1. Introduction to Climate Modeling
Zhengyu Liu

3. Why we need a Climate Model?
Quantitative Assessment of Complex System
Numerical Laboratory, especially important for geophysical field
Most credible strategy for prediction

4. Outline
Part I: A Conceptual Climate Model Radiative Equilibrium Model Prognostic Coupled Ocean-Atmosphere Model Part II: Challenges in Numerical Climate Model Numerical Stability Parameterization Interaction

5. Part I: Global mean model (0-D) Why the Earth surface temperature is about 15oC (288K)?
Equilibrium Model
Prognostic Model

6. Radiative Equilibrium ModelS
Cloud S
(1-)S
TS4
(1-)S
GHG
T4
T4
S=342Wm2
Qs = S -  T4 =0
T = (S/ )1/4 ,
T =279oK=6oC,
Too cold! ,
Qs = (1-a)S -  T4 =0
T = [(1-a)S/ ]1/4
T =255oK= -18oC
Even colder
QTOA = (1-a)S -  T4 = 0
Qs = (1-a)S +  T4 -  TS4 =0
T= ((1-a)S/ )1/4
TS =21/4T=288oK=15oC
About right…
b=dT/dS ~ 1/(4 T 3)
Climate Sensitivity

7. Climate Sensitivity
To explain the ice age, Arrhenius estimated that halving of CO2 would decrease temperatures by 4 - 5 °C (Celsius) and a doubling of CO2 would cause a temperature rise of 5 - 6 °C. In his 1906 publication, Arrhenius adjusted the value downwards to 1.6 °C (including water vapour feedback: 2.1 °C). Recent (2007) estimates from IPCC say this value (the Climate sensitivity) is likely to be between 2 and 4.5 °C. Arrhenius expected CO2 doubling to take about 3000 years; it is now estimated in most scenarios to take about a century.
Svante Arrhenius

8. Prognostic Coupled Climate System
To4
Atmosphere
Ocean
Equilibrium Model

9. in Numerical Climate Modeling
Part II: Challenges
in Numerical Climate Modeling
C1. Numerical Stability
C2. Physical Parameterization
C3. Interaction Among Components

10. C1: Numerical Stability I: Temporal Descritizationtn
tn+1 Tn
Tn+1
Stability Condition  time step limit
Implications:
too large time step leads to model “blow up”
(also poor model accuracy:
2) Very small time step is needed to simulate faster process (large a) e.g. atmosphere, and even more, small scale clouds…..

11. Implication to Coupled Climate System
Atmosphere
Ocean
Equilibrium Model
Full Model
Intermediate Model
Expensive!

12. A Numerical Demonstration

13. 3-D Climate model

14. Atmospheric General Circulation

15. The Coupled Ocean-Atmosphere SystemEQ
Transient Eddies
Stationary Waves
S. Pole
N. Pole
N. Atlantic THC ~ decadal-centennial
Southern Ocean THC ~ millennial
Thermocline circulation ~ decadal
Hadley Cell
The Coupled Ocean-Atmosphere System

16. General Circulation Model (GCM): Equations
Momentum Equations
Mass Equation
Heat Equation
Equation of State
where

17. General Circulation Model: Spatial Descritization
atmosphere
ocean

18. C1: Numerical Stability II: Spatial- Descritization
Energy balance model
Stability Condition
Implication
Double resolution requires quadrupling temporal resolution!
So, the computation is increased 2(dy)*4(dt)=8 ~ 10 times
( for 3-D model, x, y, z… , 2(dx)*2(dy)*2(dz)*4(dt)=32 times !)
Conclusion
Very expensive for high resolution modeling!

19. C2: Physics Parameterization: Subgrid-scale physics
Current Climate Model
General circulation (resolved)
Synoptic storms (resolved)
Clouds (parameterized)
Droplets (….)
Condensation nucli (…)
……….

20. Computational Implication on parameterizationS
T4
To4
Atmosphere
Ocean
Coupled Climate System
Full Model
Intermediate Model
parameterization

21. C3: Interdisciplinary Interaction:
Land
3.8°
Atmosphere
Sea Ice
x3°
Ocean
C3: Interdisciplinary Interaction:
Earth System Modeling
Earth System Model

22. Tropical Bias Observation Model (original) Model
(solar penetration parameterization)

23. Model-Data Comparison of Last Glaical Maximum Climate
LGM Climate
Obs SST
CLIMAP SST
Model SST
Model SST
Model Tair
Model Tair
1) CO2 vs. ice sheet
2) Proxy uncertainty vs. model uncertainty
Liu et al., 2002, GRL

24. C3: Earth System Model Physical Model +Ecological Model
+Biogeochemical Model
+…….
No perfect model!
A model for a purpose!

25. The End

26. Radiative Equilibrium ModelS
Qsurf = S -  T4 =0
T c= (S/ )1/4 , S=342 Wm2  T c =279oK=6oC,
Too cold! ,

27. Cloud Albedo Effect: Radiative Equilibrium ModelS
Cloud S
(1-)S
Qsurf = (1-a)S -  T4 =0
T cc= [(1-a)S/ ]1/4 ,  T cc =255oK= -18oC
Even colder

28. Greenhouse Effect Qsurf = (1-a)S +  Tg4 -  T4 =0
glass
Tg4
Qsurf = (1-a)S +  Tg4 -  T4 =0
QTOA = (1-a)S -  Tg4 = 0
Tg= ((1-a)S/ )1/4 = Tcc=255K ,
 Tcg =21/4Tg=288oK=15oC
About right…

29. How does the climate respond to global warming forcing?
CO2 induced Radiative Forcing
Climate Sensitivity

30. CO2 induced Radiative Forcing
RF= 5.25 ln (CO2) W/m (S. Arrhenius, 1900)
Examples: Present relative to 1850 (CO2 =250ppm)
RF=5.25 *ln (385 / 250 ppm) = ~2.5 W/m2
Doubliing CO2
RF=5.25 *ln (500 / 250 ppm) = ~4 W/m2
Climate Sensitivity
ΔT=b*RF
b = climate sensitivity!
= increase in temperature per unit increase
in radiative forcing

31. Climate Sesnsitivity and Global Warming Response
RF(CO2)
T4 S
Global warming prediction:
Qsurf = S +RF-  T4 =0
T = [(S+RF)/ ]1/4 ~= (S/ )1/4 +b*RF = Tc +b*RF, here RF<<S
or global warming ΔT= T- Tc = b*RF
Climate sensitivity: b=d (S/ )1/4 /dS = 1/(4 Tc 3)=0.2 K / Wm-2 
Double CO2 : ΔT = b*RF = 0.2 * 4 = 0.8oK, small??

32. The Role of Climate Model
Quantitative Assessment of Complex System
Numerical Laboratory, especially important for geophysical field
Most credible strategy for prediction