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Observations of Linear andNonlinear Dust Acoustic Waves*

Bob Merlino, Jon HeinrichSu Hyun Kim and John Meyer

Department of Physics and AstronomyThe University of Iowa, Iowa City, Iowa

*Supported by DOE and NSF

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Introduction• The DAW is the most basic dust density wave

involving motion of the dust particles• Dispersion relation: • Often reaching very high amplitudes with non-

sinusoidal waveforms, may develop into shocks• Very difficult to see the linear growth phase, except at

high neutral pressures where it is nearly quenched

• Observations discussed in this talk:– Linear growth of DAWs in a drifting dusty plasma– Nonlinear DAWs and second order wave theory– Secondary dust waves associated with nonlinear DAWs

dapdD Ck

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Dust acoustic waves (DAW)• The DAW wave is spontaneously excited in gas

discharge dusty plasmas by an ion-dust streaming instability

• Dispersion relation from fluid theory– finite Td

– Collisions of electrons, ions and dust with neutrals

– DC electric field E0

2

2 20 0

1 0, wherei e d

pjj

j j jn Tjku ku i k V

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Ion-dust streaming instability

11 3

15 3

1

0.5 , 2000, ~ 10

40, ~ 10 , 2

1.26 , 5

Parameters: d d d

i i e

r m Z n m

A n m T eV

k mm mm

P = 100 mtorr E0 = 100 V/m

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DAWs in discharge plasmas

• DAWs are often observed in discharge dusty plasmas at low neutral pressures

• Solid lines are numerical solutions of the dispersion relation for various experimental parameters

• The region below a curve signifies that the mode is unstable

• The points correspond to different experiments

• Ion drift in discharges are sufficient for instabilityPhys. Plasmas 16, 124501, 2009

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Dusty plasma device

Dust: silica microspheres (1 mm diameter)Plasma: argon, 10 – 20 Pa, ni ~ 1015 m3, Te 100 Ti 2-3 eV

CMOSCamera

Top View

B

Dust Tray

532 nmLaser

Plasma

B

Side View

Anode

g

Lens

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DAWs excited in a drifting dust cloud

• A secondary dust suspension is trappedby a biased grid 15 cm from the anode.

• When the bias on the grid is switched off, the grid returns to its floating potential, and the secondary cloud is released.

• The secondary cloud begins drifting toward the anode.

ion drift

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Drifting dust cloud and DAWs

• When the center of cloud is about 10 cm from the anode, dustacoustic waves begin to be excited in the quiescent dust cloud.• The DAWs begin being excited when they reach the point where the ion drift is sufficient to drive the ion-dust streaming instability

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Growth rate measurementn

d /

n do

Distance from anode (cm)

t = 0 st = 0 s

t = 0.03 s

t = 0.06 s

t = 0.09 s

Time (s)n

d /

nd

o

FIT

rd = 0.5 m silica microspheres

0.2d

d

n

n

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Comparison to DAW (F, K) theory

f (F)

f (K)

(F)

(K)

Fre

quen

cy (

Hz)

Grow

th rate (s1)

Wavelength (m)

Growth rate Frequency

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Nonlinear dust acoustic waves

Spontaneously excitedDA waves often growto very high amplitudes

DA waveforms are non-sinusoidal, typically with sharp wave crestsand flat wave troughs

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2nd order DA wave theory

1 2( , ) cos( ) cos 2d d dn x t n kx t n kx t

Nonlinearity generates 2nd harmonic term

2 2 2 2 2 22 2 1 1

2 2 2 21d d d d

da

n n n nA B

x tx C t x

• Simple fluid theory (Stokes’ waves in ocean wave theory)• expand (nd, ud, ) as a series in the small parameter, to second order: 012

SOLUTION

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Compare 2nd order theory to data

• The fit has a second harmonic amplitude of 30% of the first harmonic amplitude.

• 2nd order theory captures sharp crests and flat troughs.

• Higher order theory provides qualitative and quantitative corrections over linear theory – this was a first start.

Exp.

Theory

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Secondary dust density waves

• Secondary dust density waves (SDDW) were observed in the troughs of high amplitude DAWs

• The SDDW propagated in the direction opposite to the primary DAW

• SDDW grow in thedust that is displaced by the nonlinear DAW and then restored back

Primary DAW

Secondary DDW

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0 50 100 150Position (arb)

Dus

t D

ensi

ty (

arb)

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Secondary dust density waves

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Dust-dust streaming instability• We considered the possibility

that the SDDW were excited by a dust-dust streaming instability between the background dust and the restoring dust drift.

• The kinetic dispersion relation was obtained and solved for the parameters of the experiment.

• The theory give values for the frequency and wavelength (for max. growth) that fit the results

(M. Rosenberg)

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Summary

• The linear growth of DAWs was observed in a drifting dusty plasma

• The measured growth rates agreed well with the kinetic theory of DAWs

• High amplitude (nonlinear ) DAWs exhibit non-sinusoidal waveforms that seem to be accounted for by second-order DAW theory

• Secondary DDW were observed in the presence of nonlinear DAW which may be excited by a dust-dust streaming instability