Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 1 / 27. ECMWF forecast model

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1 Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical core of the ECMWF forecast model Michail Diamantakis ECMWF Symposium on Modelling and Computations in Atmospheric Sciences ACMAC, University of Crete 19 September 2013 Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 1 / 27

2 European Centre for Medium Range Weather Forecasts ECMWF: intergovernmental organization supported by 34 European states Development of numerical methods for weather prediction with emphasis on medium-range weather forecasting (EPS technique) Distribution of medium-range weather forecast data to member states Scientific and technical research related to improvement of forecasts Reconstruction of past atmospheric conditions (Reanalysis) Meteorological data archiving Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 2 / 27

3 Outline 1 Introducing the IFS model 2 Semi-Lagrangian numerics, strengths and weaknesses Weak numerical instabilities in the stratosphere The mass conservation problem 3 Global atmospheric modeling challenges for next decades Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 3 / 27

4 Outline 1 Introducing the IFS model 2 Semi-Lagrangian numerics, strengths and weaknesses Weak numerical instabilities in the stratosphere The mass conservation problem 3 Global atmospheric modeling challenges for next decades Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 4 / 27

5 Main components of a NWP system NWP system: a nonlinear initial-boundary value PDE solver! Observations Data Assimilation system Forecast Model Dynamics Parametrizations Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 4 / 27

6 The forecast model of ECMWF (IFS) Current high resolution control forecast model Hydrostatic Primitive Equation set (non-hydrostatic option built in) Pressure based hybrid vertical coordinate Space discretization: Spectral (on spherical harmonics) in the horizontal Finite element in the vertical Spectral transform on reduced Gaussian grid Two time-level, semi-lagrangian semi-implicit time-stepping T1279L137: horizontal ( 16km), 137 vertical levels, t = 600s Ensemble Prediction System to resolve initial condition uncertainty 51 members (separate forecasts with perturbed initial conditions) T639/L62 resolution (due for L90 upgrade) Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 5 / 27

7 Hydrostatic primitive equations (HPE) in IFS Horizontal momentum equation DV h = f k V h h φ R d T v h ln p + P V + K V Dt D Dt t + V h + η η V h horizontal momentum η : terrain following p-based vertical coordinate T v T [1 + (R v /R d 1) q], q : specific humidity P V forcing from subgrid-processes, K V horizontal diffusion Hydrostatic equation in terms of geopotential φ η = R dt v p p η Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 6 / 27

8 HPE in IFS (part II) Thermodynamic equation χ DT Dt = κt v ω [1 + (δ 1)q] + P T + K T P T : forcing (eg. latent heat release from condensation/evaporation) κ R d /c pd, δ c vd /c pd ω Dp/Dt, vertical velocity (diagnostic) Transport of moist species Dq χ Dt = P qχ, χ : water vapour (spec. hum.), cloud water species Continuity equation t ( p η ) ( ) p + V h + ( η p ) = 0 η η η Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 7 / 27

9 Schematic representation of the spectral transform method Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 8 / 27

10 Gaussian Grid (at coarse resolution) Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 9 / 27

11 Outline 1 Introducing the IFS model 2 Semi-Lagrangian numerics, strengths and weaknesses Weak numerical instabilities in the stratosphere The mass conservation problem 3 Global atmospheric modeling challenges for next decades Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 10 / 27

12 Semi-Lagrangian modeling in NWP Semi-Lagrangian: an established numerical technique for solving the atmospheric transport equations. Many global and regional weather & climate prediction models use a semi-lagrangian semi-implicit numerical formulation (SLSI): IFS(ECMWF)/ARPEGE(MeteoFrance)/ALADIN, UM(UKMO), HIRLAM, SL-AV(Russia) GEM(Environment Canada), GFS(NCEP) GSM(JMA) HADGEM(UK), C-CAM(Australia)... What is the reason for this? Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 10 / 27

13 Semi-Lagrangian modeling in NWP Semi-Lagrangian: an established numerical technique for solving the atmospheric transport equations. Many global and regional weather & climate prediction models use a semi-lagrangian semi-implicit numerical formulation (SLSI): IFS(ECMWF)/ARPEGE(MeteoFrance)/ALADIN, UM(UKMO), HIRLAM, SL-AV(Russia) GEM(Environment Canada), GFS(NCEP) GSM(JMA) HADGEM(UK), C-CAM(Australia)... What is the reason for this? Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 10 / 27

14 A very stable and efficient numerical method Semi-Lagrangian advection (Staniforth and Cote MWR 1991) Unconditionally stable Good phase speeds with little numerical dispersion Simplicity: the nonlinear advective terms are absorbed by the Lagrangian derivative operator and essentially the advection problem is turned to an interpolation one! Combines virtues of Lagrangian and Eulerian approach Semi-implicit time-stepping Wide (unconditional) stability for fast forcing terms: gravity + acoustic waves (in non-hydrostatic models) SL+SI t determined by desired accuracy and not limited by stability Introduction of SLSI in IFS (1991) 6 times efficiency (CPU) gain! Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 11 / 27

15 A very stable and efficient numerical method Semi-Lagrangian advection (Staniforth and Cote MWR 1991) Unconditionally stable Good phase speeds with little numerical dispersion Simplicity: the nonlinear advective terms are absorbed by the Lagrangian derivative operator and essentially the advection problem is turned to an interpolation one! Combines virtues of Lagrangian and Eulerian approach Semi-implicit time-stepping Wide (unconditional) stability for fast forcing terms: gravity + acoustic waves (in non-hydrostatic models) SL+SI t determined by desired accuracy and not limited by stability Introduction of SLSI in IFS (1991) 6 times efficiency (CPU) gain! Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 11 / 27

16 A very stable and efficient numerical method Semi-Lagrangian advection (Staniforth and Cote MWR 1991) Unconditionally stable Good phase speeds with little numerical dispersion Simplicity: the nonlinear advective terms are absorbed by the Lagrangian derivative operator and essentially the advection problem is turned to an interpolation one! Combines virtues of Lagrangian and Eulerian approach Semi-implicit time-stepping Wide (unconditional) stability for fast forcing terms: gravity + acoustic waves (in non-hydrostatic models) SL+SI t determined by desired accuracy and not limited by stability Introduction of SLSI in IFS (1991) 6 times efficiency (CPU) gain! Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 11 / 27

17 A very stable and efficient numerical method Semi-Lagrangian advection (Staniforth and Cote MWR 1991) Unconditionally stable Good phase speeds with little numerical dispersion Simplicity: the nonlinear advective terms are absorbed by the Lagrangian derivative operator and essentially the advection problem is turned to an interpolation one! Combines virtues of Lagrangian and Eulerian approach Semi-implicit time-stepping Wide (unconditional) stability for fast forcing terms: gravity + acoustic waves (in non-hydrostatic models) SL+SI t determined by desired accuracy and not limited by stability Introduction of SLSI in IFS (1991) 6 times efficiency (CPU) gain! Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 11 / 27

18 A very stable and efficient numerical method Semi-Lagrangian advection (Staniforth and Cote MWR 1991) Unconditionally stable Good phase speeds with little numerical dispersion Simplicity: the nonlinear advective terms are absorbed by the Lagrangian derivative operator and essentially the advection problem is turned to an interpolation one! Combines virtues of Lagrangian and Eulerian approach Semi-implicit time-stepping Wide (unconditional) stability for fast forcing terms: gravity + acoustic waves (in non-hydrostatic models) SL+SI t determined by desired accuracy and not limited by stability Introduction of SLSI in IFS (1991) 6 times efficiency (CPU) gain! Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 11 / 27

19 A very stable and efficient numerical method Semi-Lagrangian advection (Staniforth and Cote MWR 1991) Unconditionally stable Good phase speeds with little numerical dispersion Simplicity: the nonlinear advective terms are absorbed by the Lagrangian derivative operator and essentially the advection problem is turned to an interpolation one! Combines virtues of Lagrangian and Eulerian approach Semi-implicit time-stepping Wide (unconditional) stability for fast forcing terms: gravity + acoustic waves (in non-hydrostatic models) SL+SI t determined by desired accuracy and not limited by stability Introduction of SLSI in IFS (1991) 6 times efficiency (CPU) gain! Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 11 / 27

20 A very stable and efficient numerical method Semi-Lagrangian advection (Staniforth and Cote MWR 1991) Unconditionally stable Good phase speeds with little numerical dispersion Simplicity: the nonlinear advective terms are absorbed by the Lagrangian derivative operator and essentially the advection problem is turned to an interpolation one! Combines virtues of Lagrangian and Eulerian approach Semi-implicit time-stepping Wide (unconditional) stability for fast forcing terms: gravity + acoustic waves (in non-hydrostatic models) SL+SI t determined by desired accuracy and not limited by stability Introduction of SLSI in IFS (1991) 6 times efficiency (CPU) gain! Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 11 / 27

21 Semi-Lagrangian advection Let ρ be the air density and ρ χ the density of a tracer transported by a wind field V Continuity equation in non-conservative form Dρ χ Dt = ρ χ V, Dρ Dt Using specific ratios φ χ = ρ χ /ρ: Dφ χ Dt = ρ V, D = 0 φt+ t χ φ t χ,d t = 0 φ t+ t χ Dt = t + V = φ t χ,d d: departure point (d.p.), i.e. spatial location of field at time t A simplified linear equation: non-linear advection terms absent To compute φ t+ t : (i) find d (ii) interpolate φ t to d Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 12 / 27

22 Semi-Lagrangian advection Let ρ be the air density and ρ χ the density of a tracer transported by a wind field V Continuity equation in non-conservative form Dρ χ Dt = ρ χ V, Dρ Dt Using specific ratios φ χ = ρ χ /ρ: Dφ χ Dt = ρ V, D = 0 φt+ t χ φ t χ,d t = 0 φ t+ t χ Dt = t + V = φ t χ,d d: departure point (d.p.), i.e. spatial location of field at time t A simplified linear equation: non-linear advection terms absent To compute φ t+ t : (i) find d (ii) interpolate φ t to d Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 12 / 27

23 Basic steps in Semi-Lagrangian algorithm Solve Dφ Dt = 0 given a grid r and a velocity field V 1 Fluid parcels are always assumed to arrive on r at t + t 2 For each grid-point solve backward trajectory equation for the d.p.: Dr Dt = V(r, t) rt+ t }{{} rd t = }{{} arrival g.p. unknown d.p. t+ t 3 Remap (interpolate) φ to r d (Lagrangian grid) to obtain t V(r, t)dt φ t+ t = φ t d Usual techniques employed Midpoint rule and fixed point iteration for solving trajectory equation Cubic-Lagrange (tri-cubic) interpolation normally used to obtain φ t d Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 13 / 27

24 Basic steps in Semi-Lagrangian algorithm Solve Dφ Dt = 0 given a grid r and a velocity field V 1 Fluid parcels are always assumed to arrive on r at t + t 2 For each grid-point solve backward trajectory equation for the d.p.: Dr Dt = V(r, t) rt+ t }{{} rd t = }{{} arrival g.p. unknown d.p. t+ t 3 Remap (interpolate) φ to r d (Lagrangian grid) to obtain t V(r, t)dt φ t+ t = φ t d Usual techniques employed Midpoint rule and fixed point iteration for solving trajectory equation Cubic-Lagrange (tri-cubic) interpolation normally used to obtain φ t d Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 13 / 27

25 Computing trajectories Classic scheme: midpoint using time extrapolation Midpoint discretization: ( r + rd r r d = t V, t + t ), V = (u, v, η) Extrapolate velocity field: V t+ t/2 = 1.5V t 0.5V t t 2 r d (0) = r Interpolate V t+ t/2 to r M 0.5[r + r (l 1) d ] Update: r (l) d = r t V t+ t/2 rm Iterative algorithm converges in 2,3 iterations in atmospheric flows (provided trajectories do not cross, Smolarkiewicz & Pudikiewicz 1992), l = 1, 2,... Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 14 / 27

26 SLSI time-stepping in IFS General adiabatic prognostic equation DX Dt = F. Let F = N + L, N F L where, L: fast terms linearised on a reference state, N: nonlinear slow terms Fast linear terms L are treated implicitly Non-linear terms can be extrapolated at t + t/2 using: N t+ t/2 1.5N t 0.5N t t 2nd order two time-level discretization X t+ t X t d t = 1 2 ( L t d + Lt+ t) + N t+ t/2 M, M : trajectory midpoint Prognostic variables eliminated deriving a constant coefficient Helmholtz equation for divergence: cheap and easy to solve in spectral space Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 15 / 27

27 SLSI time-stepping in IFS General adiabatic prognostic equation DX Dt = F. Let F = N + L, N F L where, L: fast terms linearised on a reference state, N: nonlinear slow terms Fast linear terms L are treated implicitly Non-linear terms can be extrapolated at t + t/2 using: N t+ t/2 1.5N t 0.5N t t 2nd order two time-level discretization X t+ t X t d t = 1 2 ( L t d + Lt+ t) + N t+ t/2 M, M : trajectory midpoint Prognostic variables eliminated deriving a constant coefficient Helmholtz equation for divergence: cheap and easy to solve in spectral space Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 15 / 27

28 SLSI time-stepping in IFS General adiabatic prognostic equation DX Dt = F. Let F = N + L, N F L where, L: fast terms linearised on a reference state, N: nonlinear slow terms Fast linear terms L are treated implicitly Non-linear terms can be extrapolated at t + t/2 using: N t+ t/2 1.5N t 0.5N t t 2nd order two time-level discretization X t+ t X t d t = 1 2 ( L t d + Lt+ t) + N t+ t/2 M, M : trajectory midpoint Prognostic variables eliminated deriving a constant coefficient Helmholtz equation for divergence: cheap and easy to solve in spectral space Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 15 / 27

29 Outline 1 Introducing the IFS model 2 Semi-Lagrangian numerics, strengths and weaknesses Weak numerical instabilities in the stratosphere The mass conservation problem 3 Global atmospheric modeling challenges for next decades Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 16 / 27

30 Iterative Centred Implicit SLSI solver Extrapolation results to a weak instability: Bates et al (QJRMS 1995), Durran & Reinecke (MWR 2003), Cordero et al (QJRMS 2005) This is mostly noticed in strong jet areas in the stratosphere Stable alternative: iterative approach (P. Bénard, MWR 2003) Solve (as usual) to compute a predictor for prognostic variable X X (P) X t+ t, X (P) : solution at the end of an iteration Use time-interpolation (no extrapolation) for implicit RHS terms: 1. Use V t+ t/2 = 0.5 [ V (P) + V t] in trajectory iterations 2. Use N t+ t/2 = 0.5 [ N (P) + N t] in: X t+ t Xd t = 1 ( L t d t 2 + Lt+ t) + N t+ t/2 M Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 16 / 27

31 Iterative Centred Implicit SLSI solver Extrapolation results to a weak instability: Bates et al (QJRMS 1995), Durran & Reinecke (MWR 2003), Cordero et al (QJRMS 2005) This is mostly noticed in strong jet areas in the stratosphere Stable alternative: iterative approach (P. Bénard, MWR 2003) Solve (as usual) to compute a predictor for prognostic variable X X (P) X t+ t, X (P) : solution at the end of an iteration Use time-interpolation (no extrapolation) for implicit RHS terms: 1. Use V t+ t/2 = 0.5 [ V (P) + V t] in trajectory iterations 2. Use N t+ t/2 = 0.5 [ N (P) + N t] in: X t+ t Xd t = 1 ( L t d t 2 + Lt+ t) + N t+ t/2 M Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 16 / 27

32 SETTLS: stable extrapolation (Hortal, QJRMS 2002) Assume dx dt = R and expand: R d (t) {( }}){ dx X (t + t) X d (t) + t + t2 dt d 2 R(t) R d (t t) t ({}}{ d 2 ) X dt 2 X (t + t) = X d (t) + t 2 {R(t) + [2R(t) R(t t)] d } Use same formula for: Trajectories: r (l) d = r 0.5 t Non-linear RHS terms: N t+ t/2 M = 0.5 AV [ ( V t + 2V t V t t) ] d (l 1) [ N t + (2N t N t t) ] d Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 17 / 27

33 SETTLS: stable extrapolation (Hortal, QJRMS 2002) Assume dx dt = R and expand: R d (t) {( }}){ dx X (t + t) X d (t) + t + t2 dt d 2 R(t) R d (t t) t ({}}{ d 2 ) X dt 2 X (t + t) = X d (t) + t 2 {R(t) + [2R(t) R(t t)] d } Use same formula for: Trajectories: r (l) d = r 0.5 t Non-linear RHS terms: N t+ t/2 M = 0.5 AV [ ( V t + 2V t V t t) ] d (l 1) [ N t + (2N t N t t) ] d Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 17 / 27

34 Stratospheric noise Occasionally, solution becomes noisy in the upper stratosphere (near strong jets) Testing shows that noise originates from the calculation of the vertical component of the departure point Solutions: (a) short t (b) Iterative scheme (c) off-centring semi-implicit (d) smoothing of vertical velocities (e) SETTLS limiter (a) 5hPa D at T+24 (16/01/12) (b) 5hPa D at T+24 + SETTLS limiter Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 18 / 27

35 Impact of noise in satellite data assimilation instabilities noisy solution upper atmosphere rejection of satellite obs (c) Control FC (d) Smoothing η SETTLS (upper atmosphere) Limiter: ( ηd = η t + η dt, η t 2 SETTLS, t η t > β η otherwise t,t t 0 < β < 2, t η t = η t η t t (e) SETTLS traj limiter Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 19 / 27

36 Outline 1 Introducing the IFS model 2 Semi-Lagrangian numerics, strengths and weaknesses Weak numerical instabilities in the stratosphere The mass conservation problem 3 Global atmospheric modeling challenges for next decades Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 20 / 27

37 Mass Conservation Mass conservation becomes increasingly important Has always been for long (climate) integrations Moist tracer conservation important for microphysics/convection Increasing importance of environmental/chemical forecasts Eulerian flux-form (finite-volume) models Conservation straightforward: change of mass in volume A = total flux through its faces S from neighbouring grid-boxes ρdv = (ρu)dv ρ t+ t ρ t = t ρu ηds t A A A Semi-Lagrangian schemes are not formally conserving: equations applied on grid-points rather than volumes - there is no flux-counting S Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 20 / 27

38 Mass Conservation Mass conservation becomes increasingly important Has always been for long (climate) integrations Moist tracer conservation important for microphysics/convection Increasing importance of environmental/chemical forecasts Eulerian flux-form (finite-volume) models Conservation straightforward: change of mass in volume A = total flux through its faces S from neighbouring grid-boxes ρdv = (ρu)dv ρ t+ t ρ t = t ρu ηds t A A A Semi-Lagrangian schemes are not formally conserving: equations applied on grid-points rather than volumes - there is no flux-counting S Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 20 / 27

39 Mass Conservation Mass conservation becomes increasingly important Has always been for long (climate) integrations Moist tracer conservation important for microphysics/convection Increasing importance of environmental/chemical forecasts Eulerian flux-form (finite-volume) models Conservation straightforward: change of mass in volume A = total flux through its faces S from neighbouring grid-boxes ρdv = (ρu)dv ρ t+ t ρ t = t ρu ηds t A A A Semi-Lagrangian schemes are not formally conserving: equations applied on grid-points rather than volumes - there is no flux-counting S Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 20 / 27

40 Global mass conservation error in IFS In NWP not much attention has been given in mass conservation. Why? An accurate numerical scheme may not be mass conserving but its mass conservation error should be small In a 10 day IFS forecast at T1279 horizontal resolution total air mass approximately increasing by less than 0.01% of its initial value Global conservation errors in tracer advection are larger and depend on the smoothness of the field, e.g. ozone and specific humidity have much smaller errors than sharp fields (plumes,clouds) Global conservation error tends to reduce when increasing resolution but NOT when timestep is reduced Mass conserving SL schemes Global mass fixers Inherently conserving schemes Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 21 / 27

41 Global mass conservation error in IFS In NWP not much attention has been given in mass conservation. Why? An accurate numerical scheme may not be mass conserving but its mass conservation error should be small In a 10 day IFS forecast at T1279 horizontal resolution total air mass approximately increasing by less than 0.01% of its initial value Global conservation errors in tracer advection are larger and depend on the smoothness of the field, e.g. ozone and specific humidity have much smaller errors than sharp fields (plumes,clouds) Global conservation error tends to reduce when increasing resolution but NOT when timestep is reduced Mass conserving SL schemes Global mass fixers Inherently conserving schemes Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 21 / 27

42 Global mass conservation error in IFS In NWP not much attention has been given in mass conservation. Why? An accurate numerical scheme may not be mass conserving but its mass conservation error should be small In a 10 day IFS forecast at T1279 horizontal resolution total air mass approximately increasing by less than 0.01% of its initial value Global conservation errors in tracer advection are larger and depend on the smoothness of the field, e.g. ozone and specific humidity have much smaller errors than sharp fields (plumes,clouds) Global conservation error tends to reduce when increasing resolution but NOT when timestep is reduced Mass conserving SL schemes Global mass fixers Inherently conserving schemes Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 21 / 27

43 Global mass conservation error in IFS In NWP not much attention has been given in mass conservation. Why? An accurate numerical scheme may not be mass conserving but its mass conservation error should be small In a 10 day IFS forecast at T1279 horizontal resolution total air mass approximately increasing by less than 0.01% of its initial value Global conservation errors in tracer advection are larger and depend on the smoothness of the field, e.g. ozone and specific humidity have much smaller errors than sharp fields (plumes,clouds) Global conservation error tends to reduce when increasing resolution but NOT when timestep is reduced Mass conserving SL schemes Global mass fixers Inherently conserving schemes Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 21 / 27

44 Global mass conservation error in IFS In NWP not much attention has been given in mass conservation. Why? An accurate numerical scheme may not be mass conserving but its mass conservation error should be small In a 10 day IFS forecast at T1279 horizontal resolution total air mass approximately increasing by less than 0.01% of its initial value Global conservation errors in tracer advection are larger and depend on the smoothness of the field, e.g. ozone and specific humidity have much smaller errors than sharp fields (plumes,clouds) Global conservation error tends to reduce when increasing resolution but NOT when timestep is reduced Mass conserving SL schemes Global mass fixers Inherently conserving schemes Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 21 / 27

45 Global mass conservation error in IFS In NWP not much attention has been given in mass conservation. Why? An accurate numerical scheme may not be mass conserving but its mass conservation error should be small In a 10 day IFS forecast at T1279 horizontal resolution total air mass approximately increasing by less than 0.01% of its initial value Global conservation errors in tracer advection are larger and depend on the smoothness of the field, e.g. ozone and specific humidity have much smaller errors than sharp fields (plumes,clouds) Global conservation error tends to reduce when increasing resolution but NOT when timestep is reduced Mass conserving SL schemes Global mass fixers Inherently conserving schemes Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 21 / 27

46 Global Mass Fixers Low cost algorithms Easy to implement in existing models: no need to change model formulation Most sophisticated algorithms correct only where interpolation error is expected to be large Possible to construct algorithms to preserve local monotonicity or positive-definitiness Maintaining overall solution accuracy Examples: Bermejo & Conde MWR (2002), Priestley MWR (1993), Zerroukat JCP (2010), Mac Gregor s (C-CAM, CSIRO Tech Rep 70) Criticism: fixers are somewhat ad-hoc and only global in nature (local conservation is not in general satisfied) Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 22 / 27

47 Global Mass Fixers Low cost algorithms Easy to implement in existing models: no need to change model formulation Most sophisticated algorithms correct only where interpolation error is expected to be large Possible to construct algorithms to preserve local monotonicity or positive-definitiness Maintaining overall solution accuracy Examples: Bermejo & Conde MWR (2002), Priestley MWR (1993), Zerroukat JCP (2010), Mac Gregor s (C-CAM, CSIRO Tech Rep 70) Criticism: fixers are somewhat ad-hoc and only global in nature (local conservation is not in general satisfied) Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 22 / 27

48 Global Mass Fixers Low cost algorithms Easy to implement in existing models: no need to change model formulation Most sophisticated algorithms correct only where interpolation error is expected to be large Possible to construct algorithms to preserve local monotonicity or positive-definitiness Maintaining overall solution accuracy Examples: Bermejo & Conde MWR (2002), Priestley MWR (1993), Zerroukat JCP (2010), Mac Gregor s (C-CAM, CSIRO Tech Rep 70) Criticism: fixers are somewhat ad-hoc and only global in nature (local conservation is not in general satisfied) Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 22 / 27

49 Global Mass Fixers Low cost algorithms Easy to implement in existing models: no need to change model formulation Most sophisticated algorithms correct only where interpolation error is expected to be large Possible to construct algorithms to preserve local monotonicity or positive-definitiness Maintaining overall solution accuracy Examples: Bermejo & Conde MWR (2002), Priestley MWR (1993), Zerroukat JCP (2010), Mac Gregor s (C-CAM, CSIRO Tech Rep 70) Criticism: fixers are somewhat ad-hoc and only global in nature (local conservation is not in general satisfied) Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 22 / 27

50 Global Mass Fixers Low cost algorithms Easy to implement in existing models: no need to change model formulation Most sophisticated algorithms correct only where interpolation error is expected to be large Possible to construct algorithms to preserve local monotonicity or positive-definitiness Maintaining overall solution accuracy Examples: Bermejo & Conde MWR (2002), Priestley MWR (1993), Zerroukat JCP (2010), Mac Gregor s (C-CAM, CSIRO Tech Rep 70) Criticism: fixers are somewhat ad-hoc and only global in nature (local conservation is not in general satisfied) Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 22 / 27

51 Global Mass Fixers Low cost algorithms Easy to implement in existing models: no need to change model formulation Most sophisticated algorithms correct only where interpolation error is expected to be large Possible to construct algorithms to preserve local monotonicity or positive-definitiness Maintaining overall solution accuracy Examples: Bermejo & Conde MWR (2002), Priestley MWR (1993), Zerroukat JCP (2010), Mac Gregor s (C-CAM, CSIRO Tech Rep 70) Criticism: fixers are somewhat ad-hoc and only global in nature (local conservation is not in general satisfied) Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 22 / 27

52 Global Mass Fixers Low cost algorithms Easy to implement in existing models: no need to change model formulation Most sophisticated algorithms correct only where interpolation error is expected to be large Possible to construct algorithms to preserve local monotonicity or positive-definitiness Maintaining overall solution accuracy Examples: Bermejo & Conde MWR (2002), Priestley MWR (1993), Zerroukat JCP (2010), Mac Gregor s (C-CAM, CSIRO Tech Rep 70) Criticism: fixers are somewhat ad-hoc and only global in nature (local conservation is not in general satisfied) Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 22 / 27

53 Example: Bermejo & Conde fixer MWR (2002) (f) T+24 q field at 700 hpa (g) BC fixer correction qj =q j λw j, λ= δm V wdv, δm = V qdv M 0, w q H q : min q q q 2 w s.t. V qdv = M 0 Corrections are concentrated in areas of large gradients }{{} cubic interp q L }{{} linear interp Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 23 / 27

54 A note on inherently conserving schemes Two well known examples: CSLAM (P. Lauritzen JCP 2010), SLICE-3D (Zerroukat & Allen QJRMS 2012) Finite volume semi-lagrangian continuity: D ψdv = 0 ψ n+1 dv = ψ n dv, Dt V (t) V V d ψ = air/tracer density Tracer and air mass advection fully consistent and locally and globally conserving Computational expense is a limiting factor for their application in operational meteorology However, they may be competitive for the multiple-tracer advection problem Performance of such schemes in presence of complex terrain? Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 24 / 27

55 A note on inherently conserving schemes Two well known examples: CSLAM (P. Lauritzen JCP 2010), SLICE-3D (Zerroukat & Allen QJRMS 2012) Finite volume semi-lagrangian continuity: D ψdv = 0 ψ n+1 dv = ψ n dv, Dt V (t) V V d ψ = air/tracer density Tracer and air mass advection fully consistent and locally and globally conserving Computational expense is a limiting factor for their application in operational meteorology However, they may be competitive for the multiple-tracer advection problem Performance of such schemes in presence of complex terrain? Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 24 / 27

56 A note on inherently conserving schemes Two well known examples: CSLAM (P. Lauritzen JCP 2010), SLICE-3D (Zerroukat & Allen QJRMS 2012) Finite volume semi-lagrangian continuity: D ψdv = 0 ψ n+1 dv = ψ n dv, Dt V (t) V V d ψ = air/tracer density Tracer and air mass advection fully consistent and locally and globally conserving Computational expense is a limiting factor for their application in operational meteorology However, they may be competitive for the multiple-tracer advection problem Performance of such schemes in presence of complex terrain? Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 24 / 27

57 A note on inherently conserving schemes Two well known examples: CSLAM (P. Lauritzen JCP 2010), SLICE-3D (Zerroukat & Allen QJRMS 2012) Finite volume semi-lagrangian continuity: D ψdv = 0 ψ n+1 dv = ψ n dv, Dt V (t) V V d ψ = air/tracer density Tracer and air mass advection fully consistent and locally and globally conserving Computational expense is a limiting factor for their application in operational meteorology However, they may be competitive for the multiple-tracer advection problem Performance of such schemes in presence of complex terrain? Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 24 / 27

58 A note on inherently conserving schemes Two well known examples: CSLAM (P. Lauritzen JCP 2010), SLICE-3D (Zerroukat & Allen QJRMS 2012) Finite volume semi-lagrangian continuity: D ψdv = 0 ψ n+1 dv = ψ n dv, Dt V (t) V V d ψ = air/tracer density Tracer and air mass advection fully consistent and locally and globally conserving Computational expense is a limiting factor for their application in operational meteorology However, they may be competitive for the multiple-tracer advection problem Performance of such schemes in presence of complex terrain? Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 24 / 27

59 Outline 1 Introducing the IFS model 2 Semi-Lagrangian numerics, strengths and weaknesses Weak numerical instabilities in the stratosphere The mass conservation problem 3 Global atmospheric modeling challenges for next decades Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 25 / 27

60 Future challenges High resolution global NWP models on massively parallel machines As horizontal resolution increases and falls below 10km the hydrostatic approximation becomes inaccurate use non-hydrostatic dynamics CPUs do not become faster any more and massive parallelism is the only alternative (exascale computing) Current algorithms may not scale well in future architectures Efficiency of (any) advection scheme on a regular lat/lon grid on the sphere deteriorates: convergence of meridionals, anisotropy near poles [Timestepping and/or communication costs] SLSI and Eulerian SI advection schemes on quasi-uniform grids (cubed sphere, icosahedral etc) are in good position SL will probably remain a strong candidate for multi-tracer applications Spectral transform methods: one critical issue is cost of transpositions Mass conservation: an important matter for future dynamical cores Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 25 / 27

61 Future challenges High resolution global NWP models on massively parallel machines As horizontal resolution increases and falls below 10km the hydrostatic approximation becomes inaccurate use non-hydrostatic dynamics CPUs do not become faster any more and massive parallelism is the only alternative (exascale computing) Current algorithms may not scale well in future architectures Efficiency of (any) advection scheme on a regular lat/lon grid on the sphere deteriorates: convergence of meridionals, anisotropy near poles [Timestepping and/or communication costs] SLSI and Eulerian SI advection schemes on quasi-uniform grids (cubed sphere, icosahedral etc) are in good position SL will probably remain a strong candidate for multi-tracer applications Spectral transform methods: one critical issue is cost of transpositions Mass conservation: an important matter for future dynamical cores Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 25 / 27

62 Future challenges High resolution global NWP models on massively parallel machines As horizontal resolution increases and falls below 10km the hydrostatic approximation becomes inaccurate use non-hydrostatic dynamics CPUs do not become faster any more and massive parallelism is the only alternative (exascale computing) Current algorithms may not scale well in future architectures Efficiency of (any) advection scheme on a regular lat/lon grid on the sphere deteriorates: convergence of meridionals, anisotropy near poles [Timestepping and/or communication costs] SLSI and Eulerian SI advection schemes on quasi-uniform grids (cubed sphere, icosahedral etc) are in good position SL will probably remain a strong candidate for multi-tracer applications Spectral transform methods: one critical issue is cost of transpositions Mass conservation: an important matter for future dynamical cores Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 25 / 27

63 Future challenges High resolution global NWP models on massively parallel machines As horizontal resolution increases and falls below 10km the hydrostatic approximation becomes inaccurate use non-hydrostatic dynamics CPUs do not become faster any more and massive parallelism is the only alternative (exascale computing) Current algorithms may not scale well in future architectures Efficiency of (any) advection scheme on a regular lat/lon grid on the sphere deteriorates: convergence of meridionals, anisotropy near poles [Timestepping and/or communication costs] SLSI and Eulerian SI advection schemes on quasi-uniform grids (cubed sphere, icosahedral etc) are in good position SL will probably remain a strong candidate for multi-tracer applications Spectral transform methods: one critical issue is cost of transpositions Mass conservation: an important matter for future dynamical cores Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 25 / 27

64 Future challenges High resolution global NWP models on massively parallel machines As horizontal resolution increases and falls below 10km the hydrostatic approximation becomes inaccurate use non-hydrostatic dynamics CPUs do not become faster any more and massive parallelism is the only alternative (exascale computing) Current algorithms may not scale well in future architectures Efficiency of (any) advection scheme on a regular lat/lon grid on the sphere deteriorates: convergence of meridionals, anisotropy near poles [Timestepping and/or communication costs] SLSI and Eulerian SI advection schemes on quasi-uniform grids (cubed sphere, icosahedral etc) are in good position SL will probably remain a strong candidate for multi-tracer applications Spectral transform methods: one critical issue is cost of transpositions Mass conservation: an important matter for future dynamical cores Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 25 / 27

65 Future challenges High resolution global NWP models on massively parallel machines As horizontal resolution increases and falls below 10km the hydrostatic approximation becomes inaccurate use non-hydrostatic dynamics CPUs do not become faster any more and massive parallelism is the only alternative (exascale computing) Current algorithms may not scale well in future architectures Efficiency of (any) advection scheme on a regular lat/lon grid on the sphere deteriorates: convergence of meridionals, anisotropy near poles [Timestepping and/or communication costs] SLSI and Eulerian SI advection schemes on quasi-uniform grids (cubed sphere, icosahedral etc) are in good position SL will probably remain a strong candidate for multi-tracer applications Spectral transform methods: one critical issue is cost of transpositions Mass conservation: an important matter for future dynamical cores Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 25 / 27

66 Future challenges High resolution global NWP models on massively parallel machines As horizontal resolution increases and falls below 10km the hydrostatic approximation becomes inaccurate use non-hydrostatic dynamics CPUs do not become faster any more and massive parallelism is the only alternative (exascale computing) Current algorithms may not scale well in future architectures Efficiency of (any) advection scheme on a regular lat/lon grid on the sphere deteriorates: convergence of meridionals, anisotropy near poles [Timestepping and/or communication costs] SLSI and Eulerian SI advection schemes on quasi-uniform grids (cubed sphere, icosahedral etc) are in good position SL will probably remain a strong candidate for multi-tracer applications Spectral transform methods: one critical issue is cost of transpositions Mass conservation: an important matter for future dynamical cores Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 25 / 27

67 Future challenges High resolution global NWP models on massively parallel machines As horizontal resolution increases and falls below 10km the hydrostatic approximation becomes inaccurate use non-hydrostatic dynamics CPUs do not become faster any more and massive parallelism is the only alternative (exascale computing) Current algorithms may not scale well in future architectures Efficiency of (any) advection scheme on a regular lat/lon grid on the sphere deteriorates: convergence of meridionals, anisotropy near poles [Timestepping and/or communication costs] SLSI and Eulerian SI advection schemes on quasi-uniform grids (cubed sphere, icosahedral etc) are in good position SL will probably remain a strong candidate for multi-tracer applications Spectral transform methods: one critical issue is cost of transpositions Mass conservation: an important matter for future dynamical cores Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 25 / 27

68 Developments in high res global NH modeling Examples of current developments UK: GungHo project (Globally Uniform Next Generation Highly Optimized): develop a new dynamical core suitable for global, regional weather and climate simulation on massively parallel computers Germany: ICON model (non-hydrostatic, flux-form Eulerian, icosahedral grid, vertically implicit, explicit in horizontal) USA: MPAS model (non-hydrostatic, flux-form Eulerian, Voronoi (mostly hexagonal) meshes, split-explicit Runge-Kutta timestepping) Japan: NICAM model (non-hydrostatic, cloud resolving, icosahedral grid, Eulerian flux-form, split-explicit RK time-stepping) ECMWF: through CRESTA project techniques to improve scalability of IFS are being investigated Current trend in global modeling: a shift towards developing Eulerian dynamical cores Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 26 / 27

69 Developments in high res global NH modeling Examples of current developments UK: GungHo project (Globally Uniform Next Generation Highly Optimized): develop a new dynamical core suitable for global, regional weather and climate simulation on massively parallel computers Germany: ICON model (non-hydrostatic, flux-form Eulerian, icosahedral grid, vertically implicit, explicit in horizontal) USA: MPAS model (non-hydrostatic, flux-form Eulerian, Voronoi (mostly hexagonal) meshes, split-explicit Runge-Kutta timestepping) Japan: NICAM model (non-hydrostatic, cloud resolving, icosahedral grid, Eulerian flux-form, split-explicit RK time-stepping) ECMWF: through CRESTA project techniques to improve scalability of IFS are being investigated Current trend in global modeling: a shift towards developing Eulerian dynamical cores Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 26 / 27

70 Thank you for your attention! Global Numerical Weather Predictions and the semi-lagrangian semi-implicit dynamical 27 / 27