Nonlinear dust-acoustic solitary waves in strongly coupled dusty plasmas

Transcription

1 Nonlinear ust-acoustic solitary waves in strongly couple usty plasmas S. E. Cousens, S. Sultana, I. Kourakis, V. V. Yaroshenko, 2 F. Verheest, 3, 4 an M. A. Hellberg 4 Centre for Plasma Physics, Department of Physics an Astronomy, Queen s University Belfast, BT7 NN Northern Irelan, UK 2 Max-Planck-Institut für extraterrestrische Physik, 8574 Garching, Germany 3 Sterrenkunig Observatorium, Universiteit Gent, Krijgslaan 28, B-9000 Gent, Belgium 4 School of Chemistry an Physics, University of KwaZulu-Natal, Private Bag X5400, Durban 4000, South Africa Dust-acoustic waves are investigate in a three-component plasma consisting of strongly couple ust particles an Maxwellian electrons an ions. A flui moel approach is use, with the effects of strong coupling being accounte for by an effective electrostatic pressure which is a function of the ust number ensity an the electrostatic potential. Both linear an weakly nonlinear cases are consiere by erivation an analysis of the linear ispersion relation an the Korteweg-e Vries equation, respectively. In contrast to previous stuies using this moel, this paper presents the results arising from an expansion of the ynamical form of the electrostatic pressure, accounting for the variations in its value in the vicinity of the wave. I. INTRODUCTION A usty plasma is characterise by the presence of massive, charge ust particles in aition to the electron, ion an neutral components that are foun in orinary plasmas. These ust particles range in size from nanometres to millimetres, are typically billions of times more massive than protons an can have between one thousan an several hunre thousan elementary charges. The stuy of usty plasmas has become an increasingly important area of research in plasma physics; its scope encompasses a wie variety of fiels such as astrophysics, semiconuctor manufacturing an fusion reactors. It is interesting to note that the first scientific stuy performe in the International Space Station was a usty plasma physics experiment 2, an it is perhaps an inication of its perceive importance within the scientific community. The presence of a massive, charge ust component in a usual electron-ion plasma can have profoun effects on the ynamics of the system. This article theoretically investigates aspects of two such effects; namely the introuction of the ust-acoustic wave DAW as a new wave moe, an the possibility of strong coupling between the ust particles. The DAW is a very low frequency moe in which the wave is supporte by the inertia of the ust particles, with the restoring force being provie by the pressure of both the electrons an ions. The DAW was first theoretically investigate in 990 by Rao et al. 3, an experimentally observe by Barkan et al. 4, who were able to prouce fascinating images an realtime vieos of the propagation of DAWs ue to their low phase velocity an the large size of the ust particles. In aition to DAWs, there may also be the scousens0@qub.ac.uk basharminbu@gmail.com IoannisKourakisSci@gmail.com; i.kourakis@qub.ac.uk associate nonlinear structures such as ust-acoustic solitary waves, which arise ue to a balance between nonlinear effects an ispersion. Solitons are a particular type of solitary wave which maintain their shape an spee after interactions an have been extensively stuie in mathematics an physics ue to their stable structure an also because they arise as solutions to various exactly solvable moels incluing the Korteweg-e Vries KV an the nonlinear Schröinger NLS equations. In 986 Ikezi 5 preicte that a usty plasma can enter the strongly couple regime ue to the high charge an low temperature of the ust. Here, the coupling parameter, Γ where we have Γ = Z 2 e2 4πɛ 0 k B T a with Z, T an a = n /3 being the charge number, temperature an mean interparticle istance of the ust particles, respectively. This preiction was soon verifie in plasma ischarges 6 8, an strongly couple plasmas has since become a popular research area amongst plasma physicists. The large masses of the ust particles can cause complications when stuying strongly couple usty plasmas on earth-base experiments. This is because gravity can ominate over some of the more subtle interactions, masking some interesting phenomena. A series of experiments aboar the TEXUS 35 rocket flight 9 an the International Space Station 2 were able to stuy strongly couple usty plasmas in both liqui an crystalline forms in an environment with negligible gravitational effects. Among the interesting results obtaine from these experiments it was seen that a voi, a region which containe plasma but no ust particles, was forme in the centre of the ischarge ue to effects of the ion rag force. There have subsequently been many approaches use to theoretically stuy strongly couple DAWs such as the generalize thermoynamical 0, the

2 2 quasilocalize charge approximation, kinetic 2 an hyroynamical 3 moels. Another moel which has recently been applie to stuy DAWs is the flui approach presente by Gozainos et al. 4. Drawing inspiration from previous space experiments 2,9, they evelope a numerical moel to simulate crystalline usty plasmas uner microgravity conitions. In this paper they formulate an equation of state for this regime given by P N nn 3 Γk BT n + κexp κ 2 where N nn is the number of nearest neighbours that etermine the usty plasma s structure an κ is the lattice parameter, efine as the mean interparticle istance n /3, ivie by the ynamical Debye screening length, λ D such that with κ = 3 n λ D 3 ɛ 0 k B T i T e λ D = e 2 n i T e + n e T i 4 where T s an n s are the temperature an number ensity of species s = e, i, respectively. This moel, although originally evelope for crystalline plasma structures, has recently been applie as an approximation to the equation of state for strongly couple plasmas near to the liqui-crystal phase transition. This has inclue the stuy of Bohm sheaths 5, oublelayer formation 6,7, the linear DAW moe 8, an nonlinear solitary wave structures 9,20. This theory is seen to be in excellent agreement with the experimental observations of linear wave moes, as elegantly emonstrate by Yaroshenko et al., for example Fig. 5 in Ref. 8. By consiering the form of Eqns. an 2, an effective electrostatic temperature was efine such that k B T = N nnz 2 e2 2πɛ 0 3 n + κexp κ, 5 which is typically a few orers of magnitue larger than the ust kinetic temperature. In oing so, they emonstrate that this moel preicts the transition to a thermal moe at high wave numbers for ispersion curves obtaine in previous experimental stuies 2, which the ust kinetic temperature alone was not great enough to explain. Our aim here is to investigate linear an nonlinear ust-acoustic waves in strongly couple usty plasmas. We account for strong coupling between the ust grains by using the moel presente by Gozainos et al. 4, along with the electrostatic temperature approach of Yaroshenko et al. 8. The reuctive perturbation metho is employe to erive both the linear ispersion relation an the Korteweg-e Vries equation for this system. We then investigate how the properties of linear an nonlinear waves vary with respect to initial plasma parameters. We choose these parameters to reflect those typically observe in usty plasma experiments an base them on those presente by Banyopahyay et al. 22. In this paper, the electrons an ions are assume to follow the Maxwellian istribution, such that their ensities are epenent on the local electrostatic potential, which varies with the passing of the wave. The lattice parameter κ, through its epenence on the ynamically varying screening length λ D, is epenent on the ensities of the Maxwellian species an thus also varies with the electrostatic potential. The electrostatic pressure is thus a ynamically varying quantity, epening on both Φ, through the κ term, an n, both explicitly an through κ. In this paper, we therefore emphasise that P P n, Φ 6 an so the electrostatic pressure is foun to vary ue to the perturbations in n an Φ in the vicinity of the wave. In previous stuies in this area 8,20, an approximation was mae such that the electrostatic pressure is only a function of the equilibrium parameters, an so an aim of this article is to show what effects this approximation has on the results preicte by this moel. The ust charge number, Z is a function of the size of the ust particle an of the local plasma conitions. There are various theoretical moels which can be use to estimate the charge number, but here we use an orbit motion limite OML approach 23, which is base on a balance between electron an ion currents onto the ust particles. In a real physical situation, this quantity shoul vary ynamically in the vicinity of the wave, but for simplicity in this investigation, we only consier the equilibrium values in etermining the charge number of the ust, an therefore o not apply perturbation theory to Z. This article is structure as follows. In Section II, the flui equations to moel ust-acoustic waves in a strongly couple plasma are presente, using the equation of state erive by Gozainos et al. 4. This is followe in Section III, by an outline of the reuctive perturbation metho an an introuction to the concept of electrostatic temperature perturbations which are use in this paper. In Section IV, the relevant equations use in this stuy, namely the linear ispersion relation an the KV equation, are erive from the normalise flui moel. In Section V, a parametric investigation is presente for both the linear an solitary wave cases for typical plasma conitions. In Section VI we iscuss the effects of strong coupling an of the electrostatic temperature perturbations on the ynamics of the waves. A summary of the most important results an conclusions of this paper is presente in Section VII.

3 3 II. MODEL EQUATIONS A. System Description We consier linear an nonlinear acoustic waves propagating through an unboune, three-component plasma consisting of ust particles, ions an electrons. The raius, r an the mass, m of the ust particles are assume to be constant. The charge of the ust particles, q = Z e is assume to be constant, but with a value etermine by the equilibrium plasma parameters. Two forces acting on the ust particles are consiere. The first is the electrostatic force which arises from the internally generate electric fiel of the wave, while the secon is from the ust particles electrostatically repelling each other. No gravitational or externally applie electromagnetic fiels are applie to the system. In the linear case, the wave propagates ue to the inertia of the ust particles being balance by the restoring effects of the electron an ion pressures. For nonlinear wave propagation, we consier a solitary wave, which is supporte ue to a balance between nonlinear effects an ispersion. Far from the isturbing effects of the wave, the plasma is consiere to be in an equilibrium state. Here the plasma is macroscopically homogeneous, quasineutral an at rest. In the proximity of the wave, the charge ust particles are perturbe by its generate electrostatic potential. The ions an electrons, which have masses much less than that of the ust particles, are assume to instantaneously reistribute themselves accoring to the Maxwellian istribution. B. Flui Equations We use the flui approach to moel linear an weakly nonlinear ust-acoustic waves in a D strongly couple plasma, in a liqui rather than crystalline state. It woul be reasonable to ask whether the flui equations, typically use for weakly couple an iffuse plasmas are applicable in the strongly couple case. In 996 however, Wang an Bhattacharjee 24 evelope a kinetic theory incluing strong coupling from the Klimontovich equation an the Bogolyubov- Born-Green-Kirkwoo-Yvon BBGKY hierarchy an showe that the Vlasov equation, from which the flui moel may be erive, is still vali in the range Γ Γ cr, where Γ cr is the critical coupling parameter at which crystallization occurs. The number ensity of the ust particles obeys the continuity equation such that n t + x n u = 0, 7 where n an u are the ust number ensity an the ust flui velocity, respectively. The force arising ue to strong coupling between the ust particles is here moelle by an effective electrostatic pressure graient, where we have the pressure, P = n k B T with T efine in Eqn. 5. It was shown in Ref. 6 that the electrostatic temperature T is always a few orers of magnitue higher than the kinetic temperature T of the ust in the region of the crystal-liqui phase transition. In this paper we therefore follow this assumption an so neglect the effects of T on the ynamics of the system. The momentum equation for this system can therefore be written as u m n t + u u x = n Z e Φ x P x. 8 The system is then close by Poisson s equation given by 2 Φ x 2 = e ɛ 0 n i n e Z n. 9 where we have set the ion charge number, Z i = for simplicity. The ensity of the electrons an ions in the proximity of the wave is epenent on their temperature an on the electrostatic potential at that point such that eφ n e = n e0 exp 0 k B T e an n i = n i0 exp eφ. k B T i Far from the effects of the wave, at equilibrium, Poisson s equation gives the quasineutrality conition such that n i0 n e0 Z n 0 = 0. C. Normalisation We now normalise the various quantities in Eqns. 7-9 by the scalings presente in Table I to obtain a imensionless system of equations. The variables n, u, Φ, an T are written in imensionless form as n, u, φ, an, respectively. The normalise space an time quantities are enote by a tile. Eqns. 7-9 thus become an n t + nu = 0, 2 x u n t + u u = n φ x x n x 3 2 φ x 2 n + c φ + c 2 φ 2 + c 3 φ 3, 4 where for Poisson s equation the relation expmφ +mφ+mφ 2 /2+... has been use. The coefficients in Poisson s equation are calculate to be c =, 5

4 4 Parameter Characteristic Scale Temperature T 0 = Z2 n 0T i T e n i0 T e+n e0 T i Length λ D0 = ɛ0 k B T 0 n 0 Z 2 e2 Time ω p = ɛ 0 m n 0 Z 2 e2 Velocity v 0 = kb T 0 m Number ensity n 0 = n 0 Electrostatic Potential Φ 0 = k BT 0 Z e TABLE I: Characteristic scales appearing in this investigation an in general where we have c 2 = µ µθ2 2 + µθ 2, 6 c 3 = µ2 + µθ µθ 3, 7 c j = µj j + µθ j j! + µθ j, 8 µ = n e0 n i0, D. Grain Charging θ = T i T e. 9 We consier the ust particles to have a charge number, Z which epens on the equilibrium conitions of the plasma, an is constant in time throughout the system. We etermine the value of Z by using an orbit motion limite approach 23 an assuming that the electron an ion currents onto the ust particles are such that I e + I i = 0. In ong so, we obtain the following equation for Z [ ] Z = 4πɛ 0r k B T e e 2 W θ µ σ expθ θ, 20 where σ = m e /m i an W enotes the Lambert W function which has recently been applie to the problem of ust charging 25. This function is inclue in programs such as MATLAB an Mathematica an therefore allows a close form representation of the ust charge number, without having to rely on numerical methos. For a erivation of the above equation, see Appenix A. Eqn. 20 gives an expression for the charge number at equilibrium in the plasma. However, as the wave passes, the local electron an ion ensities will change, resulting in a ynamic variation in charge number. This effect is neglecte in this article for simplicity an to facilitate the focus on the variations in the electrostatic temperature in the vicinity of the wave. III. REDUCTIVE PERTURBATION METHOD The reuctive perturbation metho involves the expansion, in terms of a small parameter ɛ, of the ynamical variables of the system about their equilibrium states such that A = A 0 + ɛ j A j, 2 j= where A is the variable to be expane. Here, the normalise quantities to be expane are the ust number ensity n, the ust flui velocity u an the electrostatic potential φ such that an n = + ɛn + ɛ 2 n , 22 u = ɛu + ɛ 2 u φ = ɛφ + ɛ 2 φ From Eqn. 5 it is seen that T, is a function of n an Φ, through the κ term, an thus varies ynamically. For analytical convenience, we introuce the concept of electrostatic temperature perturbations. For etails of the erivation of the following quantities, see Appenix B. By expaning the ynamic variables in Eqn. 5, we acquire a form of the normalise electrostatic temperature, such that = 0 + ɛ + ɛ where the normalise equilibrium electrostatic ust temperature, 0 is expresse as with 0 = T 0 /T 0 26 k B T 0 = N nnz 2 e2 2πɛ 0 3 n 0 + κ 0 exp κ 0, 27 where κ 0 = / 3 n 0 λ D0. The perturbations an 2 are foun to be an = n + 2 φ 28 2 = 2 n φ n n φ + 25 φ 2 29 where we have = 2 = κ 0 + κ κ 0, 30 2 = 22 = 0 c 2 κ κ 0, 3

5 5 an 23 = 0 κ 3 0 3κ 2 0 2κ 0 2, κ 0 24 = 0 3 c κ 2 0κ κ 0 33 By then applying the reuctive perturbation metho as outline in Section III, to lowest orer in ɛ we obtain v n + u = 0, 42 β n v u α φ = 0 43 IV. 25 = 0 2 3c3 c 2 2κ 0 κ κ DERIVATION OF MODEL EQUATIONS A. Linear Dispersion Relation To obtain the linear ispersion relation for this system, the epenent variables in Eqns. 2-4 are expane in a power series of ɛ, as escribe in Section III, with terms higher than ɛ neglecte. By then assuming oscillatory solutions to the perturbe quantities we obtain the following system of equations an ωn + ku = 0, 35 β kn + ωu + α kφ = 0, 36 n + φ + k 2 = 0 37 in which ω an k are the normalise frequency an wavenumber, respectively an we have α = 2, β = By combining the above system of equations, it can be seen that we arrive at a ispersion relation ω 2 = α k 2 + k 2 + β k In the long wavelength limit, Eqn. 39 gives the normalise phase velocity v such that v = ω k = B. KV Equation To obtain the KV equation, in which we balance nonlinearity with ispersion, we first stretch the space an time coorinates in Eqns. 2-4 in the style of Washimi an Taniuti 26 such that ξ = ɛ /2 x v t, τ = ɛ 3/2 t. 4 an n + φ = By integrating Eqns , noting that the perturbations ten to zero at ξ ±, we obtain equations for the ust flui ensity an ust flui velocity perturbations in terms of the electrostatic potential perturbation such that n = φ, u = vφ 45 as well as an expression for the normalise phase velocity which is the same as Eqn. 40. To next lowest orer in ɛ, we obtain φ τ + 2vφ φ v n 2 + u 2 = 0, 46 v φ τ φ φ n 2 v u φ 2 = 0 47 an 2 φ 2 n 2 φ 2 c 2 φ 2 = 0, 48 where the relations in Eqn. 45 have been use to express the first orer perturbe quantities solely in terms of φ. By ifferentiating Eqn. 48 with respect to ξ, an equation for n 2 / can be obtaine, which can then be substitute into Eqn. 47 to get an equation for u 2 /. These expressions for both n 2 / an u 2 / can then be substitute into Eqn. 46 to obtain an equation of the form in which we have φ τ + Ãφ φ + B 3 φ = Ã = + 2v2 + 2αc 2 + 2γ, 50 2v B = α 2v, 5

6 6 where α is efine in Eqn. 38 an γ is efine as γ = Eqn. 49 can be solve by separation of variables, giving a solution of φ ξ, τ = 3Ũ Ã sech2 ξ Ũ τ BŨ where Ũ is the normalise velocity of the nonlinear wave in the moving reference frame. We now transform back to the laboratory frame, in which the solitary wave is travelling with a normalise velocity Ṽ, which is greater than the soun spee in the plasma by an amount δṽ. Taking φ to the first orer of ɛ, we then have [ ] φ x, t = φ m sech χ 2, 54 where φ m = 3δṼ Ã, 4 B = δṽ 55 are the amplitue an with of the solitary wave an χ = x Ṽ t. V. PARAMETRIC INVESTIGATION In this section, we present a parametric investigation base on the equations erive in Section IV. In the previous sections, we have emonstrate that both the linear ispersion relation an the KV equation are epenent on the following equilibrium quantities: n i0, n e0, T i, T e, σ = m e /m i, r an m. To etermine the coupling parameter of the system, so that its phase state may be estimate, the ust temperature T must be specifie. When consiering solitary wave solutions to the KV equation, we must also specify the velocity of the solitary wave relative to the soun spee in the plasma, which in imensional form may be expresse as δv = V v ph. From these nine quantities, the various attributes of the linear an nonlinear wave structures can be erive. Here, we choose to investigate the effects of varying the ensities an temperatures of the Maxwellian species an fix the other quantities. For the purposes of this section, the scalings shown in Table I are inappropriate, since they are epenent on these parameters, so in this section we analyse the imensional form of the equations. Some of the normalisation quantities presente in Table I arise naturally in the algebra however, so the notation of the various imensionless quantities will be retaine in this section. We will be using the parameters shown in Table II for this investigation which are base on those measure by Banyopahyay et al. 22. The value of the Parameter Parameter Range n i m 3 n e m 3 k BT i 0.2 ev k BT e 3 8 ev k BT 0.3 ev r 0.2 µm m 0 3 kg σ = m e/m i δv 2 mm/s TABLE II: Parameters stuie in this investigation ratio σ = m e /m i is for the commonly use argon plasma, but here we set the ion charge number, Z i = for simplicity. We choose N nn = 2, corresponing to an fcc crystal lattice. We choose the range of temperatures an ensities uner investigation such that we expect that the ust particles to be strongly couple, but the plasma is in a flui rather than crystalline state. Vaulina an Khrapak 27 foun that the coupling parameter require for crystallisation is Γ cr = 06 expκ 0 + κ 0 + κ an therefore the equation of state, Eqn. 2 is vali in the region in which we have approximately < Γ < Γ cr, so we restrict our parameters to values such that this inequality hols. Here we have use the equilibrium quantities to etermine the coupling parameter, an therefore we o not take into account the effects of the wave perturbations on the phase state of the system. As escribe in Section II D, the charge of the ust grain is epenent on the temperatures an ensities of the Maxwellian species. To etermine how the coupling parameter varies with these quantities, the ust charge number, Z shoul first be calculate, accoring to Eqn. 20. Fig. shows how the ust charge number, Z is epenent on the parameters use in this investigation, an it is seen to vary between approximately 00 an 3200 for this parameter range. Using Eqns. an 56, it can then be shown that this correspons to values of Γ which fall approximately in the esire range. The lower boun for the velocity of a solitary wave is given by the soun spee in the plasma, an so it is important to consier the linear case, even when investigating nonlinear waves. Restoring imensions to Eqn. 39 we see that the ispersion relation may be written as ω 2 = k BT 0 m αk 2 + λ 2 + βk2, 57 D0k2 which in the limit kλ D0, shows a preicte phase

7 7 FIG. : Colour online The charge number of the ust particles stuie in this investigation, as functions of a the temperatures ev, an b the ensities 0 3 m 3 of the Maxwellian species. In both plots we have m = 0 3 kg, r = 0.2 µm, k BT = 0.3 ev σ = In a we have n i0 = m 3, n e0 = m 3 an in b we have k BT i = 0.3 ev, k BT e = 8 ev. FIG. 2: Colour online The phase velocity, v ph cm/s of the ust-acoustic waves stuie in this investigation, as functions of a the temperatures ev, an b the ensities 0 3 m 3 of the Maxwellian species. In both plots we have m = 0 3 kg, r = 0.2 µm an σ = In a we have n i0 = m 3, n e0 = m 3 an in b we have k BT i = 0.3 ev, k BT e = 8 ev. velocity of v ph = ω k = k B T 0 + T 0 + T T 2 m. 58 In this an the following equations, the terms written T ij are the electrostatic temperature perturbation coefficients in imensional form, corresponing to Eqns Fig. 2 shows how the phase velocity varies with the parameters uner investigation. Restoring imensions to Eqn. 49 an rearranging, we obtain in which we have an Φ τ + AΦ Φ ξ + B 3 Φ = 0, 59 ξ3 A = Z e à 60 m k B T 0 /2 B = ɛ 0k B T 0 3/2 B. 6 Z 2e2 n 0 m /2 This gives a solution in the laboratory frame of Φx, t = Φ m sech 2 χ, 62 which is the imensional form of Eqn. 54. Since Φ m = 3δV/ A an = 4B/δV, we see that the amplitue an with of the solitary waves have two main epenences. The first is a epenence on the velocity of the solitary wave relative to the soun spee in the plasma, δv, such that the amplitue of the solitary wave is irectly proportional, an the with inversely proportional to the square root of its magnitue. This parameter is relate to the strength of the perturbation an we see that, for constant A an B, faster solitary waves will be taller an thinner than slower ones. We also see the well-known relation that for a particular plasma system, the prouct Φ m 2 is inepenent of δv. The secon epenence is on the coefficients of the KV equation, A an B which are unique to the moel being presente. By holing δv constant, we emonstrate in Figs. 3 an 4 the effects that the variation of these coefficients with equilibrium plasma parameters have on the amplitue an with of solitary waves. We emphasise that, although we have hel δv constant, the absolute spee in the laboratory frame of the solitary waves are ifferent, ue to the soun spee also being epenent on these parameters. By noting the range of phase velocities shown in Fig. 2, we see that Figs. 3 an 4 correspon to solitary waves travelling with a Mach number range of M = V/v ph =.04.5.

8 8 FIG. 3: Colour online The amplitue, Φ m mv of the potential perturbations associate with the solitary waves stuie in this investigation, as functions of a the temperatures ev, an b the ensities 0 3 m 3 of the Maxwellian species. In both plots we have m = 0 3 kg, r = 0.2 µm, σ = an δv = 2 mm/s. In a we have n i0 = m 3, n e0 = m 3 an in b we have k BT i = 0.3 ev, k BT e = 8 ev. FIG. 4: Colour online The with, mm of the potential perturbations associate with the solitary waves stuie in this investigation, as functions of a the temperatures ev, an b the ensities 0 3 m 3 of the Maxwellian species. In both plots we have m = 0 3 kg, r = 0.2 µm, σ = an δv = 2 mm/s. In a we have n i0 = m 3, n e0 = m 3 an in b we have k BT i = 0.3 ev, k BT e = 8 ev. Moel α β T T n, φ T T 0 0 T = 0 0 TABLE III: Values of α an β for the three moels uner comparison temperature is set to its equilibrium value such that T T 0, which may be foun for example in Ref. 20. These moels may be recovere from the equations presente in this text in the appropriate limits, which are iscusse in this section. A. Linear ust-acoustic waves VI. THE EFFECTS OF STRONG COUPLING BETWEEN DUST PARTICLES ON DUST-ACOUSTIC WAVES In this section, we present a iscussion on how strong coupling between the ust particles, escribe via the electrostatic temperature moel affects the attributes of the linear an nonlinear ust-acoustic waves. We o this by comparing the moel presente in this paper, which in this section we enote as the T T n, Φ case, with a moel in which we set the electrostatic temperature terms to zero such that T = 0. This moel is well ocumente, an may be foun for example in Ref., although using a scaling which is ifferent from that in this article. In aition to this, we also show the effects of the inclusion of the electrostatic temperature perturbation terms by comparison to a moel in which the electrostatic We start with the linear case, an by inspection of the form of Eqn. 39, we see that the efining terms for each moel are α an β, the values of which are isplaye in Table III. Fig. 5 presents the calculate ispersion relations for the three moels.. Short wavelength limit, kλ D0 As was emonstrate by Yaroshenko et al. 8, a feature of the ispersion relation which is inicative of strong coupling is a transition into a regime similar to that of the thermal moe at high wavenumbers. This inicates that the electrostatic interactions between the ust particles prouce an effect analogous to that of a temperature, an we see this effect clearly in Fig. 5. Here, the term responsible for this effect

9 9 FIG. 5: Colour online The normalise ispersion relation for three ifferent moels of ust-acoustic waves. Here we have n i0 = m 3, n e0 = m 3, k BT i = 0.3 ev, k BT e = 8 ev, m = 0 3 kg, r = 0.2 µm, an σ = FIG. 6: Colour online Normalise solitary wave structures for three ifferent moels. Here we have n i0 = m 3, n e0 = m 3, k BT i = 0.3 ev, k BT e = 8 ev, m = 0 3 kg, r = 0.2 µm, σ = an Ṽ =.2. is β in Eqn. 39. In the weakly couple case, since we o not consier any temperature-like terms, we have β = 0 an so the wave frequency saturates at the ust plasma frequency for high wavenumbers, as can be seen from the soli blue line in Fig. 5. For the strongly couple moels, we have non-zero β terms, so for large wavenumbers, these moels preict a transition to the thermal moe, such that ω β k. Since the value of β is larger for the T T n, Φ moel than the T T 0 case, we see that the inclusion of ust electrostatic temperature perturbations in the moel makes this effect even more pronounce. 2. Long wavelength limit, kλ D0 In the case of low wavenumbers such that kλ D0 Yaroshenko et al. 8 preicte an increase in the phase velocity of the wave ue to the effects of strong coupling. Here, we see that in this limit, v = α + β. For the scaling we have chosen, the T = 0 moel has a phase velocity of. Since the phase velocity for the T T 0 moel is + 0, we see that an effect of incluing strong coupling in the moel is an increase in the phase velocity in the long wavelength limit. In this paper, for the T T n, Φ case, we have a phase velocity of v = For the parameters consiere in Section V, we have > 2 across the entire range, so we see that the electrostatic temperature perturbations further increase the preicte phase velocity. The ifference in phase velocity between the moels is a function of the equilibrium plasma parameters. To get a sense of the magnitue that these ifferences amount to, we prouce graphs similar to Fig. 2 for the two other moels, which are exclue from this work for brevity. In oing so, it may be seen that the approach use in this paper preicts phase spees that are up to approximately 20% greater than the T = 0 moel, an up to approximately 5% greater than the T T 0 moel for the parameter range consiere in Section V. B. Nonlinear ust-acoustic solitary waves We now iscuss the nonlinear case. By consiering the form of Eqn. 55, we see there are two key factors affecting the attributes of the nonlinear solitary waves. The first is the soun spee excess, δṽ of the solitary wave, an the secon is the values of the coefficients in the KV equation, Ã an B. Both of these factors are epenent on the moel use, so we will now iscuss each of them in turn.. Differences in the calculate soun spee excess For a measure solitary wave spee in the laboratory frame Ṽ, each moel will give a ifferent value of δṽ upon which the amplitue an with of the solitary waves are epenent. For example, using the same parameters which were use for the comparison in Fig. 5, the normalise phase spees, v for the ynamically varying electrostatic temperature moel, the constant electrostatic temperature moel an the electrostatically col moel are.4,.0 an, respectively. For a solitary wave moving in the laboratory frame with a normalise velocity of Ṽ =.2, the three moels then give δṽ = Ṽ v of 0.06, 0. an 0.2, respectively. Since the amplitue is irectly proportional to, an the with is inversely proportional to the square root of δṽ, it is seen that this factor contributes to the strongly couple moels preicting shorter, but wier solitary waves. This factor is most important for lower values of Ṽ. To see this, let us consier two separate moels, with phase velocities v an v 2, respectively. One may arbitrarily choose v 2 > v. It then follows that δṽ = Ṽ v > δṽ2. The ratio δṽ/δṽ2 may then be seen to increase monotonically as Ṽ is ecrease an approaches v 2. Thus it follows that the relative ifferences in the preicte amplitues an withs of the solitary waves, arising from the eviations in δṽ

10 0 between the moels, are greatest for lower values of Ṽ. 2. Differences in the KV coefficients We now compare the à an B coefficients in the KV equation for the three moels, which leas to ifferences in the amplitue an with of the preicte solitary waves. Accounting for the ifferences in α an v = α + β between the moels, it may be seen that the KV coefficients for the T = 0 an T T 0 moels are an à = 3 + 2c 2 2 B = 2 63 à = 3 + 2c B = 2 + 0, 64 respectively. Having calculate the values of these coefficients across the parameter range escribe in Section V, we fin that the T T n, Φ moel presente in this paper preicts values of à which are greater in magnitue than those preicte by the T T 0 an T = 0 moels by up to a maximum of approximately 8% an 5%, respectively. We fin that the corresponing ecrease in B reaches approximately 3% an 8%, respectively. The percentage ifference in the coefficients preicte by each moel varies across the equilibrium parameter range, so we just give the maximum values to give a sense of the magnitue of these changes. The increase in the absolute value of the à coefficient contributes to the preiction of lower amplitue solitary waves for the strongly couple moels. The ecrease in B contributes to a narrowing of the solitary waves for the strongly couple moels, which is in contrast to the effect arising from the reuction of the value of δṽ, which leas to a wiening. The net effect on the preicte with of the solitary waves thus epens on which factor is ominant, which is etermine by the specific equilibrium plasma parameters an the solitary wave velocity, Ṽ. Fig. 6 shows solitary waves preicte by the three moels for typical plasma parameters. Here we see that the combination of the reuction of δṽ an the increase of à results in a large ecrease in the amplitue of the solitary waves for the strongly couple moels. The effect of strong coupling on the with is more subtle, but Fig. 6 shows slightly wier solitary waves for the strongly couple moels, which inicates that for these parameters the effect of the reuction in δṽ ominates over the effect of the reuction in B. VII. CONCLUSIONS In this paper we have theoretically investigate both linear an nonlinear ust-acoustic waves in a usty plasma in which the ust particles are strongly couple. We have moelle the effects of strong coupling by utilising the electrostatic temperature approach of Yaroshenko et al. 8. Uniquely, we have consiere perturbations in the electrostatic temperature, which is a function of the ust number ensity an electrostatic potential, in the locality of the wave an then emonstrate how these affect the linear ispersion relation an the Korteweg-e Vries equation. Our main objectives were to investigate how the ynamics of the waves change for a range of equilibrium plasma parameters, to escribe what effects strong coupling has on the waves an to provie an overview of how electrostatic temperature perturbations affect the moel. In summary, the main conclusions of our paper are: a The phase velocity of the ust-acoustic wave, as well as the amplitue an with of solitary waves are seen to vary significantly with the equilibrium plasma parameters. This is shown in Figs. 2, 3 an 4, respectively, where we have isplaye how the variation of ensities an temperatures of the Maxwellian species affect these attributes. b The preicte phase velocity, v is seen to be larger when strong coupling effects are accommoate in the moel via the electrostatic temperature approach. c The amplitue of the preicte nonlinear solitary waves is seen to be reuce when strong coupling is inclue in the moel, ue to an increase in the absolute value of the nonlinear coefficient, à in the KV equation. In aition to this, if we consier a solitary wave travelling with a set velocity in the laboratory frame Ṽ, the increase in phase velocity arising from strong coupling effects will result in a reuction in the preicte soun spee excess, δṽ = Ṽ v compare to the weakly couple moel. This will then result in a further reuction in the amplitue preicte by the strongly couple moels. The with of the preicte nonlinear solitary waves may be seen to either increase or ecrease with the inclusion of strong coupling effects, epening on which of two factors is ominant. The first, contributing to a narrowing of the solitary waves for the parameters uner investigation, is the reuction of the B coefficient in the KV equation. The secon factor, contributing to a wiening of the solitary waves, is the reuction in δṽ brought about by the increase phase velocity in the strongly couple moel. Which factor ominates is etermine by the specific equilibrium plasma parameters an the solitary wave velocity, Ṽ. e The inclusion of the electrostatic temperature perturbations in the moel is seen to amplify the ef-

11 fects of strong coupling mentione in points b-. We have seen that the ifference between the inclusion or exclusion of the perturbations of the electrostatic temperature affects the value of the linear phase velocity by only a small amount the magnitue of this ifference epens on the specific equilibrium plasma conitions, but is seen to be at most 5% for the parameter range uner investigation. For solitary waves however, since the soun spee excess will be change, this seemingly small moification can result in large ifferences in the amplitue an with between the two moels when combine with the changes in the KV coefficients. We show therefore that the consieration of the ynamic variation of the electrostatic temperature can provie important moifications to more simple moels. The ust charge number Z, which in this case was erive using an orbit motion limite approach, is foun to vary significantly with the equilibrium plasma parameters an this has a substantial effect on the ynamics of the wave moes. For a more complete unerstaning of the system, perturbations of the ust charge number may also show important aitions to the moel, since Z is actually epenent on the ynamically varying electron an ion ensities. Acknowlegements The work of I. Kourakis an S. Sultana was supporte by a UK EPSRC Science an Innovation awar to the Centre for Plasma Physics, Queen s University Belfast Grant No EP/D06337X/. The work of M. A. Hellberg is supporte in part by the National Research Founation of South Africa NRF. Any opinion, finings, an conclusions or recommenations expresse in this material are those of the authors an therefore the NRF oes not accept any liability in regar thereto. P. K. Shukla an A. A. Mamun, Introuction to Dusty Plasma Physics Institute of Physics, Bristol, G. E. Morfill, H. M. Thomas, B. M. Annaratone, A. V. Ivlev, R. A. Quinn, A. P. Nefeov, an V. E. Fortov, AIP Conf. Proc. 649, N. N. Rao, P. K. Shukla an M. Y. Yu, Planet. Space. Sci. 38, A. Barkan, R. L. Merlino an N. D Angelo, Phys. Plasmas 2, H. Ikezi, Phys. Fluis 29, J. H. Chu, an Lin I, Phys. Rev. Lett. 72, H. Thomas, G. E. Morfill, an V. Demmel, Phys. Rev. Lett. 73, Y. Hayashi, an K. Tachibana, Jpn. J. Appl. Phys. 33, L G. E. Morfill, H. M. Thomas, U. Konopka, H. Rothermel, M. Zuzie, A. Ivlev, an J. Goree, Phys. Rev. Lett. 83, X. Wang, an A. Bhattacharjee, Phys. Plasmas 4, M. Rosenberg, an G. Kalman, Phys. Rev. E 56, M. S. Murillo, Phys. Plasmas 5, P. K. Kaw, Phys. Plasmas 8, G. Gozainos, A. V. Ivlev, an J. P. Boeuf, New J. Phys. 5, V. V. Yaroshenko, F. Verheest, H. M. Thomas, an G. E. Morfill, New J. Phys., V. V. Yaroshenko, V. Nosenko, M. A. Hellberg, F. Verheest, H. M. Thomas, an G. E. Morfill, New J. Phys. 2, V. V. Yaroshenko, M. H. Thoma, H. M. Thomas, an G. E. Morfill, IEEE Trans. Plasma Sci. 38, V. V. Yaroshenko, V. Nosenko, an G. E. Morfill, Phys. Plasmas 7, A. A. Mamun an P. K. Shukla, Europhys. Lett. 87, A. A. Mamun, K. S. Ashrafi, an P. K. Shukla, Phys. Rev. E 82, V. Nosenko, S. K. Zhanov, S. -H. Kim, J. Heinrich, R. L. Merlino, an G. E. Morfill, Europhys. Lett. 88, P. Banyopahyay, G. Prasa, A. Sen, an P. K. Kaw, Phys. Rev. Lett. 0, J. Goree, Plasma Sources Sci. Technol. 3, X. Wang an A. Bhattacharjee, Phys. Plasmas 3, A. E. Dubinov an I. D. Dubinova J. Plasma Phys. 7, H. Washimi an T. Taniuti, Phys. Rev. Lett. 7, O. S. Vaulina an S. A. Khrapak J. Theor. Exp. Phys. 90, Appenix A: The calculation of the ust charge number via the Lambert W function Accoring to OML theory, the electron an ion currents onto the ust particle may be written as eφ I e = I e0 exp, A k B T e with I i = I i0 eφ, A2 k B T i I s0 = 8πr 2 n s0 q s kb T s m s A3 for species species s = e, i. Here Φ is the ifference between the grain an plasma potentials. Setting I i +

12 2 I e = 0 gives µ σθ exp eφ + eφ = 0 A4 k B T e k B T i where we have efine σ = m e /m i, µ = n e0 /n i0, θ = T i /T e. Multiplying through by θ exp θ eφ /k B T e an rearranging then gives θ eφ exp θ eφ θ = µ expθ. A5 k B T e k B T e σ This equation is of the form X expx = fµ, θ, the solution to which may be written as X = W fµ, θ where W enotes the Lambert W function. This equation may thus be expresse as θ eφ k B T e = W θ µ σ expθ. A6 By then substituting Φ = Z e/4πɛ 0 r into this equation an rearranging, we obtain our expression for the ust charge number, Eqn. 20. Appenix B: Derivation of electrostatic temperature coefficients In this Appenix, we outline a erivation of the electrostatic temperature coefficients which were presente in Section III. The electrostatic temperature epens on the local screening length in the plasma via the lattice parameter κ, an we use this as the starting point. Looking at the normalise Debye length, we see that /2 λ D ni0 T e + n e0 T i = λ D0 n i T e + n e T i = + µθ exp eφ k B T i + µθexp eφ k B T e /2.B We may write the electrostatic potential as Φ = φφ 0, where the normalization quantity Φ 0 = k B T 0 /e = [ µ/ + µθ]k B T i /e. The enominator of the last term in Eqn. B may then be written as µ µθ exp + µθ φ + µθ exp + µθ φ. B2 By expressing φ in a power series of ɛ an Taylor expaning, Eqn. B then simplifies to λ D λ D0 = /2 j=0 j + c. B3 j+φ j where c j refers to the j th coefficient in Poission s equation, as efine in Section II C. To erive the KV equation we only require up to the secon orer, so we take the lattice parameter, κ to be κ = λ D n /3 = κ 0 + 2c2 φ + 3c 3 φ 2 /2 n /3 B4 By expressing the ynamic variables in series of ɛ an Taylor expaning, we arrive at an equation for the perturbations in the lattice parameter, such that with an κ = κ 0 + ɛκ + ɛ 2 κ 2 κ = 3 n + c 2 φ B5 B6 κ 2 = 3 n 2 + c 2 φ n2 c 2 3 n φ + 3c 3 c 2 2 φ 2 2. B7 This equation for κ can now be substitute into Eqn. 5 to obtain the electrostatic temperature perturbation terms. The calculation can be ivie up into three main components, with n /3 n /3 0 + ɛn + ɛ 2 n 2 /3 n /3 0 + ɛ n 3 + ɛ2 9 3n 2 n 2 κ + κ + κ 0 + ɛ + ɛ 2 κ 2 + κ 0 + κ 0 an, B8 B9 exp κ exp κ 0 ɛκ ɛ 2 κ 2 = exp κ 0 exp ɛκ ɛ 2 κ 2 exp κ 0 ɛκ + ɛ 2 κ2 2κ 2 B0. 2 The prouct of these three equations, along with the constants appearing at the beginning of Eqn. 5, gives, after normalisation by T 0, the electrostatic temperature perturbations that are presente in Section III.