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Semi-Lagrangian Scheme
The Semi-Lagrangian scheme (SLS) is a numerical method that is widely used in numerical weather prediction models for the integration of the equations governing atmospheric motion. A Lagrangian description of a system (such as the atmosphere) focuses on following individual air parcels along their trajectories as opposed to the Eulerian description, which considers the rate of change of system variables fixed at a particular point in space. A semi-Lagrangian scheme uses Eulerian framework but the discrete equations come from the Lagrangian perspective. Some background The Lagrangian rate of change of a quantity F is given by \frac = \frac + (\mathbf\cdot\vec\nabla)F, where F can be a scalar or vector field and \mathbf is the velocity field. The first term on the right-hand side of the above equation is the ''local'' or ''Eulerian'' rate of change of F and the second term is often called the ''advection term''. Note that the Lagrangian rate of change is also known as the material ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Method
In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm. Mathematical definition Let F(x,y)=0 be a well-posed problem, i.e. F:X \times Y \rightarrow \mathbb is a real or complex functional relationship, defined on the Cartesian product of an input data set X and an output data set Y, such that exists a locally lipschitz function g:X \rightarrow Y called resolvent, which has the property that for every root (x,y) of F, y=g(x). We define numerical method for the approximation of F(x,y)=0, the sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ... of problems : \left \_ = \left \_, with F_n:X_n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Weather Prediction
Numerical weather prediction (NWP) uses mathematical models of the atmosphere and oceans to weather forecasting, 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, weather satellites and other observing systems as inputs. Mathematical models based on the same physical principles can be used to generate either short-term 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 significant improvements in Tropical cyclone track forecasting, tropical cyclone track and air quality forecasts; however, atmospheric models perform poorly at han ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrangian And Eulerian Specification Of The Flow Field
Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with an easier problem having an enlarged feasible set ** Lagrangian dual problem, the problem of maximizing the value of the Lagrangian function, in terms of the Lagrange-multiplier variable; See Dual problem * Lagrangian, a functional whose extrema are to be determined in the calculus of variations * Lagrangian submanifold, a class of submanifolds in symplectic geometry * Lagrangian system, a pair consisting of a smooth fiber bundle and a Lagrangian density Physics * Lagrangian mechanics, a formulation of classical mechanics * Lagrangian (field theory), a formalism in classical field theory * Lagrangian point, a position in an orbital configuration of two large bodies * Lagrangian coordinates, a way of describing the motions of par ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atmosphere
An atmosphere () is a layer of gases that envelop an astronomical object, held in place by the gravity of the object. A planet retains an atmosphere when the gravity is great and the temperature of the atmosphere is low. A stellar atmosphere is the outer region of a star, which includes the layers above the opaque photosphere; stars of low temperature might have outer atmospheres containing compound molecules. The atmosphere of Earth is composed of nitrogen (78%), oxygen (21%), argon (0.9%), carbon dioxide (0.04%) and trace gases. Most organisms use oxygen for respiration; lightning and bacteria perform nitrogen fixation which produces ammonia that is used to make nucleotides and amino acids; plants, algae, and cyanobacteria use carbon dioxide for photosynthesis. The layered composition of the atmosphere minimises the harmful effects of sunlight, ultraviolet radiation, solar wind, and cosmic rays and thus protects the organisms from genetic damage. The curr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Material Derivative
In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material derivative can serve as a link between Eulerian and Lagrangian descriptions of continuum deformation. For example, in fluid dynamics, the velocity field is the flow velocity, and the quantity of interest might be the temperature of the fluid. In this case, the material derivative then describes the temperature change of a certain fluid parcel with time, as it flows along its pathline (trajectory). Other names There are many other names for the material derivative, including: *advective derivative *convective derivative *derivative following the motion *hydrodynamic derivative *Lagrangian derivative *particle derivative *substantial derivative *substantive derivative *Stokes derivative *total derivative, although the material derivati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smoothed Particle Hydrodynamics
Smoothed-particle hydrodynamics (SPH) is a computational method used for simulating the mechanics of continuum media, such as solid mechanics and fluid flows. It was developed by Gingold and Monaghan and Lucy in 1977, initially for astrophysical problems. It has been used in many fields of research, including astrophysics, ballistics, volcanology, and oceanography. It is a meshfree Lagrangian method (where the co-ordinates move with the fluid), and the resolution of the method can easily be adjusted with respect to variables such as density. Method Advantages * By construction, SPH is a meshfree method, which makes it ideally suited to simulate problems dominated by complex boundary dynamics, like free surface flows, or large boundary displacement. * The lack of a mesh significantly simplifies the model implementation and its parallelization, even for many-core architectures. * SPH can be easily extended to a wide variety of fields, and hybridized with some other mod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contour Advection
Contour advection is a Lagrangian method of simulating the evolution of one or more contours or isolines of a tracer as it is stirred by a moving fluid. Consider a blob of dye injected into a river or stream: to first order it could be modelled by tracking only the motion of its outlines. It is an excellent method for studying chaotic mixing: even when advected by smooth or finitely-resolved velocity fields, through a continuous process of stretching and folding, these contours often develop into intricate fractals. The tracer is typically passive as in but may also be active as in, representing a dynamical property of the fluid such as vorticity. At present, advection of contours is limited to two dimensions, but generalizations to three dimensions are possible. Method First we need a set of points that accurately define the contour. These points are advected forward using a trajectory integration technique. To maintain its integrity, points must be added to or removed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trajectory (fluid Mechanics)
In fluid mechanics, meteorology (weather) and oceanography, a trajectory traces the motion of a single point, often called a parcel, in the flow. Trajectories are useful for tracking atmospheric contaminants, such as smoke plumes, and as constituents to Lagrangian simulations, such as contour advection or semi-Lagrangian schemes. Suppose we have a time-varying flow field, \vec v(\vec x,~t). The motion of a fluid parcel, or trajectory, is given by the following system of ordinary differential equations: : \frac = \vec v(\vec x, ~t) While the equation looks simple, there are at least three concerns when attempting to solve it numerically. The first is the integration scheme. This is typically a Runge-Kutta, although others can be useful as well, such as a leapfrog. The second is the method of determining the velocity vector, \vec v at a given position, \vec x, and time, ''t''. Normally, it is not known at all positions and times, therefore some method of interpolation is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Immersed Boundary Method
In computational fluid dynamics, the immersed boundary method originally referred to an approach developed by Charles Peskin in 1972 to simulate fluid-structure (fiber) interactions. Treating the coupling of the structure deformations and the fluid flow poses a number of challenging problems for numerical simulations (the elastic boundary changes the flow of the fluid and the fluid moves the elastic boundary simultaneously). In the immersed boundary method the fluid is represented in an Eulerian coordinate system and the structure is represented in Lagrangian coordinates. For Newtonian fluids governed by the Navier–Stokes equations The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician Georg ..., the fluid equations are : \rho \left(\frac + \cdot\nabla\right) = -\nabla p + \mu\, \Delta u(x, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Eulerian Lagrangian Method
In computational fluid dynamics, the Stochastic Eulerian Lagrangian Method (SELM) is an approach to capture essential features of fluid-structure interactions subject to thermal fluctuations while introducing approximations which facilitate analysis and the development of tractable numerical methods. SELM is a hybrid approach utilizing an Eulerian description for the continuum hydrodynamic fields and a Lagrangian description for elastic structures. Thermal fluctuations are introduced through stochastic driving fields. Approaches also are introduced for the stochastic fields of the SPDEs to obtain numerical methods taking into account the numerical discretization artifacts to maintain statistical principles, such as fluctuation-dissipation balance and other properties in statistical mechanics. The SELM fluid-structure equations typically used are : \rho \frac = \mu \, \Delta u - \nabla p + \Lambda Upsilon(V - \Gamma)+ \lambda + f_\mathrm(x,t) : m\frac = -\Upsilon(V - \Gam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eugenia Kalnay
Eugenia Enriqueta Kalnay (1 October 1942 – 13 August 2024) was an Argentine meteorologist and a Distinguished University Professor of Atmospheric and Oceanic Science, which is part of the University of Maryland College of Computer, Mathematical, and Natural Sciences at the University of Maryland, College Park in the United States. In 1996, Kalnay was elected a member of the National Academy of Engineering for advances in understanding atmospheric dynamics, numerical modeling, atmospheric predictability, and the quality of U.S. operational weather forecasts. Kalnay was the recipient of the 54th International Meteorological Organization Prize in 2009 from the World Meteorological Organization for her work on numerical weather prediction, data assimilation, and ensemble forecasting. As Director of the Environmental Modeling Center of the National Centers for Environmental Prediction (NCEP), Kalnay published the 1996 NCEP reanalysis paper entitled "The NCEP/NCAR 40-year reanal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
