Carbon Cycle: Definition of the problem
Inez Fung
Mean Meridional Circulation + Convection
June July August
Dec Jan Feb
90S 90N
pres
sure
Intertropical Convergence Zone (ITCZ):
•v=0: barrier tointerhemispheric transport
•Convection: rapid turbulent mixing
Atm Carbon Model
Atm _transport mixingz 0
SourcesSinChem P
kr od
s
SC ( )t
C P=+
∂+ +
∂ℑ =
oa aoextrapolation" well know of spars
ba ae bs
bon "
LFF a ( F F )ndUse ( F F )S−
= + −−+ +
Atm Carbon Models
Atm _transport mz 0
Sourixing cesSinks
SC ( Ct
)+
=∂
+ =∂
ℑ
Kalnay Eq 5.3.19
b i 1 a iT ( t ) M [T ( t )]+ =
Nychka Lecture 2
i 1 i( x ) ux G( )Φ+ = +
An Atm Carbon Cycle Model
z 0SourcesSinksAtm _transport mixing
boa ao a ab
S
S
( C )
( FLandUse ( F F
C
))
t
FF F+
=ℑ
− −
∂+ =
∂
= + + +
What we’ve got:• Sources/Sinks S known approximately or not well
constrained• Cobs (actually mixing ratios Xobs) biweekly, at ~100
stations near the surface• “Decent” transport model (winds, turbulent mixing)What we want:• where has the fossil fuel CO2 gone? {Better estimates
of the magnitude and distribution of S (e.g. land exchange)}
• How did the fossil fuel CO2 get there? {improved understanding and representation of processes, e.g.
• Fab=LUE*AvailableLight; Fba=exp(αT);
An Atm Carbon Cycle Model
z 0SourcesSinksAtm _transport mixing
boa ao a ab
S
S
( C )
( FLandUse ( F F
C
))
t
FF F+
=ℑ
− −
∂+ =
∂
= + + +
Why:• Closing the carbon budget: a challenging
scientific problem• Where to put new measurements {design optimal
observing network}• Need to predict future evolution of CO2 and climate
(global warming)• Policy: how to manage carbon (reduce sources
versus enhance sinks){how to avoid dangerous future climate}
• Policy: monitor protocol compliance
The Data: CO2 in the lower troposphereFung: Cobs, Xobs.. Kalnay: yo; Nychka: Z or Y
• In situ: tower, continuous • Flask: 2m, twice weekly• >400m Tall Tower: 11, 30, 76, 122,
244 and 396 m; Continuous• Other obs (Doney lecture Thursday’
Miller lecture Friday)
• secular trend• N-S gradient• Seasonal cycle• Interannual variations
Not-so-confusing Terminology
" well known"
Atm _transport mixing
oa aoextrapolation of sparse obs
z 0Sourc
ba ab
esSinks
Land
( C )Ct
F (F Use FS F )
S
( FF )−
=+
∂+ =
∂
= −+ + +
ℑ
−
Sources = fluxes into atmSinks = fluxes out of atm S: Fluxes = forcing term = Sources – Sinks
The Priors:
" well known"
Atm _transport mixing
oa aoextrapolation of sparse obs
z 0Sourc
ba ab
esSinks
Land
( C )Ct
F (F Use FS F )
S
( FF )−
=+
∂+ =
∂
= −+ + +
ℑ
−
Interhemispheric Mixing:Two-Box Model
N S N SN S
N SNN
S N SS
N S
N S
(
M MM St
M
M M ) M
M M2
M2 ( S S ) 0 @ SteadyState
M MS
t
t
S S
τ
τ
τ
τ
∂ − −= −
−∂= − +
∂∂ −
= + +
−
∂
=
=∂
−
+ −
MNMS
SNSS
Interhemispheric exchange time determined from inert tracers (e.g. CFC, with Ss=0): ~1-2 years
2-Box Model Applied to the Carbon Cycle
N S N S
N S
column colu
N
mnN S
sConsider the case S 6 PgC/yr; S 01 yr
Recall i1 PgC 0.5 ppmv 1 PgC 1 p
f mixed in entire atm. if mixed in a hemisphere.
M M ( S S )2
M M 3 PgC
3 p
pm
p
mv
v
τ
Χ Χ
τ= =
=→
→
− = −
− =
→
− =
→
sfc sfcN S
Guess (3D model) surface gradient 1.5x column mean gra
4.5 ppmv
dient
Χ Χ− =→
MNMS
SNSS
Atmospheric COAtmospheric CO22 distribution as simulated by NCAR distribution as simulated by NCAR CCM: Fossil fuel combustion (6 CCM: Fossil fuel combustion (6 PgCPgC/y)/y)
Latitude
Pres
sure
(mb)
Surface
Zonal mean
Eq NPSP
4 5 6 78
2-Box Model Applied to the Carbon Cycle
sfc sfcN S
N S
N s
N S
N Ss
o
fc sfcN
bs
S
M M ( S S )2
M M 3 PgC
4.5 ppm
S 6 PgC/yr; S 01
But ( ) 2.5 v
r
p
v
y
pm
τ
Χ
Χ
τ
Χ
Χ
− = −
− =
→
− =
− =
= =
=→
Forward problem: If 100% FF CO2 remained in atm
N SN S
N S
N S
obs
( M M )S S sources sinks
t( M M )
3 PgC/yrt
sources = 6 PgCSinks +Sinks = 3 Pg r
/yrC/y
∂ += + = −
∂
→
∂ +=
∂
Obs only 50% of FF CO2 remains in atm
2-Box Model Applied to the Carbon Cycle
column columnN S obs
sfc sfcN
N S
N N
N S N S
N SN S
S
S S
obs
( ) 1.7 ppmvM M 1.7 PgC
(sources sin ks ) (sources sin ks ) 3.4 PgC/
Model: M M ( S S )2
M MInvert model S S 2 3.4 P
Given:
yr(6 PgC/y
( ) 2.5
gC/y
r si
r
ppmv
n
τ
τ
Χ Χ
Χ Χ→ − =
→ − =
− = −
−→ − = =
−
−
=
− − =
−
N S
N Sks ) (0 sin ks ) 3.4 PgC/yrsinks sin ks 2.6 PgC/yr
−
=
=
−
−
→
Inverse problem
Obs Carbon Budget N SSinks +Sinks = 3 PgC/yr
Where are the Carbon Sinks?N S
N S
N S
sinks sin ks 3 PgC/yrsinks sin ks 2.6 PgC/yr
Budget Gradi
sinks 2.8PgC/yr;ent
si
n ks 0.2 PgC/yr
+ = +
− =
→ = =
Northern sinks > Southern Sinks !!!!!!!
“Data/Obs”: Huge C sink in the large expanse of southern ocean; but large uncertainty in obs
N ocn “better observed” large Northern land sink!!!
Now what?increase the number of sink terms: NLand, NOcn, SLand, Socn (or more – see David Baker exercise)
NL
N NL NO
NL SNL L
NL NL
L
N
S
NL
L
"basis function
X forward _model
Sink Sink Sink
Sink Sink
X X , etc; do same for source
( Sink
s
)
"Ψ Ψ
Ψ
=
= +
= × + ×
= ×
ksrc / sink
ksrc / sink
22k ,priorstn,obs
2 2stn src / sinks
k k
stn
k
k
tn k,prior
kks
k
k
tnk
Src / Sink
X X
Variational Approach: Find to m
X forward_model( Src / Sink )
X
inimize
( )( X )J ,
S
X
H( X ) H
where
X ) (a(
Ψ
Ψ
Ψ
Ψ
Ψ
Ψ
σ σ
=
= ×
= ×
−−= +
= ×=
∑
∑
∑ ∑
∑ pply Observations Operator)
Generalize: