Formulation of Lagrangian stochastic models for geophysical turbulent flows

Transcription

1 Formulation of Lagrangian stochastic models for geophysical turbulent flows Alberto Maurizi Institute of Atmospheric Sciences and Climate - CNR, Bologna, Italy International summer school on turbulent diffusion - Vilanova i la Geltrù p. 1

2 Introduction From general theory of Lagrangian Stochastic Modelling: Markov assumption on (x, u ẋ) (Markov-1) implies equivalence between stochastic differential or Langevin equation (LE) du i = a i dt + b ij dw j Fokker-Plank equation (FPE) (Itô s way) t p + xi (u i p) + ui (a i p) = 1 2 u i uj b ij p International summer school on turbulent diffusion - Vilanova i la Geltrù p. 2

3 the well-mixed condition Well-mixed condition (WMC; Thomson, JFM, 1987; T87) if p(x,u, t 0 ) = p E (u) unif D (x) then p(x,u, t) = p E (u) unif D (x) for t > t 0 Assuming b 2 ij = C 0εδ ij (K41), the solution for a i is given by a i = C 0ε 2 1 p E p E u i + φ i p E φ i = p E u i t u p E k x k International summer school on turbulent diffusion - Vilanova i la Geltrù p. 3

4 Determination of the diffusion coefficient b consistency with K41: b 2 = C 0 ε International summer school on turbulent diffusion - Vilanova i la Geltrù p. 4

5 Determination of b The dissipation rate of turbulent kinetic energy ε = ν ( xi u j xj u i ) is hardly accessible to direct measurements except in few cases. The Kolmogorov energy spectrum E(k) = C K ε 2/3 k 5/3 provides a mean for indirect ε estimation. If C K is known (commonly accepted value around 2), and measurements of the energy spectrum are available, ε can be retrieved by fitting the above equation on data. International summer school on turbulent diffusion - Vilanova i la Geltrù p. 5

6 Determination of b: K41 energy spectrum The figure represent a typical energy spectrum in the wavenumber space taken in free troposphere which can be fitted with K41 spectrum to give estimations of ε. International summer school on turbulent diffusion - Vilanova i la Geltrù p. 6

7 Determination of b: K41 energy spectrum Especially in the BL, energy spectra are quite complex presenting different ranges of (possibly) K41 behaviour. International summer school on turbulent diffusion - Vilanova i la Geltrù p. 7

8 Determination of b: K41 energy spectrum Fitting different spectra available in literature, we can identify two broad classes for ε values: turbulence in the boundary layer (BL) turbulence in the free troposphere (FT) As the dynamics differ in the two regions, ε values reflect the fact that in BL there is a larger injection of energy from shearing instabilities or surface heat fluxes that produces fully 3D turbulence and implies larger dissipation to maintain equilibrium. BL high frequency BL low frequency troposphere m 3 s m 3 s m 3 s 2 International summer school on turbulent diffusion - Vilanova i la Geltrù p. 8

9 Determination of b: K41 energy spectrum: caveats spectra measured at a fixed point (Eulerian spectra in frequency domain) in this case there is the need to make some assumptions: in principle Taylor frozen turbulence must apply; however a relationship not based on the mean wind can be used to move from frequency to wavenumber domain, which involves the ratio of Eulerian to Lagrangian scales β = C 2/3 K /( 2C 0 ) to transform local τ l (ν) to τ l (k). spatial temporal validity shown (and available) spectra are derived from climatological measurements (long records at a fixed point, mainly for measurements in the BL) or from averaging over large volumes/time (for spectra resulting from aircraft measurements). This contrasts to determinations of ε from similarity theory in, e.g., the neutral surface layer where ε = u 3 /(κz) which strongly depends on local flow conditions. International summer school on turbulent diffusion - Vilanova i la Geltrù p. 9

10 Determination of b: Lagrangian correlation time Another way of determining b is using Lagrangian properties and the relationship C 0 ε = 2u 2 τ L where τ L is a local measure of decay of the Lagrangian correlation function. This can be used when Lagrangian measures are accessible directly (e.g., oceanic floats, but also Particle Tracking Velocimetry in lab) or when estimations of τ L can be made available indirectly. International summer school on turbulent diffusion - Vilanova i la Geltrù p. 10

11 Determination of b: Lagrangian correlation time 1 exp(-t/τ L ) (S,F)=(0.0, 3) (0.5, 4) (1.0, 6) (1.5,10) (1.0, 3) 1 ρ(t/τ L ) 0.1 E(ω) ω -2 (S,F)=(0.0, 3) (0.5, 4) (1.0, 6) (1.5,10) (1.0, 3) t/τ L ω 1D homogeneous stationary turbulence model with different values of third- and fourth-order moments of velocity. Here, despite homogeneity, T L τ L. International summer school on turbulent diffusion - Vilanova i la Geltrù p. 11

12 Determination of b: Lagrangian correlation time EAC CG WAC all ρ t (days) 2D strongly non-homogeneous fields: surface circulation as derived from drifter measurements in the Adriatic sea. Here τ L can be defined if one looks at small time slope of log R L (t). T L does not necessarily exist (although in this case it does and T L = days while τ L = days for EAC and WAC.) International summer school on turbulent diffusion - Vilanova i la Geltrù p. 12

13 Determination of b: conclusions Summary of b determination: well theoretically founded ε can be estimated from spectra or time Lagrangian scales requires k 5/3 energy spectrum time scale represent a local decorrelation time scale (not integral) International summer school on turbulent diffusion - Vilanova i la Geltrù p. 13

14 Determination of the drift coefficient a consistency with Eulerian (one-point) statistics: a i = C 0ε 2 1 p E p E u i + φ i p E φ i u i = p E t u k p E x k International summer school on turbulent diffusion - Vilanova i la Geltrù p. 14

15 Determination of a: what PDF? Is turbulence Gaussian? No, definitely not. Not even at a first level of approximation. Some examples: Boundary layer turbulence: varying parameter: distance from the surface (laboratory data). International summer school on turbulent diffusion - Vilanova i la Geltrù p. 15

16 Determination of a: what PDF? Is turbulence Gaussian? Turbulent flow in a canopy (lab. exp.). International summer school on turbulent diffusion - Vilanova i la Geltrù p. 16

17 Determination of a: what PDF? Is turbulence Gaussian? neutral boundary layer: effect of large and small scales 100 z (mm) S(z) large scales small scales z (mm) z (mm) S(z) S(z) International summer school on turbulent diffusion - Vilanova i la Geltrù p. 17

18 Determination of a: PDF statistical moments From experimental data we can compute moments up to some order N usually very small: µ (i) k uk i = 1 M M j=1 (u (i) j µ (i) k )k for k = 0, 1, N (N < 5?). The higher the order k the larger is the sample required for statistical significance. The error on the k-order moment can be roughly estimated by σ k = α k τ/( T) where α k is a coeff dependent on k (α 2 = 6, α 3 = 34,... ), T is the sampling time and τ is the correlation time of the series. International summer school on turbulent diffusion - Vilanova i la Geltrù p. 18

19 Determination of a: PDF statistical moments If, for a centred univariate, we limit to kmax = 4, and we normalise moments for k > 1 with µ 2, we obtain µ 0 = 1 µ 1 = 0 µ 2 = 1 µ 3 = S µ 4 = K S is a measure of asymmetry K gives a measure of how much tails are higher than for Gaussian distribution International summer school on turbulent diffusion - Vilanova i la Geltrù p. 19

20 Determination of a: properties of moments space The S K parameter space is not R R statistical lower limit schematic unimodal lower limit Millionshchikov 5 4 regular unimodal pdf region non-existence of pdfs non-unimodal pdf region two-values pdfs (Representation of some disregarded results from pure statistical theory) How data are placed in the S K space? International summer school on turbulent diffusion - Vilanova i la Geltrù p. 20

21 Determination of a: properties of moments space from uni- to bi-modality International summer school on turbulent diffusion - Vilanova i la Geltrù p. 21

22 Determination of a: properties of moments space International summer school on turbulent diffusion - Vilanova i la Geltrù p. 22

23 Determination of a: properties of moments space u(z<=67.7) u(67.7<z<217.03) u(217.03<z) w(z<=67.7) w(67.7<z<217.03) w(217.03<z) v(z<=67.7) v(67.7<z<217.03) v(217.03<z) Alberghi et al (2.5(x^2+1)) 1.7*abs(x)**3+2.7 Alberghi et al (3.3(x^2+1)) 3*(x**2+1) F S International summer school on turbulent diffusion - Vilanova i la Geltrù p. 23

24 Determination of a: revised Millionshchikov hypothesis If we think of the Quasi-Normal assumption as the zero-order approximation if the S K space were linear, we can formulate an alternative zero-order assumption for 4th order moments based on the knowledge of the S K space curvature. K = 3(S 2 + 1) or, going to the first-order approximation K = α(s 1 + 1) where α can be let to include some dynamical information. Considering data from different experimental sets, α is found in the range International summer school on turbulent diffusion - Vilanova i la Geltrù p. 24

25 Determination of a: analytical PDF N = 2: Gaussian PDF: O-U process: analytical results for the stochastic process N > 2: some analytical form other that Gaussian must be used: linear combination of two normal (Gaussian) distributions has been used p(u) = a 1 G(u; m 1, s 1 ) + a 2 G(u; m 2, s 2 ) six free parameters (vs. 2 parameters of a single Gaussian) requires moments up to the fifth: too much, our experimentalist can hardly afford this additional assumptions are needed ai proportional to up- down-draft areas in CBL s1 = s 2... any additional assumption is just an assumption International summer school on turbulent diffusion - Vilanova i la Geltrù p. 25

26 Determination of a: bi-gaussian solutions 3 2 Anfossi et al. (1997) Luhar et al. (1996) 1 K 0 Thomson (1987) Weil α=5-1 Weil (1990) Baerentsen and Berkovitz (1984) Du et al. (1994) S bi-gaussian solutions (+ Grahm-Charlier: shaded) International summer school on turbulent diffusion - Vilanova i la Geltrù p. 26

27 Determination of a: Maximum Information Entropy based on pure statistical inference makes the least additional assumption (introduces the least bias) maximisation of H[p(u)] = p(u) log p(u) du if available information are the first N (with N even) moments of a PDF p(u) = exp( k λ k u k ) λ k determined from u k p(u) du = µ k International summer school on turbulent diffusion - Vilanova i la Geltrù p. 27

28 Determination of a: bi-gaussian solutions + max. entropy bi-gaussian PDF with given S = 1, K = 4 and for different values of the free parameter. Solid line is the one determined with maximum entropy criterion. International summer school on turbulent diffusion - Vilanova i la Geltrù p. 28

29 Determination of a: dispersion properties of different PDFs models After T87: diffusion coefficient D = 1 b 2 q(u) 2 p(u) du, q(u) = u vp(v) dv International summer school on turbulent diffusion - Vilanova i la Geltrù p. 29

30 Determination of a: dispersion properties as a function of S and K Lagrangian integral time T L for varying S and K. International summer school on turbulent diffusion - Vilanova i la Geltrù p. 30

31 Determination of a: dispersion properties as a function of S and K Convective Boundary Layer: Ground-level cross-wind integrated concentration as a function of S, for K = 3. International summer school on turbulent diffusion - Vilanova i la Geltrù p. 31

32 Determination of a: conclusions Summary of a determination: big unresolved issue of non-uniqueness non-gaussianity of turbulence requires PDF modelling skewness-kurtosis parameter space is structured (suggests closure) dispersion properties depend on PDF for (given the same S and K) dispersion propertis depend on non-gaussian features International summer school on turbulent diffusion - Vilanova i la Geltrù p. 32