Screened Coulomb potential in a flowing magnetized plasma
Jan-Philip Joost, Patrick Ludwig, Hanno Kählert, Christopher Arran, Michael Bonitz
SScreened Coulomb potential in a flowing magnetizedplasma
J-P Joost , P Ludwig , H Kählert , C Arran , and M Bonitz Institut für Theoretische Physik und Astrophysik, Christian-Albrechts-Universitätzu Kiel, Leibnizstr. 15, 24118 Kiel, Germany Emmanuel College, Cambridge, UKE-mail: [email protected]
Abstract.
The electrostatic potential of a moving dust grain in a complex plasmawith magnetized ions is computed using linear response theory, thereby extending ourprevious work for unmagnetized plasmas [P. Ludwig et al. , New J. Phys. , 053016(2012)]. In addition to the magnetic field, our approach accounts for a finite iontemperature as well as ion-neutral collisions. Our recently introduced code Kielstream is used for an efficient calculation of the dust potential. Increasing the magnetizationof the ions, we find that the shape of the potential crucially depends on the Machnumber M . In the regime of subsonic ion flow ( M < ), a strong magnetization givesrise to a potential distribution that is qualitatively different from the unmagnetizedlimit, while for M > the magnetic field effectively suppresses the plasma wakefield. a r X i v : . [ phy s i c s . p l a s m - ph ] J u l creened Coulomb potential in a flowing magnetized plasma
1. Introduction
Plasma wakes are a fascinating collective phenomenon, which can give rise to attractionof like charged particles in multi-component plasmas. In complex plasmas [1], dynamicalscreening and wake effects have been investigated in a large number of studies, includingexperimental [2, 3, 4, 5] as well as theoretical [6, 7, 8, 9, 10] work. ‡ Computationalapproaches include first-principle molecular dynamics simulations [13, 14, 15], fluidcodes [16, 17] and particle-in-cell (PIC) simulations, e.g. [18, 19, 20, 21].With the recent availability of superconducting magnets in several laboratories [22,23], the focus has shifted towards plasmas where the effect of strongly magnetized ions onthe dynamically screened dust potential can be studied in detail [24, 25]. Theoretically,the screening of a test charge in a magnetized plasma has been studied in variouspublications [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], but manydetails remain unclear. These studies typically found an oscillatory wake pattern, asin the unmagnetized case, but with a magnetic field-dependent oscillation period andamplitude. However, the predicted trends of an increasing magnetic induction on theamplitude and wave length are contradictory. For an increasing external magnetic fieldparallel to the ion flow direction, amplification [26, 27, 32, 34, 35, 36, 40, 41] as wellas damping [28, 31, 33, 35] of the wake oscillations has been reported. Moreover, theions were mostly treated within a fluid approach and, consequently, kinetic effects wereneglected. For typical experimental parameters ( T e /T i ≈ ), however, their influencecan be substantial. In particular, it has been shown [42, 43, 9] that Landau damping cansignificantly damp the wake oscillations in unmagnetized plasmas. Therefore, similareffects are expected for magnetized plasmas as well. Moreover, there are additionalcontributions to the dielectric function related to ion Bernstein modes [44], whichpropagate perpendicular to the magnetic field [45, 46].The main goal of this paper is a systematic description of the topology of thewake structure in real space over broad parameter ranges: (i) from the subsonic to thesupersonic regime of ion streaming, and (ii) weak to strong ion magnetization wherethe field is aligned with the flow. The calculations are based on a recently developedhigh performance linear response code that allows for an effective computation of thepotential on very large grids [47].
2. Dielectric function approach
The linear response of a partially ionized plasma to an external perturbation, suchas a moving dust particle with charge Q , can be calculated from the longitudinaldielectric function (DF) [42, 48], ε ( k , ω ) = 1 + χ e + χ i , which contains contributionsfrom the electrons and ions (susceptibilities χ e,i ). The neutral gas does not enter theDF directly but can modify the ion response considerably due to ion-neutral collisions ‡ For a more extensive list of earlier work see [11, 12]. creened Coulomb potential in a flowing magnetized plasma ˜ ν i . The real space potential at the point R = r − u d t is given by [48] Φ( R ) = (cid:90) d k π Qk e i k · R ε ( k , k · u d ) . (1)Equation (1) can account for streaming ions (via the ion velocity distribution) as well asmoving dust particles (via the argument ω = k · u d of the DF, where u d is the velocityof the dust particle).A kinetic study of waves in a magnetized Maxwellian plasma was first conductedby Bernstein [44]. The derivation of the associated dielectric function can be found inclassical textbooks [49, 50]. As in our previous work [9], we use a BGK collision termto account for ion-neutral collisions (collision frequency ˜ ν i ). The ion susceptibility canthen be written as [49] χ i ( k , ω ) = 1 k λ Di (cid:80) ∞ n = −∞ ω + i ˜ ν i ω + i ˜ ν i − nω ci e − η i I n ( η i ) ζ i,n Z ( ζ i,n )1 + (cid:80) ∞ n = −∞ i ˜ ν i ω + i ˜ ν i − nω ci e − η i I n ( η i ) ζ i,n Z ( ζ i,n ) , (2)where I n ( η i ) is the modified Bessel function of the first kind, Z ( ζ i,n ) the plasmadispersion function [51], η i = k ⊥ v th,i /ω ci , and ζ i,n = ( ω + i ˜ ν i − nω ci ) / ( √ | k z | v th,i ) § . Theion thermal velocity and the ion cyclotron frequency are given by v th,i = ( k B T i /m i ) / and ω ci = q i B/m i , respectively. The magnetic field is chosen parallel to the ionstreaming direction. The electron response is treated in the static approximation, i.e., χ e = ( kλ De ) − , where λ De = (cid:112) ε k B T e / ( n e q e ) is the electron Debye length.In the sheath region of an rf discharge, the ions stream past the dust grain with amean velocity u i , and the latter can be considered at rest, u d = 0 . On the other hand,in the rest frame of the ions, where u (cid:48) i = 0 , the dust particles move with a velocity u (cid:48) d = − u i . Thus, it appears that we can equally evaluate Eq. (1) in the rest frame ofthe ions, where the ion susceptibility [Eq. (2)] is well known. However, this apparentsymmetry is broken by the presence of the neutral gas, see Ref. [52]. Equation (2) is onlyvalid for ions in thermal equilibrium. It does not account for the fact that the ions arebeing accelerated by the sheath electric field, which causes (i) a net drift of the ions withrespect to the gas and (ii) deviations of the ion distribution function from a Maxwellian.Nevertheless, the calculation outlined above may serve as a starting point to explore theeffect of a magnetic field on the screening potential within a kinetic framework.
3. Numerical implementation
The computation of the dynamically screened ion (wake) potential in real space isbased on a numerical three-dimensional discrete Fourier transformation (3D DFT) on arelatively large grid with resolutions from × × up to × × . (cid:107) In order to handle 3D grids of this size, the previously introduced high performancelinear response program
Kielstream is used [47].
Kielstream was developed in
C++ § We note that on page 133 in [49] the modulus of the wave number is missing. (cid:107)
In order to avoid pseudo-periodical effects, the range in real space must be chosen in all dimensionswith proper size, whereby more grid points are required in the streaming direction. creened Coulomb potential in a flowing magnetized plasma R e l a t i v ee rr o r o f ε ( k , ω ) Number of summands N β = 0 . β = 0 . β = 0 . β = 1 Figure 1.
Relative error in computing the real part of the dielectric function ε ( k , ω ) = 1 + χ e + χ i depending on the total number of evaluated summands of theinfinite sum occurring in equation (2). Approaching weak magnetization, for eachgrid point in k-space several hundreds of terms are required to ensure convergence.Parameters: k = k z = k ⊥ = 50 λ − De and M = 1 , T e /T i = 100 , ν i = ˜ ν i /ω p = 0 . .The magnetization is described by the ratio of the ion cyclotron and the ion plasmafrequency, β = ω ci /ω p , see section 4. to calculate the screened plasma potential for the unmagnetized case. The program isoptimized for memory efficiency and achieves high performance by parallelization usingthe openMP library and by exploiting the radial symmetry of the problem. In addition ituses the libcerf library [51, 53] to reliably evaluate the plasma dispersion function andthe fftw3 library [54] to efficiently perform the Fourier transformation. The modifiedBessel function is evaluated using the GNU Scientific Library (GSL) [55].For the calculation of the screened potential we have to carry out two steps:(i) the population of the grid and (ii) the execution of the 3D DFT. Modificationsto
Kielstream in the context of this work are related to the first part, as the dielectricfunction for the magnetized ions, Equation (2), has to be implemented. Compared to theunmagnetized case, the key issue, from the numerical point of view, is the appearance ofthe infinite sum involving expensive multiple evaluations of the Bessel and the complexplasma dispersion functions for every specific value of k . The number of elements thatmust be summed up to ensure convergence of the real and imaginary part cruciallydepends on the magnetization β = ω ci /ω p [49]. ¶ Especially for small magnetization, β → , the sum converges very slowly. Then the complicated special functions of thesummand have to be evaluated up to several hundred times for every point on the 3D gridwhich greatly increases the complexity of the problem compared to the unmagnetizedcase, see figure 1. We note that the plasma dispersion function as well as the Bessel ¶ Note that one has to ensure the convergence of the infinite sums in equation (2) for every point onthe 3D grid since the convergence explicitly depends also on k . creened Coulomb potential in a flowing magnetized plasma k ⊥ while on the other hand the Bessel function does not depend on k z .Using large tables for the required number of summands at the characteristic k -points,the numerical effort of evaluating the special functions and therefore the time for thepopulation of the grid can be substantially reduced. Without optimizations the timeused for the population of the 3D grid dominates the overall computation time.
4. Results
The plasma wakefield depends on four dimensionless parameters: the Mach number M = u d /c s , the electron-to-ion temperature ratio T e /T i , the ion-neutral collisionfrequency ν i = ˜ ν i /ω p , as well as the magnetization of ions β = ω ci /ω p , with the plasmafrequency ω p = (cid:112) n i q i / ( ε m i ) . Here, the Mach number is defined as the ratio of theion streaming velocity u i and the ion sound speed, c s = (cid:112) k B T e /m i . Without loss ofgenerality, we consider a grain charge of Q d = − e and an electron Debye length of λ De = 0 . mm. Due to the linear response ansatz, the potential can be simply rescaledto any other grain charge of interest.In the following, we consider a fixed temperature ratio of T e /T i = 100 givingrise to pronounced wakefields. Furthermore, this value is often considered in PICsimulations [9, 20]. Two values of the ion-neutral collision frequency are studied:(i) ν i = 0 . , which applies to the collisionless case + , and (ii) ν i = 0 . , which isrepresentative for typical experimental setups. The Mach number is varied in the range M = 0 . . . . . taking into account that for very small ion streaming velocities thelinear response approach may not be applicable due to strong dust-plasma interactions.Our main interest is the dependence of the wake potential on the magnetization of theion plasma background. Therefore, a broad range of magnetic inductions β = 0 . . . isconsidered. The ion Larmor radius of gyration r L = v th,i /ω c is in units of the electronDebye length given by r L /λ De = β − (cid:112) T i /T e , e.g., r L /λ De = 1 , . , . for β = 0 . , , ,respectively. In figure 2, we present a contour plot of the dynamically screened dust potential. Notethat in the presence of streaming ions the potential has a three-dimensional conicalshape. Due to the cylindrical symmetry of the potential, the plot uses cylindricalcoordinates with the z axis being aligned with the ion streaming velocity and themagnetic field. As a reference, let us first consider the unmagnetized case, β = 0 (top row), see also Ref. [9]. Even for M = 0 . , strong deviations from the isotropicYukawa potential are apparent, for both the collisionless plasma, in the lower panel, and + We note, that some finite damping is required for numerical reasons, in order to avoid convergenceissues giving rise to pseudo-periodical effects.[12] creened Coulomb potential in a flowing magnetized plasma Figure 2.
Subsonic regime: Contour plot of plasma potential Φ( r ) , see Eq. (1), forthree different values of ion streaming velocities (from left to right: M = 0 . , . and M = 0 . ) and five different levels of the external magnetic field (increasing from topto bottom). The upper half of each panel shows the case of finite damping ( ν = 0 . )while the lower one corresponds to the (almost) collisionless case ( ν = 0 . ), wherethe plasma oscillation are only weakly damped. The dust grain is located at the origin.The ions are streaming from left to right. Yellow/red to white (blue to black) colourscorrespond to positive (negative) potential values. Equipotential lines are shown for − mV (blue), mV (dark green) and mV (orange), respectively. creened Coulomb potential in a flowing magnetized plasma ν i = 0 . ) is included, in the upper panel. In the direct vicinityof the grain, there is a strong Yukawa-like repulsive potential region. In the streamingdirection, we find an attractive potential part right behind the grain which attracts othernegatively charged grains downstream. Increasing the Mach number M , the potentialpeaks on the symmetry axis are shifted away from the grain, and the range of thewakefield increases. On the other hand, the angle of the conic wavefronts decreases. For M = 0 . , we find that the potential peaks break away from the centre axis at largedistances, z (cid:38) λ De . Generally, the extension of the wakefield increases with M .Considering now a finite magnetization, β = 0 . , we find several qualitative andquantitative differences compared to the unmagnetized limit, cf. Fig. 2 (lower rows).First, screening in streaming direction becomes stronger. This effect is even morepronounced for the collisionless case. In particular, the amplitude of the wakefieldis significantly reduced. This is clearly visible, especially in the long tail of the plasmawake. Second, the equipotential lines become strongly bent. This effect becomes moreand more pronounced for larger values of β . Independently of M , additional potentialpeaks appear off the z-axis, as seen for β = 1 , while the wake oscillations near thez-axis–and hence the attraction of downstream particles–are strongly reduced. Third,screening becomes weaker with increase of β in the upstream direction.Approaching very strong magnetization, β = 10 , the 3D wake structure createsnested half shells around the grain. That is, compared to the unmagnetized case, thedirection of the cone-structure is reversed. ∗ This effect is, however, strongly reduced inthe collisional case. We note that keeping only the n = 0 term in the ion susceptibility,Eq. (2), which corresponds to the limit β → ∞ (see also figure 1), we find that thepotential pattern changes only marginally compared to β = 10 .In order to point out the unrestricted effect of magnetization, the collision frequencyis set to ν i = 0 . in the following, i.e., the (almost) collisionless case is considered.The potential profile along the z-axis for M = 0 . is shown in figure 3. As mentionedabove, there is a clear trend that screening decreases with increasing magnetizationin the upstream direction. While the potential in this direction is purely monotonic,there are strong oscillations in the potential downstream of the grain. In particular,the position of the first peak is found to be almost independent of β [27]. However,in the limiting case of large magnetization, the peak is shifted away from the grain,which can be attributed to the fact that the off-axis extrema reconnect, see figure 2.Another notable result is that for an intermediate value of β = 1 . , there is no (negative)potential minimum in the flow direction at all. Minima are observed only off the z -axis.This anomaly, however, disappears at a higher ion streaming velocity, M = 0 . ,see figure 4. It is immediately clear that the wake structure is much more extendedsince the wavelength of the wake oscillations increase with M . Typically, the peaks areslightly shifted towards the grain as β is increased [40]. Again, a strong deviation fromthis trend is observed for the limiting case of large β , where the first potential peak is ∗ The formation of similar wake pattern behind the grain in the subsonic regime has also been describedin Ref.[34]; however, for a magnetic field of β ≈ . . creened Coulomb potential in a flowing magnetized plasma -40-200204060 -2 0 2 4 Φ / m V z/λ D e β = 0 β = 0 . β = 1 β = 1 . β = 10 Yukawa M = 0 . Figure 3.
Potential cuts through the grain ( r = 0 ) along the flow direction for differentmagnetic inductions β in the subsonic regime M = 0 . ( ν = 0 . , T e /T i = 100 ).Also shown is the Yukawa potential for the corresponding static case (black solid line, M = 0 ). Cf. figures 4, 6, and 7. -40-200204060 -4 -2 0 2 4 6 8 10 12 14 Φ / m V z/λ D e β = 0 β = 0 . β = 1 β = 1 . β = 10 Yukawa M = 0 . Figure 4.
Same as figure 3 but for M = 0 . . Note the different scaling of the z -axis. much broader and located at about twice the distance from the grain compared to theunmagnetized limit.In contrast to earlier predictions for subsonic ion flow [26, 27, 32, 34, 40], theamplitude of the oscillatory wake potential generally decreases with increasing magneticinduction. (cid:93) (cid:93) We note a deviation from this general trend for M = 0 . in the limiting case β → ∞ ; while theamplitude of the trailing peak is lower than in unmagnetized case, it is significantly larger than forintermediate values of β due to the reconnection of lateral extrema. creened Coulomb potential in a flowing magnetized plasma Figure 5.
Contour plot of plasma potential Φ( r ) for subsonic ( M = 0 . ), sonic M = 1 . , and supersonic ( M = 1 . ) ion drift (from left to right) given for five levels ofthe external magnetic field (increasing from top to bottom). For further settings seefigure 2. So far, only the subsonic regime has been discussed. Figure 5 pictures a much broaderrange of M values including M = 1 , and, as a representative supersonic streamingvelocity, M = 1 . . The subsonic case M = 0 . is shown for the sake of completenessand was already discussed in the context of figure 2.At ion sound speed, corresponding to M = 1 , and β = 0 the trend discussed beforecontinues, i.e, the wake structure extends further since the ions carry four times morekinetic energy than for M = 0 . , and their trajectories are far less affected by the grain. creened Coulomb potential in a flowing magnetized plasma -40-200204060 -4 -2 0 2 4 6 8 10 12 14 16 18 20 Φ / m V z/λ D e β = 0 β = 0 . β = 1 β = 1 . β = 10 Yukawa M = 1 Figure 6.
Same as figure 3 but for sonic ion drift M = 1 . At finite magnetization, β = 0 . , the effect of wave fronts being bent around the grainis no longer observed. Instead, at large distances from the grain, an irregular wakestructure appears due to the disturbances by the external magnetic field. Increasing β further suppresses the wake structure of the potential. Compared to the unmagnetizedlimit, the lateral extension of the (negative) potential minima becomes substantiallycompressed. Furthermore, a symmetry breaking between positive and negative extremabecomes apparent.At β = 1 . , a positive contour develops that comprises several oscillations of the ionwake. Approaching very strong magnetization, β = 10 , the wake oscillations completelydisappear, and only a single large positive potential area persists. While, in general,the topology of the wake structure does not depend on the strength of the collisionaldamping ν i , the position of this positive ion focus does crucially depend on ν i . Also,we note that in the direct vicinity of the grain, the (almost isotropic) screening of theCoulomb potential is very weak compared to the other cases considered so far. Thewake structure as a whole appears as an (asymmetric) dipole-like entity.Finally, let us consider the supersonic wake structure at M = 1 . . In theunmagnetized case, the angle of the Mach cone is further reduced. With anincreasing external magnetic field, the oscillating wake structure becomes more andmore compressed around the centre axis. The symmetry breaking between positiveand negative extrema and the development of the enveloping structure as observed for M = 1 . is, however, not apparent in the supersonic regime. Instead of a dipole-likewake structure, one finds, in the limiting case of large values of β , an isotropic grainpotential without significant wake oscillations [33, 34].The potential profile for sonic ion drift velocity, M = 1 , and r = 0 is givenin figure 6. In the upstream direction, screening is being reduced with increasingmagnetization—similar to the subsonic regime. As β is increased, the positions of the creened Coulomb potential in a flowing magnetized plasma -40-200204060 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 Φ / m V z/λ D e β = 0 β = 0 . β = 1 β = 1 . β = 10 Yukawa M = 1 . Figure 7.
Same as figure 3 but for supersonic ion drift M = 1 . . Note that the wigglesin the range (1 − λ De at β = 10 are not a numerical artifact. trailing peaks shift towards the grain. In agreement with [28, 31, 33], the amplitudeof the trailing peak is found to be strongly damped compared to the unmagnetizedcase (see Fig. 8). Most interestingly, however, the exceptionally slow decay of the peakamplitudes for β = 1 leads to far reaching dust-dust interactions. As mentioned above,for β = 10 , there exists an almost isotropically screened potential structure in the directvicinity of the grain and only a single, far-reaching positively charged ion focus regiondownstream of the grain.Considering the supersonic case M = 1 . , figure 7, screening upstream of the grainis mostly independent of β , in contrast to the previously addressed cases. As observedfor M = 0 . and . , the trailing peak shifts in the direction of the grain decreasing inamplitude. Again, the strongest peak amplitudes are observed for β = 1 . The topology of the wake structure is essentially characterized by the position and theamplitude of the wakefield extrema along the z -axis. As seen in the contour plots,figure 2, the amplitude of the off-axis extrema (in particular in the subsonic regime for β ≥ ) is considerably lower than the dominant peak on the z-axis directly behind thegrain. While the primary peaks have amplitudes far above mV, the off-centre peaksare well below mV. Therefore, we restrict the following considerations to the on-axiswake extrema.As discussed before, with increasing M , the wake structure becomes more andmore elongated, i.e., the distance of the individual peaks from the grain increases. Thisfinding is well observed for any specific magnetization β , see figure 8. We note thatin the unmagnetized case, β = 0 , the peak positions are equidistant [9], while at finitemagnetization deviations are observed. In turn, considering a fixed streaming velocity creened Coulomb potential in a flowing magnetized plasma p e a k p o s i t i o n / λ D e p e a k h e i g h t / m V β β β β st peak2 nd peak3 rd peak4 th peak5 th peak M = 0 . M = 0 . M = 1 M = 1 . M = 0 . M = 0 . M = 1 st peak2 nd peak3 rd peak M = 1 . Figure 8.
Peak positions (top panels) and peak amplitudes (lower panels) of thetrailing peaks along the z -axis as function of β for four different drift velocities M = 0 . , . , . , . . Solid (red) lines indicate positive potential maxima, whilecrosses on dashed (blue) lines mark negative potential peaks. In total, for each of thefour considered Mach numbers, results for β -values, indicated by the symbols, havebeen evaluated on an up to × × element sized grid. M , an increasing magnetization β typically leads to a shift of the peak positions towardsthe grain. This observation is more pronounced for larger values of M .However, an anomaly to this general trend is found for the third extremum (thesecond positive maximum) for M > . around β ≈ . . The reason for this unusualbehaviour lies in the fact that more distant peaks from the grain exhibit a larger gradientwith respect to β , see the peak positions in the upper panel of figure 8. This means thatpeaks may approach each other and finally merge, as can be seen for the representativeexample M = 1 . The third positive maximum approaches the second one creating acommon plateau at β = 0 . (not shown), while the second minimum vanishes. Finally,the original second and third positive peaks overlap, creating an new broad wave crestwith a particularly large peak amplitude at β = 0 . .The lower panel of figure 8 shows the peak height as a function of β . Of highestrelevance is the first trailing peak, where distinct peak heights (that may allow for creened Coulomb potential in a flowing magnetized plasma -10-5051010 8 6 4 2 0 2 4 6 8 10 Φ / m V r/λ D e β = 0 β = 0 . β = 1 β = 1 . β = 10 Yukawa M = 0 . Figure 9.
Potential cuts through the grain ( z = 0 ) perpendicular to the ion flowdirection for different magnetic inductions β at M = 0 . ( ν = 0 . , T e /T i = 100 ).The Yukawa potential is shown for the corresponding static case M = 0 (black solidline). particle attraction) are observed in the subsonic regime. In particular, for the consideredparameters, the strongest peak is found for β = 0 and M = 0 . . As a general rule,above this value of M the amplitude of the first peak decreases at constant β withincreasing M . Exceptions are found for strong magnetization β > and M (cid:29) .Considering the functional dependence on β at constant drift velocity, the first peaktypically reduces with increasing magnetization monotonically. Higher order extremashow, however, a non-monotonic behaviour, where the third peak (second maximum)exhibits a minimum around β = 0 . . Interestingly, for M = 0 . and β > . , thesecond maximum attains a larger potential height than the first (primary) peak. Asimilar effect is found for M = 1 . , . and β ≈ . , see also the on-axis potentialprofiles in figures 4,6, and 7. So far, we have discussed the potential profile along the streaming direction only.However, the question of wakefield oscillations in the perpendicular direction has alsobeen under debate, see e.g, [2, 9, 33, 34]. In figure 9, we consider the effect ofmagnetization on the lateral potential profile for M = 0 . . Without a magnetic field, β = 0 , a minor positive potential area is observed radially surrounding the grain. Underthe influence of a finite magnetic field, the equipotential lines are bent around the grain,see figure 2. This leads to strong oscillations of the potential radially outwards from thegrain [34]. These oscillations become stronger with increasing magnetization, but evenwhen approaching very strong magnetization the peak amplitude is well below mV.At a slightly higher Mach number M ≥ , figure 10, a completely different picture creened Coulomb potential in a flowing magnetized plasma -10-5051010 8 6 4 2 0 2 4 6 8 10 Φ / m V r/λ D e β = 0 β = 0 . β = 1 β = 1 . β = 10 Yukawa M = 1 Figure 10.
Same as figure 9 but for sonic ion drift M = 1 . is observed [34]: The lateral wake oscillations completely vanish and the potential decaybecomes strictly monotonic [33]. In the vicinity of the grain r/λ De < , the effectiveshielding length perpendicular to the magnetic field is found to gradually increase withthe magnetic induction.
5. Conclusion
We have presented a detailed analysis of the electrostatic potential of a charged dustgrain in the presence of a strong magnetic field in direction of the ion flow. Our analysisis based on the kinetic theory result of the ion dielectric function with collisions includedvia a BGK collision term.Our main focus was directed on the behavior of the dynamically screened grainpotential when M and β are varied. It was shown that the effect of the magneticfield on the oscillatory wake structure strongly depends on the Mach number M . In theregime of subsonic ion flow, M < , with increasing magnetization the equipotential linesare bent around the grain with additional potential peaks appearing off the center axisand the amplitude of the wake maxima on the center axis being substantially reduced.For supersonic ion flow velocities, M > , the magnetic field radially compresses theplasma wakefield to the center axis, which completely vanishes in the limit of strongmagnetization.In recent experiments strong ion magnetization parallel to the direction of the ionflow could be achieved [24, 25]. For two vertically aligned grains, a strong influence ofthe magnetic field on the ion-wake-mediated particle interaction was observed when themagnetization of the ions exceeds β ≥ . (see Fig. 5(b) in [24]). Analyzing the verticalcoupling of the particle pair, a continuous reduction of the vertical grain attraction, i.e.damping of the ion focus, with an increasing magnetic field has been reported, which is creened Coulomb potential in a flowing magnetized plasma Acknowledgments
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