overview of the numerics of the ecmwf atmospheric … · ecmwf seminar 6 sept 2004 slide 1 ecmwf...

ECMWFECMWF Seminar 6 Sept 2004 Slide 1

Overview of the Numerics of the

ECMWF Atmospheric Forecast Model

M. Hortal

ECMWF


Characteristics of the ECMWF model

Hydrostatic shallow-atmosphere approximation

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


Vertical coordinate

Pressure-basedhybrid coordinate 0

L60 L910.01

0.1

100

1000

( = 0) 0( = 1) = s

p =p pηη

( ) ( )∂ η η= +

∂η η η sp dA dB p

d d

1 1

0 0

0 ; 1η = η =η η∫ ∫

dA dBd dd d


Model equations

Momentum

Thermodynamics

Hydrostatic

ln= − × −∇ φ− ∇ + +hh h d v h V V

dV fk V R T p P Kdt

TTv KP

pqT

dtdT

++−+

=))1(1( δ

ωκ: horizontal wind vectorhV: virtual temperaturevT

: p coordinate vertical velocity−ω

pdppdd cccR /,/ v≡≡ δκ

Discretized in vector formto avoid pole problems

:"horizontal" gradienth∇

pressure-gradient

PV, PT : physical parameterization

KV, KT : horizontal diffusionηη

φφη

dpTR vds )(ln1 ∂

∂−= ∫


Model equations (cont)

Continuity equation

Humidity equation

Ozone equation

0 ∂ ∂ ∂ ∂ ∂+∇ ⋅ + η = ∂ ∂η ∂η ∂η ∂η

h hp p pV

t

qPtdqd=

3

3o

o Ptd

rd=


Vertical integration of the continuity equation

ηηη

dpVddBp

tddBp

dtd

shss )ln)(ln()(ln1

0

∇⋅+∂∂

= ∫

ηη

dpVp

pt h

ss )(1)(ln

1

0∫ ∂

∂⋅∇−=

∂∂where

pVdpV hh ∇⋅+∂∂

⋅∇−= ∫ ηη

ωη

0

)(

ηηη

ηη

dpVtpp

h )(0∫ ∂

∂⋅∇−

∂∂

−=∂∂

Needed for the energy-conversionterm in the thermodynamic eq.

Needed for the semi-Lagrangiantrajectory computation


Vertical integration (finite elements)

0

( ) ( )F f x dxη

η = ∫

can be approximated as

and, applying the Galerkin method: Basis sets

2 2

1 1 0

( ) ( )= =

≈∑ ∑ ∫K M

i i i ii K i M

C d c e x dxη

η

2 2

1 1

1 1

1 20 0 0

( ) ( ) ( ) ( ) for= =

= ≤ ≤

∑ ∑∫ ∫ ∫

xK M

i j i i j ii K i M

C t x d x dx c t x e y dy dx N j N

-1= ⇒ =AC Bc C A Bc1 -1= ≡-F P A B S f J f=F PC1= -c S f


Basis elementsfor the represen-tation of thefunction tobe integrated(integrand)

f

Cubic B-splinesfor the verticaldiscretization

Basis elementsfor the representationof the integral

F


Advantages of the finite-element scheme in the vertical

8th order accuracy using cubic basis functionsVery accurate computation of the pressure-gradient term in conjunction with the spectral computation of horizontal derivativesMore accurate vertical velocity for the semi-Lagrangian trajectory computation

Improved ozone conservationReduced vertical noise in stratosphere

Smaller error in the computation of the integrals than using the finite-element scheme in differential form ∂

=∂Ffη(Private communication by Staniforth & Wood)


The spectral horizontal representation

( , , , ) ( , ) ( , )N N

m mn n

m N n m

X t X t Y=− =

λ µ η = η λ µ∑ ∑

∫ ∫−

πλ− µλµηµλ

π=η

1

1

2

0

)(),,,(41),( ddePtXtX imm

nmn

Triangular truncation(isotropic)

Spherical harmonics

Fourier functions

FFT usingNF ≥ 2N+1points (linear grid)(3N+1 if quadratic grid)

Legendre polynomials

Legendre transformby Gaussian quadratureusing NL ≥ (2N+1)/2“Gaussian” latitudes (linear grid)((3N+1)/2 if quadratic grid)No “fast” algorithm available


Advantages of the spectral method

Spherical harmonics are eigenfunctions of the Laplaceoperator:

The solution of a Helmholtz equation is straightforwardThe application of a high-order diffusion is very easy

Horizontal derivatives are computed analyticallyThe computation of the pressure-gradient term is very accurate

But:The computational cost of the Legendre transforms increases more rapidly with resolution than the rest of the model

mn

mn Y

annY 2

2 )1( +−=∇


The full and the reduced Gaussian grids


Semi-implicit time integration

( ) 00.5 + −∆ ≡ + −tt X X X X

( ) ln= + ∆ γ∇ + ∇V tt h d r h sdV RHS T R T pdt

( )= + ∆ τT ttdT RHS Ddt

( )(ln )s p ttd p RHS Ddt

= + ∆ ν

( )1

( ) lnd rdX R X p d

d

η

η ′γ = − η′η∫

0

1( ) rr

r

dpX T X dp d

η

η

η

′τ = κ η ′η

∫

Linearized pressure-gradientusing a reference temperatureTr and a reference surface pressure(ps)r

Define

1

0

1( )

r

s r

dpX X dp d

ν = ηη∫( )2

h D D++ ∇ =I Γ

Vertically coupled set of Helmholtz equations

=>

d rR T≡ γτ + νΓ


Semi-Lagrangian advection (1)1D advection equation without RHS:

*( , ) ( , )0 0

+ ∆ −= ⇒ =

∆jx t t x td

dt tϕ ϕϕ In the Lagrangian point of view,

time is the only independent variable (position should be consistent with time)

In the Lagrangian point of view, time is the only independent variable (position should be consistent with time)

*( , ) ( , )+∆ =jx t t x tϕ ϕ

j*n

"Stability analysis:absolutely stable if the value of n(x*,t) is computed by interpolation using the surrounding grid points.

Finding the departure point x* in the linear case:

* 0= − ∆jx x U t*0 0

−= ⇒ =

∆jx xdx U U

dt t


Semi-Lagrangian advection (2)

Three time level scheme with RHS :

( ) ( ) ( )2

+∆ − −∆=

∆

A DMt t t t R t

tϕ ϕx x x x

x x x x

x x x x

A

D *

xM

Centered second order accurate scheme

Disadvantages of three-time-level schemes:• Less efficient than two-time-level schemes• Computational mode

Disadvantages of three-time-level schemes:• Less efficient than two-time-level schemes• Computational mode


Semi-Lagrangian advection (3)

Two time level second order accurate schemes :3 1( ) ( ) ( )

2 2 2∆

+ ≈ − −∆tR t R t R t t( ) ( ) ( )

2+∆ − ∆

= +∆

A DMt t t tR t

tϕ ϕ with

Unstable

Forecast 200 hPa Tfrom 1997/01/04


Stable extrapolation two-time-level semi-Lagrangian (SETTLS):2 2

2

( )( ) ( )2

∆ +∆ ≈ +∆ ⋅ + ⋅

DA D

t AV

d t dt t t tdt dtϕ ϕϕ ϕ

( ) =

DD

t

d R tdtϕ 2

2

( ) ( ) − −∆ = ≈ ∆

A D

AVAV

R t R t td dRdt dt tϕwhere and

( ) ( ) ( ( ) 2 ( ) ( ) )2∆

+ ∆ = + ⋅ + − − ∆A D A Dtt t t R t R t R t tϕ ϕ

Forecast 200 hPa Tfrom 1997/01/04using SETTLS


Spherical geometry in the semi-Lagrangian advection

Z

X

Y

xG

V

A

MD ii

Tangent plane projection

j

j j

i

Trajectory calculation


Interpolations in the semi-Lagrangian (1)quasi-monotone Lagrange quasi-cubic interpolation

4

1

( ) ( )=

=∑ i ii

x C xϕ ϕ∏

∏

≠

≠

−

−= 4

4

)(

)()(

ikki

ikk

i

xx

xxxC

x: grid pointsx: interpolation pointx

x

x

x

x

xx x xx

x x x x

x x x x

x xxx

x

x

x

x

nmaxnmin

x

with the weights

n interpolated cubically

Quasi-monotone procedure:

Quasi-monotone interpolation is used in the horizontal for all variables and in the vertical for q and rO3


Modified continuity & thermodynamic equations

][)()(ln * RHSlldtdp

dtd

s =′+≡Continuity equation

shdd

s VTR

RHSdtld

TRl φφ

∇⋅+=′

⇒−=1][where *

D+10 from 0.59 hPa to 0.02 hPa at T106L31sp∆Reduces mass loss:

ηη∂∂

−∇⋅−=− b

bhTb TTVRHS

dtTTd )(][)(

TRpT

pppT

d

s

refssb

φ⋅

∂∂

∂∂

−=with

Reduces noise over orography.

Thermodynamic equation


Interpolations in the semi-Lagrangian (2)

• Linear and smoothing interpolations:

xI

X

X

X

X

-1 0 1 2

O

*

y

xx x x

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 thestratosphere.


L

10 hPa GeopotentialDivergence at modelLevel 11 (~5 hPa)

Linear interpolationin the computationof the semi-Lagrangiantrajectory

As above butusing smoothinginterpolation forthe verticalvelocity

12 hour forecast from initial data of 2002-12-28 at 12


Treatment of the Coriolis term

• Advective treatment:

2× = Ω× ⇒hdRfk Vdt

( 2+ × → + Ω×hh h

dV dfk V V R)dt dt

• Implicit treatment :0

00.5( )+

+−= − × + +

∆…h h

h hV V fk V V

t

Physical parameterizations

Coupled with the semi-Lagrangian scheme (details in talk by A. Beljaars)


4∂= − ∇

∂X K XtHorizontal diffusion

• Implicit solution in spectral space2

, , 4, ,2

( ) ( ) ( 1)( ) ( )+ ∆ − + = − ∇ + ∆ = − + ∆ ∆

n m n mn m n m

X t t X t n nK X t t K X t tt a

, , 2

2

1( ) ( )( 1)1

+ ∆ =+ + ∆

n m n mX t t X tn nK t

a

=>

• Analytical solution in spectral space2

,,2

( 1)∂ + = − ⇒ ∂ n m

n m

X n nK Xt a

2

2( 1)

, ,( ) ( )+ − ∆

+ ∆ =n nK t

an m n mX t t X t e


Summary of the numerics in the ECMWF 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


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 !

overview of the numerics of the ecmwf atmospheric … · ecmwf seminar 6 sept 2004 slide 1 ecmwf...

Documents