2.2
Instability and Transport
Ross-Mahajan [57] and Antonsen [58] showed that collisionless drift waves in a slab geometry are stable. It was shown by Chen [50] that when resistivity is taken into account drift waves are unstable and grow in time. Following this work, he showed [12] the relation of the drift waves instability and the anomalous transport of plasma across the magnetic field. He suggested that the phase shift between the density fluctuation and the potential fluctuation causes instability. It should be mentioned, however, that the work of Landreman, Antonsen, and Dorland showed that collisionless drift waves are actually unstable [59].
A stable drift wave causes no transport. When the perturbation is purely harmonic in time and space the fluid motion will also be completely harmonic. The drift wave mode simply move the plasma fluid back and forth in a harmonic motion in time and space. Only when growth or damping occurs during the drift wave evolution does net transport take place. The relation between instability and transport will be made more apparent in the following subsections.
2.2.1 Drift Instability
The study of drift waves actually began with investigations on instabilities in plasmas. Tserkovnikov [46] investigated instabilities that arises in inhomogeneous plasmas. The drift instability, which is known in early literature as the “universal” instability [60], is responsible for the enhanced diffusion of plasma across the magnetic field. The instability is solely caused by gradients in plasmas and does not require external drive. Chen [12] described in a simple physical picture how the instability of drift waves is due to local E×B (vE), drifts that enhance the density perturbation.
Let us consider first the case of a density fluctuation accompanied by potential fluctuation as depicted in Fig. 2.3. Let us consider a point in the plasma where the potential is maximum. The electric field generated by the potential peak causes a drift, vE, of plasma downward in the region to the right of the point and upward in the left
region. Due to the density gradient this drift causes a new maximum in density to be created at a point to the left of the previous one. In response to the new maximum, a new potential peak is created at the point, and the same mechanism is repeated resulting a continuous shift of the maximum point. When we consider the point with minimum potential, it will be found that the same shift in the minimum point occurs. Hence, when they are in phase, density fluctuations and potential fluctuations conspire to generate right-propagating drift-waves that are purely oscillatory.
Drift waves become unstable when the density fluctuation leads the potential fluc- tuation. As is depicted in Fig. 2.4, the downward drift vE on the right of a potential
maximum brings more plasma into the point where the density is already at its maxi- mum. This motion causes the fluctuation to increase as it moves with its propagation speed. Thus, a phase shift between density and potential fluctuation, in particular when density leads potential,
ˇ
∇n
v
d + − E E x y z Density fluctuation Potential fluctuation B.
vEFigure 2.3: Fluctuations of densitynand potentialϕWhen they fluctuate in phase, the drift wave is purely oscillatory
∇n
v
d + − E E B.
ñ Φ vEFigure 2.4: When the density fluctuation leads the potential fluctuation there is net transfer of plasma to the density fluctuation, causing an increase in the fluctuation.
gives rise to drift instability. A decaying drift-wave occurs when the density fluctuation lags behind the potential fluctuation.
The relation in Eq. (2.13) is known as the nonadiabatic response of electrons. It has been shown in Eq. (2.4) that when electrons respond adiabatically, ˇn∼ϕˇ. Therefore, it can be said that nonadiabatic response of electron causes fluctuation growth (instabil- ity) [56, 61]. The nonadiabatic response may result from dissipation, polarization drift, or finite Larmor radius. When dissipation takes place, electrons fail to move instantly along the magnetic field to balance pressure force and maintain the Boltzmann relation [Eq. (2.3)].
Not all drift waves are unstable. Beside the factors that cause instability, a stabiliz- ing effect is also present in plasmas. It was shown by Rutherford and Frieman [62] that magnetic shear can stabilize drift waves. Magnetic shear refers to poloidal magnetic field being a function of radius,By(x), or rotational transform that differs on different
magnetic surfaces.
This linear description of drift waves predicts that drift waves can grow indefinitely, but in reality unstable drift waves must eventually stabilize. Therefore, the saturation mechanism for unstable drift waves must come from nonlinear processes. One process was suggested in 1967 by Dupree [52], namely the resonant mode coupling. In this case the energy of the unstable mode is transferred to the unstable region which will be damped. Sagdeev and Galeev in 1969 [63] suggested that the stabilization comes from the nonlinear (inverse) Landau damping of the wave due to interaction with trapped ions.
§2.2 Instability and Transport 21
2.2.2 Transport due to drift instability
Early works on drift instabilities had indicated the relation between drift instability and transport across the magnetic field in plasmas. Moiseev and Sagdeev [15] found that drift instabilities may lead to an escape of plasma of the order of the Bohm diffusion. Chen [12] discovered that the phase shift between the density and potential fluctuations accounts for the anomalous transport commonly observed in fully ionized plasmas.
The generation of vortices by the E×B drift in a magnetized plasma can explain in a simple way how transport is a consequence of instabilities. It has been mentioned that drift waves are vortex that move with the phase velocity of the wave. As the structure moves the plasma drifts alternately in thex and −x directions. A heuristic explanation of instability-induced transport is sketched in Fig. 2.5, which illustrates that a vortex growing as it propagates in the positivey direction must have a left-right asymmetry such that there will be a greater drift of plasma in the x direction than in the −x direction, producing net transport in the x direction. Hence, instability is needed for transport to take place [64–66].
E B x y z vE vE
Figure 2.5: When the vortex grow as they propagate in the y direction, there will be more plasma drifting in the x direction than in the −x direction. On average there will be a net transport of plasma in the radial direction.
A quantitative explanation of transport due to the phase shift between density and potential fluctuations can be made in the following way. Let us consider the flux of plasma through a magnetic surface due to the fluctuations. The net flux of plasma in thex-direction is
Γx =hnvExi, (2.14)
Given that the flow velocity in the radial direction isvEx =−(1/B)∂yϕ, the radial flux
is Γx =− ıky B 1 4 hnˇ∗ϕˇi − hnˇϕˇ∗i+hnˇϕˇe2ıθi − hnˇ∗ϕˇ∗e−2ıθi (2.15) where the symbol (ˇ) denotes the amplitude.
If the amplitude of density and electrostatic potential are in phase (ˇn ∼ ϕˇ), Eq. (2.15) yields zero value. Hence, stable drift waves give no transport. A real value of the net flux will be obtained if the density and the electrostatic potential amplitude are out of phase. In specific, when Eq. (2.13) holds, Eq. (2.15) yields
Γx= ky B 1 2nˇ 2δ. (2.16)
It can be shown that Eq. (2.13) gives rise to instability: If we recall that Γx ∼ h∂ne
∂ti,
the above equation implies en∝ expδt. Thus, it can be inferred that the phase shift between density and potential fluctuations results in unstable (growing) fluctuations.