Effects of heat-flux vector and Braginskii viscosity on wave dissipation and instabilities in rotating gravitating anisotropic plasmas

Abstract

The pressure anisotropy-driven magnetohydrodynamic (MHD) waves and instabilities are the significant sources of energy transfer in astrophysical outflows, such as solar wind, spiral arms of galaxies and accretion disks. The heat-flux corrections, rotation and anisotropic viscosity play an unavoidable role in the wave dissipation and instabilities in such systems. In this work, we have investigated the effects of heat-flux vector and Braginskii viscosity tensor on the low-frequency hydromagnetic Chew–Goldberger–Low (CGL) waves, firehose instability and gravitational instability in uniformly rotating, strongly magnetized and anisotropic heat-conducting plasmas. The Braginskii viscosity tensor is considered in the CGL fluid equations, including heat-flux corrections and uniform rotation, keeping in mind the actual physical conditions of spiral arms of galaxies and solar coronal heating. The linear dispersion properties of gravitational instability, firehose instability, slow and fast CGL wave dissipation have been analyzed in various parametric limits. The dynamical stability of the system is discussed using the Routh–Hurwitz criterion. It is found that in the transverse propagation, the growth rate of the gravitational instability is decreased due to the presence of viscosity, and it remains unaffected due to the heat-flux corrections. In the parallel propagation, the effects of viscosity and heat-flux parameters are found to decrease the threshold wavenumber and stabilize the growth rate of gravitational instability. The upper and lower bounds of wavenumbers that determine the system’s stability, instability and overstability are decreased due to the viscosity and rotation parameters. The numerical calculations of various parameters show that gravitational instability plays a vital role in the spiral arms of the galaxies. The present results have been applied to understand the influence of heat-flux vector, rotation and viscosity on the slow and fast magnetosonic modes in the solar coronal heating mechanism.

Appendices

A Solutions of various equations given in Sect. 2

The divergence of perturbed viscosity stress tensor Eqs. (13–15) can be written as

$$\begin{aligned}&\nabla \cdot \delta {\varvec{\Pi }} _{x} = \frac{\eta _{0}}{3}(k^{2}_{\perp }\delta v_{x}-2k_{\perp }k_{\parallel }\delta v_{z}), \end{aligned}$$

(69)

$$\begin{aligned}&\nabla \cdot \delta {\varvec{\Pi }} _{y} = 0, \end{aligned}$$

(70)

$$\begin{aligned}&\nabla \cdot \delta {\varvec{\Pi }} _{z} = -2\frac{\eta _{0}}{3}(k_{\perp }k_{\parallel }\delta v_{x}-2k^{2}_{\parallel }\delta v_{z}). \end{aligned}$$

(71)

In Eq. (19), the magnetic pressure term can be expressed as

$$\begin{aligned} \frac{1}{4\pi }[(\nabla \times \delta \mathbf {B})\times \mathbf {B_0}] = -\frac{B^{2}_{0}}{4\pi } \left( \begin{array}{l} \partial _{x}\delta b_{z} - \partial _{z}\delta b_{x}\\ \\ \partial _{y}\delta b_{z}-\partial _{z}\delta b_{y}\\ \\ 0 \end{array} \right) . \end{aligned}$$

(72)

Using Eq. (23) the pressure stress tensor term can be expressed as

$$\begin{aligned} \nabla \cdot \delta {{\mathop { \mathbf {P }}\limits ^{\leftrightarrow }}} =\left( \begin{array}{l} \partial _{x} \delta p_{\perp } ~~~+~~~ (p_{\parallel 0} - p_{\perp 0})\partial _{z} \delta b_{x} \\ \\ \partial _{y} \delta p_{\perp } ~~~+~~~ (p_{\parallel 0} - p_{\perp 0})\partial _{z} \delta b_{y}\\ \\ \partial _{z} \delta p_{\parallel } ~~~+~~~ (p_{\parallel 0} - p_{\perp 0})[\partial _{x} \delta b_{x} + \partial _{y} \delta b_{y} + 2\partial _{z} \delta b_{z}] \end{array} \right) , \end{aligned}$$

(73)

where $\partial _{x,y,x}=\partial /\partial x, \partial /\partial y,$ and $\partial /\partial z$.

The reduced form of $\delta p_{\perp }$ and $\delta p_{\parallel }$ can be obtained by using Eqs. (35) and (37), and doing further simplifications one can solve for $\delta p_{\parallel }$

$$\begin{aligned} \xi ^{-1} \frac{\delta p_{\parallel }}{\rho } = {\alpha _1} \frac{k_{\bot }\delta v_{x}}{\omega } + {\beta _1}\frac{k_{\parallel }\delta v_{z}}{\omega }, \end{aligned}$$

(74)

where

$$\begin{aligned} \begin{array}{l} \xi = \frac{ 1 }{(\omega ^{2}-3k^{2}_{\parallel }c^{2}_{s\parallel })}, ~~\alpha _1= c^{2}_{s\parallel }\omega ^{2} - 3 k^{2}_{\parallel }c^{2}_{s\parallel }( c^{2}_{s\parallel } + c^{2}_{s\perp }), ~~~\beta _1= \frac{ 8k_{\parallel } q_{\parallel } \omega ^{2}}{\rho } - 3c^{2}_{s\parallel } (k^{2}_{\parallel }c^{2}_{s\parallel } - \omega ^{2}). \end{array} \end{aligned}$$

(75)

Using Eqs. (36) and (38) and doing further simplifications, we can solve for $\delta p_{\perp }$ as

$$\begin{aligned} \varvec{\Gamma } ^{-1} \frac{\delta p_{\perp }}{\rho } = \varepsilon \frac{k_{\bot }\delta v_{x}}{\omega } + \nu \frac{k_{\parallel }\delta v_{z}}{\omega }. \end{aligned}$$

(76)

where

$$\begin{aligned} \begin{array}{l} \varvec{\Gamma } = \frac{ 1 }{(\omega ^{2}-k^{2}_{\parallel }c^{2}_{s\parallel })}, ~~ \varepsilon = \frac{k_{\parallel }q_{\perp }}{\rho }\omega + 2 c^{2}_{s\perp }\omega ^{2} - k^{2}_{\parallel }c^{2}_{s\perp }( c^{2}_{s\perp } + c^{2}_{s\parallel }), ~~ \nu = 2\frac{ k_{\parallel } q_{\perp } }{\rho }\omega - c^{2}_{s\perp } ( k^{2}_{s\parallel }c^{2}_{s\parallel } - \omega ^{2}). \end{array} \end{aligned}$$

(77)

B Solutions of the equation of motion

On substituting Eqs. (72) and (73) into Eq. (19) and doing further simplification, one can get:

The $\hat{\mathbf {x}}-$ component of equation of motion

$$\begin{aligned} \begin{aligned} \rho _{0}\frac{\partial }{\partial t}\delta v_{x}&\,= - \left( p_{\parallel }-p_{\perp }\right) \frac{\partial }{\partial z}\delta b_{x}-\frac{\partial \delta p_{\perp }}{\partial x} -\nabla .\delta \Pi _{x} \\&\quad -\frac{B^{2}_{0}}{4\pi }\left( \frac{\partial }{\partial x}\delta b_{z}-\frac{\partial }{\partial z}\delta b_{x}\right) +\rho _{0}\frac{\partial \delta \phi }{\partial x} + 2\rho \Omega _{z}\delta v_{y}.&\end{aligned} \end{aligned}$$

(78)

The $\hat{\mathbf {y}}-$ component of equation of motion

$$\begin{aligned} \begin{aligned} \rho _{0}\frac{\partial }{\partial t}\delta v_{y}&\,= - \left( p_{\parallel }-p_{\perp }\right) \frac{\partial }{\partial z}\delta b_{y}-\frac{\partial \delta p_{\perp }}{\partial y} -\nabla .\delta \Pi _{y} \\&\quad -\frac{B^{2}_{0}}{4\pi }\left( \frac{\partial }{\partial y}\delta b_{z}-\frac{\partial }{\partial z}\delta b_{y}\right) +\rho _{0}\frac{\partial \delta \phi }{\partial y} +2\rho _0 \left( \Omega _{x}\delta v_{z} - \Omega _{z}\delta v_{x}\right) .&\end{aligned} \end{aligned}$$

(79)

The $\hat{\mathbf {z}}-$ component of equation of motion

$$\begin{aligned} \rho _{0}\frac{\partial }{\partial t}\delta v_{z}= -\left( p_{\parallel }-p_{\perp }\right) \left( \frac{\partial }{\partial x}\delta b_{x}+\frac{\partial }{\partial y}\delta b_{y}\right) -\frac{\partial \delta p_{\parallel }}{\partial z} -\nabla .\delta \Pi _{z}+\rho _{0}\frac{\partial \delta \phi }{\partial z}+ 2\rho \Omega _{x}\delta v_{y}. \end{aligned}$$

(80)

After substituting the relevant perturbations and derivatives, one can obtain

$$\begin{aligned}&\begin{aligned} \omega \rho _{0}\delta v_{x} =&-k_{\parallel }\left( p_{0\parallel }-p_{0\perp }\right) \left( \frac{k_{\parallel } \delta v_{x}}{\omega }\right) + k_{\perp }\Gamma \rho _{0}\times \left( \varepsilon k_{\perp }\frac{\delta v_{x}}{\omega } + \nu k_{\parallel }\frac{\delta v_{z}}{\omega }\right) \\&-\mathrm {i}\frac{\eta _{0}}{3}\left( k^{2}_{\perp }\delta v_{x} - 2k_{\perp }k_{\parallel }\delta v_{z}\right) \\&+ \frac{B^{2}_{0}}{4\pi }\left( k^{2}_{\perp }\frac{\delta v_{x}}{\omega } + k^{2}_{\parallel }\frac{\delta v_{z}}{\omega }\right) -\frac{4\pi G\rho ^{2}_{0}}{k^{2}\omega }\times \left( k^{2}_{\perp }\delta v_{x} + k_{\perp }k_{\parallel }\delta v_{z}\right) + \mathrm {i}2\rho _{0}\Omega _{z}\delta v_{y}, \end{aligned} \end{aligned}$$

(81)

$$\begin{aligned}&\omega \rho _{0}\delta v_{y} = -k^{2}_{\parallel }\left( p_{0\parallel }-p_{0\perp }\right) \frac{\delta v_{x}}{\omega } + \frac{B^{2}_{0}}{4\pi }\left( \frac{k^{2}_{\parallel }\delta v_{y}}{\omega }\right) +\mathrm {i}2\rho _{0}\left( \Omega _{x}\delta v_{z}-\Omega _{z}\delta v_{x}\right) , \end{aligned}$$

(82)

$$\begin{aligned}&\begin{aligned} \omega \rho _{0}\delta v_{z} =&-\left( p_{0\parallel }-p_{0\perp }\right) k_{\perp }k_{\parallel }\frac{\delta v_{x}}{\omega } + k_{\parallel }\delta p_{\parallel }\\&- \frac{4\pi G \rho ^{2}_{0}}{k^{2}\omega }\left( k^{2}_{\perp }\delta v_{x} + k_{\perp }k_{\parallel }\delta v_{z}\right) + \mathrm {i}\frac{2\eta _{0}}{3}(k_{\perp }k_{\parallel }\delta v_{x} - 2k^{2}_{\parallel }) \\&+ \xi k_{\parallel }\rho _{0}(\alpha \frac{k_{\parallel }\delta v_{x}}{\omega } + \beta \frac{k_{\parallel }\delta v_{z}}{\omega }) +\mathrm {i}2\rho _{0}\Omega _{x}\delta v_{y}. \end{aligned} \end{aligned}$$

(83)

C The matrix elements of the matrix $[A_{lm}]$

$$\begin{aligned}&A_{11} = (\omega ^{2}-k^{2}_{\parallel }c^{2}_{s\parallel })\left[ \omega ^{2}+k^{2}_{\parallel }\left( c^{2}_{s\parallel }-c^{2}_{s\perp }\right) -k^{2}v^{2}_{A} + \mathrm {i}\frac{\eta _{0}}{3\rho _{0}}\omega k^{2}_{\perp } + \frac{4\pi G \rho _{0}}{k^{2}}k^{2}_{\perp } \right] \nonumber \\&\qquad \qquad - k^{2}_{\perp }\left[ 2\omega ^{2}c^{2}_{s\perp } + \frac{k_{\parallel }q_{\perp }}{\rho }\omega - k^{2}_{\parallel }c^{2}_{s\perp }\left( c^{2}_{s\perp }+ c^{2}_{s\parallel }\right) \right] ,\nonumber \\&A_{12} = -2 \mathrm {i}\omega \Omega _{z}\left( \omega ^{2}-k^{2}_{\parallel }c^{2}_{s\parallel }\right) ,\nonumber \\&A_{13} = -k_{\perp }k_{\parallel }\left[ 2 \frac{k_{\parallel }q_{\perp }}{\rho }\omega - \left( k^{2}_{\parallel }c^{2}_{s\parallel }-\omega ^{2}\right) \left( c^{2}_{s\perp } + \mathrm {i}2\frac{\eta _{0}}{3\rho _{0}}\omega - \right. \left. \frac{4\pi G\rho _{0}}{k^{2}}\right) \right] ,\nonumber \\&A_{21} = 2\mathrm {i} \omega \Omega _{z},\nonumber \\&A_{22} = \omega ^{2} - k^{2}_{\parallel }\left( v^{2}_{A}+ c^{2}_{s\perp } - c^{2}_{s\parallel }\right) ,\nonumber \\&A_{23} = -2\mathrm {i}\omega \Omega _{x},\nonumber \\&A_{31} = k_{\perp }k_{\parallel }\left[ \left( \omega ^{2} - 3k^{2}_{\parallel }c^{2}_{s\parallel }\right) \left( c^{2}_{s\parallel } - c^{2}_{s\perp } + \frac{4\pi G\rho _{0}}{k^{2}}-\mathrm {i}\frac{2\eta _{0}}{3\rho _{0}}\omega \right) -\omega ^{2}c^{2}_{s\parallel } + 3k^{2}_{\parallel }c^{2}_{s\parallel }\left( c^{2}_{s\parallel }+ c^{2}_{s\perp }\right) \right] ,\nonumber \\&A_{32} = 2\mathrm {i}\omega \Omega _{x}\left( \omega ^{2} - 3k^{2}_{\parallel }c^{2}_{s\parallel }\right) ,\nonumber \\&A_{33} = (\omega ^{2} - 3k^{2}_{\parallel }c^{2}_{s\parallel })\left[ \omega ^{2} + \frac{4\pi G\rho _{0}}{k^{2}}k^{2}_{\parallel } + \mathrm {i}\frac{4\eta _{0}}{3\rho _{0}}k^{2}_{\parallel }\omega \right] - k^{2}_{\parallel }\left[ 8\frac{k_{\parallel }q_{\parallel }}{\rho _{0}}\omega + 3c^{2}_{s\parallel }(\omega ^{2} - k^{2}_{\parallel }c^{2}_{s\parallel })\right] . \end{aligned}$$

(84)