Generalized quantitative description of climate dynamics for goals - - PowerPoint PPT Presentation

Generalized quantitative description of climate dynamics for goals of environmental design and ecological risk assessment

Vladimir Penenko Elena Tsvetova

Institute of Computational Mathematics and Mathematical Geophysics SD RAS

Goal

Development of theoretical background and computational technology for parameterization of climate dynamics in environmental and ecological applications.

The methodology is based on:

• control theory,
• sensitivity theory,
• risk and vulnerability theory,
• variational principles,
• combined use of models and observed data,
• forward and inverse modeling procedures,
• orthogonal decomposition of functional spaces

for revealing the principle components and factors for analysis of the data bases and dynamical systems

CONCEPT OF ENVIRONMENTAL MODELING

Basic elements for concept implementation:

• models of processes
• data and models of measurements
• adjoint problems
• constraints on parameters and state functions
• functionals: objective, quality, control,

restrictions etc.

• sensitivity relations for target functionals and

constraints

• feedback equations

• Extraction of multi- dimensional and

multi-component factors from data bases

• Classification of typical situations with respect

to main factors

• Investigation of variability
• Formation of “leading” spaces

Analysis of the climatic system for construction

• f long-term scenarios

Mathematical background

• Analysis of multi-dimensional vector spaces

with the help of orthogonal decomposition;

• Variational principles for joint use of

measured data and models

• Sensitivity theory

Structuring and decomposition of data bases

Initial data base

( ) ( ) ( )

T T T

S V V Z ZV V V = = Γ

{ }

( , , ) ( ) , ( )

t N t

x t Y Q D R Y R D ≡ ∈ ⊂ ∈ r r r Φ ϕ

Structured data base

{ }

1 2

1

/

, , ,

i i i N

Z z C i n R = = = ∈ ϕ ϕ Z

matrix of vectors from

n N

R R × n N × C N N ×

diagonal matrix of total energy weight of Scattering function Orthogonal decomposition of Z

• n the base of optimal properties of

( ) S V

{ }

1 , , , , , ,

p p n p N T T p q p pq p q pq

V V V R R V V p q n   = ⇒ ∈ ∈     == = =     Γ λ λ Ψ λ δ Ψ Ψ δ ϕ

Multicomponent spatiotemporal bases and factor spaces in orthogonal decomposition: focuses and applications

• data compression ( principle components and factor

bases);

• classification of situations for analysis and modeling ;
• revealing the key factors in data;
• variability studies;
• classifying the processes with respect to informativeness
• f basic functions: climatic scale, annual scale, weather

noises

• efficient reconstruction of meteorological fields on the base
• f observation;
• development of a few component models;
• construction of leading phase spaces for deterministic-

stochastic models;

• formation of subspaces for long-term climatic and

ecological scenarios;

• focus on “ activity centers”, ”hot spots” , and

“risk/vulnerability” studies

Multicomponent spatiotemporal bases and factor spaces in orthogonal decomposition: focuses and applications

number eigenlavues

10 20 30 1 2 3 4 5 6 7 8 9

number eigenlavues

10 20 30 40 1 2 3 4 5 6 7 8 9 10 11 12

number eigenlavues

10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14

36 years 46 years 56 years Separation of scales : climate/weather noise

Eigenvalues of Gram matrix as a measure of informativeness of

• rthogonal subspaces

Variability of hgt500 with respect to the leading orthogonal subspace

year principle component

1950 1960 1970 1980 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2

year principle component

1950 1960 1970 1980 1990 0.11 0.12 0.13 0.14 0.15 0.16 0.17

year principle component

1950 1960 1970 1980 1990 2000 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16

36 years 46 years 56 years Quantitative measure of variability is the value of the main principle component

Leading orthogonal subspaces

36 years, 26.66% 46 years, 26.63% 56 years,26.34%

Factor subspaces for deterministic-stochastic scenarios

• Factor subspaces

is a linear subset of the vector space X ⊂ Y

X

• arbitrary element of

X

!! is invariant at algebraic transformations in X While modeling, is leading phase space,

• generated disturbances

X

ξ r r r + = X Y

ξ r ξ r

Construction of

X

Calculation of

• 1. In deterministic case:

with the help of models of processes

• 2. In deterministic-stochastic case:

spectral methods for generation of random processes

• f fractal type with dispersion

2 2 1

1 ,

H q q

H λ   σ = ≤ ≤   λ   H is “fractal” parameter, are eigenvalues of Gram matrix “Weather noise” part of subspaces is used for randomisation

i i d i n i i i

g n n g X

d

σ λ λ max , ,

1 1

≤ ≤ ≤ Ψ =∑

=

r

ξ r

1 ≤

q

λ

x y

100 200 300 50 100 150 June 30 June 30

Sum of 16 weighted vectors

x y

100 200 300 50 100 150 June 30 June 30

The first basis vector

Composite basic vectors Comparative studies for June.

number eigenvalues

Versions

• f Euler-type models

with regard of subgrid processes

• 1. Parametrization in the frames of background model (D)
• 2. Calculation with more details in nested regions

( turbulence scales, input data, assimilation of background) (D)

• 3. Stochastic (fractal) fulfilment of deviations
• f the state functions from their background values

in nested grids

• 4. Generating fractal stochastic disturbances by Fourier

filtration/analysis methods

• 5. Combination of above-mentioned strategies

Generalized field methods

Stochastic fulfilment

• f subgrid field structure

Turbulence coefficients:

H y x

y x t x A t x ) )( , , ( ) , (

2 2 ,

∆ + ∆ = r r r ϕ µ

H z

z t x B t x ) )( , , ( ) , (

2

∆ = r r r ϕ µ z y x ∆ ∆ ∆ , ,

grid parameters

B A,

functions calculated in the background (D)-model

H

«fractal» parameter

Stochastic fractal fulfillment

Algorithm:

• 1. Generating perturbations by mean bias method
• 2. Change to the next grid scale taking into account

background flow in the mesh

• 3. Adding stochastic normal value with zero mean

and dispersion

k

σ

k k

r σ σ =

subgrid dispersion

) ; (

, z y x

µ µ σ σ =

background dispersion

k H k k

r r ) 2 /(

1 −

=

j iml

S u r         ∆ = r µ ϕ α

scale of subgridness

1 ≥ k

number of recursion stochastic parameter 1 H < <

2 0 3 0 4 0

fifrac

2 0 4 0 6 0

x c

2 0 4 0 6 0

y c Y X Z

Snapshot of stochastic fulfilment for the field of pollutant concentration in the atmosphere

2 0 3 0 4 0

fifrac

2 0 4 0 6 0

x c

2 0 4 0 6 0

yc Y X Z

h = 0 .8 , w ith o u t w in d

2 0 3 0 4 0

fifrac

2 0 4 0 6 0

xc

2 0 4 0 6 0

yc Y X Z

h = 0 .8

2 0 3 0 4 0

fifrac

2 0 4 0 6 0

x c

2 0 4 0 6 0

yc Y X Z

h = 0 .5

Algorithm for randomisation by Fourier filtration/analysis methods

• 1. Fulfilment of the process on subgrid structure with regard for

background process in the meshes

• 2. Randomisation

Generating stochastic perturbations of Fourier coefficients possessing the following properties: frequencies - uniformly distributed in [0,1]; amplitudes -normally distributed with zero mean and dispersion decreasing as where n is a number of coefficient; Reconstruction of the state function from calculated Fourier coefficients

• 3. Superposition of stochastic and deterministic parts

k

σ ) /( 1

1 2 + H

n

Version: randomisation with the use of orthogonal bases

Conclusion

• The methodology is proposed for quantification of

climatic data in terms of orthogonal subspaces for environmental studies

• The set of numerical algorithms for multicomponent 4D

factor analysis and sensitivity studies is developed

• The orthogonal bases ( principle components and EOFs)

are constructed as a result of decomposition of Reanalysis data for 56 years

Acknowledgements

The work is supported by

• RFBR

Grant 07-05-00673

• Presidium of the Russian Academy of Sciences

Program 13

• Department of Mathematical Science of RAS

Program 1.3.

Thank you for your time!