# Ion Acceleration in Multi-Fluid Plasma: Including Charge Separation Induced Electric Field Effects in Supersonic Wave Layers

^{†}

## Abstract

**:**

## 1. Introduction

#### Motivation for the Development of a Multi-Fluid Plus Kinetic Ions Plasma Model

## 2. Materials and Methods

#### 2.1. Two-Fluid Solar Wind Plus Kinetically Treated Shock Acceleration Candidate Kinetic Ions Numeric Model

**Plasma Dynamics**[21], Chapter 3 (from now on referred to as B&S), however instead of treating the system as two groups of particles, ions and electrons (labeled in B&S by ‘+’ and ‘−’, respectively), we treat the system as composed of three groups of particles, labeled {s, a, e}, which are solar wind (SW) ions, shock acceleration candidate ions and electrons, respectively. The shock acceleration candidate ions (e.g., pickup ions (PUIs)) may compose between 0 to 20% of the total ion population and are kept separate since they will be treated as SAP superparticles.

**Definition**

**1.**

#### 2.1.1. Conservation Form of the Two-Fluid Plasma Equations

#### 2.1.2. Dimensionless Forms of The Equations

#### 2.2. Special Case: Quasi-One Dimensional, Exactly Perpendicular Shock

#### Steady State Equations for 1D Case with Perpendicular Magnetic Field

## 3. Results

#### 3.1. Steady State, 1D Two-Fluid Plasma System with Cold Ions

#### 3.2. Setting the Electron Mass to Zero

#### 3.3. Discussion

## 4. Conclusions

## Funding

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

DOAJ | Directory of open access journals |

MRI | multiply reflected ion |

PUI | pickup ion |

SW | solar wind |

PIC | particle-in-cell |

MHD | magnetohydrodynamics |

ESW | electrostatic solitary wave |

ODE | ordinary differential equation |

SDA | shock drift acceleration |

SSA | shock surfing acceleration |

TS | Termination Shock |

CSP | cross-shock potential |

SAP | shock accelerated particle |

B&S | Boyd and Sanderson |

AU | astronomical unit |

1D | one dimensional |

SS | steady state |

NBS | neutral background state |

SSS | steady state system |

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**Figure 1.**Illustration of the idealized initial ion distribution, in the shock frame, before and just after reflection. (

**a**) The unreflected initial distribution is composed of a pickup ions (PUIs) which reside on a shell in velocity space of radius ${u}_{0}$ (the solar wind (SW) speed) and SW ions residing in a core located at (${u}_{0},0,0$). The dashed portion of the PI shell to the left of ${v}_{spec}$ on the ${v}_{x}$-axes (vertical dashed line) has an insufficient x-component of velocity to over come the cross-shock potential (CSP, $\mathsf{\Phi}$) and will be reflected. (

**b**) The ion distribution immediately following refection is composed of the specularly reflected fractional part of the PUI shell and the remainder which, along with the SW core, is convected downstream without reflection. Not illustrated by this figure is the stretching along motional electric field ($\overrightarrow{E}=\overrightarrow{u}\times \overrightarrow{B}$) direction that the reflected partial shell will undergo during the energization process. It is evident that the reflected PUI partial shell has gained energy after a single reflection, with respect to the fluid frame where the SW core is at the origin.

**Figure 2.**A test-particle ion (lower black curve) shock surfing through a model of the termination shock (upper blue curve). The fine structure in the ramp (not clearly visible in this figure) provides multiple possible reflection sights.

**Figure 3.**Depiction (for illustration purposes) of a perturbed quantity such as ion density, ion pressure, etc. represented along the verticle axis by ${g}_{i}(x,{t}_{0})$ at some arbitrary time ${t}_{0}$ shown together with the cross-shock potential (CSP) where x is the distance along the flow (measured along the horizontal axis.) The qualitative plots for these two functions are superimposed over a qualitative sketch of an ion trajectory. The illustration shows how some incoming ions are trapped by reflection from the CSP and accelerated in the motional electric field ($\overrightarrow{E}=-\overrightarrow{u}\times \overrightarrow{B}$) before being convected downstream. This trapping leads to a small ion density increase which is accounted for by the nonzero value of ${g}_{i}$ in a small region centered around the shock front.

**Figure 4.**(

**left**) Cold Maxwellian solar wind distribution in the shock frame. (

**middle**) Initial pickup ion filled-shell distribution in the shock frame. (

**right**) Downstream multiply reflected ion (MRI) accelerated filled-shell distribution in the fluid frame.

**Figure 5.**Left panel: level curves for the conservation of momentum Equation (81) together with the direction field for ODE set (92). The level curves are red for ${M}_{e0}=0.1$, orange for ${M}_{e0}=0.15$ and blue for ${M}_{e0}=2$. The curves are dashed on portions where ${E}_{x}^{2}<0$ and/or $U<0$. All three level curves pass through the neutral background state (NBS): ($u=U=1$, ${E}_{x}=0$). The red arrow stream-plot (appearing on the same axis with the level curves in the left panel) is the direction field for ODE set (92) with ${M}_{e0}=0.1$. Note that the direction fields swirls around the stationary point (${u}^{\gamma +1}=1/{M}_{e0}^{2},U=0$) in the fashion of a harmonic oscillator system. Right panel: direction field for the linear expansion of ODE system (92) about the fixed point so that $\Delta U=U$ and $\Delta u=u-{(1/{M}_{e0})}^{2/(\gamma +1)}$ and for ${M}_{e0}=0.1$. Inspection of the direction field, for the linearized system, indicates clearly that the fixed point is node.

**Figure 6.**Level curves for ${E}_{x}^{2}={E}_{x}^{2}\left(u\right)$, satisfying both conservation of momentum (81) and energy (82), are plotted on the same axes for several choices of Mach number. The left panel plots are for ${M}_{e0}=0.1$ (thick red curve), ${M}_{e0}=1$ (orange curve) and ${M}_{e0}=2$ (dashed blue curve). The right panel plots are for ${M}_{e0}=0.03$ (thick red curve), ${M}_{e0}=\sqrt{{\alpha}_{0}/({\alpha}_{0}+1)}=0.02333$ (orange curve) and ${M}_{e0}=0.01$ (dashed blue curve). Note that all curves pass through the neutral background state (NBS) where $u=U=1$ and ${E}_{x}=0$. Assuming that physically real solutions require ${E}_{x}^{2}>0$, then only curves which are concave up where they touch the NBS (e.g., the thick red curves) can be associated with physical solutions. This limits the range of Mach numbers that can support the physical connection of the plasma to the NBS to $\sqrt{{\alpha}_{0}/({\alpha}_{0}+1)}<{M}_{e0}<1$.

**Figure 7.**Illustrating the double valued nature of the solution for ($u\left(x\left(t\right)\right),U\left(x\left(t\right)\right),{E}_{x}\left(x\left(t\right)\right)$). The case plotted here is for an upper wave—where the velocity u rises above then returns to the NBS value, and for ${M}_{e0}=0.5716$ (i.e., ${M}_{ep}=24.5$). The red dashed curves are the corresponding linearized solutions given by Equation (84); as expected, the linearized solutions well approximate the fully nonlinear solution curves (solid curves) for values near the NBS. Although double valued, the structure of the plasma variables is compatible with the ODEs (78)–(80). For example, the slope ${u}^{\prime}\to \infty $ at the sonic points where ${u}^{\gamma +1}-1/{M}_{e0}^{2}=0$ and the slopes ${u}^{\prime}={U}^{\prime}=0$ at the point where ${E}_{x}=0$.

**Figure 8.**Illustration of how a unique, single valued solution can be constructed. The left panel gives a view of both velocities plotted on the same axis, the blue curve for $u\left(x\right)$ and the black curve for $U\left(x\right)$ (which has been nonphysically scaled up for illustration purposes). The dashed, green, vertical lines, placed on the x-axis at the sonic points where ${u}^{\gamma +1}=1/{M}_{e0}^{2}$, are tangent to the $u\left(x\right)$ solution curve and are also the locations where $U\left(x\right)$ reaches a maximum, as illustrated by Figure 5 and required by Equation (82). The associated solution for $x=x\left(t\right)$, where t is the parameterization variable (not the time), is plotted in the right panel. A horizontal dashed line has been inserted at the value of $x={x}_{c}$ which separates the curve into equal area lobes. This value ${x}_{c}$ is also the bisector between the sonic points and the location where both the velocity curves (left panel) intersect themselves. A single valued, upper solution of the SSS can be constructed by cutting off the top, closed-loop portions of the velocity curves and joining them together with an infinitesimally narrow, horizontal segment at the point where the solution curves intersect at $x={x}_{c}$. This construction requires that ${u}^{\prime}={U}^{\prime}={E}_{x}=0$ for compatibility with the SSS equations and since the insertion location ${x}_{c}$ is the only place where such a compatible insertion can be made—the solution is unique.

**Figure 9.**Upper solution where the plasma flow is above the initial velocity at the wave center for ${M}_{ep}=24.4949$, $(u\left(0\right),U\left(0\right),{E}_{x}\left(0\right))=(1.00001,1,0.0000105787)$.

**Figure 10.**Lower solution for ${M}_{ep}=1.1$, $(u\left(0\right),U\left(0\right),{E}_{x}\left(0\right))=(0.99999,0.999992,-5.38484\times {10}^{-6})$.

**Figure 11.**Lower solution for ${M}_{ep}=1.8$, $(u\left(0\right),U\left(0\right),{E}_{x}\left(0\right))=(0.99999,0.999997,-0.0000107293)$.

**Figure 12.**Lower solution for ${M}_{ep}=2$, $(u\left(0\right),U\left(0\right),{E}_{x}\left(0\right))=(0.99999,0.999998,-0.0000111723)$. Note that here the solution is becoming double valued near the center of the wave (see Figure 13), suggesting the need for a unique solution.

**Figure 13.**A close-up view of the center portion of the $u\left(x\right)$-curve from Figure 12, for ${M}_{ep}=2$. Clearly the solution is double valued near the center of the wave, suggesting that a unique solution should be inserted.

**Figure 14.**Lower solution for ${M}_{ep}=4$, $(u\left(0\right),U\left(0\right),{E}_{x}\left(0\right))=(0.99999,0.999999,-0.0000124494)$.

**Figure 15.**Upper solution, where the plasma flow is below the initial velocity at the wave center, for ${M}_{ep}=1$, $(u\left(0\right),U\left(0\right),{E}_{x}\left(0\right))=(1.00001,1.00001,3.01112\times {10}^{-7})$.

**Figure 16.**Upper solution for ${M}_{ep}=1.8$, $(u\left(0\right),U\left(0\right),{E}_{x}\left(0\right))=(1.00001,1,0.0000107289)$.

**Figure 17.**Upper solution for ${M}_{ep}=3$, $(u\left(0\right),U\left(0\right),{E}_{x}\left(0\right))=(1.00001,1,0.0000121454)$.

**Figure 18.**Level curves of Equation (98) for ${M}_{ep}=0.9$ (red curve), ${M}_{ep}=1$ (orange curve) and ${M}_{ep}=3$ (blue curve). The dashed curves correspond to (the nonphysical region where) $U<0$. The gaps in the upper $U>0$ curve branches correspond to where $E{x}^{2}<0$ (also assumed to be a nonphysical region). The blue curve for which ${M}_{ep}=3\phantom{\rule{1.em}{0ex}}(>1)$ is the only curve that passes through $u=U=1$, ${E}_{x}=0$ and thus is the only one of the three that is connected to the neutral background state (NBS). The right panel shows the stream-plot of the direction field for ODE system (103) where ${M}_{ep}=3$, on the same axes with the physical segment of the blue curve from the left panel. Comparison of this Figure to the analogous Figure 5, for nonzero electron mass case, shows that setting ${m}_{e}=0$ has effectively moved the sonic point to infinity on the u-axes.

**Figure 19.**Level curves of Equation (94) for ${M}_{ep}=0.1$ (dashed red curve), ${M}_{ep}=1$ (orange curve) and ${M}_{ep}=10$ (thick blue curve). Note that ${E}_{x}^{2}\left(u\right)$ is concave up at the point where it touches the NBS only if ${M}_{ep}>1$, thus solutions physically connected to the NBS require flow that is supersonic with respect to the collective Mach number.

**Figure 20.**Upper solution for ${m}_{e}=0$, ${M}_{ep}=1.1$, $(u\left(0\right),U\left(0\right),{E}_{x}\left(0\right))=(1.00001,1.00001,5.37831\times {10}^{-6})$. The red, dashed curves correspond to the linearized solution (Equation (84)). This figure illustrates that setting ${m}_{e}=0$ has effectively shifted the sonic point to infinity on the x-axis with the result that plasma acceleration is unbounded when the velocity growth is positive.

**Figure 21.**Lower solution for ${m}_{e}=0$, ${M}_{ep}=1.1$, $(u\left(0\right),U\left(0\right),{E}_{x}\left(0\right))=(0.99999,0.999992,-5.37818\times {10}^{-6})$.

**Figure 22.**Lower solution for ${m}_{e}=0$, ${M}_{ep}=1.8$, $(u\left(0\right),U\left(0\right),{E}_{x}\left(0\right))=(0.99999,0.999997,-0.0000107344)$.

**Figure 23.**Lower solution (weak) for ${m}_{e}=0$, ${M}_{ep}=2$, $(u\left(0\right),U\left(0\right),{E}_{x}\left(0\right))=(0.99999,0.999997,-0.0000111804)$.

**Figure 24.**Lower (weak) solution for ${m}_{e}=0$, ${M}_{ep}=3$, $(u\left(0\right),U\left(0\right),{E}_{x}\left(0\right))=(0.99999,0.999999,-0.0000121717)$.

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Burrows, R.
Ion Acceleration in Multi-Fluid Plasma: Including Charge Separation Induced Electric Field Effects in Supersonic Wave Layers. *Plasma* **2020**, *3*, 117-152.
https://doi.org/10.3390/plasma3030010

**AMA Style**

Burrows R.
Ion Acceleration in Multi-Fluid Plasma: Including Charge Separation Induced Electric Field Effects in Supersonic Wave Layers. *Plasma*. 2020; 3(3):117-152.
https://doi.org/10.3390/plasma3030010

**Chicago/Turabian Style**

Burrows, Ross.
2020. "Ion Acceleration in Multi-Fluid Plasma: Including Charge Separation Induced Electric Field Effects in Supersonic Wave Layers" *Plasma* 3, no. 3: 117-152.
https://doi.org/10.3390/plasma3030010