Electromagnetic Viscous-Resistive-Drift-Wave Instability in Burning Plasma

Abstract

Drift wave instabilities (DWI) might be take a part for anomalous transport in modern day tokamaks due to contributions of edge turbulence and zonal flows. Experimentally, it is found that micro-level turbulence (associated with two-fluid dynamics) braced with macro-level (associated with single-fluid dynamics) magnetohydrodynamics-type processes. In this paper, additional two-fluid effect viz. ion viscosity on the electromagnetic (EM) linear resistive drift wave instability is analyzed by using a two-fluid MHD model. A modify dispersion relation for EM drift modes in a nonuniform magnetized plasma is derived, taken into account equilibrium pressure gradients, effect of finite ion-sound gyro-radius associated with electron–ion decoupling and viscous effect due to ions. The dispersion relation is then analyzed both analytically as well as numerically. It is shown, how additional two-fluid effects modify the growth rate for DWI in different regimes. The standard resistive drift-Alfven mode is reproduced in the cold ion case. Moreover, it is estimated that these effects provide the possibility of novel effects on EM instabilities in a nonuniform magnetized plasma. The results should be useful in the interpretation of EM fluctuations in nonuniform magneto-plasmas in which resistivity is a key element in calculations of dissipative drift instabilities.

Appendices

Appendix 1

The linearized Eq. (2) with respect to first order perturbation proportional to $\exp \left[{- \iota \left({\omega t + ky} \right)} \right]$ is,

$$- \iota \omega \rho_{0} U = - \nabla \left({p + \frac{{B_{0} B_{1}}}{4\pi}} \right) - \nabla \cdot {\varPi}_{i} + \frac{1}{4\pi}\left({\left({B_{0} \cdot \nabla} \right)B_{1} + \left({B_{1} \cdot \nabla} \right)B_{0}} \right),$$

(11)

the linearized continuity Eq. (3) read as,

$$- \iota \omega N + N_{0} \nabla \cdot U + U_{x1} \frac{{dN_{0}}}{dx} = 0.$$

(12)

Resolving Eq. (11) into components, the x-component yields,

$$- i\omega \rho_{\left(0 \right)} U_{x1} = - \frac{\partial}{\partial x}\left({p + \frac{{B_{z\left(0 \right)} B_{z} 1 + B_{y\left(0 \right)} B_{y} 1}}{4\pi}} \right) + i\frac{{kB_{y\left(0 \right)}}}{4\pi}B_{x1} - \left({\nabla \cdot {\varPi}_{i}} \right)_{x},$$

(13)

the y-component gives,

$$- i\omega \rho_{\left(0 \right)} U_{y1} = - ik\left({p + \frac{{B_{z\left(0 \right)} B_{z1}}}{4\pi}} \right) + \frac{1}{4\pi}\frac{{\partial B_{y\left(0 \right)}}}{\partial x}B_{x1} - \left({\nabla.{\varPi}_{i}} \right)_{y},$$

(14)

and the z-component expresses as,

$$- i\omega \rho_{\left(0 \right)} U_{z1} = i\frac{{kB_{y\left(0 \right)}}}{4\pi}B_{z1} + \frac{1}{4\pi}\frac{{\partial B_{z\left(0 \right)}}}{\partial x}B_{x1}.$$

(15)

Substituting perturbation $p + B_{z\left(0 \right)} B_{z}/4\pi$ from Eq. (14) along with Eq. (12) into Eq. (13) leads to the vorticity Eq. (6). The last term in Eq. (6) is originated from components of stress tensor given by Braginskii [35].

$$\left({\nabla \cdot {\varPi}_{i}} \right)_{x} = \frac{\partial}{\partial x}{\varPi}_{ixx} + \frac{\partial}{\partial y}{\varPi}_{ixy},$$

(16)

And,

$$\left({\nabla.{\varPi}_{i}} \right)_{y} = \frac{\partial}{\partial x}{\varPi}_{iyx} + \frac{\partial}{\partial y}{\varPi}_{iyy},$$

(17)

where the gyro-viscous components are ${\varPi}_{ixy} = {\varPi}_{iyx} = nT/2\omega_{ci} \left({\partial v_{x}/\partial x - \partial v_{y}/\partial y} \right)$ and ${\varPi}_{iyy} = - {\varPi}_{ixx} = nT/2\omega_{ci} \left({\partial v_{y}/\partial x + \partial v_{x}/\partial y} \right)$ as described in Ref. [31]. Neglecting convective derivative in linear studies the opportunity of gyro-viscous cancelation is subsided.

Appendix 2

The linearized GOL Eq. (5) and putting in linearized Faraday’s induction law Eq. (4), yields,

$$\frac{{\partial B_{1}}}{\partial t} = - \nabla \left({- U_{1} \times B_{0} + \frac{1}{{N_{0} e}}\left({j_{1} \times B_{0} + j_{0} \times B_{1} - \nabla p_{e}} \right) + \eta j_{1}} \right).$$

(18)

From first order perturbation proportional to $\exp \left[{- \iota \left({\omega t + ky} \right)} \right]$, above equation gives,

$$- \iota \omega B_{1} = - B_{0} \left({\nabla \cdot U_{1}} \right) + ikB_{y\left(0 \right)} U_{1} + \frac{c\eta}{4\pi}\nabla^{2} B_{1} + \frac{1}{{N_{0} e}}\left({- \iota kB_{y\left(0 \right)} j_{1} + j_{x1} \frac{{dB_{0}}}{dx} + ikj_{y\left(0 \right)} B_{1}} \right) + \frac{1}{{N_{0} e}}\left({j_{x1} B_{0} + B_{x1} j_{0} - ikp_{e} \hat{z}} \right).$$

(19)

The components of Ampere’s law with conditions $\nabla \cdot j = 0$ and $\nabla \cdot B = 0$ gives,

$$\frac{4\pi}{c}j_{x1} = ikB_{z1},\quad \frac{4\pi}{c}j_{y1} = - \frac{\partial}{\partial x}B_{z1} \quad {\text{and}}\quad \frac{4\pi}{c}j_{z1} = - \frac{1}{\iota k}\nabla^{2} B_{x1}.$$

(20)

Using these expressions, x-component of induction Eq. (19), yields Eq. (7) and z-component of induction Eq. (19), gives

$$- i\omega B_{Z1} + B_{Z\left(0 \right)} \nabla.U - ikB_{y\left(0 \right)} U_{z1} + U_{x1} \frac{{dB_{z\left(0 \right)}}}{dx} + \frac{c}{{4\pi eN_{\left(0 \right)}}}\left({B_{x1} \frac{{d^{2} B_{y\left(0 \right)}}}{{dx^{2}}} - B_{y\left(0 \right)} \nabla^{2} B_{x1}} \right) - \frac{{c^{2} \eta}}{4\pi}\nabla^{2} B_{z1} - \frac{c}{{eN_{\left(0 \right)}^{2}}}\frac{{dN_{\left(0 \right)}}}{dx}\left({\frac{1}{c}j \times B_{\left(0 \right)} + \frac{1}{c}j_{\left(0 \right)} \times B - \nabla p} \right)_{y} = 0,$$

(21)

Neglect last term of Eq. (21) by putting its value zero to eliminate fast-magneto-acoustic (FMA) mode which is not mode of interest for present study and inserting the value of $U_{z1}$ from Eq. (15) yields Eq. (8).