ECMWF Overview. The European Centre for Medium-Range Weather Forecasts is an international. organisation supported by 23 European States.

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1 ECMWF Overview The European Centre for Medium-Range Weather Forecasts is an international organisation supported by 3 European States. The center was established in 1973 by a Convention and the real-time mediumrange forecasts were made in June The Centre has been producing operational medium-range weather forecasts since 1 August The Centre has three Fujitsu systems, a 100-processor VPP5000, a 116-processor VPP700 and a 48-processor VPP700E. The aggregate sustained performance of these three machines is about 400 Gflops (400 thousand million floating point operations per second).

2 ECMWF model is a global model which predicts the behavoiur of the atmosphere in the medium-range up to ten days ahead. The resolution of the discretization is equivalent to having gridpoints separated by about 40 km around the globe and at 60 levels in the vertical.the model forecasts the wind, the temperature and the humidity at 0,911,680 points throughout the atmosphere. With this resolution it is possible, for example, to distinguish clearly the French Massif Central from the Alps. The ECMWF model is also a spectral model with a triangular truncation of 13 waves in the horizontal and 31 levels in the vertical (T13/L31). The ECMWF model has a hybrid coordinate in the vertical, η ( pp, surf ) with η ( 0, p surf ) 0 and η ( p surf, p surf ) 1.

3 Physics of the model include: Radiation Turbulent diffusion and interaction with the surface Subgrid-scale orographic drag Convection Clouds and large-scale precipitation Land surface parameterization Methane oxidation Climatological data

4 The Continuous equations in ( λθη,, ) The momentum equations U 1 U U U U + V cos θ + η fv t a cos θ λ θ η 1 -- Φ a Rdry λ T + ( lnp ) v λ + P U + K U V 1 V V U t a cos V θ θ U V V + + cos + sin ( + ) + η + fu θ λ θ η cosθ Φ a Rdry θ T + ( lnp ) v θ + P V + K V (1) ()

5 a is the radius of the Earth, η is the η -coordinate vertical velocity ( η dη dt ), Φ is geopotential, R dry is the gas constant for dry air, and T v is the virtual temperature defined by T v T { 1 + [ R vap ( R dry 1 )]q }. P U and P V represent contributions of parameterized physical processes, while K U and K V are the horizontal diffusion terms. The thermodynamic equation T 1 T T U t a cos + V cos θ + η T κt ω v P θ λ θ η { 1 + ( δ 1 )q }p T + K T (3)

6 κ R dry c pdry,, ω is the p-coordinate vertical velocity ω dp dt and δ c pvap, c pdry,. The moist equation q 1 q q U + V cos θ q + + η P t a cos θ λ θ η q + K q (4) The continuity equation p t η V p p + + η H η η η 0 (5)

7 V H ( UV, ) is the horizontal wind. The hydrostatic equation Φ R T dry v p η p η The vertical velocity ω is given by η p ω V H d η + V η H p 0 Expressions for the rate of change of surface pressure and for vertical velocity η, (6) (7)

8 are obtained by integrating (5), using the boundary conditions, η 0 at η 0 and η 1, ( lnp t s ) 1 1 p ---- V H d η 0 η p s (8) p η η p η p V t H d η 0 η (9)

9 Vertical discretization To represent the vertical variation of the dependent variables UVT,, and q the atmosphere is divided into NLEV layers. These layers are defined by the pressure at the interfaces between them (the half-levels ), and these pressures are given by p k 1 + A k B k + 1 p s (10) p p s for 0 k NLEV and B k + 1.The A k + 1 and B k + 1 are k + 1 constants whose values effectively define the vertical coordinate. They are determined by requiring that half-level pressures coincide with those of sigma coordinates for a surface pressure of 1013.mb, that at the top interior levels (1 1 /

10 and 1 / ) are levels of constant pressure, and that the lowest two layers have half the thicness for a surface pressure of 500mb that they have for the surface pressure of 1013.mb.

11 The form of this hybrid coordinate is efficient from a computational viewpoint, and allows a direct control over the flattening of coordinate surfaces as pressure decrease, since the A s and B s may be determined by specifying the distribution of model pressure levels for a mean sea-level pressure and for a surface pressure typical of the lowest expected to be attained in the model. One disadvantage of sigma coordinate is cancellation of geopotential and pressure gradient terms over steep orography. Simmons and Burridge (1981) have compared the error in the representation of pressure gradient over steep orography for hybrid and sigma coordinates. They considered a temperature field which is a function of pressure alone and calculated an error function.

12 geostrophic wind error (m/s))

13 The prognostic variables are represented by their values at full-level pressures. p k p k Values for are not explicitly required by the model s vertical finite-difference scheme and they are calculated as p k 1 ( p k p k 1 ) (11) The discrete analogue of the surface pressure tendency equation (8) is ( lnp t s ) V H p k ( ) p s NLEV k 1 (1) where p k p k + 1 p k 1. Substituting p A + B p k k k s

14 ( lnp t s ) NLEV k D p k p k s { ( ) + ( V k ln p s )( B k )} (13) where D k is the divergence at level k, 1 D k U k, and (14) a cos θ λ θ U + cos k θ B k B k 1 + B k + 1 (15) The discrete analogue of (9) is

15 p η η k + 1 k p k + 1 ( V t j p j ) j 1 (16) and from (10) we obtain p η η k + 1 k 1 p s B k + 1 ( lnp t s ) [ D p j p j + ( V j ln p s ) B j ] s j 1 (17) p In evaluating η at time, should first be used to compute values η ps t 1 t k + 1 of p s at the following time step t + t. The new value of p may then be k + 1 calculated, and thus found using, for example a centered difference in p k 1 + t time.

16 Conservation of energy is preserved by the vertical advection terms if the advection of a variable F is such as to satisfy the finite-difference analogue of the relations 1 F p 1 p η dη F η 0 η η η η dη 0 (18) 1 F p η F dη 0 η η F p η η η dη 0 (19) This is achieved by choosing F η η k 1 p η p k η k + 1 ( F k + 1 F k ) + η p η k 1 ( F k F k 1 ) (0)

17 The discrete analogue of the hydrostatic equation is R T dry v Φk + 1 Φ k 1 p ( ) k + 1 k p k 1 ln which gives + NLEV Φk + 1 Φ s R dry T v j k+ 1 p ( ) j + 1 j ln p j 1 where is the geopotential at the surface. The full-level values of the Φs geopotential, as requiered in the momentum equations (1) and (), are given by Φk Φ k α k R dry ( T v ) k (1) () (3)

18 α k The value of must be specified, but is not necessary to specify it in order to assure conservation of energy. One choise could be to require cancellation of error in the sum of the geopotential and pressure gradient terms for a reference temperature distribution which is a function of pressure alone. Therefore, and ln (4) α 1 ln α k 1 p k 1 p k p k p k 1 Connected with the form chosen for the integration of the hydrostatic equation is the expression for full level values of the pressure gradient term ( R dry T v ) ln p, in (1) and (), in terms of the known half-level values. This expression is p k + 1 determined by requiring that the vertical difference scheme preserves the conservation of angular momentum

19 π 0 1 Φ RT p p λ p λ η ps d η + Φs d λ 0 λ This requires in finite-difference form, NLEV Φk p λ k Φ s λ k 1 ps + NLEV k 1 p p k k R T p -- λ which is satisfied if R dry ( T v ) R dry ( T v ln p ) k k p k p k + 1 ln p + α ( p ) k k k 1 Angular momentum conservation is hereby achieved without the artificial dependence of geopotential on the temperature of all model levels. This is a (5) (6) (7)

20 consequence of using half-level values of Φ in the sumation of the hydrostatic equation. To obtain a form of the energy conversion term κtω { 1+ ( 1 δ )q }p in the thermodynamic equation (3) we use the definition of vertical velocity ω to write κt v ω { 1+ ( 1 δ )q }p + κt v η p V { 1 + ( 1 δ )q }p d η 0 η κt v 1 { 1+ ( 1 δ )q } p -- p (8) Full-level values of this expression are then determined by the requirment that the difference scheme conserves the total energy of the model atmosphere for adaiabatic, frictionless motion. This is achieved by evaluating the first term at level

21 k by κ ( T v ) k ( 1 δ )q k p k p k p k 1 k 1 ln V p j j j 1 ( ) + α k ( V k p k ) and the second term as κ ( T v ) k 1 1+ ( 1 δ )q k p -- p k after substitute for α k, ( p ) and 1 k -- p we obtain p k (9) (30)

22 κt v ω { 1+ ( 1 δ )q }p κ ( T v ) k ( 1 δ )q k p k p k + 1 ln ( D j p j p k 1 k 1 j 1 + p s ( V j ln p s ) B j ) + α k ( D k p k + p s ( V k ln p s ) B k )) (31) p s p k + 1 C B k p k ln k p p ( Vk ln p ) s k k 1 where C k A k + 1 B k 1 A k 1 B k + 1. Full-level values of pressure Two different full level values of p are suggested by the finite-diference scheme. For this we go back to first term in (8) V which may be generally η written κt p 1 p d η 0

23 k 1 κ ( T v ) k p k + 1 p p V p 1 ( ) -- O p k ln + + j j k k 1 p ( V p ) k k k 1 j 1 For k 1 the simplest equivalent is kt ( p 1 ) -- ( V 1 p 1 ) These expressions suggest the full-level values , k > 1 p p k k ln( ) p k 1 + p k 1 1 p 1 -- p 1 (3) (33) (34)

24 Away from the upper boundary (34) yields 1 p k -- ( p k p k 1 ) O ( p k ) p k 1 but top-level values are given by p 1 and p 1 e. (35)