Direct Determination of Characteristic Fluctuation...
24
2005 CAARMS Meeting, Jun 21-25, UCLA-IPAM 1 Direct Determination of Characteristic Fluctuation Frequencies from Phase Space Trajectories in Turbulent Plasmas K. M. Williams and J. A. Johnson III Princeton Plasma Physics Laboratory Laboratory for Modern Fluid Physics Florida A&M University Tallahassee, FL 32310 *Re searc h support ed in part by grants from NASA Headquarters and the DOE Office of Fusion Energy Sciences.
Transcript of Direct Determination of Characteristic Fluctuation...
2005
CA
AR
MS
Mee
ting,
Jun
21-
25, U
CLA
-IP
AM
1
Dire
ct D
eter
min
atio
n of
Cha
ract
eris
tic
Flu
ctua
tion
Fre
quen
cies
from
Pha
se S
pace
T
raje
ctor
ies
in T
urbu
lent
Pla
smas
K. M
. Will
iam
s an
d J.
A. J
ohns
on II
IP
rince
ton
Pla
sma
Phy
sics
Lab
orat
ory
Labo
rato
ry fo
r M
oder
n F
luid
Phy
sics
Flo
rida
A&
M U
nive
rsity
Tal
laha
ssee
, FL
3231
0
*Re
sear
ch s
uppo
rted
in
par
t by
gra
nts
fro
m N
ASA
Hea
dqua
rter
s an
d th
e D
OE
Offi
ce o
f Fus
ion
Ene
rgy
Sci
ence
s.
2005
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2
Mot
ivat
ion
•Mol
ecul
ar s
cale
effe
cts
are
impo
rtan
t in
turb
ulen
ce
and
it is
impo
rtan
t to
sam
ple
on th
e ap
prop
riate
time
sc
ales
in o
rder
to o
bser
ve th
em.
•If t
here
is a
uni
fied
theo
ry o
f tur
bule
nce
then
it
shou
ld b
e ch
arac
teriz
ed in
term
s of
an e
nerg
y co
ncep
t and
sta
ndar
d pa
ram
eter
swhi
ch c
an b
e re
late
d to
the
phys
ics
of th
e sy
stem
.•R
ecen
t evi
denc
e su
gges
ts th
at a
ph
ase
tran
sitio
n m
odel
base
d on
the c
once
pt o
f tur
bule
nt e
nerg
ymay
be
suc
cess
ful a
s a
gene
ric tr
eatm
ent o
f tur
bule
nt
beha
vior
* .•*
Will
iam
s, P
odde
r, &
Joh
nson
III,
Phy
sics
Let
ters
A, V
ol. 3
31, I
ssue
(1-
2),p
p 70
-76
2005
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3
Arc
Driv
en S
hock
Tub
e W
ith x
-t D
iagr
am: E
lem
ents
of
Pla
sma
Cre
atio
n an
d S
hock
Wav
e E
volu
tion
�D
isch
arge
Vol
tage
Bet
wee
n 18
kV-
28kV
�M
ediu
m is
Arg
on fo
r S
hock
Tub
e
�P
MT
Vol
tage
@ -
600V
�S
hock
Wav
e ve
loci
ty>
M 2
0
�S
hock
Tub
e T
empe
ratu
re ~
3 to
4eV
�S
hock
Tub
e D
ensi
ty ~
1022
m-3
(Prim
ary)
, ~10
23 m
-3 (R
efle
cted
), α∼
0.6
α∼0.
6α∼
0.6
α∼0.
6�
Dia
gnos
tics:
Pre
ssur
e T
rans
duce
rs
and
Lase
r In
duce
d F
luor
esce
nce
2005
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4
App
lyin
g G
inzb
urg-
Land
au T
heor
y to
T
urbu
lenc
e�
The
tran
sitio
n to
turb
ulen
ce c
an b
e tr
eate
d as
a s
ymm
etry
br
eaki
ng d
isor
der-
to-o
rder
tran
sitio
n, p
rodu
cing
a n
ew s
ymm
etry
br
eaki
ng p
aram
eter
(the
ord
er p
aram
eter
, Γ ΓΓΓ)
in th
e le
ss s
ymm
etric
(t
urbu
lent
) ph
ase.
�In
the
tran
sitio
n to
turb
ulen
ce, t
he s
tate
can
cha
nge c
ontin
uous
ly
thro
ugh
a re
gim
e w
here
no
dist
inct
ionc
an b
e m
ade
betw
een
the
lam
inar
(non
-tur
bule
nt)
and
turb
ulen
t sta
te, t
he c
ritic
al tr
ansi
tion
stat
e.
�T
he th
erm
odyn
amic
reg
ime,
whe
re a
pha
se tr
ansi
tion p
rodu
ces
the
disc
ontin
uous
cha
ngein
the
tran
spor
t par
amet
ers(
Γ ΓΓΓ),
defin
es
the
criti
cal v
alue
of th
e to
tal e
nerg
y in
the
syst
em fo
r th
at
ther
mod
ynam
ic s
tate
.
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5
App
lyin
g G
inzb
urg-
Land
au T
heor
y to
T
urbu
lenc
e C
ont’d
�T
he o
rder
par
amet
er (Γ ΓΓΓ
)sh
ould
then
ser
ve a
s an
exp
ansi
on
para
met
er in
the
desc
riptio
n of
crit
ical
phe
nom
ena.
�T
he tw
o ke
y el
emen
ts in
det
erm
inin
g a
repr
esen
tation
of t
rans
ition
to
turb
ulen
ce a
s a
criti
cal p
heno
men
on a
re th
e de
termin
atio
n of
an
orde
r pa
ram
eter
and
the
dete
rmin
atio
n of
an a
ppro
pria
te fi
eld
whi
ch w
ill d
rive
the
rela
tions
hip
betw
een
the
tran
spor
t effe
cts
and
a ch
arac
teriz
ing
ener
gy fo
r th
e tu
rbul
ent s
yste
m.
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6
New
Phy
sics
:T
urbu
lent
Sys
tem
s fr
om a
2ndO
rder
G-L
P
hase
Tra
nsfo
rmat
ion
•Dis
cont
inuo
us Γ ΓΓΓ
-like
cha
nges
shou
ld b
e fo
und
in p
rinci
pal
tran
spor
t par
amet
ers
asso
ciat
ed
with
a fo
rce-
like
cons
trai
nt in
the
unde
rlyin
g dy
nam
ics.
•Sig
nific
ant c
hang
es in
the
degr
ee
of c
ompl
exity
are
exp
ecte
d.
•A t
urbu
lent
ene
rgy
conc
ept
emer
ges
with
uni
que
turb
ulen
ce
defin
ing
para
met
ers:
spe
ctra
l in
dex;
cha
ract
eris
tic fr
eque
ncy;
ch
aotic
dim
ensi
on.
The
re h
ave
now
bee
n m
any
succ
esse
sfr
om th
is a
ppro
ach.
Lam
inar
Tur
bule
nt
Uc
U
Γ ΓΓΓ
2005
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7
The
Mic
rosc
opic
App
roac
h
Fro
m th
e po
int o
f vie
w o
f kin
etic
theo
ry, c
onsi
der th
e pr
oces
s th
at fo
rces
two
mol
ecul
es A
and
B to
bec
ome,
toge
ther
, the
turb
ulent
quas
imol
ecul
e P
(w
ith
appr
opria
te c
lass
ical
sta
tistic
s) a
nd g
ener
ic p
articl
e di
strib
utio
n fu
nctio
n as
f(z)
th
en
For
lam
inar
flow
:w
here
the
abov
e co
nditi
on fo
r la
min
ar fl
ow a
lso
prod
uces
the
Bol
tzm
ann
equa
tion
and
the
stan
dard
equ
atio
n fo
r flu
id d
ynam
ics.
For
turb
ulen
t tra
nspo
rt
para
met
er Γ ΓΓΓ
, a c
hang
e in
the
tran
spor
t par
amet
er ta
kes
the
form:
Tra
nspo
rt p
aram
eter
Γ ΓΓΓis
driv
en to
turb
ulen
t beh
avio
r Γ ΓΓΓ A
ppby
non
-zer
o va
lues
of
Ψ ΨΨΨ, i
gnor
ing
quan
tum
effe
cts.
Ψ(z
,ˆ z )
≡fII
(z,ˆ
z )−
f(z)
f(ˆ z )
Ψ(z
,ˆ z )
→0
(ΓA
pp−
Γ)Γ
=Ψ
AB(z
A,ˆ
z B)
f Af B
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AM
8
The
Mac
rosc
opic
App
roac
hT
he tu
rbul
ent f
ree
ener
gy p
er u
nit v
olum
e ta
kes
the
form
:
in w
hich
the
orde
r pa
ram
eter
Γ ΓΓΓon
ly a
ppea
rs in
the
less
sy
mm
etric
pha
se, t
he s
ubsc
ripts
tand
lre
fer
to th
e tu
rbul
ent (
less
sym
met
ric)
and
lam
inar
(sy
mm
etric
) co
nditi
ons,
and
ε εεεis
the
forc
e co
njug
ate
to th
e or
der
para
met
er.
αt(
X,ε,
Γ)=
αl(
X,ε
)+α
2(X
,ε)Γ
2+
α4(X
,ε)Γ
4+
...
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9
The
Mac
rosc
opic
App
roac
h
αt(
X,ε,
Γ)=
αl(
X,ε
)+α
2(X
,ε)Γ
2+
α4(X
,ε)Γ
4+
...
Thi
s fo
rm (
with
α ααα1=
α ααα 3=
0) in
sure
s th
at th
e fr
ee e
nerg
y ca
n be
min
imiz
ed fo
r Γ ΓΓΓ=
0 be
low
the
tran
sitio
n an
d fo
r Γ ΓΓΓ≠ ≠≠≠0
ab
ove
the
tran
sitio
n. T
he fo
rm o
f α ααα2
is c
hose
n to
pro
duce
th
e cr
itica
l val
ue o
f Xcco
rres
pond
ing
to th
e tr
ansi
tion
unde
r ci
rcum
stan
ces
whe
re th
e tw
o ph
ases
can
not b
e di
stin
guis
hed.
2005
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10
Mac
rosc
opic
App
roac
h (c
ont’d
)
Usi
ng th
e G
-L th
erm
odyn
amic
s of
a p
hase
tran
sitio
n, the
tr
ansp
ort p
aram
eter
Γ ΓΓΓis
then
driv
en to
turb
ulen
t beh
avio
r Γ ΓΓΓ A
ppat
val
ues
X>
X cw
here
Γ App
−Γ(
X)=
Xc(ε
)
2
α 2(X
,ε)2
α 4(X
,ε)1
(X−
Xc(ε
))2
2005
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21-
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11
Brin
ging
Tog
ethe
r th
e M
acro
and
M
icro
App
roac
hes
We
can
now
pro
duce
a g
ener
ic r
elat
ions
hip,
igno
ring q
uant
um
effe
cts,
bet
wee
n th
e th
erm
odyn
amic
par
amet
ers
for
turbu
lenc
e as
a s
econ
d or
der
phas
e tr
ansi
tion,
the
orde
r pa
ramet
er (
whi
ch
is n
ow d
eter
min
ed to
be
deriv
able
from
the
inte
rmole
cula
r, i.
e.,
mic
rosc
opic
forc
es a
nd is
a fu
nctio
n of
a c
ompl
exity pa
ram
eter
) an
d th
e pa
rtic
le p
roba
bilit
y de
nsity
func
tions
as
follo
ws:
Xc(ε
)
2
α2(X
,ε)2
α4(X
,ε)
1
(X−
Xc(ε
))2
=Ψ
AB(z
A,ˆ
z B)
f Aˆ f B
2005
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12
Brin
ging
Tog
ethe
r th
e M
acro
and
M
icro
App
roac
hes
Xc(ε
)
2
α2(X
,ε)2
α4(X
,ε)
1
(X−
Xc(ε
))2
=Ψ
AB(z
A,ˆ
z B)
f Aˆ f B
Not
ice
that
the
chie
f sho
rtco
min
g of
this
res
ult i
s that
the
r.h.
s. is
ent
irely
ba
sed
upon
cla
ssic
al k
inet
ic th
eory
. W
e ar
e re
min
ded th
at c
lass
ical
kin
etic
th
eory
fails
to p
redi
ct th
e be
havi
or o
f tra
nspo
rt co
effic
ient
s be
caus
e of
inco
rrec
t tre
atm
ents
of i
nher
ently
qua
ntum
mec
hani
cal e
ffect
s.
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13
Cha
otic
Dim
ensi
on (
D 2)
• •••A
sta
te v
ecto
r X j
is c
onst
ruct
ed fr
om th
e sc
alar
tim
e se
ries
x j
• •••D
elay
tim
e
• •••C
orre
latio
n In
tegr
al
•F
or s
mal
l val
ues
of r
the
corr
elat
ion
inte
gral
beh
aves
as
a po
wer
of r
.
•W
here
D2
is th
e ch
aotic
dim
ensi
on, a
n in
dica
tor
of a
sys
tem
’s
com
plex
ity.
•S
ince
the
traj
ecto
ry is
cyc
lical
the
num
ber
of r
evolu
tions
are
als
o de
term
ined
.
�
X j
=x
j,x
j+l,
xj+
2l,.
....
.xj+
(m−1
)l(
)τ
=l∆
t
C(r
)=
Lim
N→
∞
1 N2
Hr
−X
j−
Xk
()
j,k
=1;j
≠k
N ∑
C(r
)∝rD
2
2005
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14
Sam
ple
Raw
Dat
a: A
rgon
II (
422.
8nm
) E
mis
sion
s in
A
DS
T P
lasm
a
2005
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21-
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AM
15
Det
erm
inat
ion
of K
ey P
aram
eter
s F
rom
Sam
ple
Dat
a
Exa
mpl
es o
f str
onge
st (
dom
inan
t or
char
acte
ristic
) fr
eque
ncy.
Tur
bule
nt e
nerg
y is
es
timat
ed fr
omE
τ=
Pi
i=1
nois
e
∑
4.2
4.0
3.8
3.6
3.4
3.2
-2.4
-2.2
-2.0
-1.8
-1.6
-1.4
Log C
Log
r
For
the
sam
ple
data
abo
ve (
200
poin
ts a
t 4 n
s/sa
mple),
the
phas
e tr
ajec
tory
sho
ws
regu
lar
beha
vior
and
D
2=0.
941;
the
gree
n ar
row
indi
cate
s th
e lin
ear
regi
me
for
fittin
g. F
requ
enci
es o
f evo
lutio
n in
pha
se s
pace
are
dete
rmin
ed b
y co
untin
g th
e nu
mbe
r of
cyc
les
mad
e in
200
poin
ts w
ith th
e co
nver
sion
one
cyc
le c
orre
spon
ds to
a
freq
uenc
y of
0.6
25M
Hz
2005
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16
Map
ping
For
a p
hysi
cal
obse
rvab
le r
epre
sent
ed i
n tim
e se
ries
data
the
pr
ogre
ssio
n of
a
nonl
inea
r sy
stem
at
a
give
n st
ate
nca
n be
st
udie
d by
inve
stig
atin
g th
e de
pend
ence
of
the
(n+
1) s
tate
on
the
nthst
ate.
For
ex
ampl
e,
xn+
1=(6
x n-4
)3is
a
map
ping
use
d to
de
scrib
e th
e sy
stem
pro
gres
sion
.• •••
A s
tate
vec
tor
X jis
con
stru
cted
from
th
e sc
alar
tim
e se
ries
x j.*
�
X j
=x
j,x
j+l,
xj+
2l,.
.....x
j+(m
−1)l
()
Poi
ncar
ése
ctio
n in
the
chao
tic r
egim
e fo
r D
uffin
g equ
atio
n
*F. T
aken
set
al,
Lect
ure
No
tes
in M
ath
em
atic
s, V
ol.
89
8 p
.366
(S
prin
ger
)
2005
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17
Cha
otic
Tra
ject
orie
s: P
rimar
y an
d R
efle
cted
Sho
ck
Reg
ions
of A
DS
TM
akin
g th
e C
ase
for
Uni
vers
ality
of T
urbu
lenc
e
The
cyc
lic b
ehav
ior
in th
e tw
o gr
aphs
indi
cate
a d
eterm
inis
tic p
roce
ss a
nd
as w
ell a
s fr
eque
ncy
driv
en b
ehav
ior.
2005
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18
Ove
rall
Com
plex
ity in
the
Ioni
zing
Sho
ck W
aves
P
lasm
as
The
cha
otic
dim
ensi
on D 2
show
s a Γ ΓΓΓ
− −−−lik
e be
havi
or w
ith in
crea
sing
tu
rbul
ent e
nerg
y.
2005
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AR
MS
Mee
ting,
Jun
21-
25, U
CLA
-IP
AM
19
Cha
ract
eris
tic F
req
vs. T
urbu
lent
Ene
rgy
The
str
onge
st fr
eque
ncy,
def
ined
as
the
char
acte
ristic fr
eque
ncy,
sh
ows
a Γ ΓΓΓ− −−−l
ike
beha
vior
with
incr
easi
ng tu
rbul
ent e
nerg
y.
2005
CA
AR
MS
Mee
ting,
Jun
21-
25, U
CLA
-IP
AM
20
Sam
ple
Com
paris
ons:
Fre
quen
cy in
the
Fou
rier
Spe
ctrum
w
ith F
requ
ency
from
Pha
se T
raje
ctor
y vs
. Tur
bule
nt
Ene
rgy
in th
e P
rimar
y S
hock
Reg
ion
The
ove
rall
tren
ds in
the
char
acte
ristic
freq
uenc
y are
mirr
ored
in d
irect
m
easu
rem
ents
of f
requ
ency
from
the
phas
e sp
ace
trajec
tory
for
the
mov
ing
plas
ma.
2005
CA
AR
MS
Mee
ting,
Jun
21-
25, U
CLA
-IP
AM
21
Sam
ple
Com
paris
ons:
Fre
quen
cy in
the
Fou
rier
Spe
ctrum
w
ith F
requ
enci
es fr
om P
hase
Tra
ject
orie
s vs
. Tur
bulen
t E
nerg
y in
the
Ref
lect
ed S
hock
Reg
ion
The
ove
rall
tren
ds in
the
char
acte
ristic
freq
uenc
y are
mirr
ored
in d
irect
m
easu
rem
ents
of f
requ
ency
from
the
phas
e sp
ace
trajec
tory
for
the
plas
ma
at r
est*
.*K
. Bel
ay,J
. Val
entin
e,R
.L.W
illia
ms,
J.A
. Joh
nson
, J. A
ppl.
Phy
s. V
81,
Iss
3 (1
997)
2005
CA
AR
MS
Mee
ting,
Jun
21-
25, U
CLA
-IP
AM
22
Sam
ple:
Ove
rall
Cor
rela
tion
of F
requ
ency
D
eter
min
atio
ns
On
the
left
is a
Fou
rier
Pow
er S
pect
rum
from
the
turbu
lent
pla
sma
behi
nd a
n io
nizi
ng s
hock
wav
e. O
n th
e rig
ht, f
or th
e sa
me
data
,is
a hi
stog
ram
of f
requ
enci
es m
easu
red
usin
g th
e ph
ase
spa
ce tr
ajec
torie
s.
The
res
ults
are
com
para
ble.
2005
CA
AR
MS
Mee
ting,
Jun
21-
25, U
CLA
-IP
AM
23
Sum
mar
y
•The
da
ta
indi
cate
th
at
the
desc
ripto
rs
of
turb
ulen
ce(t
urbu
lent
en
ergy
, ch
arac
teris
tic
freq
uenc
y,
and
chao
tic
dim
ensi
on
(D2))
succ
essf
ully
sh
ow
turb
ulen
ce i
n bo
th s
tatio
nary
and
mov
ing
plas
mas
with
the
Γ ΓΓΓ-li
kebe
havi
or
asso
ciat
ed w
ith a
seco
nd o
rder
pha
se tr
ansf
orm
atio
n, ju
stify
ing
a se
arch
for
a fo
rce-
like
cons
trai
nt.
•A d
irect
cor
rela
tion
is f
ound
bet
wee
n th
e cy
clic
reg
ular
beh
avio
rs i
n th
e ph
ase
spac
e tr
ajec
torie
s an
d th
e ch
arac
teris
tic fl
uctua
tion
freq
uenc
ies
in b
oth
mov
ing
and
stat
iona
ry p
lasm
as.
2005
CA
AR
MS
Mee
ting,
Jun
21-
25, U
CLA
-IP
AM
24
Sum
mar
y
�S
ince
thes
e reg
ular
beh
avio
rsin
pha
se s
pace
evo
lutio
ns
are
gene
rally
thou
ght t
o be
a c
onse
quen
ce o
f hid
den
un
derly
ing
dete
rmin
ism
, thi
s co
nseq
uenc
e m
ight
als
o no
w b
e as
soci
ated
with
the c
hara
cter
istic
freq
uenc
iesi
n ou
r da
ta a
s th
e ba
sis
for
cont
inui
ng s
tudi
es o
f de
term
inis
tic d
ynam
ical
und
erpi
nnin
gs in
turb
ulen
ce
phys
ics.
