Non-diffusive transport in pressure driven plasma turbulence

D. del-Castillo-Negrete

B. A. Carreras

V. Lynch

Oak Ridge National Laboratory

USA

20th IAEA Fusion Energy ConferenceVilamoura, Portugal. 1-6 Nov, 2004

Beyond the standard diffusive transport paradigm

•Experimental and theoretical evidence suggests that transport in fusion plasmas deviates from the diffusion paradigm:

t T x x, t x T S(x,t)

• Fast propagation and non-local transport phenomena• Inward transport observed in off-axis fueling experiments

•Our goal is to develop transport models that overcome the limitations of the diffusion paradigm.

•The models incorporate in a unified way non-locality, memory effects, and non-diffusive scaling.

•To motivate and test the models we consider transport in pressure driven plasma turbulence

Examples:

(1)

Non-diffusive transport

Tracers dynamics

t ˜ V 2 ˜

B0

min0rc

1

r

˜ p

1

min0 R0

||2 ˜ ||

4 ˜

t ˜ V ˜ p p

r

1

r

˜

2 ˜ p || ||

2 ˜ p

p

t

1

r

r

r ˜ V r ˜ p S0 D1

r

r

r p

r

dr

dtV

1

B2 ˜ B

r 2 ~ t4 / 3Levy

distribution

Probability density function of radial displacements

Super-diffusive scaling

Pressure gradient driven plasma turbulence

(2)

Towards an effective transport model for non-diffusive turbulent transport

dr

dt ˜ V

1

B2 ˜ B

P

t ˜ V P 0

•Individual tracers follow the turbulent field

•The distribution of tracers P evolves according to

•The idea is to construct a model that “encapsulates” the complexity of the turbulence field in an effective flux and reproduces the observed pdf

homogeneous isotropic turbulence

x P

P

t

x

Gaussianclosures

diffusivetransport

???????transport in pressure driven plasma turbulence

˜ V

(3)

Non-Gaussiandistributions

What is the origin of non-diffusive transport?

ExB flow velocity eddiesinduce large tracer trappingthat leads to temporal non-locality, or memory effects

Tracer orbits

Trapped orbit

“Levy”flight

“Avalanche like” phenomena induce largetracer displacements that lead to spatial non-locality

The combination of tracer trapping and flights leads to non-diffusivetransport

(4)

Continuous time random walk model

= jump n

nn

n = waiting time = waiting time pdf

= jump size pdf

P(x,t ) (x) (t' )dt' (t t ' ) (x x' ) P(x' , t' )dx'

0

t

t

dt'

Contribution from particles that have not moved during (0,t)

Contribution from particles located at x’ and jumping to x during (0,t)

( ) ~ e

( ) ~ e 2 / 2

No memory

No long displacements

t P x x P S

Standard diffusion model(5)

MasterEquation(Montroll-Weiss 1965)

Proposed transport model

P

t

x

w w r

t

d dy K x y;t a

x

0

t

P(y,)

( ) ~ (1)

( ) ~ ( 1)

Long waiting times

Long displacements (Levy flights)

Non-local effects due to avalanches causing Levy flights modeled with integral operators in space.

K x y;t 1

t 1 x y

Non-Markovian, memory effects due to trapping in eddies modeled with integral operators in time.

P

t

P

x

Equivalent formulationusing fractional derivatives

For pressure driven plasma turbulence

3/ 4 1/ 2(6)

Algebraic decay in space due to “Levy flights” implies that there is no characteristic transport scale

Test of fractional transport model P

t

P

x

Algebraic scaling in time caused by memory effects

Turbulence model

~ x (1)

Levy distribution at fixed time

Turbulence

~ t

model

Pdf at fixed point in space

~ t

(7)

Super-diffusive scaling

Pressure fluctuations W ~ x2

turbulence

model

r 2 ~ t2

Diffusive scaling

1/ 2Tracers

0.63 / 2/ 3 0.666Model

Time

r/a(8)

•We presented numerical evidence of non-diffusive transport i.e.,super-diffusion and Levy distributions, in plasma turbulence.

•We proposed a transport model that incorporates in a unified way space non-locality, memory effects and anomalous diffusion scaling.

•There is quantitative agreement between the model and the turbulence calculations.

•The model represents a first attempt to construct effective transport operators when the complexity of the turbulence invalidates the use of Gausian closures.

Conclusions

t ˜ V t

x

(9)

Further details: del-Castillo-Negrete, et al., Phys. of Plasmas, 11, 3854, (2004).

Fractional derivatives