Unstable drift waves in presence of dust

Abstract

The well-known stability of the drift wave in a sheared slab geometry does not hold in the presence of dust particles. Due to the presence of dust particles, the magnetic shear damping is reduced drastically. As a result, collisionless drift modes become unstable under typical parameter regimes of tokamak. Consequently, drift wave must still be considered as an underlying dynamic of anomalous transport in tokamak edges, where dust particles are found to be abundant, the same physics is expected in the spacial environments as well.

The common belief is that drift-wave turbulence cannot be the agent behind energy and particle transport in tokamak edges (1). This is because the sheared slab modes are linearly stable (2), some interplay with toroidal effects is needed to provide a nonadiabatic electron response sufficient to drive them (3) and because the toroidal driving term (being proportional $\frac{L_n}{R}$, where $L_n$ is the density scale length and $R$ is the major radius) falls towards the edge regions in contrast to the observed rise in saturated fluctuation amplitudes (4).

For considering the viability of drift waves as an agent for transport, however, one should take into account the presence of dust particles in the plasma, since the presence of dust has been confirmed in the edge of fusion devices like TEXTOR-94 (5). Although the existence of dust has been known for a long time, only recently their presence in fusion devices and its possible consequences on plasma operation and performance have begun to be addressed (5–9). Dust can be formed by evaporation and sublimation of wall material which is thermally overloaded, for example in the course of a disruption (5), or by spallation and flaking of films grown for wall conditioning (carbon-, boron- or siliconization) (10). Large molecular ions have indeed been identified during carbonization (11). Future fusion devices operating with D-T will have 3He (due to decay of T), 4He (due to neutron induced spallation reactions of Iow-Z wall material) and/or alpha particles, all of these may induce embrittlement of the near surface and lead to the ejection of grains. Furthermore, in the future long-pulse fusion devices, the growth and agglomeration of dust particles may occur via sputtering processes in the edge regions of plasma. It is believed that conditions prevailing at the edges of a detached Limiter and a detached divertor are appropriate for such sputtering process (5). Dust is therefore an important safety issue for ITER and for other future fusion reactors as many deleterious effects of dust have recently been predicted (5, 6). It is therefore extremely important to investigate what effects these dust particles have on the stability of microinstabilities and consequent plasma transport.

In this extended abstract, we revisit the theory of drift waves in a sheared slab geometry in the presence of dust. We demonstrate that the well-known stability of drift waves in a sheared slab geometry does not hold in the presence of dust particle. Due to the presence of dust, the magnetic shear damping is reduced drastically and as a result, collisionless drift waves become unstable for typical parameter regimes of tokamak operation. Consequently, drift wave must still be considered as an underlying dynamics of anomalous transport in tokamak edges.

We will first develop a nonlocal theory of collisionless drift waves in the presence of dust. We consider a multicomponent plasma of plane slab geometry with a uniform temperature but with a nonuniform density with density gradient in x direction. We also consider a sheared magnetic field, so that the magnetic field is represented by $B_0(x)=B_0({\hat{e}}_z+x/L_z{\hat{e}}_y)$, where $L_s$ is the magnetic shear length and $x$ is the distance from the mode rational surface defined by $k.B=0$. In this sheared magnetic field, the parallel wave number becomes ${\mathbf{\text{k}}}_{||}=k_{y}x/L_s$. Our plasma system consists of electrons, positive ions and negatively charged dust particles and the plasma is overall charge neutral, i.e. $n_i=z_d{n}_d+n_e$. A variety of competing processes such as photo-electric emission, secondary emission and so on determine whether a dust particle will be charged negatively or positively. In our case, keeping in mind the considerably higher mobility of the electrons with respect to the ions, we have considered dust particles to be negatively charged. We also assume that the variation of $q_d$ with $n_d$ is small in comparison to the variation of $n_d$ with $x$ (12). We further assume that the positive ions and dust particles are cold and describe them by the usual fluid equations and the electrons are assumed to follow the Boltzmann relation (13). To study the effect of electron Landau resonance and trapped electrons, we consider the so-called $\mathrm{i\delta }$ model. Dust particles are assumed to be unmagnetized. As the plasma is in homogeneous in the .r direction only, the perturbation takes the form $\phi (x,t)=\phi (x)\exp [i(k_{y}y+k_{z}z-\mathrm{\omega t})]$. The continuity equation and the equation of parallel momentum can then be written as (1) $\begin{array}{rl}& -\mathrm{i\omega }{n}_{\alpha }+{\mathrm{\nabla }}_{\mathrm{\perp }}.[n_{\alpha }(x)v_{\alpha \mathrm{\perp }}]+i{k}_{||}{n}_{\alpha }{v}_{\alpha ||}=0\end{array}$(1) (2) $\begin{array}{rl}& \omega v_{\alpha ||}=k_{||}\left(\frac{q_{\alpha }}{m_{\alpha }}\right)\phi \end{array}$(2) where (3) $\begin{array}{rl}& {\mathrm{\nabla }}_{\mathrm{\perp }}=i{k}_y{\hat{e}}_y+{\hat{e}}_x\frac{d}{\mathrm{dx}}\end{array}$(3) (4) $\begin{array}{rl}& v_{\alpha \mathrm{\perp }}=v_E+v_{\mathrm{\alpha p}}\end{array}$(4) (5) $\begin{array}{rl}& v_E=-c\frac{({\mathrm{\nabla }}_{\mathrm{\perp }}\phi \times B_0)}{B_0^2}\end{array}$(5) (6) $\begin{array}{rl}& v_{\mathrm{\alpha p}}=i\left(\frac{\mathrm{c\omega }}{B_0{\omega }_{\mathrm{c\alpha }}}\right){\mathrm{\nabla }}_{\mathrm{\perp }}\phi \end{array}$(6) where the subscript a represents the dust particles (d) and the positive ions (i). Here $n_i(n_d),m_i(m_d),v_{||}(v_{d||})$, ${\omega }_{\mathrm{ci}}(=e{B}_0/m_{i}c),{\omega }_{\mathrm{cd}}(=z_{d}e{B}_0/m_{d}c)$ are respectively the perturbed ion (dust particle) density, the ion (dust particle) mass, the parallel ion (dust particle) velocity, the ion and dust cyclotron frequency, whereas $v_{\mathrm{\alpha p}}$ represents the polarization drift. Using the quasineutrality condition, we obtain the radial eigenvalue equation for the low frequency and long wavelength limit as (14, 15) (7) $\begin{array}{r}{\rho }_H^2\left(\frac{d^2}{d{x}^2}-k_y^2\right)\phi -\left(1-\frac{{\omega }_H^{\ast }+\mathrm{i\gamma }}{\omega }-\frac{x^2}{x_s^2}\right)\phi =0\end{array}$(7) where ${\rho }_H^2=(1-\xi ){\upsilon }_s^2+\xi {\rho }_d^2,{\rho }_s^2=c_s^2/{\omega }_{\mathrm{ci}}^2,{\rho }_d^2=c_d^2/{\omega }_{\mathrm{cd}}^2,c_H^2=(1+\xi )c_s^2+\xi c_d^2,c_s^2=T_e/m_i,c_d^2=z_d{T}_e/m_d$, ${\omega }_H^{\ast }=(1+\xi ){\omega }_e^{\ast }-\xi {\omega }_d^{\ast }={\omega }_e^{\ast }[(1+\xi )-\xi L_{\mathrm{ni}}/L_{\mathrm{nd}}],{\omega }_e^{\ast }=-k_y{c}_s{\rho }_s/L_{\mathrm{ni}},{\omega }_d^{\ast }=-k_y{c}_d{\rho }_d/L_{\mathrm{nd}},\gamma ={\omega }_H^{\ast }\delta$, $x_s^2={\omega }^2/k_{||}^{\mathrm{\prime }2}{c}_H^2,k_{||}^{\mathrm{\prime }}=k_y/L_s,L_{\mathrm{ni}}^{-1}(x)=|\mathrm{dln}{n}_i/\mathrm{dx}|,L_{\mathrm{nd}}^{-1}(x)=|\mathrm{dln}{n}_d/\mathrm{dx}|,\xi =z_d{N}_d/N_e$ and $\mu =m_d/m_i$. Here, all symbols are assumed to have the usual meaning unless otherwise stated.

We consider the spatial variation of the diamagnetic drift frequency ${\omega }_H^8$ and consider the simple case in which ${\omega }_H^{\ast }$ is peaked at the mode rational surface at $x=0$ and has a parabolic profile (thereby considering the most unstable situation), i.e. ${\omega }_H^{\ast }={\omega }_{0H}^{\ast }(1-x^2/L_{0H}^2)$ where $L_{0H}$ is the density variation scale length and will be assumed to be of the order of $L_{\mathrm{ni}}$. With this, Equation (7) becomes (8) $\begin{array}{r}{\rho }_H^2\frac{d^2\phi }{d{x}^2}+(\mathrm{\Lambda }-P{x}^2)\phi =0\end{array}$(8) where (9) $\begin{array}{rl}& \mathrm{\Lambda }=\left(\frac{{\omega }_{0H}^{\ast }}{\omega }\right)-k_y^2{\rho }_H^2+\frac{\mathrm{i\gamma }}{\omega }-1\end{array}$(9) (10) $\begin{array}{rl}& P=\frac{1}{L_{0H}^2}-\frac{L_{\mathrm{ni}}^2[(1+\xi )+\xi z_d/\mu ]}{{\rho }_s^2{L}_s^2{[(1+\xi )-\xi L_{\mathrm{ni}}/L_{\mathrm{nd}}]}^2}\end{array}$(10) In deriving Equation (8) we have assumed that $\omega \approx {\omega }_{0H}^{\ast }=|k_y{\rho }_s{c}_s[(1+\xi )-\xi L_{\mathrm{ni}}/L_{\mathrm{nd}}]/L_{\mathrm{ni}}|$, which is usually true for drift type waves. Equation (8) is a simple Weber equation. Depending on the sign of P, we have two types of solutions.

For, $P>0$ i.e. for (11) $\begin{array}{r}\frac{1}{L_{0H}^2}>\frac{L_{\mathrm{ni}}^2[(1+\xi )+\xi z_d/\mu ]}{{\rho }_s^2{L}_s^2{[(1+\xi )-\xi L_{\mathrm{ni}}/L_{\mathrm{nd}}]}^2}\end{array}$(11) the solution satisfying the physical boundary conditions, i.e. $\phi \to 0$ at $x=\pm \mathrm{\infty }$ is given by $\phi (x)={\phi }_0\exp [-\frac{\sqrt{|P|}}{2{\upsilon }_H}{x}^2]$

The mode therefore decays with $x$, i.e. it does not propagate and hence, is intrinsically undamped. Now, for the opposite limit, when $P<0$, i.e. for (12) $\begin{array}{r}\frac{1}{L_{0H}^2}<\frac{L_{\mathrm{ni}}^2[(1+\xi )+\xi z_d/\mu ]}{{\rho }_s^2{L}_s^2{[(1+\xi )-\xi L_{\mathrm{ni}}/L_{\mathrm{nd}}]}^2}\end{array}$(12) The solution is $\phi (x)={\phi }_0\exp [-i\frac{\sqrt{|P|}}{2{\upsilon }_H}{x}^2]$.

Thus, in this case we get a nonlocalized mode with outgoing energy flux at $x=\pm \mathrm{\infty }$. In the absence of any energy source feeding the wave, the perturbation will decay in time, because of convective wave energy leakage. The overall stability of the system is determined by the balance between this intrinsic damping and destabilizing effects modeled by the $\mathrm{i\delta }$ term and is given by the dispersion relation $\mathrm{\Lambda }={\rho }_H\sqrt{P},$i.e. $\left(\frac{({\omega }_{0H}^{\ast }+\mathrm{i\gamma })}{\omega }-1-k_y^2{\rho }_H^2\right)=i{\rho }_H{\left(\frac{L_{\mathrm{ni}}^2[(1+\xi )+\xi z_d/\mu ]}{{\rho }_s^2{L}_s^2{[(1+\xi )-\xi L_{\mathrm{ni}}/L_{\mathrm{nd}}]}^2}-\frac{1}{L_{0H}^2}\right)}^{1/2}$ which yields the stability criterion (13) $\begin{array}{r}\gamma <\frac{{\rho }_H{\left(\frac{L_{\mathrm{ni}}^2[(1+\xi )+\xi z_d/\mu ]}{{\rho }_s^2{L}_s^2{[(1+\xi )-\xi L_{\mathrm{ni}}/L_{\mathrm{nd}}]}^2}-\frac{1}{L_{0H}^2}\right)}^{1/2}{\omega }_{0H}^{\ast }}{(1+k_y^2{\rho }_H^2)}\end{array}$(13) To analyze the stability criterion, we will first consider a plasma without any dust particle. Without the dust contribution, the condition for stability of the mode is given by (14) $\begin{array}{r}\frac{1}{L_{0H}^2}<\frac{L_{\mathrm{ni}}^2}{{\rho }_s^2{L}_s^2}\end{array}$(14) Putting representative values of the plasma parameters from TEXTOR-94, e.g.$L_s\sim R\sim 175\text{cm}\text{.},L_{0H}\sim L_{\mathrm{ni}}\sim a\sim 46\text{cm}\text{.},{\rho }_s\sim 0.1\text{cm}\text{.}$ (15, 16), it is easy to see that the above inequality is easily satisfied. In other words, we have recovered the well-known result that the drift wave is stable in a sheared slab geometry (2).

We will now consider the situation when dust particles are present. To facilitate comparison with the experiments, condition (13) is simplified (for $L_{\mathrm{nd}}\sim L_{\mathrm{ni}}$) to the condition for instability. (15) $\begin{array}{r}\frac{1}{L_{0H}^2}>\frac{L_{\mathrm{ni}}^2}{{\rho }_H^2{L}_s^2}\end{array}$(15) The experimental data of dust particles from fusion devices are, however, not in general available. We will, therefore, take the dust density as the same with the impurity density found in TEXTOR-94. The drastic reduction in the stabilizing role of the magnetic shear is, however, clear because of${\rho }_H>>{\rho }_{s.}$. For an example, we assume the following representative values for a graphite (carbon) dust in the hydrogen plasma in TEXTOR-94 like tokamak (graphite grains have indeed been detected in TEXTOR-94 (5)), $L_s=175\text{cm}\text{.},L_{\mathrm{ni}}\sim L_{\mathrm{nd}}\sim L_{0H}\sim a\sim 46\text{cm}\text{.},{\rho }_s\sim 0.1\text{cm}\text{.},N_e={10}^{13}\text{cm}{\text{.}}^{13}\text{cm}{\text{.}}^{-3},N_d={10}^{11}\text{cm}{\text{.}}^{-3}$, $m_d={10}^8{m}_i$ (15–19). With these data, we find that the inequality (15) ls satisfied! This, therefore, shows that due to the presence of dust particles the well-known stability of the slab drift wave is destroyed and the mode becomes unstable due to the drastic reduction in the magnetic shear damping. The exact growth rate is given by the dispersion relation. (16) $\begin{array}{r}\omega =\frac{{\omega }_{0H}^{\ast }-\mathrm{i\gamma }}{(1+k_y^2{\rho }_H^2)+{\rho }_H{\left(\frac{1}{L_{0H}^2}-\frac{L_{\mathrm{ni}}^2[(1+\xi )+\xi z_d/\mu ]}{{\rho }_s^2{L}_s^2{[(1+\xi )-\xi L_{\mathrm{ni}}/L_{\mathrm{nd}}]}^2}\right)}^{1/2}}\end{array}$(16) In summary, we have revisited the theory of drift waves in a sheared slab geometry in the presence of dust. We demonstrate that the well-known stability of drift waves in a sheared slab geometry does not hold in the presence of dust particle. Due to the presence of dust, the magnetic shear damping is reduced drastically and as a result, both collisionless and collisional (dissipative) drift waves become unstable for typical parameter regimes of tokamak operation. Since the presence of dust has recently been confirmed in the edge of TEXTOR-94 and since it is also likely to be formed during the D-T operation, wall conditioning (carbon-’ boron- and siliconization) phases and its growth and agglomeration may occur via sputtering process in the edge regions of future long pulse fusion devices, this excitation of drift instability by dust may be a crucial issue for ITER and other future fusion reactors. Finally, it is interesting to note that, in contrast to the excitation of toroidal Alfven eigenmodes (TAE) by alpha particles which are formed only in an ignited tokamak, the excitation of drift waves by dust particles shown here will effect the tokamak operation at a much earlier stage. This is because, the dust can be formed even during the wall conditioning phases.