Superdiffusion2005

1
?????????????? ? ??????????? ???????????????
???????????? ?????? ? ??????????? ?????
???????????? ????????? ? ? ????????????   ?.?.
??????1, ?.?. ?????2, ?.?. ???????2 1????????
???????? ???????, ??? ???????????? ???????? ,
123182, ??. ????????? 1, ??????,
?????? 2???????? ??????????? ???????????? ???,
117997, ??????, ??????
2

• ???????? ?????????????? ?????? ? ?????????
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• ?????? ???????????? ??????? ?????????????,
  ??????????? ???????, ?????? ??????????
  ?????????????? (???????????????), ??????
  ?????????????????? ???????, ??????????????????
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• ??????????? ??????????? ???????? ??????????????
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• ?????? . ???????????????? ?????? ???????? ?????,
  ??????? ????????? ???????? ??????????????
  ??????????????

3
T-10 ???????

• Circular cross-section
• R 1.5 m aL 0.3 m
• BT ? 3 T
• IP ? 500 kA
• t 1 sec
• n ? 6x1013 cm-3

ECRH system up to 1.8 MW, t 0.4 sec 4 gyrotrons
at 140 GHz 1 gyrotron at 130 GHz
4
HYBTOK-II ??????? Nagoya University

• Circular cross-section
• R 0.4 m aL 0.11 m
• BT ? 0.3 T
• IP ? 5 kA
• t 0.15 sec
• n ? 2x1012 cm-3

5
Linear Plasma Device, NAGDIS-II
linear divertor plasma simulator , n1020m-3 in
steady state. Detached plasma condition by
increasing neutral gas pressure
0.3 m
6
Large Helical Device LHD
superconducting heliotron-type device with a set
of l 2/m 10
R 3.9m, a 0.65 m, Bt lt 2.89 T, NBI power,
PNBI lt 5MW, ne lt 8 1019 m -3 - 1.5 1020
m-3 Te lt 4.4 keV , Ti lt 3.5 keV tE 0.3 s
LHD Divertor probes
7
Interball , 1998
Interball-1 500- 200000 km, Interball-2
500-20000 km
8
???????????? ??????? ????????
????????????? ??????????? ???????? ????????
????????? ???? ? ???????????? ???? ?????????????
??????, ???, ???????-3, ????????, ???-2, ????????
????? ? ????.
??1-6 ??????? ?????????????? ???? ? ?????????
0-100 ???. ???, ??-3 ????????????? ???????
??????????? ?????????? ???? (0-32 ??). ??-2
???????????? ?????? ??????????? ?????????? ???? ?
????????? 20-2000 ??. ??-1 ??????? ???? ???
????????? ???? ?????? ? ????????? 0.1-2000 ??.
9
Interball-1 OT summary

• In summer outer cusp throat (OT) is open for the
  MSH flow.TBL (turbulent boundary layer) is mostly
  in MSH.
• In winter OT is closed by smooth MP at larger
  distance. Inside MP plasma balls (few Re)
  contain reduced field, heated plasma weaker
  TBL.
• OT encounters on 98.06.19 at 10-11 UT by
  Interball-1 and Polar are shown

10

• ????????? ??????

T-10 HYBTOK-II NAGDIS-II Earths magnetopause
Magnetic field,T 2 0.3 0.1 (1-10)10-9
Core plasma density ,cm 3 6 1013 3 1012 1014
Core plasma electron temperature, eV 2000 200 10
Edge plasma density ,cm 3 3 1012 1 1012 3 1012 1-100
Edge plasma electron temperature, eV 10-50 10 1-3 20
Edge plasma ion temperature, eV 10 2-5 lt1 7
Plasma scale, m 1 0.2 0.1 108
???????? ???????????? ???????? ???????????? . . .
. ??????????? ???????- ???? ???????????
????????? ???? ?????? ???????????? ??????
??????????? ?????? (35)107 ?? /??? 5
??3 20100 ?? 10-4 ? 101 ?? 10 ?? ?????? 3-107
??/??? 1013 ??-? 20 ?? 3040 ? 50 ?? 35 ??
11
?????????? ?????? ? ???????????? ???????????
????? ? ???????????? ?????????? ? ? ????????????
?????
??????????????
???????????? ?????, Interball-I, 1998.06.19
??????? ?-10,
???????? ????????? NAGDIS-II
12
????? ??????? ? ?????????????? ???????
T-10
T-10
????? ??????? ????????????? ??????????? ??
??????? ?????????????????? ??????? C(?) ? ?
, ?-2.7?-3.4 ??? ?????????????????
?????
13
??????? ????????????? (PDF) For fully
random fluctuation (Kolmogorov-type model ) PDF
is a Gaussian Coherent events
deviation from a Gaussian ( reduce the number
of degrees of freedom , process called as
intermittent)
close to Gaussian
Intermittent bursts
PDF
Large amplitudes exhibit non-Gaussianity.
14
?????? ???????????? ???????? ??????????????

• ?????? ???????????? (??????????) ??? ????????
  ??-? ?????-??????
• ???????????????? ??????????????? ?????????
  ??????? ? ???????????? ?????????, ???????? ?
  ?.?., ???????? ????????????? ??????????????????
  ?????????????? ?????? ?? ?????????? ???????????
  ?????????????? (??????- ????????? ??????
  ????????? ? ?????????)
• ?????? ??????????? ????????????? ??????????????
  ??? ???????? ??????????, ??????????? ?
  ?????????????? ?????-??????????. ?????????
  ???????? ???????????? ?????????? ???????? ???????
  ?? ???????, ??????????? ????????, ?????????
  ????????

15
?????? ?????????????? ??????????? (K41)
? ???????????? ?? ??????????? ????????????
??????????? (? ? ???????? ? ? ??????)
???????????? / ??????????????
Fluid , Europ. Phys.J. B 8,301(1999)
T-10
????????????? ???????? ??????????????
16
?????????????? ??????????? ? ?????????
?????????????, ?????-?????? ??-?
17
??????????????

• ?????????????? ??? ????????? ?????????
  ???????????? ??????????????, ? ??????? ????????
  ??????? ???????????? ? ??????????
  (????????????????).
• ??????? ??????? ????????????? ????????? ?
  ?????????
• ?????????????? ??????????? ? ?????????????????
  ???????????? ???????? ??????????? ???????? ?
  ???????????? ????????????? ?????? ??? ? ????????,
  ??? ? ? ?????????? ??????? ??????????
• ????????? ????????? ???????? ???????????
  ???????????.

18
3D isotropic MHD turbulence, DNS Biskamp
Mueller small-scale turbulent structures
Current-density isosurfaces vorticity
isosurfaces
19
??????????????? ????????? ????????????
?????????? ??????????????????????? ?????-??????

• ? ????????????????? ???????????? ??????
  ??????????????? ????????? ??????? ????????
  ??????? ?? ??????? ???????????? ????? ? ?????
  ?????? ?? ?????????, ??? ???????? ??????????
  ???????????? ? ?????????? ?????????? (? ?????????
  ? ???????????? ???????? ????? ??????????? ?
  ????????? ???????).
• ??? ??????? ?????? ?????????? ? ???????????
  ????????? l (? ltltlltltL, L - ???????????? ???????
  ????????, ? ? - ??????? ??????????) ???????????
  ?????????????? ??????????????? ??????????.
• ????????? ??????? ????? ???????? ????????????? ?
  ??????????? (???????????????).

20
?????????? ?? ?????????????? ??????
?????????????? ?????? ? ??????????? ???????????
????????

• ??? ???????? ????????? ? ??????? ??? ???????????
  ???????? ??? ? ????????????? ????????? ? ????????
  ??????????
• ?????????????? (Intermittency) ????? ???????? ?
  ??????????????? ?????????
• ?????????????? ???? ? ?????????? ??????????????
  (???????????????) ???????? ???????? , ??????? ?
  ????????? ??????????
• ??????????? ?????????????
• ???????????????? ??????????
• ?????????????

21
?????? ???????? ?????????????? ? ???????????????

• ????????????? (Kolmogorov, 1961, statistics of
  fluctuations )
• ???? ?????? (Novikov, Stewart 1964,Frish 1978)
• ????????????????? ?????? (Frish 1985, Sreenivasan
  e.a. 1991)
• ???????????????? ?????? ???????????
  ??????????????? (extended self-similarity, She,
  Leveque, Debrulle, 1994)

22
?????? ???????????? ???????? ??????????????
??????????? ???????
??? ???????? ???????? Sqrq/3 (??????
???????????) ?????????????? ????? ????????
Sqr?(q) ?????? ????????? ?? ????????? ??-?
?????-?????? ?(3)1
23
?????? ? ???????????? ? ????????
in Frish
24
?????????????? ?????? ?????? ????????

• ????? ??????????? ????????? (rs, di,de, etc.)
• ??????????? (?????? 3D ??????????? ?? 2D)
• ????????????? ? ????????? ????
• ?????????? ? ??????? ???????? ????????
• ????????? ??????? (?????????) ? ??

25
????????? ???????? ????????????????? ??????
ln2-nL, ?lnX(t) (?i1-nWi)?LX(t)
Single exponent (dimension)
Family of exponent (spectrum of dimensions)
Monofractal isotropic process Kolmogorov type
Multifractal process long-range correlations,
memory effect
Rather considerable generalization of fractal
geometry
26
?????????? ??????????????????? (?????????????????)
PDF of increments ?lX(t)X(tl)-X(t), depends on
scale, l1-128 mcs
Coarse integral time T50-200 ?s
27
Brownian motion does not demonstrate
multifractality monofractal, trivial
self-similarity
PDFs of increments ?lX(t)X(tl)-X(t), dont
depend on scales l
l, lag
28
?????????? ?? ?????????? (????????????)???????
? ????????? ????????? ??? ????????????
??????????? ????? ???????????? ? ???????????
?????? ???????? ??????????? ??????? S(q,l)
lt??lX(t)?q gt l ? ( q), ?( q)qH - ?2 q2 , ?2
???????? ???????????????????
?2?0.03-0.05
?(q)qH ??????????????? ??????????? ????????
fBm
29
??????????? ?????????????? (Extended
Self-Similarity)
Benzy , 1993, ????????????????? ?????????
???????? ???? ??? ????????? ????? ??????????

• She, Leveque, Dubrulle, 1994, ??????
• ??????? ?????????????? ?????????
  (????????????? ??????) ??-? ?????-??????
  ???????????? ??? ????????
• ???????? ???????? ??
• ???????????????? ??????????
• ?????????? ???????? ?????????? ??????????????,
  ??????? ?????? ??????? ?????????? ????????
  (???????? ?????????)
• ??? ????????? ?- ??????? ?????????????? ?
  ?- ?????????????? ????????? ????????
  ????????????? ????????

30
??????????????, ??-????????, ???????? ?????
?????????????? ?????????? ??????????
?????????????? ???? ??? ??????????? ?.?.
??????????
31
S(q,l) S(3,l)?(q)/ ?(3) She and Leveque (SL)
?????????? 3D ?????????????? Debrulle
???????? SL ??????? ????????? ? ????????
???????? (??????????? ???????) ???????
???????? ?????????? ?????????? el ,
??????????????? ???????? ????????????? ??????????
- ???? ????????? ???????
32
Cascading in log-Poisson model of intermittency
random multiplicative process
Random multiplicative process energy dissipation
rate e at two different scales l1 and
l2 e(l2)W(l1, l2) e(l1) e(l)lt(q) ,
log(Wq(l1, l2) )/log(l2/l1)t(q)
Coherent structure of large scale
defects
The defects adds a finite amount of disorder to
the singular structure events Amplification of
modulation by defects of integer numbers
Establishing log-Poisson process ln(e) obeys
Poisson distribution
33
??????????? ???????Extended Self-Similarity
??????????? ???????? ? ???????????? ?????????
ESS ?? ???? ????????? ?????????!!!!
ESS ??????????? ? ???????? ??????????? ????????
34
???????????
???????????????? ?????? ????????????? 1D
?????????? ?????????? ????????? (?????????)
???????? ????????????? ???????????
?????????????? ???????????
35
????????? ???????????????? ??????ß ???????
??????????????,?- ????????? ?????????????
????????
36
??????-????????? ?????????? ?????????
??????????????????? ? ??????????????
? ? X(t) X(t?)-X(t)
???????? ???????????? ????????????
D ?(C (l,?))
T-10
????????????????? ?????? ??????????
????????????? ltx2gtt2-?, 0lt?lt1, tltT
NAGDIS-II
Brownian motion
???????????? ?????????? ? ???????? ?????? ????
???????? ?? ?????????? ?????????????
37
??????

• ? ???????????? ??????????? ????? ?????? ?
  ???????????? ?????????? ? ? ???????????? ?????
  ??????????? ???????? ??????????? ???????????????
• ?????????? ????????? ??????????? ??????? ?????
  ???? ??????? ? ?????? ???????????????? ??????
  ???????? ??????????????
• ?????? ?? ???????????? ???????????????? ??????
  ?????????????? ??????? ???????, ??? 1D
  ????????? - ?????????? ????????????? ????????? ?
  ???
• ???????? ???????? ??????????? ? ?
  ????????????????? ??????????????, ??? ?????????
  ?? ????????????? ???????? ?????????????? ?
  ???????? ??????????????

38
??????? ????????? ??????????? ????? (FLR) ??????
? ??? ????????? ????? ?????? ???????? P ?
????????? ???????? rdu/dt j ? ? - grad(?e
?i) ? ????? ?????????? ?????
??? ? u ? ? (j ?
? - grad ?e)/en le m0 dj /dt ?????? ????????
? ??????????????????? ??????
??? ?p, p- ????????? ?????????? ???????? ? bi
?i/?. ?
???????, ? ??????? ????????? ???? ?????????? ??
z, ?????? ????-?????? ????????? ???
???????? n ?i/2r wc, ??? ?i - ????????????????
?????? ????????
39
(No Transcript)
40
Scale invariance a feature of turbulence
The multifractal formalism for turbulent flows -
to describe the anomalous scaling properties of
turbulence at large Reynolds numbers. Scale
invariance of the Navier-Stokes equation tu u
?u - ?p/?jB/c??u ? u - velocity field,
equation is invariant with respect to the scale
transformation For viscosity ? 0 - any
a. Parisi and Frisch proposed each fluctuation
h at scale r is weighted with a probability
distribution Ph(r) r3-D(h).
Multi-scaling (multifractality)
41
Klimov, S., S.Romanov, E.Amata, J.Blecki,
J.Buechner, J.Juchniewicz, J.Rustenbach,
P.Triska, L.J.C.Woolliscroft, S.Savin,
Yu.Afanas'yev, U.de Angelis, U.Auster,
G.Bellucci, A.Best, F.Farnik, V.Formisano,
P.Gough, R.Grard, V.Grushin, G.Haerendel,
V.Ivchenko, V.Korepanov, H.Lehmann, B.Nikutowski,
M.Nozdrachev, S.Orsini, M.Parrot, A.Petrukovich,
J.L.Rauch, K.Sauer, A.Skalsky, J.Slominski,
J.G.Trotignon, J.Vojta, R.Wronowski, ASPI
Experiment Measurements of Fields and Waves
Onboard the INTERBALL-1 Spacecraft, Ann.
Geophys., v. 15, p. 514-527, (1997). Savin S.P.,
Zelenyi, L.M., Amata, E. et al., Dynanic
Interaction of Plasma Flow with Hot Boundary
Layer of Geomagnetic Trap, JETP Letters, 79,
452-456, (2004)
42
T-?2 diagrammultifractal multiplicative cascade
concentrated in closed domain
Integral time T dimensional parameter. Is it an
additional scale in the process and symmetry?
43
Power spectra S(f) ?n(f)?2
Edge, SOL

• Bandwidth of drift-wave instabilities 1-1000
  kHz
• No monochromatic modes
• No clear evidence of 1/f (Kolmogorov type)
  spectra over the whole frequency range

In SOL typical d-1.5-3, S(f)fd
44
Space plasmas
45
Chapman Ferraro, JGR, 36, 77, 1931
Axford et al., JGR, 70, 1231, 1965
Stern, JGR, 90, 10,851,1985
???????, ????????, ???????, ????????,
"???????????? ???????????? ????????????" ?.
?????, 1965
46
BIMF
Bin
Fx , u
Fz
????????? ??????? ?????????????? ?
???????????????
const u / B03
??? ru- ???????????? ?????? ??????????, ? L -
??????? ????????????. ??? b 1-10 ? ????????????
?????? ?????????????? ??????????? ? ???????????
?????????. ???????? u, ????????? ?? ????????????
?????, ????????? ???? B0 - ????? ??????? ???
??????, ?.?. ?????????? ?????????????? ????????
??????????? ?? ??????? ??????? ?????, ???
???????? ? ???????? ?????????? ??????
(????????????? ???????????????????)
47
T-?2 scales feature of multifractal
multiplicative cascade
structure function of increments has nonlinear
scaling M(q,l) ???lX ?q ?l?(q) ?(q)qH-?2q2
T-10 SOL
PDF of increments ?lXX(tl)-X(t)
multifractality parameter
PDF,l300
PDF,l1
?20.03-0.05 T?50-200 ?s for fusion devices T
?60-80 sec for space plasma
Coarse (mixing) time scale of T in the process
scale of coherent structures
48
????????? ???? ????????? ? ?????? ??????? ?
?????? ????????????, ??? ?????????? ?????????
???-???????????
? ?????????? ????????????? ?????? ??????
????????????.
49
Evidence of multifractality (multi-scaling) PDF
of increments ?lX(t)X(tl)-X(t), depends on
scale l1-128 mcs
Shuffled data become Brownian with Gaussian
increments
50
??????? ?????????????
T-10, flux

• Strong events happen more frequently than
  random
• Statistics varies strongly in space
• Resembles power law Cauchy function

T-10, density
51
?????????? ??????????????????? (?????????????????)
PDF of increments ?lX(t)X(tl)-X(t), depends on
scale, l
Non-Gausian
Lag l
Gausian
52
????? ??????? S(f) ?n(f) ?2
Edge, SOL

• Bandwidth of drift-wave instabilities 1-1000
  kHz
• No monochromatic modes
• No clear evidence of 1/f (Kolmogorov type)
  spectra over the whole frequency range

In SOL typical d-1.5-3, S(f)fd
53
Extended Self-Similarity
?????????? ?????????????? ??????????? ?
??????????? ?????????
54
PDF ????? ?????????? ?????????? ????????,
????????, ????
55
????????????? ??????????? ??????????? ?
???????? ? ???????????? ??????????? ??????????,
????? ????????? ????????? (????????????? ?
????????????????) ?????? ?? ???????? ? ???????????
56
?????? ???????????? ???????? ??????????????
??????????? ???????
??? ???????? ???????? Sqrq/3 (??????
???????????) ?????????????? ????? ????????
Sqr?(q) , ?(q) ?????????? ??????? ??????
????????? ?? ????????? ??-? ?????-?????? ?(3)1
q2 ?????? ???????? (?????), ?? ????????
????????? ??????? ???????