Kristoffer Menzel 13th WPDP 1 Modeling Dust-Density Wave Fields as a System of Coupled van der Pol Oscillators Kristoffer Ole Menzel, Tim Bockwoldt, Oliver Arp, Alexander Piel 13th WPDP, Waco May 21, 2012
Kristoffer Menzel 13th WPDP 2 Dust density wave fields Void Dust-density waves (DDWs) are attributed to a Buneman-type instability Energy gain (by streaming ions) competes with damping (neutral gas) Extended 3D wave fields can be observed under µg conditions DDWs exhibit complicated spatio-temporal dynamics and have a highly nonlinear character
Experimental setup symmetric parallel plate rf discharge (13,56 MHz) gas pressure: 5-35 Pa (Ar) rf voltage: 40-70 V pp CMOS camera (100fps) monodisperse MF particles (diameter: 6,8µm und 9,55µm) Kristoffer Menzel 13th WPDP 3
Kristoffer Menzel 13th WPDP 4 Frequency distribution 7.6 Hz 6.5 Hz 5.7 Hz frequency varies spatially Huygens generation of frequency clusters defects occur at the cluster boundaries
Kristoffer Menzel 13th WPDP 5 Frequency distribution 7.6 Hz Results suggest a new interpretation 6.5 Hz of DDWs! 5.7 Hz frequency varies spatially Huygens generation of frequency clusters defects occur at the cluster boundaries
Kristoffer Menzel 13th WPDP 6 Numerical studies - model Chain of coupled, self-excited, nonlinear oscillators with frequency gradient van der Pol oscillator x i ε 1 βx 2 i ω 0i x i + ω 2 0i x i = g x i 1 + x i+1 2x i nonlinearity amplitude saturation frequency gradient ω 0i =ω 00 + i coupling
Kristoffer Menzel 13th WPDP 7 Numerical studies frequency distribution frequency distribution 0.990 1.027 1.065 Clusters with incommensurable frequencies occur 0.952
Kristoffer Menzel 13th WPDP 8 Numerical studies frequency distribution frequency distribution 0.990 1.027 1.065 Clusters with incommensurable frequencies occur space-time diagram 0.952 Wave propagation in the direction of lower frequencies
Kristoffer Menzel 13th WPDP 9 Numerical studies frequency distribution frequency distribution 0.990 1.027 1.065 Clusters with incommensurable frequencies occur space-time diagram 0.952 Wave propagation in the direction of lower frequencies Defects occur exclusively at the cluster boundaries
Kristoffer Menzel 13th WPDP 10 Numerical studies parameter variation coupling frequency gradient nonlinearity cluster size decreases with increasing coupling decreasing frequency gradient increasing nonlinearity
Kristoffer Menzel 13th WPDP 11 Comparison between model and experiment nonlinearity p=30 Pa, U rf =65V pp, d=6,8µm p=15 Pa, U rf =70V pp, d=6,8µm ω pd = n dq d 2 ε 0 m d 1/2 density gradient
Kristoffer Menzel 13th WPDP 12 Comparison between model and experiment Model works nonlinearity fine for situations with incommensurable frequencies p=30 Pa, U rf =65V pp, d=6,8µm p=15 Pa, U rf =70V pp, d=6,8µm ω pd = n dq d 2 ε 0 m d 1/2 density gradient
Kristoffer Menzel 13th WPDP 13 Frequency distribution Pilch et al., Phys. Plasmas 16, 123709 (2009) Clusters with commensurable frequencies occur at different plasma conditions Situation is similar to that in magnetized anodic plasmas with external modulation
Kristoffer Menzel 13th WPDP 14 Numerical studies - global modulation Chain of coupled, self-excited, nonlinear oscillators with frequency gradient van der Pol oscillator x i ε 1 βx 2 i ω 0i x i + ω 2 0i x i = g x i 1 + x i+1 2x i external forcing x i ε 1 βx 2 i ω 0i x i + ω 2 0i x i = g x i 1 + x i+1 2x i + F sin (ωω)
Kristoffer Menzel 13th WPDP 15 Numerical studies - global modulation Chain of coupled, self-excited, nonlinear oscillators with frequency gradient van der Pol oscillator x i ε 1 βx 2 i ω 0i x i + ω 2 0i x i = g x i 1 + x i+1 2x i external forcing x i ε 1 βx 2 i ω 0i x i + ω 2 0i x i = g x i 1 + x i+1 2x i + F sin (ωω) parametric excitation x i ε 1 β(1 + F sin (ωω))x 2 i ω 0i x i + ω 2 0i x i = g x i 1 + x i+1 2x i
Kristoffer Menzel 13th WPDP 16 Numerical studies - global modulation external forcing parametric excitation ω eee = 1.027 harmonic synchronization superharmonic synchronization
Kristoffer Menzel 13th WPDP 17 Experiment - global modulation Plasma glow and electrode voltage are modulated significantly DDWs are modulated globally by internal sources DDWs can be treated as a self-organized system containing the dust, the plasma and the external circuit
Kristoffer Menzel 13th WPDP 18 Summary Dust density wave fields exhibit a complicated spatio-temporal behavior, which is reflected in the occurrence of frequency clusters. The experimental observations lead to the conclusion that the DDWs can be modeled as an ensemble of van der Pol oscillators. The frequency distributions show clusters of incommensurable or commensurable values depending on the system parameters. The latter case can be treated in the model by including an additional external and/or parametric excitation. Thank you for the attention! This work was supported by DLR under grant no. 50WM1139
Kristoffer Menzel 13th WPDP 19