A major candidate for an instability causing fluctuations is the so-called drift wave. This instability feeds on the energy associated with a gradient in the plasma density (and/or temperature). In this section we will initially describe the basic features of this wave, and thereafter sketch a derivation of the dispersion relation in the cold ion limit (Ti ¿Te). Finally, a
primitive estimate for the fluctuation level will be presented.
It is important to note that we here treat collisionless drift waves, where kinetic effects provide the dissipation needed to release the energy required to make drift waves unstable.
Another class of drift waves is resistive drift waves, where resistivity is responsible for the dissipation [56] [85]. We will not treat this type of drift waves below.
5.4.1 Basic mechanism
The geometry we use is called the plane plasma slab. We assume that the magnetic field is shearless and uniform in the z-direction: B=Bˆz. The equilibrium density n0(x) is non-uniform in the x-direction; that is, it has a non-zero gradient ∇n0. The equilibrium temperature T0(x) is constant,
∇T0 = 0.
The parallel equation of motion of electrons in the electrostatic approximation can be written [85]
∂vek
CHAPTER 5. TRANSPORT IN FUSION PLASMAS - 15P 62 The left-hand side of equation 5.17 (electron inertia terms) can be neglected for low frequencies, i.e. for ω ¿ωci = eB/mi. Assuming that the electrons are isothermal (∂p∂ze =Te∂ne
∂z ) the equation of motion reduces to e∂φ
∂z = Te
ne
∂ne
∂z (5.18)
Solving this for ne and introducing the equilibrium electron density ne0 we arrive at
ne=ne0eTeeφ (5.19)
Writing the density as ne=ne0+δne and expanding the exponential function we end up having the Boltzmann distribution
δne
ne0
= eφ Te0
, (5.20)
where we have assumed that Te =Te0. The electrostatic potential implies a perturbed electric field in the y-direction, directed from regions of increased density (δne>0) to those of decreased density (δne <0). The electric field (in the y-direction) in associated with an E×B-drift in the x-direction, which leads to an increasing density in the δne <0 regions and a decreasing density in the δne>0 regions. The resulting wave propagating in the y-direction is the drift wave [87].
5.4.2 Dispersion relation
We wish to find the dispersion relation ω(k) of drift waves to investigate whether these frequencies are comparable to those measured. We will use a kinetic approach and base our procedure on the one given in [30]. What this subsection contains is not a derivation, but rather an overview of the ingredients in making the derivation.
Instead of using the particle distribution function f(x,v, t) we will make use of the guiding center distribution function fgc(xgc, v⊥, vk, t), where xgc is the position of the guiding center. We again work in the plane plasma slab geometry as used in subsection 5.4.1. We from now on omit the ’gc’
subscript, and so write the electron guiding center distribution as
fe(x, v⊥, vz, t), where vk =vz for our geometry. The distribution is assumed
CHAPTER 5. TRANSPORT IN FUSION PLASMAS - 15P 63 where v =p
v⊥2 +vz2. A perturbed quantityψ can be written
ψ(x, t) = ˆψ(x)e−iωt+ikyy+ikzz, (5.22) an assumption to be used when we later linearise equations. The basic equation for the electrons is the drift-kinetic equation
∂fe
If this equation is solved, the electron density can be found by integrating over all velocities, ne =R
fed3v = 2πR
fev⊥dv⊥dvz. Treating the ions as cold, the governing ion equations are:
1. The continuity equation ∂ni/∂t+∇ ·(niu) = 0, u being the velocity.
2. The perpendicular velocityu⊥= E×BB2 + M ∂eBtE2⊥, where the second term on the right-hand side is the polarisation drift.
3. The parallel equation of motion Mdudtz = eEz.
Linearising the electron drift equation one arrives at the expression ne1 = ne0eφ
for the electron density perturbation. Here, vt,e =p
Te0/m is the electron thermal velocity and vde =−ne0TeBe0z0dndxe0 is the electron diamagnetic drift velocity. Linearising the ion equations we obtain an expression for the ion density perturbation
Te0/M and the ion Larmor radius at the electron temperature ρs=√
M Te0/eBz0. Finally, using the quasi-neutrality approximation ne1 =ni1 to equate equations 5.24 and 5.25, we reach the expression for the cold ion dispersion relation
ω(1 +k2⊥ρ2s)−kyvde− kz2Cs2 Setting the imaginary right-hand side equal to zero we see that the
equation has two solutions, the electron branch (ω and kyvde have the same sign) and the ion branch (ω and kyvde have the opposite sign). Ifk⊥ρs ¿1 and kzCs ¿kyvde, the electron branch can be written
CHAPTER 5. TRANSPORT IN FUSION PLASMAS - 15P 64
ω ≈kyvde, (5.27)
which is the electron drift wave linear dispersion relation. Keeping the two terms on the left-hand side of equation 5.26 as small corrections, we obtain a slightly more complicated dispersion relation, namely
ω = kyvde
1 +k⊥2ρ2s +kz2Cs2 kyvde
(5.28) Usually, since kz =kk is very small, we neglect the second term on the right-hand side and use
ω = kyvde
1 +k2⊥ρ2s (5.29)
as our final dispersion relation.
To obtain an expression for the growth rate γ, we let ω→ω+ iγ (ω and γ are assumed to be real, γ/ω ¿1) in equation 5.26 and equate the
imaginary parts A negative γ means damping, positive γ growth. The physical process enabling an instability to develop is inverse Landau damping by electrons [30]. The growth rate can be used to obtain a diffusion coefficient [15]
D=L2nX to be taken over all existing modes. Results akin to this motivate
experimentalists to relate the energy confinement time τE toδne/ne0. It has been found that they are related for core turbulence (τE decreases for an increased density fluctuation level) but not related for edge turbulence [19].
Since it is unlikely that the density perturbation should grow beyond the amplitude at which it reverses the local density gradient, we get the upper bound k⊥δne ≤dne0/dr, or
δne
ne0 ≤ 1 k⊥Ln
, (5.32)
known as the mixing length estimate. Inserting this into equation 5.31, we get an upper limit on the diffusion coefficient, D≤γ(k)/k2⊥.
CHAPTER 5. TRANSPORT IN FUSION PLASMAS - 15P 65 The warm ion dispersion relation (Te ∼Ti) for small scales (k⊥ρs ≥1) is quite complex (see e.g. [56] or [75]), but for k⊥ρs ≤1 one can include an additional factor in the numerator of equation 5.29 [90].