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N.B. the material derivative, rate of change following the motion:

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6 N.B. the material derivative, rate of change following the motion:







13 Approximations to full equations for use in a GCM 1.r = a + z (a=radius of Earth) z << a 2.Coriolis and metric terms proportional to can be ignored 3.For large scales, vertical acceleration is small, hence vertical component becomes:

14 Primitive Equations

15 Plus other equations Continuity equation Ideal Gas Law First Law of Thermodynamics These equations can be shown to conserve energy, angular momentum, and mass.

16 How to solve equations? Few analytical solutions to full Navier-Stokes equations, and only for fairly idealised problems. Hence need to solve numerically. At heart of all numerical schemes is a Taylor series expansion: –Suppose we have an interval L, covered by N equally spaced grid points, x j =(j-1) Δx, then

17 Re-arrange to give approximation for derivatives First order accurate: Second order accurate Fourth order accurate

18 Linear Advection Equation Differential equation becomes following difference equation Second order accurate in both space and time Centered time and space scheme

19 Numerical Stability and numerical solutions Schemes may be accurate but unstable: –e.g. simple centred difference scheme for linear advection scheme will be stable only if Courant- Friedrichs-Levy number less than 1. Many schemes can have artificial (computational mode) All schemes distort true solution (e.g. change phase and/or group speed) Some schemes fail to conserve properties of system (e.g. energy)

20 Examples of Numerical Schemes

21 More solutions

22 Staggered Grids

23 Grids on Sphere

24 Vertical Grid/Coordinates Hybrid coordinates

25 Summary so far Dynamics of atmosphere (and ocean) governed by straight forward physics Discretisation has problems but generally can be understood and quantified. NO tuneable parameters so far NO need for knowledge of past and only need present to initialise models. BUT…..

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