1. Exit times and the generalised dispersion problem Benjamin Devenish and David Thomson Met Office, UK

2. Ballistic vs diffusive process ^ T B » ( ½ ¡ 1 ) r ¾ u » ( ½ ¡ 1 ) r 2 = 3 " 1 = 3 ^ T D » ( ½ ¡ 1 ) 2 r 2 K ( r ) » ( ½ ¡ 1 ) 2 r 2 = 3 " 1 = 3 F or½ ¡ 1 ¿ 1

3. Outline of talk Theoretical results for exit times for a diffusive process Kinematic simulation DNS Lagrangian stochastic model

4. Exit time pdf for diffusive process Exit time pdf Absorbing boundary at r = R Constant flux at r = 0 Initial separation Transformed diffusion equation r 0 = R = ½ @h @ t = @ 2 h @ t 2 ¡ f ( m ; d ) @ @» µ h » ¶ f ( m ; d ) = ( d ¡ 1 + m = 2 )=( 1 ¡ m = 2 ) p E ( t ) = ¡ d d t Z j r j < R p ( r ; t ) d r h ( » ; t ) = r d ¡ 1 + m = 2 p ( r ; t ) » = r 1 ¡ m = 2 =( 1 ¡ m = 2 ) K ( r ) / r m

5. Exit time pdf II Exit time pdf (following Boffetta & Sokolov 2002) Lagrangian relative velocity decorrelation time scale Closed form solutions only for special cases p E ( t ) = 1 2 ~ T ( 2 ¡ m ) 2 ½ ¡ ( 1 ¡ d = 2 ¡ m = 2 ) 1 X k = 1 j º ; k J º ( j º ; k = ½ 1 ¡ m = 2 ) J º + 1 ( j º ; k ) exp µ ¡ 1 4 j 2 º ; k ( 2 ¡ m ) 2 t ~ T ¶ one d i mens i ona l pro bl em i n ( » ; t ) space f or d = 3, m = ¡ 4 ) f ( m ; d ) = 0 d i ® us i v i t y b a l ancescurva t ureo f sp h ere

6. Special case d=3, m=-4 Jacobi’s transformation for theta functions µ 1 ( z ; t ) = ¡ i e i z + ¼ i t = 4 µ 3 ( z + ¼ 2 + ¼ t 2 ) µ 3 ( z ; t ) = ( ¡ i t ) ¡ 1 = 2 e z 2 = ¼ i t µ 3 ¡ z t ; ¡ 1 t ¢ p E ( t ) = 9 2 ~ T s ~ T 9 ¼ t @ @ ¹ 1 X k = ¡ 1 ( ¡ 1 ) k exp à ¡ µ ¹ ¡ 1 2 + k ¶ 2 ~ T 9 t ! p E ( t ) = ¡ 9 ~ T @ @ ¹ X k ( ¡ 1 ) k s i n ·µ k ¡ 1 2 ¶ 2 ¼¹ ¸ exp à ¡ 9 µ k ¡ 1 2 ¶ 2 ¼ 2 t ~ T ! p E ( t ) = 9 2 ~ T @ @ ¹ µ 1 µ ¹ ; 9 t ~ T ¶ ¹ = 1 = ½ 3 E xpressp E ( t ) i n t ermso f t h e t a f unc t i ono f¯ rs t k i n d

7. Exit time pdf for : small times I ½ ¡ 1 ¿ 1 L e t ½ = 1 + ± an d cons i d er l i m i t ± ! 0 f or ¯ xe d t ¿ ~ T p E ( t ) = ¡ s ~ T 9 ¼ t 3 X k ( ¡ 1 ) k µ k ¡ 3 ± 2 ¶ exp à ¡ 1 9 µ k ¡ 3 ± 2 ¶ 2 ~ T t ! + O ( ± 2 ) p E ( t ) » s ^ T D 2 ¼ t 3 exp à ¡ 1 2 ^ T D t ! + O ( ± 2 ) F or ½ ¡ 1 ¿ 1, ^ T D » ± 2 ~ T p E ( t ) » ( ½ ¡ 1 ) R ½ r 1 2 ¼ K t 3 exp µ ¡ ( ½ ¡ 1 ) 2 ½ 2 R 2 2 K t ¶ S i nce ^ T D » ( ½ ¡ 1 ) 2 R 2 = 2 ½ 2 K R es t r i c tt ¿ ^ T D ; l ea d i ngor d er t ermgoverne db ym i n k j k ¡ 3 ± = 2 j

8. Exit time pdf for : small times II ½ ¡ 1 ¿ 1 p E ( t ) » ( ½ ¡ 1 ) R ½ r 1 2 ¼ K t 3 exp µ ¡ ( ½ ¡ 1 ) 2 ½ 2 R 2 2 K t ¶ ² 1 - DB rown i anmo t i on ² M o d eo f p df i s ^ T D ² F or ¯ xe d ½, p E ( t ) ! 0 as t ! 0 ² F or t À ^ T D, p E ( t ) ! 1 = t 3 = 2

9. Exit time pdf for : intermediate times C ons i d er ^ T D ¿ t ¿ ~ T f or l i m i t ± ! 0 p E ( t ) » ± ~ T X k j 2 º ; k exp µ ¡ j 2 º ; k t ~ T ¶ + O ( j 2 º ; k ± 2 ) E xponen t i a l t erm b ecomesneg l i g i bl ew h en j º ; k À ( ~ T = t ) 1 = 2 N ee d t oensure t h a t error i s b oun d e d w h en j º ; k » ( ~ T = t ) 1 = 2 R equ i re j 2 º ; k ± 2 ¿ 1 ) t À ± 2 ~ T ) t À ^ T D s i nce ^ T D » ± 2 ~ T ½ ¡ 1 ¿ 1 Taylor expansion of Bessel function L e t s = j º ; k q t = ~ T I n l i m i t d s ! 0 ( correspon d s t o t ¿ ~ T ) p E ( t ) » ± ~ T Ã ~ T t ! 3 = 2 Z 1 0 s 2 exp ( ¡ s 2 ) d s + O ( j 2 º ; k ± 2 ) Independent of d and m p E ( t ) » ± ~ T 1 X s = 1 ; d s s 2 d sexp ( ¡ s 2 ) Ã ~ T t ! 3 = 2 + O ( j 2 º ; k ± 2 )

10. Exit time pdf for ½ À 1 For small argument p E ( t ) » 1 ~ T X k j º + 1 º ; k J º + 1 ( j º ; k ) exp µ ¡ j 2 º ; k t ~ T ¶ + O ( j º + 3 º ; k ½ m ¡ 2 ) J º ( x ) » x º + O ( x º + 2 ) asx ! 0 p df i s i n d epen d en t o f ½a t l ea d i ngor d er

11. Positive moments of exit time pdf r 2 C 0 = ¡ p ( r ; 0 ) ; r 2 C n = ¡ n C n ¡ 1 f orn > 1 Closed form expressions derivable from hierarchy of Poisson equations: F or½ ¡ 1 ¿ 1 h t n i » ( ½ ¡ 1 ) ~ T n X k j ¡ 2 n º ; k + O ( j ¡ ( 2 n ¡ 1 ) º ; k ± 2 ) ) h t n i / ( ½ ¡ 1 ) f ora ll n F or½ À 1 h t n i » ~ T n X k j ¡ ( 2 n + 1 ) + º º ; k J º + 1 ( j º ; k ) + O ( j ¡ 2 n + º + 1 º ; k ½ m ¡ 2 ) i n d epen d en t o f ½ C n ( r ) = n Z 1 0 p ( r ; t ) t n d t h t n i = n Z j r j < R C n ( r ) d r :

12. Negative moments of exit time pdf F or½ ¡ 1 ¿ 1 h t ¡ n i » 1 ~ T n µ ½ ½ ¡ 1 ¶ 2 n 2 n ¡ ( n + 1 = 2 ) p ¼ ) h t ¡ n i / ( ½ ¡ 1 ) ¡ 2 n A na l y t i ca l express i onson l yposs i bl e f orspec i a l case d = 3, m = ¡ 4

13. Kinematic simulation I Linear superposition of random Fourier modes Prescribed energy spectrum Possible to represent wide range of scales Includes turbulent-like structures e.g. –eddying, straining and streaming regions

14. Kinematic simulation I amplitudes determined by prescribed energy spectrum incompressibility

15. Kinematic simulation II No coupling of modes in k.s. Particles are swept through the small eddies by the large eddies Decreased correlation time of small eddies Particles have less time to be affected by the smaller eddies ) nosweep i ngo f sma ll sca l es b y l argesca l es ) pa i rsw i ll separa t emores l ow l y

16. Kinematic simulation: phenomenology ¿ ( r ) » r U ~ T » r 1 = 3 U " 2 = 3 h t i » U r 1 = 3 " 2 = 3 Separation statistics K ( r ) » ¾ 2 u ¿ ( r ) » " 2 = 3 r 5 = 3 U Exit time statistics ) h r 2 i » " 4 t 6 U 6 ‘take off’ time f or t À ~ T Lagrangian relative velocity time scale

17. Inertial range 1200 modes Unidirectional mean flow Adaptive time step based on local decorrelation time scale Separation statistics Exit time statistics U ( 10 ; 0 ; 0 ) À ¾ u = 1 L = ´ = 10 6 ¡ 10 8

18. Mean exit time Mean square exit time Mean inverse exit time KS statistics

19. KS pdf ½ = 1 : 075

20. ½ = 2

21. Direct numerical simulation Homogeneous isotropic turbulence cubic lattice Taylor-scale Reynolds number Two million Lagrangian particles Sampling rate Data available from Cineca supercomputing centre, Bologna, Italy ¿ ´ = 3 : 3 ¢ 10 ¡ 2, T L = 1 : 2, ´ = 5 ¢ 10 ¡ 3, L = 3 : 14, " = 0 : 81, C 0 = 5 : 2

22. Mean exit time Mean square exit time Mean inverse exit time DNS statistics

23. DNS pdf ½ = 1 : 075

24. Survival probability Probability that a pair will be in sphere of radius R after time t

25. DNS pdf ½ = 2

26. Survival probability Probability that a pair will be in sphere of radius R after time t

27. DNS exit time pdfs No power law scaling for Mean exit time lies within power law scaling range for –relative velocity of average pair decreases faster than decorrelation time scale –majority of pairs separate diffusively Exponential decay of tail agrees with diffusive behaviour for –only slow separators are diffusive –observed with low probability Self-similarity of tail decreases with increasing For tail of pdf affected by and L Tail of pdf for is ‘stretched’ version of tail for

28. Richardson’s constant Scaling of exit time moments according to K41 S i nce C n ( ½ ) = F n ( ½ ) k ¡ n 0 an d g = 1144 = 81 k 3 0 wege t h t n i = C n ( ½ ) r 2 n = 3 = " n = 3 R equ i remo d e l t ore l a t e C n ( ½ ) t og R i c h ar d sons d i ® us i onequa t i onw i t h K ( r ) = k 0 " 1 = 3 r 4 = 3 g = 1144 81 r 2 " µ F n ( ½ ) h t n i ¶ 3 = n

29. Richardson’s constant from positive moments ½ = 1 : 075 ½ = 2

30. Richardson’s constant II Finite duration of simulation –slowest separators do not have time to reach large r Statistical noise Intermittency Velocity memory –little impact on higher positive moments –likely to affect negative moments ² gca l cu l a t e df rommeanex i tt i meappears t o b e i n d epen d en t o f ½ ² ´an d L e ® ec t s {ex t en t o f p l a t eau i ncreasesw i t hd ecreas i ng½ {grea t ere ® ec t f or½ À 1 t h an f or½ ¡ 1 ¿ 1 { h 1 = t 3 i i n d epen d en t o f mean d i ss i pa t i onra t e {sma llb u t a ® ec t s½ ¡ 1 ¿ 1 more t h an½ À 1 {s t a t i s t i cs f or d ecreas i ng½an d r i ncreas i ng l yno i sy

31. Richardson’s constant from negative moments C n ( ½ ) = A n ( ½ ) g ¡ n = 3 D i mens i ona l argumen t s ) C n ( ½ ) / k ¡ n 0 g = r 2 " µ h t ¡ n i A ¡ n ¶ 3 = n A ¡ n ca l cu l a t e df roms t oc h as t i c d i ® eren t i a l equa t i on correspon d i ng t o d i ® us i onequa t i on

32. Richardson’s constant from negative moments ½ = 1 : 075 ½ = 2

33. Richardson’s constant from negative moments II Exit times for DNS larger (slower) than for diffusive process Inverse exit times for DNS smaller than for diffusive process gw i lld ecreasew i t hd ecreas i ng½ ) h 1 = t i f ac t oro f ½ ¡ 1 t oosma ll g » ( ½ ¡ 1 ) 3 f or½ ¡ 1 ¿ 1 ² S i nce ^ T B i scorrec tt i mesca l e f or DNS f or ½ ¡ 1 ¿ 1 ² g ca l cu l a t e df rom h 1 = t i sca l es lik e

34. R i c h ar d sonscons t an t ca l cu l a t e df rom h 1 = t i

35. Lagrangian stochastic model Quasi-one-dimensional Magnitude of separation calculated from longitudinal relative velocity Treat r and v r jointly as continuous Markov process Assume infinite inertial subrange C 0 enters model explicitly –can study effects of velocity memory

36. Lagrangian stochastic model II Pdf of Eulerian velocity difference –weighted sum of three Gaussians –constructed such that first three moments are consistent with K41 a 0 = C 0 dl n f E d» ¡ 7 3 f E Z » ¡ 1 » 0 f E ( » 0 ) d» 0 d» = " 1 = 3 r 2 = 3 a 0 ( » ) d t + " 1 = 6 r 1 = 3 p 2 C 0 d W ( t ) d r = ( "r ) 1 = 3 »d t Drift term Diffusion term » = ( v r = r ) 1 = 3

37. Richardson’s constant from positive moments of Q1D model

38. Richardson’s constant from negative moments of Q1D model

39. ² E rror l arger f orsma ll er½ ² E rror d ecreasesmono t on i ca ll yw i t h n f or½ = 2 ² F or½ = 1 : 075 error d ecreasesmono t on i ca ll yon l y f orn > 1 ² M ean i nvar i an tt o½ ² E rror d ecreasesw i t h i ncreas i ng C 0 Richardson’s constant calculated from positive moments ² E rror l arges t f orsecon d or d ermomen t f or½ = 1 : 075

40. Richardson’s constant from positive moments II g = 1144 81 r 2 " µ F n ( ½ ) h t n i ¶ 3 = n Richardson’s constant calculated from F or d i ® us i veprocess F n ( ½ ) an d h t n i sca l e l i k e½ ¡ 1 ) g » ( ½ ¡ 1 ) 3 ( 1 ¡ n )= n F or½ ¡ 1 ¿ 1 F or b a ll i s t i cprocess h t n i sca l es l i k e ( ½ ¡ 1 ) n I n d epen d en t o f ½ f orn = 1

41. gca l cu l a t e df rom h t 2 i f or C 0 = 1

42. Conclusions Physics of separation process intimately related to spacing of thresholds Kinematic simulation reaches its diffusive limit earlier than real turbulence In real turbulence velocity memory is important Spacing of thresholds and order of moment important for calculating Richardson’s constant d i ® us i ve l i m i t reac h e d on l y f or l arge½

43. Correlation of relative velocity Level of decorrelation depends on constant in this relation S i nce K » ¾ 2 u ¿, ¾ u » " 1 = 3 r 1 = 3 an d r 2 » " t 3 ) ¿ » t corre l a t i on t i mesca l eo f or d er t h e t rave l t i me

44. Richardson’s constant K41 Richardson diffusion equation h r 2 i » " t 3 strong law weak law