Editing Tide (section)

== Physics ==
{{main|Theory of tides}}

=== Forces ===
{{main|Tidal force}}
The tidal force produced by a massive object (Moon, hereafter) on a small particle located on or in an extensive body (Earth, hereafter) is the vector difference between the gravitational force exerted by the Moon on the particle, and the gravitational force that would be exerted on the particle if it were located at the Earth's center of mass.

Whereas the [[gravitational force]] subjected by a celestial body on Earth varies inversely as the square of its distance to the Earth, the maximal tidal force varies inversely as, approximately, the cube of this distance.<ref>{{cite book |last=Young |first=C. A. |date=1889 |title=A Textbook of General Astronomy |url=https://www.gutenberg.org/files/37275/37275-pdf.pdf |page=288 |access-date=2018-08-13 |archive-date=2019-10-05 |archive-url=https://web.archive.org/web/20191005003655/http://www.gutenberg.org/files/37275/37275-pdf.pdf |url-status=live }}</ref> If the tidal force caused by each body were instead equal to its full gravitational force (which is not the case due to the [[free fall]] of the whole Earth, not only the oceans, towards these bodies) a different pattern of tidal forces would be observed, e.g. with a much stronger influence from the Sun than from the Moon: The solar gravitational force on the Earth is on average 179 times stronger than the lunar, but because the Sun is on average 389 times farther from the Earth, its field gradient is weaker. The overall proportionality is

: <math>\text{tidal force} \propto \frac{M}{d^3} \propto \rho\left(\frac{r}{d}\right)^3,</math>

where {{mvar|M}} is the mass of the heavenly body, {{mvar|d}} is its distance, {{mvar|ρ}} is its average density, and {{mvar|r}} is its radius. The ratio {{math|''r''/''d''}} is related to the angle subtended by the object in the sky. Since the Sun and the Moon have practically the same diameter in the sky, the tidal force of the Sun is less than that of the Moon because its average density is much less, and it is only 46% as large as the lunar,<!-- numbers double-checked by User:JEBrown87544, 03 Jan 2007 -->{{efn|According to [https://web.archive.org/web/19980224174548/http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/961029b.html NASA] the lunar tidal force is 2.21 times larger than the solar.}} thus during a spring tide, the Moon contributes 69% while the Sun contributes 31%. More precisely, the lunar tidal acceleration (along the Moon–Earth axis, at the Earth's surface) is about 1.1{{e|−7}}&nbsp;''g'', while the solar tidal acceleration (along the Sun–Earth axis, at the Earth's surface) is about 0.52{{e|−7}}&nbsp;''g'', where ''g'' is the [[standard gravity|gravitational acceleration]] at the Earth's surface.{{efn|See [[Tidal force#Formulation|Tidal force – Mathematical treatment]] and sources cited there.}} The effects of the other planets vary as their distances from Earth vary. When Venus is closest to Earth, its effect is 0.000113 times the solar effect.<ref name="Science Mission Directorate 2000">{{cite web | title=Interplanetary Low Tide | website=Science Mission Directorate | date=2000-05-03 | url=https://science.nasa.gov/science-news/science-at-nasa/2000/ast04may_1m | access-date=2023-06-25 | archive-date=2023-06-04 | archive-url=https://web.archive.org/web/20230604014510/https://science.nasa.gov/science-news/science-at-nasa/2000/ast04may_1m | url-status=live }}</ref> At other times, Jupiter or Mars may have the most effect.

[[File:Tidal field and gravity field.svg|thumb|The lunar [[gravity]] residual [[Vector field|field]] at the Earth's surface is known as the ''[[tide-generating force]]''. This is the primary mechanism that drives tidal action and explains two simultaneous tidal bulges; Earth's rotation further accounts for two daily high waters at any location. The figure shows both the tidal field (thick red arrows) and the gravity field (thin blue arrows) exerted on Earth's surface and center (label O) by the Moon (label S).|alt=Diagram showing a circle with closely spaced arrows pointing away from the reader on the left and right sides, while pointing towards the user on the top and bottom.]]

The ocean's surface is approximated by a surface referred to as the [[geoid]], which takes into consideration the gravitational force exerted by the earth as well as [[centrifugal force]] due to rotation. Now consider the effect of massive external bodies such as the Moon and Sun. These bodies have strong gravitational fields that diminish with distance and cause the ocean's surface to deviate from the geoid. They establish a new equilibrium ocean surface which bulges toward the moon on one side and away from the moon on the other side. The earth's rotation relative to this shape causes the daily tidal cycle. The ocean surface tends toward this equilibrium shape, which is constantly changing, and never quite attains it. When the ocean surface is not aligned with it, it's as though the surface is sloping, and water accelerates in the down-slope direction.

=== Equilibrium ===
The '''equilibrium tide''' is the idealized tide assuming a landless Earth.<ref name="AMS Glossary 2020">{{cite web |title=Equilibrium tide |website=AMS Glossary |date=2020-09-02 |url=http://glossary.ametsoc.org/wiki/Equilibrium_tide |access-date=2020-09-02 |archive-date=2020-08-01 |archive-url=https://web.archive.org/web/20200801165541/http://glossary.ametsoc.org/wiki/Equilibrium_tide |url-status=live }}</ref>
It would produce a tidal bulge in the ocean, elongated towards the attracting body (Moon or Sun).
It is ''not'' caused by the vertical pull nearest or farthest from the body, which is very weak; rather, it is caused by the tangential or [[Traction force|tractive]] tidal force, which is strongest at about 45 degrees from the body, resulting in a horizontal tidal current.{{efn|"The ocean does not produce tides as a direct response to the vertical forces at the bulges. The tidal force is only about 1 ten millionth the size of the gravitational force owing to the Earth's gravity. It is the horizontal component of the tidal force that produces the tidal bulge, causing fluid to converge at the sublunar and antipodal points and move away from the poles, causing a contraction there." (...) "The projection of the tidal force onto the horizontal direction is called the tractive force (see Knauss, Fig. 10.11). This force causes an acceleration of water towards the sublunar and antipodal points, building up water until the pressure gradient force from the bulging sea surface exactly balances the tractive force field."<ref>{{Cite web |first=LuAnne |last=Thompson |author-link=LuAnne Thompson |year=2006 |url=http://faculty.washington.edu/luanne/pages/ocean420/notes/TidesIntro.pdf |title=Physical Processes in the Ocean |access-date=2020-06-27 |archive-date=2020-09-28 |archive-url=https://web.archive.org/web/20200928191813/http://faculty.washington.edu/luanne/pages/ocean420/notes/TidesIntro.pdf |url-status=live }}</ref>}}
{{efn|"While the solar and lunar envelopes are thought of as representing the actual ocean waters, another very important factor must be recognized. The components of the tide-generating forces acting tangentially along the water surface turn out to be the most important. Just as it is easier to slide a bucket of water across a floor rather than to lift it, the horizontal tractive components move the waters toward the points directly beneath and away from the sun or moon far more effectively than the vertical components can lift them. These tractive forces are most responsible for trying to form
the ocean into the symmetrical egg-shaped distensions (the tide potential, the equilibrium tide). They reach their maximums in rings 45° from the points directly beneath and away from the sun or moon."<ref name="Hicks2006">{{cite report |last=Hicks |first=S.D. |title=Understanding Tides |publisher=[[NOAA]] |year=2006 |url=https://tidesandcurrents.noaa.gov/publications/Understanding_Tides_by_Steacy_finalFINAL11_30.pdf |language=en |access-date=2020-09-02 |archive-date=2022-01-20 |archive-url=https://web.archive.org/web/20220120232639/http://www.tidesandcurrents.noaa.gov/publications/Understanding_Tides_by_Steacy_finalFINAL11_30.pdf |url-status=live }}</ref>}}
{{efn|"... the gravitational effect that causes the tides is much too weak to lift the oceans 12 inches vertically away from the earth. It is possible, however, to move the oceans horizontally within the earth's gravitational field. This gathers the oceans toward two points where the height of the water becomes elevated by the converging volume of water."<ref>{{cite book |first=James Greig |last=Mccully |date=2006 |title=Beyond The Moon: A Conversational, Common Sense Guide To Understanding The Tides, World Scientific |publisher=World Scientific |isbn=9789814338189 |url=https://books.google.com/books?id=aKLICgAAQBAJ&q=tractal |via=[[Google Books]] |access-date=2022-01-05 |archive-date=2023-09-16 |archive-url=https://web.archive.org/web/20230916153030/https://books.google.com/books?id=aKLICgAAQBAJ&q=tractal |url-status=live }}</ref>}}<ref name="PBS LearningMedia 2020">{{cite web |title=What Physics Teachers Get Wrong about Tides! - PBS Space Time |website=[[PBS]] LearningMedia |date=2020-06-17 |url=https://www.pbslearningmedia.org/resource/what-physics-teachers-pbs-space-time/what-physics-teachers-pbs-space-time/ |access-date=2020-06-27 |archive-date=2020-10-21 |archive-url=https://web.archive.org/web/20201021020010/https://www.pbslearningmedia.org/resource/what-physics-teachers-pbs-space-time/what-physics-teachers-pbs-space-time/ |url-status=live }}</ref>

=== Laplace's tidal equations ===
Ocean depths are much smaller than their horizontal extent. Thus, the response to tidal forcing can be [[Model (abstract)|modelled]] using the [[Laplace's tidal equations|Laplace tidal equations]] which incorporate the following features:

* The vertical (or radial) velocity is negligible, and there is no vertical [[wind shear|shear]]—this is a sheet flow.
* The forcing is only horizontal ([[tangent]]ial).
* The [[Coriolis effect]] appears as an inertial force (fictitious) acting laterally to the direction of flow and proportional to velocity.
* The surface height's rate of change is proportional to the negative divergence of velocity multiplied by the depth. As the horizontal velocity stretches or compresses the ocean as a sheet, the volume thins or thickens, respectively.

The boundary conditions dictate no flow across the coastline and free slip at the bottom.

The Coriolis effect (inertial force) steers flows moving towards the Equator to the west and flows moving away from the Equator toward the east, allowing coastally trapped waves. Finally, a dissipation term can be added which is an analog to viscosity.

=== Amplitude and cycle time ===
The theoretical [[amplitude]] of oceanic tides caused by the Moon is about {{convert|54|cm|in}} at the highest point, which corresponds to the amplitude that would be reached if the ocean possessed a uniform depth, there were no landmasses, and the Earth were rotating in step with the Moon's orbit. The Sun similarly causes tides, of which the theoretical amplitude is about {{convert|25|cm|in}} (46% of that of the Moon) with a cycle time of 12 hours. At spring tide the two effects add to each other to a theoretical level of {{convert|79|cm|in}}, while at neap tide the theoretical level is reduced to {{convert|29|cm|in}}. Since the orbits of the Earth about the Sun, and the Moon about the Earth, are elliptical, tidal amplitudes change somewhat as a result of the varying Earth–Sun and Earth–Moon distances. This causes a variation in the tidal force and theoretical amplitude of about ±18% for the Moon and ±5% for the Sun. If both the Sun and Moon were at their closest positions and aligned at new moon, the theoretical amplitude would reach {{convert|93|cm|in}}.

Real amplitudes differ considerably, not only because of depth variations and continental obstacles, but also because wave propagation across the ocean has a natural period of the same order of magnitude as the rotation period: if there were no land masses, it would take about 30 hours for a long wavelength surface wave to propagate along the Equator halfway around the Earth (by comparison, the Earth's [[lithosphere]] has a natural period of about 57 minutes). [[Earth tide]]s, which raise and lower the bottom of the ocean, and the tide's own gravitational self attraction are both significant and further complicate the ocean's response to tidal forces.

=== Dissipation ===
{{See also|Tidal acceleration}}
<!--this seems misleading in terms of the oceanography. Is this global average? Cites please!: Because the lunar tidal forces drive the oceans with a period of about 12.42 hours, which is considerably less than the natural period of the oceans, complex resonance phenomena take place. This, as well as the effects of friction, gives rise to an average lag time of 11 minutes between the occurrence of high water and lunar zenith. This tidal lag time corresponds to an angle of about 3 degrees between the position of the Moon, the center of the Earth, and the location of the global average high water.

Regarding the Earth–Moon system by itself (excluding the sun for the moment) unless both bodies' spin axes align perpendicularly to the orbital plane, oscillations result. Such oscillations contribute to tidal dissipation.

Dissipation by internal fluctuating deformations of the Earth due to the lunar tidal force is small compared with the dissipation in the Earth's oceans and seas, which account for 98% of the reduction of the Earth's rotational energy.<ref name="Ray1996">{{Cite journal |last1=Ray |first1=R.D. |year=1996 |title=Detection of tidal dissipation in the solid Earth by satellite tracking and altimetry |journal=[[Nature (journal)|Nature]] |volume=381 |issue=6583 |pages=595 |doi=10.1038/381595a0 |last2=Eanes |first2=R.J. |last3=Chao |first3=B.F. |bibcode=1996Natur.381..595R}}</ref> This lack of alignment is the case for the Earth–Moon system. Thus, besides tidal bulges, opposite to each other and comparable in size, that are associated with the so-called equilibrium tide,<ref name=Boon>{{cite book |title=Secrets of the Tide: Tide and Tidal Current Analysis and Applications, Storm Surges and Sea Level Trends |last=Boon |first=John D. |url=https://books.google.com/books?id=l75xhGEZ550C&pg=PA13&dq=%22equilibrium+tide%22 |isbn=1-904275-17-6 |publisher=Hollywood Publishing |year=2004 |pages=Chapter 2 pp. 13–end |oclc=57495983 |no-pp=true |via=[[Google Books]]}}</ref> additionally, surface oscillations commonly known as the dynamical tide, characterized by a wide variety of harmonic frequencies, is established.<ref name=Toledano>{{cite journal |url=https://arxiv.org/abs/astro-ph/0610563v1 |first1=Oswaldo |last1=Toledano |first2=Edmundo |last2=Moreno |first3=Gloria |last3=Koenigsberger |first4=R. |last4=Detmers |first5=Norbert |last5=Langer |date=18 October 2006 |title=Tides in asynchronous binary systems |journal=[[Astrophysics (journal)|Astrophysics]]</ref><ref name=Lamb>{{cite book |title=Hydrodynamics |last=Lamb |first=Horace |url=https://archive.org/details/hydrodynamics02lambgoog |quote=dynamical tide. |year=1916 |publisher=[[Cambridge University Press]] |edition=4th |page=[https://archive.org/details/hydrodynamics02lambgoog/page/n359 339] |isbn=0-521-45868-4 |oclc=30070401 31079426 33629948}} </ref><ref name=Americana>{{cite book |title=The Encyclopedia Americana: A Library of Universal Knowledge |last=Harris |first=Rollin A. |publisher=Encyclopedia Americana |year=1918 |url=https://books.google.com/books?d=CF4fijqC9GgC&pg=RA1-PA613&dq=%22equilibrium+tide%22 |pages=Article on Tides, pp. 613–614 |no-pp=true |via=[[Google Books]]}}</ref>
-->

Earth's tidal oscillations introduce dissipation at an [[average]] rate of about 3.75 [[terawatt]]s.<ref name=Munk1998>{{Cite journal |last1=Munk |first1=W. |date=1998 |title=Abyssal recipes II: energetics of tidal and wind mixing |journal=Deep-Sea Research Part I |volume=45 |issue=12 |page=1977 |doi=10.1016/S0967-0637(98)00070-3 |last2=Wunsch |first2=C. |bibcode=1998DSRI...45.1977M}}</ref> About 98% of this dissipation is by marine tidal movement.<ref name="Ray1996">{{Cite journal |last1=Ray |first1=R.D. |year=1996 |title=Detection of tidal dissipation in the solid Earth by satellite tracking and altimetry |journal=[[Nature (journal)|Nature]] |volume=381 |issue=6583 |pages=595 |doi=10.1038/381595a0 |last2=Eanes |first2=R.J. |last3=Chao |first3=B.F. |bibcode=1996Natur.381..595R|s2cid=4367240 }}</ref> Dissipation arises as basin-scale tidal flows drive smaller-scale flows which experience turbulent dissipation. This tidal drag creates torque on the moon that gradually transfers angular momentum to its orbit, and a gradual increase in Earth–moon separation. The equal and opposite torque on the Earth correspondingly decreases its rotational velocity. Thus, over geologic time, the moon recedes from the Earth, at about {{convert|3.8|cm|in}}/year, lengthening the terrestrial day.{{efn|The day is currently lengthening at a rate of about 0.002 seconds per century.<ref>Lecture 2: The Role of Tidal Dissipation and the Laplace Tidal Equations by Myrl Hendershott. GFD Proceedings Volume, 2004, [[Woods Hole Oceanographic Institution|WHOI]] Notes by Yaron Toledo and Marshall Ward.</ref>}}

[[Tidal acceleration|Day length has increased]] by about 2 hours in the last 600 million years. Assuming (as a crude approximation) that the deceleration rate has been constant, this would imply that 70 million years ago, day length was on the order of 1% shorter with about 4 more days per year.

=== Bathymetry ===
[[File:Gorey Harbour at low tide.JPG|thumb|The harbour of [[Gorey, Jersey]] falls dry at low tide.]]
The shape of the shoreline and the ocean floor changes the way that tides propagate, so there is no simple, general rule that predicts the time of high water from the Moon's position in the sky. Coastal characteristics such as underwater [[bathymetry]] and coastline shape mean that individual location characteristics affect tide forecasting; actual high water time and height may differ from model predictions due to the coastal morphology's effects on tidal flow. However, for a given location the relationship between lunar [[altitude (astronomy)|altitude]] and the time of high or low tide (the [[lunitidal interval]]) is relatively constant and predictable, as is the time of high or low tide relative to other points on the same coast. For example, the high tide at [[Norfolk, Virginia]], U.S., predictably occurs approximately two and a half hours before the Moon passes directly overhead.

Land masses and ocean basins act as barriers against water moving freely around the globe, and their varied shapes and sizes affect the size of tidal frequencies. As a result, tidal patterns vary. For example, in the U.S., the East coast has predominantly semi-diurnal tides, as do Europe's Atlantic coasts, while the West coast predominantly has mixed tides.<ref name=noaa7b>{{cite web |website=U.S. [[National Oceanic and Atmospheric Administration]] (NOAA) National Ocean Service (Education section) |url=http://oceanservice.noaa.gov/education/kits/tides/media/supp_tide07b.html |title=map showing world distribution of tide patterns, semi-diurnal, diurnal and mixed semi-diurnal |access-date=2009-09-05 |archive-date=2018-08-27 |archive-url=https://web.archive.org/web/20180827211303/http://oceanservice.noaa.gov/education/kits/tides/media/supp_tide07b.html |url-status=live }}</ref><ref>{{cite book |first=H.V. |last=Thurman |date=1994 |title=Introductory Oceanography |edition=7th |location=New York |publisher=[[Macmillan Publishers]] |pages=252–276}}ref</ref><ref>{{cite book |first=D.A. |last=Ross |date=1995 |title=Introduction to Oceanography |location=New York |publisher=[[HarperCollins]] |pages=236–242}}</ref> Human changes to the landscape can also significantly alter local tides.<ref>{{cite news |last1=Witze |first1=Alexandra |title=How humans are altering the tides of the oceans |url=https://www.bbc.com/future/article/20200703-how-humans-are-altering-the-tides-of-the-oceans |access-date=8 July 2020 |work=BBC Future |publisher=[[BBC]] |date=5 July 2020 |archive-date=6 July 2020 |archive-url=https://web.archive.org/web/20200706134113/https://www.bbc.com/future/article/20200703-how-humans-are-altering-the-tides-of-the-oceans |url-status=live }}</ref>