If we want to profit from the inherent cyclotron motion of charged particles as described above, by injecting electromagnetic waves matching the cyclotron frequency, we need to know more about creation, propagation and absorption of such waves. We have to deal with the characteristics of the plasma in which the waves have to propagate, since the waves will be influenced by the presence of the plasma and not propagate in the same way as in a vacuum. Furthermore, an external antenna will have to be added, in which a certain current is exciting the waves. The appropriate equations are then Maxwell’s equations in a medium,
∇ ×
E= −
∂B ∂t (2.55a)∇ ×
B=
µ0 ∂D ∂t+
jant (2.55b)∇ ·
D=
ρant (2.55c)∇ ·
B=
0 (2.55d)2.4. Electromagnetic waves 27
related to the internal behaviour of the plasma. It equals the electric field in a vacuum.
jantis the antenna current andρantits charge density, i.e. these terms are related to the
source of the electromagnetic fields. The excitation from the antenna is periodic, and we can therefore write the temporal dependency asx
(
r, t) =
ˆx(
r)
exp(−
iωt)
. We can then define the vacuum wave number and use Eqs. (2.55a) and (2.55b) to obtain∇ × ∇ ×
E−
ω2µ0D=
iωµ0jant. (2.56)Now, we need to introduce a relation between the displacement fieldDand the electric
fieldE. This is done introducing the dielectric tensor
E
such thatD
=
e0E ·
E. (2.57)Note that this is in general a dimensionless tensor, which equals the unit dyadic in a
vacuum. The constant e0 then defines the vacuum permitivity. Now, the wave equation
for the electric field reads
∇ × ∇ ×
E−
ω2
c2
E ·
E=
iωµ0jant, (2.58)withc2
=
1/e0µ0 the vacuum speed of light. This equation can be solved for finding theelectric field once we know the dielectric tensor. As said above, the latter describes the action of the medium on the wave field, and we therefore have to derive the dielectric tensor from the description of the plasma. In Eq. (2.55b) we separated internal and external terms on the right hand side. If we now concentrate on the internal terms for finding the dielectric tensor, we have to include an internal current just as we have added the external current jant. Thus, the displacement field in Eq. (2.55b) contains a current
term
∇ ×
B=
µ0 ∂D ∂t=
µ0 e0 ∂E ∂t+
j . (2.59)Assuming again theexp
(−
iωt)
dependence, we findD
=
e0E+
i
ωj.
(2.60) We have already defined the dielectric tensor in Eq. (2.57), and looking at the last equa- tion, we see that the currentjhas to be proportional to the electric field for our definition of
E
to hold. We therefore define the conductivityσasj
=
σ·
E, (2.61)and then obtain
D
=
e0 I+
i e0ωσ·
E=
e0E ·
E, (2.62)28 2. Theoretical background
such that the dielectric tensor now takes the form
E =
I+
ie0ωσ. (2.63)
If we further decompose the electric field into a sum of plane waves exp
(−
k·
x−
iωt)
, Eq. (2.58) becomes inside the plasma (wherejant=
0)k
×
k×
E+
ω2
c2
E ·
E=
0. (2.64)Nontrivial solutions can be found by setting the determinant to zero, which yields
D
(
k, ω) =
det kk−
k2I+
ω 2 c2E
=
0. (2.65)This equation relates the wave vector kto the frequency ω. Such equations are called
dispersion relations. If we define without loss of generality the coordinates such thatkis in thexzplane, define the refraction indexn
=
kc/ωand decompose Eq. (2.65) into the parallel and perpendicular directions, we obtain− E
xyE
yx(
n2⊥− E
zz) + (
n2− E
yy)
h
n2k
E
zz+ (
n2⊥− E
zz)E
xxi
=
0. (2.66)If we know the dielectric tensor, the dispersion relation (2.66) yields the refraction index, and thus the wave vector, and we can finally find the electric field with Eq. (2.64). It is interesting to note a few properties of the dielectric tensor. First of all, the real part of the dielectric tensor is related to the propagation of the wave, whereas the imaginary part describes the absorption of the wave by the plasma. An intuitive way of seeing this is its direct relation to the wave number in Eq. (2.66). Recalling that the electric field has a dependenceexp
(
ik·
x)
, we see that the real part ofkis related to a sinusoidal behaviour (propagation), whereas the imaginary part ofkwill result in exponential growth, or rather exponential decay, which is more physical since exponential growth needs infinite energy. Decay of the wave inside the plasma means, by virtue of energy conservation, absorption of the wave by the plasma, through the so-called Landau damping, which describes the damping (decay) of the wave due to wave-particle interactions. Other interesting phenom- ena take place in the limits where n→
∞orn→
0. The first case is called resonance. This is where most wave-particle interactions happen. The latter case is called cut-off, and it means that the plasma is unable to support the wave. Propagation is impossible and the wave is reflected.If we want to find the dielectric tensor, we need to find the conductivityσ. By its defini- tion, Eq. (2.61), we have to find the relation between the current and the electric field. This can be done using the kinetic description, in particular the Vlasov equation, Eq. (2.51). As described earlier, the electric field is contained in this equation through the Lorentz force F
=
Q(
E+
v×
B)
. This description is especially useful for finding the dielectric tensor since the current is defined as the first moment of the distribution function f as given in Eq. (2.52b). Indeed, the distribution function yields the description of the plasma2.4. Electromagnetic waves 29
and Eq. (2.61) together with the current defined as a moment of the distribution function allow us to find the dielectric tensor, which in turn allows the resolution of the wave equa- tion (2.58). We will show in the Chap. 5 and Appendix A the detailed computations for the specific distribution function we implemented in the code.
2.4.1. The Ion Cyclotron Range of Frequencies (ICRF)
In order to heat the plasma to the needed temperatures for fusion devices, auxiliary heat- ing (i.e. additional to the Ohmic heating generated by the toroidal plasma current) has to be applied. One promising possibility is to profit from the cyclotron motion of the particles, as described in Sec. 2.2.1, and couple an electromagnetic wave with a similar frequency to the plasma. One advantage is that, as described in Sec. 2.1.1, the magnetic field
shows a dependence on the inverse of the major radius, B
∼
1/R, and therefore thecyclotron frequency of a given ion species (i.e. given charge over mass ratio) also has a
1/Rdependence. An injected wave with a fixed frequency will therefore interact with the resonant species at one given major radius, whereω
≈
Ω(
R)
. A localised heating of theplasma is therefore possible, which can be a great advantage.
In the ion cyclotron range of frequencies,ω
∼
Ωi=
ZeBc/mi, the electrons can shortcircuit the parallel electric field, since they move much faster than the wave, due to their
much lower mass. Here chose a system of coordinates whereB
=
B ˆzandk=
k ˆx. Thedispersion relation (2.66) then simplifies to
n2⊥
= E
yy−
n2k−
E
xyE
yxE
xx−
n2k, (2.67)
where in the cold approximation (i.e. no thermal contribution from the plasma)
E
xx= E
yy= (E
++ E
−)
/2 (2.68a)E
yx= −E
xy=
i(E
+− E
−)
/2 (2.68b)E
±=
1−∑
s ω2s ω(
ω±
Ωs)
. (2.68c)ωs and Ωs are the plasma and the cyclotron frequencies of species s. We consider
the cold approximation here just for obtaining a simple overview. The dielectric tensor developed in this thesis containing the thermal effects of a strongly anisotropic population will be shown later in Chapter 5. Stix18has shown that the electric field polarisation, i.e. if one decomposes the electric field into two circularly polarised fieldsE±
= (
Ex±
iEy)
/2,is given as E+ E−
= −
E
xx+
iE
xy−
n2kE
xx−
iE
xy−
n2k= [
cold approx.] = −E
+−
n2 kE
−−
n2 k . (2.69)One can see that at the cyclotron resonance of any species in the plasma,
E
−→
∞,and therefore E+
→
0. This is true for the cold approximation. In the warm approx-30 2. Theoretical background
behaviour is similar. Unfortunately, the left handed component of the electric field E+
is the more important one, since it rotates in the same direction as the ions around the magnetic field lines. Thus, if one tries to heat the plasma using cyclotron waves in a sin- gle species plasma atω
∼
Ω, the wave-particle interactions become very inefficient dueto the (nearly) vanishing of E+. However, there are two main solutions to the problem: One is to apply a wave with a frequency of an integer multiple of the cyclotron frequency,
ω
∼
nΩ, so-called nthharmonic heating, or one can apply the heating at the fundamen-tal harmonic (n
=
1) to a small minority species, such thatE+can be sufficiently strong at the resonance in the warm model. The first possibility has the disadvantage that at high concentrations of a second ion species, the so-called ion-ion hybrid resonance ap- pears besides the cyclotron resonance and, especially, a corresponding cut-off limits the propagation of the injected wave. If the ion-ion hybrid resonance is reached, mode con- version takes place, which is the transition of the initially injected wave into another type of wave (like e.g. Bernstein waves). We will concentrate on minority heating, where no mode conversion takes place, and no cut off exists in the plasma. Here, the left handed component E+ will still show a slight local minimum along the cold resonance, but it will also show a local maximum at the high field side of that layer (if injected from the low field side). If the minority ions are warm, and thus show a non-zero Doppler shift proportional to kkvk, maximum power deposition will be located slightly displaced from the cold res- onant layer, on the high field side. We will show that this is the case, e.g. in Fig. 5.2. One reason why we choose minority scenarios is that mode conversion can only be mod- elled with finite Larmor radius models, whereas our wave code, LEMan, is limited to a zero Larmor radius model. The advantage of minority heating is that the ion species, and thus the charge over mass ratio, can be chosen, and therefore the choice of a cyclotronresonance position ω
∼
ZeBc/mis possible not simply by choosing a wave frequency.It might be enough to locally heat the minority species using rather low power wave in- jection, e.g. for generating a localised current which acts on the safety factor profile as described in Sec. 2.1.1. Note that even if the final goal is to heat the background plasma, minority heating is a reasonable alternative, since once the power has been absorbed by the minority species, the latter are describing large orbits, and are slowing down along these orbits through Coulomb collisions with the background ions and electrons. It is then possible to heat the background plasma indirectly, and in a much larger region than the initial ion cyclotron power deposition. We will study both effects, i.e. Ion Cyclotron Current Drive (ICCD) and Ion Cyclotron Resonance Heating (ICRH) in detail in this thesis.