Frank Selten, Global Climate Dept, KNMI - Universiteit...
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Frank Selten, Global Climate Dept, KNMI Climate modelling
Transcript of Frank Selten, Global Climate Dept, KNMI - Universiteit...
Frank Selten, Global Climate Dept, KNMI
Climate modelling
Frank Selten, Global Climate Dept, KNMI
Climate modelling
What is a climate model ?
A climate model is a quantitative mathematical description of aspects of the earth’s climate system
Purpose of a climate model
A climate model is used to simulate the behavior of the earth’s climate system in order to
increase our understanding of how the climate system workssimulate past climate fluctuations to help interpret historical and paleo observationssimulate the future climate based on scenario’s for emissions of green house gasses
Present state
Specify external forcings according to the date
Model calculates rate of change (tendency)
State one timestep (30 minutes) later
Principle of a climate simulation
Earth system: our natural environment
ecosphere
anthroposphere
socio-economics
values
beliefs
biosphere geosphere
attitudes
atmospherehydrospherecryosphereastenosphere
vegetationsea biology
Earth system: our natural environment
ecosphere
anthroposphere
socio-economics
values
beliefs
biosphere geosphere
attitudes
atmospherehydrospherecryosphereastenosphere
vegetationsea biology
Energy Balance Model
Energy balance model
Total incoming radiation = Total outgoing radiation
Solar radiationThermal radiation
σT4=annual and global mean absorbed solar radiation ε
Energy balance model
σT4=annual and global mean absorbed solar radiation ε
πR S (1-α) = 4πR εσT2 40
R :S :α :ε :σ :
0
2
radius of the earth (6378 km)solar constant (1370 W/m )2
average albedo (0.32)emissivity (≈ 1)Stefan-Boltzmann constant (5.67E-8 W/m /K )2 4
T = -20 degree Celsius
Without / with Greenhouse Effect: -20 / 14 degrees Celsius
EBM: the Greenhouse effect ...
CO2H20
more
CO2
Earth is warming ...
time →1902
year
summer
autumn
spring
winter
Colder than averageWarmer than average
2002
Also in De Bilt ...
And Carbon Dioxide is to blame ...
The global energy balance ...
Δ temperatureΔ radiation
Current ‘best’ estimate:
around 0.8 K/W/m2
or
3K for CO2 doubling
Role of feedbacks
Climate sensitivity:
The warming is not uniform ...
The physical climate system ...
Global climate models solve the (thermo-)dynamical equations on a computational grid
Current resolutions: 100 km and 40 layers in the vertical
Global climate models solve the (thermo-)dynamical equations on a computational grid
Current resolutions: 100 km and 60 layers in the vertical
Primitive (hydrostatic) equations
forMomentum equations
Sub-grid model :“physics”
Numerical diffusion
hydrostatic equations
Thermodynamic equation
Moisture equation
Note: virtual temperature Tv instead of T from the equation of state.
hydrostatic equations
Continuity equation
Vertical integration of the continuity equation in hybrid coordinates
Resolution problem• So far : We derived a set of evolution equations
based on 3 basic conservation principles valid at the scale of the continuum : continuity equation, momentum equation and thermodynamic equation.• What do we want to (re-)solve in models based on
these equations?
grid
scal
e
reso
lved
scal
e (?
)
The scale of the grid is much bigger than the
scale of the continuum
Text
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~10−6 m - 1m
~107 m ~105 m
~103 m
The planetary scaleCloud cluster scale
Cloud scaleCloud microphysical scale
The climate system : A truly multiscale problem
10 m 100 m 1 km 10 km 100 km 1000 km 10000 km
turbulence Cumulusclouds
Cumulonimbusclouds
Mesoscale Convective systems
Extratropical Cyclones
Planetary waves
Large Eddy Simulation (LES) Model
Cloud System Resolving Model (CSRM)
Numerical Weather Prediction (NWP) Model
Global Climate ModelSubgrid
No single model can encompass all relevant processes
DNS
mm
Cloud microphysics
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Grid-box size is limited by computational capability
Processes that act on scales smaller than our grid box will be excluded from the solutions.
We need to include them by means of parametrization (a largely statistical description of what goes on “inside” the box).
Similar idea to molecules being summarized statistically by temperature and pressure, but much more complex!
Parametrization
What is a climate model ?
Parameterization of unresolved processes biggest source of model errorsDifferent climate models differ in these descriptionsHow to optimally combine the predictions of the different models?
A multi-model ensemble method that combines imperfect models through
learningLeonie van de Berge, Mathematical Institute UtrechtFrank Selten*, Royal Netherlands Meteorological InstituteWim Wiegerinck, Radboud University NijmegenGreg Duane, University of Colorado
Earth System Dynamics, 2011
*Global Climate dept: 12 staff 14 postdocs/phd students Climate research: 120 scientists
Multi model simulation of real complex systems
Reality
• climate system• ecological systems• human brain• organisms• economic systems
Model 1 Model 2 μοδελ 4
Observed data
Model 1data Model 2 data Model 3 data μοδελ 4 data
Combined data
Usually some form of a weighted average
{
An ensemble of “imperfect models”
Model 3
Climate models
• order 20 global coupled climate models
• have improved performance over time
• but are not perfect
• are used to simulate response to different scenarios of future emissions of greenhouse gasses
• differ in the simulation of the response
Coupled Model Intercomparison Project
Performance metricBased on mean squared errors in time mean global temperatures, winds, precipitation, ....
1995
2006
= index value based on multi model mean fields: outperforms individual models: why ?
Error in annual mean surface air temperaturesmulti model mean over all CMIP3 simulations
IPCC 2007
Spread in simulated climate change
IPCC 2007
Given present state of modeling ......
...... is this the best we can do ?
I have an idea !!!
Multi model simulation of real complex systems
Reality
Model 1 Model 2 μοδελ 4Model 3
Observed data
Model 1data Model 2 data Model 3 data μοδελ 4 data
Combined data
Usually some form of a weighted average
An ensemble of “imperfect models”
Model 3
Exchange information between models while integrating
Reality
Model 1 Model 2 μοδελ 4
Observed data
Model 1data Model 2 data Model 3 data μοδελ 4 data
Combined data
multi model averaging
An interacting ensemble of “imperfect models”
Model 3
How ?
• Example using the chaotic Lorenz 1963 model
Lorenz 1963 model
has for standard parameter values:
a chaotic solution:
Perfect model approach
• model with standard parameter values ⇒ truth
• perturb parameter values to create an ensemble of imperfect models
• exchange of information between the imperfect models takes the form of linear “nudging terms”
Interacting ensemble of imperfect models
where k indexes the imperfect models
Effectively a new dynamical system is created, “a super model”,
with adjustable connection coefficients C
For k=3 we have 18 coefficients
Use data from the truth to learn the connection coefficients
Minimize Cost function:
Imperfect models unconnected
Model 1: fixed point Model 2: fixed point Model 3: strange attractor
“a” supermodel solutionfrom two different view points
The three connected models fall into an approximate synchronized motion
Model 1 Model 2 Model 3
Put differently: the models form a consensus
Synchronization of chaotic systems is a well-known phenomenon
The vertical discretisation is given in Fig. 5. The vorticity equation is applied to the200 (level 1), 500 (level 2) and the 800 hPa level (level 3), the heat equation is applied tothe 650 and 350 hPa level. Combined these two equations yield an equation expressingthe conservation of quasi-geostrophic potential vorticity (PV) in the absence of forcingand dissipation. In discretised form it reads
q̇1
= �J ( 1
, q1
)�D1
( 1
, 2
) + S1
q̇2
= �J ( 2
, q2
)�D2
( 1
, 2
, 3
) + S2
q̇3
= �J ( 3
, q3
)�D3
( 2
, 3
) + S3
, (19)
where the index i = 1, 2, 3 refers to the pressure level. Here PV is defined as
q1
= r2 1
�R�2
1
( 1
� 2
) + f
q2
= r2 2
+ R�2
1
( 1
� 2
)�R�2
2
( 2
� 3
) + f
q3
= r2 3
+ R�2
2
( 2
� 3
) + f(1 +h
H0
), (20)
where R1
(=700 km) and R2
(=450 km) are Rossby radii of deformation appropriate tothe 200-500 hPa layer and the 500-800 hPa layer, respectively and H
0
is a scale height.In eqns.( 19), D
1
, D2
, D3
are linear operators representing the e↵ects of Newtonian relax-ation of temperature, linear drag on the 800 hPa wind (with drag coe�cient dependingon the nature of the underling surface), and horizontal di↵usion of vorticity and temper-ature. The temperature relaxation has a radiative time scale of 25 days, the linear dragdamps the low-level wind on a spin down time scale of 3 days over the oceans, about 2days over low-altitude land and about 1.5 days over mountains above 2 km; a (stronglyscale-selective) horizontal di↵usion damps harmonics of total wavenumber 21 on a 2-daytime scale.The PV source terms S
i
are calculated from observations as the opposite of the averagePV tendencies obtained by inserting observed daily winter time stream function fieldsinto a version of eqns. (19) in which these terms are omitted.A Galerkin projection of eqns. (19) onto a basis of spherical harmonics truncated attotal wavenumber 21 leads to a system of 1449 coupled ordinary di↵erential equationsfor the 483 coe�cients of the spherical harmonical functions at the three levels. Simi-lar as for the T21 barotropic model (see section 1.5) the spectral transform method isimplemented to evaluate the quadratic interaction terms in order to save computationtime. In this case the restriction to modes with m + n odd is not made and the modelsimulates the flow at the Southern hemisphere as well.
The mean state and the variability are surprisingly realistic as compared to the ob-served wintertime flow in the Northern hemisphere Corti et al (1997). The simulation isless realistic in the summer hemisphere. In addition, the model displays regime behaviornot unlike the observations (Selten and Branstator, 2004). The issues for super modelingin the context of this model concern the density and form of the connections and theability of the super model to reproduce the regime behavior.
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A global atmosphere model
where Pm,n
(µ) denote associated Legendre polynomials of the first kind. The Legendrepolynomials are normalized in the same way as in Machenhauer (1991). The sphericalharmonics are eigenfunctions of the Laplace operator:
�Ym,n
(�, µ) = �n(n + 1)Ym,n
(�, µ) (12)
The expansion is triangularly truncated at wavenumber 21:
(�, µ, t) =21X
n=1
+nX
m=�n
m+n=odd
m,n
(t)Ym,n
(�, µ) (13)
The restriction to modes with m + n odd excludes currents across the equator. Thismakes the model hemispheric. After the Galerkin projection, the discretized model con-sists of 231 coupled ordinary di↵erential equations for the coe�cients of the sphericalharmonics. The orography is also projected onto the spherical harmonics and truncatedat T21. In principle the set of ODE’s has the same structure as the 6D barotropic modelin the previous section (eqns. 8) with constant, linear and quadratic terms. However,the number of quadratic interaction terms makes tendency evaluations of this T21 modelcomputationally expensive. A cheaper solution is implemented to calculate the quadraticterms on a computational grid in physical latitude-longitude space using fast transfor-mation routines to transform the spectral representation to the grid representation andvice-versa (see Machenhauer (1991)).We used a dataset of 10 winters of ECMWF daily analysis of 300 hPa vorticity fields
to calculate the forcing. The fields were multiplied by a factor of 0.6 to approximate the500 hPa level. The dataset covers the winter months December, January and Februaryfrom 1981 to 1991. The scale height is set to 10 km and A
0
is given a value of 0.2.The Ekman damping time scale is set to 15 days and the strength of the scale selectivedamping is such that wavenumber 21 is damped at a time scale of 3 days. The forcingis calculated from the ECMWF dataset according to:
⇣⇤ = J ( cl
, ⇣cl
+ f + h) + k1
⇣cl
� k2
�3⇣cl
+ J ( 0, ⇣ 0) (14)
where cl
, ⇣cl
is the observed climatological winter mean state computed from the 10winters and 0, ⇣ 0 are deviations of the 10 days running mean from the climatologicalmean. The model climatic mean will resemble the observed climatological winter meanstate if it is capable of producing a mean transient eddy forcing similar to this observedlow frequency transient eddy forcing. With this forcing field, we integrated the modelfor 2000 days and analysed its variability. It turned out that the model variability wasrather weak compared to the observed low frequency variability. From this model runwe determined the model transient eddy forcing, with respect to the observed clima-tological mean state, and replaced in ⇣⇤ the observed transient eddy forcing with themodel transient eddy forcing. We integrated the model a second time for 2000 daysand calculated its transient eddy forcing again. We integrated the model a third time
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Solved by the spectral method using spherical harmonics as basis functions
The vertical discretisation is given in Fig. 5. The vorticity equation is applied to the200 (level 1), 500 (level 2) and the 800 hPa level (level 3), the heat equation is applied tothe 650 and 350 hPa level. Combined these two equations yield an equation expressingthe conservation of quasi-geostrophic potential vorticity (PV) in the absence of forcingand dissipation. In discretised form it reads
q̇1
= �J ( 1
, q1
)�D1
( 1
, 2
) + S1
q̇2
= �J ( 2
, q2
)�D2
( 1
, 2
, 3
) + S2
q̇3
= �J ( 3
, q3
)�D3
( 2
, 3
) + S3
, (19)
where the index i = 1, 2, 3 refers to the pressure level. Here PV is defined as
q1
= r2 1
�R�2
1
( 1
� 2
) + f
q2
= r2 2
+ R�2
1
( 1
� 2
)�R�2
2
( 2
� 3
) + f
q3
= r2 3
+ R�2
2
( 2
� 3
) + f(1 +h
H0
), (20)
where R1
(=700 km) and R2
(=450 km) are Rossby radii of deformation appropriate tothe 200-500 hPa layer and the 500-800 hPa layer, respectively and H
0
is a scale height.In eqns.( 19), D
1
, D2
, D3
are linear operators representing the e↵ects of Newtonian relax-ation of temperature, linear drag on the 800 hPa wind (with drag coe�cient dependingon the nature of the underling surface), and horizontal di↵usion of vorticity and temper-ature. The temperature relaxation has a radiative time scale of 25 days, the linear dragdamps the low-level wind on a spin down time scale of 3 days over the oceans, about 2days over low-altitude land and about 1.5 days over mountains above 2 km; a (stronglyscale-selective) horizontal di↵usion damps harmonics of total wavenumber 21 on a 2-daytime scale.The PV source terms S
i
are calculated from observations as the opposite of the averagePV tendencies obtained by inserting observed daily winter time stream function fieldsinto a version of eqns. (19) in which these terms are omitted.A Galerkin projection of eqns. (19) onto a basis of spherical harmonics truncated attotal wavenumber 21 leads to a system of 1449 coupled ordinary di↵erential equationsfor the 483 coe�cients of the spherical harmonical functions at the three levels. Simi-lar as for the T21 barotropic model (see section 1.5) the spectral transform method isimplemented to evaluate the quadratic interaction terms in order to save computationtime. In this case the restriction to modes with m + n odd is not made and the modelsimulates the flow at the Southern hemisphere as well.
The mean state and the variability are surprisingly realistic as compared to the ob-served wintertime flow in the Northern hemisphere Corti et al (1997). The simulation isless realistic in the summer hemisphere. In addition, the model displays regime behaviornot unlike the observations (Selten and Branstator, 2004). The issues for super modelingin the context of this model concern the density and form of the connections and theability of the super model to reproduce the regime behavior.
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t = 0
s, q
T,t
s, q
T,t
s, q
0 hPa
200 hPa
350 hPa
500 hPa
650 hPa
800 hPa
Ps
0
1
2
3
4
5
Figure 5: Vertical discretisation of the three-level model.
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-6
-4
-2
0
2
4
0 10 20 30 40 50
strea
mfun
ction
time [days]
model 1 NLmodel 2 NLmodel 1 SAmodel 2 SA
Synchronization between two identical baroclinic spectral T21QG models on the sphere
Streamfunctions at 500 hPaat initial time after 60 days
dq2mn
dt= .......− ci q2
mn − q1mn( )
dq1mn
dt= .......− ci q1
mn − q2mn( )
Super model solutions are not unique:cost function F(C) has isolated local minima
Cross sections of the cost function
• convergence for increasing size of training set• for some connection constants, the cost
function is flat: family of solutions
But the solutions differ in quality
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But the solutions differ in quality
Can the super model simulate climate change?
since super model is trained on present day climate ...
Doubling the parameter ρ from 28 to 58: supermodel simulates change well
Imperfect models Lorenz 1984
Model 1: periodic orbit Model 2: fixed point Model 3: periodic orbit
x: strength of westerlies
y,z: sine and cosine phase of a wave
“a” supermodel solutionfrom two different view points
Questions• Are other forms of the connections more effective?
• How many connections are required ?
• Which variables to be connected and how often ?
• How much data is needed for the learning ?
• Are there more effective learning strategies ?
• How to handle the slow oceanic time scales ?
• What if reality falls outside of the model class ?
• Does the supermodel also perform well in a changing climate ?
Questions• Can we identify identical state variables in the
different models ?
• Instead of connecting state variables, is connecting the physical tendencies a good idea?
• Do balances and conservation laws place restrictions on the connections ? Similar issues play a role in data-assimilation ...
• Is it computationally feasible to run an interconnected ensemble of climate models?
• Is it possible to choose connections on the basis of insight, without learning ?
• .........
• Can the super models synchronize with the truth?
But the solutions differ in quality
Nudging to observations
• Imperfect models do not synchronize•Perfect model synchronizes for n=3•Super models synchronize for n=11 and n=13•Distance between model and observations vary with time:
But the solutions differ in quality
