, when the major axis of each ellipsoid is aligned with the minor axis of the other, the tidal amplitude is minimal. This happens in the last quarters and the first quarters. These tides are called neap tides or square tides.

Tilt of the Earth's axis

Until now the fact that the Earth's axis of rotation is tilted at about 23.27° with respect to the ecliptic (the plane containing the orbit of the Earth and the Sun) has been ignored. Furthermore, the plane of the Moon's orbit is inclined about 5.145° with respect to the ecliptic. This means that the Sun occupies positions ranging from 23.44_Xizang "(2344) Xizang")° north of the equatorial plane to 23.44° south of the same plane. The Moon can occupy positions from 28.6° to -28.6°. The consequence of this is that the major axes of the ellipsoids that have been used rarely coincide with the plane of the Earth's equator.

In the image on the right, point A is at high tide. When the next high tide occurs, 12 hours, 25 minutes and 10 seconds later, the same point will be at B. This high tide will be lower than the previous and subsequent high tide.

This diurnal alternation between large and small high tides suggests the sum of two periodicities: one diurnal and the other semi-diurnal. We then speak of diurnal and semi-diurnal tidal waves, both lunar and solar. This corresponds to a mathematical model and not to physical reality.

Note that point u and locations further north only see one high tide per day. When they should be at low high tide, they are still on the same side of the ellipsoid. A similar situation occurs in the Southern Hemisphere. Mathematically, the amplitude of the semidiurnal wave is too small for it to create additional highs or lows.

The tides are maximum when the two high tides are equal. That only happens when the major axis of the ellipsoids is parallel to the equatorial plane. That is, when the sun is in the equatorial plane. This occurs during the equinoxes. The equinox tides are the highest of the year.

Other causes of variation

Several additional factors also contribute to tidal range:.

Influence of the continents

In the simplified calculation that has been carried out, in which the Earth does not have continents and is covered by a continuous hydrosphere, the distance between the two high tide positions is 20,000 km. The ocean area whose level is higher than the average value has a diameter of 10,000 km. This distance is greater than the distance between America and Europe or Africa and corresponds to the width of the Pacific Ocean. For an entire ocean like the Atlantic or Pacific to rise in level, its total water content would have to increase. Since the continents prevent this lateral movement of the entire ocean, the semidiurnal wave model does not correspond to reality.

In the image on the right you can see that the height of the oceans does not follow a wave that moves from right to left (towards the West). The displacement of water and maximums and minimums is much more complicated.

In a model without continents, the cotidal lines coincide with the meridians. In the color image on the right, the cotidal lines of the planisphere are represented and the color of the background corresponds to the tidal range. These cotidal lines correspond to a particular astronomical situation (waxing Moon, equinoxes, etc.) and change over time. In the two images it is observed that there are cotidal lines that converge towards amphidromic points, in which the tidal amplitude is equal to zero.

The situation is even more marked in the inland seas, whose dimensions are even smaller than those of the oceans. Thus, the Atlantic cannot fill or empty the Mediterranean Sea through the Strait of Gibraltar. The waters of the Mediterranean can only move east or west, rising at one end and falling at the other. The final result is complicated by the shape of the coasts that limit and deflect this lateral movement.

To a greater or lesser degree, all inland seas and open seas (although to a lesser degree) exhibit circular motion, both in ocean currents and tidal currents, and these currents can rotate clockwise in the intertropical latitudes of the Northern Hemisphere and counterclockwise in the temperate zone of the Northern Hemisphere. In the case of the southern hemisphere, these rotating movements are reversed, although we cannot speak of seas in this case, but it is the same situation with latitudinal portions of the oceans themselves.

The tides on the coasts

As has been seen, the tidal range on the high seas is less than 1 meter. On the other hand, near the coasts the amplitude is generally greater and in some cases reaches or exceeds 10 m. The following table shows some of the places where large tides occur.[6]​ Only one place has been listed per area.

It is now explained how a tide of less than one meter at sea can create a tide of several meters on the coast. The reason is the resonance "Resonance (mechanical)") of the water layer located on the continental shelf. This layer is shallow (less than 200 m) and, in some cases, has a large extension to the continental slope. For example, the English Channel is a sheet of water 500 km long (from the entrance to the Pas de Calais), 150 km wide and only 100 m deep. To scale, that corresponds to a body of water 50 m long and 1 cm deep. When sea level rises at the entrance, water enters the English Channel. As the extension is large and the depth small, the speed of the water increases to about 4 to 5 kn "Knot (unit)") (2 to 2.5 m/s). Reaching that speed takes time (about three hours in the case of the English Channel), but stopping also requires a similar period. Once launched, the water continues to move forward, another three hours passing until it stops and reverses its direction. The oscillatory behavior is due to the inertia and delay that the water layer has in responding to excitation: the variation in height of the ocean beyond the continental slope. The tide will be larger depending on whether the zone's own oscillation period is closer to the period of external excitation, which is 12 hours and 25 minutes.

In the image on the left you can see the cotidal lines in the English Channel. The numbers on each line correspond to the high tide delay with respect to a reference. Note that there is a 6-hour difference between the high tides at the entrance to the English Channel and the Pas de Calais. There are also six hours between the entrance of the English Channel and the Irish Sea (between Ireland and England). There is an amphidromic point (in orange) at the entrance to the North Sea, opposite the Netherlands.

The oscillation period of the Bay of Fundy in Canada is thirteen hours. As it is very close to the excitation period, the tides are very large. On the contrary, when the proper period moves away from 12.4 h, the tidal amplitudes are smaller. The proper period of oscillation depends on the shape of the coast and the depth and length of the continental shelf.

In areas close to the Earth's equator, the tides are usually very weak, almost imperceptible, except at the mouths of rivers, where the rise of sea waters can give rise to the damming of river waters, producing an upstream wave when the crests of the incoming tide break against the water of the rivers. This wave produces a characteristic noise that is called macareo in the Orinoco delta and pororoca in the Amazon River.

The reason for the low amplitude of the tides in the intertropical zone is because it is the area where the effects of the movement of the Earth's rotation are greater due to the centrifugal force generated by said movement. Due to centrifugal force, sea level is much higher at the equator than in temperate zones and, above all, in polar zones. As is obvious, the greater height of the equatorial waters due to the centrifugal force prevents the tides from being clearly noticeable since this centrifugal force is exerted equally throughout the equatorial circumference while the tides only increase that level where the passage of the Moon and the Sun is located, and it is a much smaller increase in level.

Contenido

La explicación completa del mecanismo de las mareas, con todas las periodicidades, es extremadamente larga y complicada. Así que se comenzará empleando todas las simplificaciones posibles para luego acercarse a la realidad suprimiendo algunas de estas simplificaciones.

Se considerará que la Tierra es una esfera sin continentes rodeada por una hidrosfera y que gira alrededor del Sol en una trayectoria elíptica sin girar sobre su eje. Por ahora no se tendrá en cuenta la Luna.

Cuando un astro está en órbita alrededor de otro, la fuerza de atracción gravitacional entre los dos viene dada por la ley de gravitación de Newton:.

donde:.

Esta fuerza de atracción es la fuerza centrípeta que hace que el astro describa una circunferencia.

donde:.

El valor de la aceleración de gravedad debida al Sol es exactamente el que corresponde a una órbita con la velocidad angular y con el centro de masas terrestre a una distancia del Sol. Todas las partes de la Tierra tienen la misma velocidad angular alrededor del Sol, pero no están a la misma distancia. Las que están más lejos del centro de masas estarán sometidas a una aceleración de gravedad menor y la que están a una distancia inferior, a una aceleración mayor.

Existe otra fuerza, del mismo orden de magnitud, debida al hecho de que las fuerzas de atracción convergen hacia el centro del Sol, que se encuentra situado a una distancia finita. Se describirá más adelante.

En algunas fuentes se comete el error de añadir las aceleraciones centrífugas. Si se opta por utilizar un sistema de referencia inercial (inmóvil respecto a las estrellas), no se deben tener en cuenta las fuerzas centrífugas, que son fuerzas ficticias y que sólo aparecen en sistemas de referencia acelerados. Un observador en la Tierra ve fuerzas centrífugas porque la Tierra está en caída libre hacia el Sol. En cambio, para un observador exterior fijo, solo existen las fuerzas reales, como la fuerza de atracción que constituye la fuerza centrípeta.[5]​.

El resultado de este pequeño desequilibrio de fuerzas es que el agua de los océanos situada en el lado opuesto al Sol siente una fuerza que la empuja hacia el exterior de la órbita, mientras que el agua situada en el lado orientado hacia el Sol siente una fuerza que la empuja hacia dicho astro. La consecuencia es que la esfera de agua que recubre a la Tierra se alarga ligeramente y se transforma en un elipsoide de revolución, cuyo eje mayor está dirigido hacia el Sol. Se verá que este alargamiento relativo es muy pequeño: del orden de uno entre diez millones.

solar tides

To calculate the amplitude of the solar tides, two imaginary wells are built from the surface to the center of the Earth. One is parallel to the line that joins the Earth and the Sun and the other is perpendicular.

The force and acceleration felt by the water in the perpendicular well are almost parallel to the Earth-Sun axis, but not exactly. The reason is that the Sun is a finite distance away and the forces are directed towards the center of the Sun and are not completely parallel. Let us calculate the component of the acceleration of gravity perpendicular to the Earth-Sun axis, , experienced by water located at a distance from the center of the Earth. Without further ado than projecting the acceleration vector, we arrive at:

Here, it is the acceleration due to the Sun's attraction:

In this last formula, it is the mass of the Sun and it is the distance from the Earth to the Sun. For its part, the component perpendicular to the axis is:.

This acceleration varies linearly between the center of the Earth and the surface. The mean value is obtained by replacing by , where is the radius of the Earth. This acceleration adds additional "weight" to the water column of the well and causes the pressure at the bottom to increase by an amount , where is the density of the water. This increase in pressure, transmitted to the ocean surface, corresponds to a variation in ocean level given by the formula (where is the acceleration of Earth's gravity):

The numerical calculation gives a variation of 8.14 cm.

We will now proceed to calculate the decrease in the acceleration of gravity caused by the Sun at a point located at a distance from the center of the Earth. Adding this additional distance into the gravitational acceleration formula:

The first addend corresponds to the acceleration for a body located at a distance . Therefore, the decrease in acceleration is:.

In turn, the average acceleration is:.

The pressure variation is, as in the previous case, , so:.

This acceleration gives an increase in the height of the ocean of 16.28 cm.

With the sum of the two effects, the semimajor axis of the ellipsoid is 24.4 cm greater than the semiminor axis. As the Earth rotates, a point on the equator sees the height of the sea reach a maximum (high tide) twice a day: each time that point passes through the semimajor axis. In the same way, every time the point passes through a semi-minor axis, the height of the sea passes through a minimum (low tide). Although the difference obtained between high tide and low tide is 24.4 cm, we must not forget that this is only the part due to the Sun and that the continents or the inclination of the Earth's axis of rotation have not been taken into account. The variation in sea height can be approximated by a sinusoid with a period of 12 hours.

lunar tides

The Moon revolves around the Earth, but the latter is not stationary. In reality, both the Moon and the Earth rotate around the center of mass of the two stars. This point is located approximately 4670 km from the center of the Earth, that is, about 1700 km deep, under the place on the Earth's surface that moves from west to east with the lunar translation movement, where the attraction of our satellite is greatest at any given time. The Moon has a mass kg and is at an average distance from the Earth of m.

The calculation of lunar tides is similar to the calculation of solar tides. Simply replace the mass and distance of the Sun with those of the Moon. The difference in height of the ocean due to the non-parallelism of the forces is:.

The numerical calculation gives us a variation of 17.9 cm.

The difference in height of the ocean caused by the difference in attraction due to different distances from the Moon is:.

The numerical calculation gives us a variation of 35.6 cm.

The difference in length between the semimajor axis and the semiminor axis of the ellipsoid due to lunar tides of 35.6 cm. Therefore, the amplitude of lunar tides is approximately twice that of solar tides. As for solar tides, the variation in sea height at a point on the Earth's surface can be approximated by a sinusoid. This time, the period is 12 hours, 25 minutes and 10 s.

Spring tides and neap tides

The ellipsoid due to solar tides has the long axis directed towards the Sun. The ellipsoid due to lunar tides has the long axis directed towards the Moon. As the Moon rotates around the Earth, the major axes of the ellipsoids do not rotate at the same speed. With respect to stars, the rotation period of the solar ellipsoid is one year. The ellipsoid of the Moon is 27.32 days. The result is that the axes of the two ellipsoids approach each other every 14.7652943 days. When the major axes of the two ellipsoids are aligned, the tidal amplitude is maximum and they are called spring tides or syzygy tides. This happens on new moons and full moons. On the other hand, when the major axis of each ellipsoid is aligned with the minor axis of the other, the tidal amplitude is minimal. This happens in the last quarters and the first quarters. These tides are called neap tides or square tides.

Tilt of the Earth's axis

Until now the fact that the Earth's axis of rotation is tilted at about 23.27° with respect to the ecliptic (the plane containing the orbit of the Earth and the Sun) has been ignored. Furthermore, the plane of the Moon's orbit is inclined about 5.145° with respect to the ecliptic. This means that the Sun occupies positions ranging from 23.44_Xizang "(2344) Xizang")° north of the equatorial plane to 23.44° south of the same plane. The Moon can occupy positions from 28.6° to -28.6°. The consequence of this is that the major axes of the ellipsoids that have been used rarely coincide with the plane of the Earth's equator.

In the image on the right, point A is at high tide. When the next high tide occurs, 12 hours, 25 minutes and 10 seconds later, the same point will be at B. This high tide will be lower than the previous and subsequent high tide.

This diurnal alternation between large and small high tides suggests the sum of two periodicities: one diurnal and the other semi-diurnal. We then speak of diurnal and semi-diurnal tidal waves, both lunar and solar. This corresponds to a mathematical model and not to physical reality.

Note that point u and locations further north only see one high tide per day. When they should be at low high tide, they are still on the same side of the ellipsoid. A similar situation occurs in the Southern Hemisphere. Mathematically, the amplitude of the semidiurnal wave is too small for it to create additional highs or lows.

The tides are maximum when the two high tides are equal. That only happens when the major axis of the ellipsoids is parallel to the equatorial plane. That is, when the sun is in the equatorial plane. This occurs during the equinoxes. The equinox tides are the highest of the year.

Other causes of variation

Several additional factors also contribute to tidal range:.

Influence of the continents

In the simplified calculation that has been carried out, in which the Earth does not have continents and is covered by a continuous hydrosphere, the distance between the two high tide positions is 20,000 km. The ocean area whose level is higher than the average value has a diameter of 10,000 km. This distance is greater than the distance between America and Europe or Africa and corresponds to the width of the Pacific Ocean. For an entire ocean like the Atlantic or Pacific to rise in level, its total water content would have to increase. Since the continents prevent this lateral movement of the entire ocean, the semidiurnal wave model does not correspond to reality.

In the image on the right you can see that the height of the oceans does not follow a wave that moves from right to left (towards the West). The displacement of water and maximums and minimums is much more complicated.

In a model without continents, the cotidal lines coincide with the meridians. In the color image on the right, the cotidal lines of the planisphere are represented and the color of the background corresponds to the tidal range. These cotidal lines correspond to a particular astronomical situation (waxing Moon, equinoxes, etc.) and change over time. In the two images it is observed that there are cotidal lines that converge towards amphidromic points, in which the tidal amplitude is equal to zero.

The situation is even more marked in the inland seas, whose dimensions are even smaller than those of the oceans. Thus, the Atlantic cannot fill or empty the Mediterranean Sea through the Strait of Gibraltar. The waters of the Mediterranean can only move east or west, rising at one end and falling at the other. The final result is complicated by the shape of the coasts that limit and deflect this lateral movement.

To a greater or lesser degree, all inland seas and open seas (although to a lesser degree) exhibit circular motion, both in ocean currents and tidal currents, and these currents can rotate clockwise in the intertropical latitudes of the Northern Hemisphere and counterclockwise in the temperate zone of the Northern Hemisphere. In the case of the southern hemisphere, these rotating movements are reversed, although we cannot speak of seas in this case, but it is the same situation with latitudinal portions of the oceans themselves.

The tides on the coasts

As has been seen, the tidal range on the high seas is less than 1 meter. On the other hand, near the coasts the amplitude is generally greater and in some cases reaches or exceeds 10 m. The following table shows some of the places where large tides occur.[6]​ Only one place has been listed per area.

It is now explained how a tide of less than one meter at sea can create a tide of several meters on the coast. The reason is the resonance "Resonance (mechanical)") of the water layer located on the continental shelf. This layer is shallow (less than 200 m) and, in some cases, has a large extension to the continental slope. For example, the English Channel is a sheet of water 500 km long (from the entrance to the Pas de Calais), 150 km wide and only 100 m deep. To scale, that corresponds to a body of water 50 m long and 1 cm deep. When sea level rises at the entrance, water enters the English Channel. As the extension is large and the depth small, the speed of the water increases to about 4 to 5 kn "Knot (unit)") (2 to 2.5 m/s). Reaching that speed takes time (about three hours in the case of the English Channel), but stopping also requires a similar period. Once launched, the water continues to move forward, another three hours passing until it stops and reverses its direction. The oscillatory behavior is due to the inertia and delay that the water layer has in responding to excitation: the variation in height of the ocean beyond the continental slope. The tide will be larger depending on whether the zone's own oscillation period is closer to the period of external excitation, which is 12 hours and 25 minutes.

In the image on the left you can see the cotidal lines in the English Channel. The numbers on each line correspond to the high tide delay with respect to a reference. Note that there is a 6-hour difference between the high tides at the entrance to the English Channel and the Pas de Calais. There are also six hours between the entrance of the English Channel and the Irish Sea (between Ireland and England). There is an amphidromic point (in orange) at the entrance to the North Sea, opposite the Netherlands.

The oscillation period of the Bay of Fundy in Canada is thirteen hours. As it is very close to the excitation period, the tides are very large. On the contrary, when the proper period moves away from 12.4 h, the tidal amplitudes are smaller. The proper period of oscillation depends on the shape of the coast and the depth and length of the continental shelf.

In areas close to the Earth's equator, the tides are usually very weak, almost imperceptible, except at the mouths of rivers, where the rise of sea waters can give rise to the damming of river waters, producing an upstream wave when the crests of the incoming tide break against the water of the rivers. This wave produces a characteristic noise that is called macareo in the Orinoco delta and pororoca in the Amazon River.

The reason for the low amplitude of the tides in the intertropical zone is because it is the area where the effects of the movement of the Earth's rotation are greater due to the centrifugal force generated by said movement. Due to centrifugal force, sea level is much higher at the equator than in temperate zones and, above all, in polar zones. As is obvious, the greater height of the equatorial waters due to the centrifugal force prevents the tides from being clearly noticeable since this centrifugal force is exerted equally throughout the equatorial circumference while the tides only increase that level where the passage of the Moon and the Sun is located, and it is a much smaller increase in level.