ECMWF Governing Equations 4 Slide 1 Governing Equations IV by Nils Wedi (room 007; ext. 2657) Thanks to Anton Beljaars

ECMWFGoverning Equations 4 Slide 1

Governing Equations IVby Nils Wedi (room 007; ext. 2657)

Thanks to Anton Beljaars


Introduction

Nonhydrostatic model NH - IFS

Physics - Dynamics coupling


Introduction – A historyResolution increases of the deterministic 10-day medium-range

Integrated Forecast System (IFS) over ~25 years at ECMWF:

1987: T 106 (~125km)

1991: T 213 (~63km)

1998: TL319 (~63km)

2000: TL511 (~39km)

2006: TL799 (~25km)

2010: TL1279 (~16km)

2015?: TL2047 (~10km)

2020-???: (~1-10km) Non-hydrostatic, cloud-permitting, substan-tially

different cloud-microphysics and turbulence parametrization, substantially

different dynamics-physics interaction ?


Ultra-high resolution global IFS simulations

TL0799 (~ 25km) >> 843,490 points per field/level

TL1279 (~ 16km) >> 2,140,702 points per field/level

TL2047 (~ 10km) >> 5,447,118 points per field/level

TL3999 (~ 5km) >> 20,696,844 points per field/level (world

record for spectral model ?!)


Orography – T1279Max global altitude = 6503m

Alps


Orography - T3999

Alps

Max global altitude = 7185m


H TL3999NH TL3999

Computational Cost at TL3999hydrostatic vs. non-hydrostatic IFS


Nonhydrostatic IFS (NH-IFS)

Bubnova et al. (1995); Benard et al. (2004), Benard et al. (2005), Benard et al. (2009), Wedi and Smolarkiewicz (2009),Wedi et al. (2009)

Arpégé/ALADIN/Arome/HIRLAM/ECMWF nonhydrostatic dynamical core, which was developed by Météo-France and their ALADIN partners and later incorporated into the ECMWF model and adopted by HIRLAM.


Vertical coordinate

withcoordinate transformation coefficient

hybrid vertical coordinateSimmons and Burridge (1981)

Prognostic surface pressure tendency:

Denotes hydrostatic pressure in the context of a shallow, vertically unbounded planetary atmosphere.


Two new prognostic variables in the nonhydrostatic formulation

‘Nonhydrostaticpressure departure’

‘vertical divergence’

Three-dimensional divergence writes

With residual residual

Define also:


NH-IFS prognostic equations

‘Physics’


Diagnostic relations

With


Auxiliary diagnostic relations


Numerical solution

Advection via a two-time-level semi-Lagrangian numerical technique as before.

Semi-implicit procedure with two reference states with respect to gravity and acoustic waves, respectively.

The resulting Helmholtz equation is more complicated but can still be solved (subject to some constraints on the vertical discretization) with a direct spectral method as before.

(Benard et al 2004,2005)


Hierarchy of test cases

Acoustic waves

Gravity waves

Planetary waves

Convective motion

Idealized dry atmospheric variability and mean states

Idealized moist atmospheric variability and mean states

Seasonal climate, intraseasonal variability

Medium-range forecast performance at hydrostatic scales

High-resolution forecasts at nonhydrostatic scales


Spherical acoustic wave

Tim$ ( 100.000)

-0.030 -0.018 -0.006 0.006 0.018 0.030pr$ss d$partur$

1000.00

100.00

10.00

1.00

0.10

0.01

pr$

ssur$

Tim$ ( 100.000)

-0.030 -0.018 -0.006 0.006 0.018 0.030pr$ss d$partur$

1000.00

100.00

10.00

1.00

0.10

0.01

pr$

ssur$

-0.0

04

-0.0

04

-0.0

02

-0.0

02 0.0

04

0.0

04

0.0

04

0.0

04

50°S50°S

40°S 40°S

30°S30°S

20°S 20°S

10°S10°S

0° 0°

10°N10°N

20°N 20°N

30°N30°N

40°N 40°N

50°N50°N

140°W

140°W 120°W

120°W 100°W

100°W 80°W

80°W 60°W

60°W 40°W

40°W

Friday 15 Octob$r 2004 12UTC ECMWF For$cast t+10000 VT: Tu$sday 6 D$c$mb$r 2005 04UTC Mod$l L$v$l 91 **Exp$rim$ntal product

0.001-0

.008 - 0

. 002

- 0. 0

02

50°S50°S

40°S 40°S

30°S30°S

20°S 20°S

10°S10°S

0° 0°

10°N10°N

20°N 20°N

30°N30°N

40°N 40°N

50°N50°N

140°W

140°W 120°W

120°W 100°W

100°W 80°W

80°W 60°W

60°W 40°W

40°W

Friday 15 Octob$r 2004 12UTC ECMWF For$cast t+10 VT: Friday 15 Octob$r 2004 22UTC Mod$l L$v$l 91 **Exp$rim$ntal product

0.001

Tim$ ( 100.000)

-0.030 -0.018 -0.006 0.006 0.018 0.030pr$ss d$partur$

1000.00

100.00

10.00

1.00

0.10

0.01

pr$

ssur$

Tim$ ( 100.000)

-0.030 -0.018 -0.006 0.006 0.018 0.030pr$ss d$partur$

1000.00

100.00

10.00

1.00

0.10

0.01

pr$

ssur$explicit

implicit

analytic

NH-IFS

horizontal vertical

C ~ 340m/s


Orographic gravity waves H - IFS


Orographic gravity waves – NH - IFS


“Scores”

Population: 45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,45 (av$rag$d)M$an calculation m$thod: standard

Dat$: 20070301 12UTC to 20081101 12UTCS.h$m Lat -90.0 to -20.0 Lon -180.0 to 180.0

Anomaly corr$lation for$cast500hPa G$opot$ntial

M$an curv$s

0 1 2 3 4 5 6 7 8 9 10For$cast Day

30

40

50

60

70

80

90

100

110

f35d nh-ifs

f354 h-ifs



Root m$an squar$ $rror for$cast500hPa G$opot$ntial

M$an curv$s

0 1 2 3 4 5 6 7 8 9 10For$cast Day

0

20

40

60

80

100

120

f35d nh-ifs

f354 h-ifs

TL1279 L91 ~ 16 km

NHH



Anomaly corr$lation for$cast500hPa G$opot$ntial

M$an curv$s

0 1 2 3 4 5 6 7 8 9 10For$cast Day

30

40

50

60

70

80

90

100

110

f35d nh-ifs

f354 h-ifs



Root m$an squar$ $rror for$cast500hPa G$opot$ntial

M$an curv$s

0 1 2 3 4 5 6 7 8 9 10For$cast Day

0

20

40

60

80

100

120

f35d nh-ifs

f354 h-ifs


Physics – Dynamics coupling

‘Physics’, parametrization: “the mathematical procedure describing the statistical effect of subgrid-scale processes on the mean flow expressed in terms of large scale parameters”, processes are typically: vertical diffusion, orography, cloud processes, convection, radiation

‘Dynamics’: “computation of all the other terms of the Navier-Stokes equations (eg. in IFS: semi-Lagrangian advection)”

The ‘Physics’ in IFS is currently formulated inherently hydrostatic, because the parametrizations are formulated as independent vertical columns on given pressure levels and pressure is NOT changed directly as a result of sub-gridscale interactions !

The boundaries between ‘Physics’ and ‘Dynamics’ are “a moving target” …


Different scales involved

NH-effects visible


Single timestep in two-time-level-scheme


Cost partition of a single time-step

Note: Increase in CPU time

substantial if the time step

is reduced for the ‘physics’

only.


dynamics-physics coupling

gwdragvdifcloudconvradcloudconvrad

t

PPPP

tOgtPgttPgtP

PRGGt

FF

2

1

2

1

))((),(),(2

1),(

2

1

02/1

2022

2/1

2/12/100

!!!box black anot is P


Noise in the operational forecasteliminated through modified coupling


Wrong equilibrium ?

, ( ) , .

correct steady state solution:

TD P P T gT g const

t

DT

g


Compute D+P(T) independant

1

11

1.

2. (1 )

add 1. 2. together and seek steady state solution:

explicit 0 :

implicit ( 1 : (1 ),

n nn

n nn n

n

n

T T TD D

t t

T T TgT g T T

t t

D(γ ) T

g

Dγ ) T g t

g

wrong!


Compute P(D,T)

1

11

1.

2. (1 )

seek steady state solution of 2. :

explicit 0 :

implicit ( 1 : ,

n nn

n nn n n

n

n

T T TD D

t t

T T TD gT D g T T

t t

D(γ ) T

g

Dγ ) T

g

correct!


Sequential vs. parallel split of 2 processesvdif + dynamics

12 15 18 21 24 27 30 33 36Forecast step (hours)

0

5

10

15

U (

m/s

)

parallel split (ej4k)sequential split (ej4n)bad sequential split (ej4x)sequential split, dt=5 min (ej4m)

(90 W, 60 S) T159 forecasts 2002011512, dt=60 min

parallel split

sequential split

A. Beljaars


Negative tracer concentration – Vertical diffusion

Negative tracer concentrations noticed despite a quasi-monotone advection scheme

(Anton Beljaars)


Physics-Dynamics couplingVertical diffusion

Single-layerproblem

(Kalnay and Kanamitsu, 1988)

dynamics positive definite


Physics-Dynamics couplingVertical diffusion

Two-layerproblem

Not positive definite depends on !!!

dynamics positive definite

50 55 60Model level

-1e-14

0

1e-14

2e-14

3e-14

4e-14

Aer

osol

con

cent

ratio

n

old time levelafter dynamics only new time level

72.3N/2.5E

50 55 60Model level

-1e-14

0

1e-14

2e-14

3e-14

4e-14

Aer

osol

con

cent

ratio

n

old time levelafter dynamics only new time level

72.3N/2.5E

(D+P)t+t

Dt+t

(D+P)t

= 1.5

= 1

Anton Beljaars

Negative tracer concentrationwith over-implicit formulation

Documents

ECMWF Governing Equations 4 Slide 1 Governing Equations IV by Nils Wedi (room 007; ext. 2657) Thanks to Anton Beljaars