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101-409: The Travelers Weather Research Center (1954-1961) was the world's first privately-owned research institute for the scientific study of weather . It pioneered the use of statistical methods, mathematical models, and computers to improve weather forecasting techniques, and was the first to state weather predictions in terms of probabilities ("20% chance of rain"). The Travelers Weather Research Center

202-444: A K-system . A chaotic system may have sequences of values for the evolving variable that exactly repeat themselves, giving periodic behavior starting from any point in that sequence. However, such periodic sequences are repelling rather than attracting, meaning that if the evolving variable is outside the sequence, however close, it will not enter the sequence and in fact, will diverge from it. Thus for almost all initial conditions,

303-418: A coastline's length varies with the scale of the measuring instrument, resembles itself at all scales, and is infinite in length for an infinitesimally small measuring device. Arguing that a ball of twine appears as a point when viewed from far away (0-dimensional), a ball when viewed from fairly near (3-dimensional), or a curved strand (1-dimensional), he argued that the dimensions of an object are relative to

404-483: A concrete experiment. And Boris Chirikov himself is considered as a pioneer in classical and quantum chaos. The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by hand. Electronic computers made these repeated calculations practical, while figures and images made it possible to visualize these systems. As

505-524: A dual nature of chaos and order with distinct predictability", in contrast to the conventional view of "weather is chaotic". Discrete chaotic systems, such as the logistic map , can exhibit strange attractors whatever their dimensionality . In contrast, for continuous dynamical systems, the Poincaré–Bendixson theorem shows that a strange attractor can only arise in three or more dimensions. Finite-dimensional linear systems are never chaotic; for

606-481: A dynamical system to display chaotic behavior, it must be either nonlinear or infinite-dimensional. The Poincaré–Bendixson theorem states that a two-dimensional differential equation has very regular behavior. The Lorenz attractor discussed below is generated by a system of three differential equations such as: where x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} make up

707-510: A few days (unproven); the inner solar system, 4 to 5 million years. In chaotic systems, the uncertainty in a forecast increases exponentially with elapsed time. Hence, mathematically, doubling the forecast time more than squares the proportional uncertainty in the forecast. This means, in practice, a meaningful prediction cannot be made over an interval of more than two or three times the Lyapunov time. When meaningful predictions cannot be made,

808-422: A few days in advance, weather forecasters are continually working to extend this limit through meteorological research and refining current methodologies in weather prediction. However, it is theoretically impossible to make useful day-to-day predictions more than about two weeks ahead, imposing an upper limit to potential for improved prediction skill. Weather is one of the fundamental processes that shape

909-406: A fourth or higher derivative are called accordingly hyperjerk systems. A jerk system's behavior is described by a jerk equation, and for certain jerk equations, simple electronic circuits can model solutions. These circuits are known as jerk circuits. One of the most interesting properties of jerk circuits is the possibility of chaotic behavior. In fact, certain well-known chaotic systems, such as

1010-401: A given day. Since outdoor activities are severely curtailed by heavy rain , snow and the wind chill , forecasts can be used to plan activities around these events and to plan ahead to survive through them. Tropical weather forecasting is different from that at higher latitudes. The sun shines more directly on the tropics than on higher latitudes (at least on average over a year), which makes

1111-629: A graduate student in Chihiro Hayashi's laboratory at Kyoto University, Yoshisuke Ueda was experimenting with analog computers and noticed, on November 27, 1961, what he called "randomly transitional phenomena". Yet his advisor did not agree with his conclusions at the time, and did not allow him to report his findings until 1970. Edward Lorenz was an early pioneer of the theory. His interest in chaos came about accidentally through his work on weather prediction in 1961. Lorenz and his collaborator Ellen Fetter and Margaret Hamilton were using

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1212-412: A simple digital computer, a Royal McBee LGP-30 , to run weather simulations. They wanted to see a sequence of data again, and to save time they started the simulation in the middle of its course. They did this by entering a printout of the data that corresponded to conditions in the middle of the original simulation. To their surprise, the weather the machine began to predict was completely different from

1313-523: A single jet stream near the 50th parallel north latitude, and Venus has a single jet near the equator. One of the most famous landmarks in the Solar System , Jupiter's Great Red Spot , is an anticyclonic storm known to have existed for at least 300 years. On other giant planets , the lack of a surface allows the wind to reach enormous speeds: gusts of up to 600 metres per second (about 2,100 km/h or 1,300 mph) have been measured on

1414-428: A unique evolution and is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos , or simply chaos . The theory was summarized by Edward Lorenz as: Chaos: When the present determines the future but the approximate present does not approximately determine

1515-472: A variety of disciplines, including meteorology , anthropology , sociology , environmental science , computer science , engineering , economics , ecology , and pandemic crisis management . The theory formed the basis for such fields of study as complex dynamical systems , edge of chaos theory and self-assembly processes. Chaos theory concerns deterministic systems whose behavior can, in principle, be predicted. Chaotic systems are predictable for

1616-410: A week ahead. This does not mean that one cannot assert anything about events far in the future—only that some restrictions on the system are present. For example, we know that the temperature of the surface of the earth will not naturally reach 100 °C (212 °F) or fall below −130 °C (−202 °F) on earth (during the current geologic era ), but we cannot predict exactly which day will have

1717-509: A while and then 'appear' to become random. The amount of time for which the behavior of a chaotic system can be effectively predicted depends on three things: how much uncertainty can be tolerated in the forecast, how accurately its current state can be measured, and a time scale depending on the dynamics of the system, called the Lyapunov time . Some examples of Lyapunov times are: chaotic electrical circuits, about 1 millisecond; weather systems,

1818-881: Is a second countable , complete metric space , then topological transitivity implies the existence of a dense set of points in X that have dense orbits. For a chaotic system to have dense periodic orbits means that every point in the space is approached arbitrarily closely by periodic orbits. The one-dimensional logistic map defined by x → 4 x (1 – x ) is one of the simplest systems with density of periodic orbits. For example, 5 − 5 8 {\displaystyle {\tfrac {5-{\sqrt {5}}}{8}}}  → 5 + 5 8 {\displaystyle {\tfrac {5+{\sqrt {5}}}{8}}}  → 5 − 5 8 {\displaystyle {\tfrac {5-{\sqrt {5}}}{8}}} (or approximately 0.3454915 → 0.9045085 → 0.3454915)

1919-427: Is a spontaneous order. The essence here is that most orders in nature arise from the spontaneous breakdown of various symmetries. This large family of phenomena includes elasticity, superconductivity, ferromagnetism, and many others. According to the supersymmetric theory of stochastic dynamics , chaos, or more precisely, its stochastic generalization, is also part of this family. The corresponding symmetry being broken

2020-574: Is a weaker version of topological mixing . Intuitively, if a map is topologically transitive then given a point x and a region V , there exists a point y near x whose orbit passes through V . This implies that it is impossible to decompose the system into two open sets. An important related theorem is the Birkhoff Transitivity Theorem. It is easy to see that the existence of a dense orbit implies topological transitivity. The Birkhoff Transitivity Theorem states that if X

2121-489: Is an (unstable) orbit of period 2, and similar orbits exist for periods 4, 8, 16, etc. (indeed, for all the periods specified by Sharkovskii's theorem ). Sharkovskii's theorem is the basis of the Li and Yorke (1975) proof that any continuous one-dimensional system that exhibits a regular cycle of period three will also display regular cycles of every other length, as well as completely chaotic orbits. Some dynamical systems, like

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2222-746: Is an example of a chaotic system. Topological mixing is often omitted from popular accounts of chaos, which equate chaos with only sensitivity to initial conditions. However, sensitive dependence on initial conditions alone does not give chaos. For example, consider the simple dynamical system produced by repeatedly doubling an initial value. This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points eventually becomes widely separated. However, this example has no topological mixing, and therefore has no chaos. Indeed, it has extremely simple behavior: all points except 0 tend to positive or negative infinity. A map f : X → X {\displaystyle f:X\to X}

2323-798: Is based upon convolution integral which mediates interaction between spatially distributed maps: ψ n + 1 ( r → , t ) = ∫ K ( r → − r → , , t ) f [ ψ n ( r → , , t ) ] d r → , {\displaystyle \psi _{n+1}({\vec {r}},t)=\int K({\vec {r}}-{\vec {r}}^{,},t)f[\psi _{n}({\vec {r}}^{,},t)]d{\vec {r}}^{,}} , where kernel K ( r → − r → , , t ) {\displaystyle K({\vec {r}}-{\vec {r}}^{,},t)}

2424-508: Is being developed in a branch of mathematical analysis known as functional analysis . The above set of three ordinary differential equations has been referred to as the three-dimensional Lorenz model. Since 1963, higher-dimensional Lorenz models have been developed in numerous studies for examining the impact of an increased degree of nonlinearity, as well as its collective effect with heating and dissipations, on solution stability. The straightforward generalization of coupled discrete maps

2525-514: Is generally understood to mean the weather of Earth. Weather is driven by air pressure , temperature , and moisture differences between one place and another. These differences can occur due to the Sun's angle at any particular spot, which varies with latitude . The strong temperature contrast between polar and tropical air gives rise to the largest scale atmospheric circulations : the Hadley cell ,

2626-453: Is generated by the Rössler equations , which have only one nonlinear term out of seven. Sprott found a three-dimensional system with just five terms, that had only one nonlinear term, which exhibits chaos for certain parameter values. Zhang and Heidel showed that, at least for dissipative and conservative quadratic systems, three-dimensional quadratic systems with only three or four terms on

2727-446: Is hot or cold, wet or dry, calm or stormy, clear or cloudy . On Earth , most weather phenomena occur in the lowest layer of the planet's atmosphere , the troposphere , just below the stratosphere . Weather refers to day-to-day temperature, precipitation , and other atmospheric conditions, whereas climate is the term for the averaging of atmospheric conditions over longer periods of time. When used without qualification, "weather"

2828-497: Is propagator derived as Green function of a relevant physical system, f [ ψ n ( r → , t ) ] {\displaystyle f[\psi _{n}({\vec {r}},t)]} might be logistic map alike ψ → G ψ [ 1 − tanh ⁡ ( ψ ) ] {\displaystyle \psi \rightarrow G\psi [1-\tanh(\psi )]} or complex map . For examples of complex maps

2929-424: Is said to be topologically transitive if for any pair of non-empty open sets U , V ⊂ X {\displaystyle U,V\subset X} , there exists k > 0 {\displaystyle k>0} such that f k ( U ) ∩ V ≠ ∅ {\displaystyle f^{k}(U)\cap V\neq \emptyset } . Topological transitivity

3030-409: Is the topological supersymmetry which is hidden in all stochastic (partial) differential equations , and the corresponding order parameter is a field-theoretic embodiment of the butterfly effect. James Clerk Maxwell first emphasized the " butterfly effect ", and is seen as being one of the earliest to discuss chaos theory, with work in the 1860s and 1870s. An early proponent of chaos theory

3131-643: Is the saving of Japan from invasion by the Mongol fleet of Kublai Khan by the Kamikaze winds in 1281. French claims to Florida came to an end in 1565 when a hurricane destroyed the French fleet, allowing Spain to conquer Fort Caroline . More recently, Hurricane Katrina redistributed over one million people from the central Gulf coast elsewhere across the United States, becoming the largest diaspora in

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3232-587: Is the study of short-lived atmospheric phenomena smaller than mesoscale , about 1 km or less. These two branches of meteorology are sometimes grouped together as "mesoscale and microscale meteorology" (MMM) and together study all phenomena smaller than synoptic scale ; that is they study features generally too small to be depicted on a weather map . These include small and generally fleeting cloud "puffs" and other small cloud features. On Earth, temperatures usually range ±40 °C (100 °F to −40 °F) annually. The range of climates and latitudes across

3333-438: Is the third derivative of position , with respect to time. As such, differential equations of the form are sometimes called jerk equations . It has been shown that a jerk equation, which is equivalent to a system of three first order, ordinary, non-linear differential equations, is in a certain sense the minimal setting for solutions showing chaotic behavior. This motivates mathematical interest in jerk systems. Systems involving

3434-470: Is thought to have easterly winds moving at more than 9,600 kilometres per hour (6,000 mph). Weather is not limited to planetary bodies. Like all stars, the Sun's corona is constantly being lost to space, creating what is essentially a very thin atmosphere throughout the Solar System . The movement of mass ejected from the Sun is known as the solar wind . Inconsistencies in this wind and larger events on

3535-466: Is to start with a point in the basin of attraction of the attractor, and then simply plot its subsequent orbit. Because of the topological transitivity condition, this is likely to produce a picture of the entire final attractor, and indeed both orbits shown in the figure on the right give a picture of the general shape of the Lorenz attractor. This attractor results from a simple three-dimensional model of

3636-619: The Ferrel cell , the polar cell , and the jet stream . Weather systems in the middle latitudes , such as extratropical cyclones , are caused by instabilities of the jet streamflow. Because Earth's axis is tilted relative to its orbital plane (called the ecliptic ), sunlight is incident at different angles at different times of the year. On Earth's surface, temperatures usually range ±40 °C (−40 °F to 104 °F) annually. Over thousands of years, changes in Earth's orbit can affect

3737-415: The Hadley cell while a smaller scale example would be coastal breezes . The atmosphere is a chaotic system . As a result, small changes to one part of the system can accumulate and magnify to cause large effects on the system as a whole. This atmospheric instability makes weather forecasting less predictable than tidal waves or eclipses. Although it is difficult to accurately predict weather more than

3838-717: The Julia set f [ ψ ] = ψ 2 {\displaystyle f[\psi ]=\psi ^{2}} or Ikeda map ψ n + 1 = A + B ψ n e i ( | ψ n | 2 + C ) {\displaystyle \psi _{n+1}=A+B\psi _{n}e^{i(|\psi _{n}|^{2}+C)}} may serve. When wave propagation problems at distance L = c t {\displaystyle L=ct} with wavelength λ = 2 π / k {\displaystyle \lambda =2\pi /k} are considered

3939-612: The Lorenz weather system. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because it is not only one of the first, but it is also one of the most complex, and as such gives rise to a very interesting pattern that, with a little imagination, looks like the wings of a butterfly. Unlike fixed-point attractors and limit cycles , the attractors that arise from chaotic systems, known as strange attractors , have great detail and complexity. Strange attractors occur in both continuous dynamical systems (such as

4040-696: The Virgin Islands . As of 2019, tornadoes have had the greatest impact on humans with 42 fatalities while costing crop and property damage over 3 billion dollars. The weather has played a large and sometimes direct part in human history . Aside from climatic changes that have caused the gradual drift of populations (for example the desertification of the Middle East, and the formation of land bridges during glacial periods), extreme weather events have caused smaller scale population movements and intruded directly in historical events. One such event

4141-648: The adiabatic lapse rate . In some situations, the temperature actually increases with height. This phenomenon is known as an inversion and can cause mountaintops to be warmer than the valleys below. Inversions can lead to the formation of fog and often act as a cap that suppresses thunderstorm development. On local scales, temperature differences can occur because different surfaces (such as oceans, forests, ice sheets, or human-made objects) have differing physical characteristics such as reflectivity , roughness, or moisture content. Surface temperature differences in turn cause pressure differences. A hot surface warms

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4242-602: The solar wind with the terrestrial atmosphere can produce spectacular aurorae , and can play havoc with electrically sensitive systems such as electricity grids and radio signals. Chaos theory Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors in numerical computation , can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general. This can happen even though these systems are deterministic , meaning that their future behavior follows

4343-425: The system state , t {\displaystyle t} is time, and σ {\displaystyle \sigma } , ρ {\displaystyle \rho } , β {\displaystyle \beta } are the system parameters . Five of the terms on the right hand side are linear, while two are quadratic; a total of seven terms. Another well-known chaotic attractor

4444-761: The " butterfly effect ", so-called because of the title of a paper given by Edward Lorenz in 1972 to the American Association for the Advancement of Science in Washington, D.C., entitled Predictability: Does the Flap of a Butterfly's Wings in Brazil set off a Tornado in Texas? . The flapping wing represents a small change in the initial condition of the system, which causes a chain of events that prevents

4545-440: The Earth. The process of weathering breaks down the rocks and soils into smaller fragments and then into their constituent substances. During rains precipitation, the water droplets absorb and dissolve carbon dioxide from the surrounding air. This causes the rainwater to be slightly acidic, which aids the erosive properties of water. The released sediment and chemicals are then free to take part in chemical reactions that can affect

4646-444: The Lorenz attractor and the Rössler map , are conventionally described as a system of three first-order differential equations that can combine into a single (although rather complicated) jerk equation. Another example of a jerk equation with nonlinearity in the magnitude of x {\displaystyle x} is: Here, A is an adjustable parameter. This equation has a chaotic solution for A =3/5 and can be implemented with

4747-569: The Lorenz system) and in some discrete systems (such as the Hénon map ). Other discrete dynamical systems have a repelling structure called a Julia set , which forms at the boundary between basins of attraction of fixed points. Julia sets can be thought of as strange repellers. Both strange attractors and Julia sets typically have a fractal structure, and the fractal dimension can be calculated for them. In contrast to single type chaotic solutions, recent studies using Lorenz models have emphasized

4848-585: The Middle Ages. Finland suffered a severe famine in 1696–1697, during which about one-third of the Finnish population died. Weather forecasting is the application of science and technology to predict the state of the atmosphere for a future time and a given location. Human beings have attempted to predict the weather informally for millennia, and formally since at least the nineteenth century. Weather forecasts are made by collecting quantitative data about

4949-486: The Solar System. The movement of mass ejected from the Sun is known as the solar wind . On Earth , common weather phenomena include wind, cloud , rain, snow, fog and dust storms . Less common events include natural disasters such as tornadoes , hurricanes , typhoons and ice storms . Almost all familiar weather phenomena occur in the troposphere (the lower part of the atmosphere). Weather does occur in

5050-677: The Travelers until 1970. Under Malone's direction, the center won the 1958 Gold Medal from the New York Board of Trade for its research. In the same year, he chaired a national committee that created "Preliminary Plans for a National Institute for Atmospheric Research", which resulted in establishment of the National Center for Atmospheric Research . In 1961 the Travelers Weather Research Center

5151-400: The above list. Sensitivity to initial conditions means that each point in a chaotic system is arbitrarily closely approximated by other points that have significantly different future paths or trajectories. Thus, an arbitrarily small change or perturbation of the current trajectory may lead to significantly different future behavior. Sensitivity to initial conditions is popularly known as

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5252-534: The air above it causing it to expand and lower the density and the resulting surface air pressure . The resulting horizontal pressure gradient moves the air from higher to lower pressure regions, creating a wind, and the Earth's rotation then causes deflection of this airflow due to the Coriolis effect . The simple systems thus formed can then display emergent behaviour to produce more complex systems and thus other weather phenomena. Large scale examples include

5353-450: The amount and distribution of solar energy received by Earth, thus influencing long-term climate and global climate change . Surface temperature differences in turn cause pressure differences. Higher altitudes are cooler than lower altitudes, as most atmospheric heating is due to contact with the Earth's surface while radiative losses to space are mostly constant. Weather forecasting is the application of science and technology to predict

5454-527: The amount and distribution of solar energy received by the Earth and influence long-term climate. (See Milankovitch cycles ). The uneven solar heating (the formation of zones of temperature and moisture gradients, or frontogenesis ) can also be due to the weather itself in the form of cloudiness and precipitation. Higher altitudes are typically cooler than lower altitudes, which is the result of higher surface temperature and radiational heating, which produces

5555-476: The benefit of a theory to explain what they were seeing. Despite initial insights in the first half of the twentieth century, chaos theory became formalized as such only after mid-century, when it first became evident to some scientists that linear theory , the prevailing system theory at that time, simply could not explain the observed behavior of certain experiments like that of the logistic map . What had been attributed to measure imprecision and simple " noise "

5656-432: The current state of the atmosphere and using scientific understanding of atmospheric processes to project how the atmosphere will evolve. Once an all-human endeavor based mainly upon changes in barometric pressure , current weather conditions, and sky condition, forecast models are now used to determine future conditions. On the other hand, human input is still required to pick the best possible forecast model to base

5757-686: The difference in current time and the time for which the forecast is being made (the range of the forecast) increases. The use of ensembles and model consensus helps to narrow the error and pick the most likely outcome. There are a variety of end users to weather forecasts. Weather warnings are important forecasts because they are used to protect life and property. Forecasts based on temperature and precipitation are important to agriculture, and therefore to commodity traders within stock markets. Temperature forecasts are used by utility companies to estimate demand over coming days. In some areas, people use weather forecasts to determine what to wear on

5858-421: The discrete-time case, this is true for all continuous maps on metric spaces . In these cases, while it is often the most practically significant property, "sensitivity to initial conditions" need not be stated in the definition. If attention is restricted to intervals , the second property implies the other two. An alternative and a generally weaker definition of chaos uses only the first two properties in

5959-405: The effect it has on humans in different situations, etc. Therefore, weather is something people often communicate about. The National Weather Service has an annual report for fatalities, injury, and total damage costs which include crop and property. They gather this data via National Weather Service offices located throughout the 50 states in the United States as well as Puerto Rico , Guam , and

6060-480: The following jerk circuit; the required nonlinearity is brought about by the two diodes: In the above circuit, all resistors are of equal value, except R A = R / A = 5 R / 3 {\displaystyle R_{A}=R/A=5R/3} , and all capacitors are of equal size. The dominant frequency is 1 / 2 π R C {\displaystyle 1/2\pi RC} . The output of op amp 0 will correspond to

6161-464: The forecast upon, which involves many disciplines such as pattern recognition skills, teleconnections , knowledge of model performance, and knowledge of model biases. The chaotic nature of the atmosphere, the massive computational power required to solve the equations that describe the atmosphere, the error involved in measuring the initial conditions, and an incomplete understanding of atmospheric processes mean that forecasts become less accurate as of

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6262-429: The future. Chaotic behavior exists in many natural systems, including fluid flow, heartbeat irregularities, weather and climate. It also occurs spontaneously in some systems with artificial components, such as road traffic . This behavior can be studied through the analysis of a chaotic mathematical model or through analytical techniques such as recurrence plots and Poincaré maps . Chaos theory has applications in

6363-858: The greatest snowfall in a period of twelve months occurred in Mount Rainier , Washington, US. It was recorded as 31,102 mm (102.04 ft) of snow. Studying how the weather works on other planets has been seen as helpful in understanding how it works on Earth. Weather on other planets follows many of the same physical principles as weather on Earth , but occurs on different scales and in atmospheres having different chemical composition. The Cassini–Huygens mission to Titan discovered clouds formed from methane or ethane which deposit rain composed of liquid methane and other organic compounds . Earth's atmosphere includes six latitudinal circulation zones, three in each hemisphere. In contrast, Jupiter's banded appearance shows many such zones, Titan has

6464-675: The history of the United States. The Little Ice Age caused crop failures and famines in Europe. During the period known as the Grindelwald Fluctuation (1560–1630), volcanic forcing events seem to have led to more extreme weather events. These included droughts, storms and unseasonal blizzards, as well as causing the Swiss Grindelwald Glacier to expand. The 1690s saw the worst famine in France since

6565-453: The hottest temperature of the year. In more mathematical terms, the Lyapunov exponent measures the sensitivity to initial conditions, in the form of rate of exponential divergence from the perturbed initial conditions. More specifically, given two starting trajectories in the phase space that are infinitesimally close, with initial separation δ Z 0 {\displaystyle \delta \mathbf {Z} _{0}} ,

6666-408: The importance of considering various types of solutions. For example, coexisting chaotic and non-chaotic may appear within the same model (e.g., the double pendulum system) using the same modeling configurations but different initial conditions. The findings of attractor coexistence, obtained from classical and generalized Lorenz models, suggested a revised view that "the entirety of weather possesses

6767-692: The kernel K {\displaystyle K} may have a form of Green function for Schrödinger equation :. K ( r → − r → , , L ) = i k exp ⁡ [ i k L ] 2 π L exp ⁡ [ i k | r → − r → , | 2 2 L ] {\displaystyle K({\vec {r}}-{\vec {r}}^{,},L)={\frac {ik\exp[ikL]}{2\pi L}}\exp[{\frac {ik|{\vec {r}}-{\vec {r}}^{,}|^{2}}{2L}}]} . In physics , jerk

6868-448: The lower the sun angle is, which causes those locations to be cooler due to the spread of the sunlight over a greater surface. The strong temperature contrast between polar and tropical air gives rise to the large scale atmospheric circulation cells and the jet stream . Weather systems in the mid-latitudes, such as extratropical cyclones , are caused by instabilities of the jet stream flow (see baroclinity ). Weather systems in

6969-568: The observer and may be fractional. An object whose irregularity is constant over different scales ("self-similarity") is a fractal (examples include the Menger sponge , the Sierpiński gasket , and the Koch curve or snowflake , which is infinitely long yet encloses a finite space and has a fractal dimension of circa 1.2619). In 1982, Mandelbrot published The Fractal Geometry of Nature , which became

7070-408: The one-dimensional logistic map defined by x → 4 x (1 – x ), are chaotic everywhere, but in many cases chaotic behavior is found only in a subset of phase space. The cases of most interest arise when the chaotic behavior takes place on an attractor , since then a large set of initial conditions leads to orbits that converge to this chaotic region. An easy way to visualize a chaotic attractor

7171-416: The onset of SDIC (i.e., prior to significant separations of initial nearby trajectories). A consequence of sensitivity to initial conditions is that if we start with a limited amount of information about the system (as is usually the case in practice), then beyond a certain time, the system would no longer be predictable. This is most prevalent in the case of weather, which is generally predictable only about

7272-504: The phase space, though it is common to just refer to the largest one. For example, the maximal Lyapunov exponent (MLE) is most often used, because it determines the overall predictability of the system. A positive MLE is usually taken as an indication that the system is chaotic. In addition to the above property, other properties related to sensitivity of initial conditions also exist. These include, for example, measure-theoretical mixing (as discussed in ergodic theory) and properties of

7373-417: The planet Neptune . This has created a puzzle for planetary scientists . The weather is ultimately created by solar energy and the amount of energy received by Neptune is only about 1 ⁄ 900 of that received by Earth, yet the intensity of weather phenomena on Neptune is far greater than on Earth. As of 2007 , the strongest planetary winds discovered are on the extrasolar planet HD 189733 b , which

7474-502: The planet can offer extremes of temperature outside this range. The coldest air temperature ever recorded on Earth is −89.2 °C (−128.6 °F), at Vostok Station , Antarctica on 21 July 1983. The hottest air temperature ever recorded was 57.7 °C (135.9 °F) at 'Aziziya , Libya, on 13 September 1922, but that reading is queried . The highest recorded average annual temperature was 34.4 °C (93.9 °F) at Dallol , Ethiopia. The coldest recorded average annual temperature

7575-414: The predictability of large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the overall system could have been vastly different. As suggested in Lorenz's book entitled The Essence of Chaos , published in 1993, "sensitive dependence can serve as an acceptable definition of chaos". In the same book, Lorenz defined the butterfly effect as: "The phenomenon that a small alteration in

7676-777: The previous calculation. They tracked this down to the computer printout. The computer worked with 6-digit precision, but the printout rounded variables off to a 3-digit number, so a value like 0.506127 printed as 0.506. This difference is tiny, and the consensus at the time would have been that it should have no practical effect. However, Lorenz discovered that small changes in initial conditions produced large changes in long-term outcome. Lorenz's discovery, which gave its name to Lorenz attractors , showed that even detailed atmospheric modeling cannot, in general, make precise long-term weather predictions. In 1963, Benoit Mandelbrot , studying information theory , discovered that noise in many phenomena (including stock prices and telephone circuits)

7777-601: The right conditions, chaos spontaneously evolves into a lockstep pattern. In the Kuramoto model , four conditions suffice to produce synchronization in a chaotic system. Examples include the coupled oscillation of Christiaan Huygens ' pendulums, fireflies, neurons , the London Millennium Bridge resonance, and large arrays of Josephson junctions . Moreover, from the theoretical physics standpoint, dynamical chaos itself, in its most general manifestation,

7878-677: The right-hand side cannot exhibit chaotic behavior. The reason is, simply put, that solutions to such systems are asymptotic to a two-dimensional surface and therefore solutions are well behaved. While the Poincaré–Bendixson theorem shows that a continuous dynamical system on the Euclidean plane cannot be chaotic, two-dimensional continuous systems with non-Euclidean geometry can still exhibit some chaotic properties. Perhaps surprisingly, chaos may occur also in linear systems, provided they are infinite dimensional. A theory of linear chaos

7979-537: The southwest and Beijing's Fangshan District recording a rainfall of 25 millimeters. Whereas there is inconclusive evidence for these techniques' efficacy, there is extensive evidence that human activity such as agriculture and industry results in inadvertent weather modification: The effects of inadvertent weather modification may pose serious threats to many aspects of civilization, including ecosystems , natural resources , food and fiber production, economic development , and human health. Microscale meteorology

8080-420: The state of a dynamical system will cause subsequent states to differ greatly from the states that would have followed without the alteration." The above definition is consistent with the sensitive dependence of solutions on initial conditions (SDIC). An idealized skiing model was developed to illustrate the sensitivity of time-varying paths to initial positions. A predictability horizon can be determined before

8181-425: The state of the atmosphere for a future time and a given location. Earth's weather system is a chaotic system ; as a result, small changes to one part of the system can grow to have large effects on the system as a whole. Human attempts to control the weather have occurred throughout history, and there is evidence that human activities such as agriculture and industry have modified weather patterns. Studying how

8282-404: The stratosphere and can affect weather lower down in the troposphere, but the exact mechanisms are poorly understood. Weather occurs primarily due to air pressure, temperature and moisture differences from one place to another. These differences can occur due to the sun angle at any particular spot, which varies by latitude in the tropics. In other words, the farther from the tropics one lies,

8383-536: The surface further (such as acid rain ), and sodium and chloride ions ( salt ) deposited in the seas/oceans. The sediment may reform in time and by geological forces into other rocks and soils. In this way, weather plays a major role in erosion of the surface. Weather, seen from an anthropological perspective, is something all humans in the world constantly experience through their senses, at least while being outside. There are socially and scientifically constructed understandings of what weather is, what makes it change,

8484-400: The surface of the star, such as coronal mass ejections , form a system that has features analogous to conventional weather systems (such as pressure and wind) and is generally known as space weather . Coronal mass ejections have been tracked as far out in the Solar System as Saturn . The activity of this system can affect planetary atmospheres and occasionally surfaces. The interaction of

8585-501: The system appears random. In common usage, "chaos" means "a state of disorder". However, in chaos theory, the term is defined more precisely. Although no universally accepted mathematical definition of chaos exists, a commonly used definition, originally formulated by Robert L. Devaney , says that to classify a dynamical system as chaotic, it must have these properties: In some cases, the last two properties above have been shown to actually imply sensitivity to initial conditions. In

8686-411: The tropics can occur more spontaneously compared to those at higher latitudes, where they are more tightly controlled by larger-scale forces in the atmosphere. Because of these differences, clouds and rain are more difficult to forecast in the tropics than at higher latitudes. On the other hand, the temperature is easily forecast in the tropics, because it does not change much. The aspiration to control

8787-419: The tropics warm (Stevens 2011). And, the vertical direction (up, as one stands on the Earth's surface) is perpendicular to the Earth's axis of rotation at the equator, while the axis of rotation and the vertical are the same at the pole; this causes the Earth's rotation to influence the atmospheric circulation more strongly at high latitudes than low latitudes. Because of these two factors, clouds and rainstorms in

8888-704: The tropics, such as monsoons or organized thunderstorm systems, are caused by different processes. Because the Earth's axis is tilted relative to its orbital plane, sunlight is incident at different angles at different times of the year. In June the Northern Hemisphere is tilted towards the Sun , so at any given Northern Hemisphere latitude sunlight falls more directly on that spot than in December (see Effect of sun angle on climate ). This effect causes seasons. Over thousands to hundreds of thousands of years, changes in Earth's orbital parameters affect

8989-423: The two trajectories end up diverging at a rate given by where t {\displaystyle t} is the time and λ {\displaystyle \lambda } is the Lyapunov exponent. The rate of separation depends on the orientation of the initial separation vector, so a whole spectrum of Lyapunov exponents can exist. The number of Lyapunov exponents is equal to the number of dimensions of

9090-406: The variable evolves chaotically with non-periodic behavior. Topological mixing (or the weaker condition of topological transitivity) means that the system evolves over time so that any given region or open set of its phase space eventually overlaps with any other given region. This mathematical concept of "mixing" corresponds to the standard intuition, and the mixing of colored dyes or fluids

9191-603: The weather is evident throughout human history: from ancient rituals intended to bring rain for crops to the U.S. Military Operation Popeye , an attempt to disrupt supply lines by lengthening the North Vietnamese monsoon . The most successful attempts at influencing weather involve cloud seeding ; they include the fog - and low stratus dispersion techniques employed by major airports, techniques used to increase winter precipitation over mountains, and techniques to suppress hail . A recent example of weather control

9292-411: The weather works on other planets has been helpful in understanding how weather works on Earth. A famous landmark in the Solar System , Jupiter's Great Red Spot , is an anticyclonic storm known to have existed for at least 300 years. However, the weather is not limited to planetary bodies. A star's corona is constantly being lost to space, creating what is essentially a very thin atmosphere throughout

9393-406: The x variable, the output of 1 corresponds to the first derivative of x and the output of 2 corresponds to the second derivative. Similar circuits only require one diode or no diodes at all. See also the well-known Chua's circuit , one basis for chaotic true random number generators. The ease of construction of the circuit has made it a ubiquitous real-world example of a chaotic system. Under

9494-408: Was Henri Poincaré . In the 1880s, while studying the three-body problem , he found that there can be orbits that are nonperiodic, and yet not forever increasing nor approaching a fixed point. In 1898, Jacques Hadamard published an influential study of the chaotic motion of a free particle gliding frictionlessly on a surface of constant negative curvature, called " Hadamard's billiards ". Hadamard

9595-561: Was China's preparation for the 2008 Summer Olympic Games . China shot 1,104 rain dispersal rockets from 21 sites in the city of Beijing in an effort to keep rain away from the opening ceremony of the games on 8 August 2008. Guo Hu, head of the Beijing Municipal Meteorological Bureau (BMB), confirmed the success of the operation with 100 millimeters falling in Baoding City of Hebei Province , to

9696-620: Was able to show that all trajectories are unstable, in that all particle trajectories diverge exponentially from one another, with a positive Lyapunov exponent . Chaos theory began in the field of ergodic theory . Later studies, also on the topic of nonlinear differential equations , were carried out by George David Birkhoff , Andrey Nikolaevich Kolmogorov , Mary Lucy Cartwright and John Edensor Littlewood , and Stephen Smale . Although chaotic planetary motion had not been observed, experimentalists had encountered turbulence in fluid motion and nonperiodic oscillation in radio circuits without

9797-483: Was considered by chaos theorists as a full component of the studied systems. In 1959 Boris Valerianovich Chirikov proposed a criterion for the emergence of classical chaos in Hamiltonian systems ( Chirikov criterion ). He applied this criterion to explain some experimental results on plasma confinement in open mirror traps. This is regarded as the very first physical theory of chaos, which succeeded in explaining

9898-513: Was established by J. Doyle DeWitt , president of the Travelers Insurance Company , to more accurately estimate weather phenomena with relation to property damage, crop losses, and accidents. In 1955, the center hired Dr. Thomas F. Malone from MIT to direct the center and oversee its long-term planning and research. Malone served as the center's director until 1957, and its research director from 1957-1964; he remained with

9999-423: Was patterned like a Cantor set , a set of points with infinite roughness and detail Mandelbrot described both the "Noah effect" (in which sudden discontinuous changes can occur) and the "Joseph effect" (in which persistence of a value can occur for a while, yet suddenly change afterwards). In 1967, he published " How long is the coast of Britain? Statistical self-similarity and fractional dimension ", showing that

10100-472: Was subsumed into a new Travelers subsidiary, The Travelers Research Corporation, in anticipation of forthcoming federal regulations controlling air quality. In 1969, the subsidiary was incorporated as the Research Corporation of New England, also known as TRC, which continues to this day. Weather Weather is the state of the atmosphere , describing for example the degree to which it

10201-520: Was −55.1 °C (−67.2 °F) at Vostok Station , Antarctica. The coldest average annual temperature in a permanently inhabited location is at Eureka, Nunavut , in Canada, where the annual average temperature is −19.7 °C (−3.5 °F). The windiest place ever recorded is in Antarctica, Commonwealth Bay (George V Coast). Here the gales reach 199 mph (320  km/h ). Furthermore,

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