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AUTUMN COLLEGE ON PLASMA PHYSICS8 October - 2 November 2001

Alfvenic Wave Packets-1 & II

B. Buti

California Institute of TechnologyPasadena, U.S.A.

These are preliminary lecture notes, intended only for distribution to participants.

strada costiera, I I - 34014 trieste italy - tel.+39 04022401 I I fax +39 040224163 - sci_info@ictp.trieste.it - www.ictp.trieste.it

Alfvenic Wave Packets

1. INTRODUCTION

2. LINEAR ALFVEN WAVES

3. NONLINEAR ALFVEN WAVES:

OBSERVATIONS

STABILITY

GOVERNING EQUATIONS

4. SOLITARY ALFVEN WAVES

5. CHAOTIC ALFVEN WAVES

6. DISPERSIVE MHD SIMULATIONS

7. HYBRID SIMULATIONS

NONLINEAR LANDAU DAMPING

ANOMALOUS TRANSPORT

SHML

STA6L

NOT£ \

±

/" ™ ^ g^P \ . i^-iffct „

LHP

NO

A-

»

CHAOS

RHP {P

TWO MODES OF SOLAR WIND FLOW

PROPERTY (1 AU)

Speed (V)

Density (n)

Flux (nV)

Magnetic Field(£r)

Temperatures

Coulomb collisions

Anistropies

Beams

Structure

Composition

Waves

Minor species

LOW SPEED

< 400 km/sec

- 1 0 /cm3

~ 3 x 108/cm2 sec

~ 2.8nT

Tp ~ 4 x 104 K

Te ~ 1.3 x 105K> Tp

Important

Tp isotropic

None

Filamentary, highly >

He/H ~ 1 - 30%

Both directions

riijnv variable

HIGHSPEED

700 - 900 km/sec

~ 3 /cm3

~ 2 x 108 /cm2 sec

~ 2.8 nT

Tp ~ 2 x 10s K

Te ~ 10*K< T,

Negligible

Fast ion beams

+ electron "strahl"

Uniform, slow changes

He/H ~ 5%

Outwards propagating

constant

Associated with

Sunspot minimum

Sunspot maximum

Ti~AT,

Streamers, Coronal holes

transiently open field

±15 deg from "equator" > 30 deg

Dominant at most latitudes Less frequent

EVOLUTtOM OF MOMUME.AR.

K^-k,

CQO&TICV

STOKES

OfBACKWARD

OBSERVATIONS:

Large amplitude Alfven waves observed in a variety ofplasmas:

* solar wind

* solar corona

* environment of comets

* planetary bow shocks

* interplanetary shocks etc.

Some Chaotic Features of Solar Wind:

* Intermittent Turbulence

* Fractal / Multifractal Nature (Dimension)

Solar Wind Spectral Characteristics:

* Power Law Spectra

* Increase in Spectral Index with heliocentric distance

* Break in Power Spectra; Break-point moves

towards lower frequencies with increasing heliocentric

distances

COSMIC PLASMAS:

Implications of the existence of large-amplitude Alfven

waves in many cosmic plasmas have been investigated

to model:

* turbulent heating of the solar corona

* interstellar scintillations of radio sources

* coherent radio emissions

* generation of stellar winds and extragalactic jets etc

2JIFF£R6MT APPROACHES

•# -

tsf\

* ^ V * 3 6tt«te

SOLITARY ALFVEN WAVES

10

at+ v . (Pv) = o

dvp — = — Vp + J X B

dB~dt

= V X (v X B) (V X B) X BP

B is normalized to Bo, p to po>v to FA = Bo/(4irpo)

1/2,t to O "̂1, the ion cyclotron frequencyand / toSubscript '0' refers to equilibrium quantities.

or n

-ax

f

= o

/

-f *4

LH

12

+ '— o

+ LHP

R.HP

O3NIUS

= - V s X. -t- 3tdvC' IJJS.+0

x == AMPL(T"U3>E OF

13

and

A U inPLASMAS

1 d

_ _dp _ d_BlP dt ~ dr dr 2

dt r dr

1 dx dr

, and Bl)

. pp ** = const.

&o(r) r2 =

U(r) r2 =

14

0.040 pr2=3.5Ro, r3=5Ro, T4=

0.0000 8*10

16

1010"

Frequency (f10" 10-1

17

OBSERVATIONSTURBULENCE

POWER SPECTRAetc.

18

WAVES 3mHz

Ulysses

1.0

nT

-1.0

BT, nT o.

IBI, nT o.8

0700 0710 0720 0730

Time, UT

June 3, 1994Day 154

r • • • | • i r • | i

„ . « I

i . , , , i

. . . j , . ,

(J)1

. . ,1 1 . .

• • 1 •,

©

i . 1 ll

1 '(3)

1 • . 1 .

1 1 . . .

0740 0750 0800

Figure 10. An example of an Alfv6n wave with a phase-steepened edge (rotational discontinuity [RD]). From leftto right, dashed vertical lines give the start and stop times of intervals of analyses.

221S DENSKAT AND NEUBAUER: SOLAR wj

HELIOS 7

10

10

Fig. 2. Magnetic field (vector components and magnitude) pow-er spectral densities at different heliographic distances. The 'meanfield1 coordinate system (Z axis parallel to the average direction ofthe vector magnetic field) was used throughout the paper. Thespectra were computed from 40-s averages of the magnetic field over11 1/2 hours on January 24, 1900 UT to January 25, 0623 UT-, 1976(0.97 AU) and on April 14, 2300 UT to April 15, 1023 UT, 1976 (0.29AU). Also given are the 95% confidence limits for the spectraldensity computed from an equivalent of 32 degrees of freedom.

21

Tp/Te

_J I 1

ISEE-3 Hourly averages

I

2 . 5 -

2 -

1.5-

* •.'»' ," ''

0.5-

0

beta

CHAOTIC ALFVEN WAVES

23

DRIVEN DNLS EQUATION:

id2 B+

at 4(B | B

NUMERICAL SOLUTION

Spertral-collocatiop nripthor! wif.b periodic boundary conditions used

Initial condition:(21/2 - l W 2 B eie

[2V2 CoSh (2 Vs x) -

Bmaj : is the amplitude of the soliton

0 (x, * = 0) = - ys x + 3 tan"1 [(21/2-hl) tanh(2Vs x)\

V8 is the soliton speed defined by,

v =

8 (1 -

24

* O

|B|2

0.2

•15

• 1 0

.05

n I ! J .1.1----

0 10 15 20 25 M 30

7B ss.

25

CHAOTIC AVJ

0.1

2IBI

.05

O

IBI

1 .

IBI

O.5

RHP

A ==.

. . . . . . J U~....*./ LVIA. . A . / U A . .....I I. »J

2 x 1 O 4 X 1 O 5.IW

6 x 1 O5

oRF1b 1 .5x104-

26

UUP

O.5T

O 2x1 4-x 1 O' 6x1 O'

. 5

O

= -D3

WAikiM2x10 ' Ax 1 6x1O'

.75

.25

As r=-

o 6.OX1O3 1 .2x1O4 1.8x10^

27

10 100

10

100

FIGURE Shows Spectra for magnecic field turbulencegenerated through chao» for a) RHP with A = 0.5. b)LHP with A = O.t. c) LHP with A = .03 and d) LHPwith A = .003.

28

SLOW SDLAR^ lAlNCD

« . - * • «

F A S T SOLAR.

A04

29

•*• MOJJE.LS SA.SE.3)

/

MOT

COUPLl^4.Sl

U-F

30

*- S3

BUT

HA.U-

ear/VELU , BOX I

O-f

31

MHD SIMULATIONS

32

at ox ox

8B 8 t ^ . d (ldB1_ „—(vxB — v) = —i

dt dx dx \p dx

vx) + (7 - l)/3— = 0a*

Ua. is flow velocity along direction of propaga-tion, B = (By + i Bz) and v = (vy + i vz).Adiabatic equation of state: pp~1 = const..7 is ratio of specific heats.

33

Initial condition:

B(x t = 0) =[2V2 Cosh (2 V8 x) -

is the amplitude of the soliton

$ (x, t = 0) = - Va x + 3 tan"1 [(21/2+l) tanh(2V, x)]

V, is the soliton speed defined by,

TT \ ~ A / max

0.20

0.10

0.00

(a)

•

I Ii 1 A

MM' JM

', \

0.00

0.04

0.02

0.00

-0.020.00

-0.02

-0.04

-0.06

(a)

i hr

8p

85 170 255XOOV^Qp)

0 85 170 255X(10VA/Qp)

Figuire 1. (a) The evolution of the magnetic and den-sity fluctuations, 13 and fip, respectively, for a RHP soli-ton for ft — 0.3 at /, w 40SJ~ * (da«lied line) and at 51

~l (solid line), (b) Same as (a) but for 0 = 1.5.

sf =

35

0.40

0.20

0.20

0.120.08

0.04

0.00-0.040.00

-0.02

-0.04

-0.06

. (a)

(b)

•

V(!

5p

5p

-

85 170 255 85 170 255

Figure 2. (a) Same as Fig. la but for LHP soliton.(b) Same as Fig. lb but for LHP soliton.

36

l&l

0

Bperp(t)

200 400 600

37

1.5

1.0

38

oo 0.0 200 300 400 500

39

HYBRID SIMULATIONS

COLLAPSE OF ALFVENIC WAVEPACKETS

DYNAMICAL TURNING POINT—» CHANGE OF POLARIZATION

RADIATIONS

40

Simulation Model

* One - Dimensional Hybrid Code

* Electrons as Isothermal Fluid

* Protons as Particles Maxwellian

* HIGH RESOLUTION:

Simulation Box Length 860 Ion Inertial Lengths

Number of Cells 2048

number of Particles / Cell 200

Simulation Time 1000 Ion Gyro-Periods

Time resolution 0.01

4

41

Initial condition:

B i x t = 0 ) =[2V2 cosh (2 V. x) - 1]V2

is the amplitude of the soliton

0 (x,t = 0)-•= - V, aj + 3 tan"1 [(21/2-hl) tanh(2V, x)]

V9 is the soliton speed defined by,

v _8 (1 -

COLJ-.APS&

8

oo 4

0

0

P10b

200• I I I

400 600 800

7*a •

-ro

43

CD

500

72. =

-r =- \

44

M

45

8

0 200 400 600 800

R10b

46

0.0060

0.0050

n nnAn i~CN*

_c

Q_

(DCL

0.0030

0.0020

0.0010

0.0000

i I I 1 1 1 1 1 1

- ! - = 'gCOOOOCOOOOOt tX^

AAAAAAAAAAAAAAAAA/AVA7AA?77'A"j9f A\AAAAAA^

0 200 400 600t

800

P15bc

1000

47

ION HOLES

NONLINEAR LANDAU DAMPING

48

_ I

^ o-i

~NW

yr~V'-~-*

200 400 600 800

nl i i T I .—^—;— I . > ~ — I , , , 1

0 200 400 600 800 200 400 600 300

S 4F

i—i—i—|—i—i—nr-|—i—i—i—1—i i i f~} 8 1 I I ' I ' ' ' I ' ' ' I

0 200 400 600 800 200 400 600 800 200 400 SOO 800

49

1 i B(—l—t-l—p

o-a.

•00 600 800 200 400 600 800 200 400 600 800

200 400 600 800 200 400 600 800 200 400 600 800

50

oF

cj

cor~tPRS.ssi\fe SOLITARY

SCALE

o*. J)PLA.SKA 75> -

HOLES

WHAT

I.

O HOT

II

F8cc30,

V

53

2 i

200 400 600 800 200 400 600 800 0 200 400 600 800

at—i—>—i—f—i—i—i—|

2 ;,

" • ' |

0 200 400 600 800 200 400 600 800 200 400 600 ~ 800

^ = O-l

54

= Y-x

I i . i X K- i 1 1 1—1—1—I200 400 600 BOO 200 400 600 800

200 *00 600 BOO

81 ' ' ' I ' ' ' | ' ' ' I ' ' ' I

200 *00 600 800 200 «00 600 BOO200 400 600 800

55

Bs

0.5

0.5

0.5

0.5

0.5

Bs

0.5

0.5

0.5

0.5

0.5

0.7

0.7

Ti/Te

1/7

1/3

1

3

9

Ti/Te

1/7

1/3

1

3

9

1/3

9

BP IA HOLES

Speed

0.61 Va

0.6 Va

A

A

A

B

Speed (Va)

0.83

0.75

0.66

0.56

0.32

0.3

A

FP IA HOLES

Speed

0.65 Va

0.63 Va

0.46 Va

A

A

N

Speed (Va)

0.81

A

A

0.54

0.34

A

A

56

CONCLUSIONS

* Governing Evolution Equation for Nonlinear ALFVENWaves is DNLS / MDNLS

* DNLS => Stable Solitary Waves

* Inhomogeneities destroy the coherent structures

* Chaotic Route to Turbulence with k " spectra

* For p ~ 1 and 5B / B ~ 1 Simulations needed

MHD SIMULATIONS :

* 5N and 5B coupling very significant

* Evolution of RHP Alfvenic Wave Packet very differentthan LHP Wave Packet

* LHP => Blow-upRHP => Steepening and High Frequency Radiations

57

HYBRID SIMULATIONS :

* For Large p, 8N - 5B coupling as well as Wave -Particle interactions very crucial

1. LHP EVOLUTION:

* Blow - up observed in LHP case in MHD Simulationsarrested by Kinetic Effects

* COLLAPSE => Transport of Energy to larger k

* Appearance of Turning Point due to Structural Instabilityin the evolution of LHP

* Turning Point function of p and the initial amplitude ofthe Wave Packet

* At the Turning Point LHP => RHP

* Ion Acoustic Waves (Coherent Ion Holes) generated forT > T

e 1

* For Te » Tj, forward and backward (in inertial frame)propagating Ion Holes

* Nonlinear Landau damping forTe < Ti

* Forward propagating Magnetosonic waves in some cases

58

2. RHP EVOLUTION:

* High - Frequency Radiations for relatively lower

* Only Backward Propagating Ion Holes

* No Magnetosonic waves generated

* More robust compared to LHP

59

References:

Bavassano, B., Dobrowolny, M., Mariani, F. and Ness, N.F., Radialevolution of Power Spectra of Interplanetary Alfvenic Turbulence, J.Geophys. Res., 87, 3617, 1982.

Belcher, J. W. and Davis, L, Large amplitude Alfvenic waves in theinterplanetary medium, J. Geophys. Res., 76, 3534, 1971.

Buti, B., Ion Acoustic Holes in a Two-Electron-Temperature Plasma,Phys. Lett, 76A, 251, 1980.

Buti, B., Stochastic and Coherent Processes in Space Plasmas, inCometary and Solar Plasma Physics, Ed. B. Buti, World Scientific,Singapore, 1988.

Buti, B., Nonlinear and Chaotic Alfven waves, in Solar and PlanetaryPlasma Physics, Ed. B. Buti, p 92 , World Scientific, Singapore, 1990.

Buti, B., Chaotic Alfven Waves in Multispecies Plasmas J. Geophys.Res., 97, 4229, 1992.

Buti, B., Solitary Alfven Waves in Inhomogeneous StreamingPlasmas, J. plasma Phys., 47, 39, 1992.

Buti, B., Chaos and Turbulence in Solar Wind, {\it Astrophys. SpaceSci., 243}, 33, 1996.

Buti, B., V. Jayanti, et a!., Nonlinear Evolution of Alfvenic WavePackets, Geophys. Res. Letts., 25, 2377, 1998.

Buti, B., V.L. Galinski, V.I. Shevchenko, et al., Evolution of NonlinearAlfven Waves in Streaming Inhomogeneous Plasmas, Astrophys.Jou., 523, 849, 1999.

Buti, B. and Nocera, L., Chaotic Alfven Waves in the Solar Wind inSolar Wind Nine, AIP Proceedings, 471- Eds. S. Habbal et al., p. 173,American Institute of Physics, 1999.

60

Buti, B., M. Velli, P.C. Liewer, et al., Hybrid Simulations of Collapseof Alfvenic Wave Packets, Phys. Plasmas, 7, 3998, 2000.

Buti, B., B.E. Goldstein and P.C. Liewer, Ion Holes in the Solar Wind:Hybrid Simulations, Geophys. Res. Letts. , 28, 91, 2001.

Dawson, S.P. and Fontan, C.F., Soliton Decay of Nonlinear AlfvenWaves: Numerical Studies, Phys Fluids, 31, 83, 1988.

Hada, T., Kennel, C.F. and Buti, B., Stationary Nonlinear AlfvenWaves and solitons, J. Geophys. Res., 94, 65, 1989.

Hada, T., Kennel, C.F., Buti, B. and Mjolhus, E., Chaos in DrivenAlfven systems, Phys. Fluids, B2, 2581, 1990.

Inhester, B., A Drift-Kinetic Treatment of the Parametric decay oflarge-Amplitude Alfven Waves, Jou. Geophys. Res., 95, 10525,1990.

Kaup, D.J., and Newell, A.C., An exact Solution for a DerivativeNonlinear Schroedinger Equation, J. Math. Phys., 19, 798, 1978.

Kennel, C. F., Buti, B., Hada, T. and Pellat, R., Nonlinear,Dispersive, EllipticallyPolarized Alfven Waves, Phys. Fluids, 31,1949, 1988.

Verheest, F. and Buti, B., Parallel Solitary Alfven Waves inWarmMultispecies Beam-Plasma Systems J. Plasma Phys., 47, 15,1992.

Velli, M., Buti, B., Goldstein, B.E. and Grappin, R., Propagation andDisruption of Alfvenic solitons in the Expanding Solar Wind in SolarWind Nine, AIP Proceedings, 47_1, Eds. S. Habbal et al., p. 445,American Institute of Physics, 1999.

Winske, D. and M.M. Leroy, Hybrid Simulation Techniques applied tothe Earth's Bow Shock in Computer Simulations of Space Plasmas ,Eds. H. Matsumoto and T. Sato (D. Reidel, Hingham, Mass., 1985),p. 25.

61

Review Article:

Buti, B., Chaos in Magnetoplasmas, Nonlinear Processes^ rrrGeophysics, 6^129, 1999.

62