Long-period tides

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Long-period tides are gravitational tides with periods longer than one day, typically with amplitudes of a few centimeters or less. Long-period tidal constituents with relatively strong forcing include the lunar fortnightly (Mf) and lunar monthly (Ms) as well as the solar semiannual (Ssa) and solar annual (Sa) constituents.

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An analysis of the changing distance of the Earth relative to Sun, Moon, and Jupiter by Pierre-Simon de Laplace in the 18th century showed that the periods at which gravity varies cluster into three species: the semi-diurnal and the diurnal tide constituents, which have periods of a day or less, and the long-period tidal constituents.

In addition to having periods longer than a day, long-period tidal forcing is distinguished from that of the first and second species by being zonally symmetric.[ clarification needed ] The long period tides are also distinguished by the way in which the oceans respond: forcings occur sufficiently slowly that they do not excite surface gravity waves. The excitation of surface gravity waves is responsible for the high amplitude semi-diurnal tides in the Bay of Fundy, for example. In contrast, the ocean responds to long period tidal forcing with a combination of an equilibrium tide along with a possible excitation of barotropic Rossby wave normal modes [1]

Types of tides Tide type.svg
Types of tides

Formation mechanism

Gravitational Tides are caused by changes in the relative location of the Earth, sun, and moon, whose orbits are perturbed slightly by Jupiter. Newton's law of universal gravitation states that the gravitational force between a mass at a reference point on the surface of the Earth and another object such as the Moon is inversely proportional to the square of the distance between them. The declination of the Moon relative to the Earth means that as the Moon orbits the Earth during half the lunar cycle the Moon is closer to the Northern Hemisphere and during the other half the Moon is closer to the Southern Hemisphere. This periodic shift in distance gives rise to the lunar fortnightly tidal constituent. The ellipticity of the lunar orbit gives rise to a lunar monthly tidal constituent. Because of the nonlinear dependence of the force on distance additional tidal constituents exist with frequencies which are the sum and differences of these fundamental frequencies. Additional fundamental frequencies are introduced by the motion of the Sun and Jupiter, thus tidal constituents exist at all of these frequencies as well as all of the sums and differences of these frequencies, etc. The mathematical description of the tidal forces is greatly simplified by expressing the forces in terms of gravitational potentials. Because the Earth is approximately a sphere and the orbits are approximately circular it also turns out to be very convenient to describe these gravitational potentials in spherical coordinates using spherical harmonic expansions.

Oceanic response

Several factors need to be considered in determine the ocean's response to tidal forcing. These include loading effects and interactions with the solid Earth as the ocean mass is redistributed by the tides, and self-gravitation effects of the ocean on itself. However the most important is the dynamical response of the ocean to the tidal forcing, conveniently expressed in terms of Laplace's tidal equations. Because of their long periods surface gravity waves cannot be easily excited and so the long period tides were long assumed to be nearly in equilibrium with the forcing in which case the tide heights should be proportional to the disturbing potential and the induced currents should be very weak. Thus it came as a surprise when in 1967 Carl Wunsch published the tide heights for two constituents in the tropical Pacific with distinctly nonequilibrium tides. [2] More recently there has been confirmation from satellite sea level measurements of the nonequilibrium nature of the lunar fortnightly tide (GARY D. EGBERT and RICHARD D. RAY, 2003: Deviation of Long-Period Tides from Equilibrium: Kinematics and Geostrophy, J. Phys. Oceanogr., 33, 822-839), for example in the tropical Atlantic. Similar calculations for the lunar monthly tide show that this lower frequency constituent is closer to equilibrium than the fortnightly.

A number of ideas have been put forward regarding how the ocean should respond to long period tidal forcing. Several authors in the 1960s and 1970s had suggested that the tidal forcing might generate resonant barotropic Rossby Wave modes, however these modes are extremely sensitive to ocean dissipation and in any event are only weakly excited by the long period tidal forcing (Carton,J.A.,1983: The variation with frequency of the long-period tides. J. Geophys. Res.,88,7563–7571). Another idea was that long period Kelvin Waves could be excited. [3] More recently Egbert and Ray present numerical modeling results suggesting that the nonequilibrium tidal elevation of the lunar fortnightly is more closely connected to the exchange of mass between the ocean basins.

Effect on lunar orbit

The effect of long-period tides on lunar orbit is a controversial topic, some literatures conclude the long-period tides accelerate the moon and slow down the earth. [4] [5] However Cheng [6] found that dissipation of the long-period tides brakes the moon and actually accelerates the earth's rotation. To explain this, they assumed the earth's rotation depends not directly on the derivation of the forcing potential for the long period tides, so the form and period of the long-period constituents is independent of the rotation rate. For these constituents, the moon (or sun) can be thought of as orbiting a non-rotating earth in a plane with the appropriate inclination to the equator. Then the tidal "bulge" lags behind the orbiting moon thus decelerating it in its orbit (bringing it closer to the earth), and by angular momentum conservation, the earth's rotation must accelerate. But this argument is qualitative, and a quantitative resolution of the conflicting conclusions is still needed. [1]

Pole tide

One additional tidal constituent results from the centrifugal forces due, in turn, to the so-called polar motion of the Earth. The latter has nothing to do with the gravitational torques acting on the Earth by the Sun and Moon, but is "excited" by geophysical mass transports on or in the Earth itself given the (slight) oblateness of the Earth's figure, which actually gives rise to an Euler-type rotational motion with a period of about 433 days for the Earth known as the Chandler wobble (after its first discoverer Seth Chandler in the early 1900s). Incidentally the Eulerian wobble is analogous to the wobbling motion of a spinning frisbee thrown not-so-perfectly. Observationally, the (excited) Chandler wobble is a major component in the Earth's polar motion. One effect of the polar motion is to perturb the otherwise steady centrifugal force felt by the Earth, causing the Earth (and the oceans) to deform slightly at the corresponding periods, knowns as the pole tide. Like the long-period tides the pole tide has been assumed to be in equilibrium and an examination of the pole tide at ocean-basin scales seems to be consistent with that assumption. [7] The equilibrium amplitude of the pole tide is about 5 mm at it maximum at 45 degrees N. and S. latitudes; it is most clearly observed in satellite altimetry maps of sea surface height. [8] At regional scales, though, the observational record is less clear. For example, tide gauge records in the North Sea show a signal that seemed to be non-equilibrium pole tide which Wunsch has suggested is due to a resonance connected with the excitation of barotropic Rossby waves, but O'Connor and colleagues suggest it is actually wind-forced instead. [9]

Usage

The long-period tides are very useful for geophysicists, who use them to calculate the elastic Love number and to understand low frequency and large-scale oceanic motions.

Related Research Articles

Moon Earths natural satellite

The Moon is Earth's only natural satellite. Together with Earth it forms the Earth-Moon satellite system. It is about one-quarter of Earth in diameter. In the Solar System it is the fifth largest satellite, larger than any of the known dwarf planets and the largest satellite of a planet relative to the planet. The Moon is a planetary-mass object that formed a differentiated rocky body, making it a satellite planet under the geophysical definitions of the term. It lacks any significant atmosphere, hydrosphere, or magnetic field. Its surface gravity is about one-sixth of Earth's. Jupiter's moon Io is the only satellite in the Solar System known to have a higher surface gravity and density.

Tidal acceleration Natural phenomenon due to which tidal locking occurs

Tidal acceleration is an effect of the tidal forces between an orbiting natural satellite and the primary planet that it orbits. The acceleration causes a gradual recession of a satellite in a prograde orbit away from the primary, and a corresponding slowdown of the primary's rotation. The process eventually leads to tidal locking, usually of the smaller body first, and later the larger body. The Earth–Moon system is the best-studied case.

Tide Rise and fall of the sea level under astronomical gravitational influences

Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the Moon and the Sun, and the rotation of the Earth.

Tidal force A gravitational effect also known as the differential force and the perturbing force

The tidal force is a gravitational effect that stretches a body along the line towards the center of mass of another body due to a gradient in gravitational field from the other body; it is responsible for diverse phenomena, including tides, tidal locking, breaking apart of celestial bodies and formation of ring systems within the Roche limit, and in extreme cases, spaghettification of objects. It arises because the gravitational field exerted on one body by another is not constant across its parts: the nearest side is attracted more strongly than the farthest side. It is this difference that causes a body to get stretched. Thus, the tidal force is also known as the differential force, as well as a secondary effect of the gravitational field.

Geophysics Physics of the Earth and its vicinity

Geophysics is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term geophysics sometimes refers only to solid earth applications: Earth's shape; its gravitational and magnetic fields; its internal structure and composition; its dynamics and their surface expression in plate tectonics, the generation of magmas, volcanism and rock formation. However, modern geophysics organizations and pure scientists use a broader definition that includes the water cycle including snow and ice; fluid dynamics of the oceans and the atmosphere; electricity and magnetism in the ionosphere and magnetosphere and solar-terrestrial physics; and analogous problems associated with the Moon and other planets.

Tidal locking Situation in which an astronomical objects orbital period matches its rotational period

Tidal locking between a pair of co-orbiting astronomical bodies occurs when one of the objects reaches a state where there is no longer any net change in its rotation rate over the course of a complete orbit. In the case where a tidally locked body possesses synchronous rotation, the object takes just as long to rotate around its own axis as it does to revolve around its partner. For example, the same side of the Moon always faces the Earth, although there is some variability because the Moon's orbit is not perfectly circular. Usually, only the satellite is tidally locked to the larger body. However, if both the difference in mass between the two bodies and the distance between them are relatively small, each may be tidally locked to the other; this is the case for Pluto and Charon. Alternative names for the tidal locking process are gravitational locking, captured rotation, and spin–orbit locking.

An equatorial bulge is a difference between the equatorial and polar diameters of a planet, due to the centrifugal force exerted by the rotation about the body's axis. A rotating body tends to form an oblate spheroid rather than a sphere.

The Chandler wobble or Chandler variation of latitude is a small deviation in the Earth's axis of rotation relative to the solid earth, which was discovered by and named after American astronomer Seth Carlo Chandler in 1891. It amounts to change of about 9 metres (30 ft) in the point at which the axis intersects the Earth's surface and has a period of 433 days. This wobble, which is an astronomical nutation, combines with another wobble with a period of one year, so that the total polar motion varies with a period of about 7 years.

Libration Apparent oscillation of a minor body seen from the major body it orbits

In lunar astronomy, libration is the wagging or wavering of the Moon perceived by Earth-bound observers and caused by changes in their perspective. It permits an observer to see slightly different hemispheres of the surface at different times. It is similar in both cause and effect to the changes in the Moon's apparent size due to changes in distance. It is caused by three mechanisms detailed below, two of which cause a relatively tiny physical libration via tidal forces exerted by the Earth. Such true librations are known as well for other moons with locked rotation.

Amphidromic point Location at which there is little or no tide

An amphidromic point, also called a tidal node, is a geographical location which has zero tidal amplitude for one harmonic constituent of the tide. The tidal range for that harmonic constituent increases with distance from this point, though not uniformly. As such, the concept of amphidromic points is crucial to understanding tidal behaviour. The term derives from the Greek words amphi ("around") and dromos ("running"), referring to the rotary tides which circulate around amphidromic points.

Physical oceanography Study of physical conditions and physical processes within the ocean

Physical oceanography is the study of physical conditions and physical processes within the ocean, especially the motions and physical properties of ocean waters.

Rossby waves, also known as planetary waves, are a type of inertial wave naturally occurring in rotating fluids. They were first identified by Sweden-born American meteorologist Carl-Gustaf Arvid Rossby. They are observed in the atmospheres and oceans of planets owing to the rotation of the planet. Atmospheric Rossby waves on Earth are giant meanders in high-altitude winds that have a major influence on weather. These waves are associated with pressure systems and the jet stream. Oceanic Rossby waves move along the thermocline: the boundary between the warm upper layer and the cold deeper part of the ocean.

Polar motion Motion of Earths rotational axis relative to its crust

Polar motion of the Earth is the motion of the Earth's rotational axis relative to its crust. This is measured with respect to a reference frame in which the solid Earth is fixed. This variation is a few meters on the surface of the Earth.

Walter Munk American oceanographer

Walter Heinrich Munk was an American physical oceanographer. He was one of the first scientists to bring statistical methods to the analysis of oceanographic data. His work won awards including the National Medal of Science, the Kyoto Prize, and induction to the French Legion of Honour.

Orbit of the Moon The Moons circuit around the Earth

The Moon orbits Earth in the prograde direction and completes one revolution relative to the Vernal Equinox and the stars in about 27.32 days and one revolution relative to the Sun in about 29.53 days. Earth and the Moon orbit about their barycentre, which lies about 4,670 km (2,900 mi) from Earth's centre, forming a satellite system called the Earth–Moon system. On average, the distance to the Moon is about 385,000 km (239,000 mi) from Earth's centre, which corresponds to about 60 Earth radii or 1.282 light-seconds.

Earth tide is the displacement of the solid earth's surface caused by the gravity of the Moon and Sun. Its main component has meter-level amplitude at periods of about 12 hours and longer. The largest body tide constituents are semi-diurnal, but there are also significant diurnal, semi-annual, and fortnightly contributions. Though the gravitational force causing earth tides and ocean tides is the same, the responses are quite different.

Theory of tides Scientific interpretation of tidal forces

The theory of tides is the application of continuum mechanics to interpret and predict the tidal deformations of planetary and satellite bodies and their atmospheres and oceans under the gravitational loading of another astronomical body or bodies.

King tide Colloquial term for an especially high spring tide, such as a perigean spring tide.

A king tide is an especially high spring tide, especially the perigean spring tides which occur three or four times a year. King tide is not a scientific term, nor is it used in a scientific context.

Internal tides are generated as the surface tides move stratified water up and down sloping topography, which produces a wave in the ocean interior. So internal tides are internal waves at a tidal frequency. The other major source of internal waves is the wind which produces internal waves near the inertial frequency. When a small water parcel is displaced from its equilibrium position, it will return either downwards due to gravity or upwards due to buoyancy. The water parcel will overshoot its original equilibrium position and this disturbance will set off an internal gravity wave. Munk (1981) notes, "Gravity waves in the ocean's interior are as common as waves at the sea surface-perhaps even more so, for no one has ever reported an interior calm."

Tides in marginal seas are tides affected by their location in semi-enclosed areas along the margins of continents and differ from tides in the open oceans. Tides are water level variations caused by the gravitational interaction between the moon, the sun and the earth. The resulting tidal force is a secondary effect of gravity: it is the difference between the actual gravitational force and the centrifugal force. While the centrifugal force is constant across the earth, the gravitational force is dependent on the distance between the two bodies and is therefore not constant across the earth. The tidal force is thus the difference between these two forces on each location on the earth.

References

  1. 1 2 Wunsch, Carl, Haidvogel D.B., Iskandarani M. (1997). "Dynamics of the long-period tides" (PDF). Progress in Oceanography. 40 (1): 81–108. Bibcode:1997PrOce..40...81W. doi:10.1016/S0079-6611(97)00024-4.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  2. ↑ Wunsch C (1967). "The long-period tides". Rev. Geophys. 5 (4): 447–475. Bibcode:1967RvGSP...5..447W. doi:10.1029/RG005i004p00447.
  3. ↑ Miller A.J.; Luther D.S.; Hendershott M.C. (1993). "The fortnightly and monthly tides: resonant Rossby waves or nearly equilibrium gravity waves?" (PDF). Journal of Physical Oceanography. 23 (5): 879–897. Bibcode:1993JPO....23..879M. doi:10.1175/1520-0485(1993)023<0879:TFAMTR>2.0.CO;2.
  4. ↑ Christodoulidis, D.C.; Smith, D.E.; Williamson, R.G.; Klosko S.M. (1988). "Observed tidal braking in the Earth/Moon/Sun system". Journal of Geophysical Research. 93 (B6): 6216–6236. Bibcode:1988JGR....93.6216C. doi:10.1029/JB093iB06p06216. hdl: 2060/19890002733 .
  5. ↑ Marsh, J.G.; Lerch, F.J.; Putney, B.H.; Felsentreger, T.L.; Sanchez, B.V.; Klosko, S.M.; Patel, G.B.; Robbins, J.W.; Williamson, R.G.; Engelis, T.E. (1990). "The GEM‐T2 Gravitational Model". Journal of Geophysical Research: Solid Earth. 95 (B13): 22043–22071. Bibcode:1989gem..rept.....M. doi:10.1029/JB095iB13p22043. hdl: 2060/19900003668 .
  6. ↑ Cheng, M.K.; Lanes, R.J.; Tapley, B.D. (1992). "Tidal deceleration of the Moon's mean motion". Geophysical Journal International. 108 (2): 401–409. Bibcode:1992GeoJI.108..401C. doi:10.1111/j.1365-246X.1992.tb04622.x.
  7. ↑ Desai S.D. (2002). "Observing the pole tide with satellite altimetry" (PDF). J. Geophys. Res. 107 (C11): 3186. Bibcode:2002JGRC..107.3186D. doi: 10.1029/2001JC001224 .
  8. ↑ "5.2.2.3.2 Pole tides – Radar Altimetry Tutorial and Toolbox". Radar Altimetry Tutorial and Toolbox – A collaborative portal for Altimetry users. Retrieved 2021-06-28.
  9. ↑ O’Connor, William P.; Chao, Benjamin Fong; Zheng, Dawei; Au, Andrew Y. (2000-08-01). "Wind stress forcing of the North Sea 'pole tide'". Geophysical Journal International. 142 (2): 620–630. Bibcode:2000GeoJI.142..620O. CiteSeerX   10.1.1.619.5066 . doi:10.1046/j.1365-246x.2000.00184.x. ISSN   0956-540X.
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