Rayleigh–Taylor instability

The Rayleigh–Taylor instability, or RT instability, is an instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid. Examples include the behavior of water suspended above oil in the gravity of Earth, mushroom clouds like those from volcanic eruptions and atmospheric nuclear explosions, supernova explosions in which expanding core gas is accelerated into denser shell gas, instabilities in plasma fusion reactors and inertial confinement fusion. Water suspended atop oil is an everyday example of Rayleigh–Taylor instability, it may be modeled by two plane-parallel layers of immiscible fluid, the more dense on top of the less dense one and both subject to the Earth's gravity; the equilibrium here is unstable to any perturbations or disturbances of the interface: if a parcel of heavier fluid is displaced downward with an equal volume of lighter fluid displaced upwards, the potential energy of the configuration is lower than the initial state.

Thus the disturbance will grow and lead to a further release of potential energy, as the more dense material moves down under the gravitational field, the less dense material is further displaced upwards. This was the set-up as studied by Lord Rayleigh; the important insight by G. I. Taylor was his realisation that this situation is equivalent to the situation when the fluids are accelerated, with the less dense fluid accelerating into the more dense fluid; this occurs deep underwater in a nuclear explosion. As the RT instability develops, the initial perturbations progress from a linear growth phase into a non-linear growth phase developing "plumes" flowing upwards and "spikes" falling downwards. In the linear phase, the fluid movement can be approximated by linear equations, the amplitude of perturbations is growing exponentially with time. In the non-linear phase, perturbation amplitude is too large for a linear approximation, non-linear equations are required to describe fluid motions. In general, the density disparity between the fluids determines the structure of the subsequent non-linear RT instability flows.

The difference in the fluid densities divided by their sum is defined as the Atwood number, A. For A close to 0, RT instability flows take the form of symmetric "fingers" of fluid; this process is evident not only in many terrestrial examples, from salt domes to weather inversions, but in astrophysics and electrohydrodynamics. For example, RT instability structure is evident in the Crab Nebula, in which the expanding pulsar wind nebula powered by the Crab pulsar is sweeping up ejected material from the supernova explosion 1000 years ago; the RT instability has recently been discovered in the Sun's outer atmosphere, or solar corona, when a dense solar prominence overlies a less dense plasma bubble. This latter case is a clear example of the magnetically modulated RT instability. Note that the RT instability is not to be confused with the Plateau–Rayleigh instability of a liquid jet; this instability, sometimes called the hosepipe instability, occurs due to surface tension, which acts to break a cylindrical jet into a stream of droplets having the same total volume but higher surface area.

Many people have witnessed the RT instability by looking at a lava lamp, although some might claim this is more described as an example of Rayleigh–Bénard convection due to the active heating of the fluid layer at the bottom of the lamp. The evolution of the RTI follows four main stages. In the first stage, the perturbation amplitudes are small when compared to their wavelengths, the equations of motion can be linearized, resulting in exponential instability growth. In the early portion of this stage, a sinusoidal initial perturbation retains its sinusoidal shape. However, after the end of this first stage, when non-linear effects begin to appear, one observes the beginnings of the formation of the ubiquitous mushroom-shaped spikes and bubbles; the growth of the mushroom structures continues in the second stage and can be modeled using buoyancy drag models, resulting in a growth rate, constant in time. At this point, nonlinear terms in the equations of motion can no longer be ignored; the spikes and bubbles begin to interact with one another in the third stage.

Bubble merging takes place, where the nonlinear interaction of mode coupling acts to combine smaller spikes and bubbles to produce larger ones. Bubble competition takes places, where spikes and bubbles of smaller wavelength that have become saturated are enveloped by larger ones that have not yet saturated; this develops into a region of turbulent mixing,which is the fourth and final stage in the evolution. It is assumed that the mixing region that develops is self-similar and turbulent, provided that the Reynolds number is sufficiently large; the inviscid two-dimensional Rayleigh–Taylor instability provides an excellent springboard into the mathematical study of stability because of the simple nature of the base state. This is the equilibrium state that exists before any perturbation is added to the system, is described by the mean velocity field U = W ( x, z


Under the Whyte notation for the classification of steam locomotives by wheel arrangement, a 4-6-2+2-6-4 is a Garratt or Union Garratt articulated locomotive using a pair of 4-6-2 engine units back to back, with the boiler and cab suspended between them. The 4-6-2 wheel arrangement of each engine unit has four leading wheels on two axles in a leading bogie, six powered and coupled driving wheels on three axles, two trailing wheels on one axle in a trailing truck. Since the 4-6-2 type is known as a Pacific, the corresponding Garratt type is known as a Double Pacific; the Double Pacific type was common for Garratt locomotives those intended for faster passenger service. The first of the type was the Class GF, built by Hanomag for the South African Railways in 1927; the first to be built by Beyer and Company, the owner of the Garratt patent, was the G class for the New Zealand Railways Department in 1928. Beyer, Peacock built the last Double Pacific in 1943, for the Nigerian Railways; the South African Railways operated a Double Pacific version of the Union Garratt articulated locomotive.

The Union Garratt was a hybrid locomotive Modified Fairlie and Garratt. The front end was of a typical Garratt arrangement, with a water tank mounted on the front engine unit’s frame, while the rear end was constructed in the Modified Fairlie fashion, with the coal bunker mounted on a rigid extension of the locomotive’s main frame and with the pivoting rear engine unit positioned beneath the coal bunker, it had an additional large underbelly water tank under the boiler. The main frame therefore carried the smokebox, firebox, coal bunker, as well as the underbelly water tank; as a result, like the Modified Fairlies, the Union Garratts were prone to metal fatigue and cracking of the frames. In the case of the Union Garratts, this occurred at the rear, brought about by the long frame overhang laden with the coal bunker. In the late 1960s, four South African Class GF 4-6-2+2-6-4 Garratt locomotives were sold to the Caminhos de Ferro de Moçambique, where they became the Series 911. Three six-cylinder G class Garratt locomotives were built for the New Zealand Railways Department by Beyer and Company in 1928.

They were the only six-cylinder simplex Garratts used in New Zealand and used Walschaerts valve gear to operate the outside cylinders, while Gresley conjugated valve gear operated the inner third cylinders. The locomotive was equipped with a mechanical stoker and was of an unusual design, with the coal bunker mounted on an extension to the main frame behind the cab instead of being mounted as a coal-and-water bunker on the rear engine unit’s frame, as was the usual practice on Garratt locomotives. Unlike a Union Garratt, the rear water tank was still mounted on the rear engine unit; the Garratts were acquired in response to traffic growth over the heavy grades of the North Island Main Trunk and to eliminate the use of banking locomotives on steep gradients. However, since they proved to be unsuccessful and problem-ridden in service, they were withdrawn in 1937 and rebuilt to six three-cylinder 4-6-2 Pacific locomotives by the NZR’s Hillside Workshops. In 1927, the South African Railways placed 37 Class GF Garratt locomotives in service.

They were the first of the 4-6-2+2-6-4 type and were built by Hannoversche Maschinenbau AG in Germany. The locomotive specifications were prepared by Colonel F. R. Collins DSO, Chief Mechanical Engineer of the SAR from 1922 to 1929, who designed it as a mixed traffic locomotive for use on branch lines and secondary mainlines throughout the country. A second order was placed with Henschel & Son for eighteen locomotives which were delivered in 1928; these were followed by a third order, placed with Maffei in 1928, for a final batch of ten locomotives which were delivered in that same year. They had bar frames and used Walschaerts valve gear, its good turn of speed and reasonably high tractive effort made the Class GF Double Pacific a good utility locomotive and, with 65 units, the class was the most numerous Garratt type in SAR service until the arrival of the Class GMA and Class GMAM in 1954. In 1928, the SAR placed two Class GH passenger train versions of its Class U Union Garratt in service and built by Maffei, to work the named fast passenger trains of the era, the Union Express and Union Limited, forerunners of the Blue Train.

They had Walschaerts valve gear, bar frames and mechanical stokers. They made several trips working the two Union trains out of Cape Town, but they were not as successful as had been hoped and they were soon taken off that duty, their mechanical stokers proved to be troublesome and were removed. Both remained in service until 1958. Six Garratt locomotives of this wheel arrangement, built by Euskalduna in 1931, were used on the 5 ft 6 in gauge Ferrocarril Central de Aragón in Spain, they worked the heaviest passenger trains on the steepest gradients from Valencia to Zaragoza. After the FCA was integrated into RENFE in 1941, the locomotives were converted to oil-burners and worked in the Tarragona-Valencia section on the line between Barcelona and Seville, until they were replaced by diesel traction in 1967 and retired


Kitadaitōjima spelled as Kita Daitō, Kita-Daitō-shima, Kitadaitō, is the northernmost island in the Daitō Islands group, located in the Philippine Sea southeast of Okinawa, Japan. It is administered as part of the village of Shimajiri District, Okinawa; the island is cultivated for agriculture, although it lacks freshwater sources. The island has an airport for local flights. Kitadaitōjima is a isolated coralline island, located 9 kilometres north of Minamidaitōjima, the largest island of the archipelago, 360 kilometres from Naha, Okinawa; as with the other islands in the archipelago, Kitadaitōjima is an uplifted coral atoll with a steep coastal cliff of limestone, a depressed center. The island is oval in shape, with a circumference of about 13.52 kilometres, length of 4.85 kilometres and an area of 11.94 square kilometres. The highest point is 74 metres above sea level; the 660 inhabitants live in a village in the center of the island. Kitadaitōjima has a humid subtropical climate with warm summers and mild winters.

Precipitation is significant throughout the year. The island is subject to frequent typhoons. Gallery It is uncertain, it is the most that their first sighting was by the Spanish navigator Bernardo de la Torre in 1543, in between 25 September and 2 October, during his abortive attempt to reach New Spain from the Philippines with the San Juan de Letran. It was charted, together with Kitadaitōjima, as Las Dos Hermanas. There is little doubt that Minamidaitōjima and Kitadaitōjima were again sighted by the Spanish on 28 July 1587, by Pedro de Unamuno who named them Islas sin Probecho. However, on 2 July 1820, the Russian vessel Borodino surveyed the two Daitō islands and named the south as "South Borodino Island"; the island remained uninhabited until claimed by the Empire of Japan in 1885. In 1900, a team of pioneers from Hachijōjima, an island located 287 kilometres south of Tokyo led by Tamaoki Han'emon, who had pioneered settlement on Minamidaitōjima, became the first human inhabitants of the island, started the cultivation of sugar cane from 1903.

During this period until World War II, Kitadaitōjima was owned in its entirety by Dai Nippon Sugar, which operated mines for the extraction of guano for use in fertilizer. Many of the inhabitants were seasonal workers from Taiwan. After World War II, the island was occupied by the United States; the use of increased mechanization increased phosphate yields marginally until the deposits were exhausted by the mid-1950s. The island was returned to Japan in 1972. Official home page