Toroidal ITG mode

There are several branches of the ITG mode, such as the slab, impurity and toroidal[32]. We focus on the toroidal version, as it is the most relevant in tokamaks, and present here only an outline of the mechanism[33]. For a more in-depth derivation and discussion, see [25, 32].

Figure 3.1 shows a poloidal cross-section of a tokamak with both temperature and density gradients. The red region represents a hot, dense core, with temperature

T1 and densityn1, while the blue region is cooler and rarefied, with its correspond-

ing temperature and density T2, n2, and with T1 > T2, n1 > n2. As the magnetic

drifts go as the kinetic energy,vD ∼v2, an initial temperature perturbation results

in a differential drift across the perturbation. The ions in the cooler (negative) perturbation have a smaller drift than those in the hotter (positive) perturbation,

μΒxΒ B B +++ - - - ExB +++ - - - B B

Figure 3.1: Schematic of toroidal ITG mode mechanism. The∇B drift is dependent on the particle’s velocity, so those particles in the hotter, denser inner region have a larger drift than those in the cooler, rarefied outer region. This leads to a build up of charge on the edges of the perturbation, in turn leading to anE×B drift radially outwards on the outboard side, and inwards on the inboard side. The E×B drift enhances the perturbation on the outboard side, as it brings hotter plasma into the positive perturbation, and vice versa on the inboard side.

resulting in a build up of ions on edge of the positive perturbation and a deficit on the other side. This density build up/deficit leads to a potential perturbation as the adiabatic electrons stream in to balance the density and preserve quasineutrality. The potential sets up an E×B drift which drags hot plasma into the positive per- turbation and cool plasma into the negative perturbation, enhancing both of them. This feedback loop is what drives the instability, causing it to undergo exponential growth.

This mechanism not only applies on the outboard, low-field side of the toka- mak, but also on the inboard side, where∇B is pointing in the opposite direction to∇p. All the drifts are in the same directions, but now theE×B drift drags cool plasma into thepositive perturbation and hot plasma into thenegative perturbation, suppressing the instability. For this reason, the inboard side is often called the “good curvature” region, as the curvature drift acts to damp instabilities; correspondingly, the outboard side is known as the “bad curvature” region.

effects. The finite size of the gyroradius means that particles only feel the gyroav- eraged electrostatic potential perturbations with length scales smaller than the gy- roradius. This leads to a suppression of the instability for modes with kθρi1.

Neglecting FLR effects, the growth rate of the ITG mode is proportional tokθρi -

the inclusion of FLR is responsible for the turn over in the growth rate spectrum as the mode wavelength approaches the Larmor radius.

B

B

Figure 3.2: Magnetic shear damps modes with parallel variation.

Figure 3.2 illustrates the reason for instabilities in tokamaks to align them- selves along the magnetic field[34], that is, align themselves such that k·B = 0. The blue sheet represents the surface of constant perturbed potential in a sheared magnetic field. The mode has kk = 0 along the field line labelled B1, and this is

roughly true for a small radial excursion from this flux surface. However, because the magnetic field is sheared, at some nearby flux surface, the mode must have

kk 6= 0 alongB2. The rapid parallel dynamics act to quickly damp the perturbation

at this flux surface2. Hence the tendency for microinstabilities and turbulence to align with the field lines. We can be more precise than this. Given a mode with toroidal and poloidal mode numbers n and m respectively, we can state that we expect to see modes[25] with

m=−nq(ψ), (3.10)

whereq(ψ) is the safety factor on a flux surface ψ, defined in section 1.2.2. Modes that follow eq. (3.10) will find that they havek·B= 0 across the whole flux surface, on average.

The electron drift wave discussed in section 3.1 does not lead to any radial

2

Of course, this is only true for those instabilities susceptible to parallel stabilisation, which is the case for the microinstabilities discussed in this thesis.

flux. This can be seen immediately from the formula for radial particle flux:

Γ =δnδv⊥, (3.11)

where the overline indicates an average over time, andv⊥ =vE×B·eψ is the radial

component of theE×B velocity. Given that the time-average of a single fluctuating quantity is zero (as the mean is zero), and thatδnandδv⊥ are out of phase byπ/2

(asδnand φare in phase), then eq. (3.11) must also be zero. In fact, the version of the ITG mode presented here also lacks a radial particle flux, but it does lead to a time-average radial heat flux:

Q=δT δv⊥. (3.12)