Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 1/27
Numerical modeling, weather predictability and forecast
Philippe Drobinski
Laboratoire de Météorologie Dynamique
Ecole Polytechnique
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 2/27
Outline
� Elements of numerical modelingModeling turbulent flow
Subgrid scale parameterization
Initial and boundary conditions
� Theories on atmospheric predictabilityLorenz attractor
Initial condition problem
Sensitivity to parameterization
� Weather forecast and climate predictionA historical perspective
Deterministic and ensemble forecasts
Weather forecast and climate projections
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 3/27
∆=100 km
Large-scale numerical weather prediction
(NWP) models
∆=10 km
Meso-scale numerical weather prediction
(NWP) models
∆<2 km
Large-eddy simulation
(LES) models
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
Which scales are resolved?
Modeling turbulent Subgrid scale parameterizationInitial and boundary conditions
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 4/27
'uu~u iii +=
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
Separation into large and small scales
Modeling turbulent Subgrid scale parameterizationInitial and boundary conditions
Reynolds averaged equations: ξ = ξ + ξ’
j
ij
j3ij
ij
ij
i
xuf
x
p
x
uu
t
u
∂τ∂
−ε+∂∂
−=∂∂+
∂∂
j
j
j
*
j
V
pj
jx
q
x
QEL
C
1
xu
t ∂∂
−
∂∂
+ρ
−=∂
θ∂+∂θ∂
'u'u jiij =τ
''uq jj θ=
SGS stress
SGS flux
(SGS: sub-grid scale)
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 5/27
Limited-area models (LAM)General circulation models (GCM)
Types of models
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
Discretization of the conservation equations (partial differential equations)
Finite difference method Finite element methodFinite volume methodBoundary element method
Finite element method suited for complex geometryFinite difference method easy to implement
More details are out of the scope of this course
Modeling turbulent Subgrid scale parameterizationInitial and boundary conditions
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 6/27
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
Modeling turbulent Subgrid scale parameterizationInitial and boundary conditions
15103rd order
1062nd order
631st order
Number of unknowns
Number of equations
EquationMomentPrognostic equation for:
j'j
'ii x/uut/U ∂∂−=∂∂ L
k'k
'j
'i
'j
'i x/uuut/uu ∂∂−=∂∂ L
m'm
'k
'j
'i
'k
'j
'i x/uuuut/uuu ∂∂−=∂∂ L
iU
'j
'iuu
'k
'j
'i uuu
Need for closure equations referred as parameterization
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 7/27
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
Modeling turbulent Subgrid scale parameterizationInitial and boundary conditions
Data assimilation principle:Estimate the atmospheric state of day D-1 (first guess)Perform a simulation from day D-1 yesterday to day DQuantify model/measurement differenceModify the first guess in order to decrease the differenceRe-iterate while necessary
Reconstruct 3D meteorological information from incomplete dataset � data assimilation
Data assimilation
3D state of the atmosphere is needed to provide initial and boundary conditions to the NWP
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 8/27
Numerical approximation of convection equation
Boussinesq equations
Lorenz attractorInitial condition problemSensitivity to parameterization
( )[ ]
Θ
∂∂+
∂∂κ=Θ
∂∂+
∂∂ν+−α−−
∂∂
ρ−=
∂∂+
∂∂ν+
∂∂
ρ−=
=∂∂+
∂∂
2
2
2
2
2
2
2
2
0
0
2
2
2
2
0
zxdt
d
wzx
TT1gz
p1
dt
dw
uzxx
p1
dt
du
0z
w
x
u
∂∂−
∂∂
∂∂+
∂∂ν+
∂Θ∂α=
∂∂−
∂∂
z
u
x
w
zxxg
z
u
x
w
dt
d2
2
2
2
( )[ ]00 TT1 −α−ρ=ρ
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
Boussinesq equations with
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 9/27
ψ
∂∂+
∂∂σ+
∂θ∂σ=ψ
∂∂+
∂∂
θ
∂∂+
∂∂+
∂ψ∂=θ
2
2
2
2
2
2
2
2
2
2
2
2
2
zxxzxdt
d
zxxR
dt
d
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
( ) ( )L
Ha;numberRayleigh
HgR;numberandtlPr
3
=κν
∆Τα=
κν=σ
Non-dimensional equations
Solutions in a periodic box of size 2L×2H (Saltzman, 1962)
with
( ) ( ) ( )
( ) ( ) ( )∑
∑+π
+π
θ=θ
ψ=ψ
n,m
nzamxi
n,m
n,m
nzamxi
n,m
etˆt,z,x
etˆt,z,x
Lorenz attractorInitial condition problemSensitivity to parameterization
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 10/27
bZXYt
Z
YrXXZt
Y
YXt
X
−=∂∂
−+−=∂∂
σ+σ−=∂∂
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
Lorenz model (Lorenz, 1963; Saltzman 1962 simplification)
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )z2sintZzsinaxcos2tYt,z,xR
zsinaxsin2tXt,z,x1a
a
1
c
2
π−ππ=θπ
ππ=ψ+−
Equilibrium states
1a
4b;
R
Rr;
2
c +==
κν=σwith
Perturbation equations
( ) 1rZ;1rbYX
0ZYX
−=−±==
===
( )
−−−−
σσ−=
∂∂
0
0
0
0
0
0
z
y
x
bXY
X1Zr
0
z
y
x
t
a problem with 3 degrees of freedom
Unstable for σ=10,b=8/3 and r=28
Lorenz attractorInitial condition problemSensitivity to parameterization
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 11/27
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
Lorenz attractorInitial condition problemSensitivity to parameterization
Ensemble forecast on Lorenz attractor
Example of how forecast uncertainty can vary depending on the location of the initial state
http://www.wmo.ch/pages/prog/www/DPS/TC-DPFS-2002/Papers-Posters/Keynote-Richardson.pdf
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 12/27
( )
( ) 2
2
2
6
222
1
6
111
q,Jt
q
q,Jt
q
ψ∇κ−ψ∇ν−=ψ+∂
∂
ψ∇ν−=ψ+∂
∂
Quasi-geostrophic model
Haidvogel and Held (1980)
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
Lorenz attractorInitial condition problemSensitivity to parameterization
x
y
« First » numerical weather prediction model
Anticyclones
Cyclones
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 13/27
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
Initial error growth – Lyapunov exponent – predictability time
( )qft
q=
∂∂ ( )
( ) qq0tq
q0tq
0
0ref
δ+====
2 simulations: qref, q
Omrani et al. (2011)
Lyapunov exponent λPredictability time τP ~ 1/λ
( ) ( ) t
ref e~tqtq λ−
( ) ( ) ?tqtq ref =−
qref(t) q(t)≠≠≠≠t>>τp
Lorenz attractorInitial condition problemSensitivity to parameterization
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 14/27
Measured fluxes
Eddy diffusion
Non linear
Mixed
Evaluation of closure model sensitivity in off-line mode
( )
k
j
k
i2
nl
ij
2
S
mix
ij
x
u~
x
u~C
S~
S~
C2
∂∂
∂∂∆+
∆−=τ
Non linear models
Non-linear models are unstable (simulations blow up) when applied aloneThey are used in linear combination with eddy-diffusion model
k
j
k
i2
nl
nl
ijx
u~
x
u~C
∂∂
∂∂∆=τ
Mixed models
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
Lorenz attractorInitial condition problemSensitivity to parameterization
( ) ij
2
sij S~
S~
C2 ∆−=τEddy-diffusivity model
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 15/27
Bjerknes, V. (1904). Das Problem der Wettervorhersage, betrachtet vom Standpunkte der Mechanik und der Physik. Meteorologische Zeitschrift, 21, 1-7 Vilhelm Bjerknes
1862-1951
If, as every scientifically inclined individual believes, atmospheric conditions develop according to natural laws from their precursors, it follows that the necessary and sufficient conditions for a rational solution of the problems of meteorological prediction are:
the condition of the atmosphere must be known at a specific time with sufficient accuracythe laws must be known, with sufficient accuracy, which determine the development of one weather condition from another.
Lewis Fry Richardson1881-1953
First numerical integration … by hand in 1922 by L.W. Richardson � complete failure. Richardson estimates that a real-time weather forecast needs 30000 people making calculation simultaneously
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
A historical perspectiveDeterministic and ensemble forecastsWeather forecast and climate projections
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 16/27
Hans Ertel1904-1971
H. Ertelfinds out the potential vorticity and its conservation.develops the quasi-geostrophic theory of mid-latitude atmospheric dynamics.
J.G. Charney, R. Fjørtoft, and J. von Neumann, 1950: Numerical integration of the barotropic vorticity equation. Tellus, 2, 237–254
First « successfull » numerical weather forcast with « general circulation model » in 1950 at ENIAC (J. Charney, P. Thompson, L. Gates, R. Fjörtoft).
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
A historical perspectiveDeterministic and ensemble forecastsWeather forecast and climate projections
Jule Gregory Charney1917-1981
Ertel, H., 1942: Ein neuer hydrodynamischerErhaltungssatz. - Meteorol. Z. 59, 277–281
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 17/27
Group photo of the 1st International Symposium on Numerical Weather Prediction held in Tokyo on 78-13 November 1960 (Syono, 1962)
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
A historical perspectiveDeterministic and ensemble forecastsWeather forecast and climate projections
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 18/27
Two similar initial states can lead to different forecasts !
(Lorenz, 1982)
Growth of forecast error for operational 10-day forecast at ECMWF avearged over a 100-day period). Global root-mean-square 500-hPa height difference (m) between j-day and k-day forecast.
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
A historical perspectiveDeterministic and ensemble forecastsWeather forecast and climate projections
Deterministic forecast
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 19/27
Importance of observations (inaccurate and unevenly spaced)
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
A historical perspectiveDeterministic and ensemble forecastsWeather forecast and climate projections
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 20/27
Visual analysis of initial conditions
Production of new initial conditions
Example of Klaus windstorm (January, 23rd, 2009)
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
A historical perspectiveDeterministic and ensemble forecastsWeather forecast and climate projections
Importance of the forecasters
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 21/27
Example of Klaus windstorm (January,
23rd, 2009)
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
A historical perspectiveDeterministic and ensemble forecastsWeather forecast and climate projections
Impact on meteorological forecast
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 22/27
(Shapiro and Thorpe, 2004)
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
A historical perspectiveDeterministic and ensemble forecastsWeather forecast and climate projections
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 23/27
Ensemble forecast
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
A historical perspectiveDeterministic and ensemble forecastsWeather forecast and climate projections
Distribution of initial condition uncertainty
Deterministic forecast
Real state of the atmosphere
Forecast of the uncertainty
Principle
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 24/27
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
A historical perspectiveDeterministic and ensemble forecastsWeather forecast and climate projections
Methods
Multi-analyses (analyses from different weather services, e.g. Météo-France, ECMWF, NCEP; perturbed initial conditions)
Multi-models
Need for a synthetic probabilitic representation
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 25/27
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
A historical perspectiveDeterministic and ensemble forecastsWeather forecast and climate projections
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 26/27
Climate = statistical distribution of timeA climate model is a tool for scientific investigation for:
• understanding of the past and current climates on Earth, Mars, Venus ...• investigating the causes of its variations (forcings / internal variability) • producing projections for the future
• Providing an accurate forecast up to 10 days (typical predictability time) does not mean we can not produce reliable projections for the future, which are statistical representation of the atmospheric and/or oceanic states
Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction
A historical perspectiveDeterministic and ensemble forecastsWeather forecast and climate projections
Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 27/27
Next time…
Thank you for your attentionThank you for your attention………… any questions?any questions?