Dr. A A Mamun Professor of Physics Jahangirnagar University Dhaka, Bangladesh

Size: px

Start display at page:

Download "Dr. A A Mamun Professor of Physics Jahangirnagar University Dhaka, Bangladesh"

Rosamund Fowler
5 years ago
Views:

1 SOLITARY AND SHOCK WAVES IN DUSTY PLASMAS Dr. A A Mamun Professor of Physics Jahangirnagar University Dhaka, Bangladesh

2 OUTLINE Introduction Static Dust: DIA Waves DIA Solitary Waves DIA Shock Waves Mobile Dust: DA Waves DA Solitary Waves DA Shock Waves Conclusion

3 Introduction What is dusty plasma? Dusty plasma: : plasma (electrons + ions + neutrals) with dust having size: µm - mm: not constant in general mass: billions times heavier than ions: not const. in general charge: not neutral: - or + depending on charging processes: electronic or protonic charge: not const. in general. The addition of such dust makes a plasma system very complex. This is why, it is also known as a complex plasma. Ref: Review articles: Goertz (1989); Mendis & Rosenberg (1994) books: Bouchoule (1999); Verheest (2000); Shukla & Mamun (2002)

4 What are exp. discoveries in dusty plasma physics? DUST CRYSTAL 1994 Thomas et al Chu & Lin I Hayashi & Tachibana Γ>172 IKezi 1986 DA WAVES?? 1995 Barkan et al Rao, Shukla & Yu 1990 DIA WAVES?? 1996 Barkan et al Shukla & Silin 1992 DL WAVES 1997 Morfill et al Homann et al Melandsø 1996 MACH CONES 1999 Samsonov et al Havnes et al 1996 DUST VOIDS 2001 Thomas et al Morfill et al 1999

5 What are main research areas in dusty plasma physics?

6 What are new waves in dusty plasmas? There are two new normal modes (waves) which are experimentally observed in DPs: 1. Static Dust: DIA Waves: ω k = ( n n i0 e0 )( T m e i ) The restoring force comes from the electron thermal pressure and the inertia is provided by the ion mass. The dust charge maintains only the equilibrium charge neutrality. The phase speed of the DIA waves can be times higher than the ion-acoustic speed C i =(T e /m i ) 1/2 because of the factor n i0 /n e0 (=1+Z d n d0 /n e0 ) which can be for many space and laboratory dusty plasma situations. DIA waves has been theoretically predicted by Shukla & Silin (1992) and experimentally observed by D Angelo et al (1996).

2. Mobile Dust: DA Waves: ω k ( Z d n n i 0 d 0 )( Z d m T d i ) The restoring force mainly comes from ion thermal pressure and the inertia is provided by dust mass: very low

7 2. Mobile Dust: DA Waves: ω k ( Z d n n i 0 d 0 )( Z d m T d i ) The restoring force mainly comes from ion thermal pressure and the inertia is provided by dust mass: very low frequency mode (V~9 cm/s, λ~0.6 cm, f~15 Hz): visible with naked eyes: DA waves has been theoretically predicted by Rao et al (1990) and experimentally observed by Barkan et al (1995).

8 What are solitary waves? How do they form? How do they convert to shock-like structures? Solitary waves are hump/dip shaped nonlinear waves of permanent profile due to balance between nonlinearity and dispersion (without dissipation). However, when dissipation is important, shock waves are formed. The formation of arbitrary amplitude stationary solitary and shock waves y V(y) 0 A m y y A m 0 x 0 x

dispersive term is much more important than the dissipative term: K-K dv equation: When

9 Evolution equation for small amplitude solitary/shock waves: K-dVK dv-burgers equation (Karpman( 1975): The term containing B (C) is the dispersive (dissipative) term When the dispersive term is much more important than the dissipative term: K-K dv equation: When the dissipative term is much more important than the dispersive term we have: : Burgers equn:

10 Static Dust: DIA Waves Model: We consider an unmagnetized DP with static dust [Shukla[ & Silin 1992]. The dynamics of 1D DIA waves: where n i, u i,,, t, z are normalized by n i0, C i, T e /e, 1/ω pi, μ 1/2 respectively, and μ=n e0 /n i0 = 1-Z1 d n. d /n i0 =n e0 /n i0 /n i0 1/2 λ de,

11 DIA SWs: Small Amplitude: RPM [Washimi[ & Taniuti 1966] reduces (1) - (3) to a K-dVK Eq: where ζ=ε 1/2 (z - v 0 t), τ= ε 3/2 t, a s =(3-1/ 1/μ) μ 1/2 /2, and b s =(2μ) -3/2 The stationary SW solution of this K-dVK Eq: 3/2. where m =3u 0 /a s and Δ s =(4b s /u 0 ) 1/2. Since a s is positive (negative) when μ>(<)1/3, small amplitude DIA SWs with >(<)0 exist when μ>(<)1/3.

12 Arbitrary Amplitude: SPA [Sagdeev[ 1966] reduces (1) - (3) to an energy integral: where ξ=z - Mt and V( ) ) reads [Bharuthram[ & Shukla 1992]: It is clear: V( V( ) ) = dv( )/d )/d = = 0 at =0. So SW solution of this energy integral exists if (d 2 V/d 2 ) =0 =0< < 0 and (d 3 V/d 3 =0>(<) 0 for >(<)0. The vanishing of quadratic term: M c =μ -1/2. At M=M c the cubic term of V( ) ) becomes (3-1/ 1/μ) μ 1/2 1/2 /2= a s. This means that arbitrary amplitude DIA SWs with >(<)0 exist when μ>(<)1/3.

Effects of Non-Planar Geometry The dynamics of DIA waves in planar (ν=0; r=z) or non-planar [cylindrical (ν=1) or spherical (ν=2)] geometry: RPM [Maxon & Vieceli 1974] reduces these Eqs.

13 Effects of Non-Planar Geometry The dynamics of DIA waves in planar (ν=0; r=z) or non-planar [cylindrical (ν=1) or spherical (ν=2)] geometry: RPM [Maxon & Vieceli 1974] reduces these Eqs. to a mk-dv Eq: where ζ=-ε 1/2 (r + v 0 t), τ= ε 3/2 t. The 2 nd term arises due to the effects of non-planar geometry. We have numerically solved, and studied the effects of cylindrical (ν=1) and spherical (ν=2) geometries on the

14 geometries on time-dependent DIA SWs. The numerical results are as follows [Mamun & Shukla 2002]: ν=1 red:τ=-9 green:τ=-6 blue:τ=-3 ν=2 red:τ=-9 green:τ=-6 blue:τ=-3 μ=0.4 μ=0.4 μ=0.2 μ=0.2

15 Effects of Ion-Temperature To examine the effects of ion-temperature we use Eqs. (1), (3) and P i is normalized by n i0 T i and σ i =T i /T e. The RPM reduces (1), (3)-(5) to

16 The numerical analysis of V(,σ i ) or the EI: as we increase ion fluid temperature, amplitude of DIA SWs decreases, but their width increases. The SW solutions of the EI [Mamun 1997] for M=1.45: σ i =0 μ=0.2 σ i =0.1 σ i =0.2 σ i =0 σ i =0.1 σ i =0.2 μ=0.4

17 DIA Shock Waves Theoretical Model: We consider an unmagnetized, dissipative DP [Nakamura et al 1999; Shukla 2000] which can be described by (1), (3) and (6) RPM and η i =ε 1/2 η 0 reduce (1), (3) and (6) to a K-dV-Bergers Eq: (7) where η=η 0 /2V 0. Eq. (7) has monotonic shock solution for η 2 >4U 0 b s (where U 0 =shock speed) and oscillatory shock solution for η 2 <4U 0 b s [Karpman 1975; Shukla & Mamun 2001]. For very small η, i.e. η 2 <<4U 0 b s, shock waves will have oscillatory profiles in which first few oscillations will be close to solitons moving with U 0.

18 Experimental Observation DIA shocks were experimentally observed by Nakamura et al (1999). The plasma parameters used for this experiment: ne 10 8 cm-3, T e 10 4 K, T i 0.1T e, Z d 10 5 for n d <10 3 cm -3 and Z d 10 2 for n d <10 5 cm -3. The oscillatory shock waves were excited in a plasma first without dust and then with dust: as n d is increased, oscillatory wave structures behind the shock decreases and completely disappears at n d =10 3 cm -3.

19 A source of Dissipation: Dust Charge Fluctuation To find an alternate mechanism for the formation of DIA shock waves, we consider charge fluctuating dust grains. Then the dynamics of DIA waves can be described by (1), (2) and (8) Z d (normalized by Z d0 ) is not constant, but varies according to (9) where η=[αm e (1-μ)/2m i ] 1/2, β=(r d /a d )1/2, β i =β(πm e /m i ) 1/2, a d =n -1/3 d0 α=z d0 e 2 /T e r d. At equilibrium we have and where u (normalized by C 0 i ) is the streaming speed of ions.

20 RPM, η c =ε 1/2 η c0 reduce (1), (2), (8), (9) to K-dV-Bergers Eq. (10) where [Mamun & Shukla 2001]

21 Two situations for analytic solution of Eq. (10): C 2 >4U 0 B: This corresponds to monotonic shock solution of Eq. (10). For C 2 >>4U 0 B its monotonic shock solution [Karpman 1975] is C 2 <4U 0 B: This corresponds to oscillatory shock solution of Eq. (10). For C 2 <<4U 0 B its oscillatory shock solution [Karpman 1975] is where 0 = (ζ=0) and K is a constant. This solution means that for very small C, shock waves will have oscillatory profiles in which first few oscillations will be close to solitons moving with U 0. Therefore, dust charge fluctuation can act as a source of dissipation which may responsible for the formation of monotonic or oscillatory DIA shock profiles depending on the plasma parameters.

22 Mobile Dust: DA Waves Model: We consider an unmagnetized DP with mobile dust [Rao et al 1990]. The dynamics of 1D DA waves: (11) (12) (13) where n d, u d,, t, z are normalized by n d0, C d, T i /e, 1/ω pd, C d /ω pd, respectively, μ e =1/(δ-1), μ i =δ/(δ-1), δ=n i0 /n e0, σ i =T i /T e.

23 DA SWs: Small Amplitude: RPM [Washimi & Taniuti 1966] reduces (11)- (13) to a K-dV Eq. and. The stationary SW solution of this K-dV Eq. where m =3u 0 /a s and Δ s =(4b s /u 0 ) 1/2. Since δ>1 and σ i >0, i.e. a s is always negative, small amplitude DA SWs with <0 can only exist [Mamun 1999].

24 Arbitrary Amplitude: SPA [Sagdeev 1966] reduces (11) - (13) to an energy integral: 1999]: where V( ) reads [Mamun where ξ=z - Mt and V( ) reads [Mamun 1992]: V( ) = dv( )/d = 0 at =0. So SW solution of this energy integral exists if (d 2 V/d 2 ) =0 < 0 and (d 3 V/d 3 =0 >(<) 0 for >(<)0. The vanishing of quadratic term: M c =[(δ-1)/(δ+σ i )] 1/2. At M=M c the cubic term of V( ):

Effects of Trapped Ion Distribution The trapped ion distribution of Schamel reduces the ion number density n i for <<1 to a simple form [Schamel 1975] : (14) where σ it =T it /T i. Eq.

25 Effects of Trapped Ion Distribution The trapped ion distribution of Schamel reduces the ion number density n i for <<1 to a simple form [Schamel 1975] : (14) where σ it =T it /T i. Eq. (14) is valid for both σ it <0 and σ it >0. Now, using RPM and Eq. (11) - (13) with the replacement of Boltzmann ion distribution [exp(- )] by the trapped ion distribution [Eq. (14)] we obtain This is a modified KdV equation with stronger nonlinearity: larger amplitude, smaller width and larger propagation speed.

$Effects of Nonthermal Ion Distribution The nonthermal ion distribution of Cairns et al reduces the ion number density n i to a form [Cairns et al 1995)] : (15) where α determines the fraction of fast$

26 Effects of Nonthermal Ion Distribution The nonthermal ion distribution of Cairns et al reduces the ion number density n i to a form [Cairns et al 1995)] : (15) where α determines the fraction of fast particles present. Now, using SPA and Eq. (11) - (13) with the replacement of Boltzmann ion distribution [exp(- )] by the nonthermal ion distribution [Eq. (15)] we obtain: where V( ) is [Mamun et al 1996]: and α 0 =4α/(1+3α). The analysis of V( ) predicts that the presence of nonthermal (fast) ions [with α>0.155] supports coexistence of DA SWs with <0 and >0.

27 These results are clearly observed from the following plots: δ=10 σ i =0.1 δ=10 σ i =0.1 δ=10 σ i =0.1 δ=10 σ i =0.1

28 Effects of Positive Dust Component Model: We consider an unmagnetized DP with dust of opposite polarity [Sayed & Mamun 2007]. The dynamics of 1D DA waves: where subscript 1 (2) corresponds to - (+) dust. N 1 (N 2 ) are normalized by n 10 (n 02 ), U 1 and U 2 are normalized by C 1 =(Z 1 k B T i /m 1 ) 1/2,

29 ψ is normalized by k B T i /e, α=z 2 /Z 1, β=m 1 /m 2, µ e =n e0 /Z 1 n 10, µ i =n i0 /Z 1 n 10, and σ=t i /T e. RPM [Washimi & Taniuti 1966] reduces (1) - (5) to a K-dV Eq: (6) where A and B are given by (7) The steady state solution of K-dV Eq. (6) is (8)

30 where the amplitude ψ m and the width Δ are given by (9) Eqs. (7) (9) implies that the solitary potential profile is + (-) if A> (<) 0. So, we have numerically analyzed A and obtain A=0 curves displayed in figure below: Β + (-) solitary profiles exist for the parameters whose values lie above (below) these A=0 curves, where σ=0.5, α=0.01, µ e =0.1 (solid curve), µ e =0.15 (dotted curve) and µ e =0.2 (dash curve) Μ i

31 We have graphically shown how amplitude of + (corresponding to parameters whose values lie above A=0 curves) and (corresponding to the parameters whose values lie below A=0 curves) solitary potential profiles vary with µ i. α=0.01, σ=0.5, µ e =0.2, β=150 (solid curve), β=160 (dotted curve), and β=170 (dash curve). α=0.01, σ=0.5, µ e =0.2, β=5 (solid curve), β=10 (dotted curve), and β=15 (dash curve). m Μ i m Μ i

32 We have graphically shown how width of + (corresponding to parameters whose values lie above A=0 curves) and (corresponding to the parameters whose values lie below A=0 curves) solitary potential profiles vary with µ i. α=0.01, σ=0.5, µ e =0.2, β=150 (solid curve), β=160 (dotted curve), and β=170 (dash curve). α=0.01, σ=0.5, µ e =0.2, β=5 (solid curve), β=10 (dotted curve), and β=15 (dash curve) Μ i Μ i

33 DA Shock Waves We consider an unmagnetized strongly coupled DP described by GH equations [Kaw & Sen 1998]: Eqs. (11), (13) and (16) where is the normalized longitudinal viscosity coefficient, τ m is the normalized visco-elastic relaxation time, η b and ζ b are shear and bulk viscosity coefficients, ν dn is the normalized dust-neutral collision frequency. All transport coefficients are well defined by Kaw & Sen (1999), Mamun et al (2000), Shukla & Mamun (2001), etc.

34 Using RPM, η d =ε 1/2 η 0 and Eqs. (11), (13) & (16) we have: (17) where It is obvious that for a collision-less limit (ν dn =0) A<0, B>0, and C>0, but for a highly collision limit (ν n τ m >2, since a δσ 2 for δ=10 and σ i =1) A>0, B>0 and C>0.

There are two situations for analytic solutions of Eq. (17): C 2 >4U 0 B: This corresponds to monotonic shock solution of Eq. (17). For C 2 >>4U 0 B its monotonic shock solution [Karpman 1975] is C 2 <4U 0 B: This corresponds to oscillatory shock solution of Eq.

35 There are two situations for analytic solutions of Eq. (17): C 2 >4U 0 B: This corresponds to monotonic shock solution of Eq. (17). For C 2 >>4U 0 B its monotonic shock solution [Karpman 1975] is C 2 <4U 0 B: This corresponds to oscillatory shock solution of Eq. (17). For C 2 <<4U 0 B its oscillatory shock solution [Karpman 1975] is where 0 = (ζ=0) and K is a constant. This solution means that for very small η d shock waves will have oscillatory profiles in which first few oscillations will be close to solitons moving with U 0. Therefore, strong correlation of dust can act as a source of dissipation which may responsible for the formation of monotonic or oscillatory DA shock profiles depending on the plasma parameters.

36 Conclusion The dust particle does not only modify the existing plasma waves, but also introduces a number of new eigen modes, e.g. DIA, DA, DL, etc., which in nonlinear regime form different types of interesting coherent structures. The basic features of DIA & DA SWs [Mamun et al 1996; Mamun & Shukla 2001, 2002] in comparison with IA SWs: IA SWs (V Te >V P >V Ti Ti ) DIA SWs (V Te >V P >V Ti Ti ) DA SWs (V Ti >V P >V Td Td ) >0 (only) >0 when μ>1/3 <0 when μ<1/3 <0 (only) n e >0 n i >0 V>C i C i =(T e /m i ) 1/2 n e >(<)0 when μ>(<)1/3 n i >(<) 0 when μ>(<)1/3 V>C i μ -1/2 V> V> 1/2 μ=n e0 /n i0 n e <0; n i >0 n d <0 V>C d C d =(Z d T i /m d ) 1/2

37 Effect of ion (dust) fluid temperature on DIA and DA SWs: : As ion (dust) fluid temperature increases, amplitude of DIA (DA) SWs decreases, but their width increases [Mamun 1997; Mamun & Shukla 2000]. Effect of a non-planer geometry on DIA and DA SWs: : It reduces to a modified K-dVK equation containing an extra-term term (ν/2t) with ν=1 (2) is for cylindrical (spherical) geometry [Mamun & Shukla 2001, 2002]. Effect of dust charge fluctuation on DIA SWs: : The dust grain charge fluctuations do not only change the amplitude and width of DIA SWs,, but also provide a source of dissipation, and may be responsible for the formation of DIA shock waves [Mamun & Shukla 2002]. Effect of fast ions on DA SWs: : The presence of fast ions may allow compressive and rarefactive SWs to coexist: electrostatic SWs observed by Freja and Viking spacecrafts [Mamun et al 1996].

38 Effects of trapped ion distribution on DA SWs: : it gives rise to a modified K-dVK equation exhibiting stronger nonlinearity: smaller width & larger propagation speed [Mamun et al 1996; Mamun 1997]. Effects of positive dust component on DA SWs: : The consideration of positive dust gives rise to the existence of positive solitons for some parametric regimes as well as it significantly modify the basic properties (amplitude, width & Mach number) of DA solitary waves [Mamun and Shukla 2002; Sayed & Mamun 2007]. Effect of strong dust correlation on DA SWs: : It provides a source of dissipation, and is responsible for the formation of DA shock waves [Shukla & Mamun 2001]. The combined effects of strong dust correlation and trapped ion distribution reduce to a modified K-dVK dv-burgers equation with some new features [Mamun et al 2004]

39 Because of time limit, I confined my talk to an unmagnetized dusty plasma. However, a number of investigations [Mamun 1998a; Mamun et al 1998, Mamun & Hassan 1999] on DA SWs in a magnetized dusty plasma have been made: (i) external magnetic field makes ES-SWs SWs more spiky and (ii) ES-SWs SWs becomes unstable: multi-dimensional instability [Mamun 1998b, 1998c]. Also because of time limit, I confined my talk to ES SWs. However, a number of investigations [Mamun 1999; Mamamun & Gebreel 2000; Mamun & Shukla 2003] on dust EM SWs in a magnetized dusty plasma have been made. The physics of nonlinear waves that we have discussed must play a significant role in understanding the properties of localized electrostatic as well as electromagnetic solitary structures in space & laboratory dusty plasmas.

40 Last but, of course, not the least:

Effect of Positive Dust on Non-linear Properties of Ion-acoustic Waves

Effect of Positive Dust on Non-linear Properties of Ion-acoustic Waves Sanjit Kumar Paul Department of Basic Sciences and Humanities, University of Asia Pacific,Green Road, Dhaka-1215, Bangladesh. Abstract-The