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## On the Time Step of Semi-Lagrangian Semi-Implicit Atmospheric Model

HU Zhi-Jin and SHI Yue-QinChinese Academy of Meteorological Sciences,Beijing 100081
Semi-Lagrangian Semi-Implicit(SLSI) atmospheric model has some advantages and a longer time step is usually used.The limitation of the time step for SLSI model is discussed.The source-sink term in SLSI model is usually calculated by the average value at departure and arrived time-space points.Serious computation error is shown in a stationary point source case with a constant advection speed u_(0) by using a usual SLSI model if time step Δt is significantly greater than ΔX/u_(0),where Δt is time step and ΔX is grid spacing.These errors could not be reduced by using the weighted average value of the source-sink term.It is shown in this simple case that the accurate results can be gotten if an accurate form of the source-sink term is used.To ensure the accuracy of SLSI model for calculating the nonliner source-sink term case there is the alternative of the simplified scheme with a shorter time step or the sophisticated scheme with a longer time step.Atmospheric waves also have a nonliner source-sink term.It is shown in a non-advection wave case that the absolute value of source term by using the averaged value at beginning and end points is always less than the accurate ones.The computation error grows with Δt,and becomes serious for ΔtT_w/4,where T_w is time period of the wave.The solution becomes a stationary one when Δt=T_w/2.Better results are given by using the sophisticated source scheme for ΔtT_w/4.But significant error appears when ΔtT_w/4,and simulated wave becomes a standing one for Δt=T_w/2.Generally speaking,minimum period and length(T_(min),L_(min)) of atmospheric waves,which may be simulated by the model,are determined by the time step and grid length(Δt,ΔX),such as T_(min)=4Δt,L_(min)=4ΔX.Serious error occurring in calculating the shorter wave(T4Δt) is inevitable as shown in the above case.T_(min) will match with L_(min) when Δt=ΔX/C_0,where C_(0) is the wave speed.If ΔtΔX/C_0,only waves with L4Δt·C_0 can be accurately simulated by the model,but waves with 4ΔXL4Δt·C_0 will be misrepresented.So the Courant number N_C=C·Δt/ΔX is a useful index for determining the Δt value in all atmospheric models.Various atmospheric phenomena have different typical life time(T_(a)) and size(L_(a)).Δt T_(a) /4 and ΔX L_(a) /4 are essential for correct simulation.The SLSI model has better computation stability even if a longer time step Δt is used.But the characteristic time(T_(0)) of cloud or atmospheric chemical processes must be considered in calculating the positively defined variables.The calculated value may be negative when ΔtT_0 for explicit model or Δt2T_0 for the SLSI model,which leads to an unreasonable result.
【Fund】： 国家重点基础研究发展规划项目G1998040912、2004CB418306;; 国家自然科学基金资助项目40305001
【CateGory Index】： P435
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