Fractional calculus: basic theory and applications (Part III) Diego del-Castillo-Negrete Oak Ridge National Laboratory Fusion Energy Division P.O. Box 2008, MS 6169 Oak Ridge, TN 37831-6169 phone: (865) 574-1127 FAX: (865) 576-7926 e-mail:
[email protected]
Lectures presented at the Institute of Mathematics UNAM. August 2005 Mexico, City. Mexico
5.b) Applications to transport in fusion plasmas Here we present phenomenological models of plasma transport based on the use of fractional diffusion operators. In particular, we extend the fractional diffusion models discussed before, by incorporating finite-size domain effects, boundary conditions, sources, spatially dependent diffusivities, and general asymmetric fractional operators. These additions are critical in order to order to go beyond tracers transport calculations. We show that the extended fractional model is able to reproduce within a unified framework some of the phenomenology of non-local, nondiffusive transport processes observed in fusion plasmas, including anomalous confinement time scaling, “up-hill” transport, pinch effects, and on-axis peaking with off-axis fuelling. The discussion presented here is based on [del-Castillo-Negrete-etal,2005a] and [del-Castillo-Negreteetal,2005b] where further details can be found.
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) Examples: •Anomalous confinement time scaling • Fast propagation and non-local transport phenomena • Inward transport observed in off-axis fueling experiments •To overcome the limitations of the diffusion paradigm, we have proposed transport models based on fractional derivative operators that incorporate in a unified way non-locality, memory effects, and non-diffusive scaling.
Towards an effective transport model for non-diffusive turbulent transport r dr ˜ 1 ˜ r = V = 2 "# $ B dt B
•Individual tracers follow the turbulent field •The distribution of tracers P evolves according to
!P ˜ + V " #P = 0 !t
!
•The idea is to construct a model that “encapsulates” the complexity of the turbulence field V˜ in an effective flux !, and reproduces the observed pdf homogeneous isotropic turbulence
diffusive transport
transport in pressure driven plasma turbulence
Gaussian closures
Non-Gaussian distributions
!P !# =" !t !x
! = "# $ x P
% = $ " #!x P
Riemann-Liouville derivatives n = [" ] + 1
a
x
b
["]=integral part of "
!
1 &n D # = Left derivative a $(n % " ) & x n " x
Right derivative !
" b
xD # =
n
&n %(n $ " ) & x n
($1)
x
' a
# ( u)
( x % u)
b
' x
" %n +1
# ( u)
(u $ x )
" $n +1
du
du
!
Some limitations of the RL definition Although a well-defined mathematical object, the Riemman-Liouville definition of the fractional derivative has some problems when it comes to apply it in physical problems. In particular: The derivative of a constant is not zero
a
Dx" A =
A 1 #(1$ " ) ( x $ a)"
limx "a a Dx# $ = %
The RL derivative is in general!singular at the lower limit
unless
" (k ) (a) = 0
This might be an issue when using RL operators for writing evolution equations This might be an issue when applying boundary conditions
!
The Laplace transform ! n$1 of the RL " " ˆ L 0 Dt # = s # (s) $ % sk derivative k= 0 depends on the fractional derivative at zero
[
!
]
[
" $k$1 # ] t= 0 0 Dt
This might be an Issue when solving Initial value problems
Behavior near the lower terminal $
#
(k )
" (a) k " (x) = $ ( x % a) k! k= 0
Let
# (k ) (a) k&" " a Dx ( x & a) k! k= 0
aD # =%
" x
using
! D" ( x # a) k = a x
! k#" $( k + 1) ( x # a) $( k + 1# " ) m&1
a
limx "a a D $ = limx "a ' k= 0
!
" (k ) (a) = 0
# x
limx "a a D $ = % unless
!
m "1 # $ < m
k = 1,Lm "1
!
!
!
D
( x $ a) # = ' # (a) %( k + 1$ " ) k= 0 (k )
$ (k ) (a) 1 ! %( k + 1& # ) ( x & a)# &k
# x
!
k$"
&
" x
Caputo fractional derivative These problems can be resolved by defining the fractional operators in the Caputo sense. Consider the case 1 < " < 2 k +2$"
& # (k +2) (a) ( x $ a) # (a) 1 # '(a) 1 $ =' aD # $ %(1$ " ) (x $ " )" %(2 $ " ) (x $ " )" $1 k= 0 %( k + 3 $ " ) !
" x
singular terms
regular terms k +2$"
# (k +2) (a) ( x $ a) %( k + 3 $ " )
&
! a
" x
D
[# (x) $ # (a) $ #
Define the Caputo derivative by subtracting the ! singular terms
C 0
(1)
(a)] = ' k= 0
Dx" # = a Dx" [# (x) $ # (a) $ # (1) (a)]
x
using
' # ""(u) # (k +2) (a) du = ( & ! a ( x $ u)% $1 k! k= 0
C a
!
Dx" # =
1 $(2 % " )
x
' a
x
k
( x $ a) & x $ u % $1 du ) a (
# &&(u) " %1 du ( x % u)
In general
C a
1 D #= $(n % " ) " x
($1) n D #= %(n $ " )
C x
" b
!
C a
and as expected:
!
x
' a
x
' a
& un # " %n +1 du ( x % u) & un # " $n +1 du x $ u ( )
Dx" A = 0
limx "a a Dx# $ = 0 n$1
L[ 0 D # ] = s #ˆ (s) $ % sk
!
" t
"
k= 0
[
0
# (" $k$1) ] t= 0
!
Note that in an infinite domain
!C D$ % = "# x
"#
C x
Dx$ %
!
D"# $ = x D"# $
!
Nonlocal transport model "T " =# [ ql + qr + qG ] + P(x,t) "t "x qG = " # G $x T
Diffusive flux
!
Left fractional flux
Regularized finite-size left derivative
!T ql = " l (2 " # ) $ l 0 Dx# "1
!
0
D
1 T= $( 3 # " )
! x
ql
" #1 1
D
1 T= $( 3 # " )
qr
Nonlocal fluxes
0 !
x
& ( x # y)
2#"
T %% dy
0
Regularized finite-size right derivative
Right fractional flux
qr = r (2 " # ) $ r x D1# "1 T
!
" #1 x
!
x
!
1
1
& ( y # x) x
2#"
T %% dy
Nonlocal transport model "T " =# [ ql + qr + qG ] + P(x,t) "t "x Diffusive flux
!
qG = " # G $x T
ql = " l (2 " # ) $ l 0 Dx# "1 T
Nondiffusive ! fluxes
qr = r (2 " # ) $ r x D1# "1 T
! Boundary conditions
! “core” "G
Finite size domain
Diffusive transport
x " (0,1) !
0
ql (0) + qr (0) + qG (0) = 0
"G , " l , " g Non-diffusive transport
!
!
T "(0) = T(1) = 0
1 !
!
Local and non-local transport #
2L
! L (x) =
1 2L
" ! ( x + s) ds L
L=“radius of influence” x Local diffusive transport Non-local diffusive transport
!t " = D! x2 " = 6D
!t " + ! x Q = 0
"
#"
L
2
L
as L ! 0
Q = ! D "x #
Q (x,t) = ! D $ G(x ! x' ) " x' # (x' ,t) dx'
Perturbative transport experiments have shown evidence of non-local diffusion in fusion plasmas
Nonlocal transport ! (x, t) = $ dx' P( x " x' ;t ) !(x' ,t = 0) #
"#
lim P ~ e # x
2
lim P ~ x #( 1+ $ )
x !"
x !"
local transport
P(x ! x' ,t)
nonlocal transport
! (x' ,t = 0)
P(x ! x' ,t)
x'
! (x, t) " 0
! (x' ,t = 0)
x'
! (x, t) " 0
Due to the algebraic tail of the propagator, fractional diffusion leads to nonlocal transport
Nonlocal transport Edge temperature rise Fractional diffusion
T (r = a)
Gaussian diffusion
Nonlocal transport and global diffusive coupling %"h# ! # = lim " h$ 0 h " x
Grunwald-Letnikov definition
N j %" ( !"h #(x) = * ($1) ' # ( x $ j h) & j) j=0
" x
! #k $
(% # ) " h
"
h
k
k
1 = " h
" j
&w
#k ' j
j=0
# ! + 1& ! w !j = % 1 " w j "1 $ j '
w0! = 1
Finite difference approximation
Global coupling coefficients
For " =1 and 2 usual nearest neighbor coupling of Laplacian operator For 1 < " < 2 fractional diffusion implies global coupling
Steady state and anomalous confinement time scaling Uniform heating
On axis heating
"r = 0 "l # 0
!
Blue "=2 Green " =1.75 Red " =1.5 Black " =1.25
Steady state solution "= $ L" ' 1 , $ x ' / T = P0 & ) .1+ & ) 1 % # l ( *(" + 1) - % L ( 0 "
!
! !
# #
L 0 L 0
T dx
Confinement time
P dx
# L# "= $ (2 + # ) % l
Anomalous scaling 1
Confinement time scaling " = Confinement time #" & L* = % d ( $ "a '
L = Domain size
!
! "d
!
= Standard diffusivity
Diffusion + Right fractional
1 2)*
1
$ " # ' 2*# T = & d2 ) % "a (
" L %2 ~$ *' #L &
*
" a = Anomalous diffusivity !
! !
!
!
Diffusion + Symmetric fractional
Diffusion + Left fractional " L %( ~$ *' #L &
" L %2 ~$ *' #L &
" L %( ~$ *' #L &
!
" L %2 ~$ *' #L &
!
!
!
“Up-hill” transport and pinch velocity Fractional diffusion
Standard diffusion
r>0 l=0
!
qG = " # G $x T
!
qr = r x D1" #1 T Fractional Flux
Gaussian Flux
!
Up-hill Transport zone
Fractional flux asymmetry and pinch velocity Symmetric fractional diffusion
Asymmetric fractional diffusion
r=l r>l ! !
Flux Right flux
Right flux
qr
qr
ql Left flux !
!
Left flux
ql
! !
“Cold-pulse” fast propagation Standard diffusion
time
time
Fractional diffusion
Pulse Steady state x
x
Off-axis heating Standard diffusion
Fractional diffusion
"R
!
Flux
Flux qG qT = qG + qR
qG !
qR !
! !
5.c) Application to reactiondiffusion systems Here we discuss the role of fractional diffusion in reaction-diffusion systems. In particular, we present a numerical and analytical study of front propagation in the fractional Fisher-Kolmogorov equation. The discussion presented here is based on [del-Castillo-Negrete-etal,2003] where further details can be found.
Reaction diffusion models of front dynamics ! =1
Fisher-Kolmogorov equation
stable
$ t ! = #$ x2! +" ! (1 % ! )
c
diffusion
! =0
reaction
c =2 "!
unstable
•Plasma physics •Model of genes dynamics •Model of logistic population growth with dispersion •Other applications include combustion and chemistry.
The fractional Fisher-Kolmogorov equation $ t ! =# &' Dx% ! +" ! (1 & ! )
1
1 & (y ) ' x2 ! dy #" D & = $(2 # % ) #" (x # y )% #1 % x
The asymmetry in the fractional derivative leads to an asymmetry in the fronts Self-similar dynamics, exponential tails, constant front velocity
“left moving”
Quasi-self-similar dynamics, algebraic tails, front acceleration “right moving”
Asymmetric front dynamics ! = 1.25 Exponential decaying constant speed fronts ! (x, 0) =
Algebraic decaying accelerated fronts
1' # x " x0 % * 1 + tanh $ 2 )( 2W & ,+
!
1' # x " x0 % * ! (x, 0) = 1" )1 + tanh $ 2( 2W & ,+
!
x
x
Asymmetric front dynamics ! = 1.25 Exponential decaying constant speed fronts ! (x, 0) =
Algebraic decaying accelerated fronts
1' # x " x0 % * 1 + tanh $ 2 )( 2W & ,+
1' # x " x0 % * ! (x, 0) = 1" )1 + tanh $ 2( 2W & ,+
t
t
x
x
Leading edge calculation for algebraic fronts $ 1 x
