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.
The Lagrangian rate of change of a quantity is given by
where can be a scalar or vector field and 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 and the second term is often called the advection term. Note that the Lagrangian rate of change is also known as the material derivative.
It can be shown that the equations governing atmospheric motion can be written in the Lagrangian form
where the components of the vector are the (dependent) variables describing a parcel of air (such as velocity, pressure, temperature etc.) and the function represents source and/or sink terms.
In a Lagrangian scheme, individual air parcels are traced but there are clearly certain drawbacks: the number of parcels can be very large indeed and it may often happen for a large number of parcels to cluster together, leaving relatively large regions of space completely empty. Such voids can cause computational problems, e.g. when calculating spatial derivatives of various quantities. There are ways round this, such as the technique known as Smoothed Particle Hydrodynamics, where a dependent variable is expressed in non-local form, i.e. as an integral of itself times a kernel function.
Semi-Lagrangian schemes avoid the problem of having regions of space essentially free of parcels.
The Semi-Lagrangian schemeedit
Semi-Lagrangian schemes use a regular (Eulerian) grid, just like finite difference methods. The idea is this: at every time step the point where a parcel originated from is calculated. An interpolation scheme is then utilized to estimate the value of the dependent variable at the grid points surrounding the point where the particle originated from. The references listed contain more details on how the Semi-Lagrangian scheme is applied.
semi, lagrangian, scheme, numerical, method, that, widely, used, numerical, weather, prediction, models, integration, equations, governing, atmospheric, motion, lagrangian, description, system, such, atmosphere, focuses, following, individual, parcels, along, . 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 Contents 1 Some background 2 The Semi Lagrangian scheme 3 See also 4 External links 5 ReferencesSome background editThe Lagrangian rate of change of a quantity F displaystyle F nbsp is given byDFDt F t v F displaystyle frac DF Dt frac partial F partial t mathbf v cdot vec nabla F nbsp where F displaystyle F nbsp can be a scalar or vector field and v displaystyle mathbf v nbsp 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 displaystyle F nbsp and the second term is often called the advection term Note that the Lagrangian rate of change is also known as the material derivative It can be shown that the equations governing atmospheric motion can be written in the Lagrangian formDVDt S V displaystyle frac D mathbf V Dt mathbf S mathbf V nbsp where the components of the vector V displaystyle mathbf V nbsp are the dependent variables describing a parcel of air such as velocity pressure temperature etc and the function S V displaystyle mathbf S mathbf V nbsp represents source and or sink terms In a Lagrangian scheme individual air parcels are traced but there are clearly certain drawbacks the number of parcels can be very large indeed and it may often happen for a large number of parcels to cluster together leaving relatively large regions of space completely empty Such voids can cause computational problems e g when calculating spatial derivatives of various quantities There are ways round this such as the technique known as Smoothed Particle Hydrodynamics where a dependent variable is expressed in non local form i e as an integral of itself times a kernel function Semi Lagrangian schemes avoid the problem of having regions of space essentially free of parcels The Semi Lagrangian scheme editSemi Lagrangian schemes use a regular Eulerian grid just like finite difference methods The idea is this at every time step the point where a parcel originated from is calculated An interpolation scheme is then utilized to estimate the value of the dependent variable at the grid points surrounding the point where the particle originated from The references listed contain more details on how the Semi Lagrangian scheme is applied See also editLagrangian and Eulerian specification of the flow field Contour advection Trajectory fluid mechanics External links editctraj C trajectory library including semi Lagrangian tracer codes References editE Kalnay Atmospheric Modeling Data Assimilation and Predictability Chapter 3 Section 3 3 3 Cambridge University Press Cambridge 2003 A Persson User Guide to ECMWF forecast products Section 2 1 3 http www ecmwf int sites default files User Guide V1 2 20151123 pdf D A Randall Atmospheric Modeling AT604 Chapter 5 Section 5 11 http kiwi atmos colostate edu group dave at604 html Retrieved from https en wikipedia org w index php title Semi Lagrangian scheme amp oldid 854040414, wikipedia, wiki, book, books, library,
