CALCULATION OF TURBULENT DIFFUSION
FOR THE CHIRIKOV-TAYLOR MODEL
A.
B. Rechester
Plasma Fusion Center
M.I.T.
R.
B.
White
Plasma Physics Laboratory
Princeton University
Submitted for publication in Phys. Rev. Lett.
PFC/JA-80-12
5/80
Calculation of Turbulent Diffusion for the Chirikov-Taylor Model
A.
B.
Rechester
Plasma Fusion Center,
Massachusetts Institute of Technology
Cambridge, Massachusetts
02139
and
R. B. White
Plasma Physics Laboratory;
-
Princeton University
Princeton, New Jersey
08540
A probabilistic method for the solution of the Vlasov equation has
been applied to the Chirikov-Taylor model.
The analytical solutions
for the probability function and its second velocity moment have been
obtained..
Good agreement between the theory and numerical computa-
tions has been found.
-2The purpose of this letter is to present an analytical solution
This model has been studied ex-
for the Chirikov-Taylor model1,2
years in reference to many different plasma
tensively in the last ten
physics problems
17-6
Such an interest is due to the fact that this
simple model exhibits chaotic or turbulent dynamics manifested in the
region above the so-called stochastic transition
We will intro-
,.
duce the method of solution and the underlying physical arguments
through the differential equation which describes the motion of charged
The equation of
particles in a field of electrostatic plane waves.
motion is:
2
d
~
d~
=
(1)
e E(x,t)
dt
where
is
the
mass, e:is the charge of the particle, and the
electric field is given by the equation
E(K, t)
Er
=
exp[i(kC
-
m,n
W
assume here
periodic- boundary
'nt)]
conditions
+ c.c.
in
the
(2).
interval
0'< x < a, km = 23rm/a, m integer, and a discrete spectrum in &a
Consider the case:
k
M
2=a
w
im
n
=27
0
n
with n=0,+l,+2,...4N, and E1 ,n = E/(2i) is constant.
mn is a Kronecker function.
rescaling
distance
Using
with a /(2-A)
takes the form
dx
dt
dimensionless
and time with .2T
The symbol
variables
*
,
by
Eq.(l)
-3dv
dt
sin ut
sinx
c = (27r) 3
and
(5)
sin[wt(2N + 1)]
1.
N
e/a
o
0/
Consider the phase space x, v
distribution
of
phase-space
and
points
the
introduce
f(x, v, t = 0).
initial
The time
evolution of f is described by a Vlasov equation
-
+ V af
dva
This is just a continuity equation
flow
(6)
= 0 .
of
phase-space
flow.
This
deterministic, that is, the position of every point can
is
be predicted uniquely for an arbitrary large time t.
nonlinear
complicated
character
of Eq.
(5),
Due to
some phase points
which are initially separated can come arbitrarily close to
other if the time of evolution is long enough.
f can become arbitrarily large-because-phase
each
This implies that
space
density
is
We may say that
along the orbits (Liouville theorem).
conserved
.the
long time solutions are intrinsically singular.
This is
why
it
is impossible to find an analytic solution of. the.Valsov equation
described
even for the simple motion
other
hand,
there
are
always
some
by
Eqs. (4), (5).
processes
which
On
the
are not
included in these -equations, including snall errors introduced in
a
numerical solution.
In spite of being very small they do play
an important role in the limit of large
later..
A
times
as
we
will
see
simple way of modeling such effects is to introduce a
random velocity term in Eq.
(4) or a random force
term
in
Eq.
(5).
We have done these numnerically.
view, it
From the analytic point of
is more convenient to treat these as additional
or velocity diffusion terms in the Vlasov equation.
spatial
The analysis
happens to be much easier for the case of spatial diffusion.
The
Vlasov equation in this case takes the form:
Thus
the
I".I-~aX;t
a
--
(7)
deterministic description given by the Vlasov equation
is substituted by a probabilistic One with
probability
P(x,v,t)
We consider 6
,
which we normalize to unity .
As we will see later it
singular
equation.
nature
of
is
the
the
diffusion
long
time
which
solutions
function
<<
resolves
of
the
1.
the
Vlasov
The effect of weak diffusion has been discussed qualita-
tively for the similar problem of electron heat transport in tokamaks
with destroyed magnetic surfaces 7.
The initial -condition for P we
take of the form:
P(x, V, 0)
2r
6(v
-
v o))
.
The Chirikov-Taylor model corresponds to the limit N+
sin(Tt(2N+1))/sin(Trt)
can be approximated by
(t-i).
.
Then
I
-5-
Consider now the time evolution of the probability function
< t < i+0 it can be written as
P. In the interval i-O
= P(x,v+ e sinx,
P(x,v,i+ 0)
Here
i is
an
integer
and
+0
is
(9)
i -0)
an arbitrary small number.
Evolution in the interval i+0 < t < i+1
0
-
is
given
by
the
formula
P (x, v,(Xj t)
J) -
dv1,
=
-(
G (vvtt)P
(x-x 1 ,v,vj
1 t-t
1
,y 1 1 t )dx 1
)P(x 1 ,V
10)
where G is a solution of the equation
G
e (t-t)
G=
-
2
a G
2t23x 2
6 (v-v)
)
.V2 Ta (t-t
Here
3G
(t)
)
6 (x
-
x )6 (v
1)1
1
v )6(t
1
2a(t
n=- D
is the Heavyside function.
-
t
+21nJ 2
(t-tl)
-[x-xl-v
ex-
-
ti)
(12)
-6The
appears
smiation
conditions in the interval 0 ( x ( 2
the periodic boundary
of
because
We
.
can
now
write
a
formal solution to Eq.(7) for initial probability given by Eq.(8)
t=T,
for arbitrary time t. For simplicity we consider
a large
integer.
P(xT v,T)
CO T-1
...E
=
n
f 27r
dxi
n =-o i=O
-
(13)
T
6(v-v
-T
-S
0-
T
) ex-x
p1:-
(x--
i
-v-S
0
+2rn ) 2
j-J.
j
and
j
S.
J
e
P=O
sin x
.
It is easy to check by direct. integration that P(x ,v,T) is
correctly normalized,
dx P(x,
dv
--
V, T)
= 1
0
To calculate the diffusion rate we use the formula
(14)
-7-
D = lim
T+-
2O
1
1
(15)
2
(v-+v
0 )
P (x, v,'T) d x dv
2T
With the use of the identity
1
rn) 2
(y+
exp
exp (-a/2 m2 + m)
c
M=-.Co
(16)
we can write Eq.(15) as
CO
D = 1imE
T
S
exp
I
m1=_CO i=O
MT =-C>
T+i
dx.
27
T.
(jzj (a/2)m2
+ im. (x
-
x._
sin x
S.=EZ
-v
-s
(17)
p
p=o
In the case m. = 0 the calculations are trivial
I1
and we find
-8(18)
2
D
QL
This case corresponds to the
made
in
quasilinear
C4
random
theory.
phase
approximation
7b evaluate D for m1
often
0 we make
use of the identity
exp(±iz sinx) =
Jn (z) exp(±inx)
(19)
n=
Here .J (Z) are Bessel functions and z>0.
Making use of Eq.(19) it
involves products of Bessel functions.
Eq. (17)
6
large
is easy to see that the series
the Bessel functions decay as
few low order terms in Eq. (17) .
-
-
1_J
2
D
)e
J2
which
J
is. obtained
three of the mr #
arises
o
,
We
have
functions.
from m
neglected
Our
(c)
e
e
2
3
+
3
a]
-3
and all others are zero.
from terms with m.
comes
The expression obtained for D is
(20)
from Eq(17) from those terms in which two or
term is similar, with
term
In the region of
so we can keep just a
2
-
in
= +1, m.
M
=
±1,
=-m
m,
products
= -2mn
=-M
,
The
1=2,3,...T.
, I=3,4,...T, and
, m-
involving
-=
three
m
or
J
term
The J
the
,T.
more. Bessel
nisnerical computations suggest that convergence
-9of the series approximated by Eq(20) in
much
faster
exp(-
than
nm,
/2).
that
The
due
simply
possible
the
>>
region
1
is
to the exponential factors
relation
between
this
fast
convergence and exponential orbit divergence for Eqs(4),(5) which
takes place in the region 4 >> 1
)
instability
has
not
yet
( the
been
so
called
examined.
stochastic
The
leading
asymptotic terms in Eq(20) are
D
D
-
t. - 1/25 cos E
-a2
2e
(21)
-5-
QL
Thus D approaches DL
pointed -"out
is
the
E4 00 .
We recall here
the -limit
LV0 -i
unit frequency
->
LO
1.
It
that
was
the. spectrum
assumed to be continuous in quasilinear theory.
our -model a continuous spectrum can be
E
>>
limit
to us by a referee that the usual quasilinear limit
corresponds to
W
in
0.
bandwith
0 and
E
achieved
Within
considering
Theri in drder that the power density per
remain
4
constant
(see Eq.
we
J
even bigger than DL for some values of e
need
to
rescale
)
The contribution- to D from terms with m.
QL
by
of
d 0 can be comparable to or
(see also Figure 1).
This could
have practical implications especially for the problems of radiofrequency
heating
6
We will address this question in future publications.
We have made extensive numerical study of Eqs.
(4),
(5),
intro-
ducing a small random step in x with a normal distribution of mean a.
Some
of the results of these computations are shown in Figure 1 for the case a
To obtain sufficient accuracy in the evaluation of D we used the number
of
particles
N
p
= 3000,
=
10-5
-10-the fluctuations in D behaving as l/vT
T = 50.
p
, and advancing the equations to
We also verified that intorducing a random step in v instead of in x
leads to essentially the same results for small a and large e.
tions in D/D
The oscilla-
were apparently first noted by Chirikov , but were independently
found by Kyriakos Hizanidis
and pointed out to the present authors.
note-from this figure that D approaches DQL in the limit of large C.
plotted on Figure 1 is the function given by Eq. (20).
there is good agreement for large C,
We also
Also
It is clear that
but for c of the order of unity we need
to retain more. terms in the evaluation of Eq. (17).
We think that the method
outlined may prove useful in other problems were apparently chaotic behavior
arises from deterministic equations.
ACINqLEDGEIS
We are very grateful to M.
T.
IDjpree, K.
discussions
of
Molvic, K.
this
paper.
Rosenbluth ,J. M.
N.
Hizanidis, and
A.
T.
H.
Stix
for
Rechester also would like to
thank the Institute for Advance Study for its hospitality,
the final part of this work was completed.
supported by the I. S.
Greene,
where
This work was
Department of Energy contracts No.
EY-76-C-02-3073 and DE-AS02-78ET53074.
-12REFERENCES
1.
B. V. Chirikov ,Physics Reports 52,263 (1979)
2.
G.M. Zaslavsky and B. V. Chirikov, Usp. Fiz. Nauk 14,
195 (1972)
(Sov. Phys.
Usp. 14, 549 (1972))
3.
M.N. Rosenbluth, Phys. Rev.
4.
M. A. Lieberman and A. J. Lichtenberg, Plasma Physics
-
5.
6.
15, 125 (1973)-
Lett 29,408 (1972)
~
J. M. Greene , J. Math. Phys. 20,
T. H.
1183 (1979)
Stix, The Role of Stochasticity in Radiofrequency
Plasma Heating,
Proceedings of Joint Varenna-Grenoble International
Symposium on Heating in Toroidal Plasmas, Grenoble,
France, v II, 363-376
see also Princeton Plasma Physics Lab. report PPPL 1539 (1979).
7.
A.B. RechesterandM.
8.
Kyriakos Hizanidis,
N. Rosenbluth,
Phys. Rev. Lett. 40,
private communication.
38 (1978).
(1978).
2.5
2.0D/DL
1.0
0
0.5-S
0
120
3
*
S o
05
* 0
FIGURE CAPTIONS
V Fig.
1.
The ratio of the numerically obtained diffusion
quasilinear value as a function of
E
.
Here
to
C
the
=
Also plotted is the analytic expression given
by Eq.(20).