Collisionless Dissipative Nonlinear Alfven Waves: Nonlinear Steepening, Compressible Turbulence, and Particle Trapping

10 Pages·0.23 MB·English

by  M. V. Medvedev

Collisionless Dissipative Nonlinear Alfv´en Waves: Nonlinear Steepening,
Compressible Turbulence, and Particle Trapping ∗
M.V. Medvedev †
Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138
tospheres of planets and magnetized stars, etc.. It is in-
The magnetic energy of nonlinear Alfv´en waves in com- teresting that in incompressible plasma, v = 0, fluid
pressible plasmas may be ponderomotively coupled only to ∇·
and magnetic nonlinearities in a finite-amplitude Alfv´en
ion-acoustic quasi-modeswhichmodulate thewavephaseve-
wavecanceleachotherexactly,sothatitobeysthe same
locity and cause wave-front steepening. In the collisionless
dispersionrelationasthelinearwaveofinfinitesimalam-
plasma withβ 6=0,thedynamicsof nonlinearAlfv´enwaveis
plitude.
alsoaffectedbytheresonantparticle-waveinteractions. Upon
9 relatively rapid evolution (compared to the particle bounce Such a mutual cancellation of Reynolds and magnetic
9
time),thequasi-stationarywavestructures,identicaltotheso stresses in the wave breaks down when plasma is com-
9
called (Alfv´enic) Rotational Discontinuities, form, the emer- pressible. Indeed, the magnetic field pressure, B˜2/8π,
1
gence and dynamicsof which hasnot been previously under- associated with the wave field ponderomotively modu-
n
stood. Collisionless (Landau) dissipation of nonlinear Alfv´en lates the local plasma density, n, which, in turn, affects
a
J waves is also a plausible and natural mechanism of the solar the phase velocity ofthe wave. This processmay alsobe
9 wind heating. Considering a strong, compressible, Alfv´enic viewed as an effective parametric coupling of an Alfv´en
turbulence as an ensemble of randomly interacting Alfv´enic
1 wave to acoustic waves in plasma. Such acoustic modes,
discontinuitiesandnonlinearwaves,itisshownthatthereex-
however,arenotplasmaeigenmodes. Theyaredrivenby
isttwodistinctphasesofturbulence. Whatphaserealizesde-
1
the Alfv´en wave and, hence, propagate with the Alfv´en
v pendson whetherthiscollisionless dampingisstrongenough
speed,whichis,ingeneral,differentfromthesoundspeed
7 toprovideadequateenergysinkatallscalesand,thus,tosup-
5 port a steady-state cascade of the wave energy. In long-time in plasma. The nonlinear coupling of the phase speed of
2 asymptotics,however,theparticledistributionfunctionisaf- the wave to its amplitude in the nonlinear Alfv´en waves
1 fectedbythewavemagneticfields. Inthisregimeofnonlinear is ultimately responsible for the wave-front steepening
0 Landaudamping,resonantparticlesaretrappedinthequasi- andformationofcollisionlessshocksinastrophysicaland
9
stationary Alfv´enicdiscontinuities, giving rise toa formation space plasmas.
9
/ of a plateau on the distribution function and quenching col- Instudiesofnonlinearwavesitiscustomarytoassume
h lisionless damping. Usingthevirialtheorem for trappedpar-
‘weak nonlinearity’,i.e., the waveamplitude is finite but
p ticles, it is analytically demonstrated that their effect on the
still small compared to the ambient field. This means
- nonlinear dynamics of such discontinuities is non-trivial and
o that a waveis linear in the leading order of expansionin
forces a significant departure of the theory from the conven-
r wave amplitude. All nonlinear effects appear as a grad-
t tional paradigm.
s ual, large-scale modulation of the wave amplitude. In
a
: PACS numbers: 52.35.Mw, 52.35.Ra, 96.50.Bh, 96.50.Ci otherwords,onestudies slowevolutionofanenvelopeof
v
a wave packet.
i
X It is well known that in collisionless, incompressible
r plasma, low-frequency Alfv´en waves are undamped or
a I. INTRODUCTION
very weakly damped since collisionless dissipation is in-
efficient. Ion-cyclotron damping is always weak for low-
Alfv´en wavesare the most ‘robust’ plasma oscillations frequency waves and Landau damping, which could be
in magnetized systems. They are encountered in a wide important, is absent because there are no longitudinal
variety of astrophysical objects such as stellar and solar electric and magnetic field perturbations in the Alfv´en
winds, interstellar medium, hot accretion flows, magne- wave. Equivalently, an Alfv´en wave is strictly transverse
inthedriftapproximationandthuscannotaffectthepar-
ticle motions along the field lines. (Note that weak Lan-
dau dissipation of obliquely propagating Alfv´en waves
∗Invited talk on the American Physical Society, Division of occurs on scales comparable to the ion Larmor radius,
PlasmaPhysicsmeeting,1998. IncollaborationwithP.H.Di- ρ , where the drift approximation breaks down. This
i
amond,V.I.Shevchenko,M.NRosenbluth,andV.L.Galinsky
damping is really weak since typically k ρ 1.)
⊥ i
(UCSD). ≪
In contrast, a nonlinear Alfv´en wave in compressible
†Also at the Institute for Nuclear Fusion, RRC “Kurchatov
plasma is nonlinearly coupled to acoustic-type oscilla-
Institute”, Moscow 123182, Russia;
tions which are accompanied (in a two-fluid model) by
E-mail: [email protected]; URL: http://cfa-
longitudinal electric fields and are damped via the Lan-
www.harvard.edu/∼mmedvede/
1
dau mechanism. Hence, dissipation enters the dynam- function(PDF)attheresonantvelocity,f′′(v ),andnot
0 A
ics of nonlinear Alfv´en waves nonlinearly, even in the to its slope, f′(v ), as in linear theory. (3) The phase-
0 A
regime when the resonantparticle response is calculated space densityoftrappedparticlesiscontrolledbyplasma
in the linear approximation, i.e., neglecting trapping of β. Thus, the structure of quasi-stationary Alfv´enic soli-
resonantparticles. Anotherimportantandquiteunusual tonsanddiscontinuitiesmaybemodifiedinτ limit.
→∞
feature is the nonlocality of the damping. It is associ- The rest of this paper is organized as follows. In Sec.
ated with the finite transit time of a resonant particle II asemi-qualitative derivationof the envelopeevolution
through the wave envelope modulation and represented equation for finite-amplitude Alfv´en waves, referred to
by an integral operator in the wave evolution equation. as the Derivative Nonlinear Schr¨odinger (DNLS) equa-
Unusual properties of dissipation in the nonlinear tion, wave is presented. In Sec. III we generalize it to
Alfv´en waves naturally result in the following peculiar include the resonant particles effect (Landau damping)
features of these waves. First of all, it was shown that and obtain the kinetically modified derivative nonlinear
Alfv´en waves do not always damp to zero. Instead, Schr¨odinger equation (KNLS). We discuss numerical so-
they evolve into the localized, coherent, quasi-stationary lutions for this equationwhichappear to be the Alfv´enic
structuresreferredtoasdirectionalandrotationaldiscon- Rotational Discontinuities that observed in situ in the
tinuities. These are the narrow regions where the wave solar wind. We also discuss the energy dissipation time
magnetic field vector rapidly rotates through some an- scalesforvarioustypesofnonlinearAlfv´enwaves. InSec.
gle and, respectively, are or are not accompanied by the IVthenoisy-KNLSmodelofstrong,compressiblemagne-
wave amplitude variations. These discontinuities con- tohydrodynamic(MHD)Alfv´enicturbulenceisdiscussed.
nect the parts of a wave packetof different polarizations In Sec. V a self-consistent treatment of the nonpertur-
or wave phases. Second, among the Alfv´enic discontinu- bative effects of particle response (particle trapping) on
ities, there are solutions which constitute a new class of the nonlinear Alfv´en wave dynamics is presented.
shock waves,— the collisionless dissipative shocks. Such
shocks form when nonlinear wave steepening is balanced
by collisionless damping process (and not by the wave II. NONLINEAR ALFVE´N WAVES: THE DNLS
EQUATION
dispersion, as in usual collisionless shocks). Due to scale
invariance of Landau damping, the shock width is de-
termined by dispersion, however,which is the only char- The usual derivation of the evolution equation of the
acteristic scale in the problem. It should be emphasized nonlinearAlfv´enwavesisbasedonthemultipletimescale
thatAlfv´enicrotationaldiscontinuitieshavebeendiscov- expansion. It assumes that the envelope of an Alfv´en
ered by in situ measurements [1–3] of magnetic fields in wavevariesonmuchlongertime-scale,τ =(B0/B˜⊥)2t
≫
the solar wind. Despite intense observational and theo- t, than the linear wave period, t = ω−1 = (kkvA)−1,
retical efforts these years, a comprehensive quantitative where B and B˜ are the unperturbed magnetic field and
0
theory which explains the emergence and dynamics of itsperturbations,v =B /√4πm n isthe Alfv´enspeed,
A 0 i
Alfv´enicdiscontinuitieswasproposedonlyrecently[4–6]. m is the ion mass, and and designate components
i
k ⊥
All the results are in excellent agreement with the ob- parallelandperpendiculartotheambient(unperturbed)
servational data and numerical hybrid simulations [7,8]. magnetic field. Sucha derivationhas been performedby
Third, preliminary study of a one-dimensional model of different authors and we refer the reader to the original
strong,compressibleAlfv´enic turbulence [9] suggeststhe literature [11,12]. To provide physical insights, we give
existence of two distinct phases or states of the turbu- here a transparent, semi-qualitative derivation instead.
lence,dependingonhowstrongthecollisionlessdamping AwaveequationofasmallamplitudeAlfv´enwavemay
is. Atlast,resonantparticles,whichwereresponsiblefor be written as
the wave damping in the linear Landau process, appear
to be trapped in the wave potential at large times and ω kkvA 1 kkvA/2Ωi B˜⊥ =0, (1)
− ±
the linear theory of Landau damping is invalid. Qual-
where Ω (cid:0)=eB /m(cid:2)c is the ion-cyc(cid:3)l(cid:1)otronfrequency. The
itatively, the damping rate oscillates with roughly the i 0 i
last term is simply weak dispersion, k v /Ω 1, due
bounce frequency and gradually decreases to zero un- k A i ≪
toafiniteLarmorradius. Aponderomotiveforcecreates
til resonant particles phase mix and form a plateau on
density fluctuations, δn, so that the Alfv´en speed is no
the distribution function. A recently proposed asymp-
longer constant:
totic theory of particle trapping [10] predicts non-trivial
nonlinearcontributionsofresonantparticlestononlinear B 1δn
0
v = v 1 . (2)
Alfv´en wave dynamics. They are the following. (1) The A 4πmi(n0+δn) ≃ A0(cid:18) − 2n0(cid:19)
poweroftheKNLSnonlinearityassociatedwithresonant
particles increases to fourth order when trapped parti- We assumepthatthe wavepropagatesalongthe magnetic
cles are accounted for. (2) The effective coupling is now field in z-direction (to the right) and its amplitude is
proportionalto the curvature of the particle distribution changing gradually, B˜ = B˜ (z v t,τ). Going back
⊥ ⊥ A0
−
2
to real-space representation, this suggests the following Later, at τ 5, when the width of the front becomes
∼
replacement: of order Ω /v dispersion limits nonlinear steepening
i A
andproducessmall-scale,oscillatorywave-likestructures
∂ ∂ ∂
ω iv +i , k i . (3) with significant small-scale energy component. At later
→− A0∂z ∂τ k →− ∂z times, the magnetic field is highly irregular indicating
strong nonlinear Alfv´en wave turbulence. Note that no
Wecanneglectthevariationofv inthesmalldispersion
A
(quasi-stationary) Alfv´enic discontinuities emerge.
term as a next order effect. From Eq. (1) it is easy to
write:
∂ v ∂ δn v2 ∂2 III. NONLINEAR ALFVE´N WAVES WITH
B˜ A B˜ i A B˜ =0. (4)
∂τ ⊥− 2 ∂z n ⊥ ± 2Ω ∂z2 ⊥ LANDAU-TYPE DISSIPATION: THE KNLS
(cid:18) 0 (cid:19) i EQUATION
Hereafter we omit subscript “0” in v . The perturbed
A
magnetic field here is a complex quantity and is defined It is known that kinetic effects like Landau damp-
asB˜⊥ =B˜x iB˜y. Thedensityperturbationsarecaused ing are absent from the MHD approximation. To in-
±
by the ponderomotive force exerted on plasma by the clude them into the nonlinear Alfv´en wave model self-
fluctuatingmagneticfieldofthewave. Theyareobtained consistently,afullykinetictreatmentisneeded[14]. The
from the continuity, momentum balance equations, and resultsareoftentoocomplicatedandobscuring,however.
equationofstatewhichyieldtheequationforion-acoustic There are two ways to go around the difficulties. First
waves with source: way is to use a hybrid MHD-kinetic approach in which
oneperformstheadhoccalculationsthequantitieswhich
∂2 ∂2 δn v2 ∂2 B˜ 2
c2 = A | ⊥| , (5) are known to be affected by kinetics (i.e., the density
∂t2 − s∂z2 n − 2 ∂z2 B2
(cid:18) (cid:19) 0 0 perturbations, δn) and plugs them into the MHD model
[15–17]. Second way is to make some self-consistent clo-
wherec2 =γ(T +T )/m istheion-acousticspeed,γ =3
s e i i sures in deriving the MHD equations to modify them so
isthe polytropicconstant,andp=n(T +T )andm
e i e ≪ thattomimiccollisionlessprocesses[18]. Thenthesenew
m wereusedinderivation. Since,ion-acousticwavesare
i MHD equations may be used to re-derive the nonlinear
driven by the Alfv´en wave, the density perturbation is a
functionofz v t. Thus ∂2 v2 ∂2 andEq.(5)yields: Alfv´en wave equation [5]. Here we again give a simple,
− A ∂t2 ≡ A∂z2 semi-qualitative derivation. For rigorousanalysis, we re-
fer reader to the original literature [5,15–17].
δn 1 B˜ 2
= | ⊥| , (6) Toincludekinetic effects,weneedtomodifythe equa-
n −2(1 β) B2
0 − 0 tion for ion-acoustic waves only, while Eq. (4) remains
unchanged. The equation of damped ion-acousticmodes
where β =c2/v2 is roughly the ratio of plasma pressure
s A may be formally written as
tomagneticpressure. Substituting the densityperturba-
tionintotheequationforthewavemagneticfield(4)and ∂2 ∂ ∂2 δn v2 ∂2 B˜ 2
defining b=B˜ /B , we obtain the DNLS equation: +γˆ c2 = A | ⊥| , (8)
⊥ 0 ∂t2 ∂t − s∂z2 n − 2 ∂z2 B2
(cid:18) (cid:19) 0 0
∂b 1 ∂ v2 ∂2b
+ b2b i A =0. (7) whereγˆ isadampingrate. Ifthedampingrateisafunc-
∂τ 4(1 β)∂z | | ± 2Ωi∂z2 tion of k in Fourier space, γˆ in real space is represented
−
(cid:0) (cid:1) by an operator acting on a wave field. We now recall
This equation describes evolution of planar nonlinear
that in k-space Landau damping rate is proportional to
Alfv´en waves propagating in one dimension along the
1/k . Thus in real space we write
ambient magnetic field. It was shown that this equa- | k|
tion is solvable [13] and admits soliton and cnoidal wave
∂
solutions. γˆ =vAχˆ , (9)
H∂z
A numerical solution of the initial value problem for
the DNLS is shown in Fig. 1a. It is natural to assume where ˆ[f]= 1 ∞ P f(z′)dz′ is the integral Hilbert
that nonlinear waves emerge from small-amplitude (lin- operatoHr actinπg o−n∞az′f−uznction f ( denotes principal
ear) ones. Thus the most general class of initial pro- value integrationR), which is equalPto (ik /k )f in
k k k
| |
files are the finite-amplitude, periodic waves of different Fourier space and, thus, ensures correct k-scaling of col-
helicities (polarizations). In the case of Fig. 1a, an ini- lisionlessdamping. Hereχisaparameterwhichcontains
tiallylinearlypolarized,sinusoidalinitialwaveprofilehas all insignificant physics such as the structure of a PDF,
been chosen. (The range of dimensionless coordinate, etc., and depends on the closure chosen. In the model
0 ζ 1024correspondstothe numberharmonicsused of Ref. [5], χ has the physical meaning of a thermocon-
≤ ≤
in calculation, 512 k 512.) One can see that wave ductivity coefficient. Again, we assume a traveling wave
front indeed st−eepens≤at≤early times, τ 2 400Ω−1.
∼ ∼ i
3
solutionfor the density perturbations,δn=δn(z v t). nonlinear Alfv´en wave evolution is affected by resonant
A
−
Than we have particles and is described by the second nonlinear, inte-
gral term in Eq. (11). Unlike the DNLS equation, the
δn δn 1
(1 β) +χˆ = b2. (10) KNLS equation is not integrable because of dissipation.
− n H n −2| |
0 (cid:20) 0(cid:21) Numerical solution of this integro-differential equation
for the same initial conditions as in Fig. 1a and β = 1,
To solve this equation we use the identity ˆ ˆ = 1,
HH − reveals different wave dynamics, as shown in Fig. 1b.
whichhasaphysicalmeaningoftime-reversibilityofLan-
Instead of irregular, fluctuating wave fields, localized,
dau damping [5]. Acting on Eq. (10) with ˆ, excluding
HKˆN[δLnS/ne0q]uaantidons:ubstituting into Eq. (4), wHe obtain the asqerueeanstiht-orsetfaeotrpimoanrvaaermyryewtreaarvpsei(fdoinlrym,awdsdi(tAihtiliofnvn´eτnt<∼oicβ5dia=sncod1n0Tt3iΩn/−iuTi1t.)ieTws)hhaeicrrhee
e i
control the wave dynamics as well as specify the type of
∂b ∂ v2 ∂2b
+v m bb2+m bˆ[b2] +i A =0, discontinuitywhichresults. These are(i) the wavehelic-
∂τ A∂z 1 | | 2 H| | 2Ω ∂z2
i ity(i.e.,thepolarizationtype),(ii)theobliquityangleΘ,
(cid:16) (cid:17)
(11) and the angle between the polarizationplane (for nearly
linear polarizations) and k–B plane.
0
wherem =(1 β)/4∆, m = χ/4∆, ∆=(1 β)2+χ2,
1 2 There are three general types of quasi-stationary
− − −
are coefficients, which are model-dependent. The rigor-
Alfv´enic discontinuities, discriminated by the obliquity
ous derivation [5] yields
and phase jump (i.e., angle though which the magnetic
fieldvectorrotates). First,thereisawideclassofthearc-
1 (1 β∗)+χ2(1 β∗/γ)
m = − − , (12a) type rotational discontinuities which emerge in oblique
1 4(1−β∗)2+χ2(1−β∗/γ)2 propagation, Θ > 10◦, independent of the wave helic-
1 χβ∗(γ 1)/γ ity(polarization)∼. Theyarecharacterizedbyanarc-type
m = − , (12b)
2 −4(1 β∗)2+χ2(1 β∗/γ)2 shape diagram in the hodograph (the b –b diagram),
− − x y
as shown in the “snap-shot”, Fig. 3. The wave phase
where β2 =(T +T )β/T (T /T )β for T <T , and χ
e i i e i i e jumpthroughthediscontinuityis∆φ<π. Second,there
isthebestfitparameterof≃theclosuretothee∼xactkinetic
is a class of the S-type directional discontinuities which
particleresponse[18]withatemperaturecorrection,χ= emerge in quasi-parallel propagation, Θ 0◦, only from
8β/πγexp 1/β (T /T )3/2exp (T T )/2T . The ∼
e i i e i linearly and almost linearly polarized initial perturba-
{− } { − }
dependence of m and m vs. 1/β is drawn in Fig. 2.
p 1 2 tions. They are characterized by a remarkable S-shaped
Negative sign of m indicates damping. The damping of
2 hodograph, as shown in Fig. 4. They are called direc-
anAlfv´enwaveisstrongestinwarm,isothermalplasmas,
tional (not rotational) discontinuities because the phase
β 1, T T . The KNLS equation straightforwardly
e i jump is accompanied by moderate amplitude variation.
∼ ∼
generalizes to the case of obliquely propagating waves.
The wavephasechangesby ∆φ=π throughthe the dis-
In fact, Eq. (11) remains unchanged, but the wave field
continuity. Third, there is a narrow class of arc-type ro-
is re-defined as follows: b = (B˜ +iB˜ +B sinΘ)/B ,
x y 0 0 tational discontinuities propagating parallel to the mag-
where Θ is the obliquity angle, sinΘ k B .
0 neticfield. Theyemergefromwaveswithellipticalpolar-
∝ ·
Clearly, m and m are functions of T /T and β. It
1 2 e i izationonly. The wavephase changeis ∆φ=π. Parallel
can be shown that if either T /T 1, or β 1, or
e i propagatingcircularlypolarizedwaves(helicityequalsto
≫ ≫
β 1, the Landau damping of nonlinear Alfv´en waves
unity)donotevolveto discontinuitiesandaredecoupled
≪
becomes very weak, m 0. Then the DNLS nonlin-
2 from dissipation.
→
earity dominates and m (1 β)−1, i.e., changes sign
1
→ −
at β 1. In this regime, other dissiaption mechanisms
≃
(e.g., ion-cyclotron damping of the linear Alfv´en wave B. Comparison with observations
[carrier]) may become important, see Sec. IIIB.
There are two regimes of the Alfv´en wave evolution,
depending on the values of T /T and β. Although there
e i
A. Alfv´enic discontinuities
is no sharp boundary between the regimes, a qualita-
tive insight may be gained from Fig. 6, where the curve
The KNLS equation (11) reduces to the DNLS for
m = m is plotted. Kinetic dampingis importantfor
1 2
β 0. Collisionless damping of nonlinear Alfv´en waves | | | |
T /T and β from the region labeled “hydrodynamic”.
→ e i
vanishes in this case. The wave dynamics is thus decou-
There m < m and Alfv´enic rotational discontinu-
1 2
pled from resonant particle effects and Alfv´enic discon- | | | |
ities rapidly form. We emphasize the remarkably good
tinuities do not emerge (unless one artificially plugs in
correspondenceoftheKNLSsolutionsandthesolarwind
diffusion, finite plasma conduction, or other collisional observational data [1–3]. Recent 21 hybrid code simula-
2
effects, as done in some simulations). For finite β’s, the
tions are also in excellent agreement with the results of
4
the KNLS theory [7,8] and thus support the idea that nonlinear Alfv´en waves a plausible (and natural) mech-
the mechanismofformationofAlfv´enicdiscontinuitiesis anism of the solar wind heating within one astronomical
thecombinedeffectofresonantwave-particleinteractions unit.
and nonlinear wave steepening.
In the opposite case, m >> m (labeled “bursty”
1 2
in Fig. 6), the Landau|dam| ping|is |weak and may be IV. COMPRESSIBLE TURBULENCE OF
NONLINEAR ALFVE´N WAVES
neglected compared to the ion-cyclotron one. Nonlin-
ear wave steepening cascades wave energy to the scales
k max m , m b2Ω /v , as estimated from Eq. Therearetwoapproachesinthe studies ofturbulence,
1 2 i A
∼ {| | | |}| |
(11). Thus, the number of particles being in the cy- cascade and structure-based. The cascade-type theories
clotron resonance, k(v v) Ω = 0, may be large usually treat turbulence as a “soup of eddies”, — a col-
A i
− −
if m 0, β 1. Nonlinear dynamics in this case is lection of plasma excitations with random phases which
2
→ ≃
governed by the so called DNLS-Burgers equation [19]. interactwitheachother(usuallyonmatchingscales)and
This theory explains well the emergence and evolution produce a cascade of energy. The structure-based-type
of Alfv´en shock trains (also referred to as “shocklets”) approach,oneoftheexamplesofwhichisthenoisyBurg-
[20],which were observedupstreamof the bow-shocksof ers model in hydrodynamics, is based on the study of
planets and comets (see e.g., Ref. [21]). nonlinear evolution equations with external noise drive.
Itthusstudiestheturbulenceofcoherentstructures,such
as shocks, solitons, nonlinear (cnoidal) waves, etc., gen-
C. Energy dissipation eratedandinteractingwitheachotherrandomly. Sucha
process may be referredto as “coherentcascade”,mean-
Resonant particle effects make the otherwise almost ing that modes (eddies) at all scales interact coherently
non-dissipativeAlfv´enwavestodamp. Fig.5showstem- (inphase)and,thus,experiencestrongerinteractionsdue
poral evolution of magnetic energy associated with dif- to the cumulative effect. Its is known that the first ap-
ferent structures (Alfv´enic discontinuities). Clearly, the proach is more suitable for studies of weak turbulence
energydissipationrateinthecaseofquasi-parallelpropa- and the second one better describes the strong, highly
gationissensitivetotheinitialpolarization,i.e.,helicity, nonlinear turbulence.
which is, roughly, a measure of asymmetry of a spec- The analytical study of the structure-based noisy-
trum between +k and k harmonics. Crude estimates KNLS model as a generic model of collisionless, large-
−
[6]ofthetypicaldampingtimeare: (i)forlinearlypolar- amplitude, compressible MHD (Alfv´enic) turbulence is
izedwaves(zerohelicity),e.g.,theS-typediscontinuities, presented in Ref. [9]. Stationarity is maintained via
τ m −1 andthe damping is algebraic, b2 τ−1/2, the balance of noise and dissipative nonlinearity. The
lin 2
∝| | | | ∝
rather than exponential, (ii) for elliptically polarized Fourier-transformed KNLS equation (11) with noise
waves τ m ∆ −1, where ∆ b 2 b 2 source reads
ell 2 s s +k −k
∝ | | ∼ | | −| |
is a measure of spectrum asymmetry, and (iii) for circu-
P(cid:0) (cid:1) iω+iv k+iµ k2 b +iλk b b b
ldaorlynoptodlaarmizped. wInavtehse(hcaelsiecitoyf oubnliitqyu)eτcpirro→pag∞atiaosn,ththeye (cid:0)− 0 0 (cid:1) ωk ωkX′′,,kω′′′′ ωk′′ ωk′′′′ ωk−−ωk′′−−kω′′′′
ambient field enters the damping rate and for all wave [m +im sgn(k k′)]=f , (14)
helicities τobl m2sin2Θ−1. From Fig. 5 one sees × 1 2 − ωk
∝ | |
that the a typical time-scale of the wave damping is
where v and µ are the bare phase velocity (it is nec-
τthe<∼so1l0a0r∼win1d04cΩo−ind1,itwiohnisc.h is (roughly) a few hours for edsisspareyrsfioo0rn,aλsel=f-0c1onissisttheentstraenndoramrdalpizearttiuornbaatniaolnyspisa)raamnd-
Given the spectrum of magnetic field energy pertur- eter, and the function sgn(x) = x/x. Here we omit-
bations, Ek = (b2)k, the instantaneous rate of energy ted the subscript by k. The one| l|oop renormaliza-
changeis calcula|te|dexactly [6,24]fromEq.(11)to yield: k
tion group analysis of the problem with zero mean, δ-
correlated noise with white k-spectrum yields complex-
∂E ∞ 2
∂τ =m2 Ekk |kk|dkk . (13) valued renormalized coefficients vturb, µturb. The ‘Re’
Z−∞ and ‘Im’ parts of µturb are simply turbulent dispersion
(cid:12) (cid:12)
Note that this equation is(cid:12)equ(cid:12)ally applicable to the sin- and turbulent viscosity. The real part of vturb may be
interpretedaswavemomentumlossviainteractionswith
glewavecaseaswellasto thecompressibleAlfv´enictur-
resonant particles and its imaginary part manifests fast,
bulence in general, provided the turbulence spectrum is
exponentialdampingdue tophasemixing. Furtheranal-
known.
ysis showsthe existence of twodifferent phasesofturbu-
Typical dissipation timescale is several hours for the
lence. The bifurcation occurs at the point
solar wind conditions. Hence, collisionless dissipation of
m /m 1. (15)
1 2
| |≃
5
If m /m < 1, Landau dissipation is efficient to sink mine the PDF. The generalized KNLS again is Eq. (4)
1 2
all|the inje|cted energy during the “coherent cascade” so withδn=δn +δn ,whereδn andδn χˆ[U(z)]
NR R NR R
∝ K
that the hydrodynamic (ω 0, k 0) regime with are the non-resonant (bulk) and resonant particle re-
no sharp fronts realizes. In t→his regim→e, the strongly in- sponses, respectively. Here ˆ is a new kinetic operator
teracting Alfv´enic discontinuities dominate in the tur- (which replaces ˆ) acting oKf the wave field which con-
H
bulence. In the opposite case, nonlinearity overcomes tains all the information about resonant (trapped) par-
damping and no stationary, hydrodynamic turbulence is ticle trajectories. It is interesting that the very possi-
predicted. Small-scale, bursty turbulence is expected in bility of writing the generalized KNLS equation in this
this regime. Given the weak cyclotron damping, such a formreliesontheintrinsictimereversibilityoftheVlasov
regime corresponds to the Alfv´enic shocklet turbulence. equation, linear or nonlinear: ˆˆ = 1, see Ref. [5].
KK −
Recalling that m and m are functions of β and T /T , The resonant particle response is calculated using Li-
1 2 e i
a “diagram of state” can be drawn, as shown in Fig. 6. ouville’s theorem (“the PDF is constant along particle
trajectories”):
V. ASYMPTOTIC, τ →∞, DYNAMICS OF f(v,z,t)=f v E,z± , (16)
0 ± 0
DISSIPATIVE NONLINEAR ALFVE´N WAVES
where z± = z±(z,t,E;U((cid:0)z)) i(cid:0)s the i(cid:1)n(cid:1)itial coordinate of
0 0
The linear Landau damping theory used in derivation a particle of total energy E which at time t is at the
of the KNLS assumes a time-independent (Maxwellian) pointz andhasavelocityv±(E,z). BydefinitionδnR =
PDF. It is clear that, for a finite-amplitude wave, parti- dv f f(t=0) , so that
cleswhicharenearresonancewiththewave,v v ,will ∆vres − 0
cbaeutsreatphpeeidr kbiynetthiceewnaevrgeyp(omteenatsiuarleUd(izn)t≡heB˜w⊥2a/v≃8eπfrnAa0mbee)-, RδnR = (cid:16) f0 v(cid:17)± E,z0± −f0(v±(E,z)) dE. (17)
2m (E U(z))
12m(v−vA)2, is less than the potential barrier, |Um| = X(±)Z∆Eres (cid:0) (cid:0) i(cid:1)(cid:1) −
maxU(z), as in Fig. 7. Such particles experience re- p
| | Herethesumisoverparticlesmovingtotheright(+)and
flections at turning points z and z exchanging en-
1 2
to the left (–), as in Fig. 7. The integration is over the
ergy with the wave and significantly modify the PDF
resonant (negative) energies of trapped particles, U
near resonance. The damping rate of a wave oscil- m
≤
E 0 with U being the amplitude of the potential. In
lates in time with gradually decreasing amplitude un- m
≤
the short-time limit, τ ), Eq. (17) reduces to the
til phase mixing results in flattening of the PDF (for
KNLS case with ˆ = ˆ→and∞χ=πv2f′(v )/m n .
resonant velocities) and formation of a plateau; then K H A 0 A i 0
Totreattheτ limitweemploythevirialtheorem,
the damping rate vanishes [22]. Thus, the linear cal-
→∞
whichrelatestheaveragekineticandpotentialenergiesof
culation of Landau dissipation, while correct for times
particles trapped in an adiabatically changing potential:
short compared to the typical bounce (trapping) time,
τ τ (k U/m )−1 (k v b)−1, fails for quasi-
≪ tr ≃ k i ≃ k A| | 2 K(z) =n U˜(z) . (18)
stationary waveforms (Alfv´enic discontinuities) on times h i h i
τno>∼nliτnNeaLr≫wavτteprp[τroN−fiL1le≃evmol1uktkiovnA(tB˜im⊥2e/]B. 02H) eisncteh,eLtaynpdicaaul HitseraergU˜u(mz)en=t oUf(ozr)d−erUnm, ii.se.,aU˜ho(amzo)ge=neaonuUs˜(fzu)n.ctDioinrecotf
dissipationshouldbecalculatednon-perturbativelytode-
integration of Eq. (17) yields
termine the resonant particle response to the nonlinear
wave.
2 n
ThenonlinearLandaudampingproblemis,ingeneral, δn f′′(v ) U(z)
not analytically tractable, as it requires explicit expres- h Ri(cid:12)τ→∞ ≃ 0 A sm3i | |(cid:20)n+2
(cid:12) p
sions for all particle trajectories as a function of initial (cid:12) 2
(cid:12) (U U(z)) U(z) . (19)
position andtime. Such trajectoriescannotbe explicitly × | m|−| | − 3(n+2)| |
(cid:21)
calculated for a potential of arbitrary shape. Usually, a
fuuslelfuplarttoicaleppsrimoxuimlataitoentihserweqauvierepdr.ofiIlnessohmapeecbasyesa, sitimis- tNhoetedtahmaptitnhgeitserambs∝entf.0′(vSAi)ncveanisˆhesˆide=nticˆaˆlly,=henc1e,
hKihKi 6 KK −
ple analytic expression which may be assumed to per- we can only estimate the coupling constant to be χ
∼
sist, while the wave amplitude varies, to calculate the f0′′(vA) vA3/n0 2/m3i. The index n is formally not de-
trajectories [23]. This is not the case for our problem fined for an arbitrary potential, but it can be estimated
(cid:2) (cid:3)p
because nonlinear Landau damping controls the profile numerically, given the wave profile [10].
of the Alfv´en wave. Clearly, in the asymptotic limit, τ τtr, the damp-
≫
A nonperturbative, self-consistent theory of nonlinear ing rate vanishes due to phase mixing. Nevertheless, the
wave-particleinteractionswasconstructedintheasymp- resonant particles still contribute the wave dynamics, in
totic limit, τ , applying the virialtheoremto deter- that
→∞
6
hδnRi∼f0′′(vA)|b|3, (20) [11] R.H. Cohen and R.M. Kulsrud, Phys. Fluids 17, 2215
(1974).
thus determining a new nonlinear wave equation. [12] C.F. Kennel,B.Buti, T.Hada, andR.Pellat, Phys.Flu-
Simple Bernstein-Green-Kruskal type analysis shows ids 30, 1949 (1988).
that the ‘height’ of the asymptotic plateau on the PDF [13] D.J. Kaup and A.C. Newell, J. Math. Phys. 19, 798
depends on the plasma parameters through m and m (1978).
1 2
[14] A. Rogister, Phys.Fluids 14, 2733 (1971).
[10]. The difference between the plateau and the initial,
[15] E. Mjølhus and J. Wyller, J. Plasma Phys. 40, 299
Maxwellian PDF evaluated at v is
A
(1988).
m b [16] S.R. Spangler, Phys. Fluids B 1, 1738 (1989).
1
fplat− fMaxw|vA = m v| |. (21) [17] S.R. Spangler, Phys. Fluids B 2, 407 (1989).
2 A [18] G.W. Hammett and F.W. Perkins, Phys. Rev. Lett. 64,
Thus we conclude that there is under-population of 3019 (1990).
[19] M.A. Malkov, R.Z. Sagdeev, and V.D. Shapiro, Phys.
trapped particles (void on the PDF, f < f ) at
plat Maxw
Lett. A 151, 505 (1990).
low β’s and over-population (bump, f > f ) at
plat Maxw [20] M.A.Malkov,C.F.Kennel,C.C.Wu,R.Pellat,andV.D.
high β’s.
Shapiro, Phys.Fluids B 3, 1407 (1991).
Finally, assumingweak damping ofa nonlinear Alfv´en
[21] B.T. Tsurutaniand E.J. Smith,Geophys.Res.Lett. 13,
wave (e.g., ion-cyclotron damping), one can see [10]
263 (1986).
from the conservation of the parallel adiabatic invariant [22] V.D. Shapiro, Sov.Phys. JETP 17, 416 (1963).
J = pkdz thattrappedparticleswillcondensenearthe [23] T.M. O’Neil, Phys.Fluids 8, 2255 (1965).
bottomofthewavepotential: ∆v b,formingaclump [24] E. Mjølhus and J. Wyller, Phys. Scr. 33, 442 (1986).
k
H ∝| |
on the PDF at v v .
A
≃
FIG.1. Waveprofileevolutionofalinearlypolarizedwave
VI. ACKNOWLEDGMENTS with a sinusoidal initial profile propagating parallel to the
magnetic field for Te ≃Ti and β=0 (a) and β=1 (b).
It is a pleasure to thank Patrick Diamond, Valentin
Shevchenko, Marshall Rosenbluth, and Vitaly Galinsky
FIG.2. ThecoefficientsM1 =2m1 andM2 =2m2 vs. 1/β
for valuable contributions to this work. We also thank
for Te =Ti.
Roald Sagdeev, Shadia Habbal, Vitaly Shapiro, Ramesh
Narayan, Eliot Quataert, and Sally Ride for interesting
discussions. FIG.3. The arc-type rotational discontinuity propagating
obliquely to the magnetic field, Θ∼30◦, with ∆φ<π, (a) -
amplitudeand phase profiles, (b) - hodograph.
FIG.4. Same as Fig. 3 for the S-typedirectional disconti-
nuitypropagating parallel to thefield.
[1] R.P.LeppingandK.W.Behannon,J.Geophys.Res. 91,
8725 (1986).
[2] M. Neugebauer, Geophys. Res.Lett. 16, 1261 (1989). FIG.5. Waveenergy evolution.
[3] B.T.Tsurutani,C.M.Ho,J.K.Arballo,E.J.Smith,B.E.
Goldstein, M. Neugebauer, A. Balogh, and W.C. Feld-
man, J. Geophys. Res. 101, 11027 (1996). FIG. 6. The β-Te/Ti-diagram of state. The region inside
[4] M.V. Medvedev, P.H. Diamond, V.I. Shevchenko, and thecurvecorrespondstohighlydampedturbulence. Nosteep
V.L. Galinsky,Phys. Rev.Lett. 78, 4934 (1997). frontsappear. Therearesteepfrontsintheregionoutsidethe
[5] M.V.MedvedevandP.H.Diamond,Phys.Plasmas3863 curve.
(1996).
[6] M.V. Medvedev, V.I. Shevchenko, P.H. Diamond, and
V.L. Galinsky,Phys. Plasmas, 4 1257 (1997).
FIG.7. Trapping potential.
[7] B.J.VasquezandP.G.Cargill,J.Geophys.Res.98,1277
(1993).
[8] B.J. Vasquez and J.V. Hollweg, J. Geophys. Res. 101,
13,527 (1996).
[9] M.V. Medvedev and P.H. Diamond, Phys. Rev. E 56,
R2371 (1997).
[10] M.V. Medvedev, P.H. Diamond, M.N. Rosenbluth, and
V.I.Shevchenko,Phys. Rev.Lett. 81, 5824 (1998).
7
NOTE TO EDITOR:
ALL FIGURES SHOULD BE SCALED TO THE SIN-
GLE COLUMN FORMAT.
.
.
.
Fig. 1:.
τ=0 (a) (b)
τ=0
τ=1
0.4
|b|
τ=2 τ=5
τ=5
0.2
τ=40
0.0
0 500 1000 0 500 1000
ζ ζ
Fig.2:
1
0.8
M1
0.6
0.4
0.2
2 0
M
M1, -0.2
-0.4
-0.6
-0.8
-1 M2
-1.2
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
1/beta
4 0.4
τ=15 A B C D (a) τ=15 (b)
0.3
2 |b| 0.2
ase 0 phase 0.2 |b C B 0.0 b
h | y
p D A
π
0.1
−2 −0.2
−4 0.0 −0.4
0 500 1000 −0.4 −0.2 0.0 0.2 0.4
ζ b
Fig. 3
x
5 0.4
τ=40 (a) τ=40 C (b)
0.4
3 A B C D 0.2
D
|b| 0.3
e
s
a 1 |b 0.0 b
ph phase 0.2 | A y
−1 −0.2
0.1 B
−3 0.0 −0.4
−500 0 500 −0.2 0.0 0.2 0.4
ζ b
Fig. 4
x
2.0
circular harmonic
1.5
y arc−RD,
g ∆φ=π
r 1.0
e arc−RD,
n ∆φ<π
e
0.5 S−type DD,
∆φ=π
intemittent structures
0.0
0 10 20 30 40
τ
Fig. 5
Fig. 6:
10
8
T
e
T
i 6
bursty
4
2
hydrodynamic
bursty
0
1 2 3 4 5
β
Fig. 7:
U(z)
v
- v
+
z
0
z z
1 2
E<0
U
m