Exit times and the generalised dispersion problem

Benjamin Devenish and

David Thomson

Met Office, UK

Ballistic vs diffusive process

T̂B » (½¡ 1)r¾u

» (½¡ 1)r 2=3

"1=3

T̂D » (½¡ 1)2r 2

K (r )

» (½¡ 1)2r 2=3

"1=3

For ½¡ 1¿ 1

Outline of talk

• Theoretical results for exit times for a diffusive process

• Kinematic simulation

• DNS

• Lagrangian stochastic model

Exit time pdf for diffusive process

• Exit time pdf

• Absorbing boundary at r = R

• Constant flux at r = 0

• Initial separation

• Transformed diffusion equation

r0 = R=½

@h@t =

@2h@t2

¡ f (m;d)@@»

µh»

¶

f (m;d) = (d¡ 1+ m=2)=(1¡ m=2)

pE (t) = ¡ddt

Z

jr j<Rp(r; t) dr

h(»;t) = rd¡ 1+m=2p(r;t)

»= r1¡ m=2=(1¡ m=2)

K (r) / rm

Exit time pdf II

• Exit time pdf (following Boffetta & Sokolov 2002)

• Lagrangian relative velocity decorrelation time scale

• Closed form solutions only for special cases

pE (t) =1

2~T(2¡ m)2½¡ (1¡ d=2¡ m=2)

1X

k=1

j º;kJ º (j º;k=½1¡ m=2)

J º+1(j º;k)exp

µ¡

14j 2º;k(2¡ m)2 t

~T

¶

one dimensional problem in (»;t) space

for d = 3, m = ¡ 4 ) f (m;d) = 0di®usivity balances curvature of sphere

Special case d=3, m=-4

Jacobi’s transformation for theta functions

µ1(z;t) = ¡ iei z+¼it=4µ3(z + ¼2 + ¼t

2 )

µ3(z;t) = (¡ it)¡ 1=2ez2=¼itµ3

¡zt ; ¡ 1

t

¢

pE (t) =9

2~T

s~T

9¼t@@¹

1X

k=¡ 1

(¡ 1)k exp

Ã

¡µ

¹ ¡12

+k¶2 ~T

9t

!

pE (t) = ¡9~T

@@¹

X

k

(¡ 1)k sin· µ

k ¡12

¶2¼¹

¸exp

Ã

¡ 9µ

k ¡12

¶2 ¼2t~T

!

pE (t) =9

2~T

@@¹

µ1

µ¹ ;

9t~T

¶¹ = 1=½3

Express pE (t) in terms of theta function of ¯rst kind

Exit time pdf for : small times I

½¡ 1¿ 1

Let ½= 1+±and consider limit ±! 0 for ¯xed t ¿ ~T

pE (t) = ¡

s~T

9¼t3

X

k

(¡ 1)k

µk ¡

3±2

¶exp

Ã

¡19

µk ¡

3±2

¶2 ~Tt

!

+O(±2)

pE (t) »

sT̂D

2¼t3exp

Ã

¡12

T̂D

t

!

+O(±2)

For½¡ 1¿ 1, T̂D » ±2~T

pE (t) »(½¡ 1)R

½

r1

2¼K t3exp

µ¡

(½¡ 1)2

½2

R2

2K t

¶Since T̂D » (½¡ 1)2R2=2½2K

Restrict t ¿ T̂D ; leading order term governed by minkjk ¡ 3±=2j

Exit time pdf for : small times II

½¡ 1¿ 1

pE (t) »(½¡ 1)R

½

r1

2¼K t3exp

µ¡

(½¡ 1)2

½2

R2

2K t

¶

² 1-D Brownian motion

² Modeof pdf is T̂D

² For ¯xed ½, pE (t) ! 0 as t ! 0

² For t À T̂D , pE (t) ! 1=t3=2

Exit time pdf for : intermediate times

Consider T̂D ¿ t ¿ ~T for limit ±! 0

pE (t) »±~T

X

k

j 2º;k exp

µ¡ j 2

º;kt~T

¶+ O(j 2

º;k±2)

Exponential term becomes negligible when j º;k À ( ~T=t)1=2

Need to ensure that error is bounded when j º;k » ( ~T=t)1=2

Require j 2º;k±2 ¿ 1 ) t À ±2 ~T ) t À T̂D since T̂D » ±2 ~T

½¡ 1¿ 1

Taylor expansion of Bessel function

Let s = j º;k

qt=~T

In limit ds ! 0 (corresponds to t ¿ ~T)

pE (t) »±~T

Ã~Tt

! 3=2 Z 1

0s2 exp(¡ s2) ds +O(j 2

º;k±2)Independent of d and m

pE (t) »±~T

1X

s=1;ds

s2dsexp(¡ s2)

Ã~Tt

! 3=2

+ O(j 2º;k±2)

Exit time pdf for

½À 1

For small argument

pE (t) »1~T

X

k

j º+1º;k

J º+1(j º;k)exp

µ¡ j 2

º;kt~T

¶+ O(j º+3

º;k ½m¡ 2)

J º (x) » xº + O(xº +2) as x ! 0

pdf is independent of ½at leading order

Positive moments of exit time pdf

r 2C0 = ¡ p(r;0); r 2Cn = ¡ nCn¡ 1 for n > 1

Closed form expressions derivable from hierarchy of Poisson equations:

For ½¡ 1¿ 1

htn i » (½¡ 1) ~TnX

k

j ¡ 2nº;k + O(j ¡ (2n¡ 1)

º ;k ±2)

) htn i / (½¡ 1) for all nFor ½À 1

htn i » ~TnX

k

j ¡ (2n+1)+ºº;k

J º +1(j º ;k)+ O(j ¡ 2n+º +1

º;k ½m¡ 2)

independent of ½

Cn(r) = nZ 1

0p(r;t)tndt htn i = n

Z

jr j<RCn(r) dr:

Negative moments of exit time pdf

For ½¡ 1¿ 1

ht¡ n i »1~Tn

µ½

½¡ 1

¶2n 2n¡ (n + 1=2)p

¼

) ht¡ n i / (½¡ 1)¡ 2n

Analytical expressions only possible for special case d = 3, m = ¡ 4

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

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

) no sweeping of small scales by large scales

) pairs will separate more slowly

Kinematic simulation: phenomenology

U

r

2/3

1/3

rr

U

1/3 1/3 r

¿(r) » rU

~T » r 1=3U"2=3

hti » Ur 1=3

"2=3

Separation statistics

K (r) » ¾2u¿(r)

»"2=3r5=3

U

Exit time statistics

) hr2i »"4t6

U6

‘take off’ time

for t À ~T

Lagrangian relative velocity time scale

• 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 ¾u = 1

L=́ = 106 ¡ 108

Mean exit time Mean square exit time

Mean inverse exit time

KSstatistics

KS pdf

½= 1:075

KS pdf

½= 2

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

07.0

31024

280R

¿́ = 3:3¢10¡ 2, TL = 1:2, ´ = 5¢10¡ 3, L = 3:14, " = 0:81, C0 = 5:2

Mean exit time Mean square exit time

Mean inverse exit time

DNS statistics

DNS pdf

½= 1:075

Survival probability

• Probability that a pair will be in sphere of radius R after time t

DNS pdf

½= 2

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

2

075.1

2

075.12

075.1

Richardson’s constantScaling of exit time moments according to K41

Since Cn(½) = Fn(½)k¡ n0 and g= 1144=81k3

0 weget

htn i = Cn(½)r2n=3="n=3

Require model to relate Cn(½) to g

Richardsons di®usion equation with K (r) = k0"1=3r4=3

g =114481

r2

"

µFn(½)htn i

¶3=n

Richardson’s constantfrom positive moments

½= 1:075

½= 2

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 g calculated from mean exit timeappears to be independent of ½

² ´ and L e®ects

{ extent of plateau increases with decreasing ½

{ greater e®ect for ½À 1 than for ½¡ 1¿ 1

{ h1=t3i independent of mean dissipation rate{ small but a®ects ½¡ 1¿ 1 more than ½À 1

{ statistics for decreasing ½and r increasingly noisy

Richardson’s constant from negative moments

Cn(½) = An(½)g¡ n=3

Dimensional arguments ) Cn(½) / k¡ n0

g=r2

"

µht¡ n iA¡ n

¶3=n

A¡ n calculated from stochastic di®erential equationcorresponding to di®usion equation

Richardson’s constantfrom negative moments

½= 1:075

½= 2

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

g will decreasewith decreasing ½

) h1=ti factor of ½¡ 1 too small

g» (½¡ 1)3 for ½¡ 1¿ 1

² Since T̂B is correct timescale for DN S for ½¡ 1¿ 1

² g calculated from h1=ti scales like

Richardsons constant calculated from h1=ti

Lagrangian stochastic model

• Quasi-one-dimensional

• Magnitude of separation calculated from longitudinal relative velocity

• Treat r and vr jointly as continuous Markov process

• Assume infinite inertial subrange

• C0 enters model explicitly

– can study effects of velocity memory

Lagrangian stochastic model II

• Pdf of Eulerian velocity difference– weighted sum of three Gaussians– constructed such that first three

moments are consistent with K41

a0 = C0dlnf E

d»¡

73f E

Z »

¡ 1»0f E (»0) d»0

d»="1=3

r2=3a0(»)dt +

"1=6

r1=3

p2C0dW(t)

dr = ("r)1=3»dt

Drift term Diffusion term

»= (vr =r)1=3

Richardson’s constant from positive moments of Q1D model

Richardson’s constant from negative moments of Q1D model

² Error larger for smaller ½² Error decreases monotonically with n for ½= 2

² For ½= 1:075 error decreases monotonically only for n > 1

² Mean invariant to ½

² Error decreases with increasing C0

Richardson’s constantcalculated from positivemoments

² Error largest for second order moment for ½= 1:075

Richardson’s constant from positive moments II

g =114481

r2

"

µFn(½)htn i

¶3=n

Richardson’s constant calculated from

For di®usiveprocess Fn(½) and htn i scale like½¡ 1

) g» (½¡ 1)3(1¡ n)=n

For ½¡ 1¿ 1

For ballistic process htn i scales like (½¡ 1)n

Independent of ½for n = 1

g calculated fromht2i for C0 = 1

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

di®usive limit reached only for large½