Main factors of climate variability and their application for environment protection problems in Siberia Vladimir i Penenko & Elena Tsvetova Institute of Computational Mathematics and Mathematical Geophysics SD RAS Novosibirsk
Algorithms for revealing climatic variability Singular vectors (SV) for forward tangent operator of dynamical models and the use of SV-decomposition for scenario construction and errors analysis ( uncertainty reducing); ensembles of prognostic scenarios with generation of perturbations ( breeding cycle, Lyapunov s vectors); Monte-Carlo methods for scenario construction Stochastic-dynamic moment equations and Liouville equations ICMMG technology Oth Orthogonal ldecomposition of fthe phase spaces of non-linear dynamical systems for formation of informative basis subspaces; Minimization of uncertainties with respect to given criteria of prognosis quality ( + data assimilation if any)
Scenarios construction and adaptive monitoring with SV t A( ) 0 t Tangent linearization about A L 0 ( x,t) ( x, 0 ) (a priori) ( x,t) L ( x, 0), x D,t 0,t L( x, t) - forward tangent propagator about ( x,t) x * ( x, 0) L (,t),t t 0 D t
Basic relations and patterns for SVs (t) (t), (t) t L ( 0), L ( 0) ( 0), LL * ( 0) ( 0 ), ( 0 ) t D evaluation domain at t t D target area at t=0 0 0,t optimal time interval ( 48 h)
Partial eigenproblem for SVs i i * 2 LLVi i Vi i K, ( ),V singular values and vectors of L(SEVs SEVs, SVs) Lan osh algorithm Lanzosh algorithm Ortogonal decomposition of perturbation spaces Optimal construction of perturbations with respect to rapidly growing SVs
Structuring and decomposition of data bases Initial data base ( xty,, ) QD ( ) R, Y RD ( ) Initial data base t N t Structured data base 1/ 2 Zz 1 C i n R i i,,, i N Z nn matrix of vectors from Rn RN C N N diagonal matrix of total energy weight of Scattering function T T T Sv vzzv vv ( ) ( ) ( )
Orthogonal decomposition of Z on the base of optimal properties of Sv v v p, vprn, pr N T T vv p q p pq, p q pq, pq, 1, n V v p, diag 0, p p N, p,, nn matrices R p 1 n nn matrix ( ) ZV 1 decomposition algorithm T Z V reconstruction ti algorithm
Factor subspaces for deterministic- stochastic scenarios Factor spaces r j r = + x X 0 is a linear subset of the vector space X Ì DAT A x r is arbitrary element from!! Algebraic operations in X leave X 0 invariant X 0 is r the leading phase space, are generated perturbations x r X
Construction of the vector set X 0 n d ì X = ï 0 íå c i Y i, n d n,0 c i max s ïî 1 Formation of vectors 1. Deterministic case: calculation by means of the process models 2Dt 2. Deterministic-stochastic i ti t ti case: c i generation by means of the stochastic processes of the fractal type described by gaussian process with variance 2 2H q q, 0 H 1 H is a parameter of the fractal size, l are the eigenvalues of the Gram matrix q x r ü ï ý þ ï
Forming the guiding phase space with allowance for observation data on the subdomain m Z (x, ) measured data; x, t n a m Z x, t a x, x t, x, x t D, n n p1 n a p 2 basis p p t a min Z x, a x,, x, D, n n a m p m m m p p a p1 m D m 1 m m a F a a, p, n p a 1 a m, W, p, q 1, n m m F m p, W Z m m m pq p q D a n p1 D m If t then Z(x, t) is forecast!
Winter pattern of the global 500-hPa geopotential height (the 1st main factor) 150-2005 January 15 Level value: 1-0.85 3-0.1 5-0.56-0.42-0.28-0. 0.01 15 0.16 1 0.30 1 0.45 21 0.5 180 150 21 1 15 5 21 120 15 1 15 1 1 y, deg 0 60 30 15 15 15 15 0 0 30 60 0 120 150 180 210 240 20 300 330 Longitude
Winter pattern of global circulation (the 1st main factor) 150-2005 January 15 150 y 100 50 0 0 100 200 300 x
Summer pattern of the global 500-hPa geopotential height (the 1st main factor) 150-2005 July 15 Level value: 1-0.64 3-0.4 5-0.35-0.20-0.06 0.0 0.23 15 0.38 1 0.52 1 0.66 21 0.81 y 180 150 120 0 5 3 5 3 60 30 3 1 5 5 3 0 0 30 60 0 120 150 180 210 240 20 300 330 Longitude 15 1
Summer pattern of global circulation (the 1st main factor) 150-2005 June 15 150 y 100 50 0 0 100 200 300 x
East Siberia Region 0-140 E, 45-65 N June 150-2005
Variability of the phase spaces with respect to the first main factors Eigenvectors N1, June, 150-2005 Global, 1% Eastern Siberia, 1,%
Quantification of subspace scales Eigenvalues of Gram matrices, June, 500-hPa, 150-2005 Global scale Regional scale
500-hPa
Monthly risk functions for Lake Baikal July December
Conclusion The set of numerical algorithms for orthogonal decomposition of the phase spaces of dynamical system evolution is developed for climate and ecology studies The methods are applied dfor construction of long-term scenarios for risk assessment with respect to anthropogenic impact This allows us to take into account climatic data for environmental studies of global and regional scale
Acknowledgements The work is supported by RFBR Grant 0-05-0063 05 0063 Presidium of the Russian Academy of Sciences Program 16 Department of Mathematical Science of RAS Program 1.3.