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Adiabatic formulation of the ECMWF model1 Agathe Untch (office 11)

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Adiabatic formulation of the ECMWF model2 Introduction Step by step guide through the decisions to be taken / choices to be made when designing the adiabatic formulation of a global Numerical Weather Prediction (NWP) model. In the process we are constructing the dynamical core of the ECMWF operational NWP Model.

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Adiabatic formulation of the ECMWF model3 Introduction (cont.) A numerical model has to be: –stable –accurate –efficient No compromise possible on stability! The relative importance given to accuracy versus efficiency depends on what the model is intended for. –For example: an operational NWP model has to be very efficient to allow the running of all applications (data-assimilation, forecasts, ensemble prediction system) in a tight daily schedule. a research model might not have to be so efficient but cant compromise on accuracy.

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Adiabatic formulation of the ECMWF model4 Introduction (cont.) Essential to the performance of any NWP or climate-prediction model are a.) the form of the continuous governing equations (approximated or full Euler equations?) b.) boundary conditions imposed (conservation properties depend on these). c.) the numerical schemes chosen to discretize and integrate the governing equations.

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Adiabatic formulation of the ECMWF model5 Euler Equations for a moist atmosphere on a rotating sphere 3D momentum equation Continuity equation Thermodynamic equation Equation of state Humidity equation Transport equations of various physical/chemical species

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Adiabatic formulation of the ECMWF model6 Notations: total time derivative specific volume q specific humidity X i mass mixing ratios of physical or chemical species (e.g. aerosols, ozone) g gravity = gravitation g* + centrifugal force L latent heat Spherical geopotential approximation is made: neglect Earths oblateness (~0.3%). => spherical geometry assumed!

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Adiabatic formulation of the ECMWF model7 With Euler Equations in spherical coordinates x Momentum equations in spherical coordinates :

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Adiabatic formulation of the ECMWF model8 Continuity equation in spherical coordinates With Thermodynamic equation in spherical coordinates With

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Adiabatic formulation of the ECMWF model9 Shallow atmosphere approximation a a a z 1. Replace r by the mean radius of the earth a andby where z is height above mean sea level., 2. Neglect vertical and horizontal variations in g. 3. Neglect all metric terms not involving 4. Neglect the Coriolis terms containing (resulting from the horizontal component of ).). a

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Adiabatic formulation of the ECMWF model10 Euler Equations in shallow atmosphere approximation For n=1 we refer to these equations as Non-hydrostatic equations in Shallow atmosphere Approximation (NH-SA) n is the tracer for the hydrostatic approximation: n=0 => vertical momentum equation = hydrostatic eq. is horizontal velocity

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Adiabatic formulation of the ECMWF model11 Choice of predicted variables Combine continuity, thermodynamic & gas equation to obtain a prognostic equation for p. => Predicted variables: or ? Form of NH-SA equations more commonly used in meteorology: (Allows enforcement of mass conservation)(Thermodyn. Computations are simpler)

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Adiabatic formulation of the ECMWF model12 Hydrostatic Approximation (n=0) Benefits from hydrostatic approximation –Vertical momentum equation becomes a diagnostic relation (=> one prognostic variable ( w ) less!) –Vertically propagating acoustic waves are eliminated (these are the fastest waves in the atmosphere, causing the biggest stability problems in numerical integrations!) Drawbacks of hydrostatic approximation –Not valid for short horizontal scales (for mesoscale phenomena) –Short gravity waves are distorted in the hydrostatic pressure field. Operational version of the ECMWF model is a hydrostatic model. –operational horizontal resolution ~25km (T799), so hydrostatic approximation is (still) OK.

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Adiabatic formulation of the ECMWF model13 Hydrostatic shallow atmosphere equations (Hydrostatic Primitive Equations (HPE)) p is monotonic function of z and can be used as vertical coordinate. (Eliassen (1949))

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Adiabatic formulation of the ECMWF model14 Choice of vertical coordinate Height above mean sea level z: - most natural vertical coordinate Pressure p (isobaric coordinate): - has advantages for thermodynamic calculations - makes the continuity equation a diagnostic relation in the hydrostatic system - can be extended for use in non-hydrostatic models Potential temperature (isentropic coordinate): - good coordinate where atmosphere is stably stratified (potential temperature increases monotonic with z). - adiabatic flow stays on isentropic surfaces (2D flow) - good coord. in stratosphere, not very good in troposphere Most commonly used vertical coordinates:

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Adiabatic formulation of the ECMWF model15 Choice of vertical coordinate (cont.) Generalized vertical coordinate s: Any variable s which is a monotonic single-valued function of height z can be used as a vertical coordinate. (Kasahara (1974), Staniforth & Wood (2003)) Coordinate transformation rules (from z to any vertical coordinate s):

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Adiabatic formulation of the ECMWF model16 Pressure p as vertical coordinate in the hydrostatic system Coordinate transformation rules (from z to p): => with geopotential Hydrostatic relation between p and z:

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Adiabatic formulation of the ECMWF model17 Hydrostatic Primitive Equations with pressure as vertical coordinate Pressure gradient replaced by geopotential gradient (at constant pressure). Continuity eq. is a diagnostic eq. in p-coordinates. Number of prognostic variables reduced to 3 (horizontal winds & T)! Geopotential computed from hydrostatic equation. pressure vertical velocity

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Adiabatic formulation of the ECMWF model18 Hydrostatic pressure as vertical coordinate for a non-hydrostatic shallow atmosphere model Introduced by Laprise (1991) In the hydrostatic system with pressure as vertical coordinate the continuity equation is a diagnostic equation. The idea is to find a vertical coordinate for the NH system which makes the continuity equation a diagnostic equation. Continuity equation in generalized vertical coordinate s: (Kasahara(1974)) For every s for which=> continuity is diagnostic eq.!

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Adiabatic formulation of the ECMWF model19 Hydrostatic pressure as vertical coordinate for a non-hydrostatic shallow atmosphere model (cont.) Chooseand denote with the coordinate s for which i.e. For is the weight of a column of air (of unit area) above a point at height z, i.e. hydrostatic pressure.

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Adiabatic formulation of the ECMWF model20 Hydrostatic pressure as vertical coordinate for a non-hydrostatic shallow atmosphere model (cont.) => D3D3 in Z

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Adiabatic formulation of the ECMWF model21 Boundary conditions Governing equations have to be solved subject to boundary conditions. The lower boundary of the atmosphere (surface of the earth) is a material boundary (air parcel cannot cross it!) velocity component perpendicular to surface has to vanish (e.g. at a flat and rigid surface vertical velocity w = 0) Unfortunately, the topography of the earth is far from flat, making it quite tricky to apply the lower boundary condition. Solution: Use a terrain-following vertical coordinate. For example: traditional sigma-coordinate (Phillips, 1957)

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Adiabatic formulation of the ECMWF model22 Terrain-following vertical coordinate Therefore, we look to create a vertical coordinate which makes the upper and lower boundaries flat. That is, s is constant following the shape of the boundary (i.e. the boundary is a coordinate surface). We are looking for a vertical coordinate s which makes it easy to apply the condition of zero velocity normal to the boundary, even for very complex boundaries like the earths topography. The easiest case is a flat and rigid boundary where the boundary condition simply is: at the topat the bottom &e.g. at the boundary.

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Adiabatic formulation of the ECMWF model23 Terrain-following vertical coordinate (cont.) A simple function that fulfils these conditions is is a monotonic single-valued function of hydrostatic pressure and also depends on surface pressure in such a way that is the pressure at the top boundary.)(Where For this is the traditional sigma-coordinate of Phillips (1957) The ECMWF model uses a terrain-following vertical coordinate based on hydrostatic pressure. The principle will be explained based on hydrostatic pressure :

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Adiabatic formulation of the ECMWF model24 Sigma-coordinate (First introduced by Phillips (1957)) Drawback: Influence of topography is felt even in the upper levels far away from the surface. Remedy: Use hybrid sigma-pressure coordinates x

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Adiabatic formulation of the ECMWF model25 Hybrid vertical coordinate First introduced by Simmons and Burridge (1981). The functions A and B can be quite general and allow to design a hybrid sigma-pressure coordinate where the coordinate surfaces are sigma surfaces near the ground, gradually become more horizontal with increasing distance from the surface and turn into pure pressure surfaces in the stratosphere (B=0). The difference to the sigma coordinate is in the way the monotonic relation between the new coordinate and hydrostatic pressure is defined: In order that the top and bottom boundaries are coordinate surfaces (=> easy application of boundary condition), A and B have to fulfil: at the surface at the top

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Adiabatic formulation of the ECMWF model26 Comparison of sigma-coordinates & hybrid η-coordinates sigma-coordinateη-coordinate Coordinate surfaces over a hill for

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Adiabatic formulation of the ECMWF model27 Non-hydrostatic equations in hybrid vertical coordinate prognostic continuity eq.

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Adiabatic formulation of the ECMWF model28 Hydrostatic Primitive Equations in hybrid η vertical coordinate Continuity equation is prognostic again because the (hydrostatic) pressure is not the vertical coordinate anymore. In addition to the geopotential gradient term, a pressure gradient term again!

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Adiabatic formulation of the ECMWF model29 Hydrostatic equations of the ECMWF operational model (incorporating moisture)

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Adiabatic formulation of the ECMWF model30 Notations: p-coordinate vertical velocity q specific humidity X mass mixing ratio of physical or chemical species (e.g. aerosols, ozone) virtual temperature gas constant of dry air,gas constant of water vapour specific heat of dry air at constant pressure specific heat of water vapour at constant pressure contributions from physical parametrizations horizontal diffusion terms

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Adiabatic formulation of the ECMWF model31 From the continuity equation with boundary conditions: we can derive (by vertical integration) the following equations: Needed for the energy-conversion term in the thermodynamic equation Needed for the semi-Lagrangian advection Prognostic equations for surface pressure B from def. of vert. coord.

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Adiabatic formulation of the ECMWF model32 Prognostic equations of the ECMWF hydrostatic model These equations are discretized and integrated in the ECMWF model.

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Adiabatic formulation of the ECMWF model33 Discretisation in the ECMWF Model Space discretisation –In the horizontal: spectral transform method –In the vertical: cubic finite-elements Time discretisation –Semi-implicit semi-Lagrangian two-time-level scheme

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Adiabatic formulation of the ECMWF model34 Horizontal discretisation in grid-point space only (grid-point model) –finite-difference, finite volume methods (in spectral space only) in both grid-point and spectral space and transform back and forth between the two spaces (spectral transform method, spectral model) –Gives the best of both worlds: Non-local operations (e.g. derivatives) are computed in spectral space (analytically) Local operations (e.g. products of terms) are computed in grid-point space –The price to pay is in the cost of the transformations between the two spaces in finite-element space (basis functions with finite support) Options for discretisation are:

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Adiabatic formulation of the ECMWF model35 Horizontal discretisation (cont.) ECMWF model uses the spectral transform method Representation in spectral space in terms of spherical harmonics: m: zonal wavenumber n: total wavenumber λ= longitude μ= sin(θ) θ: latitude P n m : Associated Legendre functions of the first kind Ideally suited set of basis functions for spherical geometry (eigenfunctions of the Laplace operator).

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Adiabatic formulation of the ECMWF model36 The horizontal spectral representation FFT (fast Fourier transform) using N F 2N+1 points (linear grid) (3N+1 if quadratic grid) Legendre transform by Gaussian quadrature using N L (2N+1)/2 Gaussian latitudes (linear grid) ((3N+1)/2 if quadratic grid) No fast algorithm available Triangular truncation (isotropic) Spherical harmonics associated Legendre polynomials Fourier functions m Triangular truncation: n N m = -N m = N

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Adiabatic formulation of the ECMWF model37 Grid-points in longitude are equidistantly spaced (Fourier) points 2N+1 for linear grid 3N+1 for quadratic grid Grid-points in latitude are the zeros of the Legendre polynomial of order N G Gaussian latitudes N G (2N+1)/2 for the linear grid. Horizontal discretisation (cont.) Representation in grid-point space is on the reduced Gaussian grid: Gaussian grid: grid of Guassian quadrature points (to facilitate accurate numerical computation of the integrals involved in the Fourier and Legendre transforms) - Gauss-Legendre quadrature in latitude: N G (3N+1)/2 for the quadratic grid. - Gauss-Fourier quadrature in longitude:

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Adiabatic formulation of the ECMWF model38 The Gaussian grid Full grid Reduced grid Associated Legendre functions are very small near the poles for large m About 30% reduction in number of points

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Adiabatic formulation of the ECMWF model39 T799 T km grid-spacing ( 843,490 grid-points) Current operational resolution 16 km grid-spacing (2,140,704 grid-points) Future operational resolution (from end 2009)

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Adiabatic formulation of the ECMWF model40 Spectral transform method Grid-point space -semi-Lagrangian advection -physical parametrizations Fourier Space Spectral space -horizontal gradients -semi-implicit calculations -horizontal diffusion FFT LT Inverse FFT Inverse LT Fourier Space FFT: Fast Fourier Transform, LT: Legendre Transform

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Adiabatic formulation of the ECMWF model41 Horizontal discretisation (cont.) Advantages of the spectral representation: a.) Horizontal derivatives are computed analytically => pressure-gradient terms are very accurate => no need to stagger variables on the grid b.) Spherical harmonics are eigenfunctions of the the Laplace operator => Solving the Helmholtz equation (arising from the semi-implicit method) is straightforward. => Applying high-order diffusion is very easy. Disadvantage: Computational cost of the Legendre transforms is high and grows faster with increasing horizontal resolution than the cost of the rest of the model.

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Adiabatic formulation of the ECMWF model42 Comparison of cost profiles at different horizontal resolutions

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Adiabatic formulation of the ECMWF model43 Cost of Legendre transforms T511 T799 T1279 T2047

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Adiabatic formulation of the ECMWF model44 Profile for T2047 on IBM p690+ (768 CPUs) Legendre Transforms ~17% of total cost of model Physics ~36% of total cost

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Adiabatic formulation of the ECMWF model45 L91 L60 Vertical resolution of the operational ECMWF model: 91 hybrid η-levels resolving the atmosphere up to 0.01hPa (~80km) (upper mesosphere) Vertical discretisation Variables are discretized on terrain-following pressure based hybrid η-levels.

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Adiabatic formulation of the ECMWF model46 Vertical discretisation (cont.) Choices: - finite difference methods - finite element methods Operational version of the ECMWF model uses a cubic finite-element (FE) scheme based on cubic B-splines. No staggering of variables, i.e. all variables are held on the same vertical levels. (Good for semi-Lagrangian advection scheme.) Inspection of the governing equations shows that there are only vertical integrals (no derivatives) to be computed (if advection is done with semi-Lagrangian scheme).

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Adiabatic formulation of the ECMWF model47 Prognostic equations of the ECMWF hydrostatic model These equations are discretized and integrated in the ECMWF model. Reminder: slide 32

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Adiabatic formulation of the ECMWF model48 From the continuity equation with boundary conditions: we can derive (by vertical integration) the following equations: Needed for the energy-conversion term in the thermodynamic equation Needed for the semi-Lagrangian advection Prognostic equations for surface pressure B from def. of vert. coord. Reminder: slide 31

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Adiabatic formulation of the ECMWF model49 Vertical integration in finite elements can be approximated as Applying the Galerkin method with test functions t j => Basis sets A ji B ji

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Adiabatic formulation of the ECMWF model50 Vertical integration in finite elements Including the transformation from grid-point (GP) representation to finite-element representation (FE) and the projection of the result from FE to GP representation one obtains Matrix J depends only on the choice of the basis functions and the level spacing. It does not change during the integration of the model, so it needs to be computed only once during the initialisation phase of the model and stored.

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Adiabatic formulation of the ECMWF model51 Cubic B-splines for regular spacing of levels (Prenter (1975))

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Adiabatic formulation of the ECMWF model52 Cubic B-splines as basis elements Basis elements for the represen- tation of the function to be integrated (integrand) f Basis elements for the representation of the integral F

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Adiabatic formulation of the ECMWF model53 Benefits from using the finite-element scheme in the vertical High order accuracy (8 th order for cubic elements) 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 the stratosphere No staggering of variables required in the vertical: good for semi-Lagrangian scheme because winds and advected variables are represented on the same vertical levels.

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Adiabatic formulation of the ECMWF model54 Discretisation in time Discretize on -three time-levels (e.g. leapfrog scheme) - produce a computational mode (time-filtering needed) -two time-levels - more efficient than three-time-level schemes - (less stable) Decisions to be taken: How to treat the advection: - in Eulerian way - in semi-Lagrangian way How to discretize the right-hand sides of the equations in time: - explicitly - implicitly - semi-implicitly

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Adiabatic formulation of the ECMWF model55 Operational version of the ECMWF model uses - Two-time-level scheme - Semi-implicit treatment of the right-hand sides - Semi-Lagrangian advection Discretization in time (cont.)

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Adiabatic formulation of the ECMWF model56 Time discretisation of the right-hand sides Discretisation of the right-hand side (RHS) of the equations: - RHS taken at the centre of the time interval: explicit (second order) discretisation. Stability is subject to CFL-like criterion - RHS average of its value at initial time and at final time: implicit discretisation (generally stable) => leads to a difficult system to solve (iterative solvers) - treat only some linearized terms of RHS implicitly (semi-implicit discretisation)

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Adiabatic formulation of the ECMWF model57 Semi-implicit time integration Notations: X : advected variable RHS: right-hand side of the equation L: part of RHS treated implicitly Superscripts: 0 indicates value for explicit discretiz, - indicates value at start of time step + indicates value at end of time step For compact notation define: implicit correction term L=RHS => implicit scheme L= part of RHS => semi-implicit => Benefit: slowing-down of the waves too fast for the explicit CFL cond. Drawback: overhead of having to solve an elliptic boundary-value prob.

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Adiabatic formulation of the ECMWF model58 Semi-implicit time integration (cont.) Choice of which terms in RHS to treat implicitly is guided by the knowledge of which waves cause instability because they are too fast (violate the CFL condition) and need to be slowed down with an implicit treatment. In a hydrostatic model, fastest waves are horizontally propagating external gravity waves (long surface gravity waves), Lamb waves (acoustic wave not filtered out by the hydrostatic approximation) and long internal gravity waves. => implicit treatment of the adjustment terms. L= linearization of part of RHS (i.e. terms supporting the fast modes) => good chance of obtaining a system of equations in the variables at + that can be solved almost analytically in a spectral model.

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Adiabatic formulation of the ECMWF model59 Semi-implicit time integration (cont.) semi-implicit equations semi-implicit corrections

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Adiabatic formulation of the ECMWF model60 Semi-implicit time integration (cont.) semi-implicit equations Where: Reference state for linearization: ref. temperature ref. surf. pressure => lin. geopotential for X=T => lin. energy conv. term for X=D

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Adiabatic formulation of the ECMWF model61 Semi-implicit time integration (cont.) semi-implicit equations Reference state for linearization: ref. temperature ref. surf. pressure By eliminating in above system all but one of the unknowns (D + ) => operator acting only on the vertical unity operator

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Adiabatic formulation of the ECMWF model62 Semi-implicit time integration (cont.) Vertically coupled set of Helmholtz equations. Coupling through Uncouple by transforming to the eigenspace of this matrix gamma (i.e. diagonalize gamma). Unity matrix I stays diagonal. => One equation for each In spectral space (spherical harmonics space): because Once D + has been computed, it is easy to compute the other variables at +.

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Adiabatic formulation of the ECMWF model63 Semi-Lagrangian advection Semi-Lagrangian (SL) schemes are more efficient & more stable than Eulerian advection schemes. Coupling SL advection with semi-implicit treatment of the fast modes results in a very stable scheme where the timestep can be chosen on the basis of accuracy rather than for stability. Disadvantage: Lack of conservation of mass and tracer concentrations. (More difficult to enforce conservation than in Eulerian schemes)

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Adiabatic formulation of the ECMWF model64 Semi-Lagrangian advection (cont) x x A D * x M Centred second order accurate scheme Three time-level scheme: Ingredients of semi-Lagrangian advection are: 1.) Computation of the departure point (tajectory computation) 2.) Interpolation of the advected fields at the departure location All equations are of this (Lagrangian) form: (See slide 32)

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Adiabatic formulation of the ECMWF model65 Prognostic equations of the ECMWF hydrostatic model These equations are discretized and integrated in the ECMWF model. Reminder: slide 32

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Adiabatic formulation of the ECMWF model66 Semi-Lagrangian advection (cont) Unstable! => noisy forecasts Two-time-level second order accurate schemes : Forecast of temperature at 200 hPa (from 1997/01/04) with Extrapolation in time to middle of time interval

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Adiabatic formulation of the ECMWF model67 Stable extrapolating two-time-level semi-Lagrangian (SETTLS): Forecast 200 hPa T from 1997/01/04 using SETTLS With and Taylor expansion to second order

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Adiabatic formulation of the ECMWF model68 Interpolation in the semi-Lagrangian scheme with the weights ECMWF model uses quasi-monotone quasi-cubic Lagrange interpolation xxxx x xxxx xxxx xxxx x x x x y x Number of 1D cubic interpolations in two dimensions is 5, in three dimensions 21! To save on computations: cubic interpolation only for nearest neighbour rows, linear interpolation for rest => quasi-cubic interpolation => 7 cubic + 10 linear in 3 dimensions. Cubic Lagrange interpolation:

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Adiabatic formulation of the ECMWF model69 Interpolation in the semi-Lagrangian scheme (cont) x: grid points x: interpolation point quasi-monotone procedure: Quasi-monotone interpolation is used in the horizontal for all variables and also in the vertical for humidity and all tracers ( e.g. ozone, aerosols ). x x x x x x interpolated cubically Quasi-monotone interpolation: Has a detrimental effect on conservation, but prevents unphysical negative concentrations.

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Adiabatic formulation of the ECMWF model70 => Reduced mass loss/gain during a forecast. Modified continuity & thermodynamic equations Continuity equation Accuracy of cubic interpolation is much reduced when the field to be interpolated is rough (e.g. surface pressure over orography) Idea by Ritchie & Tanguay (1996): Subtract a time-independent term from the surface pressure which contains a large part of the orographic influence on surface pressure, advect the rest (smoother term) and treat the advection of the rough term with the right-hand side of the continuity equation [RHS].

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Adiabatic formulation of the ECMWF model71 Modified continuity & thermodynamic equations (cont) with Reduces noise levels over orography in all fields, but in particular in vertical velocity. Thermodynamic equation Similar idea for thermodynamic equation: (Hortal &Temperton (2001)) Approximation to the change of T with height in the standard atmosphere.

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Adiabatic formulation of the ECMWF model72 Trajectory calculation M i j A i j D i j Tangent plane projection Semi-Lagrangian advection on the sphere X Y Z A V x D Momentum eq. is discretized in vector form (because a vector is continuous across the poles, components are not!) Trajectories are arcs of great circles if constant (angular) velocity is assumed for the duration of a time step. Interpolations at departure point are done for components u & v of the velocity vec- tor relative to the system of reference local at D. Interpolated values are to be used at A, so the change of reference system from D to A needs to be taken into account.

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Adiabatic formulation of the ECMWF model73 Treatment of the Coriolis term Implicit treatment : Advective treatment: In three-time-level semi-Lagrangian: In two-time-level semi-Lagrangian: treated explicitly with the rest of the RHS Extrapolation in time to the middle of the trajectory leads to instability (Temperton (1997)) Two stable options: Helmholtz eqs partially coupled for individual spectral components => tri-diagonal system to be solved. (Vector R here is the position vector.)

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Adiabatic formulation of the ECMWF model74 Summary of the adiabatic formulation of the operational ECMWF atmospheric model Hydrostatic shallow-atmosphere equations with pressure-based hybrid vertical coordinate Two-time-level semi-Lagrangian advection –SETTLS (Stable Extrapolation Two-Time-Level Scheme) –Quasi-monotone quasi-cubic Lagrange interpol. at departure point –Linear interpolation for trajectory computations and RHS terms –Modified continuity & thermodynamic equations to advect smoother fields (net of the orographic roughness) Semi-implicit treatment of linearized adjustment terms & Coriolis terms Cubic finite elements for the vertical integrals Spectral horizontal Helmholtz solver (and derivative computations) Uses the linear reduced Gaussian grid

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Adiabatic formulation of the ECMWF model75 Thank you very much for your attention

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