Low energy Langmuir cavitons: asymptotic limit of weak turbulence

2.5 MB·English

by  P. Henri

Low energy Langmuir cavitons:
asymptotic limit of weak turbulence
P. Henri1,2, F. Califano1,2, C. Briand2, A. Mangeney2
1Dip. Fisica,Universit`adiPisa;LargoPontecorvo3,56127Pisa,Italy
3
1 2LESIA,ObservatoiredeParis,CNRS,UPMC,Universit´eParisDiderot;5PlaceJ.Janssen,92190Meudon,France
0 Keywords: Plasma turbulence, Turbulent flows: coherent structures, Electrostatic waves and oscillations.
2

n
a Abstract
J
Langmuir turbulence is an archetype of wave turbulence in plasma physics. By means of 1D-1V
5
Vlasov-Poissonsimulations,weshowthatcoherentstructures,calledLangmuircavitons,aregenerated
1
by the long time evolution of Langmuir weak turbulence, thus illustrating the breakdown of a weak

turbulence regime. These structures correspond to an equilibrium between the pressure forces and
]
h the ponderomotive force resulting from high frequency Langmuir oscillations. Langmuir cavitons are
p typicalfeaturesofstrongLangmuirturbulenceexpectedtobegeneratedathighenergyandtosaturate
-
when Langmuir energy is of the order of the plasma thermal energy. Despite this wide-spread belief,
m
here we observe that cavitons, emerging from weak Langmuir turbulence evolution, saturate at much
s lower energies. We show that these Langmuir coherent structures are characterized by a much larger
a
l length scale with respect to the Debye length. This gives evidence that ”large” and ”shallow” stable
p cavitons should be seen in space plasma observations. The transition toward strong turbulence is
.
s showntobeaconsequenceofaninitialweakturbulentinversecascade. Finally,theeffectiveequation
c of state for ion acoustic oscillations is tested numerically from the kinetic model.
i
s
y 1 Introduction
the fields and (b) the bandwidth of the phenom-
h
p ena under consideration [1, 2, 3]. At finite but
[ The nonlinear evolution of waves is usually classi- small energy, the nonlinear dynamics is described

fied in terms of ”weak” and ”strong” turbulence. by three-waves or four-waves interactions in the
2
The meaning of these terms is not well established RPA through the kinetic wave equations, whereas
v
0 although the major conceptual difference between at higher energies intermittency dominates the dy-
9 weak and strong turbulence is the presence of a namics, through the apparition of coherent struc-
0 characteristic dimensionless parameter ε charac- tures.
3
terizing the level of nonlinearity, as the ratio be-
. To summarize, weak turbulence is based on the
1 tweenatypicallinear(ordispersive)time(theelec-
0 following points: (i) linear dispersion still holds;
tron plasma period in Langmuir turbulence) and
3 (ii) statistical homogeneity in space holds; (iii)
the nonlinear time, depending on the wave ampli-
1 wave particle interaction is described by resonant
: tudes. Turbulence is then considered to be ”weak”
v quasilinear theory. On the other hand, strong
if ε (cid:28) 1; otherwise it is ”strong”. A second as-
i turbulence is based on the following: (i) linear
X sumption is usually made: weak turbulence results
dispersion no longer holds; (ii) strong statistical
r from a superposition of finite, but weak amplitude
inhomogeneity in space; (iii) wave particle interac-
a waves, obeying the linear dispersion relation, but
tion become complex because of wave (in density
with randomly distributed phases. Starting with
cavities) and particle trapping.
(almost)randomphases,suchrandomnessmustbe
preserved over the nonlinear evolution time. Two
key points are decisive for the validity of Random Onlyafewattemptshavebeenmadetocompare
Phase Approximation (RPA): (a) the amplitude of weak turbulence theory with numerical results.
1
The three dimensional equations for capillary ofvariations,0.01<δn/n<0.80[11]. Thesestruc-
water waves has been solved numerically [4] and tures have been observed in the laboratory in sev-
a Zakharov & Filonenko power-law spectrum [5] eralplasmaexperiments[12]whileinnaturalspace
was observed. Weak turbulence theory may be plasmasonlyinactiveionosphericexperiments[13].
valid in some spectral range but fails in others Cavitons at ”high” energies have also been inten-
[6]. Recently, the problem of breakdown of weak sivelystudiedthroughnumericalexperimentsofthe
turbulence by intermittent events associated Zakharovequations[14,15,16,17]andtheVlasov-
with coherent structures has been addressed Poisson equations [18, 19, 20]. For moderate forc-
for different models [7, 6] and an illustration ing, it has been shown that weak turbulence and
of the modulational instability influence on the strong turbulence features can coexist [14].
breakdown of weak turbulence resulting from an The Langmuir cavitons canonical scenario,
inverse cascade has been given for the nonlinear starting from an initial long wavelength Langmuir
Schrdinger equation [8]. However, these results spectrum, can be summarized as follows: (a)
havebeenestablishedforfluidmodels,i.e. without parametric cascading leading to Langmuir conden-
kinetic effects, whose role in the phase randomiz- sation and increase in wave intensity; (b) cavity
ing is unknown. Therefore, the validity of weak formation ((cid:15) E2/2nk T > k2λ2 ); (c) caviton
0 B D
turbulence theory in a full kinetic regime is still collapse (k → ∞); (d) particles acceleration and
an open problem to be investigated by numerical cavity emission of ion sound waves; (e) start of a
simulations. ”caviton nucleation cycle”.
By considering here the limit of 1D Langmuir
1.1 Langmuir turbulence turbulence, an archetype of wave turbulence in
plasma physics, we show the transition from weak
We hereafter concentrate on the specific case of to strong turbulence through the formation of co-
electrostatic Langmuir turbulence, an archetype of herent structures (cavitons), independently of the
wave turbulence in plasma physics. In the weak initial level of coherence of Langmuir oscillations.
turbulence regime, the 3-wave (resp. 4-wave) evo- The resulting cavitons may saturate at ”low” en-
lution dominates for large (resp. small) wave num- ergy levels (electric energy orders of magnitude
berskL >kMI (resp. kL <kMI)throughthedecay lower than thermal energy) generally considered
(resp. modulational) instability, which typically to belong to the weak Langmuir turbulence regime
transfers the L-wave energy towards smaller (resp. and remain then relatively stable. We obtained a
larger) wave vectors. The transition wavenum- power law governing the relation between the scale
ber is kMIλD = 1/3 cs/vth,e with λD the Debye length of the structures and their energy to be di-
length, cs the ion sound speed and vth,e the elec- rectly tested on space plasma data. Finally, the
tron thermal velocity [9]. Weak Langmuir turbu- nonlineartimescaleneededtoreachthestrongtur-
lence is mainly driven by the electrostatic decay bulenceregimeisfoundtoscaleastheinverseofthe
of a Langmuir wave (hereafter L-wave) into an- initial Langmuir energy.
other L-wave and an ion acoustic wave (hereafter Theseresultsareobtainedwithakineticdescrip-
IA), while several processes associated to the pon- tion of 1D electrostatic plasma. To our knowledge,
deromotive force, as modulational instability, os- no similar work on the breakdown of weak turbu-
cillating two stream instability, soliton formation, lence has been done using a fluid model (Zakharov
Langmuir collapse, are considered to dominate the equations).
strong turbulence dynamics [1, 2]. In the specific
caseofLangmuirturbulence,theturbulenceisusu-
ally considered ”strong” for an electric-to-thermal 2 Model
energy ratio larger than the electron-to-ion mass
ratio: W =(cid:15) E2/nk T >m /m . Wesolvethe1D1VVlasov-Poissonsystemofequa-
0 B e i
Langmuircavitonsarelocalizedelectricfieldsos- tions for the electron and proton distribution func-
cillating at the plasma frequency self-consistently tion f , f and the self-consistent electric potential
e p
associatedtodensitycavities[10]withalargerange and electric field, φ and E. All equations are nor-
2
malized by using electron quantities, the electron the interval 2·10−4 < W < 10−1, here again cor-
charge, mass and thermal velocity, e, m and v , responding to the transition from weak to strong
e th,e
theplasmafrequencyω ,theDebyelengthλ and Langmuir turbulence regime. Finally, the initial
pe D
a characteristic density and electric field, n¯ and random density noise is δn/n = 10−5. In case (ii),
e
E¯ =m v ω /e. we exclude the forcing from the beginning and let
e th,e pe
We define W = 0.5×E2 as the electric energy the system to evolve starting with an electron den-
density normalized to the electron kinetic energy. sityrandomnoisecorrespondingtoaflatspectrum
Then, the dimensionless Vlasov equations read: in the electric field. The r.m.s. electron density
amplitude is 10−3 (cid:46) δn /n (cid:46) 10−2, correspond-
e e
∂fe +v∂fe −(E+E )∂fe =0 (1) ing to a r.m.s. electric field 10−2 (cid:46) E (cid:46) 10−1, i.e.
∂t ∂x ext ∂v an energy range 2·10−4 <W <2·10−2.
Weuseakineticmodeltofullytakeintoaccount
∂f ∂f 1 ∂f
p +u p + E p =0 (2) all processes, as the wave-particle interactions,
∂t ∂x µ ∂u
that could limit the development of Langmuir
where v and u are the electron and ion velocity, turbulence by extracting electric energy and con-
µ = m /m = 1/1836 the electron-to-proton mass verting it into kinetic energy. The phase velocity
e p
ratio and Eext an ”external” driver acting on the vφ of the initial L-waves is much larger than vth,e,
electrons only that can be switched on or off dur- so that the Landau damping of Langmuir waves is
ing the runs. The details of the external forcing inefficient. Ontheotherhand, Tp =Te (acommon
can be found in Appendix 1 of Ref. [21]. Finally, situation in space plasmas like the solar wind)
the Vlasov equations are self-consistently coupled so that the generated IA waves are efficiently
to the Poisson equation. Landau damped during the transient part of the
simulation, corresponding to a weak turbulence
∂2φ (cid:90) (cid:90) ∂φ evolution. This kinetic damping could limit, first
= f dv− f du ; E =− (3)
∂x2 e p ∂x the Langmuir cascade, second, the generation
of density fluctuations that are a seed for the
WeuseanumericalboxoflengthLx =5000λD and subsequent generation of density inhomogeneities
a velocity range −5 ≤ v/vth,e ≤ +5 for electrons observed in the asymptotic part of the simulation,
and −5 ≤ u/uth,i ≤ +5 protons. The spatial and corresponding to the strong turbulence evolution.
velocity mesh grid is dx=λ , dv =0.04 v and
D th,e
du = 0.04 u where u is the proton thermal
th,i th,i
velocity. Periodic boundary conditions are used
in the spatial direction. The initial electron and 3 Numerical results
proton velocity distributions are Maxwellian with
equaltemperatures,T =T ,atypicalconditionin We first discuss the results of a simulation from
p e
solar wind plasma, so that IA fluctuations are effi- the first set of numerical experiments, starting
cientlydampedout. Weaddatt=0adensityran- with a monochromatic L-wave with E = 0.06
L
dom noise in the wavelength range 30<λ<500. and λ = 100 λ . The time evolution of the
L D
We consider two different initial conditions, (i) electric energy and the ion density are shown in
coherentand(ii)incoherentL-waveswithaninitial Fig. 1. The L-wave first undergoes electrostatic
electric energy level varying in the range 10−4 (cid:46) parametric instability (Langmuir electrostatic
W (cid:46) 10−2, corresponding to the transition from decay) during the period 3 · 104 < t < 6 · 104.
weak to strong turbulence regime. In case (i), the In the mean time, IA fluctuations are generated.
external driver E , oscillating at ω , excites a Then, ion cavities start to form at t (cid:39) 105. These
ext pe
monochromatic L-wave with wavelength λ and cavities are filled by electric energy in the form
L
propagating in one direction only. The forcing is of an electrostatic field oscillating at the plasma
switched off when the L-wave reaches the desired frequency. The ion density fluctuations δn/n and
amplitude E . The initial wave then evolves self the Langmuir energy W are shown in Fig. 2,
L L
consistently according to the Vlasov-Poisson sys- top panel, (blue and black lines respectively) for
tem. ThecorrespondingLangmuirenergyrangesin t > 1.5×105. The ion density fluctuations as well
3
Figure1: LongtimeevolutionofelectricenergyW
L Figure 2: Top panel: ion cavitons with the as-
(top panel) and ion density δn/n (bottom panel)
sociated ion density fluctuations δn/n (blue line)
in the (x,t) plane, starting from a monochromatic
andelectricenergyW (blackline). Bottompanel:
Langmuirwave. Noteinthebottompanelthegen- L
Thepressureandtheponderomotiveforce(redand
erationof(i)ionacousticwavesfromtheLangmuir
black line, respectively). The ion density fluc-
electrostatic decay between 3×104 < t < 6×104;
tuations as well as the envelop of the Langmuir
(ii) ion cavities filled with electric energy for t >
electric energy remain practically constant from
105. t>1.5×105.
as the envelope of the Langmuir electric energy
Then cavitons are formed and remain stable un-
remain practically constant from t > 1.5×105 to
til the end of the simulation. In case (ii) the ini-
the end of the numerical simulation. The cavities
tial electron density fluctuation self-organises into
result from an equilibrium between the total pres-
a large spectrum of Langmuir noise. The ”sea”
sure, −∇(P + P ) and the ponderomotive force,
e i of L-waves that fills the simulation box then col-
−e2/(4m ω2 ) ∂ E2, associated to high frequency
i pe x lapses into stable low energy cavitons. In all cases
Langmuiroscillations(seeFig.2,bottompanel,red
thenonlinearstructuresaresimilar,indicatingthat
andblackline,respectively). Thesecoherentstruc-
the asymptotic behavior of Langmuir turbulence
tures, identified as ”cavitons”, are the signature of
is independent from the initial level of coherence
atransitiontoastrongLangmuirturbulentregime.
of Langmuir oscillations. As expected, regardless
of the initialization of the turbulence, the larger
We have performed many simulations starting the initial L-waves, the sooner Langmuir cavitons
with(i)coherentand(ii)incoherentinitialL-waves. aregenerated. Foreachcaviton, wehavemeasured
In all cases, the system freely evolves until it even- the depth of the ion density hole δn/n, the length
tually jumps to a strong turbulence state charac- of the structure L (width of the ion cavity at the
terized by the presence of Langmuir cavitons. The heightof1/eofthemaximumdepth)andthemax-
formation of cavitons does not need a high level of imum Langmuir electric energy W that sustains
L
L-waves if the system evolves ”long enough”. In the density hole. The results are shown in Fig. 3
case (i), as discussed, parametric decay occurs first on a W interval ranging over three decades. It
L
and saturates in the first part of the simulation. is worth noticing that Langmuir cavitons are ob-
4
served also at remarkably low electric energy val-
ues, W ∼ 10−3. The depth of the ion cavity
L
δn/n is of the order of the Langmuir electric en-
ergydensity(asexpectedforhighenergycavitons)
over three decades:
δn/n=(δn/n) Wα (4)
0 L
with(δn/n) =0.28±0.06andα=1.13±0.06the
0
fitting parameters. Fit results are given with their
respective3σerrors. Thewidthofthecavitonsalso
scales on the Langmuir electric density energy:
L=L Wβ (5)
0 L
with L = 18 ± 4 and β = −0.47 ± 0.05. As
0
expected, Langmuir cavitons have a larger scale
length when the Langmuir electric energy is lower.
Anewunexpectedandimportantresultisthegen-
eration of stable Langmuir caviton with a scale
length of many hundred of Debye lengths. Finally,
Figure 3: Depth expressed in relative density δn/n
thepowerlawbetweenthedepthandthelengthof
(toppanel)andwidthLexpressedinDebyelength
theioncavitiesassociatedwiththecavitonsresults
(bottom panel) of cavitons measured in the sim-
as:
L=L (δn/n)γ (6) ulations according to the associated Langmuir en-
0 ergy W . Each diamond represents a single cavi-
L
with L =10±3 and γ =−0.42±0.05. ton. The line shows the power law fit.
0
standing structures with electric oscillations that
4 Discussion
self-consistentlysustainthedensitycavityofwhich
they are eigenmodes.
We hereafter discuss, first, the difference between
the cavitons observed in our simulations and
the Langmuir solitons also associated to strong
Langmuir turbulence. Then we investigate the 4.2 Transition from weak to strong
transition from weak to strong turbulence and its turbulence
observed timescale.
In order to identify the mechanism responsible for
the transition from weak to strong turbulence, we
have studied the temporal evolution of the electric
4.1 Langmuir solitons and cavitons
and ion density spectra. In the first part of the ex-
Interestingly, the scaling laws obtained here for periments, successive Langmuir decay instabilities
the Langmuir cavitons, Eq. 4-6, are similar to generatesmallerandsmallerwavenumbers(3-wave
those of Langmuir solitons [22]. In particular, inverse cascade characteristic of weak turbulence).
the typical depth of the ion cavities is of the The 3-wave cascade spontaneously develops for
order of the Langmuir electric energy, obtained the first series of numerical experiments starting
in the simulation over three orders of magnitude. from a monochromatic L-wave. This is shown in
We recall that Langmuir cavitons, or Langmuir Fig. 4 (initial wave vector k = 2π/50) where we
L
standing solitons,arepropagatingstructuresmain- draw the development of the Langmuir turbulence
taining their shape through the balance between (top panel) and the excitation of daughter S-waves
dispersive and nonlinear effects, while cavitons are (bottom panel) driven by the successive decays of
5
Fourierspace. Thiseffectiveequationofstate, well
known in the linear regime with T >> T , is thus
e i
extendedtoT ∼T intheweakturbulenceregime.
e i
This result is confirmed by a scattered plot of the
ion pressure vs ion density during the weak turbu-
lent stage of the simulation (not shown here).
Once this weak turbulent cascade has devel-
oped, cavitons are systematically generated (ap-
parition of a large spectrum in both panels) when
k λ (cid:39)10−2 corresponding to a critical wave vec-
L D
tor k λ = 1/3 c /v (blue dashed line in
MI D s th,e
Fig. 4) for which the growth rate of the modula-
tional instability, a known precursor of strong tur-
bulence, overcomes the growth rate of the decay
instability [9]. This is confirmed by other simula-
tions starting with different initial wave vectors.
To sum up, the weak turbulent inverse cas-
cade brings the fluctuations to smaller wave vec-
tors, where different nonlinear processes generate
the coherent structures typical of a strong turbu-
lent regime. In this picture, the strong turbulence
Figure 4: Evolution of the electric field and
regime is the natural asymptotic limit of weak tur-
ion density spectrum (top and bottom pan-
bulence.
els respectively). Black dashed lines: ex-
pected wavenumber from 3-wave cascade in- Finally, a series of experiments has been carried
teractions. Blue dashed line: transition out with an initial wave vector kL <kMI, in order
wavenumber k between decay and mod- to directly trigger the modulational instability in-
ulational instabMiIlities. stead of the decay instability. No L-wave cascade
is observed, but similar cavitons are indeed gen-
erated, in agreement with the previously exposed
the L-waves. The L-wave cascade follows the path picture.
in Fourier space expected for such 3-wave cascade The detailed mechanism for the generation of
process (the theoretical wave numbers are shown thecavitonsisstillunclear,evenifitisknownthat
in dotted lines). As previously pointed out [21], cavitons can be by-products of the development
the decay product S-waves are generated over a of the modulational instability. This point will be
finite band of wave numbers and do not survive investigated in future works.
outside the coupling region with L-waves (bottom
panel) because of strong Landau damping. Note
that there is no cascade on the S-wave: instead
4.3 Evaluation of the nonlinear
the harmonics of the S-waves are temporarily ex-
timescale
cited as long as a pump L-wave injects energy into
a fundamental S-wave, but the high kinetic damp-
Tocompletethispreviouspicture,weshowinFig.5
ingpreventsanyfurtherdevelopmentofaninertial
the characteristic nonlinear time scale τ corre-
range on these S-waves. NL
sponding to the formation time of the first caviton
Interestingly,thedevelopmentoftheL-wavecas-
in a simulation. We see that τ scales as the in-
cadeallowstoidentifyisothermalelectronsandadi- NL
verse of the initial Langmuir energy W :
abaticionsastheeffectiveequationofstateforion L,init
iascoguivsetnicbflyucct2u=ati(oγnsT. H+erγeTth)e/mion, swohuenrde sγpe=edSc1s τNL (cid:39)τNL0 WLη,init (7)
s e e i i i e
and γ = 5/3. A slightly different definition of c with τ = 178±4 and η = −1.0±0.1. Since
i s NL0
would lead to an incorrect path of the cascade in the growth rate for Langmuir decay scales as
6
temperatures. It is given by isothermal electrons
and adiabatic ions; this extends the result known
in the limit T >> T . Furthermore, electrostatic
e i
coherent structures of typical width much greater
thatafewDebyelengthsaregeneratedbythelong
timeevolutionofaninitialrelativelymoderateam-
plitude turbulence. These results have been ob-
tained for a 1D kinetic description of electrostatic
plasmas and should be extended to multidimen-
sional studies in order to test the stability of the
observed coherent structures.
Figure 5: The nonlinear time corresponding to the
The breakdown of weak Langmuir turbulence
formation of Langmuir cavitons for different initial
and the existence of large coherent structures can
Langmuir energies W . Diamond: simulation
have an important impact on the interpretation
L,init
results, blue line: fit.
of space plasma data. The authors are thus
confident that these new insights in Langmuir
turbulence may encourage the space physics
W [23], τ is interpreted as the sum of
L,init NL community to revisit the admitted conclusion
a succession of Langmuir decay time scales to
that strong turbulent Langmuir structures are
reach the critical wave vector k for which the
c
formed at too high energies to be relevant in space
modulational instability becomes dominant and
plasma environments. This last result should be
can generate the observed coherent structures
directly tested on waveforms data in space plasma
(cavitons). Therefore, in a plasma like the solar
environments.
wind, since in the L-waves are typically observed
in the range 10−5 < Wobs < 10−2, the time scale
L
to reach a strong turbulence regime assuming
Wobs ∼ 10−2 would be about 10 sec (with f 10
pe
L
kHz).
Acknowledgments
Wearegratefultotheitaliansuper-computingcen-
5 Conclusions ter CINECA (Bologna) where part of the calcu-
lations where performed. We also acknowledge
In summary, we have shown that L-wave turbu- Dr. C. Cavazzoni for discussions on code perfor-
lence is expected to breakdown for times larger mance.
than a typical nonlinear time scale, in agreement
withRefs.[7,6]. Thedistinctionbetweenweakand References
strong turbulence thus looses part of its significa-
tion. These results are independent from the ini-
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