Unstable Electrostatic Waves Associated with Density and Temperature Gradients in an Inhomogeneous Plasma

Abstract

A study is carried out on the amplification of electrostatic Bernstein waves in the presence of drift wave turbulence in an inhomogeneous plasma. We have considered the Vlasov-Poisson system of equations for the interaction among the waves. In this investigation, we have considered a particle distribution model in which an external force due to density and temperature gradients present in the system is incorporated. The resonant mode of drift wave turbulence interacts with plasma particles through resonant interactions. These accelerated particles transfer their energy to Bernstein waves through a modulated field. This nonlinear wave interaction process is based on weak turbulence theory and is known as the Plasma maser effect. In this process, we have evaluated the fluctuating parts of the distribution function owing to the presence of turbulent resonant waves, the modulation field of interacting waves, and the non-resonant Bernstein waves by integrating along the unperturbed particles orbit from a set of linearized Vlasov equations. The nonlinear dispersion relation for Bernstein waves is presented to analyse the influence of gradient parameters.

1. Introduction

The Bernstein waves were first observed and studied in the 1950s by an American physicist named Ira Bernstein, who made significant contributions to the field of plasma physics. These waves play a crucial role in understanding various phenomena in astrophysics and fusion research. In the last decade, the Bernstein waves have been studied in a wide range of linear and nonlinear phenomena, and they still represent a major area of interest in fundamental research. Satellite observations of space plasmas have been interpreted using the findings of such investigations. Additionally, strategies for heating plasmas and diagnosing plasma behaviour have been developed in this regard.

Observations of Bernstein waves have been made both in laboratory experiments and in measurements of magnetospheric plasma carried out by spacecrafts. Stix [1] identified that there were two kinds of Bernstein waves: one was electron mode and the other was ion mode. It is interesting to note that electron Bernstein waves have been observed in both Earth’s and Jupiter’s magnetospheres. Kurth et al. [2] and Stix [1] were the first to observe these waves in Earth’s magnetosphere, while Grimaldo et al. [3] more recently observed them in Earth’s plasma. Experimentally, the electron Bernstein wave has been studied in tokamak plasma, especially in the spherical tokamak [4]. It seems that current research on electron cyclotron wave heating and current drive in magnetically confined thermonuclear fusion is focused on conventional aspect ratio tokamaks, with particular emphasis on ITER (International Thermonuclear Experimental Reactor). Electrostatic Bernstein waves have been studied in depth in various contexts, including the solar atmosphere, the auroral region, and through analytical and numerical methods. Additionally, the generation of electrostatic Bernstein modes through nonlinear wave-particle interaction in the presence of drift wave turbulence has been explored by several authors [5, 6, 7, 8]. Deka [9] investigated the amplification of the electrostatic Bernstein mode in the presence of ion-acoustic turbulence, while Singh [10] studied its generation in the presence of kinetic Alfven wave turbulence and a density gradient in the medium.

Singh and Deka [11] investigated the amplification of the electrostatic Bernstein mode in the presence of drift wave turbulence, and Singh [8] investigated the amplification in the presence of a density gradient in the media. In order to study Bernstein-ring instability, Yoon et al. [12] considered a more accurate electron distribution function. They indicate that the absence of a strong perpendicular velocity gradient in the model distribution suggests that the electrostatic Bernstein waves cannot be excited. In the Saturnian magnetosphere, Henning et al. [13] studied the electrostatic Bernstein waves in plasmas with dual kappa electron distributions.

We have investigated the instability of Bernstein waves propagating perpendicular to the external magnetic field under the influence of two types of waves: [14] one is the resonant low-frequency drift wave, satisfying the Cherenkov resonance condition, and the other is a non-resonant high frequency electromagnetic extraordinary mode wave, satisfying neither the Cherenkov resonance condition nor the nonlinear scattering conditions. They have found that when both waves are present, a new maser-like emission occurs, which is called the plasma maser effect [15, 16, 17].

We demonstrate how the above mentioned process causes energy upconversion from the resonant drift mode to the non-resonant Bernstein mode wave. It is illustrated that the nonlinear dielectric constant is made up of two parts: direct coupling terms and polarisation mode coupling terms.

The chapter is structured as follows: In Section 2, the geometrical framework and mathematical formulation of our model is outlined, providing a solid foundation for the subsequent analysis. Following this, Section 3 presents the nonlinear dispersion relation of the Bernstein wave. The results and discussions are given in Section 4. Finally, in Section 5, we delve into a conclusion of the analytical investigations conducted in this chapter and explore potential applications that can arise from them.

2. Basic framework of the problem

2.1 Mathematical formulation

In this chapter, we have considered an inhomogeneous plasma that supports drift motions and the turbulence that results from the drift waves. For this framework, we consider a particle distribution function [18] involving density and temperature gradient parameters connected to the external force F. The gradients of density and temperature are taken along the y-direction (Figure 1).

Figure 1.

Geometrical framework of the model: K = (0, 0, K_║) indicates the propagation vector of the Bernstein wave. $\mathit{k}=(k_{\perp },\mathit{0},k_{||})$ indicates the propagation vector of the drift wave. $\mathbf{\nabla }n\left(y\right)$ indicates the density gradient along the negative y-direction. B(y) indicates the magnetic field along z-direction. $\mathbf{\nabla }T\left(y\right)$ indicates the magnetic field gradient along the +ve y-direction.

$$f_j\left(T_j,y,\mathit{v}\right)={\left(\frac{m}{2\pi T_{0j}}\right)}^{\frac{3}{2}}\left[1+\lambda \left(y+\frac{v_x}{{\mathrm{\Omega }}_j}\right)\right]\mathit{exp}\left[-\left(\frac{m{\mathit{v}}^2}{2{\mathit{T}}_{0j}}-\frac{\mathit{F}y}{{\mathit{T}}_{0j}}\right)\right]$$E1

Where

$$\lambda ={\left[\left(\frac{\partial }{\partial T_j}\frac{d{T}_j}{\mathit{dy}}\right)-\left(\frac{1}{f_j}\frac{d{f}_j}{\mathit{dy}}\right)\right]}_{y=0}-\frac{\mathit{F}}{T_{0j}}$$E2

The interaction of high-frequency electrostatic Bernstein wave with drift wave turbulence is governed by Vlasov-Poisson system of equations.

$$\left[\frac{\partial }{\partial t}+\mathit{v}.\frac{\partial }{\partial \mathit{r}}+\left\{\frac{e}{m_j}\left(\mathit{E}+\frac{\mathit{v}\times {\mathit{B}}_{\mathbf{0}}}{c}\right)-\frac{\mathit{F}}{m_j}\right\}.\frac{\partial }{\partial \mathit{v}}\right]F_{0j}\left(\mathit{r},\mathit{v},t\right)=0$$E3

$$\nabla .\mathit{E}=-4e{n}_j\pi \int f_{0j}\left(\mathit{r},\mathit{v},t\right)d\mathit{v}$$E4

The linear response theory [18] of a turbulent plasma describes the unperturbed electron distribution function and the unperturbed electric field as:

$$F_{0j}=f_{0j}+\epsilon f_{1j}+{\mathrm{\epsilon }}^2{f}_{2j}$$E5

$${\overrightarrow{E}}_{0l}=\epsilon {\overrightarrow{E}}_l+{\epsilon }^2{\overrightarrow{E}}_2$$E6

where $f_{0j}$ is the space and time averaged component of the distribution function, $f_{1j}$ and $f_{2j}$ are fluctuating parts due to the drift wave turbulence of the distribution function, and $\epsilon$ is an extremely small parameter associated with the low frequency drift wave turbulence.

To the order of $\epsilon$, we obtain from Eq. (3)

$$\left[\frac{\partial }{\partial t}+\mathit{v}.\frac{\partial }{\partial \mathit{r}}+\left\{\frac{e}{m_j}\left(\epsilon {\mathit{E}}_l+{\epsilon }^2{\mathit{E}}_2+\frac{\mathit{v}\times {\mathit{B}}_0}{c}\right)-\frac{\mathit{F}}{m_j}\right\}.\frac{\partial }{\partial \mathit{v}}\right]f_{1j}\left(\mathit{r},\mathit{v},t\right)=\frac{e_j}{m_j}\left({\mathit{E}}_{\mathit{l}}.\frac{\partial }{\partial \mathit{v}}{f}_{0j}\right)$$E7

By employing the Fourier transform in the several quantities according to

$$A\left(\mathit{r},\mathit{v},t\right)=\sum _{k,\omega }A\left(\mathit{k},\omega ,\mathit{v}\right)exp\left[i\left(\mathit{k}\mathbf{.}\mathit{r}-\mathit{\omega t}\right)\right]$$E8

We perturb the quasi-steady state with the test Bernstein wave field $\mathit{\mu \delta }\overrightarrow{E_h}$ and wave vector $K=\left(0,0,K_{\parallel }\right)$. The overall perturbed electric field as well as the electron distribution function as a result of this perturbation are as follows:

$$\delta \mathit{E}={\mu }^{\prime }\delta {\mathit{E}}_h+{\mu }^{\prime }\mathit{\epsilon \delta }{\mathit{E}}_{\mathit{lh}}+{\mu }^{\prime }{\epsilon }^2∆\mathit{E}$$E9

$${\mathit{\delta f}}_j={\mu }^{\prime }\delta f_h+{\mu }^{\prime }\mathit{\epsilon \delta }{f}_{\mathit{lh}}+{\mu }^{\prime }{\epsilon }^2∆f$$E10

where ${\mu }^{\text{'}}$ is another small parameter $\left({\mu }^{\text{'}}\ll \epsilon \right)$ and ${\delta \mathit{E}}_{\mathit{lh}}$, $∆\mathit{E}$,$\delta f_{\mathit{lh}}$ and $∆f$are the mixed mode perturbation parameters. The omission of second order field quantities is justified by random phase approximations. To the order of $\mu ,\mathit{\mu \epsilon }$ and ${\mathit{\mu \epsilon }}^2$ we get

$$\mathit{P\delta }{f}_h=\frac{e_j}{m_j}\left({\mathit{E}}_h+\frac{\mathit{v}\times \delta {\mathit{B}}_h}{c}\right).\frac{\partial f_{0j}}{\partial \mathit{v}}$$E11

$$\mathit{P\delta }{f}_{\mathit{lh}}=\frac{e}{m_j}\delta {\mathit{E}}_h.\frac{\partial f_{1j}}{\partial v}+\frac{e}{m_j}\frac{\mathit{v}\times {\delta \mathit{B}}_h}{c}.\frac{\partial f_{1j}}{\partial \mathit{v}}+\frac{e}{m_j}\delta {\mathit{E}}_{\mathit{lh}}.\frac{\partial f_{0j}}{\partial \mathit{v}}+\frac{e}{m_j}\delta {\mathit{E}}_l.\frac{\partial f_h}{\partial \mathit{v}}+\frac{e}{m_j}\frac{\mathit{v}\times {\delta \mathit{B}}_{\mathit{lh}}}{c}.\frac{\partial f_{0j}}{\partial \mathit{v}}$$E12

$$P\mathrm{\Delta }f=\frac{e}{m_j}\left[\delta {\mathit{E}}_{\mathit{lh}}.\frac{\partial f_{1j}}{\partial \mathit{v}}+\frac{\mathit{v}\times {\delta \mathit{B}}_{\mathit{lh}}}{c}.\frac{\partial f_{1j}}{\partial \mathit{v}}+{\mathit{E}}_l.\frac{\partial \delta f_{\mathit{lh}}}{\partial \mathit{v}}\right]$$E13

where $P=\left[\frac{\partial }{\partial t}+\mathit{v}.\frac{\partial }{\partial \mathit{r}}-\left\{\frac{e}{m_j}\left(\frac{\mathit{v}\times {\mathit{B}}_0}{c}\right)-\frac{F}{m_j}\right\}.\frac{\partial }{\partial \mathit{v}\mathbf{.}}\right]$

Eqs. (11), (12) and (13) are the fundamental equations for the induced Bremsstrahlung interaction, which results from the electron acceleration driven by nonlinear forces.

3. Non-linear dispersion relation of Bernstein wave

The fluctuating parts of the low frequency turbulent field $f_{1j}$ are now obtained from Eq. (8)

$$f_{1j}=\underset{-\infty }{\overset{0}{\int }}\frac{e}{m}\overrightarrow{E_l}.\frac{\partial f_{0j}}{\partial \overrightarrow{v}}exp\left[i.\left\{\overrightarrow{k.}\left({\overrightarrow{r}}^{\text{'}}-\overrightarrow{r}\right)-\mathit{\omega \tau }\right\}\right]\mathit{d\tau }$$E14

Which takes the form

$$f_{1j}=\frac{\mathit{ie}}{m}\frac{E_{l\parallel }}{\omega -k_{\parallel }{v}_{\parallel }+i{.0}^{+}}\frac{\partial f_{0j}}{\partial v_{\parallel }}$$E15

Where $i{0}^{+}$ is a small imaginary part of $\omega$, $\overrightarrow{k}$ is the wave number of the drift wave, and $\parallel$ means parallel to the magnetic field.

We obtain other fluctuating parts of the perturbed distribution function from Eqs. (11) to (13) as follows:

$$\begin{array}{l}\delta f_h=\frac{e}{m}\underset{-\infty }{\overset{0}{\int }}\delta E_h\cdot \frac{\partial f_{0j}}{\partial \overrightarrow{v}}exp\left[i\left\{\overrightarrow{K}.\left({\overrightarrow{r}}^{\text{'}}-\overrightarrow{r}\right)-\mathrm{\Omega }\left(t^{\text{'}}-t\right)\right\}\right]{\mathit{dt}}^{\text{'}}\\ =\frac{e}{m}\underset{-\infty }{\overset{0}{\int }}\left(\delta E_{\mathit{hy}}\frac{\partial }{\partial v_y^{\text{'}}}+\delta E_{\mathit{hz}}\frac{\partial }{\partial v_z^{\text{'}}}\right)f{}_{0j}\times exp\left[i\left\{K_{\perp }\left(y^{\text{'}}-y\right)+K_{\parallel }\left(z-z^{\text{'}}\right)-\mathrm{\Omega }\tau \right\}\right]\mathit{d\tau }\\ =\frac{e}{m}\underset{-\infty }{\overset{0}{\int }}\left\{\delta E_{\mathit{hy}}\left(-\frac{m}{T_j}{v}_y^{\text{'}}+\frac{\lambda }{{\mathrm{\Omega }}_j}\right)+\delta E_{\mathit{hz}}\frac{\partial }{\partial v_{\parallel }^{\text{'}}}\right\}{f}_{0j}\times exp\left[i\left\{K_{\perp }\left(y^{\text{'}}-y\right)+K_{\parallel }\left(z^{\text{'}}-z\right)-\mathrm{\Omega }\tau \right\}\right]\mathit{d\tau }\\ \delta f_h=\frac{e}{m}{f}_{0j}\delta E_{\mathit{hy}}\underset{-\infty }{\overset{0}{\int }}\left(-\frac{m}{T_j}{v}_y^{\text{'}}+\frac{\lambda }{{\mathrm{\Omega }}_j}\right)\times exp\left[i\left\{K_{\perp }\left(y^{\text{'}}-y\right)+K_{\parallel }\left(z^{\text{'}}-z\right)-\mathrm{\Omega }\tau \right\}\right]\mathit{d\tau }\\ +\frac{e}{m}\delta E_{\mathit{hz}}\frac{\partial }{\partial v_{\parallel }}{f}_{0j}\underset{-\infty }{\overset{0}{\int }}exp\left[i\left\{K_{\perp }\left(y^{\text{'}}-y\right)+K_{\parallel }\left(z^{\text{'}}-z\right)-\mathrm{\Omega }\tau \right\}\right]\mathit{d\tau }\end{array}$$E16

We can now write the expression $\delta f_h\left(\overrightarrow{K},\mathrm{\Omega }\right)$ in the form

$$\delta f_h\left(K,\mathrm{\Omega }\right)=\frac{\mathit{ie}}{m}\frac{\delta E_h}{\left|\overrightarrow{K}\right|}\frac{m}{T_j}\left[1+\left(\mathrm{\Omega }-\frac{\lambda T_j{K}_{\perp }}{m{\mathrm{\Omega }}_j}\right)\sum _a\sum _b\frac{J_a\left(\alpha \right)J_b\left(\alpha \right)exp\left\{i\left(a-b\right)\left(\theta -\frac{\pi }{2}\right)\right\}}{K_{\parallel }{v}_{\parallel }-\mathrm{\Omega }-{\lambda }^{\text{'}}\frac{v_{\perp }^2{K}_{\perp }}{2{\mathrm{\Omega }}_j}+a{\mathrm{\Omega }}_j}\right]\times f_{0j}$$E17

Next, we evaluate Eq. (12) to obtain $\delta f_{\mathit{lh}}$ as follows.

We write

$$\delta f_{\mathit{lh}}\left(\overrightarrow{K},\mathrm{\Omega }\right)=I_1^{\mathit{th}}+I_2^{\mathit{th}}+I_3^{\mathit{th}}$$E18

The first part is

$$\begin{array}{c}{I}_1^{\mathit{th}}=\frac{e}{m}\underset{-\infty }{\overset{0}{\int }}{E}_{l\parallel }\frac{\partial }{\partial v_{\parallel }}\left(\delta f_h\right)exp\left[i\left\{\left(\overrightarrow{K}-\overrightarrow{k}\right).\left(\overrightarrow{r^{\text{'}}}-\overrightarrow{r}\right)-\left(\mathrm{\Omega }-\omega \right)\tau \right\}\right]\mathit{d\tau }\\ =\frac{-\mathit{ie}}{m}{E}_{l\Vert }\frac{\partial }{\partial v_{\Vert }}[\frac{\mathit{ie}}{m}\frac{\delta E_h}{\left|\overrightarrow{K}\right|}\frac{m}{T_j}{f}_{0j}\{1+\sum _a\sum _b\frac{J_a\left(\alpha \right)J_b\left(\alpha \right)exp\left\{i\left(a-b\right)\left(\theta -\frac{\pi }{2}\right)\right\}}{K_{\Vert }{v}_{\Vert }-\mathrm{\Omega }-\frac{{\lambda }^{\text{'}}{v}_{\perp }^2{K}_{\perp }}{2{\mathrm{\Omega }}_j}+a{\mathrm{\Omega }}_j}\\ \times \left(\mathrm{\Omega }-\frac{\lambda T_j{K}_{\perp }}{m{\mathrm{\Omega }}_j}\right)\}]\times \left(\sum _s\sum _t\frac{J_s\left(\alpha \right)J_t\left(\alpha \right)exp\left\{i\left(s-t\right)\left(\theta -\frac{\pi }{2}\right)\right\}}{\left(K_{\Vert }-k_{\Vert }\right)v_{\Vert }-{\mathrm{\Omega }}^{\text{'}}-\frac{{\lambda }^{\text{'}}{v}_{\perp }^2{K}_{\perp }}{2{\mathrm{\Omega }}_j}+s{\mathrm{\Omega }}_j}\right)\end{array}$$E19

The second part is

$$\begin{array}{l}{I}_2^{\mathit{lh}}=\frac{e}{m}\underset{-\infty }{\overset{0}{\int }}\left(E_{\mathit{hy}}\frac{\partial }{\partial v_y^{\text{'}}}+\frac{\partial }{\partial v_z^{\text{'}}}\right)f_{1j}\times exp\left[i\left\{\left(\overrightarrow{K}-\overrightarrow{k}\right).\left({\overrightarrow{r}}^{\text{'}}-\overrightarrow{r}\right)-\left(\mathrm{\Omega }-\omega \right)\tau \right\}\right]\mathit{d\tau }\\ =\frac{e}{m}\underset{-\infty }{\overset{0}{\int }}[E_{\mathit{hy}}\left\{\frac{\mathit{ie}}{m}\frac{E_{l\Vert }}{{\omega }^{\text{'}}-k_{\Vert }^{\text{'}}{v}_{\Vert }+i{0}^{+}}\frac{\partial }{\partial v_{\Vert }}\left(\frac{\partial }{\partial v_y^{\text{'}}}{f}_{0j}\right)\right\}\\ +E_{\mathit{hz}}\left\{\frac{\partial }{\partial v_{\Vert }}\left(\frac{\mathit{ie}}{m}\frac{E_{l\Vert }}{{\omega }^{\text{'}}-k_{\Vert }{v}_{\Vert }+i{0}^{+}}\frac{\partial }{\partial v_{\Vert }}{f}_{0j}\right)\right\}]\\ \times exp\left[i\left\{\left(\overrightarrow{K}-\overrightarrow{k}\right).\left({\overrightarrow{r}}^{\text{'}}-\overrightarrow{r}\right)-\left(\mathrm{\Omega }-\omega \right)\tau \right\}\right]\mathit{d\tau }\\ =\frac{e}{m}\underset{-\infty }{\overset{0}{\int }}\frac{\delta E_h}{\left|\overrightarrow{K}\right|}[\left\{K_{\perp }\frac{E_{l\Vert }}{{\omega }^{\text{'}}-k_{\Vert }{v}_{\Vert }+i{0}^{+}}\left(\frac{\partial }{\partial v_{\Vert }}{f}_{0j}\right)\times \left(-\frac{m}{T_j}{v}_y^{\text{'}}+\frac{\lambda }{{\mathrm{\Omega }}_j}\right)\right\}\\ +\left\{K_{\Vert }\frac{\partial }{\partial v_{\Vert }}\frac{\mathit{ie}}{m}\frac{E_{l\Vert }}{{\omega }^{\text{'}}-k_{\Vert }{v}_{\Vert }+i{0}^{+}}\left(\frac{\partial }{\partial v_{\Vert }}{f}_{0j}\right)\right\}]\\ \times exp\left[i\left\{\left(\overrightarrow{K}-\overrightarrow{k}\right).\left({\overrightarrow{r}}^{\text{'}}-\overrightarrow{r}\right)-\left(\mathrm{\Omega }-\omega \right)\tau \right\}\right]\mathit{d\tau }\end{array}$$E20

$$\begin{array}{l}{I}_2^{\mathit{lh}}=\frac{-\mathit{ie}}{m}\frac{\delta E_h}{\left|\overrightarrow{K}\right|}[-\frac{m}{T_j}\left(\frac{\mathit{ie}}{m}\frac{E_{l\Vert }}{{\omega }^{\text{'}}-k_{\Vert }^{\text{'}}{v}_{\Vert }+i{0}^{+}}\right)\{1+\left({\mathrm{\Omega }}^{\text{'}}-v_{\Vert }{K}_{\Vert }^{\text{'}}\right)\\ \times \sum _a\sum _b\frac{J_a\left(\alpha \right)J_b\left(\alpha \right)exp\left\{i\left(a-b\right){\theta }^{\text{'}}\right\}}{a{\mathrm{\Omega }}_j-\frac{K_{\perp }\lambda v_{\perp }^2}{2{\mathrm{\Omega }}_j}+v_{\Vert }{K}_{\Vert }^{\text{'}}-{\mathrm{\Omega }}^{\text{'}}}\}\frac{\partial }{\partial v_{\Vert }}{f}_{0j}\\ +K_{\Vert }\sum _a\sum _b\frac{J_a\left(\alpha \right)J_b\left(\alpha \right)exp\left\{i\left(s-t\right){\theta }^{\text{'}}\right\}}{s{\mathrm{\Omega }}_j-\frac{K_{\perp }{\lambda }^{\text{'}}{v}_{\perp }^2}{2{\mathrm{\Omega }}_j}+v_{\Vert }{K}_{\Vert }^{\text{'}}-{\mathrm{\Omega }}^{\text{'}}}\times \frac{\partial }{\partial v_{\Vert }}\left(\frac{\mathit{ie}}{m}\frac{E_{l\Vert }}{{\omega }^{\text{'}}-k_{\Vert }^{\text{'}}{v}_{\Vert }+i{0}^{+}}\right)\frac{\partial }{\partial v_{\Vert }}{f}_{0j}]\end{array}$$E21

and

The third part is

$$\begin{array}{l}{I}_3^{\mathit{lh}}=\frac{e}{m}\underset{-\infty }{\overset{0}{\int }}\delta {\overrightarrow{E}}_{\mathit{lh}}.\frac{\partial }{\partial \overrightarrow{v}}{f}_{0j}\times exp\left[i\left\{\left(\overrightarrow{K}-\overrightarrow{k}\right).\left({\overrightarrow{r}}^{\text{'}}-\overrightarrow{r}\right)-\left(\mathrm{\Omega }-\omega \right)\tau \right\}\right]\mathit{d\tau }\\ =\frac{e}{m}\underset{-\infty }{\overset{0}{\int }}\delta {\overrightarrow{E}}_{\mathit{lh}}\left(-\overrightarrow{v}\frac{m}{T_j}+\overrightarrow{y}\frac{\lambda }{{\mathrm{\Omega }}_j}\right)f_{0j}\times exp\left[i\left\{\left(\overrightarrow{K}-\overrightarrow{k}\right).\left({\overrightarrow{r}}^{\text{'}}-\overrightarrow{r}\right)-\left(\mathrm{\Omega }-\omega \right)\tau \right\}\right]\mathit{d\tau }\\ I_3^{\mathit{lh}}=\frac{e}{m}\underset{-\infty }{\overset{0}{\int }}\frac{\delta {\overrightarrow{E}}_{\mathit{lh}}.\left(\overrightarrow{K}-\overrightarrow{k}\right)}{{\left|\overrightarrow{K}-\overrightarrow{k}\right|}^2}\left\{-\frac{m}{T_j}\left(\overrightarrow{K}-\overrightarrow{k}\right).\overrightarrow{v}+\overrightarrow{y}\left(\overrightarrow{K}-\overrightarrow{k}\right)\frac{\lambda }{{\mathrm{\Omega }}_j}\right\}{f}_{0j}\\ \times exp\left[i\left\{\left(\overrightarrow{K}-\overrightarrow{k}\right).\left({\overrightarrow{r}}^{\text{'}}-\overrightarrow{r}\right)-\left(\mathrm{\Omega }-\omega \right)\tau \right\}\right]\mathit{d\tau }\\ I_3^{\mathit{lh}}=\frac{e}{\mathit{im}}\frac{\delta {\overrightarrow{E}}_{\mathit{lh}}}{\left|\overrightarrow{K}-\overrightarrow{k}\right|}\left(-\frac{m}{T_j}\right)f_{0j}\underset{-\infty }{\overset{0}{\int }}[1+\left(\mathrm{\Omega }-\omega -\frac{\lambda K_{\perp }{T}_j}{{\mathrm{\Omega }}_{j}m}\right)\\ \times exp\left[i\left\{\left(\overrightarrow{K}-\overrightarrow{k}\right).\left({\overrightarrow{r}}^{\text{'}}-\overrightarrow{r}\right)-\left(\mathrm{\Omega }-\omega \right)\tau \right\}\right]]\mathit{d\tau }\\ =\frac{\mathit{ie}}{m}\frac{\delta {\overrightarrow{E}}_{\mathit{lh}}}{\left|\overrightarrow{K}-\overrightarrow{k}\right|}\frac{m}{T_j}{f}_{0j}\left\{1+\left({\mathrm{\Omega }}^{\text{'}}-\frac{\lambda K_{\perp }{T}_j}{{\mathrm{\Omega }}_{j}m}\right)\sum _{a,b}\frac{J_a\left(\alpha \right)J_b\left(\alpha \right)exp\left\{i\left(a-b\right){\theta }^{\text{'}}\right\}}{a{\mathrm{\Omega }}_j-\frac{K_{\perp }{\lambda }^{\text{'}}{v}_{\perp }^2}{2{\mathrm{\Omega }}_j}-v_{\parallel }{K}_{\parallel }^{\text{'}}{\mathrm{\Omega }}^{\text{'}}}\right\}\end{array}$$E22

Finally, the high fluctuating part is obtained as follows:

We write

$$\mathrm{\Delta }f=I_1^{\mathrm{\Delta }f}+I_2^{\mathrm{\Delta }f}$$E23

The first part is evaluated as

$$\begin{array}{c}{I}_1^{\mathrm{\Delta }f}=\frac{e}{m}{E}_{l\parallel }\underset{-\infty }{\overset{0}{\int }}\frac{\partial }{\partial v_{\parallel }}\left(\delta f_{\mathit{lh}}\right)exp\left[i\left\{\overrightarrow{K}.\left({\overrightarrow{r}}^{\text{'}}-\overrightarrow{r}\right)-\mathrm{\Omega }\tau \right\}\right]\mathit{d\tau }\\ =\frac{-\mathit{ie}}{m}{E}_{l\parallel }\sum _{a,b}\frac{J_a\left(\alpha \right)J_b\left(\alpha \right)exp\left\{i\left(a-b\right){\theta }^{\text{'}}\right\}}{a{\mathrm{\Omega }}_j-\frac{K_{\perp }{\lambda }^{\text{'}}{v}_{\perp }^2}{2{\mathrm{\Omega }}_j}-\mathrm{\Omega }}\frac{\partial }{\partial v_{\parallel }}\delta f_{\mathit{lh}}\end{array}$$E24

The second part is

$$\begin{array}{l}{I}_2^{\mathrm{\Delta }f}=\frac{e}{m}\underset{-\infty }{\overset{0}{\int }}\delta {\overrightarrow{E}}_{\mathit{lh}}.\frac{\partial }{\partial \overrightarrow{v}}\left(\frac{\mathit{ie}}{m}\frac{E_{l\parallel }\frac{\partial }{\partial v_{\parallel }}{f}_{0j}}{{\omega }^{\text{'}\text{'}}-k_{\parallel }^{\text{'}\text{'}}{v}_{\parallel }+i{0}^{+}}\right)exp\left[i\left\{\overrightarrow{K}.\left({\overrightarrow{r}}^{\text{'}}-\overrightarrow{r}\right)-\mathrm{\Omega }\tau \right\}\right]\mathit{d\tau }\\ =\frac{e}{m}\underset{-\infty }{\overset{0}{\int }}\left[\delta E_{\mathit{lhy}}\frac{\partial }{\partial v_y^{\text{'}}}\left(\frac{\mathit{ie}}{m}\frac{E_{l\parallel }\frac{\partial }{\partial v_{\parallel }}{f}_{0j}}{\omega -k_{\parallel }{v}_{\parallel }+i{0}^{+}}\right)+E_{\mathit{lhz}}\frac{\partial }{\partial v_{\parallel }}\left(\frac{\mathit{ie}}{m}\frac{E_{l\parallel }\frac{\partial }{\partial v_{\parallel }}{f}_{0j}}{\omega -k_{\parallel }{v}_{\parallel }+i{0}^{+}}\right)\right]\times exp\left[i\left\{\overrightarrow{K}.\left({\overrightarrow{r}}^{\text{'}}-\overrightarrow{r}\right)-\mathrm{\Omega }\tau \right\}\right]\mathit{d\tau }\end{array}$$E25

Which takes the form

$$\begin{array}{l}{I}_2^{\mathrm{\Delta }f}=\frac{e}{m}\frac{\delta E_{\mathit{lh}}}{\left|\overrightarrow{K}-\overrightarrow{k}\right|}\underset{-\infty }{\overset{0}{\int }}\left[K_{\perp }\left\{\frac{\mathit{ie}}{m}\frac{E_{l\parallel }}{\omega -k_{\parallel }{v}_{\parallel }+i{0}^{+}}\frac{\partial }{\partial v_{\parallel }}\left(\frac{\partial }{\partial v_y^{\text{'}}}{f}_{\mathit{oj}}\right)\right\}+\left(K_{\parallel }-k_{\parallel }\right)\frac{\partial }{\partial v_{\parallel }}\left\{\frac{\mathit{ie}}{m}\frac{E_{l\parallel }}{\omega -k_{\parallel }{v}_{\parallel }+i{0}^{+}}\frac{\partial }{\partial v_{\parallel }}{f}_{\mathit{oj}}\right\}\right]\\ \times exp\left[i\left\{\overrightarrow{K}.\left({\overrightarrow{r}}^{\text{'}}-\overrightarrow{r}\right)-\mathrm{\Omega }\tau \right\}\right]\mathit{d\tau }\\ =\frac{-e}{m}\frac{\delta E_{\mathit{lh}}}{\left|\overrightarrow{K}-\overrightarrow{k}\right|}\left[\left(\frac{\mathit{ie}}{m}\frac{E_{l\parallel }}{\omega -k_{\parallel }{v}_{\parallel }+i{0}^{+}}\frac{\partial }{\partial v_{\parallel }}{f}_{\mathit{oj}}\right)\left[1+\left(\mathrm{\Omega }-K_{\parallel }{v}_{\parallel }-\frac{\lambda T_j}{m{\mathrm{\Omega }}_j}\right)\times \sum _{a,b}\frac{J_s\left(\alpha \right)J_t\left(\alpha \right)exp\left\{i\left(s-t\right){\theta }^{\text{'}}\right\}}{s{\mathrm{\Omega }}_j-\frac{K_{\perp }{\lambda }^{\text{'}}{v}_{\perp }^2}{2{\mathrm{\Omega }}_j}+K_{\parallel }{v}_{\parallel }-\mathrm{\Omega }}+\right]\right.\\ \left.\left(K_{\parallel }-k_{\parallel }\right)\frac{\mathit{ie}}{m}\frac{\partial }{\partial v_{\parallel }}\times \left(\frac{E_{l\parallel }}{\omega -k_{\parallel }{v}_{\parallel }+i{0}^{+}}\frac{\partial }{\partial v_{\parallel }}{f}_{\mathit{oj}}\right)\sum _{a,b}\frac{J_s\left(\alpha \right)J_t\left(\alpha \right)exp\left\{i\left(s-t\right){\theta }^{\text{'}}\right\}}{s{\mathrm{\Omega }}_j-\frac{K_{\perp }{\lambda }^{\text{'}}{v}_{\perp }^2}{2{\mathrm{\Omega }}_j}+K_{\parallel }{v}_{\parallel }-\mathrm{\Omega }}\right]\end{array}$$

We can thus write the expression $\mathrm{\Delta }f\left(K,\mathrm{\Omega }\right)$ in the form of

$$\begin{array}{l}\mathrm{\Delta }f=\frac{-\mathit{ie}}{m}{E}_{l\parallel }\sum _a\sum _b\frac{J_a\left(\alpha \right)J_b\left(\alpha \right)exp\left\{i\left(a-b\right){\theta }^{\text{'}}\right\}}{a{\mathrm{\Omega }}_j-\frac{K_{\perp }{\lambda }^{\text{'}}{v}_{\perp }^2}{2{\mathrm{\Omega }}_j}-\mathrm{\Omega }}\frac{\partial }{\partial v_{\parallel }}\delta f_{\mathit{lh}}+\frac{-e}{m}\frac{\delta E_{\mathit{lh}}}{\left|\overrightarrow{K}-\overrightarrow{k}\right|}\left[\left(\frac{\mathit{ie}}{m}\frac{E_{l\parallel }}{\omega -k_{\parallel }{v}_{\parallel }+i{0}^{+}}\frac{\partial }{\partial v_{\parallel }}{f}_{\mathit{oj}}\right)\right.\\ \left[1+\left(\mathrm{\Omega }-K_{\parallel }{v}_{\parallel }-\frac{\lambda T_j}{{\mathrm{\Omega }}_{j}m}\right)\times \sum _{s,t}\frac{J_s\left(\alpha \right)J_t\left(\alpha \right)exp\left\{i\left(s-t\right){\theta }^{\text{'}}\right\}}{s{\mathrm{\Omega }}_j-\frac{K_{\perp }{\lambda }^{\text{'}}{v}_{\perp }^2}{2{\mathrm{\Omega }}_j}+K_{\parallel }{v}_{\parallel }-\mathrm{\Omega }}\right]+\\ \left.K_{\parallel }^{\text{'}}\frac{\mathit{ie}}{m}\frac{\partial }{\partial v_{\parallel }}\times \left(\frac{E_{l\parallel }}{\omega -k_{\parallel }{v}_{\parallel }+i{0}^{+}}\frac{\partial }{\partial v_{\parallel }}{f}_{0j}\right)\sum _{s,t}\frac{J_s\left(\alpha \right)J_t\left(\alpha \right)exp\left\{i\left(s-t\right){\theta }^{\text{'}}\right\}}{s{\mathrm{\Omega }}_j-\frac{K_{\perp }{\lambda }^{\text{'}}{v}_{\perp }^2}{2{\mathrm{\Omega }}_j}+K_{\parallel }{v}_{\parallel }-\mathrm{\Omega }}\right]\end{array}$$E26

where ${\mathrm{\Omega }}^{\text{'}}=\mathrm{\Omega }-\omega ,K_{\parallel }^{\text{'}}=K_{\parallel }-k_{\parallel }$.

The nonlinear dielectric function of the test high frequency Bernstein wave in the presence of drift wave turbulence is then calculated using the Vlasov Poisson system of equations.

$$\nabla .\delta {\overrightarrow{E}}_h=-4\pi {\mathit{jen}}_j\int \left(\delta f_{\mathit{hj}}+\Delta f_j\right)\mathit{dv}$$E27

which becomes

$$\delta E_h\left(\overrightarrow{K},\mathrm{\Omega }\right){\epsilon }_h\left(\overrightarrow{K},\mathrm{\Omega }\right)=0$$E28

We write the dispersion relation ${\epsilon }_h\left(\overrightarrow{K},\mathrm{\Omega }\right)$ in the form

$${\epsilon }_h\left(\overrightarrow{K},\mathrm{\Omega }\right)={\epsilon }_0\left(\overrightarrow{K},\mathrm{\Omega }\right)+{\epsilon }_d\left(\overrightarrow{K},\mathrm{\Omega }\right)+{\epsilon }_p\left(\overrightarrow{K},\mathrm{\Omega }\right)$$E29

Where ${\epsilon }_0\left(\overrightarrow{K},\mathrm{\Omega }\right)$ is the linear part, ${\epsilon }_d\left(\overrightarrow{K},\mathrm{\Omega }\right)$ is the direct coupling part and ${\epsilon }_p\left(\overrightarrow{K},\mathrm{\Omega }\right)$ is the polarisation coupling part.

4. Conclusions

In this chapter, we want to propose a nonlinear dispersion relation of electrostatic Bernstein mode waves that is established in the presence of drift wave turbulence, which is a common phenomenon in an inhomogeneous plasma. The fluctuating parts $f_{1j}$, $\delta f_h$, $\delta f_{\mathit{lh}}$ and, $\mathrm{\Delta }f$, as outlined in Eqs. (15), (17), (18), and (26) respectively, have been obtained. The fluctuating part $f_{1j}$ is a result of the drift wave turbulent field, which is characterised by its linear nature. Furthermore, $\delta f_h$ denotes the fluctuating part of the particle distribution function resulting from the perturbed electrostatic Bernstein mode, whereas $\delta f_{\mathit{lh}}$ and $\mathrm{\Delta }f$ represents the nonlinear fluctuating parts of the distribution function. The expressions of $\mathrm{\Delta }f$ includes polarisation coupling and direct coupling terms. The nonlinear dispersion relation will be given by

${\epsilon }_h\left(\overrightarrow{K},\mathrm{\Omega }\right)={\epsilon }_0\left(\overrightarrow{K},\mathrm{\Omega }\right)+{\epsilon }_d\left(\overrightarrow{K},\mathrm{\Omega }\right)+{\epsilon }_p\left(\overrightarrow{K},\mathrm{\Omega }\right)$ which includes the linear part ${\epsilon }_0\left(\overrightarrow{K},\mathrm{\Omega }\right)$, the direct coupling part ${\epsilon }_d\left(\overrightarrow{K},\mathrm{\Omega }\right)$, and the polarisation coupling part ${\epsilon }_p\left(\overrightarrow{K},\mathrm{\Omega }\right)$.

5. Discussion and conclusion

The expression of the nonlinear dispersion relation involves gradient parameters associated with density and temperature gradients, along with the external force term. The growth of Bernstein mode can be predicted from this nonlinear dispersion relation. It is clearly observed that the growth of Bernstein mode is connected with drift wave turbulent field energy, and this growth may be influenced by the parameter λ associated with the density gradient and temperature gradient.

In this present investigation, the external force field F is made to be involved in the dispersion relation and in the growth of Bernstein mode. In all previous investigations on the growth of Bernstein mode, the external force field F was not considered.

Conflict of interest

The authors declare no conflict of interest.