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MODELS

Specialty Definition: MODELS

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Specialty Definition: Climate model

(From Wikipedia, the free Encyclopedia)

Climate models are quantitative methods of representing the interactions of the atmosphere, oceans, land surface, and ice. Models can range from relatively simple to quite comprehensive.

Climate models can be ordered into a rough hierarchy of complexity:

• Simple back-of-the-envelope calculations of the radiative temperature treat the earth as a single point
• this can be expanded vertically (radiative-convective models), or horizontally (energy balance models)
• finally, (coupled) atmosphere-ocean-seaice global climate models discretize and solve the full equations for fluid motion.

This is not a full list; fox example "box models" can be written to treat flows across and within ocean basins.

Zero-dimensional models

It is possible to obtain a very simple model of the radiative equilibrium of the Earth by writing

(1-a)Sπr² = 4πr²sT⁴

where

• The left hand side represents the incoming energy from the Sun
• The right hand side represents the outgoing energy from the Earth, calculated from Stefan-Boltzmann law assuming a constant radiative temperature, T, that is to be found,

and

• S is the Solar Constant - the incoming solar radiation per unit area - about 1367 Wm^-2
• a is the Earth's average albedo, approximately 0.37 to 0.39
• r is Earth's radius - approximately 6.371×10⁶m
• &pi is well known, approximately 3.14159
• s is the Stefan-Boltzmann constant - approximately 5.67×10^-8 JK^-4m^-2s^-1

Note that the factor of πr² can be factored out, giving

(1-a)S = 4sT⁴

which gives a value of 246 to 248 kelvin - about -27 to -25 °C - as the Earth's average temperature T. This is approximately 35 degrees colder than the average surface temperature of 282 K. This is because the above equation attempts to represent the radiative temperature of the earth, and the average radiative level is well above the surface. The difference between the radiative and surface temperatures is the natural greenhouse effect.

This very simple model is quite instructive, and the only model that could fit on a page. But it produces a result we are not really interested in - the radiative temperature - rather than the more useful surface temperature. It also contains the albedo as a specified constant, with no way to "predict" it from within the model.

Radiative-Convective Models

The zero-dimensional model above predicts the temperature of an imaginary layer where long wave radiation is emitted to space. This can be extended in the vertical to a one dimensional radiative-convective model, which simplifies the atmosphere to consider only two processes of energy transport:

• upwelling and downwelling radiative transfer through atmospheric layers
• upwards transport of heat by convection (especially important in the lower troposphere).

The radiative-convective models have advantages over the simple model: they can tell you the surface termperature, and the effects of varying greenhouse gas concentrations on the surface temperature. But they need added parameters, and still represent by one point the horizontal surface of the earth.

Links:

• http://www.giss.nasa.gov/gpol/abstracts/1980/WangStone.html
• http://members.fortunecity.com/yhu/paper/scat/node4.html
• http://www.grida.no/climate/ipcc_tar/wg1/258.htm

Energy Balance Models

Alternatively, the zero-dimensional model may be expanded horizontally to consider the energy transported - ahem - horizontally in the atmosphere. This kind of model may well be zonally meaned. This model has the advantage of allowing a plausible dependence of albedo on temperature - the poles can be allowed to be icy and the equator warm - but the lack of true dynamics means that horizontal transports have to be specified.

• http://www.shodor.org/master/environmental/general/energy/application.html

GCM's (Global Climate Models or General Circulation Models)

Three (or more properly, four) dimensional GCM's discretise the equations for fluid motion and integrate these forward in time. They also contain parametrisations for processes - such as convection - that occur on scales too small to be resolved directly. More sophisticated models may include representations of the carbon and other cycles.

Atmospheric GCMs (AGCMs) model the atmosphere (and typically contain a land-surface model as well) and impose sea surface temperatures. A large amount of information including model documentation is available from AMIP [1]. They may include atmospheric chemistry. AGCMs consist of a dynamical core, which integrates the equations of fluid motion for, typically:

• surface pressure
• horizontal components of velocity in layers
• temperature and moisture in layers

and parametrisations which handle other processes: these include

• radiation (solar/short wave and terrestrial/infra-red/long wave)
• convection
• land surface processes and hydrology

The method by which AGCMs discretise the fluid equations may be the familiar finite difference method or the somewhat harder to understand spectral method. Typical AGCM resolution is between 1 and 5 degrees in latitude or longitude: the Hadley Centre model HadAM3, for example, uses 2.5 degrees in latitude and 3.75 in longitude, giving a grid of 73 by 96 points; and has 19 levels in the vertical.

Oceanic GCMs (OGCMs) model the ocean (with fluxes from the atmosphere imposed) and may or may not contain a sea ice model.

Coupled atmosphere-ocean GCMs (AOGCMs) combine the two models. They thus have the advantage of removing the need to specify fluxes across the interface of the ocean surface. These models are the basis for sophisticated model predictions of future climate, such as are discussed by the IPCC.

AOGCMs represent the pinnacle of complexity in climate models and internalise as many processes as possible. They are the only tools that could provide detailed regional predictions of future climate change. However, they are still under development. The simpler models are generally susceptible to simple analysis and their results are generally easy to understand. AOGCMs, by contrast, are often as hard to analyse as the real climate system.

The most modern AOGCMs simulate the observed warming over the past 150 years, when forced by observed changes in "Greenhouse" gases and aerosols [1] [1].

Note that global climate models, whilst very similar in structure to (and often sharing computer code with) numerical weather prediction models are nonetheless logically distinct: see weather vs climate for details.

See also

climate change, global warming

References

• A global distributed GCM model, to run on your desktop computer.
• (IPCC 2001 section 8.3) - on model hierarchy
• (IPCC 2001 section 8) - much information on coupled GCM's
• - Hadley Centre general info on their models
• Coupled Model Intercomparison Project

  Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Climate model."

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Model

(From Wikipedia, the free Encyclopedia)

This article is about models in science and technology, for models in art, fashion and cosmetics, see model (person) or supermodel

The word model is used in various contexts meaning something (abstract or physical) that represents 'the real thing'. That entity may be anything from a single item or object (for example, a bolt) to a complete system of any size (for example, the Solar System). In general, a model is an object which we study, not for its intrinsic interest, but because it is a formalized or simplified representation of a class of phenomena which can be studied easily.

Modeling: the process of generating a model

Modeling refers to the process of generating a model as an abstract representation of some real world entity. Typically a model will contain only the significant features or aspects of the item in question, and two models of the same item may differ quite significantly. This may be due to differing requirements of the model's end user (one user may be interested in aspects of the item which are quite separate from those of another user) or, perhaps more simply, this may be due to the difference in perception of that item by the modeller and decisions made during the modelling process. This is why it is critically important for any end user to understand the original purpose or application for the model.

Models in science

Models have applications throughout science with variations according to the subject matter under discussion. Abstract models such as statistical models and mathematical models are used throughout the natural sciences including physics, chemistry, biology and economics. (See also: model theory, applied mathematics.) A model is often a simplified version (cognitive model), but it can also mean an especially good or useful example of a process, or for studying a process such as a model organism in developmental biology.

Physical models

A physical model of something large is usually smaller, and of something very small is larger. A physical model of something that can move, like a vehicle or machine, may be completely static, or have parts that can be moved manually, or be powered. A physical model may show inner parts that are normally not visible. The purpose of a physical model on a smaller scale may be to have a better overview, for testing purposes, as hobby or toy. The purpose of a physical model on a larger scale may be to see the structure of things that are normally too small to see properly or to see at all, for example a model of an insect or of a molecule.

A physical model of an animal shows how it is built without it walking or flying away, and without danger, and if the real animal is not available. A soft model of an animal is popular among children and some adults as cuddly toy. A model of a human may be a doll or a statue.

See also

• Model organism
• Model nation
• Herbert Simon

Modeling languages

• UML for software systems
• Role Activity Diagram and IDEF for processes
• VRML for 3-D models designed particularly with the World Wide Web in mind.

Physical models

• Model airplane
• Model car
• Model radio-controlled car
• Model railway
• Model rocket
• Model ship

  Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Model."

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Model (person)

(From Wikipedia, the free Encyclopedia)

A model is somebody who acts as a human prop for purposes of art, photography, pornography, fashion, advertising etc. Modelling is distinguished from other types of public performance such as an acting, dancing or mime artistry, although the boundary is probably not well defined. Appearing in a movie or a play is generally not considered to be modelling, irrespective of the nature of the role, so many models can also describe themselves as actors. Some models have acquired the status of sex symbol, and a highly paid model is sometimes known as a "supermodel". Supermodels are celebrities who may appear in commercials endorsing products, and often parlay their fame into acting careers.

"Catwalk modelling" (also known as "runway modelling") is displaying fashion, "glamour modelling" usually includes elements of nudity or eroticism, while "nude modelling" describes any kind of modelling that is performed without clothes. Art school modelling involves posing for students of art.

For notable models, see supermodel and glamour photography.

Some who became better known for other things also worked as professional models:

• Quentin Crisp - "gay icon"
• Emmanuelle Seigner - probably best known as an actress
• Koo Stark - dated a member of the British royal family

See also: Cover girl

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Model (person)."

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Model theory

(From Wikipedia, the free Encyclopedia)

In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. It assumes that there are some pre-existing mathematical objects out there, and asks questions regarding how or what can be proven given the objects, some operations or relations amongst the objects, and a set of axioms.

The independence of the axiom of choice and the continuum hypothesis from the other axioms of set theory (proven by Paul Cohen and Kurt Godel) are the two most famous results arising from model theory. It was proven that both the axiom of choice and its negation are consistent with the Zermelo-Fraenkel axioms of set theory; the same result holds for the continuum hypothesis. These results are a part of Axiomatic Set Theory, a particular application of Model Theory.

In the case of the real numbers, one would start with a set of individuals, where each individual is a real number, and a set of relations and/or functions, such as {×,+,-,.,0,1}. If we ask a question such as "∃ x (x × x = 1 + 1)" in this language, then it's clear that the sentence is true for the reals - there is such a real number x; for the rationals, however, the sentence is false. Conversely, "∃ x (x × x = 0 - 1 - 1)" is false in the reals - to make it true we can add a constant symbol i and a new axiom "i × i = 0 - 1", which gives us the complex numbers.

Model theory is then concerned with what is provable within given mathematical systems, and how these systems relate to each other. It is particularly concerned with what happens when we try to extend some system by the addition of new axioms or new language constructs.

A model is formally defined in context of some language L, following Tarski's concept of truth. The model consists of two things:

1. A universe set U which contains all the objects of interest (the "domain of discourse"), and
2. a mapping from L to U (called the evaluation mapping or interpretation function) which has as its domain all constant, predicate and function symbols in the language.

A theory is defined as a set of sentences which is consistent; often it is also defined to be closed under logical consequence. Under this definition a theory is thus a maximally consistent set of sentences. For example, the set of all sentences true in some particular model (e.g. the reals) or class of models is a theory.

Completeness in model theory is defined as the property that every statement in a language or its opposite is provable from some theory. Complete theories are desirable since they describe fully some model.

The compactness theorem states that a set of sentences S is satisfiable, i.e., has a model, if every finite subset of S is satisfiable. In the context of proof theory the analogous statement is trivial, since every proof can have only a finite number of antecedents used in the proof; in the context of model theory however, this proof is somewhat more difficult. There are two well known proofs, one by Gödel (which goes via proofs) and Malcev (which is more direct).

Model theory is concerned with first order logic, and to first order logic all cardinals look the same. This is expressed in the Lowenheim-Skolem theorems - which state that any theory with an infinite model A has models of all infinite cardinalities (greater than that of the language) which agree with A on all sentences - they are "elementarily equivalent".

So in particular, set theory (whose language is countable) has a countable model - this is known as Skolem's Paradox, even though it's true! To see why it was thought paradoxical, consider that there are sentences in set theory which postulate the existence of uncountable sets - and these sentences are true in our countable model.

TODO - Vaught's test. Extensions, Embeddings and Diagrams. To give a flavor, mentioning the hyperreals would be good. (All of these need substantial filling out)

Note: The term 'mathematical model' is also used informally in other parts of mathematics and science.

See also:

• Proof theory
• Hyperreals
• Compactness theorem
• Descriptive complexity

  Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Model theory."

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Abbreviations & Acronyms: MODELS

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