Study of nonlinear 3-D evolution of kinetic Alfvén wave and fluctuation spectra

Abstract

Waves and instabilities play a very crucial role in astrophysical plasmas e.g. solar wind, Geospace etc. The main objective of current study is to investigate the importance of nonlinear processes associated with kinetic Alfvén waves (KAWs) in order to understand the physical mechanism behind the magnetopause turbulence. Numerical simulation of the coupled equations guiding the dynamics of three dimensionally propagating kinetic Alfvén wave (KAW) and slow magnetosonic wave has been performed for intermediate beta plasma (i.e. $m_{e}/m_{i} \ll \beta < 1$, where $\beta$ is thermal to magnetic pressure ratio) applicable to the magnetopause. A simplified semi-analytical model based on paraxial approach has also been developed. We have examined the field localization and associated power spectrum of 3-D kinetic Alfvén wave for this nonlinear interaction. Governing dynamical equations of KAW and slow magnetosonic wave get coupled when the ponderomotive force arising due to pump KAW is taken into account while studying the slow magnetosonic wave dynamics. The numerical prediction of power law scaling is just consistent with the observation of THEMIS spacecraft in the magnetopause.

Appendix A

Ponderomotive forces for electrons due to 3D KAW are as follows:

$$ F_{ex} = \frac{k_{0 \bot}^{2}}{16\pi n_{0}}\frac{\partial \vert A_{z} \vert ^{2}}{\partial x} $$

(A.1)

$$ F_{ey} = \frac{k_{0 \bot}^{2}}{16\pi n_{0}}\frac{\partial \vert A_{z} \vert ^{2}}{\partial y} + \frac{m_{e}c^{2}\alpha^{2}k_{0x}k_{0y}}{4B_{0}^{2}} \frac{\partial \vert A_{z} \vert ^{2}}{\partial x} $$

(A.2)

and

$$ F_{ez} = - \frac{m_{e}c^{2}k_{0 \bot}^{4}}{64\pi^{2}e^{2}n_{0}^{2}}\frac{\partial \vert A_{z} \vert ^{2}}{\partial z} $$

(A.3)

Similarly for ions, ponderomotive force components are given as:

$$ \begin{aligned}[b] F_{ix} = - \frac{m_{i}P^{2}c^{2}k_{0y}^{2}\alpha^{2}}{4B_{0}^{2}}\frac{\partial \vert A_{z} \vert ^{2}}{\partial x} + \frac{eQck_{0x}k_{0z}\alpha^{2}}{4\omega_{0}B_{0}}\frac{\partial \vert A_{z} \vert ^{2}}{\partial z} \end{aligned} $$

(A.4)

$$ \begin{aligned}[b] F_{iy} & = \frac{m_{i}P^{2}c^{2}k_{0x}k_{0y}\alpha^{2}}{4B_{0}^{2}}\frac{\partial \vert A_{z} \vert ^{2}}{\partial x} + \frac{e^{2}k_{0z}\alpha}{ 4m_{i}\omega_{0}c} \frac{\partial \vert A_{z} \vert ^{2}}{\partial y} \\ &\quad {}+ \frac{eQck_{0y}k_{0z}\alpha^{2}}{4\omega_{0}B_{0}}\frac{\partial \vert A_{z} \vert ^{2}}{\partial z} \end{aligned} $$

(A.5)

and

$$ \begin{aligned}[b] F_{iz} & = \frac{eQck_{0x}k_{0z}\alpha}{4\omega_{0}B_{0}}\frac{\partial \vert A_{z} \vert ^{2}}{\partial x} + \frac{eQck_{0y}\alpha}{ 4cB_{0}} \frac{\partial \vert A_{z} \vert ^{2}}{\partial y} \\ &\quad {}- \frac{e^{2}k_{0z}^{2}\alpha^{2}}{4m_{i}\omega_{0}^{2}}\frac{\partial \vert A_{z} \vert ^{2}}{\partial z} \end{aligned} $$

(A.6)