A Two-Box Climate Model: EPcm
Environmental Physics 2007-2008
Model documentation –v2
(February 2008)
http://www.sp.ph.ic.ac.uk/~arnaud/EP_ClimateModel.html
Motivation
The following is a simple Two-Box Climate model, designed for pedagogical purposes.
It aims at predicting the time dependent response of Tropical and Extra-Tropical surface
temperatures to a given time-dependent change in atmospheric greenhouse gas
concentration. The model includes a representation of atmospheric and oceanic heat
transport and storage, and a single positive feedback: the water vapour feedback. The
model
is
coded
with
MATLAB
(see
http://www.sp.ph.ic.ac.uk/~arnaud/EP_ClimateModel for download)
References
-Emanuel, K., 2002: A simple model of multiple climate regimes, J. Geophys. Res., vol
107.
-Czaja A., and J. C. Marshall, 2006: The partitioning of heat transport between the
ocean and atmosphere, J. Atm. Sci., vol 63, 1498-1511.
Structure
1. Model variables
2. Model equations
3. Model control simulation
4. 2XCO2 experiments
5. MATLAB code
Version 2 improvements
-Time dependent atmospheric CO2 concentration
-New parameterization of atmospheric circulation strength
-New graphics
-Boolean variable for Water Vapour Feedback turned on/off
Model variables
Average atmospheric temperature (whole column of air)
Ta 
1
Ps

Ps
0
TdP
in which Ps  105 Pa is surface pressure.
Average oceanic temperature (upper ocean)
To 
1
ho
0
 Tdz
 ho
in which ho  500m is the warm layer (“thermocline”) thickness.
Surface temperature (assumed to be ocean)
Ts  To  To in which To  5K measures a fixed (vertical) oceanic temperature gradient
Atmospheric and oceanic heat transport
H A  C A A TA & H A  CO O TO
in which C A,O is a heat capacity,  A,O measures the strength of the circulation and T is the
temperature difference between Tropics (Eq-30N) and Extra-Tropics (30N-90N).
TA1
Atmosphere
HA
TS 1
HO
TA2
TS 2
TO1
TO 2
Tropics
Extra
Tropics
Ocean
Model Equations
Top of the atmosphere radiative fluxes (Box 1&2)
FT 1  TE41  ( 1TA41  (1   1 )TS41 )
[ FT ]  Wm 2
FT 2  TE42  ( 2TA42  (1   2 )TS42 )
1  1  e (CO q ) with q1  1000  RH 1qsat (
2
1
TS 1  TA1
,750mb)
2
 2  1  e (CO q ) with q2  1000  RH 2 qsat (
2
2
TS 2  TA1
,750mb)
2
Surface fluxes (vertical convection parameterization)
FS1  (TS1  TA1  TZ ) if (TS1  TA1  TZ )  0 ,
FS 1  0 otherwise
[ FS ]  Wm 2
FS 2  (TS 2  TA2  TZ ) if (TS 2  TA2  TZ )  0 , FS 2  0 otherwise
Atmospheric circulation strength (diffusive parameterization)
 A  K A (TS1  TS 2 )
[ A ]  kgs 1
Heat transports (Ocean & Atmosphere)
H A  C A A (TA1  TA2 )
[H A ]  W
FA  H A / R 2
[ FA ]  Wm 2
H O  CO O (TO1  TO 2 )
[HO ]  W
FO  H O / R 2
[ FO ]  Wm 2
Energy conservation for Atmosphere (Box 1&2)
CA
PS dTA1
 ( FT 1  FS 1 )  FA
g dt
CA
PS dTA2
 ( FT 2  FS 2 )  FA
g dt
Energy conservation for Ocean (Box 1&2)
 O CO hO
dTO1
 ( FS 1  FO )
dt
 O CO hO
dTO 2
  FS 2  FO
dt
Surface temperature (Box 1&2)
TS1  TO1  TO
TS 2  TO 2  TO
Control Climate parameter values
Ocean:
Thickness of thermocline layer
ho  500m
Vertical temperature gradient
To  5K
Atmosphere:
Relative humidity (Box 1&2)
RH 1  RH 2  0.5
Effective heat capacity
C A  2C p  2000Jkg 1 K 1
Critical vertical temperature gradient
TZ  40K
Circulation strength parameter
K A  100 / 15 SvK 1
Emissivity parameter for water vapour
  1.25
Emissivity parameter for carbon dioxyde
  1.5  10 3
Carbon concentration
CO2  280 ppm
Ocean / Atmosphere coupling:
 O / A  0.25
Ratio of mass transport
Solar input:
Emission temperature (Box 1)
TE1  268K
Emission temperature (Box 2)
TE 2  240K
Fudge factor
Large number for convective parameterizations
  100
Control Climate results
Surface temperature of Tropics
TS1  298.56 K
Surface temperature of Extra-Tropics
TS 2  282.38K
Global surface temperature
TS  290.47K
Atmospheric circulation strength
 A  108  109 kgs1
Atmospheric heat transport
H A  3.29PW
Oceanic heat transport
H O  1.74 PW
Total Heat transport
H A  H O  5.04 PW
2XCO2 Experiment minus Control
Surface temperature of Tropics
TS1  2.17 K
Surface temperature of Extra-Tropics
TS 2  2.38K
Global surface temperature
TS  2.28K
Atmospheric circulation strength
 A  20  109 kgs1
Atmospheric heat transport
H A  0.6PW
Oceanic heat transport
H O  0.35PW
Total Heat transport
( H A  H O )  0.95PW
2XCO2 Experiment minus Control (no water vapour feedback)
Surface temperature of Tropics
TS1  1.45K
Surface temperature of Extra-Tropics
TS 2  1.59 K
Global surface temperature
TS  1.52 K
Atmospheric circulation strength
 A  14  109 kgs1
Atmospheric heat transport
H A  0.43PW
Oceanic heat transport
H O  0.24 PW
Total Heat transport
( H A  H O )  0.67 PW
Implied water vapour feedback
f H 2O  1  1.52 / 2.28  0.33
MATLAB code
Generalities
The Two-Box climate model is modular. The main program is TwoBox_Main.m. In it, you decide the
length of integration and the model forcing (emission temperatures TE1 , TE 2 ; CO2 concentration). The
other important routine is TwoBox_Param.m in which you set the value of the model parameters.
These are the two main routines you are likely to change when playing with the model. Some
experiments are already stored on the website and can be used to initialize the model (MATLAB
binary files):
(i)
Output_CTRLv2.mat (the model control simulation as described in this document)
(ii)
Outputv2_2XCO2step_nonoise.mat: a 300-yr long integration starting from the control
state with a step increase of CO2 to 560ppm.
(iii)
Outputv2_2XCO2ramp25yr_nonoise.mat: a 300-yr long integration starting from the
control state with a slow increase of CO2 up to 560ppm.
(iv)
Outputv2_2XCO2step_nonoise_nowavafdk.mat: same as (ii) but with the water vapour
feedback artificially removed.
(v)
Outputv2_2XCO2ramp25yr_nonoise_nowavafdk.mat: same as (iii) but with the water
vapour feedback artificially removed.
Model Architecture:
TwoBox_Main.m
TwoBox_Param.m
set model parameters
TwoBox_Init.m
initial conditions before time integration (default is CTRL run)
TwoBox_Run.m
time-loop over Nt timesteps
calc_EPSA.m
compute emissivities
calc_Psi.m
compute strength of atmospheric & oceanic circulations
calc_Fs.m
compute net surface heat flux
calc_Ft.m
compute net TOA radiative flux
calc_Fa.m
compute Atmospheric heat transport
calc_Fo.m
compute Oceanic heat transport
TwoBox_Diagnos.m
a few diagnostics (global average, etc)
TwoBox_Plots.m
displays model outputs
TwoBox_Save.m
store model outputs in MATLAB binary file
Running the model
Just decide the length of integration and the CO2 forcing in TwoBox_Main.m and then type
>> TwoBox_Main
on the MATLAB window. Enjoy!