Tide

for the hypothetical case of an ocean of constant depth without land, and on the assumption that Earth is not rotating; otherwise there is a lag angle. Solar tides not shown.|alt= (U.S.), low tide occurs roughly at moonrise and high tide with a high Moon, corresponding to the simple gravity model of two tidal bulges; at most places however, the Moon and tides have a phase shift. Tides are the periodic rise and fall of sea level resulting from the differential gravitational forces exerted primarily by the Moon and the Sun, combined with inertial effects associated with the Earth–Moon system’s orbital motion and the Earth's rotation. While these astronomical forcings generate the fundamental tidal potential, actual observed tides are strongly modified by terrestrial factors, including the geometry of ocean basins, continental boundaries, bathymetry, the coriolis effect, frictional dissipation within shallow seas and the tidal resonance of coastlines. Tides vary on timescales ranging from hours to years due to a number of factors, which determine the lunitidal interval. To make accurate records, tide gauges at fixed stations measure water level over time. Gauges ignore variations caused by waves with periods shorter than minutes. These data are compared to the reference (or datum) level usually called mean sea level.…

for the hypothetical case of an ocean of constant depth without land, and on the assumption that Earth is not rotating; otherwise there is a lag angle. Solar tides not shown.|alt= (U.S.), low tide occurs roughly at moonrise and high tide with a high Moon, corresponding to the simple gravity model of two tidal bulges; at most places however, the Moon and tides have a phase shift. Tides are the periodic rise and fall of sea level resulting from the differential gravitational forces exerted primarily by the Moon and the Sun, combined with inertial effects associated with the Earth–Moon system’s orbital motion and the Earth's rotation. While these astronomical forcings generate the fundamental tidal potential, actual observed tides are strongly modified by terrestrial factors, including the geometry of ocean basins, continental boundaries, bathymetry, the coriolis effect, frictional dissipation within shallow seas and the tidal resonance of coastlines. Tides vary on timescales ranging from hours to years due to a number of factors, which determine the lunitidal interval. To make accurate records, tide gauges at fixed stations measure water level over time. Gauges ignore variations caused by waves with periods shorter than minutes. These data are compared to the reference (or datum) level usually called mean sea level. While tides are usually the largest source of short-term sea-level fluctuations, sea levels are also subject to change from thermal expansion, wind, and barometric pressure changes, resulting in storm surges, especially in shallow seas and near coasts. Tidal phenomena are not limited to the oceans, but can occur in other systems whenever a gravitational field that varies in time and space is present. For example, the shape of the solid part of the Earth is affected slightly by Earth tide, though this is not as easily seen as the water tidal movements. s, curved arrows show the direction of the tides, each indicating a synchronized 6-hour period. Tidal ranges generally increase with increasing distance from amphidromic points. Tide waves move around these points, generally counterclockwise in the N. Hemisphere and clockwise in the S. Hemisphere |alt=Map showing relative tidal magnitudes of different ocean areas Because the M 2 tidal constituent dominates in most locations, the stage or phase of a tide, denoted by the time in hours after high water, is a useful concept. Tidal stage is also measured in degrees, with 360° per tidal cycle. Lines of constant tidal phase are called cotidal lines, which are analogous to contour lines of constant altitude on topographical maps, and when plotted form a cotidal map or cotidal chart. High water is reached simultaneously along the cotidal lines extending from the coast out into the ocean, and cotidal lines (and hence tidal phases) advance along the coast. Semi-diurnal and long phase constituents are measured from high water, diurnal from maximum flood tide. This and the discussion that follows is precisely true only for a single tidal constituent. For an ocean in the shape of a circular basin enclosed by a coastline, the cotidal lines point radially inward and must eventually meet at a common point, the amphidromic point. The amphidromic point is at once cotidal with high and low waters, which is satisfied by zero tidal motion. (The rare exception occurs when the tide encircles an island, as it does around New Zealand, Iceland and Madagascar.) Tidal motion generally lessens moving away from continental coasts, so that crossing the cotidal lines are contours of constant amplitude (half the distance between high and low water) which decrease to zero at the amphidromic point. For a semi-diurnal tide the amphidromic point can be thought of roughly like the center of a clock face, with the hour hand pointing in the direction of the high water cotidal line, which is directly opposite the low water cotidal line. High water rotates about the amphidromic point once every 12 hours in the direction of rising cotidal lines, and away from ebbing cotidal lines. This rotation, caused by the Coriolis effect, is generally clockwise in the southern hemisphere and counterclockwise in the northern hemisphere. The difference of cotidal phase from the phase of a reference tide is the epoch. The reference tide is the hypothetical constituent "equilibrium tide" on a landless Earth measured at 0° longitude, the Greenwich meridian. In the North Atlantic, because the cotidal lines circulate counterclockwise around the amphidromic point, the high tide passes New York Harbor approximately an hour ahead of Norfolk Harbor. South of Cape Hatteras the tidal forces are more complex, and cannot be predicted reliably based on the North Atlantic cotidal lines. === Equilibrium theory === The simplest model, due to Isaac Newton that explains two daily tides is known as equilibrium theory. Equilibrium theory makes three simplifications: 1) ignore Earth's land, 2) ignore the viscosity of water so it can respond to gravity instantly, 3) ignore friction between the Earth and water. In a coordinate system rotating with the Earth-Moon pair, the distance between the Earth and Moon is constant: they are in equilibrium. This equilibrium can be explained as a balance of the force of the Moon's gravity and centrifugal force from rotation. At the center of the Earth, the forces are equal and opposite. For other points the forces do not exactly balance and the residual force is called the tide-generating force. For points on Earth's surface but closest to the Moon, gravity is very slightly stronger; or points farthest away, centrifugal force is slightly stronger. On the poles away from the Earth-Moon line, the small net force points into the Earth. The ocean water is barely affected by these forces. In between the poles and the equator, a component of the small force points horizontal to the surface of the Earth and towards the equator. No force opposes this small force. The ocean water flows in response to this force, leaving the poles and accumulating near the equator. The result is a double tidal bulge along the Earth-Moon axis, somewhat larger on the side closer to the Moon. As the Earth rotates on its axis, different points on Earth move through these bulges, roughly explaining the daily double tides. Both the oceans' water and the solid Earth experience these differences in pull, but the rigid Earth resists deformation and keeps its roughly spherical shape, while the fluid redistributes to match the imbalance, forming the bulges. The equilibrium tide is the idealized tide assuming a landless Earth. residual 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). Note that this illustration exaggerates the differences in the actual Earth-Moon system, for clarity.|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. is the phase offset with regard to the astronomical state at time t = 0. There is one term for the Moon and a second term for the Sun. The phase of the first harmonic for the Moon term is called the lunitidal interval or high water interval. The next refinement is to accommodate the harmonic terms due to the elliptical shape of the orbits. To do so, the value of the amplitude is taken to be not a constant, but varying with time, about the average amplitude . To do so, replace in the above equation with where is another sinusoid, similar to the cycles and epicycles of Ptolemaic theory. This gives A(t) = A_o\bigl(1 + A_a \cos(\omega_a t + p_a)\bigr), which is to say an average value with a sinusoidal variation about it of magnitude , with frequency and phase . Substituting this for in the original equation gives a product of two cosine factors: A_o \bigl( 1 + A_a \cos(\omega_a t + p_a)\bigr) \cos(\omega t + p). Given that for any and \cos x \cos y = \tfrac{1}{2} \cos(x + y) + \tfrac{1}{2} \cos(x - y), it is clear that a compound term involving the product of two cosine terms each with their own frequency is the same as three simple cosine terms that are to be added at the original frequency and also at frequencies which are the sum and difference of the two frequencies of the product term. (Three, not two terms, since the whole expression is (1 + \cos x) \cos y .) Consider further that the tidal force on a location depends also on whether the Moon (or the Sun) is above or below the plane of the Equator, and that these attributes have their own periods also incommensurable with a day and a month, and it is clear that many combinations result. With a careful choice of the basic astronomical frequencies, the Doodson Number annotates the particular additions and differences to form the frequency of each simple cosine term. .|alt=Graph showing one line each for M  2 , S  2 , N  2 , K  1 , O  1 , P  1 , and one for their summation, with the X axis spanning slightly more than a single day Remember that astronomical tides do not include weather effects. Also, changes to local conditions (sandbank movement, dredging harbour mouths, etc.) away from those prevailing at the measurement time affect the tide's actual timing and magnitude. Organisations quoting a "highest astronomical tide" for some location may exaggerate the figure as a safety factor against analytical uncertainties, distance from the nearest measurement point, changes since the last observation time, ground subsidence, etc., to avert liability should an engineering work be overtopped. Special care is needed when assessing the size of a "weather surge" by subtracting the astronomical tide from the observed tide. Careful Fourier data analysis over a nineteen-year period (the National Tidal Datum Epoch in the U.S.) uses frequencies called the tidal harmonic constituents. Nineteen years is preferred because the Earth, Moon and Sun's relative positions repeat almost exactly in the Metonic cycle of 19 years, which is long enough to include the 18.613 year lunar nodal tidal constituent. This analysis can be done using only the knowledge of the forcing period, but without detailed understanding of the mathematical derivation, which means that useful tidal tables have been constructed for centuries. The resulting amplitudes and phases can then be used to predict the expected tides. These are usually dominated by the constituents near 12 hours (the semi-diurnal constituents), but there are major constituents near 24 hours (diurnal) as well. Longer term constituents are 14 day or fortnightly, monthly, and semiannual. Semi-diurnal tides dominated coastline, but some areas such as the South China Sea and the Gulf of Mexico are primarily diurnal. In the semi-diurnal areas, the primary constituents M 2  (lunar) and S 2  (solar) periods differ slightly, so that the relative phases, and thus the amplitude of the combined tide, change fortnightly (14 day period). In the M 2 plot above, each cotidal line differs by one hour from its neighbors, and the thicker lines show tides in phase with equilibrium at Greenwich. The lines rotate around the amphidromic points counterclockwise in the northern hemisphere so that from Baja California Peninsula to Alaska and from France to Ireland the M 2 tide propagates northward. In the southern hemisphere this direction is clockwise. On the other hand, M 2 tide propagates counterclockwise around New Zealand, but this is because the islands act as a dam and permit the tides to have different heights on the islands' opposite sides. (The tides do propagate northward on the east side and southward on the west coast, as predicted by theory.) The exception is at Cook Strait where the tidal currents periodically link high to low water. This is because cotidal lines 180° around the amphidromes are in opposite phase, for example high water across from low water at each end of Cook Strait. Each tidal constituent has a different pattern of amplitudes, phases, and amphidromic points, so the M 2 patterns cannot be used for other tide components. , U.S., during a 50-hour period.|alt=Graph with a single line rising and falling between 4 peaks around 3 and four valleys around −3 Because the Moon is moving in its orbit around the Earth and in the same sense as the Earth's rotation, a point on the Earth must rotate slightly further to catch up so that the time between semi-diurnal tides is not twelve but 12.4206 hours—a bit over twenty-five minutes extra. The two peaks are not equal. The two high tides a day alternate in maximum heights: lower high (just under three feet), higher high (just over three feet), and again lower high. Likewise for the low tides. When the Earth, Moon, and Sun are in line (Sun–Earth–Moon, or Sun–Moon–Earth) the two main influences combine to produce spring tides; when the two forces are opposing each other as when the angle Moon–Earth–Sun is close to ninety degrees, neap tides result. As the Moon moves around its orbit it changes from north of the Equator to south of the Equator. The alternation in high tide heights becomes smaller, until they are the same (at the lunar equinox, the Moon is above the Equator), then redevelop but with the other polarity, waxing to a maximum difference and then waning again. ====Cook Strait==== The tidal flow through Cook Strait between the two main islands of New Zealand is particularly interesting, as the tides on each side of the strait are almost exactly out of phase, so that one side's high water is simultaneous with the other's low water. Strong currents result, with almost zero tidal height change in the strait's center. Yet, although the tidal surge normally flows in one direction for six hours and in the reverse direction for six hours, a particular surge might last eight or ten hours with the reverse surge enfeebled. In especially boisterous weather conditions, the reverse surge might be entirely overcome so that the flow continues in the same direction through three or more surge periods. The character of the Cook Strait's tidal cycle also differs between coasts. On the west coast and Tasman/Golden Bay, tidal ranges tend to follow a classic fortnightly spring–neap cycle, with larger ranges at springs and smaller ranges at neaps, driven by the alignment of the Moon and Sun. In contrast, on parts of the east coast, such as around Wellington and Napier, the tidal pattern shows a stronger monthly modulation related to perigean–apogean signals (Moon’s varying distance), and does not exhibit as pronounced a fortnightly spring–neap signal. The graph of Cook Strait's tides shows separately the high water and low water height and time, through November 2007; these are not measured values but instead are calculated from tidal parameters derived from years-old measurements. Cook Strait's nautical chart offers tidal current information. For instance the January 1979 edition for (northwest of Cape Terawhiti) refers timings to Westport while the January 2004 issue refers to Wellington. Near Cape Terawhiti in the middle of Cook Strait the tidal height variation is almost nil while the tidal current reaches its maximum, especially near the notorious Karori Rip. Aside from weather effects, the actual currents through Cook Strait are influenced by the tidal height differences between the two ends of the strait and as can be seen, only one of the two spring tides at the north west end of the strait near Nelson has a counterpart spring tide at the south east end (Wellington), so the resulting behaviour follows neither reference harbour. ====Nantucket Shoals==== In the Nantucket Shoals region of the Atlantic Ocean, tidal currents are rotary in character, meaning the direction of flow gradually changes through all compass directions over a tidal cycle rather than simply reversing back and forth along the same line. Over the shoals, the currents tend to turn clockwise, with typical peak velocities in the range of 1.5 to 2.5 knots, and minimums around 0.5 knot, though velocities can vary significantly with position and tidal phase. Because the current direction steadily rotates rather than abruptly reversing, there is no distinct slack water period as found in simple reversing currents; instead, the current speed waxes and wanes throughout the cycle while its direction continuously shifts. ==In culture== Tides have long held cultural as well as scientific significance, shaping mythology, literature, language, and human worldviews. Writers and historians note that the rhythmic rise and fall of the sea has inspired metaphors of time, change, and human experience across traditions, while authors of cultural history have traced how tidal phenomena intersect with folklore and literary expression. In many ancient societies, before scientific explanations were developed, tidal movements were often explained through cosmology or divine forces, reflecting both reverence and mystery in human understanding of the ocean's rhythms. Artists and poets across periods have drawn on tidal imagery to explore themes of transformation and reflection. Tides appear in folklore and literary metaphors representing cycles of life, fate, and emotional ebb and flow. The moon's influence appears in coastal ritual practices and seafaring lore, underscoring the interplay between natural rhythms and cultural narratives. Gaylord Johnson "How Moon and Sun Generate the Tides" Popular Science, April 1934

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