Gauging anisotropic scaling in
plasma turbulence and
topological change at the ion
scales
Khurom H. Kiyani
Romain Meyrand, Fouad Sahraoui, Sandra Chapman
Theme issue on the topic of plasma turbulence
Downloaded from http://rsta.royalsocietypublishing.org/ on May 28, 2015
Volume 373 | Issue 2041 | 13 May 2015
Contents
Theme issue: Dissipation and heating in solar wind turbulence
Article ID
INTRODUCTION
Dissipation and heating in solar wind turbulence:
from the macro to the micro and back again
KH Kiyani, KT Osman and SC Chapman
Short-wavelength plasma turbulence and
temperature anisotropy instabilities: recent
computational progress
SP Gary
20140149
A dynamical model of plasma turbulence in the solar
wind
GG Howes
20140145
20140154
Kinetic scale turbulence and dissipation in the solar
wind: key observational results and future outlook
ML Goldstein, RT Wicks, S Perri and F Sahraoui
20140147
20140148
Dynamic properties of small-scale solar wind
plasma fluctuations
MO Riazantseva, VP Budaev, LM Zelenyi,
GN Zastenker, GP Pavlos, J Safrankova,
Z Nemecek, L Prech and F Nemec
20140150
Generation of magnetic holes in fully kinetic
simulations of collisionless turbulence
V Roytershteyn, H Karimabadi and A Roberts
20140151
20140152
Turbulent reconnection and its implications
A Lazarian, G Eyink, E Vishniac and G Kowal
20140144
20140155
ARTICLES
Intermittency, nonlinear dynamics and dissipation
in the solar wind and astrophysical plasmas
WH Matthaeus, M Wan, S Servidio, A Greco,
KT Osman, S Oughton and P Dmitruk
The role of turbulence in coronal heating and solar
wind expansion
SR Cranmer, M Asgari-Targhi, MP Miralles, JC
Raymond, L Strachan, H Tian and LN Woolsey
Third-moment descriptions of the interplanetary
turbulent cascade, intermittency and back transfer
JT Coburn, MA Forman, CW Smith, BJ Vasquez and
JE Stawarz
Anisotropy in solar wind plasma turbulence
S Oughton, WH Matthaeus, M Wan and KT Osman
Article ID
20140146
Theme issue on the topic of plasma turbulence
ISSN 1364-503X | Volume 373 | Issue 2041 | 13 May 2015
Dissipation and heating in solar wind turbulence
Theme issue compiled and edited by Khurom H. Kiyani, Kareem T. Osman and Sandra C. Chapman
http://
www.warwick.ac.uk/fac/
sci/physics/research/
cfsa/people/kiyani/
philstransa.zip
Typical spectral density of solar wind turbulence
Ѥc~10NP
transition region
ï(
10
WUDFHSRZHUVSHFWUDOGHQVLW\>Q7Hz-1]
ѩeaNP
ѩL~10NP
10
ï(
10
0
10
f -1 range
inertial range
sub-ion range
ï
10
ï(
ï
10
$&(0),>GD\V@
$&(0),>KUV@
&OXVWHU)*067$))ï6&>PLQV@
ï
10
ï
10
ï
10
ï
10
ï
10
ï
10
ï
10
spacecraft frequency [Hz]
0
10
1
10
2
||
PSD /PSD
−2
2
−4
0
−6
−2
−8
−4
−0.4
−0.6
−6
−0.8
−1
−1.2
−8
⊥
log
2
−1
PSDlog
/PSDPSD [nT2 Hz−1log
PSD
[nT
Hz
]
]
10
|| 10 ⊥
10
0
power spectral
density {component anisotropy}
−0.4
−0.6
−0.8
−1
kρi � 1
parallel
transverse
search coil noise
kρe � 1
isotropy
parallel
transverse
search coil noise
Cluster FGM and
STAFF search-coil
−2
0
1
isotropy −1
log10 spacecraft frame frequency [Hz]
Kiyani et al. ApJ (2013)
adaptive local scale-dependent fluctuations and
background guide field
background field
fluctuations
small
scales
large
scales
local scale-dependent fluctuations and background
field using low/high pass filters
f1
spectral power
b1
fluctuations
background field
frequency
local scale-dependent fluctuations and background
field using low/high pass filters
spectral power
b1
b2
f1
f2
fluctuations
background field
filter bank
implementation
using quadrature
mirror filters
b3 f3
frequency
adaptive local scale-dependent fluctuations and
background guide field
background field
fluctuations
small
scales
large
scales
Each band (fluctuations) has its corresponding background field
angles of measurement w.r.t. B
2πfsc
k=
VSW
Taylor frozen-in approx,
for low-frequency
phenomena
-VSW
VA
<1
MHD V
SW
else
Vφ
<1
VSW
Anisotropy (angle between B and V)
4
9
x 10
k oblique
8
k oblique
k Perp
7
Histogram count
6
DR
IR
5
4
3
2
1
0
20
30
40
50
60
70
80
B.V angle (degrees)
90
100
110
120
Magnetic Compressibility and linear wave modes
S P. Gary and C. W. Smith,
J. Geophys. Res 114, A12105 (2009).
magnetic compressibility
N
2
�
δB
1
� (tj , f )
C� (f ) =
2 (t , f )
N j=1 δB�2 (tj , f ) + δB⊥
j
0
Magnetic Compressibility C||
10
isotropy
ï1
10
local field 55º-65º
local field 65º-75º
local field 75º-85º
local field 85º-95º
global field
ï2
10 ï2
10
ï1
0
1
10
10
10
Normalised wavenumber kli
Kiyani et al.
ApJ (2013)
kinetic Alfvén wave fluctuations
N
2
�
δB
1
� (tj , f )
C� (f ) =
2 (t , f )
N j=1 δB�2 (tj , f ) + δB⊥
j
0
Magnetic Compressibility C||
10
Kiyani et al.
ApJ (2013)
isotropy
−1
10
−2
10 −2
10
local field 55º-65º
local field 65º-75º
local field 75º-85º
local field 85º-95º
global field
linear Vlasov 85º
linear Vlasov 89.99º
−1
0
1
10
10
10
Normalised wavenumber kρi
Gary & Smith (2009)
Salem et al. (2012)
TenBarge et al. (2012)
Hall physics
Hall-MHD model
∂B
1
= ∇ × (v × B) +
∇ × (B · ∇B) + η∇2 B
∂t
µene
inertial range
kρi <<1
convective term
dominates
[B-field frozen to ions]
dissipation range
kρi >1
Hall term dominates
[ions demagnetise]
Hall physics
∂B
1
2
= ∇ × (v × B) +
∇ × (B · ∇B) + η∇ B
∂t
µene
assume in the inertial range
∇ㅗ >> ∇|| [ kㅗ >> k|| ]
and
δBㅗ >> δB||
supported by observations, simulations and theory
Hall physics
∂B
1
2
= ∇ × (v × B) +
∇ × (B · ∇B) + η∇ B
∂t
µene
�
∂B� ��
∼
∇
×
(B
·
∇
∇δB
)
+
∇
×
(B
·
∇δB
∇
)
⊥1
⊥2
⊥2
⊥1
∂t �Hall
Hall physics
∂B
1
2
= ∇ × (v × B) +
∇ × (B · ∇B) + η∇ B
∂t
µene
�
∂B� ��
∼
∇
×
(B
·
∇
∇δB
)
+
∇
×
(B
·
∇δB
∇
)
⊥1
⊥2
⊥2
⊥1
∂t �Hall
The Hall term isotropises components.
transition range
at kρi =1
Convective 50%
Hall 50%
0
Magnetic Compressibility C||
10
isotropy
ï1
10
local field 55º-65º
local field 65º-75º
local field 75º-85º
local field 85º-95º
global field
ï2
10 ï2
10
ï1
0
1
10
10
10
Normalised wavenumber kli
C� � 0.5 × 0.33 + 0.5 × 0.1 � 0.22
higher order scaling
(a i.)
&'()*)+,-+./0+/.),1/20+342-
C!
"!
%$
%!
$
!
!$
!
<'.'**)*
%!
;
*45 6: 9
%!
"$
<'.'**)*
;
%$
%!
"!
,
+.'2-().-)
+.'2-().-)
"$
*45 6: 9
(b i.)
&'()*)+,-+./0+/.),1/20+342-
$
!
!$ ,
"
#
*45 6!,7-)0-89
!"
(b ii.)
@2).+3'*,.'25),-0'*325,)A<42)2+-
%=#
,
<'.'**)*
+.'2-().-)
%="
"6;9
!=>
F3--3<'+342,.'25),-0'*325,)A<42)2+,
$
<'.'**)*
GH!=?I
#
+.'2-().-)
GH!=J>
C
!=#
%
!="
!,
!
"
B4;)2+,;
#
!,
!
"
B4;)2+,;
note: Inertial range
data is from ACE
Also see:
K. H. Kiyani et al.
Phys. Rev. Lett.
103, 075006 (2009)
"
DE:D,-+/11
"6;9
%
!=?
"
�m
N �
�
�
�
δB
(t
,
τ
)
1
j
�(⊥)
m
�
�
√
S�(⊥) (τ ) =
�
�
N j=1
τ
%!
%!
(a ii.)
!
*45 6!,7-)0-89
Structure Functions
#
standardised probability densities
0
−1
parallel
transverse
0
Inertial
Range
log10 Ps(δ Bs)[σ nT−1]
s
−1
log10 Ps(δ B )[σ nT ]
1
−2
−3
−4
−5
−6
−20
−10
0
10
δ Bs [nT/σ]
parallel
transverse
Dissipation
Range
−3
−4
−5
−6
−20
0
δ Bs [nT/σ]
20
1
isotropy
)[σ nT−1]
Parallel
Fluctuations
s
−1
)[σ nT ]
s
−2
−7
20
1
0 dissipation
range
−1
inertial
−2
range
−1
0 dissipation
range
−1
inertial
−2
range
Kiyani et al.
ApJ (2013)
Transverse
Fluctuations
Effects of varying plasma beta [ERMHD]
�
�
∂B⊥
c
=−
∇⊥ × B · ∇B�
∂t
4πene
∂B�
βi (1 + Z/τ )
c
=−
∇⊥ × [B · ∇B⊥ ]
∂t
2 + βi (1 + Z/τ ) 4πene
includes density variations in a linear way
Schekochihin (2009) and Cho & Lazarian (2009)
Survey study
Interval
Date/Time (UT)
SC
|B| [nT]
σ|B|
βi
σβi
ni [cm−3 ]
σni
Te /Ti
σTe /Ti
Ti! /Ti⊥
σTi! /Ti⊥
|Vsw | [km s−1 ]
σVsw
θBV [deg]
σθBV
Sample Size
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
2006/03/07 14:00-14:32 UT
2006/03/07 14:32-14:53 UT
2006/03/07 14:53-15:04 UT
2006/03/07 15:04-15:40 UT
2007/01/27 18:00-19:50 UT
2007/01/27 21:20-21:50 UT
2007/01/27 21:50-23:20 UT
2004/02/22 03:10-04:20 UT
2004/02/22 04:20-04:50 UT
2004/02/24 17:10-18:10 UT
2004/02/21 17:24-17:50 UT
2004/02/21 18:15-19:30 UT
2007/02/05 15:40-19:10 UT
2007/02/22 12:10-13:30 UT
2007/02/22 13:30-14:20 UT
2007/02/22 14:20-17:00 UT
2004/01/27 00:00-02:00 UT
2007/02/23 01:50-02:50 UT
2007/01/30 05:20-08:10 UT
2002/02/19 00:32-01:10 UT
2002/02/19 01:10-02:45 UT
2002/02/19 02:45-03:30 UT
2001/04/05 22:35-23:32 UT
2002/02/11 19:19-20:29 UT
2007/01/30 00:00-01:20 UT
2007/01/20 12:10-13:20 UT
2007/01/20 13:20-14:10 UT
2007/03/28 04:00-04:40 UT
[3,3,2]
[3,3,2]
[3,3,2]
[2,3,2]
[3,1,2]
[3,3,2]
[3,3,2]
[3,3,2]
[3,3,2]
[3,1,NA]
[3,3,2]
[3,3,2]
[3,3,1]
[4,3,1]
[4,3,1]
[4,3,1]
[3,1,1]
[1,1,1]
[1,1,2]
[3,1,2]
[3,1,2]
[3,1,2]
[1,1,2]
[3,1,2]
[4,1,4]
[3,1,4]
[3,1,4]
[4,1,4]
3.01
2.42
1.71
3.06
4.84
4.34
4.78
8.75
8.18
5.57
8.49
8.45
4.93
3.24
2.23
4.14
9.87
3.62
6.01
9.18
7.45
10.18
7.29
7.45
4.52
3.83
3.45
4.06
12.35
9.81
28.46
10.27
2.36
4.33
5.25
2.32
2.80
1.40
2.06
4.02
5.46
5.75
29.42
5.81
3.84
3.06
4.98
8.43
11.31
2.66
5.29
2.52
7.30
3.33
3.71
4.44
3.17
5.21
12.45
2.89
1.08
2.00
1.63
0.57
0.74
0.93
0.43
0.49
1.00
1.84
6.68
1.11
0.44
1.65
1.54
0.94
1.64
0.33
0.75
0.77
2.58
2.10
2.54
2.68
46.67
33.27
75.86
39.32
31.86
29.68
28.04
26.78
25.65
45.45
25.15
30.06
22.71
29.23
51.40
25.56
14.32
21.73
44.87
46.14
47.89
34.57
36.91
36.32
51.48
53.64
67.42
54.38
6.42
6.79
6.89
6.74
11.96
15.41
15.25
12.39
12.89
7.64
9.12
9.69
8.49
11.41
12.88
9.15
7.87
9.01
2.75
27.97
27.79
13.40
3.10
3.55
2.57
1.75
1.88
2.32
4.87
4.18
4.23
5.38
5.17
3.64
3.88
3.84
5.06
4.36
3.81
7.50
4.47
8.11
10.37
9.48
7.96
2.94
7.95
7.76
11.58
7.00
18.63
6.40
6.90
9.06
7.28
6.56
2.17
2.15
2.34
2.37
2.71
2.22
2.21
2.76
2.50
NA
2.98
2.78
2.53
3.37
2.59
2.65
1.82
2.68
0.41
1.53
1.46
3.30
0.85
1.23
0.45
0.44
0.56
0.49
28.82
30.02
31.53
26.04
30.81
28.35
24.99
25.75
22.34
NA
23.02
24.26
14.63
19.00
15.95
15.46
12.38
17.64
43.48
46.24
42.18
32.71
42.45
32.90
41.92
51.58
70.28
57.37
1.16
1.08
1.05
1.07
1.29
1.10
1.04
1.28
1.32
0.91
1.46
1.31
1.08
1.00
0.82
0.93
0.81
0.84
0.67
1.18
0.90
0.85
0.53
0.91
0.75
0.54
1.16
0.86
46.31
40.23
44.19
42.45
46.65
44.52
38.97
47.11
43.51
62.11
35.89
45.16
24.05
35.23
28.55
26.45
20.58
27.93
61.00
67.59
55.03
52.45
52.84
49.12
58.89
59.57
57.71
55.87
384.32
382.15
380.00
376.95
301.65
309.75
309.42
383.45
374.27
388.67
376.24
368.87
331.67
304.50
305.62
296.07
428.90
288.81
649.29
353.42
364.10
356.48
533.51
569.55
666.52
588.85
577.89
568.74
0.78
0.81
0.71
3.01
1.81
0.65
0.83
1.90
1.55
1.57
0.73
1.46
0.92
3.79
2.39
4.50
1.40
0.67
2.75
0.98
1.00
1.98
1.69
2.33
2.35
2.43
3.03
2.79
79.00
92.00
82.00
96.00
140.00
86.00
99.00
73.00
87.00
71.00
66.00
85.00
97.00
110.00
111.00
109.00
102.00
NaN
77.00
116.00
80.00
70.00
95.00
82.00
75.00
78.00
59.00
66.00
18.99
15.22
25.61
10.42
10.00
5.81
9.09
10.96
11.49
14.08
7.58
15.29
8.25
12.73
8.11
5.50
6.86
NaN
20.78
6.90
23.75
11.43
7.37
13.41
24.00
14.10
21.53
22.73
43,905
31,499
16,498
49,906
151,406
40,905
130,906
100,905
40,906
81,810
38,999
108,406
306,812
108,518
74,999
235,907
171,812
81,812
246,818
56,999
142,500
63,405
67,027
96,812
2,151,828
1,890,002
1,350,005
1,071,818
Table A1
Plasma parameters, along with their variability/errors for all 28 Cluster intervals used in this study: SC is the particular Cluster spacecraft used for the [FGM/STAFF-SC,CIS,PEACE] instrument respectively in
the interval; |B| is the magnetic field magnitude; βi , ni and Vsw are the ion plasma beta, number density and solar wind speed respectively; Te and Ti are the electron and ion temperatures respectively; θBV is the
angle between the local background magnetic field and the mean solar wind velocity vectors; and ! and ⊥ designate parallel and perpendicular to the magnetic field vector respectively. All these quantities are
mean averages over the interval. All ‘errors’ (labelled with σ) are percentage errors, calculated by dividing the typical variability – represented by the standard deviation – of the relevant quantity in the interval by
the mean value. Intervals 25-28 are intervals where the instruments were operating in burst mode i.e. where the magnetic field sampling (after joining the FGM and STAFF-SC signals) is approximately 450 Hz;
the rest of the intervals are sampled in normal mode with nominal magnetic field sampling at approximately 25 Hz.
28 data intervals from the cluster mission
5
4
4.5
3.5
4
βi
3.5
3
Vsw
3
2.5
2
2.5
2
1.5
1.5
1
1
0.5
0.5
0
2
4
6
8
10
12
5
0
300
350
400
450
500
550
600
650
6
|B|
4
3
θBV
5
4
3
2
2
1
1
0
2
4
6
8
10
0
60
70
80
90
100
stationary plasma parameters
110
120
130
140
results: spectral slopes (trace)
Trace spectral exponents
7
inertial range
sub−ion range
6
counts
5
4
3
2
1
0
1.5
2
2.5
3
see also: Smith et al. (2006) and Sahraoui et al. (2013)
Magnetic compressibility variation with beta
0
10
C||
isotropy
ï1
10
ï2
10
ï2
10
ï1
10
0
kli
10
1
10
see also:
Smith, Vasquez and
Hamilton, JGR (2006)
Magnetic compressibility variation with beta
0
10
C||
isotropy
ï1
10
β↑
ï2
10
ï2
10
ï1
10
0
kli
10
1
10
see also:
Smith, Vasquez and
Hamilton, JGR (2006)
Magnetic compressibility variation with beta
−0.5
isotropy
∼
−0.6
0.1
βtotal
0.3
∼ βtotal
log10 C||
−0.7
−0.8
−0.9
−1
C||(kρi=0.1) [inertial range]
Corr(βtotal,C||)=0.564; P=0.002
−1.1
C||(kρi=3.2) [sub−ion range]
Corr(βtotal,C||)=0.579; P=0.002
−1.2
0
0.2
0.4
0.6
0.8
1
log10 βtotal
1.2
1.4
1.6
1.8
results: scaling
all monoscaling at sub-ion scales
Sub−ion scale monoscaling exponents
10
9
HA
8
H ||
7
counts
6
5
4
H⊥ > H||
3
2
1
0
0.5
0.6
0.7
0.8
0.9
1
1.1
results: intermittency and heavy tails
log10 Ps(δ Bs)[σ nT−1]
0
−1
−2
6
parallel
transverse
Dissipation
Range
4
−3
3
−4
2
−5
−6
1
−7
0
−20
0
s
δ B [nT/σ]
20
1
nT−1]
5
0 dissipation
range
−1
inertial
Transverse
Fluctuations
6
8
10
12
14
kurtosis
16
18
results: solar wind speed variation
3.3
sub−ion spectral slope
3.2
Pearson correlation coefficient = 0.58
P−value = 0.001
3.1
3
2.9
2.8
2.7
2.6
2.5
2.4
300
350
400
450
500
550
solar wind speed [km/s]
600
650
results: solar wind speed variation
3.3
sub−ion spectral slope
3.2
Pearson correlation coefficient = 0.58
P−value = 0.001
3.1
3
2.9
2.8
2.7
2.6
2.5
2.4
300
350
400
450
500
550
600
650
solar wind speed [km/s]
Macbride, Smith & Foreman (2008) and
Smith et al. (2006)
Scaling, fractals and physics
Monofractal dissipation
Single exponent H
living on a fractal set of
dimension D where
dissipation occurs
Global Scale Invariance
Cantor ‘dust’ courtesy of Andrew Top: http://www.andrewtop.com/IFS3d/
IFS3d.html
Multifractal dissipation
Courtesy of R. Roemer (Warwick): multifractal electronic
wavefunction at metal insulator transition in 3D Anderson model
Multifractal dissipation
multiple exponents h
living on a fractal sets of
dimension D(h) where
dissipation occurs
Local Scale Invariance
Speculation on the physics
see also Chevillard, Castaing, and Levˆeque, Eur. Phys. J. B 45, 561–567 (2005)
Using fluid models to explore structural changes
Use simulations to see if the scaling can be
replicated by fluid models containing
relevant physics and to explore the
topology of structures involved
Using fluid models to explore structural changes
MHD
EMHD
Structure functions
Structure functions
10
10
log10(Sm)
15
log10(Sm)
15
5
0
5
0
ï3
ï2
ï1
log10(1/k)
0
ï3
ï2
ï1
log10(o [secs])
Scaling exponents
Scaling exponents
2
(|Z+||Zï|)m/2œ
3
p/8 + 1 ï (1/4)p/2
2
c(m)
c(m)
3
1
0
Slope=
0.64728
±0.011401
1
0
1
2
3
Moment m
4
5
0
0
0
1
2
3
Moment m
4
5
LIC visualisation -- mostly current sheets in MHD
MHD with strong guide field [15362 X 128 decaying]
LIC visualisation of filamentary structures in EMHD
Electron MHD [10243 decaying]
visualisation of filamentary structures in EMHD -Beltrami fields [(Curl B) X B=0] & Hall physics
take home message
•Enhanced magnetic compressibility @ sub-ion scales.
With strong beta variation.
•Near component isotropy at sub-ion scales.
•PDFs show higher levels of intermittency at such scales;
ubiquitous multi->monoscaling transition in all intervals.
• Simulations reproduce this scaling behaviour and -using them as a proxy for what is happening -- we see
that there is a transition from a current sheet dominated
regime to one which is filament dominated.