http://en.wikipedia.

org/wiki/Numerical_weather_prediction Numerical weather prediction uses mathematical models of the atmosphere and oceans to predict the weather based on current weather conditions. Though first attempted in the 1920s, it was not until the advent of computer simulation in the 1950s that numerical weather predictions produced realistic results. A number of global and regional forecast models are run in different countries worldwide, using current weather observations relayed from radiosondes or weather satellites as inputs to the models. Mathematical models based on the same physical principles can be used to generate either shortterm weather forecasts or longer-term climate predictions; the latter are widely applied for understanding and projecting climate change. The improvements made to regional models have allowed for significant improvements in tropical cyclone track and air quality forecasts; however, atmospheric models perform poorly at handling processes that occur in a relatively constricted area, such as wildfires. Manipulating the vast datasets and performing the complex calculations necessary to modern numerical weather prediction requires some of the most powerful supercomputers in the world. Even with the increasing power of supercomputers, the forecast skill of numerical weather models extends to about only six days. Factors affecting the accuracy of numerical predictions include the density and quality of observations used as input to the forecasts, along with deficiencies in the numerical models themselves. Although post-processing techniques such as model output statistics (MOS) have been developed to improve the handling of errors in numerical predictions, a more fundamental problem lies in the chaotic nature of the partial differential equations used to simulate the atmosphere. It is impossible to solve these equations exactly, and small errors grow with time (doubling about every five days). In addition, the partial differential equations used in the model need to be supplemented with parameterizations for solar radiation, moist processes (clouds and precipitation), heat exchange, soil, vegetation, surface water, and the effects of terrain. In an effort to quantify the large amount of inherent uncertainty remaining in numerical predictions, ensemble forecasts have been used since the 1990s to help gauge the confidence in the forecast, and to obtain useful results farther into the future than otherwise possible. This approach analyzes multiple forecasts created with an individual forecast model or multiple models. 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 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 [20] 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 a fixed receiver, as well as from weather satellites. The World Meteorological Organization acts to standardize the instrumentation, observing practices [21] and timing of these observations worldwide. Stations either report hourly in METARreports, or [22] every six hours in SYNOP reports. These observations are irregularly spaced, so they are processed by data assimilationand objective analysis methods, which perform quality control and [23] obtain values at locations usable by the model's mathematical algorithms. Some global models use finite differences, in which the world is represented as discrete points on a regularly spaced grid [24] of latitude and longitude; other models use spectral methods that solve for a range of wavelengths. [25] 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 [26] the stratosphere. Information from weather satellites is used where traditional data sources are not [27] available. Commerce provides pilot reports along aircraft routes and ship reports along shipping [28] routes. Research projects use reconnaissance aircraft to fly in and around weather systems of [29][30] interest, such as tropical cyclones. Reconnaissance aircraft are also flown over 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 [31] [32] 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 [33] higher latitudes of the Pacific. [edit]Computation Main article: Atmospheric model

A prognostic chart of the 96-hour forecast of 850mbar geopotential height and temperature from the Global Forecast System

An atmospheric model is a computer program that produces meteorological information for future times at given locations and altitudes. Within any modern model is a set of equations, known as [34] the primitive equations, used to predict the future state of the atmosphere. These equationsalong with the ideal gas laware used to evolve the density, pressure, and potential temperature scalar fields and the airvelocity (wind) vector field of the atmosphere through time. Additional transport equations for pollutants and other aerosols are included in some primitive-equation high-resolution [35] models as well. The equations used are nonlinear partial differential equations which are [36] impossible to solve exactly through analytical methods, with the exception of a few idealized [37] cases. Therefore, numerical methods obtain approximate solutions. Different models use different solution methods: some global models and almost all regional models use finite difference methods for all three spatial dimensions, while other global models and a few regional models [36] use spectral methods for the horizontal dimensions and finite-difference methods in the vertical. These equations are initialized from the analysis data and rates of change are determined. These rates of change predict the state of the atmosphere a short time into the future; the time increment for this prediction is called 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 step into the future. This time stepping is repeated until the solution reaches the desired forecast time.

The length of the time step chosen within the model is related to the distance between the points on [38] the computational grid, and is chosen to maintain numerical stability. Time steps for global models [39] are on the order of tens of minutes, while time steps for regional models are between one and four [40] minutes. The global models are run at varying times into the future. The UKMET Unified Model is [41] run six days into the future, while the European Centre for Medium-Range Weather Forecasts' Integrated Forecast System and Environment Canada's Global Environmental Multiscale [42] Model both run out to ten days into the future, and the Global Forecast System model run by [43] the Environmental Modeling Center is run sixteen days into the future. The visual output produced [44] by a model solution is known as a prognostic chart, orprog.

http://climateaudit.org/2005/12/22/gcms-and-the-navier-stokes-equations/
Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations. Fluids are important and hard to understand. There are many fascinating problems and conjectures about the behaviour of solutions of the Navier-Stokes equations. Since we dont even know whether these solutions exist, our understanding is at a very primitive level. Standard methods from PDE appear inadequate to settle the problem. Instead, we probably need some deep, new ideas. In May 2002, the Clay Mathematics Institute (CMI) of Cambridge, Massachusetts, in an initiative to further the study of mathematics, allocated a $7m prize fund for the solution of seven Millennium Problems, focusing on important classic questions that have resisted solution over the years. One of the $1m problems stands out for its massive practical importance: the solution of the Navier-Stokes equations (NSEs) for fluid flow. Although there are many named variants and special cases, the fundamental equations are the incompressible Navier-Stokes for Newtonian fluids. In their most compact form, they comprise a pair of vector partial differential equations (PDEs): one expresses the forces acting (pressure, viscosity and body forces); the other is the continuity equation, which says that divergence of the velocity field is zero for an incompressible fluid (that is, what comes in, goes out). The NSEs are among the most-studied partial differential systems, the subject of around 15-20 published papers a week. Nevertheless, theyre among the least understood at a theoretical level. In short, A GCM "control run", is essentially one numerical run from a hugely complicated Navier-Stokes equation, the deep mathematical properties of which mathematicians say they know very little. Climatologists on the other hand appear to know the results to high degres of certainty remarkable.

http://plus.maths.org/content/os/issue25/features/budd/index You may not have heard of the Navier-Stokes equations, but you encounter them every day. These are the equations that describe the weather! We all know that the weather is important to our lives, that it can sometimes be predictable and other times very unpredictable. It seems strange that all of the different types of behaviour we associate with the weather can be described by a single set of equations, but we believe that this is the case. The subject of the sixth problem is a first, and vital, link in the chain of reasoning that we hope will establish exactly this fact. We start by thinking about exactly what we mean by weather. Weather is the combination of the motion of the atmosphere, coupled to the motion of the oceans, the transport of moisture within the atmosphere, all coupled to changes in the pressure and temperature of the air. It is possible to write down (partial differential) equations which describe all of this mathematically. In their totality these equations are rather complicated, however at the guts of them are the equations that describe the underlying motion of the air on its own. These equations are the same for air as they are for water or any other fluid and they were derived in the 19th century by the two mathematicians Navier and Stokes, hence their name the Navier-Stokes equations. Imagine that you are looking at a point in the atmosphere. At this point the air will have a velocity and apressure . These are all related together by the Navier-Stokes equations which describe how changes to the velocity intime are related to changes in the velocity and the pressure in space. Brace yourselves...here they come!

Unfortunately there is some bad news to come. The first piece of bad news is that the Navier-Stokes equations are very, very hard to solve. We only know of a few

exact solutions (that is, solutions which we can write down using a formula), usually for problems which are of little or no physical interest. A lot of work has been done on finding approximate solutions which work for certain important physical situations, such as the flow of water in a pipe. The procedures for finding such solutions dominate the subject calledfluid mechanics, which you may meet in a university course in applied mathematics, physics or engineering (especially aeronautical engineering). Fortunately it is possible to write computer programmes which can find numerical solutions to these equations. Indeed there is a huge industry called computational fluid dynamics devoted to this task. It is computer programs of this sort which are used by the meteorological office to help predict the weather. They are also used in the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of the effects of pollution, the study of the insides of a star, calculations of climate change and, in a notable success story, in the design of Thrust 2, the first supersonic racing car. (This calculation was done by the computational fluid dynamics team at the University of Swansea). The next piece of bad news is that even in the best of circumstances these programs can take even the fastest computer an enormous time to run, and these computers can often only solve relatively simple problems. They are also quite unable to deal with the phenomenon called turbulence. Turbulence is the complex behaviour that fluids show on small length scales. (An understanding of turbulence is one of the great problems in physics for the new millennium.) You can see or feel turbulence every time that you look at a cloud, examine the motion of the water in a waterfall or stick your head out of a car window. No computer programme on earth can simulate this behaviour exactly, and at the present all that we have are rough approximations. It is a sobering thought that these approximations have to be used every time a simulation is made of a safety-critical situation (such as the effects of a fire or of a coolant leak in a nuclear power station). The approximations are not bad (giving errors of around 20%), but this situation is hardly satisfactory. All of the above issues are very important to the way that we use the Navier-Stokes equations to help us to understand the physical world around us, but they pale into insignificance when compared to the subject of the sixth Millennium Prize Problem. This is not whether we can solve the Navier-Stokes equations (either exactly or using a computer), but whether they have any solutions at all. You may feel that this is an unimportant question - after all it is obvious - isn't it? that the equations must have a solution. However there are plenty of examples in mathematics of equations which don't have solutions. For example, before the invention of negative numbers, the equation x+1=0 had no solution. The Greeks thought that all numbers could be expressed in terms of fractions (rational numbers) and had a very deep shock when they discovered that the equation x2=2 did not have a solution which could be expressed as a fraction. Similarly, if you only knew about real numbers then it would not be possible to solve the equation x2=-1. It is quite possible that a situation could occur in which a possible solution of the NavierStokes equations starts by being completely physical, but quickly becomes infinite and fails to represent anything corresponding to the physical situation that the equations are trying to describe.

The present situation is that noone has managed to show that the solutions of the Navier-Stokes equations correspond to real physical solutions for all time. Conversely, noone has found a "solution" of the Navier-Stokes equations that becomes infinite and loses its physical meaning. If such a solution were to be found, would it really be nonsense or might it give us some insight into the problem of turbulence (the latter being the view of the author)? We don't know! What we do know is that for the cases that we can compute, the Navier-Stokes equations do seem to give a very accurate description of the motion of fluids and that they also seem to be uniquely hard. Make a small change to the equations and we can answer all of the questions, but return to the physically motivated equations and nothing is certain. Mathematicians seem evenly divided as to whether solutions exist or not and the question of the existence of solutions seems likely to stay unresolved for a long time.