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86-777: Forecasts are computed using mathematical equations for the physics and dynamics of the atmosphere. These equations are nonlinear and are impossible to solve exactly. Therefore, numerical methods obtain approximate solutions. Different models use different solution methods. Global models often use spectral methods for the horizontal dimensions and finite-difference methods for the vertical dimension, while regional models usually use finite-difference methods in all three dimensions. For specific locations, model output statistics use climate information, output from numerical weather prediction , and current surface weather observations to develop statistical relationships which account for model bias and resolution issues. The main assumption made by

172-450: A 0,0 which will be equal to the mean of the resolution. This corresponds to choosing the integration constant. To turn this into an algorithm, only finitely many frequencies are solved for. This introduces an error which can be shown to be proportional to h n {\displaystyle h^{n}} , where h := 1 / n {\displaystyle h:=1/n} and n {\displaystyle n}

258-430: A Runge Kutta technique) to find a solution. The nonlinear term is a convolution , and there are several transform-based techniques for evaluating it efficiently. See the references by Boyd and Canuto et al. for more details. One can show that if g {\displaystyle g} is infinitely differentiable, then the numerical algorithm using Fast Fourier Transforms will converge faster than any polynomial in

344-591: A conscientious objector , though this subsequently disqualified him from having any academic post. Richardson worked from 1916 to 1919 for the Friends' Ambulance Unit attached to the 16th French Infantry Division. After the war, he rejoined the Meteorological Office but was compelled to resign on grounds of conscience when it was amalgamated into the Air Ministry in 1920. He subsequently pursued

430-702: A dimensionless parameter of the theory of turbulence, is named for him. He famously summarised turbulence in rhyming verse in Weather Prediction by Numerical Process (p 66): Big whirls have little whirls that feed on their velocity, [A play on two lines of Augustus De Morgan 's poem Siphonaptera : "Great fleas have little fleas upon their backs to bite 'em, / And little fleas have lesser fleas, and so ad infinitum ." ( A Budget of Paradoxes , 1915). De Morgan's lines themselves reword two lines of Jonathan Swift 's 1733 satirical poem "On Poetry: A Rapsody".] One of Richardson's most celebrated achievements

516-463: A spectral element method is a finite element method of very high order, there is a similarity in the convergence properties. However, whereas the spectral method is based on the eigendecomposition of the particular boundary value problem, the finite element method does not use that information and works for arbitrary elliptic boundary value problems . Lewis Fry Richardson Lewis Fry Richardson , FRS (11 October 1881 – 30 September 1953)

602-416: A career on the fringes of the academic world before retiring in 1940 to research his own ideas. His pacifism influenced his research interests. According to Thomas Körner , the discovery that his meteorological work was of value to chemical weapons designers caused him to abandon his efforts in this field and destroy findings he had not yet published. Richardson's interest in meteorology led him to propose

688-498: A class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations . The idea is to write the solution of the differential equation as a sum of certain " basis functions " (for example, as a Fourier series which is a sum of sinusoids ) and then to choose the coefficients in the sum in order to satisfy the differential equation as well as possible. Spectral methods and finite-element methods are closely related and built on

774-518: A compatible global model for initial conditions of the edge of their domain. Uncertainty and errors within LAMs are introduced by the global model used for the boundary conditions of the edge of the regional model, as well as within the creation of the boundary conditions for the LAMs itself. The vertical coordinate is handled in various ways. Some models, such as Richardson's 1922 model, use geometric height ( z {\displaystyle z} ) as

860-488: A few notions by quantitative techniques in the hope of reaching a reliable answer." In Statistics of Deadly Quarrels Richardson presented data on virtually every war from 1815 to 1950, which he categorized using a base 10 logarithmic scale based on the number of battle deaths a conflict produced. In this way, he was the first to observe that the sizes of wars appeared to follow a highly right-skewed Pareto distribution , in which while small conflicts are relatively common,

946-582: A fixed receiver, as well as from weather satellites . The World Meteorological Organization acts to standardize the instrumentation, observing practices and timing of these observations worldwide. Stations either report hourly in METAR reports, or every six hours in SYNOP reports. These observations are irregularly spaced, so they are processed by data assimilation and objective analysis methods, which perform quality control and obtain values at locations usable by

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1032-441: A huge 145 hectopascals (4.3 inHg) rise in pressure over six hours when the pressure actually was more or less static. But detailed analysis by Lynch has shown that the cause was a failure to apply smoothing techniques to the data, which rule out unphysical surges in pressure. When these are applied, Richardson's forecast is essentially accurate—a remarkable achievement considering the calculations were done by hand while Richardson

1118-732: A meteorologist and director general and chief executive of the British Meteorological Office from 1992 to 1997. A great-niece of the same line of descent is the former politician Virginia Bottomley , now Baroness Bottomley. Since 1997, the Lewis Fry Richardson Medal has been awarded by the European Geosciences Union for "exceptional contributions to nonlinear geophysics in general" (by EGS until 2003 and by EGU since 2004). Winners have been: Since 1959, there has been

1204-423: A scale of less than 1 kilometre (0.62 mi), and would require a grid even finer than this to be represented physically by the equations of fluid motion. Therefore, the processes that such clouds represent are parameterized , by processes of various sophistication. In the earliest models, if a column of air in a model gridbox was unstable (i.e., the bottom warmer than the top) then it would be overturned, and

1290-406: A scheme for weather forecasting by solution of differential equations , the method used nowadays, though when he published Weather Prediction by Numerical Process in 1922, suitable fast computing was unavailable. He described his ideas thus: After so much hard reasoning, may one play with a fantasy? Imagine a large hall like a theatre, except that the circles and galleries go right round through

1376-464: A three-dimensional global climate model that gave a roughly accurate representation of the current climate. Doubling CO 2 in the model's atmosphere gave a roughly 2 °C rise in global temperature. Several other kinds of computer models gave similar results: it was impossible to make a model that gave something resembling the actual climate and not have the temperature rise when the CO 2 concentration

1462-408: Is a known, complex-valued function of two real variables, and g is periodic in x and y (that is, g ( x , y ) = g ( x + 2 π , y ) = g ( x , y + 2 π ) {\displaystyle g(x,y)=g(x+2\pi ,y)=g(x,y+2\pi )} ) then we are interested in finding a function f ( x , y ) so that where the expression on

1548-480: Is a research department, where they invent improvements. But there is much experimenting on a small scale before any change is made in the complex routine of the computing theatre. In a basement an enthusiast is observing eddies in the liquid lining of a huge spinning bowl, but so far the arithmetic proves the better way. In another building are all the usual financial, correspondence and administrative offices. Outside are playing fields, houses, mountains and lakes, for it

1634-492: Is considered an element of the beginning of the modern study of fractals . Benoît Mandelbrot quotes Richardson's research in his 1967 paper How Long Is the Coast of Britain? Richardson identified a value (between 1 and 2) that describes the changes (with increasing measurement detail) in observed complexity for a particular coastline; this value served as a model for the concept of fractal dimension . In April 1912, soon after

1720-458: Is coordinated by an official of higher rank. Numerous little "night signs" display the instantaneous values so that neighbouring computers can read them. Each number is thus displayed in three adjacent zones so as to maintain communication to the North and South on the map. From the floor of the pit a tall pillar rises to half the height of the hall. It carries a large pulpit on its top. In this sits

1806-515: Is drawn up into the adjacent atmosphere. Thus, they are important to parameterize. The horizontal domain of a model is either global , covering the entire Earth, or regional , covering only part of the Earth. Regional models also are known as limited-area models, or LAMs. Regional models use finer grid spacing to resolve explicitly smaller-scale meteorological phenomena, since their smaller domain decreases computational demands. Regional models use

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1892-524: Is his retroactive attempt to forecast the weather during a single day—20 May 1910—by direct computation. At the time, meteorologists made forecasts principally by looking for similar weather patterns from records and extrapolating. Richardson attempted to use a mathematical model of the principal features of the atmosphere, and use data taken at a specific time (7 AM) to calculate the weather six hours later ab initio . As meteorologist Peter Lynch makes clear, Richardson's forecast failed dramatically, predicting

1978-522: Is legitimate if we assume for instance that f has a continuous second derivative. By the uniqueness theorem for Fourier expansions, we must then equate the Fourier coefficients term by term, giving which is an explicit formula for the Fourier coefficients a j , k . With periodic boundary conditions, the Poisson equation possesses a solution only if b 0,0 = 0. Therefore, we can freely choose

2064-654: Is mentioned in John Brunner 's book Stand on Zanzibar , where Statistics of Deadly Quarrels is used as an argument that wars are inevitable. Richardson's work is also mentioned in Poul Anderson 's speculative fiction novelette Kings Who Die . Richardson is mentioned in Charlie Kaufman 's 2020 novel Antkind . Richardson's couplet "Big whirls have little whirls that feed on their velocity; little whirls have lesser whirls & so on to viscosity"

2150-543: Is mentioned in Lupe Fiasco 's song "Dots & Lines". In 1909, Richardson married Dorothy Garnett, daughter of the mathematician and physicist William Garnett . They were unable to have children due to an incompatibility of blood types, but adopted two sons and a daughter between 1920 and 1927. Richardson's nephew Ralph Richardson became a noted actor. His great-nephew (through his wife Dorothy's eldest brother, (James Clerk) Maxwell Garnett ), Julian Hunt , became

2236-428: Is more physically based, they form when the relative humidity reaches some prescribed value. Still, sub grid scale processes need to be taken into account. Rather than assuming that clouds form at 100% relative humidity, the cloud fraction can be related to a critical relative humidity of 70% for stratus-type clouds, and at or above 80% for cumuliform clouds, reflecting the sub grid scale variation that would occur in

2322-655: Is run out to 10 days into the future, while the Global Forecast System model run by the Environmental Modeling Center is run 16 days into the future. The equations used are nonlinear partial differential equations which are impossible to solve exactly through analytical methods, with the exception of a few idealized cases. Therefore, numerical methods obtain approximate solutions. Different models use different solution methods: some global models use spectral methods for

2408-663: Is sought in simple domains with smooth solutions. However, because of their global nature, the matrices associated with step computation are dense and computational efficiency will quickly suffer when there are many degrees of freedom (with some exceptions, for example if matrix applications can be written as Fourier transforms ). For larger problems and nonsmooth solutions, finite elements will generally work better due to sparse matrices and better modelling of discontinuities and sharp bends. Here we presume an understanding of basic multivariate calculus and Fourier series . If g ( x , y ) {\displaystyle g(x,y)}

2494-430: Is the highest frequency treated. Since we're only interested in a finite window of frequencies (of size n , say) this can be done using a fast Fourier transform algorithm. Therefore, globally the algorithm runs in time O ( n log n ). We wish to solve the forced, transient, nonlinear Burgers' equation using a spectral approach. Given u ( x , 0 ) {\displaystyle u(x,0)} on

2580-489: The Hadley Centre for Climate Prediction and Research 's HadCM3 model, are being used as inputs for climate change studies. Air pollution forecasts depend on atmospheric models to provide fluid flow information for tracking the movement of pollutants. In 1970, a private company in the U.S. developed the regional Urban Airshed Model (UAM), which was used to forecast the effects of air pollution and acid rain . In

2666-476: The National Weather Service for their suite of weather forecasting models. The United States Air Force developed its own set of MOS based upon their dynamical weather model by 1983. Model output statistics differ from the perfect prog technique, which assumes that the output of numerical weather prediction guidance is perfect. MOS can correct for local effects that cannot be resolved by

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2752-463: The curl of the Euler equations reduces into the barotropic vorticity equation . This latter equation can be solved over a single layer of the atmosphere. Since the atmosphere at a height of approximately 5.5 kilometres (3.4 mi) is mostly divergence-free, the barotropic model best approximates the state of the atmosphere at a geopotential height corresponding to that altitude, which corresponds to

2838-536: The orthogonality relation ⟨ e i l x , e i k x ⟩ = 2 π δ l k {\displaystyle \langle e^{ilx},e^{ikx}\rangle =2\pi \delta _{lk}} where δ l k {\displaystyle \delta _{lk}} is the Kronecker delta , we simplify the above three terms for each k {\displaystyle k} to see Assemble

2924-456: The 1970s and 1980s for individual forecast points (locations). Even with the increasing power of supercomputers, the forecast skill of numerical weather models only extends to about two weeks into the future, since the density and quality of observations—together with the chaotic nature of the partial differential equations used to calculate the forecast—introduce errors which double every five days. The use of model ensemble forecasts since

3010-440: The 1990s helps to define the forecast uncertainty and extend weather forecasting farther into the future than otherwise possible. The atmosphere is a fluid . As such, the idea of numerical weather prediction is to sample the state of the fluid at a given time and use the equations of fluid dynamics and thermodynamics to estimate the state of the fluid at some time in the future. The process of entering observation data into

3096-573: The Fourier– Galerkin method , choose both and where u ^ k ( t ) := 1 2 π ⟨ u ( x , t ) , e i k x ⟩ {\displaystyle {\hat {u}}_{k}(t):={\frac {1}{2\pi }}\langle u(x,t),e^{ikx}\rangle } . This reduces the problem to finding u ∈ U N {\displaystyle u\in {\mathcal {U}}^{N}} such that Using

3182-459: The Pacific. A model is a computer program that produces meteorological information for future times at given locations and altitudes. Within any model is a set of equations, known as the primitive equations , used to predict the future state of the atmosphere. These equations are initialized from the analysis data and rates of change are determined. These rates of change predict the state of

3268-525: The United Kingdom in 1972 and Australia in 1977. The development of global forecasting models led to the first climate models. The development of limited area (regional) models facilitated advances in forecasting the tracks of tropical cyclone as well as air quality in the 1970s and 1980s. Because the output of forecast models based on atmospheric dynamics requires corrections near ground level, model output statistics (MOS) were developed in

3354-469: The air in that vertical column mixed. More sophisticated schemes add enhancements, recognizing that only some portions of the box might convect and that entrainment and other processes occur. Weather models that have gridboxes with sides between 5 kilometres (3.1 mi) and 25 kilometres (16 mi) can explicitly represent convective clouds, although they still need to parameterize cloud microphysics . The formation of large-scale ( stratus -type) clouds

3440-400: The atmosphere a short time into the future, with each time increment known as a time step. The equations are then applied to this new atmospheric state to find new rates of change, and these new rates of change predict the atmosphere at a yet further time into the future. Time stepping is repeated until the solution reaches the desired forecast time. The length of the time step chosen within

3526-447: The atmosphere's 500 mb (15 inHg) pressure surface. Hydrostatic models filter out vertically moving acoustic waves from the vertical momentum equation, which significantly increases the time step used within the model's run. This is known as the hydrostatic approximation . Hydrostatic models use either pressure or sigma-pressure vertical coordinates. Pressure coordinates intersect topography while sigma coordinates follow

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3612-564: The better known regional numerical models are: Because forecast models based upon the equations for atmospheric dynamics do not perfectly determine weather conditions near the ground, statistical corrections were developed to attempt to resolve this problem. Statistical models were created based upon the three-dimensional fields produced by numerical weather models, surface observations, and the climatological conditions for specific locations. These statistical models are collectively referred to as model output statistics (MOS), and were developed by

3698-435: The complete continuity equation for air and is fully compressible. Nonhydrostatic models use altitude or sigma altitude for their vertical coordinates. Altitude coordinates can intersect land while sigma-altitude coordinates follow the contours of the land. The history of numerical weather prediction began in the 1920s through the efforts of Lewis Fry Richardson who utilized procedures developed by Vilhelm Bjerknes . It

3784-549: The constantly improving dynamical model guidance made possible by increasing computational power, it was not until the 1980s that numerical weather prediction (NWP) showed skill in forecasting the track of tropical cyclones. And it was not until the 1990s that NWP consistently outperformed statistical or simple dynamical models. Predicting the intensity of tropical cyclones using NWP has also been challenging. As of 2009, dynamical guidance remained less skillful than statistical methods. Spectral method Spectral methods are

3870-439: The contour of the land. Its hydrostatic assumption is reasonable as long as horizontal grid resolution is not small, which is a scale where the hydrostatic assumption fails. Models which use the entire vertical momentum equation are known as nonhydrostatic. A nonhydrostatic model can be solved anelastically, meaning it solves the complete continuity equation for air assuming it is incompressible, or elastically, meaning it solves

3956-507: The direction and speed of the geostrophic wind are independent of height. In other words, no vertical wind shear of the geostrophic wind. It also implies that thickness contours (a proxy for temperature) are parallel to upper level height contours. In this type of atmosphere, high and low pressure areas are centers of warm and cold temperature anomalies. Warm-core highs (such as the subtropical ridge and Bermuda-Azores high) and cold-core lows have strengthening winds with height, with

4042-476: The early 21st century, Richardson's work on conflict enjoyed a revival among conflict scholars, as his power-law distribution pattern was found in the statistics of several other kinds of conflict, including terrorism and violent mobs, and his work has informed the debate over the durability of the " Long Peace " since 1946. Modern statisticians have shown that while Richardson's analyses were not rigorous by modern standards, his statistical conclusions largely hold up:

4128-469: The grid size h. That is, for any n>0, there is a C n < ∞ {\displaystyle C_{n}<\infty } such that the error is less than C n h n {\displaystyle C_{n}h^{n}} for all sufficiently small values of h {\displaystyle h} . We say that the spectral method is of order n {\displaystyle n} , for every n>0. Because

4214-439: The horizontal dimensions and finite difference methods for the vertical dimension, while regional models and other global models usually use finite-difference methods in all three dimensions. The visual output produced by a model solution is known as a prognostic chart , or prog . Weather and climate model gridboxes have sides of between 5 kilometres (3.1 mi) and 300 kilometres (190 mi). A typical cumulus cloud has

4300-443: The left denotes the second partial derivatives of f in x and y , respectively. This is the Poisson equation , and can be physically interpreted as some sort of heat conduction problem, or a problem in potential theory, among other possibilities. If we write f and g in Fourier series: and substitute into the differential equation, we obtain this equation: We have exchanged partial differentiation with an infinite sum, which

4386-488: The loss of the ship Titanic , Richardson registered a patent for iceberg detection using acoustic echolocation in air. A month later he registered a similar patent for acoustic echolocation in water, anticipating the invention of sonar by Paul Langevin and Robert Boyle 6 years later. A fictional version of Richardson, named Wallace Ryman, plays a pivotal role in Giles Foden 's novel Turbulence . Richardson

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4472-407: The man in charge of the whole theatre; he is surrounded by several assistants and messengers. One of his duties is to maintain a uniform speed of progress in all parts of the globe. In this respect he is like the conductor of an orchestra in which the instruments are slide-rules and calculating machines. But instead of waving a baton he turns a beam of rosy light upon any region that is running ahead of

4558-686: The mid- to late-1970s, the United States Environmental Protection Agency took over the development of the UAM and then used the results from a regional air pollution study to improve it. Although the UAM was developed for California , it was during the 1980s used elsewhere in North America, Europe, and Asia. The Movable Fine-Mesh model, which began operating in 1978, was the first tropical cyclone forecast model to be based on atmospheric dynamics . Despite

4644-513: The model due to insufficient grid resolution, as well as model biases. Forecast parameters within MOS include maximum and minimum temperatures, percentage chance of rain within a several hour period, precipitation amount expected, chance that the precipitation will be frozen in nature, chance for thunderstorms, cloudiness, and surface winds. In 1956, Norman Phillips developed a mathematical model that realistically depicted monthly and seasonal patterns in

4730-519: The model is related to the distance between the points on the computational grid, and is chosen to maintain numerical stability . Time steps for global models are on the order of tens of minutes, while time steps for regional models are between one and four minutes. The global models are run at varying times into the future. The UKMET Unified model is run six days into the future, the European Centre for Medium-Range Weather Forecasts model

4816-578: The model to generate initial conditions is called initialization . On land, terrain maps available at resolutions down to 1 kilometer (0.6 mi) globally are used to help model atmospheric circulations within regions of rugged topography, in order to better depict features such as downslope winds, mountain waves and related cloudiness that affects incoming solar radiation. The main inputs from country-based weather services are observations from devices (called radiosondes ) in weather balloons that measure various atmospheric parameters and transmits them to

4902-681: The model's mathematical algorithms. The data are then used in the model as the starting point for a forecast. A variety of methods are used to gather observational data for use in numerical models. Sites launch radiosondes in weather balloons which rise through the troposphere and well into the stratosphere . Information from weather satellites is used where traditional data sources are not available. Commerce provides pilot reports along aircraft routes and ship reports along shipping routes. Research projects use reconnaissance aircraft to fly in and around weather systems of interest, such as tropical cyclones . Reconnaissance aircraft are also flown over

4988-424: The open oceans during the cold season into systems which cause significant uncertainty in forecast guidance, or are expected to be of high impact from three to seven days into the future over the downstream continent. Sea ice began to be initialized in forecast models in 1971. Efforts to involve sea surface temperature in model initialization began in 1972 due to its role in modulating weather in higher latitudes of

5074-447: The periodic domain x ∈ [ 0 , 2 π ) {\displaystyle x\in \left[0,2\pi \right)} , find u ∈ U {\displaystyle u\in {\mathcal {U}}} such that where ρ is the viscosity coefficient. In weak conservative form this becomes where following inner product notation. Integrating by parts and using periodicity grants To apply

5160-433: The rate of a nation's armament buildup is directly proportional to the amount of arms its rival has and also to the grievances felt toward the rival, and inversely proportional to the amount of arms it already has. Solutions of this system of equations yield insightful conclusions about the nature, and the stability or instability, of various hypothetical conditions that might obtain between nations. Richardson also originated

5246-501: The real world. The amount of solar radiation reaching ground level in rugged terrain, or due to variable cloudiness, is parameterized as this process occurs on the molecular scale. Also, the grid size of the models is large when compared to the actual size and roughness of clouds and topography. Sun angle as well as the impact of multiple cloud layers is taken into account. Soil type, vegetation type, and soil moisture all determine how much radiation goes into warming and how much moisture

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5332-424: The rest, and a beam of blue light upon those who are behindhand. Four senior clerks in the central pulpit are collecting the future weather as fast as it is being computed, and despatching it by pneumatic carrier to a quiet room. There it will be coded and telephoned to the radio transmitting station. Messengers carry piles of used computing forms down to a storehouse in the cellar. In a neighbouring building there

5418-453: The reverse true for cold-core highs (shallow arctic highs) and warm-core lows (such as tropical cyclones ). A barotropic model tries to solve a simplified form of atmospheric dynamics based on the assumption that the atmosphere is in geostrophic balance ; that is, that the Rossby number of the air in the atmosphere is small. If the assumption is made that the atmosphere is divergence-free ,

5504-516: The ruler, the longer the resulting coastline. It might be supposed that these values would converge to a number representing the coastline's true length, but Richardson demonstrated that this is not so: the measured length of coastlines, and other natural features, increases without limit as the unit of measurement is made smaller. This is known as the Richardson effect . At the time, the scientific community ignored Richardson's research. Today it

5590-436: The same ideas; the main difference between them is that spectral methods use basis functions that are generally nonzero over the whole domain, while finite element methods use basis functions that are nonzero only on small subdomains ( compact support ). Consequently, spectral methods connect variables globally while finite elements do so locally . Partially for this reason, spectral methods have excellent error properties, with

5676-400: The scientific analysis of conflict—an interdisciplinary topic of quantitative and mathematical social science dedicated to systematic investigation of the causes of war and conditions of peace. As he had done with weather, he analysed war using mainly differential equations and probability theory. Considering the armament of two nations, Richardson posited an idealised system of equations whereby

5762-473: The single-layer barotropic model, used a single pressure coordinate at the 500-millibar (15 inHg) level, and thus was essentially two-dimensional. High-resolution models—also called mesoscale models —such as the Weather Research and Forecasting model tend to use normalized pressure coordinates referred to as sigma coordinates . Some of the better known global numerical models are: Some of

5848-455: The sizes and frequencies of armed conflicts plausibly follow a power-law pattern, and the rate of new wars is well-modeled by a Poisson distribution. Richardson searched for a relation between the probability of two countries going to war and the length of their common border. While collecting data, he found that there was considerable variation in the various published lengths of international borders. For example, that between Spain and Portugal

5934-589: The so-called "exponential convergence" being the fastest possible, when the solution is smooth . However, there are no known three-dimensional single-domain spectral shock capturing results (shock waves are not smooth). In the finite-element community, a method where the degree of the elements is very high or increases as the grid parameter h increases is sometimes called a spectral-element method . Spectral methods can be used to solve differential equations (PDEs, ODEs, eigenvalue, etc) and optimization problems . When applying spectral methods to time-dependent PDEs,

6020-768: The solution is typically written as a sum of basis functions with time-dependent coefficients; substituting this in the PDE yields a system of ODEs in the coefficients which can be solved using any numerical method for ODEs . Eigenvalue problems for ODEs are similarly converted to matrix eigenvalue problems . Spectral methods were developed in a long series of papers by Steven Orszag starting in 1969 including, but not limited to, Fourier series methods for periodic geometry problems, polynomial spectral methods for finite and unbounded geometry problems, pseudospectral methods for highly nonlinear problems, and spectral iteration methods for fast solution of steady-state problems. The implementation of

6106-539: The space usually occupied by the stage. The walls of this chamber are painted to form a map of the globe. The ceiling represents the north polar regions, England is in the gallery, the tropics in the upper circle, Australia on the dress circle and the Antarctic in the pit. A myriad computers [people who compute] are at work upon the weather of the part of the map where each sits, but each computer attends only to one equation or part of an equation. The work of each region

6192-440: The spectral method is normally accomplished either with collocation or a Galerkin or a Tau approach . For very small problems, the spectral method is unique in that solutions may be written out symbolically, yielding a practical alternative to series solutions for differential equations. Spectral methods can be computationally less expensive and easier to implement than finite element methods; they shine best when high accuracy

6278-563: The theory that the propensity for war between two nations is a function of the length of their common border. And in Arms and Insecurity (1949), and Statistics of Deadly Quarrels (1960), he sought to analyse the causes of war statistically. Factors he assessed included economics, language, and religion. In the preface of the latter, he wrote: "There is in the world a great deal of brilliant, witty political discussion which leads to no settled convictions. My aim has been different: namely to examine

6364-433: The thermotropic model is that while the magnitude of the thermal wind may change, its direction does not change with respect to height, and thus the baroclinicity in the atmosphere can be simulated using the 500  mb (15  inHg ) and 1,000 mb (30 inHg) geopotential height surfaces and the average thermal wind between them. Barotropic models assume the atmosphere is nearly barotropic , which means that

6450-625: The three terms for each k {\displaystyle k} to obtain Dividing through by 2 π {\displaystyle 2\pi } , we finally arrive at With Fourier transformed initial conditions u ^ k ( 0 ) {\displaystyle {\hat {u}}_{k}(0)} and forcing f ^ k ( t ) {\displaystyle {\hat {f}}_{k}(t)} , this coupled system of ordinary differential equations may be integrated in time (using, e.g.,

6536-524: The troposphere. This was the first successful climate model . Several groups then began working to create general circulation models . The first general circulation climate model combined oceanic and atmospheric processes and was developed in the late 1960s at the Geophysical Fluid Dynamics Laboratory , a component of the U.S. National Oceanic and Atmospheric Administration . By 1975, Manabe and Wetherald had developed

6622-480: The vertical coordinate. Later models substituted the geometric z {\displaystyle z} coordinate with a pressure coordinate system, in which the geopotential heights of constant-pressure surfaces become dependent variables , greatly simplifying the primitive equations. This follows since pressure decreases with height through the Earth's atmosphere . The first model used for operational forecasts,

6708-492: The very largest conflicts are orders of magnitude larger than the "typical" conflict. While conflicts' sizes can be predicted ahead of time, Richardson showed that the number of international wars per year follows a Poisson distribution . On a smaller scale he showed a similar pattern for gang murders in Chicago and Shanghai, and hypothesized that a universal rule connected the frequency and the size of all "deadly quarrels". In

6794-436: Was an English mathematician , physicist , meteorologist , psychologist , and pacifist who pioneered modern mathematical techniques of weather forecasting, and the application of similar techniques to studying the causes of wars and how to prevent them. He is also noted for his pioneering work on fractals and a method for solving a system of linear equations known as modified Richardson iteration . Lewis Fry Richardson

6880-651: Was increased. By the early 1980s, the U.S. National Center for Atmospheric Research had developed the Community Atmosphere Model (CAM), which can be run by itself or as the atmospheric component of the Community Climate System Model . The latest update (version 3.1) of the standalone CAM was issued on 1 February 2006. In 1986, efforts began to initialize and model soil and vegetation types, resulting in more realistic forecasts. Coupled ocean-atmosphere climate models, such as

6966-534: Was not until the advent of the computer and computer simulation that computation time was reduced to less than the forecast period itself. ENIAC created the first computer forecasts in 1950, and more powerful computers later increased the size of initial datasets and included more complicated versions of the equations of motion. In 1966, West Germany and the United States began producing operational forecasts based on primitive-equation models, followed by

7052-515: Was serving with the Quaker ambulance unit in France. Richardson also applied his mathematical skills in service of his pacifist principles, in particular in understanding the basis of international conflict. For this reason, he is now considered the initiator, or co-initiator (with Quincy Wright and Pitirim Sorokin as well as others such as Kenneth Boulding , Anatol Rapaport and Adam Curle ), of

7138-668: Was taught physics in the natural sciences tripos by (among others) J. J. Thomson and graduated with a first-class degree in 1903. At age 47 he received a doctorate in mathematical psychology from the University of London . Richardson's working life represented his eclectic interests: In 1926, he was elected to the Fellowship of the Royal Society . Richardson's Quaker beliefs entailed an ardent pacifism that exempted him from military service during World War I as

7224-699: Was the youngest of seven children born to Catherine Fry (1838–1919) and David Richardson (1835–1913). They were a prosperous Quaker family, David Richardson operating a successful tanning and leather-manufacturing business. At age 12 he was sent to a Quaker boarding school , Bootham School in York , where he received an education in science, which stimulated an active interest in natural history . In 1898 he went on to Durham College of Science (a college of Durham University ) where he took courses in mathematical physics , chemistry , botany , and zoology . He proceeded in 1900 to King's College, Cambridge , where he

7310-467: Was thought that those who compute the weather should breathe of it freely. (Richardson 1922) In 1950, when Richardson received news of the first weather forecast by the first modern computer, ENIAC , he called it an "enormous scientific advance". The first calculations for a 24-hour forecast took ENIAC nearly 24 hours to produce. Richardson was also interested in atmospheric turbulence and performed many terrestrial experiments. The Richardson number ,

7396-489: Was variously quoted as 987 or 1,214 km, and that between the Netherlands and Belgium as 380 or 449 km. The reason for these inconsistencies is the " coastline paradox ". Suppose the coast of Britain is measured using a 200–km ruler, specifying that both ends of the ruler must touch the coast. Now cut the ruler in half and repeat the measurement, then repeat: [REDACTED] [REDACTED] [REDACTED] The smaller

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