Oblique magnetic field influence on the wakefield in complex plasmas

The wake features around a grain were explored with coptic particle-in-cell simulations [28]. The simulated system consists of a stationary charged grain in the presence of streaming ions under the influence of an external magnetic field. Besides providing comparatively accurate solutions for even non-linear regime, the importance of the code lies in the fact that it can be customized to resolve the particle in the near neighborhood of grain by orders of magnitude than elsewhere by imposing a non-uniform grid in the vicinity of the grain. The simulation set-up is similar to the one considered in the recent works [6, 11, 23, 29] except that we have a magnetic field component perpendicular as well as parallel (few cases) to the flow direction. Further, numerical detail and fundamentals can be learned from the paper with descriptions about coptic [28, 30].

coptic facilitates the incorporation of point-charge grains as well as finite-sized objects. The grain considered in the present work is point-charge and is solved using particle-particle particle-mesh (P³ M) technique. The ion dynamics in six-dimensional phase space in the presence of the self-consistent electric field −∇φ of plasma, an optional external force D [30] (this extra force $\mathbf{D}$ is zero in our simulations for the shifted Maxwellian distribution) and an external magnetic field $\mathbf{B}$ is delineated by the equation

$$\begin{equation} {m_i} \frac{d \mathbf{v}}{dt} = q \left[- \nabla {\phi} + \mathbf{v} \times \mathbf{B} \right]+ \mathbf{D}, \end{equation} \tag{ 1 }$$

here q and m_i denote the ion charge and ion mass respectively.

To introduce a constant flux of ions, we used shifted Maxwellian distribution with a given value of the ions drift velocity (Much number). In order to solve the Poisson equation, a second-order accurate finite-difference-scheme equivalent to Shortley-Weller approximation has been adopted. The solution of this Poisson equation,

$$\begin{equation} \nabla^2{\phi} = (e n_e-qn_i)/\epsilon_0, \end{equation} \tag{ 2 }$$

endows us with the resultant potential where electron and ion number densities are represented by n_e and n_i respectively,. The interpolation of the electric field and the solution of the Poisson equation is incorporated with higher precision and convergence as it uses compact difference stencil which is specialized in dealing with arbitrary oblique boundaries for Cartesian mesh as well. On the outer mesh-edge, the potential gradient along $\hat{z}$ is set to zero.

At considered ordinary laboratory dusty plasma parameters, the charge of the dust particle is about $10^4~e_0$ (with e₀ being the electron charge) [1]. Therefore, the dust particle is considered to be strongly electron repellent, i.e. the ion-collecting probe approximation is applied. In this regime, the integration of the steady-state parallel momentum equation shows that the electrons are Boltzmann distributed [31–34]:

$$\begin{equation} n_e = n_{e \infty}\exp(e\phi/T_e). \end{equation} \tag{ 3 }$$

The combination of equation (1) with B ≠ 0 for ions and of equation (3) for electrons is the basis of the hybrid Boltzmann-particle-in-cell (pic) codes with a background magnetic field [32, 33, 35]. Patacchini et al [35] showed the validity of such hybrid approach for the entire range of ion magnetization for the strongly electron repellent dust particle (probe) by making an explicit comparison of the results computed using full pic (electron and ion particle) code calculations with the hybrid approach. The hybrid approach considered here has also been used previously, e.g. to study wave dispersion relations of magnetized dusty plasmas [34, 36].

In the code, collisions are included according to the constant velocity-independent collision frequency Poisson statistical distribution which is similar to the BGK-type collisions. Predominant collision in the system is the ion-neutral charge-exchange collision. This charge-exchange collision is implemented in the code by the mutual exchange of the ion velocity with the neutral velocity randomly drawn from neutral velocity distribution.

We also performed few simulations with grids of even higher resolution and non-uniform mesh spacing to resolve the dynamics in the vicinity of the grain [28]. Simulation is progressed in time for 1000-2000 time steps (with units discussed below) by which it usually reaches steady-state.

The most important dimensionless parameters are as follows:

• (a)  
  The orientation of the magnetic field is characterized by the angle α between magnetic field induction vector and streaming (an external electric field) direction (which is along z axis) as illustrated in figure 1. For 0 ≤ α ≤ π/2, the absolute transverse magnetic field and absolute longitudinal magnetic field cases correspond to α = π/2 and α = 0, respectively.
• (b)  
  The strength of the magnetic field is conveniently and conventionally characterized by the parameter $\beta = \omega_{\rm ci}/\omega_{\rm pi}$ defined as the ratio of cyclotron frequency of an ion to the plasma frequency of ions. Dimensionless parameters α and β are sufficient to completely describe the magnetic field.
• (c)  
  Mach number M describing ionic streaming speed has been defined as $M = v_d/c_s$, where $c_s = \sqrt{T_e/m_i}$ is the ion sound speed and v_d is ionic streaming velocity. Thermal Mach number M_th shares the relation with Mach number M according to the relation $M = v_d/c_s = \sqrt{T_i/T_e}\, M_{th}$.

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Besides, we follow the standard normalization, as described in [28], i.e. the space coordinate is normalized as $r\rightarrow r/r_0$, velocity as $v \rightarrow v/c_s$, and potential as $\phi \rightarrow \phi/( T_e/e)$, where $r_0 = ( \lambda_{De}/5)$ is the normalizing scale length and c_s is unity in normalized units. Time units as $\nu/(c_s/r_0) \sim 0.2(\nu/\omega_{pi})$ is used to normalize collision frequency ν, where ω_pi is the ion plasma frequency. The normalized grain charge, Q_d, is represented as ${\bar{Q_d}} = {Q_d e}/({4\pi\epsilon_0 \lambda_{De} T_e})$, where e₀ is the unit electron charge. We considered a cell grid of 64 × 64 × 128 with more than 60 million ions and grid side length of 15 × 15 × 20 Debye lengths for simulation purposes.

A summary of plasma parameters are presented in table 1. For typical dusty plasma parameters (e.g. an electron Debye length of λ_de = 845 µm and an electron temperature of 2.585 eV), the normalized grain charge ${\bar{Q_d}} = 0.01$ redacted in units of electron charge is approximately $7.5\times 10^3 e_0$ and β ≤ 1.0 is the magnetization parameter corresponding to $\mathbf{B}\lt 150~\mathrm{mT}$. These parameters are chosen in accordance with those observed in dusty plasma experiments [25, 37–39].

Table 1. List of plasma and simulation parameters.

Physical property	Parameter range
Magnetization β	0.0-1.0
Temperature ratio $T_e/T_i$	100
Mach Number M	0.5 - 1.5
Collision frequency ν/ωpi	0.002
Normalized grain charge $\bar{Q}_d$	0.01

Formation of wake behind grain due to ion focusing followed by defocusing for the unmagnetized case has been well investigated. A comparison of such a wake without magnetic field with that in the presence of magnetic field (two extreme cases - one magnetic field along flow and the other perpendicular to flow) is first presented to elucidate the impact of orientatation and strength of magnetic field on wake structure formation. In figure 2, the contour plots of the potential for three distinct cases: (a) without magnetic field (β = 0), (b) magnetic field applied along flow ($\vec{B}\parallel \vec{v}$), and (c) magnetic field in a direction transverse to the flow ($\vec{B} \perp \vec{v}$) are illustrated. Figure 2 demonstrated that the considered three cases have different wake structures. In the well studied magnetic field free case, figure 2(a), we see a V-shape wakefield with interchanging maxima and minima along flow direction. The wakefield with an external magnetic field applied along streaming velocity—another in detail investigated situation [10, 11]—shows no such oscillatory pattern with 'candle flame' shape (see figure 2(b)).

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The rotation of the magnetic field orientation from longitudinal to transverse direction restores oscillatory pattern of the wake field as illustrated in subplot (c) of figure 2. The noticeable differences of the latter case, figure 2 (c) from the magnetic field free case, figure 2 (a) are a stronger localization of the ion focusing and depletion regions around maxima and minima accompanied by a weaker manifestation of the V-shape, and a weakened damping of these oscillations. We also notice the sustained long range oscillatory structures with reduced effective wake wavelength in the case of grain in transverse to flow magnetic field (cf subplot (c)) and will be discussed later. In dimensional units, for typical dusty plasma parameters, e.g. T_e = 2.585, β = 1.0, and M = 1, the first wake peak would correspond to a potential of 60.62 mV for magnetic field transverse to flow and 25.98 mV for magnetic field along flow [11].

Next, in figure 3, we consider the wakefield for the transverse to flow magnetic field with different magnetization parameters at subsonic, sonic, and supersonic speeds. The three rows of the figure denote the cases with three different Mach numbers M = 0.5, M = 1, and M = 1.5 (from top to bottom), while the four columns show the cases with increasing strengths of magnetic fields β = 0.05, β = 0.1, β = 0.2, and β = 0.4 (from left to right). For M = 0.5 (top row), magnetic field decreases the oscillations amplitude and the following recurrent maxima and minima. The amplitude of wake oscillation is so small that no explicit trend is observed. Further increasing the magnetic field strength up to β = 0.4, one can observe an extended positive potential region which completely wipes out the secondary peaks at subsonic flows. Increasing the flow speed to M = 1 (middle row) and M = 1.5 (bottom row), pronounced wake oscillations behind the grain is observed. With increase in β, the effect of magnetic field manifests in the increase of the number of oscillations and decrease in the distance between subsequent potential maxima and minima. Additionally, the aforementioned stronger localization of the wake pattern around maxima and minima is distinctly visible. This can be understood by the characteristic trajectory opted by ions in weak and moderate magnetic field strengths. In the presence of magnetic field, the flow acquires anisotropy which leads to different plasma dynamics along and perpendicular to flow. At very weak magnetization ions follow the path as they do in the magnetic field free case. However, at moderately strong magnetic field strength, the ions get magnetized and the motion of ions gets more restricted around magnetic field induction lines. This leads to a stronger localization of the charges behind grain and a shortened characteristic wake oscillation wavelength.

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To further investigate the influence of the transverse magnetic field, the wake potential along the streaming axis at M = 0.5, 1, and 1.5 along with almost unmagnetized or very feeble magnetic field case (β = 0.05) is shown in figure 4. It was found that β = 0.05 case provides similar wake potential characteristics as the magnetic field free case already reported in previous works [6, 10, 11, 40]. At subsonic and sonic speeds (top and middle panel respectively), the transverse magnetic field with β = 0.1 leads to the substantial deviation of the potential from the case β = 0.05. However, in the supersonic regime (bottom panel), data for β = 0.1 case weakly differs from β = 0.05 case. With further increase in the magnetic field strength to β = 0.2 and β = 0.4, a strong impact of the transverse magnetic field in all three subsonic, sonic and supersonic regimes is observed. In this case, the transverse magnetic field decreases the oscillations amplitude as well as the distance between maxima and minima of the wakefield.

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In figure 5, the potential profile along transverse direction with z = 0 is shown. This figure shows that the transverse magnetic field with β ≤ 0.4 does not change the potential profile in perpendicular direction to z axis at z = 0. This is an important information, as in experiments, the dust particles are often located on a single plane perpendicular to the flow direction. Moreover, this behavior is in contrast with the longitudinal magnetic field case reported in previous works [10, 11, 23] where it was reported that the longitudinal magnetic field substantially affects the potential profile perpendicular to z axis (streaming direction) at the grain location (z = 0).

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For completeness, the wake peak height dependence on β is given in figure 6 for transverse to flow magnetic field. In agreement with the above presented wake potential data, the wake peaks are substantially smaller at M = 0.5 compared to sonic and supersonic cases. An increase in the flow speed leads to an increase in the wake peak amplitude (M = 1) followed by a significant decrease (M = 1.5) at even higher streaming speed. This behavior is due to the competition between increase in the number of influx ions and higher escape ability of ions with increase in streaming speed. Magnetic field applied perpendicular to the flow does not alter this non-monotonic trend with respect to change in the flow speed. The data for stronger magnetic field exhibits the smaller wake peak amplitude in accordance with those obtained for magnetic field applied along flow [11]. However, the relative damping of the oscillations is weaker in downstream direction compared to the longitudinal magnetic field case.

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In figure 7, the variation of the first maxima for the wake potential amplitudes as a function of Mach number, M, at α = π/2 is elucidated. In figure 7 the comparison of the magnetized wake potential peak (with β = 1) to the wake potential peak without an external magnetic field is shown. In the latter case the value of the potential is significantly larger than in the magnetized case. For illustrative comparison of these two cases, the value of the magnetized wake potential peak is multiplied by the factor 10. From figure 7 we see that the wake potential maxima exhibits a non-monotonic curve with change in ion flow speed. In the magnetized case, the characteristic trend is qualitatively similar to the one observed without magnetic field, however, the quantitative values are significantly different. In the absence of the magnetic field, the wake peak first increases with increase in the flow strength up to Mach number M ≃ 0.5 and then decreases with further increase in flow strength. Similar behavior exhibits the wake peak with magnetic field (β = 1), but with maxima around M = 1.0 and with much slower decline after maxima with further increase in the ion flow velocity. The increase of the wake peak potential value with increasing the ion streaming velocity at small M is due to enhancement in ion focusing downstream grain and followed decline in the peak strength at large M is because ions can effectively escape the attraction by negatively charged grain.

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The presence of recurring oscillations downstream grain with similar shaped potential maxima and minima is illustrated in figure 8 where β = 0.125, α = π/2, and M = 1. Here, the wake potential has oscillatory pattern extending up to $20~\lambda_{\rm De}$ in downstream region. It reveals that wake of purely oscillatory character in downstream region with a length of order ten λ_de can be achieved with modification in the strength and orientation of magnetic field.

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The ion density contour profile also supports the discussed wakefield variation in the presence of transverse magnetic field and is illustrated in figure 9 for flow speed M = 1.5 at β = 0.05, 0.1, 0.2, and 0.4. It is known from the work by Sundar et al [11] that the presence of longitudinal magnetic field induces 'candle flame' like ion density distribution in downstream region. However, there is no visible 'candle flame' like protruding structure in the density contour in the case of the magnetic field applied perpendicular to the flow direction. Additionally, transverse magnetic field makes the density fluctuation propagate farther from the grain thereby extending the range of wake oscillations. This supports our observation of enhancement in the number of wake potential oscillations behind grain at sonic and supersonic speeds.

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It is shown above that the wake behavior for the case of longitudinal magnetic field is completely different from that of magnetic field applied perpendicular to the flow. Therefore, it is interesting to inquire the intermediate case, i.e. when both longitudinal and transverse components of the magnetic field are non-zero. The consideration of the intermediate configuration provides the implicit detail about the influence of deviation of magnetic field orientation from the longitudinal direction on the wake features.

Figure 10 depicts the influence of magnetic field orientation on the wake potential at the subsonic (top row), sonic (middle row), and supersonic (bottom row) speeds. Here, to explicitly reveal the part played by the orientation of the magnetic field on the wake pattern, the strength of the overall magnetic field is kept intact (with β = 1.0) and variation in $\mathbf{B_\parallel}$ and $\mathbf{B_\perp}$ components is considered. In figure 10, from left to right, the strength of the magnetic field component $\mathbf{B_\parallel}$ (along the ion flux) reduces and consequently, the strength of the transverse component $\mathbf{B_\perp}$ increases. Accordingly, the angle between the ion flux direction and magnetic field induction direction, α, changes from 0^° [the left column] to 36^° (0.64 rad)[the right column] with intermediate value corresponding to 10^° (0.2 rad) [the middle column]. It can be observed that a change in the orientation of magnetic field modifies the wake behavior substantially. For the sonic and supersonic cases, a small deviation of the magnetic field direction from the purely longitudinal case has a very strong effect on the plasma dynamics around grain which eventually changes the wakefield pattern (compare the left pair of columns). This is especially strongly manifested in the supersonic regime (M = 1.5, see the bottom row) where even a 10^° deviation of the magnetic field induction from the longitudinal direction creates absolutely different pattern of the wake field in comparison with the purely longitudinal case (i.e. $\mathbf{B} = \mathbf{B_\parallel}$). In contrast to the sonic and supersonic cases, in the subsonic regime the wake field modifies smoothly with the rotation of the magnetic field induction direction from transverse to longitudinal configuration.

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In figure 11, the variation of the position and amplitude of the first wakefield maximum with orientation of magnetic field is illustrated for M = 1. From top subplot of figure 11, one can see that at $\alpha\lt \pi/12$ the position along z axis of the first potential peak remains almost unchanged with increase in α. Then at α > π/12, z coordinate significantly drops meaning that the first positive peak appears closer to the grain. At $\pi/6\lt \alpha\lt \pi/2$, further deviation of the magnetic field induction direction from the longitudinal to flow orientation does not change the position of the peak. The amplitude of the potential peak is shown in bottom subplot of figure 11. The value of the potential peak first slightly decreases with increase in α at $\alpha\lesssim \pi/12$. At $\pi/12\lt \alpha\lesssim \pi/4$, further deviation from the longitudinal direction results in a drastic increase in the potential peak value. For the range $\pi/4 \lt \alpha\lt \pi/2$, the potential peak amplitude remains approximately unaffected with variation in α. A non-monotonic behavior of the potential peak value and its location with change in magnetic field direction reaffirms that the deviation of the magnetic field orientation from the parallel to ion flow direction affects the wake peak potential, peak position and structure substantially.

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Interestingly, when both longitudinal and transverse components of the magnetic field are non-zero, the formation of two positive peaks split in transverse direction behind grain can be seen in figure 12 at β = 1 and α = 0.21. Two peak formation downstream grain in the presence of oblique to flow magnetic field is not a generic feature rather unique as it depends on the interplay of the perpendicular $B_\perp$ as well as parallel $B_\parallel$ magnetic field effects. The potential distribution in transverse direction at the location of two peaks is illustrated explicitly in figure 13 where comparison with the case z = 0 is also shown. In figures 12 and 13, there are no positive peaks at the location of the grain (z = 0), but the two peaks are clearly visible in the vicinity of the grain in the downstream region. This feature is qualitatively similar to the two positive peaks observed in ultracold dusty plasmas [41]. This observation is interesting in the sense that certain characteristics observed by lowering the temperature can be achieved by changing the strength and orientation of magnetic field [41, 42]. Nevertheless, there are differences in the two cases in the mechanism of wake formation and peak splitting. Here, the two positive potential peaks are separated by negative potential region and these are followed by further oscillations downstream whereas in the ultracold dusty plasmas there is a molecular-like structure with only two positive peaks behind grain.

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