## Fundamentals of numerical weather prediction by Jean Coiffier

By Jean Coiffier

''Numerical types became crucial instruments in environmental technology, fairly in climate forecasting and weather prediction. This booklet presents a finished review of the innovations utilized in those fields, with emphasis at the layout of the latest numerical types of the ambience. It provides a brief historical past of numerical climate prediction and its evolution, ahead of describing a few of the modelRead more...

content material: desktop generated contents observe: Foreword to the French variation; Foreword to the English variation; Preface; Acknowledgements; record of symbols; 1. part a century of numerical climate prediction; 2. climate prediction equations; three. Finite adjustments; four. Spectral equipment; five. the consequences of discretization; 6. Barotropic versions; 7. Baroclinic version equations; eight. a few baroclinic types; nine. actual parameterizations; 10. Operational forecasting; Appendix A. Examples of non-hydrostatic versions; additional studying; References; Index.

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2) and gives a 2nd-order accuracy approximation: A( x + Δx) − A( x − Δx) 2 Δx = ∂A ∂x + Δx 2 ∂3 A 3! ∂ x 3 + Ο ( Δx3 ). 3) By using a combination of central differences calculated over the intervals 2Δx and 4Δx, we get a 4th-order accuracy approximation, which is written: 4 A( x + Δx) − A( x − Δx) 3 2Δx − 1 A( x + 2Δx) − A( x − 2Δx) 3 4Δx = ∂A ∂x − 4 Δx 4 ∂ 5 A 5 ! ∂x 5 + Ο ( Δx5 ). Generally, the truncation error of an approximation is defined as the difference between the value of an expression calculated numerically and its exact value.

Based on the geometrical construction, the map scale factor is easily calculated from a displacement along a meridian and is: m=− dρ a dϕ = − L 1 + tan 2 θ dθ 2 −2 a d θ . The map scale factor calculated from a displacement along a parallel is: θ θ L tan 1 + tan 2 2 2 m= = . θ a cos ϕ d λ 2a tan 2 ρ d λ It is verified that the projection is conformal since the expression of the map scale factor is the same in both cases. Taking into account the definition of the co-latitude θ, its expression simplifies as: θ 2 1 + tan 2 1 + sin ϕ 0 m = (1 + sin ϕ 0 ) .

X 2 We verify therefore that: A( x + Δx) − A( x) Δx = ∂A ∂x + Δx ∂ 2 A 2 ! ∂x 2 + Ο ( Δx 2 ). 1) This evaluation, called a forward difference, is a 1st-order accuracy approximation of the derivative of A at point x. A similar evaluation may be made with a backward difference which gives: A( x) − A( x − Δx) Δx = ∂A ∂x − Δx ∂ 2 A 2 ! ∂x 2 + O( Δx 2 ). 2) In this way we get 1st-order accuracy evaluations by approximating the behaviour of A in the vicinity of x to that of a 1st-degree polynomial function of x.