Dynamic Systems Sea-Level Rise Model: Interpretation of the Forcing Function

Jiabao Guan, Biao Chang and Mustafa M. Aral Multimedia Environmental Simulations Laboratory School of Civil and Environmental Engineering Georgia Institute of Technology

Climate Change / Sea-level Rise DATA on: Temperature Sea-Level Rise Radiative forces (CO2 Emissions)

Global Temperature Change:

Global Sea-Level Rise:

Data on CO2 Emissions:

Sea Level Rise (m at 2090-2099 relative to 1980-1999)

Case Model-based range excluding future rapid dynamical changes in ice flow

B1 scenario 0.18 – 0.38 A1T scenario 0.20 – 0.45 B2 scenario 0.20 – 0.43 A1B scenario 0.21 – 0.48 A2 scenario 0.23 – 0.51 A1FI scenario 0.26 – 0.59

The Summary for Policy Makers (SPM) released recently provide the following table of sea level rise projections (IPCC4th Framework Report, 2007):

The IPCC Study:

IPCC estimates:

Semi-Empirical Models: Rahmstorf’s Study (Science, Vol. 315 pp.19, 2007) and others:

{ }0.5 1.4m−

Semi-Empirical Models: Rahmstorf’s Study (Science, Vol. 315 pp.19, 2007) and others.

Sea-level Rise Interval Predicted above 1990 level:

IPCC Interval Predicted above 1990 level:

{ }0.09 0.88m−

Major limitations of the previous empirical models

No feedback of SLR on temperature change

Zero-dimensional, thus does not capture spatial variations in SLR

Impact of external forcing is not considered

A Dynamic Systems Model

Earlier studies showed that the relationship between T & H is linear.

Our Hypothesis: (Dynamic Systems Model):

Temperature:

Sea-Level:

( )1 1, , ,T f T H U c=

( )2 2, , ,H f T H U c=

Expected Relationship:

11 12 1

21 22 2

( ) ( ) ( )

( ) ( ) ( )

dT t a T t a H t cdt

dH t a T t a H t cdt

= + +

= + +

Initial Model Proposed:

(Simplified)

( ) ( ) ( ),d t dT t dH tdt dt dt

τ =

X

( ) ( )d t tdt

= +X AX C

11 12

21 22

a aa a

=

A 1

2

cc

=

C

Initial Model Proposed:

(Simplified)

Initial Model Proposed:

(Simplified)

Dynamic Systems Model:

( )tU

( )tXddtX

( )tU

( )tX ddtX

System

( )

( )

11 12 13 1

21 22 23 2

( ) ( ) ( )

( ) ( ) ( )

i ii

i ii

dT t a T t a H t a U t cdt

dH t a T t a H t a U t cdt

= + + +

= + + +

∑

∑

Proposed Model:

more flexible; may answer more questions; may provide control analysis perspective; and, hopefully will be more useful. potential drawback may require more data.

The Proposed model is:

( ) ( ) ( ) ( )d t t t tdt

= + + +X AX BU C w

11 12

21 22

a aa a

=

A 11 12 1

21 22 2

m

m

b b bb b b

=

B

1

2

cc

=

C

Proposed Model:

( 1) ( ) ( ) ( )k k k k+ = + + + 1XΦX ΓU Ω w

; ;t t t= + ∆ = ∆ = ∆Φ I A Γ B Ω C

Discrete form of the Proposed Model:

( ) ( ){ }* minimizei

i i i iF τ= − −φ

YΛφ Y Λφ

LSM Model to determine aij , bij & ci :

Confidence Interval:

( )( )( )( )

12/2, 4

12/2, 4

ˆ ˆ ˆ ˆˆ( ) ( ) 1 ( ) ( )

ˆ ˆ ˆ ˆˆ( ) ( ) 1 ( ) ( )

CI n T

CI n H

T k T k t Z k Z k

H k H k t Z k Z k

τ τα

τ τα

σ

σ

−

−

−

−

= ± +

= ±

+

Λ Λ

Λ Λ

(Wadsworth, 1998)

Application:

2-year moving average outcome is used for

both state variables.

Temperature Data:

Sea-Level Data:

Resulting Matrix Coefficients:

System System matrix Control matrix

Constant vector

Discrete

Continuous

0.75042 -0.000530.41015 0.99568

0.002450

0.857440.25863−

-0.24958 -0.000530.41015 -0.00432

0.002450

0.857440.25863−

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Tem

pera

ture

(o C)

ObservationReconstructionConfidence interval

1880 1900 1920 1940 1960 1980 2000Year

-20

-15

-10

-5

0

5

10

Sea

-leve

l ris

e (c

m)

ObservationReconstructionConfidence interval

(a)

(b)

Global CO2 Impact: Calibration

R2 = 0.80

R2 = 0.97

-1

-0.5

0

0.5

1

1.5

2

2.5Te

mpe

ratu

re (o C

)

ObservationProjectionConfidence interval

1880 1900 1920 1940 1960 1980 2000 2020 2040 2060 2080 2100Year

-20

-10

0

10

20

30

40

50

60

70

Sea

-leve

l ris

e (c

m)

ObservationProjectionConfidence intervalRahmstorf's results

(a)

(b)

Global CO2 Impact: Prediction for “2 oC” Scenario (Scenario is designed to limit global warming in 2100 2oC above the temperature in 2000 )

(Hansen et al., 2000)

1880 1900 1920 1940 1960 1980 2000 2020 2040 2060 2080 2100Year

0

200

400

600

800

1000

1200

CO

2 e

mis

sio

n (

pp

m)

SCENARIOSA1FIA1BA1TA2B1B2

IPCC Global CO2 Emission Scenarios:

-1

0

1

2

3

4

5

6

Tem

pera

ture

(o C

)

ObservationA1FIA1BA1TA2B1B2

1880 1900 1920 1940 1960 1980 2000 2020 2040 2060 2080 2100Year

-20

0

20

40

60

80

100

120

Sea

-leve

l ris

e (c

m)

ObservationA1FIA1BA1TA2B1B2

(a)

(b)

Global CO2 Impact: Comparison of results

Solid lines are Dyn. Sys. Model results. Dashed lines are IPCC results.

For example: If we want to restrict the global temperature and sea-level rise to zero-growth, the global CO2 emission should be controlled with the relationship given by

Decision Making on Global CO2 Emissions:

2( ) 101.87 ( ) 0.22 ( ) 349.98COu t T t H t= + +

2( )COu t represents the amount of yearly CO2 emission (ppm).

where

maral@ce.gatech.edu

MESL @ GT