**8. Introduction to electromagnetic instabilities and their dispersion relations**

The resistive instability is driven by coupling to a dissipative process. Due to the collisions between the electrons and neutral particles, resistive instability can occur in the acceleration channel of the Hall thruster. This instability occurs due to the interaction of the wave with the electrons' flow in the presence of electron collisions. The two major problems in plasma confinement are equilibrium and stability of plasma. The system in an equilibrium state if all the forces, which act on a system are balanced but stability and instability of the equilibrium can be assured by giving some perturbations to the equilibrium state. The stable and unstable state of equilibrium depend on small perturbation whether the perturbations are damped or amplified. For small perturbations, equilibrium is non- linear but stability can be linearized.

## **9. Plasma model and basic equations for purely azimuthal waves**

For the case of small amplitude perturbations and wavelengths much smaller than the scale lengths of inhomogeneities, the analysis of linearization can be applied. In the simplest approach, we consider the x*-* axis is taken along the axis of the thruster and the z*-*axis is taken along the radius of the thruster. The direction

the magnetic field *B* ! is along the z- axis. The y*-* axis is taken correspond to the azimuthal direction. Consistent to the fluid approach, we write below the basic fluid equations

$$\frac{\partial \mathfrak{n}\_j}{\partial t} + \overrightarrow{\nabla} \cdot \left(\overrightarrow{\nu}\_j \mathfrak{n}\_j\right) = \mathbf{0} \tag{11}$$

where *n <sup>j</sup>* is the mass density and *υ* ! *<sup>j</sup>* is the velocity of species j (j = i, e). The momentum equations for electrons and ions are

$$\frac{\partial \overrightarrow{\boldsymbol{\nu}\_{\varepsilon}}}{\partial t} + \left(\overrightarrow{\boldsymbol{\nu}\_{\varepsilon}} \cdot \overrightarrow{\boldsymbol{\nabla}}\right) \overrightarrow{\boldsymbol{\nu}\_{\varepsilon}} + \boldsymbol{v} \overrightarrow{\boldsymbol{\nu}\_{\varepsilon}} = -\frac{e\overrightarrow{E}}{m} - \frac{e}{m} \left(\overrightarrow{\boldsymbol{\nu}\_{\varepsilon}} \times \overrightarrow{\boldsymbol{B}}\right) \tag{12}$$

$$\frac{\partial \overrightarrow{\boldsymbol{\nu}}\_{i}}{\partial t} + \left(\overrightarrow{\boldsymbol{\nu}}\_{i} \cdot \overrightarrow{\boldsymbol{\nabla}}\right) \overrightarrow{\boldsymbol{\nu}}\_{i} = -\frac{e\overrightarrow{E}}{M} \tag{13}$$

### **9.1 Linearization of fluid equations**

To linearize all the equations, let us write *ni* ¼ *n*<sup>0</sup> þ *ni*1, *υ* !*<sup>i</sup>* ¼ *υ* !*i*<sup>1</sup> þ *υ* !0, *B* ! ¼ *B* ! <sup>1</sup> þ *B* ! <sup>0</sup> and *E* ! ¼ *E* ! <sup>1</sup> þ *E* ! 0. The unperturbed density is taken as *n*0, electric field *Numerical Investigations of Electromagnetic Oscillations and Turbulences in Hall… DOI: http://dx.doi.org/10.5772/intechopen.99883*

(magnetic field) as *E* ! 0(*B* ! 0) and the perturbed value of the electric field (magnetic field) is taken as *E* ! 1(*B* ! 1). Here, we consider the perturbed densities for ions and electrons as *ni*<sup>1</sup> and *ne*<sup>1</sup> velocities as *υ* ! *<sup>i</sup>*<sup>1</sup> and *υ* ! *<sup>e</sup>*<sup>1</sup> indicated by subscript 1 along with their unperturbed values as *υ*<sup>0</sup> and *u*<sup>0</sup> in the x- and y-directions respectively. In view of small variations of both the density and magnetic field along the channel, the plasma inhomogeneities are neglected. The oscillations of the perturbed on and electron densities are taken small enough ð Þ *ni*1, *ne*<sup>1</sup> < < *n*<sup>0</sup> so that the collisional effect due to the velocity perturbations dominate over the one due to the density perturbation. Since *υ* !<sup>0</sup> and *u*<sup>0</sup> are constant, the terms *υ* !<sup>0</sup> � ∇ ! *n*0, *n*<sup>0</sup> ∇ ! � *υ* !0 and *n*<sup>1</sup> ∇ ! � *υ* ! 0 are equal to be zero. Further the terms *<sup>υ</sup>* ! <sup>1</sup> � ∇ ! *n*1, and *n*<sup>1</sup> ∇ ! � *υ* ! 1 are neglected as they are quadratic in perturbation.

The linearizations of the above eqation leads to

$$\left(\frac{\partial}{\partial t} + \mu\_0 \frac{\partial}{\partial \boldsymbol{\rho}} - \boldsymbol{\nu}\right) \overrightarrow{\boldsymbol{\nu}}\_{\epsilon 1} + \frac{\boldsymbol{e}}{m} \left(\overrightarrow{E}\_1 + \overrightarrow{\boldsymbol{\nu}}\_{\epsilon 1} \times \overrightarrow{\boldsymbol{B}} + \overrightarrow{\boldsymbol{\mu}}\_0 \times \overrightarrow{\boldsymbol{B}}\_1\right) = \mathbf{0} \tag{14}$$

$$
\left(\frac{\partial}{\partial t} + \nu\_0 \frac{\partial}{\partial \mathbf{x}}\right) \vec{\nu}\_{i1} - \frac{e\vec{E}\_1}{M} = \mathbf{0} \tag{15}
$$

$$\frac{\partial n\_{\epsilon1}}{\partial t} + \mu\_0 \frac{\partial n\_{\epsilon1}}{\partial \mathbf{y}} + n\_0 \left(\overrightarrow{\nabla} \cdot \overrightarrow{\nu}\_{\epsilon1}\right) = \mathbf{0} \tag{16}$$

$$\frac{\partial n\_{i1}}{\partial t} + n\_0 \frac{\partial n\_{i1}}{\partial \mathbf{x}} + n\_0 \left(\vec{\nabla} \cdot \vec{\nu}\_{i1}\right) = \mathbf{0} \tag{17}$$

### **9.2 Dispersion equation and growth rate of azimuthal waves**

We take the variation of oscillating quantities in azimuthal direction as � exp ð Þ *iωt* � *iky* . The linearized Eqs. (14)–(17) are used and the density and velocity perturbations are expressed in terms of the electric field as

$$\upsilon\_{\rm ex1} = \frac{eE\_{\mathcal{V}}}{m\Omega} + -i\frac{eE\_{\mathcal{X}}\hat{o}}{m\Omega^2} \tag{18}$$

$$\upsilon\_{\varepsilon\gamma1} = \frac{u\_0 B\_1}{B} - \frac{eE\_\infty}{m\Omega} - \frac{ieE\_\gamma \hat{o}\nu}{m\Omega^2} \tag{19}$$

From Eq. (15)

$$
\rho\_{\rm ix1} = -\frac{ieE\_{\rm x}}{M\alpha} \tag{20}
$$

$$
\rho\_{\dot{\nu}1} = -\frac{ieE\_{\dot{\nu}}}{Ma} \tag{21}
$$

From Eqs. (13) and (14)

$$n\_{\epsilon1} = -\frac{ekn\_0 \left(i\hat{\alpha}E\_y + \Omega E\_x\right)}{m\Omega^2(\omega - ku\_0)}\tag{22}$$

$$n\_{i1} = -\frac{ien\_0kE\_y}{Ma^2} \tag{23}$$

The x-component of the perturbed current density

$$J\_x = em\_0(v\_{i\ge 1} - v\_{\varepsilon x1}) + en\_{i1}v\_0e\tag{24}$$

and the y-component of the perturbed current density

$$J\_{\gamma} = e n\_0 \left( \nu\_{i\gamma 1} - \nu\_{e\gamma 1} \right) - e n\_{e1} \mu\_0 \tag{25}$$

### **9.3 Conductivity tensor**

From the Maxwell's equation, we have

$$
\nabla \times \overrightarrow{B} = \mu\_0 \overrightarrow{J} + \mu\_0 \varepsilon\_0 \frac{\partial \overrightarrow{E}}{\partial t} \tag{26}
$$

or

$$
\overrightarrow{J}^{\rightarrow} = \frac{\nabla \times \overrightarrow{B}}{\mu\_0} - \frac{\overrightarrow{\partial E}}{\partial t} \tag{27}
$$

Faraday's law gives the relationship between changing electric and magnetic field as

$$
\nabla \times \overrightarrow{E} = -\frac{\partial \overrightarrow{B}}{\partial t} \tag{28}
$$

For a plane electromagnetic wave

$$
\overrightarrow{E} = \overrightarrow{E\_0} \, e^{\frac{}{\hbar}(\overrightarrow{\vec{k}} \cdot \overrightarrow{r} - \alpha t)} \tag{29}
$$

Where *E* ! is the electric field, *B* ! is the magnetic field, j represents the complex number, *k* ! is the wave vector, t is the time, *E*<sup>0</sup> ! and *B*<sup>0</sup> ! is complex magnitude of electric and magnetic field. Then ∇ operation gives ik and *<sup>∂</sup> <sup>∂</sup><sup>t</sup>* gives �*iω*. Substituting these operators Faraday's law become

$$
\overrightarrow{\vec{k}} \times \overrightarrow{E} = a\overrightarrow{B}
$$

or

$$
\overrightarrow{B} = \frac{\overrightarrow{k} \times \overrightarrow{E}}{\rho} \tag{30}
$$

### **9.4 Permittivity tensor**

From Maxwell's equations, we get

$$
\overrightarrow{D} = -\frac{\overrightarrow{k} \times \overrightarrow{k} \times \overrightarrow{E}}{a^2 \mu\_0} \tag{31}
$$

*Numerical Investigations of Electromagnetic Oscillations and Turbulences in Hall… DOI: http://dx.doi.org/10.5772/intechopen.99883*

which on expansion will give

$$
\begin{bmatrix} D\_{\mathbf{x}} \\ D\_{\mathbf{y}} \\ D\_{\mathbf{z}} \end{bmatrix} = \frac{1}{\alpha^2 \varepsilon\_0 \mu\_0} \begin{bmatrix} k^2 - k\_{\mathbf{x}}^2 & -k\_{\mathbf{x}} k\_{\mathbf{y}} & -k\_{\mathbf{x}} k\_{\mathbf{z}} \\ -k\_{\mathbf{y}} k\_{\mathbf{x}} & k^2 - k\_{\mathbf{y}}^2 & -k\_{\mathbf{y}} k\_{\mathbf{z}} \\ -k\_{\mathbf{z}} k\_{\mathbf{x}} & -k\_{\mathbf{z}} k\_{\mathbf{y}} & k^2 - k\_{\mathbf{z}}^2 \end{bmatrix} \begin{bmatrix} E\_{\mathbf{x}} \\ E\_{\mathbf{y}} \\ E\_{\mathbf{z}} \end{bmatrix} \tag{32}
$$

Then permittivity tensor will be

$$\varepsilon\_{r} = \frac{1}{a\nu^{2}\varepsilon\_{0}\mu\_{0}} \begin{bmatrix} k^{2} - k\_{x}^{2} & -k\_{x}k\_{\mathcal{Y}} & -k\_{x}k\_{x} \\ -k\_{\mathcal{Y}}k\_{\mathcal{X}} & k^{2} - k\_{\mathcal{Y}}^{2} & -k\_{\mathcal{Y}}k\_{x} \\ -k\_{x}k\_{\mathcal{X}} & -k\_{x}k\_{\mathcal{Y}} & k^{2} - k\_{x}^{2} \end{bmatrix}.$$

In general

$$
\varepsilon\_{r\ddot{\eta}} = \frac{1}{\alpha^2 \varepsilon\_0 \mu\_0} \left[ k^2 \delta\_{\ddagger} - k\_i k\_j \right] \tag{33}
$$

But for a pure dielectric medium *<sup>k</sup>* <sup>¼</sup> *<sup>ω</sup> c* ffiffiffiffi *εr* p and we will get

$$
\varepsilon\_r = \begin{bmatrix}
\varepsilon\_{rr} - \varepsilon\_{r\chi} & -\varepsilon\_{r\chi}\varepsilon\_{\eta\chi} & -\varepsilon\_{r\chi}\varepsilon\_{r\chi} \\
\end{bmatrix}.
$$

Then, the Maxwell's equations are used in view of the perturbed electric and magnetic fields of the electromagnetic wave, and the plasma dielectric tensor *εij* is obtained as

$$
\varepsilon\_{\vec{i}\vec{j}} E\_j = E\_j \delta\_{\vec{i}\vec{j}} + \frac{j\_i \left(E\_j\right)}{i\alpha\varepsilon\_0} \tag{34}
$$

Finally, the wave equation *k*<sup>2</sup> *<sup>δ</sup>ij* � *kik <sup>j</sup>* � *<sup>ω</sup>*<sup>2</sup> *<sup>c</sup>*<sup>2</sup> *<sup>ε</sup>ij* � �*<sup>E</sup> <sup>j</sup>* <sup>¼</sup> 0 is written in the following form

$$
\hbar^2 \varepsilon\_{\gamma\gamma} - \frac{\alpha^2}{c^2} \left( \varepsilon\_{\infty} \varepsilon\_{\gamma\gamma} - \varepsilon\_{\infty} \varepsilon\_{\gamma\infty} \right) = \mathbf{0} \tag{35}
$$

Where, the components of the dielectric tensor are obtained from Eq. (34) with the help of the Eqs. (24) and (25)

$$\varepsilon\_{\text{xx}} = \frac{(\alpha - k u\_0) \alpha\_\epsilon^2}{\alpha \Omega^2} + \mathbf{1} - \frac{\alpha\_i^2}{\alpha^2} \tag{36}$$

$$
\varepsilon\_{\text{xy}} = \frac{i\alpha\_{\epsilon}^{2}}{a\Omega} - \frac{\alpha\_{\text{i}}^{2}k\nu\_{0}}{a^{3}} \tag{37}
$$

$$
\varepsilon\_{\rm px} = \frac{\alpha\_{\epsilon}^{\prime}}{i\alpha\Omega} + \frac{\alpha\_{\epsilon}^{\prime}k u\_{0}\Omega}{i\Omega^{2}\alpha(\omega - k u\_{0})} \tag{38}
$$

$$\varepsilon\_{\mathcal{V}} = \frac{a\iota\_{\epsilon}^{2}ku\_{0}\hat{\alpha}k}{\Omega^{2}o(\omega - ku\_{0})} - \frac{a\iota\_{\imath}^{2}}{o(\omega - ku\_{0})} + \frac{a\iota\_{\epsilon}^{2}\hat{\alpha}}{o\Omega^{2}} + \mathbf{1} \tag{39}$$

By substituting these components into Eq. (35) we get the following cumbersome analytical expression of the dispersion relation of electromagnetic waves propagating in magnetized plasma [40].

$$\begin{split} \frac{k^{2}c^{2}}{\alpha^{2}} &= \frac{(\boldsymbol{\alpha} - \boldsymbol{k}u\_{0})\boldsymbol{\alpha}\_{\varepsilon}^{2}}{\alpha\boldsymbol{\alpha}\Omega^{2}} + \boldsymbol{1} - \frac{\boldsymbol{\alpha}\_{\varepsilon}^{2}}{\alpha^{2}} \\ &+ \frac{\left\{\frac{\boldsymbol{\alpha}\_{\varepsilon}^{2}}{i\alpha\Omega} + \frac{\boldsymbol{\alpha}\_{\varepsilon}^{2}k\boldsymbol{u}\_{0}\Omega}{i\Omega^{2}\boldsymbol{\alpha}(\boldsymbol{\alpha} - \boldsymbol{k}u\_{0})}\right\} \left\{\frac{\boldsymbol{\alpha}\_{\varepsilon}^{2}}{i\alpha\Omega} - \frac{\boldsymbol{\alpha}\_{i}^{2}k\boldsymbol{v}\_{0}}{\alpha^{3}}\right\} \\ &+ \left\{\frac{\boldsymbol{\alpha}\_{\varepsilon}^{2}k\boldsymbol{u}\_{0}\boldsymbol{\alpha}k\boldsymbol{k}}{\Omega^{2}\boldsymbol{\alpha}(\boldsymbol{\alpha} - \boldsymbol{k}u\_{0})} - \frac{\boldsymbol{\alpha}\_{i}^{2}}{\boldsymbol{\alpha}(\boldsymbol{\alpha} - \boldsymbol{k}\_{\gamma}\boldsymbol{\mu}\_{0})} + \frac{\boldsymbol{\alpha}\_{\varepsilon}^{2}\hat{\boldsymbol{\alpha}}}{\boldsymbol{\alpha}\boldsymbol{\alpha}\Omega^{2}} + 1\right\} \end{split} \tag{40}$$

### **9.5 Typical parameters of Hall thrusters**

The values and ranges of some typical parameters are given in **Table 2**.

### **9.6 Numerical results and discussion**

We solve the dispersion Eq. (40) to find out complex root by using typical values of the magnetic field, azimuthal wave number, collision frequency, electron drift velocity, ion drift velocity and electron temperature. Then the effect of these parameters on the growth *γ* of the electromagnetic wave is studied in **Figures 1–4**.

**Figure 2** shows that the growth rate of the wave get enhanced for the larger values of the drift velocity of the electrons [64]. Since the drift velocity can be correlated with the discharge voltage, so it can be said that the growth rate is increased with the discharge voltage. Esipchuk and Tilinin [83] also reported the proportionality of the frequency of drift instability to the discharge voltage. The increase in the growth rate may be attributed to the strong coupling between the electric field and electron current [84].

**Figure 3** shows the variation of growth rate under the effect of collisional frequency. It is seen that the wave grows at a faster rate in the presence of more electron collisions. Since the stronger resistive coupling of the oscillations to the electrons' closed drift requires the electron collisions, it is obvious that this instability grow faster in the presence of higher collision frequency. Similar results were also reported in the simulation studies of resistive instability by Fernandez *et al.* [41] that the growth rate is directly proportional to the square root of the collision frequency.

The dependence of growth rate on the magnetic field is shown in **Figure 4**, where it is observed that the wave grows faster in the presence of strong magnetic


### **Table 2.**

*Typical values of parameters used in Hall thruster.*

*Numerical Investigations of Electromagnetic Oscillations and Turbulences in Hall… DOI: http://dx.doi.org/10.5772/intechopen.99883*

**Figure 2.** *The dependence of growth rate with electron drift velocity.*

**Figure 3.** *Variation of growth rate with collision frequency.*

**Figure 4.** *Variation of growth rate with magnetic field.*

**Figure 5.** *Dependence of growth on ion drift velocity.*

field. The growth rate of the wave gets suppressed under the larger drift of the ions as shown in the **Figure 5**. Since in the presence of their large drift, the ions try to diminish the transverse oscillations of the electrons in the x-direction.

*Numerical Investigations of Electromagnetic Oscillations and Turbulences in Hall… DOI: http://dx.doi.org/10.5772/intechopen.99883*
