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.

CONCEPT OF ENVIRONMENTAL MODELING 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

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

Analysis of the climatic system for construction of long-term scenarios • 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

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 { } r r r Φ ≡ ϕ ∈ ⊂ ∈ Initial data base ( , , ) x t Y Q D ( ) R Y , R D ( ) t N t { } = = ϕ = ϕ ∈ 1 2 / , , , Z z C i 1 n R Structured data base i i i N × × R R Z n N matrix of vectors from n N ϕ × N N diagonal matrix of total energy weight of C = = Γ T T T Scattering function S V ( ) ( V Z ZV ) ( V V ) Orthogonal decomposition of Z on the base of optimal properties of S V ( ) { }   Γ = λ ⇒ λ ∈ Ψ ∈ V V , V R , R   p p n p N   == λ δ Ψ Ψ = δ =  T T  V V , , , p q 1 , n   p q p pq p q pq

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 of basic functions: climatic scale, annual scale, weather noises

Multicomponent spatiotemporal bases and factor spaces in orthogonal decomposition: focuses and applications • efficient reconstruction of meteorological fields on the base of 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

Eigenvalues of Gram matrix as a measure of informativeness of orthogonal subspaces 9 8 7 eigenlavues 6 36 years Separation of scales : climate/weather noise 5 4 3 12 2 11 10 1 9 0 10 20 30 number 8 eigenlavues 46 years 7 6 14 5 13 4 12 3 11 2 10 1 eigenlavues 9 10 20 30 40 8 number 7 56 years 6 5 4 3 2 1 0 10 20 30 40 50 number

Variability of hgt500 with respect to the leading orthogonal subspace 0.2 Quantitative measure of variability is 0.19 the value of the main principle component 0.18 principle component 0.17 0.16 0.15 0.17 0.14 0.16 0.13 principle component 56 years 0.15 1950 1960 1970 1980 year 0.14 36 years 0.13 0.16 0.12 0.15 0.11 principle component 0.14 1950 1960 1970 1980 1990 year 0.13 46 years 0.12 0.11 0.1 0.09 1950 1960 1970 1980 1990 2000 year

Leading orthogonal subspaces 56 years,26.34% 36 years, 26.66% 46 years, 26.63%

Factor subspaces for deterministic-stochastic scenarios • Factor subspaces r r r = + ξ Y X 0 is a linear subset of the vector space X ⊂ Y r ξ X - arbitrary element of X !! is invariant at algebraic transformations in X 0 X is leading phase space, While modeling, 0 r ξ - generated disturbances

X Construction of 0 λ n r = ∑ d Ψ ≤ ≤ ≤ σ i X g , n n , 0 g max λ 0 i i d i i = r i 1 1 ξ 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 of fractal type with dispersion 2 H λ   σ = ≤ ≤ q 2   , 0 H 1 λ q   λ ≤ 1 1 H is “fractal” parameter, are eigenvalues of Gram matrix q “Weather noise” part of subspaces is used for randomisation

Composite basic vectors Comparative studies for June . The first basis vector Sum of 16 weighted vectors June June 30 30 June June 30 30 150 150 100 100 y y 50 50 6 0 0 0 100 200 300 0 100 200 300 x x 5 eigenvalues 4 3 2 1 10 20 30 40 50 number

Versions of Euler-type models Generalized field methods 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 of 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

Stochastic fulfilment of subgrid field structure Turbulence coefficients: µ r = ϕ r r ∆ + ∆ 2 2 H ( x , t ) A ( , x , t )( x y ) x , y µ r = ϕ r r ∆ 2 H ( , ) ( , , )( ) x t B x t z z ∆ ∆ ∆ grid parameters x , y , z functions calculated in the background A , B (D)-model «fractal» parameter H

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 σ and dispersion k σ = σ r subgrid dispersion k 0 k σ = σ µ µ ( ; ) background dispersion 0 0 x , y z = H k r r /( 2 ) scale of subgridness − k k 1 j   µ   = α ϕ r   stochastic parameter r ∆ 0   u S iml < < ≥ 0 H 1 number of recursion k 1

Z Y X 4 0 fifrac 3 0 6 0 2 0 4 0 y c 0 2 0 2 0 x c 4 0 6 0 0 Snapshot of stochastic fulfilment for the field of pollutant concentration in the atmosphere

Z Y X 4 0 fifrac 3 0 6 0 2 0 4 0 yc 0 2 0 2 0 4 0 x c h = 0 .8 , w ith o u t w in d 6 0 0

Z Y X 4 0 fifrac 3 0 6 0 2 0 4 0 yc 0 2 0 2 0 xc 4 0 h = 0 .8 0 6 0

Z Y X 4 0 fifrac 3 0 6 0 2 0 4 0 yc 0 2 0 2 0 4 0 x c h = 0 .5 6 0 0

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 + σ 2 H 1 1 /( n ) and dispersion decreasing as k where n is a number of coefficient; Reconstruction of the state function from calculated Fourier coefficients 3. Superposition of stochastic and deterministic parts 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.

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