Overview of the Numerics of the ECMWF. Atmospheric Forecast Model

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1 Overview of the Numerics of the Atmospheric Forecast Model M. Hortal Seminar 6 Sept 2004 Slide 1

2 Characteristics of the model Hydrostatic shallow-atmosphere approimation Pressure-based hybrid vertical coordinate Two-time-level semi-lagrangian semi-implicit time integration scheme Spectral horizontal representation (spherical harmonics) Pseudo-spectral (finite-element) vertical representation Transform method for computing non-linear terms using non-staggered grid Fourth order horizontal diffusion Seminar 6 Sept 2004 Slide 2

3 Vertical coordinate Pressure-based hybrid coordinate L60 L91 p ( η =0) = p( η =1)= p s p da( η) db( η) = + η dη dη p s 1 1 da db dη= 0; η= 1 η d d dη Seminar 6 Sept 2004 Slide 3

4 Model equations Momentum dv h pressure-gradient = fk V φ R T ln p + P + K h h d v h V V Discretized in vector form to avoid pole problems Thermodynamics dt = Hydrostatic κ Tv ω ( 1+ ( δ 1) q) p η + φ = φ s R d T v (ln p) η 1 P T + dη K T V h : horizontal wind vector : virtual temperature T v h :"horizontal" gradient ω : p coordinate vertical velocity κ R / c d / c pd, δ c pv P V, P T : physical parameterization pd K V, K T : horizontal diffusion Seminar 6 Sept 2004 Slide 4

5 Model equations (cont) Continuity equation p p p + + η h Vh = t η η η η Humidity equation d q = P q 0 Ozone equation d r o 3 = P o 3 Seminar 6 Sept 2004 Slide 5

6 Vertical integration of the continuity equation d (ln 1 db db ps ) = ( (ln ps ) + Vh ln ps ) dη dη t dη 0 where t (ln 1 1 p p V p η dη s ) = ( h ) s 0 η ω = 0 ( V h p ) dη + V η h p Needed for the energy-conversion term in the thermodynamic eq. p η = η p t η 0 ( V p dη η h ) Needed for the semi-lagrangian trajectory computation Seminar 6 Sept 2004 Slide 6

7 Vertical integration (finite elements) can be approimated as η F( η ) = f( ) d K 0 M 2 2 Cd( η) c e( ) d i i i i i= K i= M η and, applying the Galerkin method: Basis sets K 1 M 1 C t( d ) ( d ) = c t( ) e( ydy ) d for N j N 2 2 i j i i j i i= K1 0 i= M AC = Bc C = A Bc c = -1 S f F = PC -1-1 F= P A BS f Jf 1 2 Seminar 6 Sept 2004 Slide 7

8 Cubic B-splines for the vertical discretization Basis elements for the representation of the function to be integrated (integrand) f Basis elements for the representation of the integral F Seminar 6 Sept 2004 Slide 8

9 Advantages of the finite-element scheme in the vertical 8 th order accuracy using cubic basis functions Very accurate computation of the pressure-gradient term in conjunction with the spectral computation of horizontal derivatives More accurate vertical velocity for the semi-lagrangian trajectory computation Improved ozone conservation Reduced vertical noise in stratosphere Smaller error in the computation of the integrals than using the finite-element scheme in differential form (Private communication by Staniforth & Wood) f F = η Seminar 6 Sept 2004 Slide 9

10 The spectral horizontal representation X N m m X( λ, µη,, t) = X ( η, t) Y ( λµ, ) m n 1 2π N m= N n= m 1 m imλ ( η, t) = X( λ, µ, η, t) Pn ( µ ) e dλdµ 4π 1 0 n n Triangular truncation (isotropic) Spherical harmonics Fourier functions Legendre transform by Gaussian quadrature using NL (2N+1)/2 Gaussian latitudes (linear grid) ((3N+1)/2 if quadratic grid) No fast algorithm available FFT using NF 2N+1 points (linear grid) (3N+1 if quadratic grid) Legendre polynomials Seminar 6 Sept 2004 Slide 10

11 Advantages of the spectral method Spherical harmonics are eigenfunctions of the Laplace operator: 2 m n( n+1 m Yn = ) Y 2 n a The solution of a Helmholtz equation is straightforward The application of a high-order diffusion is very easy Horizontal derivatives are computed analytically But: The computation of the pressure-gradient term is very accurate The computational cost of the Legendre transforms increases more rapidly with resolution than the rest of the model Seminar 6 Sept 2004 Slide 11

12 The full and the reduced Gaussian grids Seminar 6 Sept 2004 Slide 12

13 Semi-implicit time integration + 0.5( ) 0 Define X tt X + X X dv = RHS + { γ + ( ln )} V tt ht RdTr h ps η dt d = RHS + ( τ ) T tt D ( γ X) η = RdX ( ln pr) dη dη 1 η 1 dp d r ( τ X) (ln ps) = RHS p + tt( νd η =κtr X d ) η p r dη η 0 => ( 2 ) + I + Γ h D = D Vertically coupled set of Helmholtz equations Seminar 6 Sept 2004 Slide 13 Linearized pressure-gradient using a reference temperature T r and a reference surface pressure (p s ) r ν 1 dpr X = X d ( p ) dη η s r 1 0 Γ γτ + RdTrν

14 Semi-Lagrangian advection (1) 1D advection equation without RHS: ϕ ϕ (, t+ t) ϕ (, t) = = t d j * 0 0 ϕ(, t+ t) = ϕ(, t) j * Stability analysis: absolutely stable if the value of n( *,t) is computed by interpolation using the surrounding grid points. In In the the Lagrangian point point of of view, view, time time is is the the only only independent variable variable (position should should be be consistent with with time) time) * " n j Finding the departure point * in the linear case: d = U = U t j * 0 0 = U t * j 0 Seminar 6 Sept 2004 Slide 14

15 Semi-Lagrangian advection (2) Three time level scheme with RHS : D * M A A D ϕ ( t+ t) ϕ ( t t) = R 2 t M () t Centered second order accurate scheme Disadvantages of of three-time-level schemes: Less Less efficient than than two-time-level schemes Computational mode Seminar 6 Sept 2004 Slide 15

16 Semi-Lagrangian advection (3) Two time level second order accurate schemes : A D ϕ ( t+ t) ϕ ( t) M t = R ( t+ ) t 2 with t 3 1 Rt ( + ) Rt () Rt ( t) Unstable Forecast 200 hpa T from 1997/01/04 Seminar 6 Sept 2004 Slide 16

17 Stable etrapolation two-time-level semi-lagrangian (SETTLS): 2 2 A D dϕ ( t) dϕ ϕ ( t+ t) ϕ () t + t + 2 t 2 where ϕ D d 2 ϕ dr and 2 t AV AV d = D R () t D AV A D R () t R ( t t) = t A D t A D ϕ ( t + t) = ϕ ( t) + ( R ( t) + {2 R( t) R( t t)} ) 2 Forecast 200 hpa T from 1997/01/04 using SETTLS Seminar 6 Sept 2004 Slide 17

18 Spherical geometry in the semi- Lagrangian advection Z G V j j j X Y D i M i A i Trajectory calculation Seminar 6 Sept 2004 Slide 18 Tangent plane projection

19 Interpolations in the semi-lagrangian (1) quasi-monotone Lagrange quasi-cubic interpolation 4 = Ci i i= 1 ϕ() () ϕ with the weights C ( ) i 4 k i = 4 k i ( ( i k k ) ) n interpolated cubically Quasi-monotone procedure: n ma n min : grid points : interpolation point Quasi-monotone interpolation is used in the horizontal for all variables and in the vertical for q and r O3 Seminar 6 Sept 2004 Slide 19

20 Modified continuity & thermodynamic equations Continuity equation where l * φs = R T d d (ln d * ps ) ( l + l ) = dl = [ RHS] + 1 R T d [ RHS] V h φ s Reduces mass loss: D+10 p s from 0.59 hpa to 0.02 hpa at T106L31 Thermodynamic equation d( T T b ) = [ RHS] T ( V h T b Tb ) η η with T b = p s p p s T p ref φs R T d Reduces noise over orography. Seminar 6 Sept 2004 Slide 20

21 Interpolations in the semi-lagrangian (2) Linear and smoothing interpolations: y X X O * X X I d Linear interpolation is applied to the velocities needed in the trajectory computation and to the RHS of the forecast equations. Smoothing interpolation is applied to the vertical velocity in the stratosphere. Seminar 6 Sept 2004 Slide 21

22 Divergence at model Level 11 (~5 hpa) 10 hpa Geopotential Linear interpolation in the computation of the semilagrangian trajectory L As above but using smoothing interpolation for the vertical velocity 12 hour forecast from initial data of at 12 Seminar 6 Sept 2004 Slide 22

23 Treatment of the Coriolis term Advective treatment: dr fk Vh = 2Ω dv h d + fk Vh ( Vh + 2Ω R) Implicit treatment : V + h V t 0 h + 0 = fk 0.5( Vh + Vh ) + Physical parameterizations Coupled with the semi-lagrangian scheme (details in talk by A. Beljaars) Seminar 6 Sept 2004 Slide 23

24 Horizontal diffusion X t = K 4 X Implicit solution in spectral space 2 Xnm, ( t+ t) Xnm, ( t) 4 nn ( + 1) = K Xnm, ( t+ t) = K 2 Xnm, ( t+ t) t a => X ( t+ t) = X ( t) nm, nm, 2 nn ( + 1) 1+ K t 2 Analytical solution in spectral space 1 a X + = X 2 nm, nn ( 1) K 2 t a nm, X ( t+ t) = X ( t) e nm, nm, nn ( + 1) K t 2 a 2 Seminar 6 Sept 2004 Slide 24

25 Summary of the numerics in the atmospheric model Two-time-level semi-lagrangian advection SETTLS scheme Quasi-monotone quasi-cubic interpolation Linear and smoothing interpolation for trajectories Modified continuity & thermodynamic equations Semi-implicit treatment of linearized adjustment terms Cubic finite elements for the vertical integrals Spectral horizontal Helmholtz solver (and derivative comp.) Linear reduced Gaussian grid Semi-Lagrangian coupling of physics and dynamics Seminar 6 Sept 2004 Slide 25

26 Future developments Non-hydrostatic version of the model Improve semi-lagrangian interpolation Spectral representation by double Fourier series Improve conservation of advected quantities Study the influence of boundary conditions for the semi-lagrangian advection for the vertical finite-element representation Investigate noise on vorticity over orography (aliasing?) THANK YOU for your attention! Seminar 6 Sept 2004 Slide 26

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