Apparent cross-field superslow
propagation in coronal
magnetic structures
T. Kaneko1, M. Goossens2, R. Soler3, J. Terradas3,
T. V. Doorsselaere2, T. Yokoyama1, A. N. Wright4
1. UTokyo, 2. KU Leuven, 3. UIB, 4. St. Andrews
ISSI Beijing
2015/12/14
Aim
Kaneko et al. 2015, ApJ, 812, “Apparent Cross-field Superslow
Propagation of Magnetohydrodynamic Waves in Solar Plasmas”
Aim of this study
• Theoretical formulation and numerical modeling of
apparent cross-field propagation by phase mixing of
Alfven/slow mode
-
Apparent “superslow” propagation.
Application to coronal structures
(prominence / coronal potential arcade)
• To show that apparent propagation can be a useful
tool for prominence/coronal seismology
-
Phase speed depends on profile of Alfven frequency
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2015/12/14
Outline
Introduction
• Cross-field superslow propagation
Application of apparent propagation model
• Formulation of apparent wave length & phase velocity
• Application to the waves in coronal structures
- 1: prominence in flux rope
- 2: coronal potential arcade field
Group velocity of apparent propagation
Conclusion
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2015/12/14
Cross-field superslow waves: Observation
Cross-field superslow propagation in prominence
(Schmieder et al., 2013)
𝑩
SDO/AIA 304Å
Hinode/SOT Ca II H line
• THEMIS/MTR spectropolarimeter (He D3 line) :
𝐵 ≈ 7.5 G, 𝐵 is parallel to solar limb and in the plane of sky.
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2015/12/14
Distance (arcsec)
Cross-field superslow waves: Observation
Time-distance intensity map along slit (a)
16
12
8
5 ± 2 km/s
1000
500
1500
2000
Time (sec)
2500
Fast speed: 𝑣𝑓 ≈ 75 ~ 750 km/s
(𝑇 = 8000K, 𝐵 = 7.5 G, 𝑛𝑒 = 109−11 cm−3 )
𝑩
Hinode/SOT Ca II H line
The propagation speed was even less than
sound speed.
Fast mode ?
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2015/12/14
Cross-field superslow waves: Simulation
Simulation of prominence formation
(Kanekok & Yokoyama, 2015)
• Cross-field propagation
Property of fast mode
• Superslow propagation
Propagation speed 3 km s
≪ fast mode speed 160 km s
Fast mode ?
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2015/12/14
Cross-field superslow waves: Simulation
Propagation speed 3 km s
≪ fast mode speed 160 km s
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2015/12/14
Prominence Formation
What is the origin of cool dense plasmas ?
Radiative condensation (thermal nonequilibrium)
Coronal plasmas are cooled down and condensed by radiative
cooling (thermal instability).
(Karpen et al., 2007; Xia et al.,2012; Kaneko & Yokoyama,2015)
Injection, Levitaion model
Chromospheric plasmas are lifted up to coronal height by jet or
emerging flux.
(Chae et al., 2003; Okamoto et al.,2007,2008; Deng et al.,2000 )
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2015/12/14
Our model
(a)
(b)
Initial state
stratified atmoshere
thermal equilibrium
(c)
formation of
flux rope
condensation
•relatively dense plasmas at the bottom (strong cooling)
•closed field line (reduction of conduction effect)
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2015/12/14
Numerical setting 1/4
Basic equations:
𝜕𝜌
+ 𝒗 ∙ 𝛻𝜌 = −𝜌𝛻 ∙ 𝒗,
𝜕𝑡
radiative cooling
5
𝜕𝑒
+ 𝒗 ∙ 𝛻𝑒 = − 𝑒 + 𝑝 𝛻 ∙ 𝒗 + 𝛻 ∙ 𝜅𝑇 2 𝒃𝒃 ∙ 𝛻𝑇 − 𝑛2 𝛬 𝑇 + 𝐻,
𝜕𝑡
𝑝
𝑒=
,
𝛾−1
𝑚𝑝
𝑇=
,
𝑘𝐵 𝜌
thermal conduction
heating
𝜕𝒗
1
1
+ 𝒗 ∙ 𝛻𝒗 = − 𝛻𝑝 +
𝛻 × 𝑩 × 𝑩 + 𝐠,
𝜕𝑡
𝜌
4𝜋𝜌
𝜕𝑩
= −𝑐𝛻 × 𝑬,
𝜕𝑡
1
4𝜋𝜂
𝑬 = − 𝒗 × 𝑩 + 2 𝑱,
𝑐
𝑐
𝑐
𝑱 = − 𝛻 × 𝑩.
4𝜋
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2015/12/14
Numerical setting 2/4
Energy equation:
5
𝜕𝑒
2
2
+ 𝒗 ∙ 𝛻𝑒 = − 𝑒 + 𝑝 𝛻 ∙ 𝒗 − 𝑛 𝛬 𝑇 + 𝐻 + 𝜂𝐽 + 𝛻 ∙ 𝜅𝑇 2 𝒃𝒃 ∙ 𝛻𝑇
𝜕𝑡
Λ(T) erg cm3 s-1
radiative cooling
10
21
cooling loss function
1022
1023
1024
10 4
105
106
107
temperature K
108
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2015/12/14
Numerical setting 3/4
Energy equation:
5
𝜕𝑒
2
2
+ 𝒗 ∙ 𝛻𝑒 = − 𝑒 + 𝑝 𝛻 ∙ 𝒗 − 𝑛 𝛬 𝑇 + 𝐻 + 𝜂𝐽 + 𝛻 ∙ 𝜅𝑇 2 𝒃𝒃 ∙ 𝛻𝑇
𝜕𝑡
Λ(T) erg cm3 s-1
background heating
10
10
21
cooling loss function
𝐻 ∝ 𝑃𝑚
Assumption:
coronal heating related to
magnetic energy
22
1023
10
𝑇cutoff
24
10 4
105
106
when 𝑇 < 𝑇cutoff
107
temperature K
108
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𝐻 = 𝑛2 Λ(𝑇)
2015/12/14
Numerical setting 4/4
gravity
Initial condition
temperature: 106 K (uniform)
density:2 × 109 cm−3 (stratified)
field strength: 3G
mechanical & thermal equilibrium
Numerical scheme
MHD part: RCIP-MOCCT
(Kudoh et al.,1999; Xiao et al., 1996)
Anisotropic thermal conduction:
Slope-limiting method
(Sharma & Hammett 2007)
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2015/12/14
Result
flux rope formation
thermal imbalance
in thermally isolated
closed loops
radiative condensation
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2015/12/14
Cross-field Superslow Waves: Simulation
Radiative condensation
Excitation of waves
• Cross-field propagation
Property of fast mode
• Propagation speed 3 km s
≪ fast mode speed 160 km s
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2015/12/14
Mechanism of Apparent Propagation 1/2
Alfven/slow standing wave on
each magnetic loop with
individual frequency
𝑟
e.g.
Alfven velocity: 𝑣𝐴 = const.
Alfven frequency: 𝜎𝐴
𝑣𝐴
𝜎𝐴 = 𝑛
2𝜋𝑟
Each magnetic loop oscillates
independently.
𝜎 𝑟 − ∆𝑟 > 𝜎 𝑟 > 𝜎(𝑟 + ∆𝑟)
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2015/12/14
Mechanism of Apparent Propagation 2/2
Each magnetic loop oscillates independently.
𝑟
𝑟𝜑=𝜋
2
𝑟 + ∆𝑟
e.g. 𝜎 𝑟 − ∆𝑟 > 𝜎 𝑟 > 𝜎(𝑟 + ∆𝑟)
Phase lag is generated among magnetic surfaces.
phase mixing
3𝜋
𝜑=
2
𝜑=
5𝜋
2
7𝜋
𝜑=
2
𝜑=
9𝜋
2
𝑟
𝑟 − ∆𝑟
Apparent propagation
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𝑡
2015/12/14
Apparent wavelength, phase speed
Apparent propagation in the 𝑟-direction
𝜉𝑦,𝑛 𝑡, 𝑟 = exp 𝜎𝐴 𝑟 𝑡
𝜎𝑎𝑝
phase: 𝜑 = 𝜎𝐴 𝑟 𝑡
𝜕𝜑
=
= 𝜎𝐴 𝑟 ,
𝜕𝑡
𝑟
𝜕𝜑
𝑑𝜎𝐴
𝑘𝑟 ,𝑎𝑝 = −
= −𝑡
𝜕𝑟
𝑑𝑟
𝑣𝑝ℎ,𝑎𝑝
𝜎𝑎𝑝
𝜎𝐴 (𝑟)
=
=−
𝑑𝜎𝐴
𝑘𝑟,𝑎𝑝
𝑡
𝑑𝑟
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general formulation
→Kaneko et al. 2015
2015/12/14
Application to Simulation Result 1/5
𝑣𝑎𝑝 = 1 − 5 km/s
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2015/12/14
Application to Simulation Result 2/5
1. The flux rope is regarded as a concentric cylinder.
𝑑𝜎𝐴
𝑘𝑎𝑝 (𝑟, 𝑡) = −𝑡
𝑑𝑟
𝜎𝐴 (𝑟)
𝑣𝑎𝑝 (𝑟, 𝑡) = −
𝑑𝜎
𝑡 𝐴
𝑑𝑟
𝜎𝐴 = 2𝜋𝑣𝐴 𝐿 = 𝑣𝐴 /𝑟
(𝐿 = 2𝜋𝑟)
𝑘𝑎𝑝
𝑣𝑎𝑝
𝑡 𝑣𝐴 𝑑𝑣𝐴
𝑟, 𝑡 =
−
𝑟 𝑟
𝑑𝑟
1 𝑣𝐴 (𝑟)
𝑟, 𝑡 =
𝑡 𝑣𝐴 − 𝑑𝑣𝐴
𝑟
𝑑𝑟
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𝒓
𝐿 𝑟 = 2𝜋𝑟
2015/12/14
Application to Simulation Result 3/5
2. Gradient of Alfven velocities is negligible.
𝑘𝑎𝑝
𝑣𝑎𝑝
𝑡 𝑣𝐴 𝑑𝑣𝐴
𝑟, 𝑡 =
−
𝑟 𝑟
𝑑𝑟
1 𝑣𝐴 (𝑟)
𝑟, 𝑡 =
𝑡 𝑣𝐴 − 𝑑𝑣𝐴
𝑟
𝑑𝑟
𝑑𝑣𝐴
=0
𝑑𝑟
harmonic mean Alfven velocity
of each mag. surface along 𝑟
𝑑𝑣𝐴
=0
𝑑𝑟
𝑣𝐴 (𝑟)
𝑘𝑎𝑝 𝑟, 𝑡 = 𝑡 2
𝑟
𝑟
𝑣𝑎𝑝 𝑟, 𝑡 =
𝑡
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2015/12/14
Application to Simulation Result 4/5
𝑟
= 𝑣𝑎𝑝 ,
dashed lines:
𝑡 − 𝑡𝑖
𝑣𝑎𝑝 = 1 − 5 km/s
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𝑡𝑖 = 3000 s
2015/12/14
Application to Simulation Result 5/5
wavelength @ 𝑟=4 Mm
Dots:
wave length in 𝑟-direction
Solid line:
apparent wavelength
computed by
𝜆𝑎𝑝
1 2𝜋𝑟 2
2𝜋
=
𝑟, 𝑡 =
𝑘𝑎𝑝 𝑡 − 𝑡𝑖 𝑣𝐴 (𝑟)
where
𝑟 = 4 Mm,
𝑡𝑖 = 3000 s,
𝑣𝐴 = 70 km s
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2015/12/14
Distance (arcsec)
Applicability to Observation
Schmieder et al. (2013): cross-field superslow propagation
16
Intensity
Ca II H line
12
5 ± 2 km/s
8
500
1000
Fast mode model:
𝑣𝑓 =
𝐶𝑠2
+
𝑉𝐴2
1500
2000
Time (sec)
Apparent propagation model:
≈ 75 ~ 750 km/s
𝐶𝑠 = 10 km/s (𝑇 = 8000K)
𝑉𝐴 = 70~750 km/s
(𝐵 = 7.5 G, 𝑛𝑒 = 109−11 cm−3 )
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𝜎
𝑟
𝑣𝑎𝑝 =
≈ = 3 ~ 6 km/s
𝑑𝜎 𝑡
𝑡
𝑑𝑟
𝑟 = 4 arcsec, 𝑡 = 500~1000 s
Order-of-estimate is O.K.
2015/12/14
Application to coronal potential arcade
Potential arcade field
in stratified atmosphere
parameter
Λ𝐵
𝛿=
Λ
𝜌 = 𝜌0 exp(−𝑦/Λ)
𝐵 = 𝐵0 exp(−𝑦/Λ𝐵 )
10
Eigen frequency (fundamental)
𝛿=6
𝜎𝐴 𝐿/𝑣𝐴
8
6
𝛿=4
4
2
Oliver et al.
0 1993
0.0 0.2
0.4
𝛿=2
𝛿=0
0.6 0.8
𝑥0 /𝐿
𝑥0
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1.0
2015/12/14
Application to coronal potential arcade
phase speed:
𝛿<2
Everywhere upward
𝛿>3
Upward and downward
coexist.
10
Eigen frequency (fundamental)
8
𝜎𝐴 𝐿/𝑣𝐴
𝑣𝑝ℎ,𝑎𝑝
𝜎𝐴 (𝑥0 )
=−
𝑑𝜎
𝑡 𝐴
𝑑𝑥0
𝛿=6
upward
downward
6
𝛿=4
4
2
0
0.0
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𝛿=2
0.2
0.4
𝛿=0
0.6 0.8
𝑥0 /𝐿
1.0
2015/12/14
Demonstration (δ=1)
𝛿=1
Everywhere upward
Alfven mode (Oliver et al.,1999)
𝑑 2 𝑣𝑦 𝜔2
+
𝐾𝑣 = 0
𝑑𝑥 2 𝑣𝐴0 𝑦
𝑥
cos Λ0
𝐵
𝐾=
𝑥
cos Λ
𝐵
𝛿
cos −2
𝑥0
Λ𝐵
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2015/12/14
Demonstration (δ=6)
𝛿=6
Upward and downward
coexist.
Alfven mode (Oliver et al.,1999)
𝑑 2 𝑣𝑦 𝜔2
+
𝐾𝑣 = 0
𝑑𝑥 2 𝑣𝐴0 𝑦
𝑥
cos Λ0
𝐵
𝐾=
𝑥
cos Λ
𝐵
𝛿
cos −2
𝑥0
Λ𝐵
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2015/12/14
Discussion
𝛿
Λ𝐵 =
Λ
derived by apparent phase speed
derived by temperature
where Λ:pressure scale height
Λ𝐵 :magnetic pressure scale height
We can know magnetic pressure scale height Λ 𝐵 by
the analysis of phase speed of apparent propagation.
What can we do by knowing Λ𝐵 ?
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2015/12/14
Discussion
• Torus instability
criteria: 𝑛𝑐𝑟
where
𝑑 log 𝐵𝑝
𝑦
<𝑛=−
=
𝑑 log ℎ
Λ𝐵
𝑛𝑐𝑟 = 1 − 2
(e.g. Kliem & Torok, 2006; Fan & Gibson, 2007)
𝑦 > 𝑦𝑐𝑟 = 𝑛𝑐𝑟 Λ𝐵 :critical height
Fan 2010
If Λ𝐵 is derived from the apparent phase speed,
we can know critical height of torus instability.
Flare/CME prediction is possible ?
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2015/12/14
Summary for application to potential arcade
• Phase velocity of apparent propagation depends on
𝛿 (= Λ𝐵 /Λ).
• We can know 𝛿 or Λ𝐵 from phase speed of apparent
propagation.
• Only from the direction of propagation, we can
estimate how large 𝛿 is (whether lager than 2 or not).
• Hopefully, applicable to flare/CME prediction
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2015/12/14
(Apparent) Group velocity
𝜎𝑎𝑝
𝑑𝜎
𝜕𝜑
=
= 𝜎 𝑧 , 𝑘𝑎𝑝 = −𝑡
𝑑𝑧
𝜕𝑡
𝑑𝑘𝑎𝑝
𝑑2𝜎
𝑑𝜎
=−
𝑑𝑡 − 𝑡 2 𝑑𝑧
𝑑𝑧
𝑑𝑧
𝜕𝜎𝑎𝑝
𝑣𝑔 =
𝜕𝑘𝑎𝑝
𝜕𝑧
𝑑2𝜎
=− 𝑡 2
𝜕𝑘𝑎𝑝
𝑑𝑧
−1
0
𝑑𝜎
𝜕𝜎𝑎𝑝 𝜕𝑡
𝜕𝜎𝑎𝑝 𝜕𝑧
=
+
= − 𝑑𝑧2 ≠ 0
𝑑 𝜎
𝜕𝑡 𝜕𝑘𝑎𝑝
𝜕𝑧 𝜕𝑘𝑎𝑝
𝑡 2
𝑑𝑧
• We can mathematically derive group velocity.
• The group velocity is not zero across mag. surfaces…
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2015/12/14
(Apparent) Group velocity
Questions:
• Does the group velocity have physical meaning?
Can we find physical variables propagating with the
group velocity?
Answer: Yes. Shown later.
• Does it mean energy propagation across mag. surfaces?
Answer: ?? (depends on interpretation?)
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2015/12/14
Example
example:
𝜎 𝑧 = 5 1 + cos
𝜋
𝑧
2
𝑑𝜎
5𝜋
𝜋
𝑧 =−
sin 𝑧
𝑑𝑧
2
2
𝑣𝑝ℎ
𝜎 𝑧
=−
> 0,
𝑑𝜎
𝑡
𝑑𝑧
𝑑2 𝜎
5𝜋 2
𝜋
𝑧
=
−
cos
𝑧
2
𝑑𝑧
4
2
𝑑𝜎
𝑣𝑔 = − 𝑑𝑧2 < 0
𝑑 𝜎
𝑡 2
𝑑𝑧
In this particular case, the phase velocity and group velocity are
easily distinguished by the direction of propagation.
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2015/12/14
Example: current wave
|𝐽𝑥 |
𝐵𝑦
𝑣𝑝ℎ
𝜎 𝑧
=−
>0
𝑑𝜎
𝑡
𝑑𝑧
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𝑑𝜎
𝑣𝑔 = − 𝑑𝑧2 < 0
𝑑 𝜎
𝑡 2
𝑑𝑧
2015/12/14
Example:current wave
𝐵𝑦
𝐽𝑥
𝜎 𝑧
>0
𝑑𝜎
𝑡
𝑑𝑧
Phase propagates upward.
𝑣𝑝ℎ = −
𝑑𝜎
𝑣𝑔 = − 𝑑𝑧2 < 0
𝑑 𝜎
𝑡 2
𝑑𝑧
Isocontour of |𝐽𝑥 |
propagates downward
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2015/12/14
Discussion 1/3
 Proof: current wave propagates with the group velocity.
𝑘𝑎𝑝
𝑑𝜎
= −𝑡
𝑑𝑧
𝑑𝑘𝑎𝑝 = 0
𝑑𝑘𝑎𝑝
𝑑𝜎
𝑑2𝜎
=−
𝑑𝑡 − 𝑡 2 𝑑𝑧
𝑑𝑧
𝑑𝑧
𝑑𝜎
𝑑𝑧
= − 𝑑𝑧2 = 𝑣𝑔
𝑑 𝜎
𝑑𝑡
𝑡 2
𝑑𝑧
The group velocity represents the propagation speed of
constant apparent wave number.
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2015/12/14
Discussion 2/3
 Proof: current wave propagates with the group velocity.
𝐵𝑦 = sin 𝜎 𝑧 𝑡
𝐽𝑥
𝜕𝐵𝑦
𝑑𝜎
= −𝑡
cos 𝜎 𝑧 𝑡
𝐽𝑥 =
𝑑𝑧
𝜕𝑧
= 𝑘𝑎𝑝 cos 𝜎 𝑧 𝑡
Envelope of current is
proportional to 𝑘𝑎𝑝 .
Envelope is 𝑘𝑎𝑝
Envelope of current moves with the group velocity 𝑣𝑔 .
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2015/12/14
Discussion 3/3
𝑑𝜎
𝑣𝑔 = − 𝑑𝑧2
𝑑 𝜎
𝑡 2
𝑑𝑧
𝜕𝑘𝑎𝑝
𝑑𝜎
=−
𝜕𝑡
𝑑𝑧
𝑑𝜎
𝑑𝑧
𝑡 = 𝑡3
k
𝑣𝑔
k
𝑡 = 𝑡2 𝑣𝑔
𝑡 = 𝑡2
1
monotonically
decrease
𝑡 = 𝑡3
k
𝑡 = 𝑡1
𝑡 = 𝑡1
0
𝑑𝜎
𝑑𝑧
𝑑𝜎
constant
𝑑𝑧
monotonically
increase
z
1
0
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𝑡 = 𝑡1
z
0
1
2015/12/14
z
Conclusion
• Phase mixing of Alfven/slow mode wave can be
observed as cross-field superslow propagation.
- Apparent wavelength and phase velocity depends on
gradient of Alfven frequencies.
• In our prominence model, apparent wavelength and
phase velocity agrees with theoretical values
computed by in-plane Alfven velocity.
• In the case of coronal potential arcades, the phase
velocity depends on the ratio between mag. scale
height and pressure scale height.
• Group velocity exists. It corresponds to apparent
propagation speed of current density.
ISSI Beijing
2015/12/14