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: