Environmental forecasting on the base of online-integrated modeling technology Vladimir Penenko Elena Tsvetova Institute of Computational Mathematics and Mathematical Geophysics SD RAS
Online-integrated modeling technology • Models of hydrodynamics • Models of atmospheric chemistry • Data of observations / assimilation • Technology of modeling • New algorithms • New mathematics/numerics (tera- (12), peta-(15), exa- (18)
Model of atmospheric dynamics ¡ u 1 p ∂ ∂ v u fv kw F + ⋅∇ − + = − + u t x ∂ ρ ∂ v 1 p ∂ ∂ v v fu F + ⋅∇ + = − + v t y ∂ ρ ∂ w 1 p ∂ ∂ v w ku g F + ⋅∇ − = − − + w t z ∂ ρ ∂ c c p ⎛ ⎞ ∂ + p p v p ( v ) p 1 c F f ⋅∇ + ∇⋅ = − ρ + ⎜ ⎟ p p p t c c ∂ ⎝ ⎠ v v c T R ∂ p d v T ( (1 ) v ) T F f + ⋅∇ + + α ∇⋅ = + T T t c c ∂ v v ∂ ρ 1 − v 0, p R (1 ) T ( ) + ∇⋅ ρ = ρ = + α a d t ∂
¡ Transport and transformation of humidity ¡ q ∂ ( ) v v q S S F + ⋅∇ = − + + v l f q t ∂ v q ∂ c v q S F + ⋅∇ = + c c q t ∂ c q 1 ∂ ∂ l v q q v S F + ⋅∇ + z ρ = + l l lT l q t ∂ ρ ∂ l q 1 ∂ ∂ f v q q v S F + ⋅∇ + z ρ = + f f fT f q t f ∂ ρ ∂
Transport and transformation model of gas pollutants and aerosols ∂ πϕ πϕ i L div ( u grad ) ((S ) f (x,t) r ) 0, ( ) ϕ ϕ ≡ + π π ϕ − µ ϕ + π ϕ − − = i i i i i i i t ∂ Operators of transformation R U ⎧ ⎫ i r − s (q ) { } ( ) S ( ) P( ) ( ) k(q) s (q ) s (q ) − + ∑ ∏ j ϕ ϕ = ϕ ϕ − Π ϕ ≡ − ϕ ⎨ ⎬ g i i j i ⎩ ⎭ q 1 j 1 = = σ M M 1 ⎡ ⎤ ⎛ ⎞ ( ) S K , d ( ) ∑ ( ) ∑ ( ) ( ) ϕ σ = ∫ γ ϕ σ α σ − σ σ ϕ σ − σ σ ⎢ ⎥ ⎜ ⎟ a i ik k 1 km 1 1 k 1 1 2 ⎝ ⎠ ⎣ ⎦ k 1 m 1 = = σ 0 σ 2 M M ⎛ ⎞ ∂ ∂ K , d e ( ) ( ) ∑ ( ) ( ) ( ) − ϕ σ ∫ σ σ β ϕ σ σ − ⎡ ϕ σ ⎤ + ⎡ ν ϕ σ ⎤ ⎜ ⎟ ⎣ ⎦ ⎣ ⎦ i 1 ik k 1 1 i i i i 2 ∂ σ ∂ σ ⎝ ⎠ k 1 = σ 0 ) R Q , i 1, M ( ) ( ) − ϕ σ + σ = i i i
Goals • Development of methodology for construction of integrated models of atmospheric dynamics and air quality in on(off)-line regimes with account of all accessible observational data • Design of adequate/open modeling system
Variational principle as a tool for model integration It presents the equations and relations describing the development of multi-component and multi-scale processes in a simple invariant form; It holds the synthesis of continuum and discrete presentations of mathematical models and state functions;
Variational principle as a tool for model integration It provides consistency for all technology stages of mathematical modeling; • Euler, Lagrange, Jacobi, and Hamilton principles are interconnected among themselves: the stationary value of some definite integral (functional) is considered in each of them.
A CONCEPT OF ENVIRONMENTAL MODELING � • variational principles • combined use of models and observed data, • forward and inverse modeling procedures, • methodology for description of links between regional and global processes ( including climatic changes) by means of orthogonal decomposition of functional spaces for analysis of data bases and phase spaces of dynamical systems Theoretical background • control theory, • sensitivity theory, • risk and vulnerability theory
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 for inverse problems
Statement of the problem • Mathematical model ∂ ϕ B G ( , Y ) f r 0 + ϕ − − = , t ∂ 0 0 , Y Y ϕ = ϕ + ξ = + ς ; 0 0 ( D ) is the state function , ϕ ∈ ℑ t Y ( D ) is the parameter vector . ∈ ℜ t G is the “space” operator of the model , • A set of measured data D m Ψ on ϕ m m t [ H ( )] Ψ = ϕ + η , m m [ H ( )] ϕ is the model of observations . m r , , , • ξ ς η are the terms describing Desired! uncertainties and errors of the corresponding objects.
General form of functionals r r r ( ) F ( ) ( , ) x t dDdt F , , k 1,..., K ( ) Φ ϕ = ∫ ϕ χ ≡ χ = k k k k k D t F are the Lipschitz's functions of the given form, differentiable, bounded k dDdt D , * ( D ) χ χ ∈ ℑ are Radon’s or Dirac’s measures on . k t k t Quality functionals r r r r T ( ) ( H ( )) M ( H ( )) ( , ) x t dDdt , Φ ϕ = Ψ− ϕ Ψ− ϕ χ ∫ k m m k D t “Measurement” functionals K r r r r r m ( ) [ H ( ) ] ( x x ) dDdt , x D ∑ ∫ Φ ϕ = ϕ δ − ∈ mk mk t mk k 1 = D t Restriction functionals Differentiability ( x , t ) N , ( ( x , t ) 0 ϕ ≤ ϑ ϕ ≤ distributive constraints k in extended sense ( ) ( ( ) ( ) ) ( x , t ) dDdt Φ ϕ = ϑ ϕ + ϑ ϕ χ ∫ k k k k D t
Variational form of model ’ s set: hydrodynamics+ chemistry+ hydrological cycle ⎧ ⎫ n ⎪ ⎪ ( ) ( ) I Y , ((S ) f (x,t) r ) dDdt ∗ ∗ ∑ ∗ ∗ ∗ ϕ , ϕ ≡ Λϕ , ϕ + ∫ π ϕ − − ϕ + ⎨ ⎬ i i i i ⎪ ⎪ i 1 = ⎩ ⎭ D t { } ( * * ) ( * * ) pp div u pdiv u pp TT div u dDdt − + α + α + ∫ p T D t { T } * pu d dt 0 W dDdt ( ) ( ) Ω = ∫ α α ρ ρ − ρ ρ − ρ + ∫ n a a a ρ D Ω t t
Variational form of convection-diffusion operators ⎛ ⎧ 1 ∗ ⎡ ⎤ ⎛ ⎞ ∂ πϕ πϕ ∂ πϕ πϕ ⎪ ( ) ( ) div u div u ∗ ∗ ∗ ∗ ∗ ∗ Λϕ , ϕ ≡ ⎝ ∫ ϕ − ϕ + ϕ πϕ − ϕ πϕ ⎜ ⎨ ⎢ ⎥ ⎜ ⎟ 2 t t ⎜ i ∂ ∂ ⎪ ⎝ ⎠ ⎣ ⎦ ⎩ D t 1 } t grad grad ∗ ∗ dDdt ∗ ∗ dD + π µ ϕ ϕ + ∫ ϕϕ π + 0 ϕ 2 D ⎞ 1 ⎡ ∂ ϕ ⎤ ⎛ ⎞ u R q ∗ d dt ( ) ϕ − µ + α ϕ − ϕ π Ω ∫ ⎟ ⎜ ⎟ ⎢ ⎥ n b b b 2 n ⎟ ∂ ⎝ ⎠ ⎣ ⎦ ⎠ Ω t i R q = 0 boundary conditions on ϕ − ϕ − Ω b b t
Augmented functional for construction of optimal algorithms and uncertainty assessment r r r r r r % { 1 h h T T ( ) ( ) 0.5 ( M ) ( r M r ) Φ ϕ = Φ ϕ + α η η + α k 1 2 2 m h D D t t r r r r r r r h I ( , , Y ) } ∗ T T h ⎡ ⎤ + ⎣ ϕ ϕ ( M ) ( M ) + α ξ ξ + α ζ ζ ⎦ 3 3 4 4 h h h h D R ( D ) D t t M i , ( i 1 , 4 ) = are the weight matrices, 4 0 , 1 are the weight coefficients, ∑ α ≥ α = i i i 1 ϕ , = are the solutions of the direct and adjoint problems generated from ∗ ϕ Y h I ( , , ) 0 ∗ ϕ ϕ = Additive aggregation of the functionals for decomposition
Variation of augmented cost functional. Consistency and optimization of algorithms r r % % h h r r r ( ) ( ) ⎛ ⎞ ⎛ ⎞ δ Φ ϕ δ Φ ϕ % h * ( ) k , k , r r δ Φ ϕ = δϕ + δϕ + ⎜ ⎟ ⎜ ⎟ k * ∂ ϕ ∂ ϕ ⎝ ⎠ ⎝ ⎠ r r r % % % h h h ( ) ( ) ( ) ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ δ Φ ϕ δ Φ ϕ δ Φ ϕ k k k , r , , Y + δ + δξ + δ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ r Y ∂ ∂ ξ ∂ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
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