Institute of Computational Mathematics and Mathematical Geophysics SD RAS,Novosibirsk

Mathematical modelsfor ecological prognosis,

design and monitoringV.V. Penenko

What is the role of atmospheric chemistry in amplifying or damping climate change?

How will human activities transform the dynamical and chemical properties of the future atmosphere?

How will quality of life change?

System organizationof environmental modeling

Models of processes•hydrodynamics•transport and transformation of pollutants

Data basesModels of observations

FunctionalsQuality, observations,restrictions, control, cost,etc.

extended:functional +model as integral identity

Solution of forward problems Solution of adjoint problems

Calculation of sensitivity functionsand variations of functionals

Analysis of sensitivity relationsrisk/vulnerability, observability, sources

System of decision making, design

Identification of parameters,decrease of uncertainties,data assimilation, monitoring

Analysis of the climatic system for constructionof long-term scenarios:

• Extraction of multi- dimentional and multi-component factors from data bases

• Classification of typical situations with respectto main factors

• Investigation of variability

• Formation of “leading” spaces

Approaches and tools

Scenario approach

Models of hydrodynamics

Models of transport and transformation of pollutants (gases and aerosols)

Sensitivity and observability algorithms

Combination of forward and inverse techniques

Joint use of models and data

Nested models and domains

M a t h e m a t i c a l m o d e l

0

rfY),( G

tB ,

000

0 YY, ;

)( tD i s t h e s t a t e f u n c t i o n ,

)(Y tD i s t h e p a r a m e t e r v e c t o r . G i s t h e “ s p a c e ” o p e r a t o r o f t h e m o d e l

A s e t o f m e a s u r e d d a t a m , m o n m

tD ,

mm H )]([

i s t h e m o d e l o f o b s e r v a t i o n s .

,,r, a r e t h e t e r m s d e s c r i b i n gu n c e r t a i n t i e s a n d e r r o r s o f t h ec o r r e s p o n d i n g o b j e c t s .

,01

u

i

i

Fxx

mlvdtdu

,01

v

i

i

Fyy

nludtdv

,0

i

iRT

.0)(

ii Lt

TTBTB ppppp ;/)(;/)(

SBBSB pppppp );1/()(;/)(

)( 1D

)( 2D

Model of hydrodynamics

ya

myxa

nxmn

F ayiaxi

i

sa

a

F aii

va

1

amav

ynau

xmnaL i

iii

)(

Transformation of moisture and pollutants

Gases and aerosols

•interaction with underlying surface •dry and wet deposition•condensation and evaporation•coagulation

Model of atmospheric chemistryModel of aerosol dynamicsModel of moisture transformation•water vapour •cloud water•rain water

Model of transport and transformation

if i s t h e s o u r c e t e r m

iS )( i s t h e t r a n s f o r m a t i o n o p e r a t o r ,};0{ DxttD t .

Boundary and initial conditions

)(x,,)( tii tqR (x))(x, 00 .

0)(x,)(

)gradu(div

iii

iii

rtfSt

Li

{ , , , , , 1, }v c r iq q q c m is the function of pressure

Model of aerosol dynamics

1 1 1 11 10

( ) 1( ) ( , ) ( )

2

q N Ni

ik k km mk m

qq K q q q q q dq

t

)()(),()( qrq

dqqqqKq ii

q N

kkiki

M

1

0 111

),()()( tqQqRqq iiiii 2

2

11 NNi ;,

},),,({ Nitqi 1 -concentration of particles in volume ;, qqq

),( 1qqK - coagulation kernel;),( tqri - rates of condensation and evaporation;),( tqi - coefficients of diffusive change of particles;),( tqRi - removal parameter;),( tqQi -source term;

},,,{,,, Nmkiikikik 1-parameters of collective interaction of particles

Hydrological cycle of atmospheric circulation for studying aerosols

If supersaturation -->condensation

rcv qqq ,, - content of water vapor, cloud water and rain water in respectively

Notations:

RCP - autoconversion of cloud water to rain water (*dt)

RAP - accretion of cloud droplets by rain drops (*dt)

REP - evaporation of rain water(*dt)

CONP - condensation (evaporation)(*dt)

3gm

?0cq ?0cc qq 0RCP

0

0

RA

RC

P

P

Hydrological cycle

RC

calculate

P

REcalculate P ?0rq ?0rq RAcalculateP

0RAP

?cRARC qPP c

RC RA

redistribute q

between P and P?rRE qP

rRE qP 0REP

no no

no

yes

no

yes yes

yes

yes

yes

output

input

no

yes

G e n e r a l f o r m o f f u n c t i o n a l s

KkdDdttF

tD

kkk ,...,,)(x,)()( 1

kF a r e t h e f u n c t i o n s o f t h e g i v e n f o r m ,

dDdtk a r e R a d o n ’ s m e a s u r e s o n tD , )(*

tk D .

Q u a l i t y f u n c t i o n a l s

KkdDdttMtD

kmT

mk ,...,,)(x,)()()( 1

“ M e a s u r e m e n t ” f u n c t i o n a l s

mtmkmk

D

K

k

DdDdttt

x,)x(x)(x,)( 1

R e s t r i c t i o n f u n c t i o n a l s

0 )(x,(,)(x, tNt k d i s t r i b u t i v e c o n s t r a i n t s

dDdttkk

D

kk

t

)(x,))()(()(

Functionals of measurements

dDdttxHMH mm

T

Dmm

t

),()()()( 111

dDdttxM mm

T

D

mm

t

),()( 222

2 1( ) ( ) ( , ) , 1,t

m c o

D

x t dDdt nr

2 2( ) ( ) ( , ) , 1,t

m c o

D

x t dDdt nr

, if 0( )

0, otherwise

A AA

The structure of the source term

K

kkkk kXxtQf

1

1),,()(

)(tQksource power

),( kk Xx source shape

kX reference point of the source

Particular case

))((),( tXxXx kkk

Functionals for assessment of source parameters

tD

aT

a dtdDffMffJ 11111 )()(

dtMQQJ ak

K

k

t

k 122

1 0

2 )(

dtMXXJK

k

takk 13

1 0

2

3

dtMXXdtd

JK

k

takk 14

1 0

2

4 )(

0)(

,)0(

:0

dtXXd

XX

takka

kk

T h e v a r i a t i o n a l f o r m u l a t i o n o f t h e p r o b l e mI n t e g r a l i d e n t i t y

0 )f,Y),(()Y,,( G

tBI

)(),( tt DD ),( ba i s i n n e r p r o d u c t o n ],0[ tDD t .

0)Y,,( I ( e n e r g y b a l a n c e e q u a t i o n )

C o n s t r u c t i o n o f t h e d i s c r e t e a p p r o x i m a t i o n s

)()(~ hk

h htD

hI )Y,,( i s e x t e n d e d f u n c t i o n a l .

S t a t i o n a r y c o n d i t i o n s

)(allfor,~

* ht

hhh

DI

0 ( d i r e c t p r o b l e m )

)(allfor,~

ht

hh

D

0 ( a d j o i n t p r o b l e m s )

E x t e n d e d f u n c t i o n a l f o r t h e m a i n a l g o r i t h m o f i n v e r s e m o d e l i n g

ht

mt D

TD

Thk

h MM r)(r)(.)()(~221150

hDR

TD

Tht

hh MM)(

)()( 4433

htD

hI )Y,,(

)4,1(, iM i a r e t h e w e i g h t m a t r i c e s ,

1,04

1

i

ii a r e t h e w e i g h t c o e f f i c i e n t s ,

, a r e t h e s o l u t i o n s o f t h e d i r e c t a n d a d j o i n t p r o b l e m s

0 )Y,,( hI

The basic algorithm of inverse modeling

0

rfY),(~

ht

hk GB

0

kkT

kT

t

hk dAB

Y),()(~

)),(.)(( 1150 Md Th

kk

0

ttk (x)001

300 tM ka ),(x,

),(x,)r(x, * tMt k1

2 ka M 14

YY

)Y,,(Y

k

hk I

0

Y),(Y),( hGA

t i s t h e a p p r o x i m a t i o n o f t i m e d e r i v a t i v e s

I n i t i a l g u e s s : aa YY,,r )()()( 00000 0

0

)Y,Y,(Y),()( kh

khk I

0

)Y,Y,(

Y kh

k I

The main sensitivity relations

The algorithm for calculation of sensitivity functions

}{ kik are the sensitivity functions}{Y iY are the parameter variations

NiKk ,1,,1

NNNdtdY

k ,,1,

The feed-back relations

Factor analysis ( global scale). Reanalysis 1960-1999

hgt 500, june

West Siberia region

60-105 E, 45-65 N

June, 1960-1999

1960 1970 1980 1990

0.1555

0.156

0.1565

0.157

0.1575

0.158

0.1585

0.159

Eigenvector N1 (97%), June, Western Siberia1960 1970 1980 1990

0.156

0.1565

0.157

0.1575

0.158

0.1585

0.159

0.1595

Eigenvector N1 (97%), June, Eastern Siberia

West Siberia, 97%

Global, 17%

Eigenfunction N1, June, 1960-1999

Novosibirsk

East Siberia Region

90-140 E, 45-65 N

June,1960-1999

1960 1970 1980 1990

0.156

0.1565

0.157

0.1575

0.158

0.1585

0.159

0.1595

Eigenvector N1 (97%), June, Eastern Siberia1960 1970 1980 1990

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

Eigenvector N1 (17%), June, global scale

Global, 17%

East Siberia, 97%

Eigenvectors N1, June, 1960-1999

Irkutsk

. June 16 2003 г

18.06.2003

n

i D

ikiikiy

ikix

t

ccyc

yc

xc

xc

1

[{

dtdcnc

dDdtcfcS iki

nikiiki

t

])(

}]))(([ dtdcqRccudDcc iki

D

iikintiki

t

00

Kk ,1

Simbol denotes variations of corresponding functions,and coefficients at are the sensitivity functions )(k

Y)),(()( hkY

hk grad

Sensitivity relation for estimation of risk/vulnerability and observability

Sensitivity function for estimation of risk/vulnerabilitydomains for Lake Baikal

Conclusion

Combination of• forward and inverse modeling• factor and principle component analysis• sensitivity theory on the base of variational principles

gives the possibility for coordinated solutionof the variety of environmental problems, such as

• diagnosis• prognosis• monitoring (mathematical background)• design