DYNAMIC-PROBABILISTIC MODELING OF UPWARES IN ATMOSPHERE WITH USE - - PowerPoint PPT Presentation

DYNAMIC-PROBABILISTIC MODELING OF UPWARES IN ATMOSPHERE WITH USE SEMI- LAGRANGIAN MODEL OF TRANSPORT OF SUBSTATION A.V.Protasov

Institute of Computational Mathematics and Mathematical Geophysics of SB RAS

Ensemble of realizations

( )

{ , 1,2,...}

i n i

ξ = r

, Where

( )

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

i i i i i Т n j j j j j

u X v X X T X j n X Spatial time punct of region ξ τ = = − − r K K

The sample was carried out for the given local area of Northern hemisphere by the

• ×

10 10 size with the centre in a point with coordinates

• 60,56 of Northern lattitude and 77,7
• East

longitude. The problem is considered in , , x y p system

• f coordinates in area, which bottom basis is a

rectangular on a tangential plane in this central

• point. For meshgrid construction the resolution

24 20 ×

• n x and y with steps

23,85 km x ∆ = and 58,74 km y ∆ = respectively was chosen

Method of statistical modeling Let R- a many-dimensional correlation matrix.

Shall present its spectral decomposition as

T

R W W = Λ , where W - matrix of eigenvectors of R, and Λ- matrix of the eigenvalues. The further step is

1 1 2 2 T

R W W = Λ

,

Then one of methods of statistical modeling can define as

1 ( ) ( ) 2 ( )

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

i i Т n j j j j j j j j

D R x y p t x y p t i

ξ

ξ ψ ξ = + = r r r K where

( )(

, , , ) , ( 1,2, )

i Т j j j j

x y p t i ψ = r K -a Gaussian stochastic vector with individual variance and zero average, Dξ - diagonal variance matrix, ( , , , )

j j j j

x y p t ξ r

• vector of

average.

The input data:

• The NCEP/NCAR Reanalysis Project

at the NOAA-CIRES Climate Diagnostics Center

• Fields of temperature
• The period: 1949 -2005 .
• the resolution: 2,5 x 2,5 degrees,
• 17 of standard standard isobaric surfaces
• 6 hours - time interval

Table 1. One of vertical sections of a correlation

matrix designed on the reanalysis data (black colour) and appropriate correlation matrix from Gandin (lilac colour) are given.

P (mb) 1000 850 700 500 400 300 250 200 150 100 1000 1.00 0.67 0.57 0.47 0.45 0.34 - -0.27 - -0.14 850 0.80 1.00 0.74 0.68 0.57 0.29 - -0.45 - -0.45 700 0.70 0.91 1.00 0.76 0.67 0.44 - -0.49 - -0.46 500 0.58 0.78 0.91 1.00 0.94 0.53 - -0.56 - -0.61 400 0.52 0.69 0.82 0.94 1.00 0.67 - -0.55 - -0.70 300 0.23 0.17 0.21 0.32 0.51 1.00 - -0.02 - -0.46 250 -0.13 -0.34 -0.38 -0.34 -0.23 0.63 - - - - 200 -0.21 -0.40 -0.44 -0.43 -0.37 0.35 0.90 1.00 - 0.51 150 -0.17 -0,27 -0.28 -0.25 -0.20 0.35 0.78 0.93 1.00 - 100 -0.09 -0.15 -0.13 -0.10 -0.06 0.36 0.66 0.79 0.93 1.00

Eigen values of the correlation matrix

• 3. VARIATIONAL ASSIMILATION

The numerical model in the operator form (3.1) is the state vector; is the vector of the parameters; is the finite difference operator determined by system

• f the equations of considered process and the appropriate

boundary conditions in the area .

( , ) 0, Ф A Y Ф Ф t ∂ + = ∂ r r r r

Ф r

t

Y Ф

=

= r r

( , ) A Y Ф r r

ˆ [0, ]

t

G G t = ⊗

The problem consists, that among this set of the solutions, determined by the vector , to find a closest solution to concrete realization from (2.2) in sense of some quality functional is the scalar product in space the of the measured data , as which the appropriate values of fields of dimension in the points on time from ensemble of realization (2.2) are used; is the appropriate operator of interpolation, is the solution of the problem (3.1) at the moment of time . It is necessary to find a minimum of the functional relatively a vector of parameters at the restrictions (3.1).

Y r

1 ( , ) , 2

S

j k j k S S D k

J LФ Ф LФ Ф = − −

∑

r r r r

( , )

s

D

k S

Ф r

( ) i n

ξ r

S

N

k

t L

j

Ф r

j

t

J

Y r

For the solving of this problem the iterative method of gradient descent based on the Lagrangian method and the solution of the direct and adjoint problems is used. As a result of the solution of a sequence of tasks of assimilation we receive new ensemble of realizations

( )

{ , 1,2,...}

i n i

ξ = r %

Climatic trajectory

1, если ( , ) 0, если f lev f lev f lev χ ≥  =  < 

, where f - value of tested function, and lev- some given number. The values of this function equal to unit in the considered area, from a averaged sequence of realizations from ensemble of an pollution define the approached climatic trajectory Tr of distribution of an pollution at the given threshold value lev.

In numerical experiments as an initial field

• f an pollution determining an instant source of

pollution, the modelling field of an pollution located in some area 5 5 × of points at a level 850 мб p = with the maximal value at this centre, equal 1 was chosen. The value of size lev was chosen completely equal with 0.001 to 0.05. The results are edentical.

We shall consider the estimation of a degree of a geometrical belonging of a concrete trajectory

• f distribution of the pollution climatic
• trajectory. We consider that as the relation of

number of points of crossing with a climatic trajectory to common number of points concrete trajectory of distribution of an pollution.

Figure 1. The diagram of stochastic value alf and histogram of its distribution (use of model Van Leer).

Figure 3. The diagram of stochastic value alf (use of Lagrangian model ).

Sample function of distribution

• f stochastic value alf

Sample function of distribution of stochastic value alf (Lagrange variant)

Figure 2. A mutual spatial arrangement of a climatic trajectory of distribution of an pollution and trajectory of an pollution to emission (B). The point A corresponds to the centre of a subarea of a source

• f

pollution. Along axes

• f

coordinates the appropriate numbers of meshgrid are postponed.

Figure 3. A mutual spatial arrangement of a climatic trajectory of distribution of an pollution (B) and trajectory of an pollution to emission. The point A corresponds to the centre of a subarea of a source of

• pollution. Along axes of coordinates the appropriate

numbers of meshgrid are postponed.

CONCLUSION

• 1. On the basis of spectral decomposition of a correlation matrix

the method of statistical modeling of the many-dimensional fields of meteorological elements is suggested and realized. 2. Is shown that within the framework of the given problem there is a brightly expressed climatic spatial - time trajectory of distribution of an pollution in an atmosphere and this trajectory has steady enough character in relation to threshold values. 3. The concept of rejection of a trajectory is entered and on modelling samples the estimations of the appropriate probability of this rejection are result.