Rule of twelfths

The rule of twelfths is an approximation to a sine curve. It can be used as a rule of thumb for estimating a changing quantity where both the quantity and the steps are easily divisible by 12. Typical uses are predicting the height of the tide or the change in day length over the seasons.

Rule of twelfths
Graph showing relationships between the rule of twelfths (coloured bars), a sine wave (dashed blue curve) and a clockface, if high tide occurs at 12:00.

The rule

The rule states that over the first period the quantity increases by 1/12. Then in the second period by 2/12, in the third by 3/12, in the fourth by 3/12, fifth by 2/12 and at the end of the sixth period reaches its maximum with an increase of 1/12. The steps are 1:2:3:3:2:1 giving a total change of 12/12. Over the next six intervals the quantity reduces in a similar manner by 1, 2, 3, 3, 2, 1 twelfths.

Period Rule or
actual
values
Increment Cumulative
Exact value Decimal Relative size Exact value Decimal Relative size
1 Rule 1 / 12 0.08333
 
1 / 12 0.08333
 
Actual (cos 0° - cos 30°) / 2 0.06699
 
(1 - cos 30°) / 2 0.06699
 
2 Rule 2 / 12 0.16667
 
3 / 12 0.25
 
Actual (cos 30° - cos 60°) / 2 0.18301
 
(1 - cos 60°) / 2 0.25
 
3 Rule 3 / 12 0.25
 
6 / 12 0.5
 
Actual (cos 60° - cos 90°) / 2 0.25
 
(1 - cos 90°) / 2 0.5
 
4 Rule 3 / 12 0.25
 
9 / 12 0.75
 
Actual (cos 90° - cos 120°) / 2 0.25
 
(1 - cos 120°) / 2 0.75
 
5 Rule 2 / 12 0.16667
 
11 / 12 0.91667
 
Actual (cos 120° - cos 150°) / 2 0.18301
 
(1 - cos 150°) / 2 0.93301
 
6 Rule 1 / 12 0.08333
 
12 / 12 1
 
Actual (cos 150° - cos 180°) / 2 0.06699
 
(1 - cos 180°) / 2 1
 

Applications

In many parts of the world the tides approximate to a semi-diurnal sine curve, that is there are two high- and two low- tides per day. As an estimate then each period equates to an hour, with the tide rising by 1, 2, or 3 twelfths of its total range in each hour. In places where there is only one high and one low water per day, the rule can be used by assuming the steps are 2 hours. If the tidal curve does not approximate to a sine wave then the rule cannot be used.[1][2] This is important when navigating a boat or a ship in shallow water, and when launching and retrieving boats on slipways on a tidal shore.[3]

The rule is also useful for estimating the monthly change in sunrise/set and day length. Given the midsummer and midwinter day lengths the day length at any intervening month can be estimated. Alternatively, given the times of either sunrise of sunset and the two solstices the time of rise and set can be found approximately for any day.

Example calculations

Tides

If a tide table gives the information that tomorrow's low water would be at noon and that the water level at this time would be two metres above chart datum, and that at the following high tide the water level would be 14 metres, then the height of water at 3:00 p.m. can be calculated as follows:

  • The total increase in water level between low and high tide would be: 14 - 2 = 12 metres.
  • In the first hour the water level would rise by 1 twelfth of the total (12 m) or: 1 m
  • In the second hour the water level would rise by another 2 twelfths of the total (12 m) or: 2 m
  • In the third hour the water level would rise by another 3 twelfths of the total (12 m) or: 3 m
  • This gives the increase in the water level by 3:00 p.m. as 6 metres.

This represents only the increase - the total depth of the water (relative to chart datum) will include the 2 m depth at low tide: 6 m + 2 m = 8 metres.

The calculation can be simplified by adding twelfths together and reducing the fraction beforehand:

Rise of tide in three hours

Daylength

If midwinter sunrise and set are at 09:00 and 15:00, and midsummer at 03:00 and 21:00, the daylight duration will shift by 0:30, 1:00, 1:30, 1:30, 1:00 and 00:30 over the six months from one solstice to the other. Likewise the day length changes by 0:30, 1:00, 1:30, 1:30, 1:00 and 00:30 each month. More equatorial latitudes change by less, but still in the same proportions; more polar by more.

Caveats

The rule is a rough approximation only and should be applied with great caution when used for navigational purposes. Officially produced tide tables should be used in preference whenever possible.

The rule assumes that all tides behave in a regular manner, this is not true of some geographical locations, such as Poole Harbour[4] or the Solent[5] where there are "double" high waters or Weymouth Bay[4] where there is a double low water.

The rule assumes that the period between high and low tides is six hours but this is an underestimate and can vary anyway.

References

  1. ^ "Rule of Twelfths for quick tidal estimates". DIY Wood Boat. Retrieved 19 December 2017.
  2. ^ Getchell, David R. The Outboard Boater's Handbook: Advanced Seamanship and Practical Skills. International Marine. p. 195. ISBN 978-0-07-023053-8.
  3. ^ Sweet, Robert J. The weekend navigator: simple boat navigation with GPS and electronics. p. 162. ISBN 978-0-07-143035-7.
  4. ^ a b Heritage, Trevor. "Poole Harbour and its tides" (PDF). Shrimperowners. Retrieved 19 December 2017.
  5. ^ Ridge, M J, FRICS MCIT. "English Channel double tides". Bristol Nomads windsurfing club. Retrieved 19 December 2017.
Bahama Banks

The Bahama Banks are the submerged carbonate platforms that make up much of the Bahama Archipelago. The term is usually applied in referring to either the Great Bahama Bank around Andros Island, or the Little Bahama Bank of Grand Bahama Island and Great Abaco, which are the largest of the platforms, and the Cay Sal Bank north of Cuba. The islands of these banks are politically part of the Bahamas. Other banks are the three banks of the Turks and Caicos Islands, namely the Caicos Bank of the Caicos Islands, the bank of the Turks Islands, and wholly submerged Mouchoir Bank. Further southeast are the equally wholly submerged Silver Bank and Navidad Bank north of the Dominican Republic.

Carbonate platform

A carbonate platform is a sedimentary body which possesses topographic relief, and is composed of autochthonic calcareous deposits. Platform growth is mediated by sessile organisms whose skeletons build up the reef or by organisms (usually microbes) which induce carbonate precipitation through their metabolism. Therefore, carbonate platforms can not grow up everywhere: they are not present in places where limiting factors to the life of reef-building organisms exist. Such limiting factors are, among others: light, water temperature, transparency and pH-Value. For example, carbonate sedimentation along the Atlantic South American coasts takes place everywhere but at the mouth of the Amazon River, because of the intense turbidity of the water there. Spectacular examples of present-day carbonate platforms are the Bahama Banks under which the platform is roughly 8 km thick, the Yucatan Peninsula which is up to 2 km thick, the Florida platform, the platform on which the Great Barrier Reef is growing, and the Maldive atolls. All these carbonate platforms and their associated reefs are confined to tropical latitudes. Today's reefs are built mainly by scleractinian corals, but in the distant past other organisms, like archaeocyatha (during the Cambrian) or extinct cnidaria (tabulata and rugosa) were important reef builders.

List of submarine volcanoes

A list of active and extinct submarine volcanoes and seamounts located under the world's oceans. There are estimated to be 40,000 to 55,000 seamounts in the global oceans. Almost all are not well-mapped and many may not have been identified at all. Most are unnamed and unexplored. This list is therefore confined to seamounts that are notable enough to have been named and/or explored.

Michael Carson (author)

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Oceanic plateau

An oceanic or submarine plateau is a large, relatively flat elevation that is higher than the surrounding relief with one or more relatively steep sides.There are 184 oceanic plateaus covering an area of 18,486,600 km2 (7,137,700 sq mi), or about 5.11% of the oceans. The South Pacific region around Australia and New Zealand contains the greatest number of oceanic plateaus (see map).

Oceanic plateaus produced by large igneous provinces are often associated with hotspots, mantle plumes, and volcanic islands — such as Iceland, Hawaii, Cape Verde, and Kerguelen. The three largest plateaus, the Caribbean, Ontong Java, and Mid-Pacific Mountains, are located on thermal swells. Other oceanic plateaus, however, are made of rifted continental crust, for example Falkland Plateau, Lord Howe Rise, and parts of Kerguelen, Seychelles, and Arctic ridges.

Plateaus formed by large igneous provinces were formed by the equivalent of continental flood basalts such as the Deccan Traps in India and the Snake River Plain in the United States.

In contrast to continental flood basalts, most igneous oceanic plateaus erupt through young and thin (6–7 km (3.7–4.3 mi)) mafic or ultra-mafic crust and are therefore uncontaminated by felsic crust and representative for their mantle sources.

These plateaus often rise 2–3 km (1.2–1.9 mi) above the surrounding ocean floor and are more buoyant than oceanic crust. They therefore tend to withstand subduction, more-so when thick and when reaching subduction zones shortly after their formations. As a consequence, they tend to "dock" to continental margins and be preserved as accreted terranes. Such terranes are often better preserved than the exposed parts of continental flood basalts and are therefore a better record of large-scale volcanic eruptions throughout Earth's history. This "docking" also means that oceanic plateaus are important contributors to the growth of continental crust. Their formations often had a dramatic impact on global climate, such as the most recent plateaus formed, the three, large, Cretaceous oceanic plateaus in the Pacific and Indian Ocean: Ontong Java, Kerguelen, and Caribbean.

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Physical oceanography

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

Physical oceanography is one of several sub-domains into which oceanography is divided. Others include biological, chemical and geological oceanography.

Physical oceanography may be subdivided into descriptive and dynamical physical oceanography.Descriptive physical oceanography seeks to research the ocean through observations and complex numerical models, which describe the fluid motions as precisely as possible.

Dynamical physical oceanography focuses primarily upon the processes that govern the motion of fluids with emphasis upon theoretical research and numerical models. These are part of the large field of Geophysical Fluid Dynamics (GFD) that is shared together with meteorology. GFD is a sub field of Fluid dynamics describing flows occurring on spatial and temporal scales that are greatly influenced by the Coriolis force.

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St.GIGA (Japanese: セント・ギガ, Hepburn: Sento.GIGA) was a satellite radio company that was formed as a subsidiary of satellite television company WOWOW and later became semi-independent, forming a keiretsu with its parent. Using the BS network to broadcast digital radio via direct broadcast satellite as a test on November 30, 1990, St.GIGA became the world's first Satellite Digital Audio Broadcast Corporation. Regular broadcasting began March 30, 1991, and by September 1, St.GIGA adopted the commercial-free concept proposed by producer Hiroshi Yokoi and began to charge a broadcasting subscription fee.Following a period of financial difficulties and as part of an agreement with Nintendo, from between April 1995 and June 2000, St.GIGA broadcast digitally encoded video games to owners of Super Famicoms with the Satellaview attachment. Satellaview broadcasts were limited in distribution to Japan alone, however through St.GIGA's services Nintendo broadcast a large number of rare ura and gaiden versions of some of its most popular franchises such as The Legend of Zelda, Mario, and Kirby. With the exception of BS Fire Emblem: Akaneia Senki, these games have never been re-released and can only be played today via incomplete emulation.

Tide

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.

Tide tables can be used for any given locale to find the predicted times and amplitude (or "tidal range"). The predictions are influenced by many factors including the alignment of the Sun and Moon, the phase and amplitude of the tide (pattern of tides in the deep ocean), the amphidromic systems of the oceans, and the shape of the coastline and near-shore bathymetry (see Timing). They are however only predictions, the actual time and height of the tide is affected by wind and atmospheric pressure. Many shorelines experience semi-diurnal tides—two nearly equal high and low tides each day. Other locations have a diurnal tide—one high and low tide each day. A "mixed tide"—two uneven magnitude tides a day—is a third regular category.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 forces such as 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.

Tide table

Tide tables, sometimes called tide charts, are used for tidal prediction and show the daily times and levels of high and low tides, usually for a particular location. Tide heights at intermediate times (between high and low water) can be approximated by using the rule of twelfths or more accurately calculated by using a published tidal curve for the location. Tide levels are typically given relative to a low-water vertical datum, e.g. the mean lower low water (MLLW) datum in the US.

Undersea mountain range

Undersea mountain ranges are mountain ranges that are mostly or entirely underwater, and specifically under the surface of an ocean. If originated from current tectonic forces, they are often referred to as a mid-ocean ridge. In contrast, if formed by past above-water volcanism, they are known as a seamount chain. The largest and best known undersea mountain range is a mid-ocean ridge, the Mid-Atlantic Ridge. It has been observed that, "similar to those on land, the undersea mountain ranges are the loci of frequent volcanic and earthquake activity".

Wave base

The wave base, in physical oceanography, is the maximum depth at which a water wave's passage causes significant water motion. For water depths deeper than the wave base, bottom sediments and the seafloor are no longer stirred by the wave motion above.

Waves
Circulation
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Ocean zones
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