EGU2011-10300

Effects of density inhomogeneities on the statistics of Langmuir waves in the solar wind

P. Guio1 and A. Zaslavsky2

1 University College London, UK 2 Harvard-Smithsonian Center for Astrophysics, USA

Email contacts: p.guio@ucl.ac.uk, azaslavsky@cfa.harvard.edu

Download poster from http://www.ucl.ac.uk/˜ucappgu

EGU2011-10300

Abstract

Large-amplitude and spatially localised Langmuir waves are frequently observed in the solar wind, usually destabilisedby energetic electron beams. Recent modelling using the high-frequency component of the Zakharov equations withterms describing the beam and a prescribed inhomogeneous density background, as in the solar wind plasma, are in goodagreement within situ observations by the TDS instrument on board the STEREO spacecraft (Zaslavsky et al., 2010).This study showed that the presence of strong density fluctuations could notably modify the observed waveforms andtheir amplitude. Here we go further in this study, describing the background solar wind plasma as a turbulent time- andspace-fluctuating medium, and show how the phenomenon of Langmuir wave reflection by the density inhomogeneitiescan affect the statistical properties of the waveform and the envelope field.

Introduction

Solar type III radio bursts are intense radio emissions produced by energetic electrons streamingalong open magnetic field lines from the low corona to distances that can reach severalAU. Suchevents are sometimes observed in-situ, and the beam electrons are then observed together with theLangmuir waves they produce, which are at the origin of the radio emission.On the other hand, the solar wind medium in which the beam propagates is known to be a stronglyturbulent medium, in which large amplitude density fluctuations are continuously present, despitetheir strong thermal damping. It has been hypothesised thatthese are the product of the decay of thelarge amplitude Alfven waves observed in the solar wind (Celnikier et al., 1987).If these density fluctuations are large enough (or equivalently if the Langmuir wave vectorkb issmall enough) the waves can be reflected by the density fluctuations, and the physics of the beamplasma interaction is then modified (Kellogg et al., 1999). In particular, peaks of the electrostaticfield appear near the reflection points, generating the localised structures observed by e.g. S/WAVES.Here we perform simulations enabling the study of the electric field statistics, and show how theyare modified in the regime where wave reflection is important.

Type III parameters: when do reflection effects appear?

The reflection of the Langmuir waves by the density fluctuations is possible when3k2λ2D < δn/n0,

whereλD is Debye length andδn/n0 is relative density fluctuations. Under this condition, the WKBsolution for the wave evolution is not valid at the reflectionpoints. In the case of the beam-plasmainstability, the resonant wave vectorkb≃ωe/vb for electron plasma frequencyωe and beam velocityvb. Thus for a given density profile the possibility of reflection will be determined by the velocityof the incoming electrons. Fig. 1 shows that both regimes, i.e. WKB and reflection, are possiblefor typical solar wind and type III beam parameters. We performed simulations for both regimes,Simulation 1 (S1) for no reflection and Simulation 2 (S2) whenreflection is important.

Fig. 1: The threshold value of the density fluctuationδn/n0 for which reflection effects start to ap-pear is plotted as a function of the Langmuir wave vectork. Below the line, there is no reflection(regime of S1) whereas above the line, reflection is important (regime of S2).

Numerical model and simulations

The Zakharov system of equations (ZSE) (Zakharov, 1972) in one-dimensional space(

i∂t + iνe× +3

2ωeλ

2D∂2

x

)

Ee =ωe

2

δn

n0Ee + SE, (1)

(

∂2t + 2νi×∂t − C2

s∂2x

)

δn =ε0

4mi∂2x |Ee|

2 + Sn, (2)

are integrated numerically with periodic boundary condition using a pseudo-spectral method (Guioand Forme, 2006).The drive term modelling the large scale density fluctuations Sn is defined ink-space bySn with aGaussian distribution modulus maximum atλc and extending to small scaleλm, i.e. with a spread∆k = 2π(1/λm−1/λc)

∣

∣

∣Sn(k)

∣

∣

∣

n0∼ exp

(

−

(

k − 2π/λc

∆k

)2)

. (3)

Eachk-modeSn(k) is advanced in time along an independent and stochastic pathof its phase, aspreviously done for the modelling of thermal fluctuations inGuio and Forme (2006). This simpleparametrisation allows to model a prescribed slowly changing and large scale fluctuating densityfield.

The parameters for the background plasma of the solar wind aren0=5·106 m−3, Te=50·103 K andTi=50·103 K givingλD∼7 m. The parameters for the beam correspond to a typical type III solar burstevent and arenb/n0=10−4, vb=18·106 m s−1 and∆vb/vb=0.2. kb∼7·10−3 m−1 andkbλD∼0.05 Thesimulations length isL=90 km∼13000λD, the density fluctuations parameters areλc=45 km andλm=15 km.

x [km]

τ [m

s]

S1: δn(x,τ)/n0 [%]

-40 -20 0 20 400

200

400

600

800

1000

1200

-0.2

-0.1

0

0.1

0.2

x [km]

τ [m

s]

S1: log10

(|Ee(x,τ)|) [mVm-1]

126 km/s151 km/s

-40 -20 0 20 400

200

400

600

800

1000

1200

-8

-6

-4

-2

0

2

x [km]

τ [m

s]

δn/n0(S2)-10×δn/n

0(S1)

-40 -20 0 20 400

200

400

600

800

1000

1200

-0.06

-0.04

-0.02

0

x [km]

τ [m

s]

S2: log10

(|Ee(x,τ)|) [mVm-1]

-500 km/s

-500 km/s

-40 -20 0 20 400

200

400

600

800

1000

1200

-8

-6

-4

-2

0

2

Fig. 2: Numerical solutions of Eqs. (1–2) for small (δnmax/n0∼0.4%, S1) and large fluctuations(δnmax/n0∼4%, S2). Upper left: fluctuation densityδn/n0 from S1. Lower left: difference betweenδn/n0 from S2 and (scaled to S2 by×10) δn/n0 from S1. Nonlinear feedback of the electric fieldon the density (by mean of the ponderomotive force) created two clear narrow density depletionssurrounded by denser regions. Upper and lower right: envelope of the electric field |Ee| from S1and S2. Larger density fluctuations reduce the growth rate of the linear beam instability (Escandeand de Genouillac, 1978). Waves beating structures in|Ee| move roughly at the group velocityvg(kb)=3kbv

2e/ωe∼126 km s−1, more accurately the most unstable mode givesvg(k)∼151 km s−1 as

seen in the upper right panel as segments corresponding to a travel time of400 ms. Interferencepatterns due to reflection and refraction in the electric field are seen unambiguously in S2.

-40 -30 -20 -10 0 10 20 30 40-0.2

-0.1

0

0.1

0.2

x [km]

δn(x

,τ=65

5 m

s)/n 0 [%

])

-40 -30 -20 -10 0 10 20 30 400

10

20

30

40

50

|Ee(x

,τ=

655

ms)

| [m

Vm-1

]

-40 -30 -20 -10 0 10 20 30 40-2

-1

0

1

2

x [km]

δn(x

,τ=65

5 m

s)/n 0 [%

])

-40 -30 -20 -10 0 10 20 30 400

10

20

30

40

50

|Ee(x

,τ=

655

ms)

| [m

Vm-1

]

Fig. 3: Snapshot ofδn/n0 and |Ee| at τ=655 ms from S1 (left) and S2 (right). For larger densityfluctuations the electric field is maximum in the positive density gradient region on the right of adensity minimum, i.e. at the reflection points of the waves destabilised inside the hole, as seen in theright panel from S2 atx∼−15 km.

log10

(|Ee| [mV/m])

τ [m

s]

S1: log10

[P(log10

(|Ee(τ)|)]

-8 -6 -4 -2 0 20

200

400

600

800

1000

1200

-3.5

-3

-2.5

-2

-1.5

-1

log10

(|Ee| [mV/m])

τ [m

s]

S2: log10

[P(log10

(|Ee(τ)|)]

-8 -6 -4 -2 0 20

200

400

600

800

1000

1200

-3.5

-3

-2.5

-2

-1.5

-1

Fig. 4: Temporal probability distribution of the amplitude ofEe from S1 (left) and S2 (right). Thedensity fluctuations clearly reduces the growth rate of the linear instability, broadens the statistics ofthe electric field with fine structures associated to localisation of Langmuir waves.

10-2

100

102

10-6

10-4

10-2

log10

(|Ee| [mV/m])

log 10

(P(lo

g 10(|

E e|)))

τ(S1: δn/n0=0.4%)>600 ms, τ(S2: δn/n

0=4%)>600 ms

S1: δn/n

0=0.4%

S2: δn/n0=4%

S1: s=1.5597

S2: s=1.8997

Fig. 5: Time-averaged probability distribution of|Ee| for saturated state with (rms) electrostatic-to-thermal energy densityW=10−6 (τ>600 ms, see Fig. 4). The distribution for S2 is flattened andbroadened compared to S1.q are exponents of power law fit (dashed lines) at small amplitude ofEe.

0 10 20 30 40 50 60

−20

0

20

t [ms]

E [m

V/m

]

0 10 20 30 40 50 60

−50

0

50

t [ms]

E [m

V/m

]

Fig. 6: Upper panel: S/WAVES TDS waveform from 18th August 2010 during a type III solar burstevent. Lower panels: two waveforms computed from S2 assuming the solar wind speed relativeto S/Cvsw=500 km s−1 and same sampling parameters as the S/WAVES waveform. The synthetictrajectory of the S/C for each waveform is seen as a solid line in the lower right panel in Fig. 2.

10-4

10-2

100

102

10-6

10-4

10-2

log10

(|Ee| [mV/m])

log 10

[P(lo

g 10(|

E e(v(τ

-τs),

τ)|)

)]

S1: δn/n

0=0.4%

S2: δn/n0=4%

S1: q=1.6324

S2: q=1.5034

10-4

10-2

100

102

10-6

10-4

10-2

log10

(|E| [mV/m])

log 10

[P(lo

g 10(|

E(v

(τ-τ

s),τ)

|))]

S1: δn/n

0=0.4%

S2: δn/n0=4%

S1: p=1.0257

S2: p=0.97878

Fig. 7: Probability distributions of the amplitude ofEe (left) and the waveformE (right) a S/C wouldobserve crossing S1 and S2 withvsw=500 km s−1. Each distribution is based on 98 waveforms oflength180 ms sampled at∼10 µs in the region between the dotted lines seen in the lower right panelin Fig. 2. The distributions for S2 are flattened and broadened compared to S1. Forboth S1 and S2,the exponentsq andp of the fitted power law at low field are of the order of the values (q∼2 andp∼1)predicted by stochastic growth theory (Li et al., 2010). But while for S1q andp remain constant asfunction of the relative speed (vsw=300–500 km s−1), for S2q andp decrease as the relative speeddecreases. This might be due to longer time spent in clumped structures asv decreases.

Conclusion

We have presented simulations of Langmuir wave generation in a turbulent medium. We showed thattwo regimes of the instability can exist for typical Type III burst parameters. In the “WKB” case,the waves are only slightly modulated by the turbulent background, and the electric field statisticsare roughly the same as in an homogeneous plasma. In the “strong wave reflection” case, the field isspatially clumped, and its probability distribution is broadened and flattened.

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