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Statistics of eddy transport

Madalina Vlad, Florin Spineanu

National Institute of Laser, Plasma and Radiation Physics

Magurele-Bucharest, Romania

Mail: [email protected]

Non-equilibrium Statistical Mechanics and the

Theory of Extreme Events in Earth Science

29 October 2013 - 1 November 2013

Isaac Newton Institute for Mathematical Sciences, Cambridge

Outline

• We present a semi-analytical study of the test particle eddy diffusion in stochastic

velocity fields

• We have developed a semi-analytical approach for the case of tokamak plasmas

that applies to the nonlinear process (the decorrelation trajectory method DTM).

• We have shown that the nonlinear effects are non-trivial in many situations.

• There are conditions that make the eddy diffusion a process produced by rare

events or even by extreme events

Aim To develop this statistical method for the more complicated case of a model

for clouds and to find if the nonlinear effects are as strong as in the case

of fusion plasmas

Content

1. Introduction : diffusion by continuous movements

2. The decorrelation trajectory method

3. The model for vapor eddy diffusion in clouds

4. Eddy transport in 2-dimensional model; effect of molecular diffusivity

5. Eddy transport in 3-dimensional model

6. Conclusions

References

‘Diffusion in disordered media’ , ‘stochastic advection’ (many applications in fluid and plasma turbulence, in solid state physics, polymers, etc)

Non-linear stochastic equation:

where is a continuous field in each realization. It is statistically described as a stochastic velocity field with Gaussian distribution and known Eulerian correlation [a deterministic equation for each realisation of v(x,t) with an unique solution x(t)] To determine: The statistical properties of the trajectories

( )( ( ), ) (0) 0

dx tv x t t x

dt

( , )v x t

1 1 1 1( , ) ( , ) ( , )ij i jE x t v x t v x x t t

dt

txdtDtx

)(

2

1)(,)(

2

2


V

VK c

fl

fl

c

c

c

,

• V the amplitude

• the correlation length

• the correlation time

The Kubo number :

c

c

The Eulerian correlation EC of the stochastic field:

cc

cc

ij

txf

f

txfVtxE

,,0

,1)0,0(

,,),( 2

-The nonlinear character the space dependence of the EC

-The time dependence determines a decorrelation mechanism (measured by K)

fl

ddd K

,

- Other decorrelation mechanisms (collisions, average

velocity, etc.). Each has a characteristic time and

defines a dimensionless parameter (similar with K)


The correlation of the Lagrangian velocity (LVC):

)),(()0,0()( ttxvvtL jiij

,')'()(0

t

xx dttLtD ,')'('2)(0

2

t

xx dttLtttx

One hundred year old problem

Taylor formulas for diffusion by continuous movements :

o Many cases: weak nonlinear effects (numerical factors are changed but not the dependence on parameters)

o Special cases: strong nonlinear effects (parameter dependence is completely changed; subdiffusion or superdiffusion)

Essentialy due to the existence of conservation laws in the individual dynamics

(invariance of the stream function on each trajectory for 2d stochastic zero divergence

velocity fields)

0

')'( dttLD xx

2. The statistical methods

What can be done with test particle?

Studies of turbulent transport, with specific aims, complementary to the self consistent

approach (usually based on simulations)

Start from a model of turbulence (EC, parameters, other components of the motion)

Determine the transport as function on the turbulence parameters and of the EC (spectrum).

Transport regimes

The same results with those based on turbulent fluxes are obtained if the space

and time scales of the average quantities are much larger than that of the

fluctuations

To determine:

The statistical properties of the trajectories: MSD, D(t), the probability.

Knowing : The statistical description of the velocity field : the probability, the EC txv ,

2. The statistical methods

• Hamiltonian system

• A typical trajectory for large K: random sequence of

trapping events and long jumps

Trapping event

Long jump

a)

b)

( , ) 0

( , ) z

v x t

v x t e


Special case: 2d divergenceless

velocity fields

2K 01K

Trapping is generic; the fraction of trapped trajectories increases with K

Trapping determines increased coherence of the motion and generation of

quasi-coherent trajectory structures


o All the theoretical methods (Corrsin approximation,

direct interaction approximation – DIA, renormalization group technique, etc)

do not describe trajectory trapping.

o In particular, they are not in agreement with the condition

and lead to diffusion

),(0 dcKwhenD

KwhenctD .


Semi-analytical statistical methods

The decorrelation trajectory method (DTM)

M.Vlad, F. Spineanu, J. H. Misguich, R. Balescu, “Diffusion with intrinsic trapping in 2-d

incompressible stochastic velocity fields”, Physical Review E 58 (1998) 7359

The nested subensemble method (NSM)

M. Vlad, F. Spineanu, “Trajectory structures and transport”, Physical Review E 70 (2004)

056304(14))

DTM is shown to be the first order in a systematic expansion

DTM and NSM are based on a set of simple trajectories determined from the

Eulerian correlation EC of the stochastic potential,

the decorrelation trajectories

The main idea of this approach is to determine the Lagrangian averages not on

the whole set of trajectories but to group together trajectories that are similar,

to average on them and then to perform averages of these averages.


• Similar trajectories are obtained by imposing suplementary initial conditions

besides the necessary one.

• Particularly important initial conditions are provided by the conserved

quantities (the stream function in this case).

.

- The Eulerian stream function and velocity in the subensemble (Sn) have average

values that depend on the EC of the potential and of the parameters of the

subensemble. The fluctuation amplitudes in (Sn) are much smaller than in R.

- The approximation : neglect trajectory fluctuations in (Sn) [super-determined

trajectories in (S) and small fluctuations of the Eulerian velocity]

The subensembles (S1) : 0)0,0( vv

,)0,0( 0

The subensembles (S2) : , …. (Sn)

The space of realization (R) = Σ subensembles (S1),

Each subensemble (S1) = Σ subensembles (S2), …


The subensembles (S) : ,)0,0( 0vv

0)0,0(

Sjiij ttxvvvPvddtL )),((),()( 00000

The LVC is the sum of the contributions of each subensemble (S):

SjiSji ttxvvttxvv )),(()),(()0,0( 0

The invariance property of the Lagrangian potential

t

ttx

t

ttx

x

ttxttxv

dt

ttxd

i

i

)),(()),(()),(()),((

)),((

The LVC (2-point average) the average Lagrangian velocity (1-point average)

The average Lagrangian potential in (S) can be determined for frozen turbulence

,)0())0(())(( 0 xtx

S

tx ))((

DTM method implies one level of subensembles

2. The decorrelation trajectory method – 2d zero divergence velocity

The Eulerien potential and velocity fields in (S) are non-stationary and non-homogeneous Gaussian fields with space-time dependent acerages

2020 //),(),( aEVEvtxtx ii

S

S

2020 //),(),( aEVEvtxVtxv jiji

S

jSj

0)0,0( S 0),( txS and as ,x

t

0)0,0( vV S 0),( txV S

and as ,x

t

),()0,0(),(,),()0,0(),(,),()0,0(),( txvtxEtxvvtxEtxtxE iijiij

are determined by the EC of the potential as: .,,21

2

122

2

2

11 etcExx

EEx

E

),(,),(12

txxx

txV SS

Zero-divergence average velocity in (S)

These subensemble (conditional) averages describe the structure of the correlated zone

ijE


The average Lagrangian velocity in (S)

),(),12

XVXXXdt

Xd SS

The average trajectory in (S) is the solution of the Hamiltonian system:

0);0( SX

));(())(( StXVtxv S

S

The approximation of the decorrelation trajectory method:

The decorrelation trajectories

2. The decorrelation trajectory methd 2. The decorrelation trajectory method – 2d zero divergence velocity

• The decorrelation trajectories (DT) are determined from the EC of the stochastic fields.

• DT’s are smooth, simple trajectories much different from particle trajectories.

• DT’s represent the characteristics of the transport.

• Gaussian transport (standard diffusion by continuous movements):

the DT’s have straight line paths along the initial velocity and saturate at

a distance where is a decorrelation time.

dvL 0 d

d

LD

2

...,2.0,1.0,00

• transport determined by 2d divergenceless

velocity in the nonlinear regime :

the DT’s have closed paths and trajectories wind

with decaying velocity

They represent

trapping of eddying type


The LVC and the diffusion coefficient

(stationary, homogeneous stochastic potential)

)()(')( 2 thtKFtL ijij

( ) ( )D t F t

),(12

exp2

1)(

0 0

22

3 ptXpu

ududptF

The function F is :

,/, 00 upvu

t

thdtt0

)'(')(

is the decorrelation trajectory in (S) along for the static turbulence

),( puX 0v

h(t) - the time dependence of the EC of the potential;

- describes the decorrelation due to the time variation of

F(t) - determined by the nonlinearity (by the space-dependece )

- describes the trapping in the structure of the stochastic field

)(x

),tx

D(t) results from a competition between trapping and release of the trajectories:

)()(),( thxtxE


00 0

22

3 ),(2

1),(1

2exp

2

1)(

pfdppuXp

uududpF

f(p,τ)

• Static case:

-The grouth of the MSD is determined at

large τ by a small number of particles which

are not yet trapped; they are on the large

contour lines of the potential (p around 0).

- D(τ) is determined at large time by

rare events

• Time-dependent case:

The large size structures are destroyed and the

corresponding trajectories are released.

They determine a finite diffusion coefficient.

DTM is able to obtain effects produced by

very small fractions of trajectories p


The function F - describes trajectory trapping, effect of the non-linearity

The function F decays because

the contribution of the small paths

is progresivelly eliminated;

(In the absence of trapping

the function F saturates at τ>1 and

the diffusion coefficient is of Bohm type.

2/1

1)(

2xx

orDtDtF B

static

fl /)(/ BDKDKF /)(The function F gives :

• the time-dependence of the diffusion coefficient in frozen turbulence; • the K-dependence of the asymptotic diffusion coefficient for time dependent turbulence


Memory effects due to trapping; Long-time Lagrangian correlation

L(t) has long negative tail of power law type; positive and negative parts compensate.

Unstable system: any weak perturbation has a strong influence on the

transport and anomalous regimes are obtained.

fl

c

t

xatxE

exp

2/1

1),(

2

2

BDD /Subdiffusive transport


Trapping determines nontrivial (unexpected) nonlinear effects in more

complicated systems.

This special shape of the Lagrangian velocity correlation in the static potential explains

these results. Various perturbations can be produced by other components of the motion

(particle collisions, average velocities, parallel motion, gradients of the confining

magnetic field, toroidal geometry, etc). They strongly modify the transport.


3. Model for the stochastic motion in clouds

Cloud – stochastic blobs of buoyant air at scales much smaller than the cloud

• We define a statistical ensemble of parcels by the statistical description of

temperature fluctuations and of the vapor mixing ratio at small scales.

• A pressure gradient appears due to the z variation of the buoyancy, the

buoyancy pressure-gradient acceleration (BPGA). It determines a vortical

(accelerated) motion (a ring vorticity) and leads to the motion of the air

around the parcel (it is pushed away by the ascending parcel and replaces the

air behind the parcel). It also determines the decrease of the vertical velocity

of the parcel.

• Stochastic motion of the air inside the cloud at small scale

• We study eddy transport of the vapors inside the cloud using test particle

approach

• The entrainment-detrainment and dilution are stochastic processes that

appear in the volume of the cloud

• The turbulent motion is 3-dimensional

• The velocity field is describe by a vector potential

• The function f determines horizontal rings of vorticity; they are

perturbed by the function φ, which determines a vertical vorticity

• Vortical components of the velocity appear in each plane

• The components of A represent partial stream functions for each plane

• An average vertical velocity produced by buoyancy Vd

,,,

,,

yyxxyzxxzy

xy

ffffAv

ffA

xy

yyyz

xxxz

yxplane

ffzyplane

ffzxplane

,:),(

,:),(

,:),(


• The correlations of the functions f and φ are taken Gaussian (because the blobs

are separated), with horizontal correlation length much smaller than the vertical

correlation

• The functions f and φ are statistically independent

• We focus on the study of diffusion (mixing) of vapors and liquid water in this

type of 3-dim turbulence

• The entrainment is determined by the increase of the size of the blobs as they

move vertically. It also depends on the liquid mixing ratio that exists in the

cloud due to the contribution of previous blobs.

We present:

1. The 2d case corresponding to functions f and φ dependent on x and z

(the effects of the average velocity produced by the buoyancy)

2. The effects of the molecular diffusivity on eddy diffusion in 2d

3. Eddy diffusion in 3d velocity field

? Are the nonlinear effects as strong in this 3d velocity fields ?


Effects of an average flow on eddy diffusion

Dimensionless parameter: Flow velocity:

and characteristic time (the drift-time over the distance )

V

VV d

d

1

2

,d

dx dV

dt dx

2

1

dx d

dt dx

Constant average gradient of the stochastic stream function:

Particle motion :

.0),(,),(,),( 21 txvVtxvyVtx dd

d

dV


Hamiltonian system

The average flow has a strong influence on the diffusion coefficient

even if it is not sheared. Trapping effects appear for 1,1 dVK

, 0,xz xxv f f

Very large amplification of the

diffusion along the average

velocity:

- ballistic transport for static

potential

- D ~ K at large K

Strong decrease of the diffusion

in the perpendicular direction

at large K

K

3.0dV


( , )dD K V( ), 0, 0.3dD K V

)()( KFDKD B

1dVKNo dependence on

Isotropic diffusion dV

1dVQuasilinear regime 1,1 dd VVK

Trapping regime

Decrease of the radial diffusion and

strong increase of the poloidal one

dc V

VKD

2

11

c

cd

d

V

VD

'

122

Diffusion regimes

( ),

4,20,100

dD V

K


Physical explanation: the average velocity determines an average stream function

which modifies the configuration of the total stream function

o For vertical strips of open contour lines appear

o For all the contour lines are opened and there is no trapping.

A part / all trajectories are released by the average velocity

1dV

1dV


V

VV d

d


The probability of vertical displacements splits in two parts:

- Trapped particles remain around z=0

- Free particles move along the opened lines of the stream function

1dV

1dV

This is not a dynamical effect, but it is determined by the selection of the velocity on the free paths

• exactly compensates the trapped particles:

dff VVntxv ))((

df VV fV

The statistical invariance of the Lagrangian velocity determines the

“acceleration” of the free particles on the opened lines of the stream function


df d

f

VV V

n

The eddy diffusion is produced by

rare and extreme events


( ( )) 0 0tr d f fv x t n V n V

The velocity field is advected in the clouds with the average vertical velocity

(Galilean transformation; the diffusion coefficients are not changed)

Opposite flows of trapped and free particles

are generated in the presence of trapping

Trapped particles move with the average velocity and the free particles have an

opposite average motion that compensates the flux of trapped particles

Stochastic downdrafts appear due to vapor trapping in the turbulent velocity

field advected by the buoyancy

),(1

2

1 tdx

d

dt

dx

)(2

1

2 tdx

d

dt

dx

)(ti Gaussian white noise

),'()'()( tttt yx 2

2

2

thV

is the collisional diffusivity

Dimensionless parameter : Péclet Number

fl

collBDP

/2

ccoll

(the diffusion time over the correlation length,

the collisional decorrelation time)

Characteristic time (decorrelation mechanism)

Effects of collisions (molecular diffusion, random noncorrelated perturbation)

4. Eddy transport in 2-dimensional model – Molecular diffusion effects

The asymptotic diffusion coefficient ),( PKDDeff

trapping

Strong

nonlinear increase

)(),( KFDPKD B

KP )( ccoll

No dependence on P

1P

Quasilinear regime

,),( PDPKD B

1 PK

,),( 2.0KDPKD B

)( flcollc

( , )effD D K P

4. Eddy transport in 2-dimensional model – Molecular diffusion effects

• 3-dimensional turbulent motion

• The velocity field is describe by a vector potential

• The system is not Hamiltonian

• But, the components of A represent partial stream functions for each

plane, perturbed by the other components

Study of eddy diffusion in this 3d velocity field

,,,

,,

yyxxyzxxzy

xy

ffffAv

ffA

( , ) ,

( , ) ,

( , ) ,

x fx xz xx

y fx yz yy

y x

plane x z f leads to v f f

plane y z f leads to v f f

plane x y leads to v


Statistical study using the DTM

- Large number of parameters

- The subensembles are defined with the initial values of the components

of A and of the velocity

- The average velocities in the subensemble are calculated and the then the

decorrelation trajectories. They are much more complicated than in 2-d.

Trapping exists, but with more complicated topology, thus the mixing

ratio for these “particles” is not changed during the rise of the buble


2 1 2 1 2

1 2, , / , / , , ,II IIK a

2)2(1)1(

1

1)1( )0,0,0(,)0,0,0(,)0,0,0(:)( aAaAaAS zi

• The decorrelation is produced by the average vertical motion (buoyancy)

• “particles” from the central part of the bubles escape lateraly (at distances

larger that the buble size and they move down (relative to the buble).

Detrainment appears in this model



a is the ratio of the amplitude of f(x,y,z) over the amplitude of φ(x,y,z)

Increase of D as the relative

amplitude of the component

f increases

• Trapping effects can be

statistically relevant in 3d,

thus undiluted parts of the

blobs persists

• Increased dilution in 3d

compared to the 2d case

• An intrinsic decorrelation

mechanism in 3d, which

eliminate the subdiffusive

regime

The decorrelation trajectory method (DTM) and the NSA

• It appears to be a statistical method that is able to describe the strong

nonlinear effects that can be generated in eddy transport

• It warks also for the conditions in which the process is essentially

produced by rare and/or extrem events.

The results on vapor eddy diffusion in a cloud

• Eddy diffusion in the small scale turbulence inside of a cloud produce the

dilution of only a part of each blob (trapped particles do not diffuse)

• Stochastic downdrafts are generated inside clouds

• Minimum eddy diffusion (thus dilution) appears when the geometry is

approximately 2d

• The 3d stochastic velocity fields contain an intrinsic decorrelation

mechanism, which increses the diffusion coefficients

Many possible developments and applications exists

6. Conclusions

Main publications on test particle transport based on DTM and NSA

•M. Vlad, F. Spineanu, J.H. Misguich, R. Balescu, ”Diffusion with intrinsic trapping in 2-d incompressible velocity

fields”, Physical Review E 58 (1998) 7359-7368.

•M. Vlad, F. Spineanu, “Trajectory structures and transport”, Physical Review E 70 (2004) 056304(14))

•M. Vlad, F. Spineanu, J.H. Misguich, R. Balescu, “Collisional effects on diffusion scaling laws in electrostatic

turbulence”, Physical Review E 61 (2000) 3023-3032.

• M. Vlad, F. Spineanu, J.H. Misguich, R. Balescu, “Diffusion in biased turbulence”, Physical Review E 63 (2001)

066304. • M. Vlad, F. Spineanu, J.H. Misguich, R. Balescu, “Reply to ‘Comment on Diffusion in biased turbulence’”, Physical Review E 66 (2002) 038302.

• M. Vlad, F. Spineanu, J.H. Misguich, R. Balescu, “Electrostatic turbulence with finite parallel correlation length and

radial diffusion”, Nuclear Fusion 42 (2002), 157-164.

•M. Vlad, F. Spineanu, J.H. Misguich, R. Balescu, “Magnetic line trapping and effective transport in stochastic magnetic

fields”, Physical Review E 67 (2003) 026406.

• M.Vlad, F. Spineanu, J. H. Misguich, J.-D. Reusse, R. Balescu, K. Itoh, S. –I. Itoh, “Lagrangian versus Eulerian

correlations and transport scaling”, Plasma Physics and Controlled Fusion 46 (2004) 1051-1063.

• R. Balescu, M. Vlad, F. Spineanu, International Journal of Quantum Chemistry 98 (2004) 125-130,

Special Issue: Complexity: Microscopic and Macroscopic Aspects; issue edited by Ioannis Antoniou, Albert

Goldbeter and René Lefever

• M.Vlad, F. Spineanu, J. H. Misguich, J.-D. Reuss, R. Balescu, K. Itoh, S. –I. Itoh, ‘Turbulence spectrum and

transport scaling’, Journal of Plasma and Fusion Research Series 6 (2004) 249-252

•Vlad M. and Spineanu F., Plasma Phys. Control. Fusion 47 (2005) 281.

• M. Vlad, F. Spineanu, S.-I. Itoh, M. Yagi, K. Itoh , “Turbulent transport of the ions with large Larmor radii”,

Plasma Physics and Controlled Fusion 47 (2005) 1015

•M. Vlad, F. Spineanu, S. Benkadda, “Impurity pinch from a ratchet process”, Physical Reviews Letters 96

(2006) 085001.

• M. Vlad, F. Spineanu, S. Benkadda, “Collision and plasma rotation effects on ratchet pinch”, Physics of

Plasmas 15 (2008) 032306.

• M. Vlad, F. Spineanu, S. Benkadda, “Turbulent pinch in non-homogeneous confining magnetic field”,

Plasma Physics Controlled Fusion 50 (2008) 065007.

• M. Vlad, F. Spineanu, “Vortical structures of trajectories and transport of tracers advected by turbulent

fluids”, Geophysical and Astrophysical Fluids Dynamics 103 (2009) 143.

• M. Vlad, F. Spineanu, „Trapping, anomalous transport and quasi-coherent structures in magnetically

confined plasmas”, Plasma and Fusion Research 4 (2009) 053

• M. Vlad, F. Spineanu, “Ion transport in drift type turbulence”, Physics of Plasmas, submitted 2013

Examples of other people work with the DTM method M. Neuer, K. Spatschek, Phys. Rev. E 74 (2006) 036401

T. Hauff, F. Jenko, Physics of Plasmas 13 (2006) 102309

Petrisor I., Negrea M., Weyssow B., Physica Scripta 75 (2007) 1.

Negrea M., Petrisor I., Balescu R., Physical Review E 70 (2004) 046409.

Balescu R., Physical Review E 68 (2003) 046409.

Balescu R., Petrisor I., Negrea M., Plasma Phys. Control. Fusion 47 (2005) 2145.

Still many other applications and developments to be done (plasmas, fluids, astrophysics,…)

First results on turbulence evolution (magnetically confined plasmas) • M. Vlad, „Ion stochastic trapping and drift turbulence evolution”, Physical Review E 87 053105 (2013).

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