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Adiabatic formulation of the ECMWF model 1 Adiabatic formulation of the ECMWF model Agathe Untch e-mail: [email protected] (office 11)

Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: [email protected] (office 11)

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Page 1: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 1

Adiabatic formulation of the ECMWF model

Agathe Untche-mail: [email protected]

(office 11)

Page 2: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 2

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.

Page 3: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 3

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 can’t compromise on accuracy.

Page 4: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 4

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.

Page 5: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 5

Euler Equations for a moist atmosphere on a rotating sphere

rFgpVDt

VD

1

2

00)(

VDt

DV

t

QDt

Dp

Dt

DqL

Dt

DTcv

RTp

3D momentum equation

Continuity equation

Thermodynamic equation

Equation of state

xii P

Dt

DX

qPDt

Dq Humidity equation

Transport equations ofvarious physical/chemicalspecies

Page 6: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 6

Notations:

VtDt

D total time derivative

1

specific volume

q specific humidity

Xi 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 Earth’s oblateness (~0.3%). => spherical geometry assumed!

Page 7: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 7

),,cos(),,(Dt

Dr

Dt

Dr

Dt

Drwvu

w

v

u

Fgr

pu

r

vu

Dt

Dw

Fp

ru

r

vw

r

u

Dt

Dv

Fp

rwv

r

uw

r

uv

Dt

Du

1cos2

1sin2

tan

cos

1cos2sin2

tan

22

2

rw

r

v

r

u

trr

tDt

D

cos

With

Euler Equations in spherical coordinates

x

),,( rMomentum equations in spherical coordinates :

Page 8: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 8

Continuity equation in spherical coordinates

0coscos 322 Drr

Dt

D

r

r

r

v

r

u

Dt

Dr

rDt

D

Dt

DD

cos3With

Thermodynamic equation in spherical coordinates

QDt

Dp

Dt

DTcv

rw

r

v

r

u

trr

tDt

D

cos

With

Page 9: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 9

Shallow atmosphere approximation

w

v

u

Fgr

pu

r

vu

Dt

Dw

Fp

ru

r

vw

r

u

Dt

Dv

Fp

rwv

r

uw

r

uv

Dt

Du

1cos2

1sin2

tan

cos

1cos2sin2

tan

22

2

a

a

a

z

r

z1. Replace r by the mean radius of the earth a and by

where z is height above mean sea level.,

2. Neglect vertical and horizontal variations in g.

tan3. Neglect all metric terms not involving

cos4. Neglect the Coriolis terms containing (resulting from

the horizontal component of ).

a

Page 10: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 10

Euler Equations in shallow atmosphere approximation

RTp

wDt

D

c

Qw

c

RT

Dt

DT

nFz

pg

Dt

Dwn

FpkfDt

D

vv

rw

rz

0,v

,v

1

1v

v

h

h

vhh

For n=1 we refer to these equations asNon-hydrostatic equationsin Shallow atmosphere Approximation (NH-SA)

n is the tracer for thehydrostatic approximation:n=0 => vertical momentumequation = hydrostatic eq.

hv

is horizontal velocity

Page 11: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 11

Choice of predicted variables

Combine continuity, thermodynamic & gas equation to obtain a prognostic equation for p.

RT

p

Tc

Qpw

c

pc

Dt

Dp

c

Q

Dt

Dp

cDt

DT

Fnz

pg

Dt

Dwn

FpkfDt

D

vh

v

p

pp

rw

rzhh

,v

1

1

1v

vv

RTp

wDt

D

c

Qw

c

RT

Dt

DT

nFz

pg

Dt

Dwn

FpkfDt

D

vv

rw

rz

0,v

,v

1

1v

v

h

h

vhh

=>

Predicted variables: ),,,,( Twvu ),,,,( pTwvuor ?

Form of NH-SA equations more commonly used in meteorology:

),,,,( Twvu ),,,,( pTwvu(Allows enforcement of mass conservation) (Thermodyn. Computations are simpler)

Page 12: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 12

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.

Page 13: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 13

Hydrostatic shallow atmosphere equations(Hydrostatic Primitive Equations (HPE))

RT

p

Tc

Qpw

c

pc

Dt

Dp

c

Q

Dt

Dp

cDt

DT

z

pg

FpkfDt

D

vh

v

p

pp

rzhh

,v

1

01

1v

vv

p is monotonic function of z and can be used as vertical coordinate. (Eliassen (1949))

Page 14: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 14

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):pc

R

p

pT

0

- 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:

Page 15: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 15

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):

ss

zz

ss

z

z

ssz

1

1

Page 16: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 16

Pressure p as vertical coordinate in the hydrostatic system

Coordinate transformation rules (from z to p):

ss

zz

ss

z

z

ssz

1

1

ppp

zz

pg

pp

z

z

ppppz

1

1

=>

with geopotential

gz

p

Hydrostatic relation between p and z:

Page 17: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 17

Hydrostatic Primitive Equations with pressure as vertical coordinate

pdzgd

RT

p

p

c

Q

cDt

DT

FkfDt

D

p

p

s

z

z

s

hp

pp

rphh

ss

1

0v

vv

v

Pressure gradient replaced by geopotential gradient (at constantpressure).

Continuity eq. is a diagnostic eq. in p-coordinates. Number of prognostic variables reduced to 3 (horizontal winds & T)!

Geopotential computed from hydrostatic equation.

Dt

Dp pressure vertical velocity

gz

p

Dt

Dp

ptDt

Dph

p

,v

Page 18: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 18

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 thecontinuity 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))

0v

s

z

Dt

Ds

ss

z

s

z

t hs

s

For every s for which consts

z

=> continuity is diagnostic eq.!

Page 19: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 19

Hydrostatic pressure as vertical coordinate for a non-hydrostatic shallow atmosphere model (cont.)

Choose 1 gconst and denote with the coordinate s for which

),,,(

),,,(),,,(

tz

zdtzgtz

TT

z

z

T

T

1

gs

z

gz

gz

1i.e.

For 0TT andz

is the weight of a column of air (of unit area) above a point at height z, i.e. hydrostatic pressure.

Page 20: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 20

Hydrostatic pressure as vertical coordinate for a non-hydrostatic shallow atmosphere model (cont.)

wg

RT

p

Tc

Qp

c

pc

Dt

Dp

c

Q

Dt

Dp

cDt

DT

Fnp

gDt

Dwn

Fp

pkfDt

D

hh

h

vv

p

pp

rw

rhh

vvD

0v

D

1

1

1v

v

3

3

v

RT

p

Tc

Qpw

c

pc

Dt

Dp

c

Q

Dt

Dp

cDt

DT

Fnz

pg

Dt

Dwn

FpkfDt

D

vh

v

p

pp

rw

rzhh

,v

1

1

1v

vv

=>

D3

in Z in

Page 21: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 21

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.

sp

pFor example: traditional sigma-coordinate

(Phillips, 1957)

Page 22: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 22

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 earth’s topography.

The easiest case is a flat and rigid boundary where the boundary condition simply is: 0

Dt

Dss

1s 0s at the top at the bottom &e.g.

at the boundary.

Page 23: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 23

Terrain-following vertical coordinate (cont.)

A simple function that fulfils these conditions is

Ts

T

ts

),,(

),( ss is a monotonic single-valued function of hydrostatic pressure and also depends on surface pressure in such a way that

s1),(0),( sssT sands

T is the pressure at the top boundary.)(Where

TTs tsts )),,((),,,(

For this is the traditional sigma-coordinate of Phillips (1957)0T

s

The ECMWF model uses a terrain-following vertical coordinate based on hydrostatic pressure. The principle will be explained based on hydrostatic pressure :

Page 24: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 24

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-pressurecoordinates

),(),,( txtx s

),( txs

x

Page 25: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 25

Hybrid vertical coordinate

),,()()(),,,( tBAt s

First introduced by Simmons and Burridge (1981).

1)1(,0)1( BA

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 thesurface 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 isdefined:

In order that the top and bottom boundaries are coordinate surfaces (=> easy application of boundary condition), A and B have to fulfil:

0)0(,)0( BA T

10

at the surface

at the top

Page 26: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 26

Comparison of sigma-coordinates & hybrid η-coordinates

sigma-coordinate η-coordinate

Coordinate surfaces over a hill for

),()()(),,( txBAtx s ),(),,( txtx s

Page 27: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 27

wg

RT

p

t

Tc

Qp

c

pc

Dt

Dp

c

Q

Dt

Dp

cDt

DT

Fnp

gDt

Dwn

Fp

pkfDt

D

hh

h

vv

p

pp

rw

rhh

11

3

3

1

v

1

vvD

0v

D

1

1

1v

v

Non-hydrostatic equations in hybrid vertical coordinate

wg

RT

p

Tc

Qp

c

pc

Dt

Dp

c

Q

Dt

Dp

cDt

DT

Fnp

gDt

Dwn

Fp

pkfDt

D

hh

h

vv

p

pp

rw

rhh

vvD

0v

D

1

1

1v

v

3

3

v

inin

prognostic continuity eq.

Page 28: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 28

Hydrostatic Primitive Equations in hybrid η vertical coordinate

UUVaUUatU cos1cos12 UUvd KPpTRafV )(ln1

VVUaVVaVUatV 222 costancos1cos1

VVvd KPpTRafU )(lncos

dp

RT

p

ppp

t

c

Q

cDt

DT

FpkfDt

D

s

s

h

pp

rhh

1

0v

1v

vv

Continuity equation is prognosticagain because the (hydrostatic) pressure is not the vertical coordinate anymore.

In addition to the geopotential gradient term, a pressure gradient term again!

Dt

Dp

ptDt

Dph

p

,v

Page 29: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 29

x

q

vds

h

v

vdhh

PDt

DX

PDt

Dq

dp

TR

ppp

t

KPpq

T

Dt

DT

KPpTRkfDt

D

ln

0v

)1(1

lnvv

1

TT

vv

Hydrostatic equations of the ECMWF operational model (incorporating moisture)

Page 30: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 30

Notations:

Dt

Dp p-coordinate vertical velocity

q specific humidity

X mass mixing ratio of physical or chemical species (e.g. aerosols, ozone)

vT virtual temperature

q

R

RTT

d

vv 11

pdppdd cccR /,/ v dR gas constant of dry air, vR gas constant of water vapour

pdc specific heat of dry air at constant pressure

vpc specific heat of water vapour at constant pressure

contributions from physical parametrizationsxqT PPPP ,,,v

TKK ,v horizontal diffusion terms

Page 31: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 31

0

pp

Vp

t h

0

pp

Vp

t h

From the continuity equation

pVdp

V hh

0

)(

dpV

d

dBp

td

dBp

Dt

Dshss )ln)(ln()(ln

1

0

dp

Vp

pt h

ss )(

1)(ln

1

0

dp

Vt

pph )(

0

1 0 0 atandatwith boundary conditions:

we can derive (by vertical integration) the following equations:

Needed for the energy-conversionterm in the thermodynamic equation

Needed for the semi-Lagrangianadvection

Prognostic equations for surfacepressure

spBAp )()( B from def. of vert. coord.

Page 32: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 32

Prognostic equations of the ECMWF hydrostatic model

dp

TR

PDt

DX

PDt

Dq

dpd

dBp

td

dBp

Dt

D

KPpq

T

Dt

DT

KPpTRkfDt

D

vds

x

q

sss

v

vdhh

ln

)lnv)(ln()(ln

)1(1

lnvv

1

h

1

0

TT

vv

These equations are discretized and integrated in the ECMWF model.

Page 33: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 33

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

Page 34: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 34

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:

Page 35: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 35

Horizontal discretisation (cont.)

ECMWF model uses the spectral transform method

Representation in spectral space in terms of spherical harmonics:

immn

mn ePY )(),( m: zonal wavenumber

n: total wavenumber

λ= longitudeμ= sin(θ) θ: latitudePn

m: Associated Legendre functions of the first kind

)()(

0;)1()1(!2

1

)!(

)!()12()( 22/2

mn

mn

nmn

mnm

nm

n

PP

md

d

nmn

mnnP

Ideally suited set of basis functions for spherical geometry (eigenfunctions of the Laplace operator).

Page 36: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 36

The horizontal spectral representation

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

m mn n

m N n m

X t X t Y

1

1

2

0

)(),,,(4

1),(

ddePtXtX imm

nmn

FFT (fast Fourier transform)usingNF 2N+1points (linear grid)(3N+1 if quadratic grid)

Legendre transformby Gaussian quadratureusing NL (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

Page 37: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 37

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 NG 0)(0 m

NGP Gaussian latitudes

NG (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:

NG (3N+1)/2 for the quadratic grid.

- Gauss-Fourier quadrature in longitude:

Page 38: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 38

The Gaussian grid

Full gridReduced grid

• Associated Legendre functions are very small near the poles for large m

About 30% reduction in number of points

Page 39: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 39

T799 T1279

50°N 50°N

Orography at T1279

10

50

100

150

200

250

300

350

400

450

500

550

600

650

684.1

50°N 50°N

Orography at T799

10

50

100

150

200

250

300

350

400

450

500

550

600

634.0

25 km grid-spacing( 843,490 grid-points)Current operational resolution

16 km grid-spacing(2,140,704 grid-points)Future operationalresolution (from end 2009)

Page 40: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 40

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

Page 41: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 41

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.

mn

mn Y

a

nnY

22 )1(

Page 42: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 42

Comparison of cost profilesat different horizontal resolutions

0

5

10

15

20

25

30

35

40

45

50

no

rmal

ized

% c

ost

T511

T799

T1279

T2047

Page 43: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 43

Cost of Legendre transforms

T511T799

T1279

T2047

Page 44: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 44

Profile for T2047on IBM p690+ (768 CPUs)

Legendre Transforms~17% of total cost of model

Physics ~36% of total cost

Page 45: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 45

L91L600.01

0.02

0.03

0.050.07

0.1

0.2

0.3

0.5

0.7

1

2

3

5

7

10

20

30

50

70

100

200

300

500

700

1000

Pres

sure

(hPa

)

60 levels 91 levels

1

2

3

4

5

6

7

8

9

10

12

14

16

18

20

25

30

35

40455055606570

91

Leve

l num

ber

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 onterrain-following pressurebased hybrid η-levels.

Page 46: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 46

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 samevertical 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).

) ( knownte and is l coordianhe verticaition of t the defincomes fromd

dB

Page 47: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 47

Prognostic equations of the ECMWF hydrostatic model

dp

TR

PDt

DX

PDt

Dq

dpd

dBp

td

dBp

Dt

D

KPpq

T

Dt

DT

KPpTRkfDt

D

vds

x

q

sss

v

vdhh

ln

)lnv)(ln()(ln

)1(1

lnvv

1

h

1

0

TT

vv

These equations are discretized and integrated in the ECMWF model.

Reminder: slide 32Reminder: slide 32

Page 48: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 48

0

pp

Vp

t h

0

pp

Vp

t h

From the continuity equation

pVdp

V hh

0

)(

dpV

d

dBp

td

dBp

Dt

Dshss )ln)(ln()(ln

1

0

dp

Vp

pt h

ss )(

1)(ln

1

0

dp

Vt

pph )(

0

1 0 0 atandatwith boundary conditions:

we can derive (by vertical integration) the following equations:

Needed for the energy-conversionterm in the thermodynamic equation

Needed for the semi-Lagrangianadvection

Prognostic equations for surfacepressure

spBAp )()( B from def. of vert. coord.

Reminder: slide 31Reminder: slide 31

Page 49: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 49

Vertical integration in finite elements

0

( ) ( )F f x dx

can be approximated as

Applying the Galerkin method with test functions tj =>

-1 AC Bc C A Bc

2 2

1 1 0

( ) ( )

K M

i i i ii K i M

C d c e x dx

Basis sets

2 2

1 1

1 1

1 2

0 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

Aji Bji

Page 50: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 50

Vertical integration in finite elements

1 -c S f

F PC

1 -1 -F P A B S f J f

Including the transformation from grid-point (GP) representation tofinite-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 computedonly once during the initialisation phase of the model and stored.

Page 51: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 51

Cubic B-splines for regular spacing of levels (Prenter (1975))

0

)(

)(3)(3)(3

)(3)(3)(3

)(

4

1)(

32

31

211

23

31

211

23

32

3

i

iii

iii

i

i

hhh

hhh

hB

otherwise

for

for

for

for

ii

ii

ii

ii

21

1

1

12

Page 52: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 52

Cubic B-splines as basis elements

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

f

Basis elementsfor the representationof the integral

F

Page 53: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 53

Benefits from using the finite-element scheme

in the vertical• High order accuracy (8th 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.

Page 54: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 54

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

Page 55: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 55

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.)

Page 56: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 56

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)

Page 57: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 57

Semi-implicit time integration

xRHSDt

DX

000 )(5.0)(5.0)( LLLRHSLLLRHSDt

DXxx

Notations:X : advected variableRHS: right-hand side of the equationL: part of RHS treated implicitlySuperscripts: “0” indicates value for explicit discretiz,

“-” indicates value at start of time step “+” indicates value at end of time step

0)(5.0 LLLLtt For compact notation define:

“implicit correction term”

L=RHS => implicit schemeL= part of RHS => semi-implicit

LRHSDt

DXttx 0=>

Benefit: slowing-down of the waves too fast for the explicit CFL cond.Drawback: overhead of having to solve an elliptic boundary-value prob.

Page 58: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 58

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.

Page 59: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 59

Semi-implicit time integration (cont.)

1

0

h

TT

vv

)v(1

)(ln

)1(1

lnvv

dp

pp

t

KPpq

T

Dt

DT

KPpTRkfDt

D

ss

v

vdhh

DRHSpt

DRHSDt

DT

pTRTRHSDt

D

ttps

ttT

srdtth

)(ln

lnγv

v

semi-implicitequations

semi-implicit corrections

Page 60: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 60

Semi-implicit time integration (cont.)

DRHSpt

DRHSDt

DT

pTRTRHSDt

D

ttps

ttT

srdtth

)(ln

lnγv

v

DRHSpt

DRHSDt

DT

pTRTRHSDt

D

ttps

ttT

srdtth

)(ln

lnγv

v

1

0

0

1

1

γ

dd

dpX

pX

dd

dpX

p

TX

dd

dp

p

XRX

r

sr

r

r

r

r

r

d

semi-implicitequations

Where:

Reference state for linearization:

rT ref. temperature

srp ref. surf. pressure

=> lin. geopotential for X=T

=> lin. energy conv. term for X=D

Page 61: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 61

Semi-implicit time integration (cont.)

DRHSpt

DRHSDt

DT

pTRTRHSDt

D

ttps

ttT

srdtth

)(ln

lnγv

v

DRHSpt

DRHSDt

DT

pTRTRHSDt

D

ttps

ttT

srdtth

)(ln

lnγv

v

semi-implicitequations

Reference state for linearization:

rT ref. temperaturesrp ref. surf. pressure

By eliminating in above system all but one of the unknowns (D+) => DD

~I 2

rdTR γ operator acting only on the vertical

I unity operator

Page 62: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 62

Semi-implicit time integration (cont.)

DD~

I 2

rdTR γ

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. =>

DDi

~1 2

One equation for each LevNi 1

In spectral space (spherical harmonics space):

mn

mni DD

a

nn ~)1(1

2

mn

mn Y

a

nnY

22 )1(

because

Once D+ has been computed, it is easy to compute the other variables at “+”.

Page 63: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 63

Semi-Lagrangian advection

Semi-Lagrangian (SL) schemes are more efficient & more stablethan Eulerian advection schemes.

Coupling SL advection with semi-implicit treatment of the fastmodes results in a very stable scheme where the timestep canbe 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)

Page 64: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 64

Semi-Lagrangian advection (cont)

x x x x

x x x x

x x x x

A

D *

xM( ) ( )

( )2

A DMt t t t

R tt

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

),( tRDt

D All equations are of this (Lagrangian) form:

(See slide 32)

Page 65: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 65

Prognostic equations of the ECMWF hydrostatic model

dp

TR

PDt

DX

PDt

Dq

dpd

dBp

td

dBp

Dt

D

KPpq

T

Dt

DT

KPpTRkfDt

D

vds

x

q

sss

v

vdhh

ln

)lnv)(ln()(ln

)1(1

lnvv

1

h

1

0

TT

vv

These equations are discretized and integrated in the ECMWF model.

Reminder: slide 32Reminder: slide 32

Page 66: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 66

Semi-Lagrangian advection (cont)

Unstable! => noisy forecasts

Two-time-level second order accurate schemes :

Forecast of temperatureat 200 hPa(from 1997/01/04)

( ) ( )( )

2

A DMt t t t

R tt

3 1( ) ( ) ( )

2 2 2

tR t R t R t twith

Extrapolation in time to middle of time interval

Page 67: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 67

Stable extrapolating two-time-level semi-Lagrangian (SETTLS):

2 2

2

( )( ) ( )

2

DA D

t AV

d t dt t t t

dt dt

Forecast 200 hPa Tfrom 1997/01/04using SETTLS

( )

DD

t

dR t

dt

2

2

( ) ( )

A D

AVAV

R t R t td dR

dt dt t

With and

Taylor expansion to second order

( ) ( ) ( ( ) {2 ( ) ( )} )2

A D A Dt

t t t R t R t R t t

Page 68: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 68

Interpolation in the semi-Lagrangian scheme

4

1

( ) ( )

i ii

x C x

4

4

)(

)(

)(

ikki

ikk

i

xx

xx

xCwith the weights

ECMWF model uses quasi-monotone quasi-cubic Lagrange interpolation

xx x xx

x x x x

x x x x

x xxx

x

x

x

x

y

x

Number of 1D cubic interpolationsin two dimensions is 5, in three dimensions 21!

To save on computations:cubic interpolation only for nearestneighbour rows, linear interpolation for rest => “quasi-cubic interpolation”=> 7 cubic + 10 linear in 3 dimensions.

Cubic Lagrange interpolation:

Page 69: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 69

Interpolation in the semi-Lagrangian scheme (cont)

x: grid points

x: interpolation pointquasi-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).

xx

x

x

x

x

interpolated cubically

maxmin

Quasi-monotone interpolation:

Has a detrimental effect on conservation, but prevents unphysicalnegative concentrations.

Page 70: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 70

shdd

s VTR

RHSDt

lD

TRl

1

][where *

=> Reduced mass loss/gain during a forecast.

Modified continuity & thermodynamic equations

][)()(ln * RHSllDt

Dp

Dt

Ds Continuity equation

Accuracy of cubic interpolation is much reduced when the fieldto be interpolated is rough (e.g. surface pressure over orography)

Idea by Ritchie & Tanguay (1996): Subtract a time-independentterm 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].

Page 71: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 71

Modified continuity & thermodynamic equations(cont)

b

bhTb T

TVRHSDt

TTD

)(][)(

TRp

T

p

ppT

d

s

refssb

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.

Page 72: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 72

Trajectory calculation

M i

j

Ai

j

Di

j

Tangent plane projection

Semi-Lagrangian advection on the sphere

X

Y

Z

A

Vx

D

Momentum eq. is discretized in vector form (because a vector is continuous acrossthe poles, components are not!)

Trajectories are arcs of great circles if constant (angular) velocity is assumedfor the duration of a time step.

Interpolations at departure point are donefor components u & v of the velocity vec-tor relative to the system of reference localat 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.

Page 73: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 73

Treatment of the Coriolis term

• Implicit treatment :0

00.5( )

h h

h h

V Vfk V V

t

• Advective treatment:

2

h

dRfk V

dt( 2

h

h h

dV dfk V V R)

dt dt

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 hereis the position vector.)

Page 74: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 74

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

Page 75: Adiabatic formulation of the ECMWF model1 Agathe Untch e-mail: Agathe.Untch@ecmwf.int (office 11)

Adiabatic formulation of the ECMWF model 75

Thank you very much for your attention