Ion cyclotron wave propagation and accessibility

RADIOFREQUENCY WAVES, HEATING AND CURRENT DRIVE IN MAGNETICALLY CONFINED PLASMAS

6.2. THEORY OF RF WAVE PROPAGATION IN A MAGNETIZED PLASMA The theory of wave propagation in magnetized plasmas has been

6.2.3. Ion cyclotron wave propagation and accessibility

The ion cyclotron wave, often called the “magnetosonic wave“, or simply the “fast wave” (FW) has been used successfully to heat plasmas for decades [6.2]. Note that on the CMA diagram (Fig. 6.2) this is the R-x wave, which propagates across the ion cyclotron resonance in a single ion species plasma without significant absorption. In contrast, the slow “ion cyclotron wave”, which is an extension of the shear Alfvén wave, is left hand circularly polarized and is absorbed at the ion cyclotron frequency as it propagates across the ion cyclotron frequency. In large machines at high densities it is not practical to launch such waves with external antennas and the waves would mainly propagate in the edge of the plasma. However, a version of ion cyclotron waves may be generated by mode conversion processes, and this will be discussed in subsequent sections.

The dispersion relationship of the fast magnetosonic wave in the ICRF (ion cyclotron range of frequencies) may be obtained from Eqs (6.30) and (6.31), and it is given by

2 2

2 2

( R)( L)

S

N N

N N

 

 

 



(6.64) where the appropriate expressions for L, R and S in the ICRF are as follows:

2

R 1 ( )

pi

ci ci

i



  

   (6.65)

2

L 1 ( )

pi

ci ci

i



  

   (6.66)

2

2 2

S 1 ( )

pi i ci



 

   (6.67)

Here pi is the angular ion plasma frequency and ci is the angular ion cyclotron frequency. Equation (6.64) predicts a rather complex behaviour for fast wave propagation and accessibility from the plasma edge, especially in the case of multi-ion species plasmas. In addition toRefs [6.41–6.43], an excellent summary of such phenomena can be found in Ref. [6.44]. Here we simply want to point out the salient features of Eq. (6.64).

The region of N2Rcorresponds to the right hand cut-off layer (N2 0), and the fast wave is evanescent at densities lower than this critical density, nR. The right hand cut-off layer always exists in the plasma, regardless of the relative value of / ci. Consequently, the wave has to “tunnel through” an evanescent layer in the plasma periphery and the reflected RF power must be prevented from

getting back into the RF source (usually a high power tetrode) by an external tuning (matching) network. The N2 Llayer is also a cut-off layer and in a single ion species plasma, this layer occurs only if   ci. The critical densities are

2 2

R ( 1) ci( ci) / ( i / i 0)

n  N     q m (6.68)

2 2 2 2

S ( 1)( ci ) / ( i / i 0)

n  N    q m (6.69)

2 2

L ( 1) ci( ci ) / ( i / i 0)

n  N     q m (6.70)

where nL and nR are the densities corresponding to the left and right hand cut-off layers and nS is the density corresponding to the resonance layer where N2 S and N2  . For   ci the relative densities are nLnSnR and the three densities are called the cut-off-resonance-cut-off triplet [6.3]. At the resonance layer, finite temperature effects may have to be included and mode conversion into the surface wave, or the “kinetic shear Alfvén wave”, may take place [6.3].

The inhomogeneous magnetic field of a tokamak will only quantitatively change this picture. If   cieverywhere in the plasma column, and if we consider regions of fast wave propagation well away from any cut-off or resonance layer, the fast wave dispersion can be approximated from Eq. (6.50) by the following simple relationship:

2 2 2 1/2

v (1A / pi)

k c k

     (6.71)

where vA cci /pi is the Alfvén speed and where usually N2N2 so that kvA. If   ci everywhere in the plasma, one needs to be concerned only with the right hand cut-off layer, which is usually in the scrape-off layer, in front of the antenna, and around the plasma cross-section, at a density of nR (see Fig. 6.5).

Next, we need to consider the importance of wave polarization here. In general, the fast wave is elliptically polarized, but at the majority ion cyclotron resonance it is purely right hand polarized in the cold plasma limit and thus we do not expect any absorption by ions. This can be obtained by inspecting the second row of Eq. (6.9), from which we can obtain the ratio of the left hand circularly polarized component of the electric field E+, which in the ICRF regime reduces to

/ y ( x y) / y s

E E  iE E E  (6.72)

where s = 0, 1 and 1/9 for / ci 1, 2 and 4/3, respectively. The first value corresponds to the fundamental majority cyclotron resonance frequency in a pure deuterium plasma, the second value corresponds to the harmonic of the deuterium cyclotron frequency (case of H minority cyclotron resonance in a majority deuterium plasma) and the third value corresponds to the cyclotron resonance of

a few per cent 3He minority species in a deuterium majority plasma (or second harmonic tritium minority in a deuterium–tritium plasma).

Since waves are absorbed mainly by the left hand circularly polarized component of an electromagnetic wave at its cyclotron frequency (unless the temperature of the plasma ions is very high, of the order of 100 keV or more), it will be necessary to seed the pure deuterium target plasma with a second ion species (minority species) for its cyclotron absorption by the wave. If a minority ion species (or a second majority ion species) is present in the plasma, the mode conversion layer will be affected by the second ion species. For example, if

  cmnear the centre of the tokamak plasma column, then once more, the

2 L

N  and N2 S layers will move into the plasma column and be located on the high field side of the lighter minority species cyclotron resonance layer. The

2 R

N  layer will remain near the plasma periphery, maintaining the presence of the evanescent layer. This is shown in Fig. 6.5 in the form of the plasma cross-section, and in Fig. 6.6 we show the various modes that can propagate and their relative perpendicular index of refraction. It should be noted that in a two ion species plasma corresponding to the above situation the N2 S resonance layer may occur in the plasma core and approximately we have the so-called ion-ion hybrid resonance occurring, where we balance the two (or more) ion species, and in the cold plasma limit

2

2 2 0

( )

pi

ii ci

i



  

 (6.73)

where ii stands for the ion-ion hybrid resonance frequency. This process was discovered early in fusion research and formed the basis of later experiments in TFR and PLT and others more recently, and the results will be discussed in Section 6.3.4. The efficiency of mode conversion processes in the absence of dissipation were discussed in the 1980s in Ref. [6.42].

FIG. 6.5. Plasma cross-section in the ICRF regime showing the R and L cut-offs and the S resonance region and their relationship to the proton cyclotron resonance layer. CD, CH are deuterium and hydrogen cyclotron frequencies [6.41]. Reprinted from Ref. [6.41].

Copyright (2011) by the American Physical Society.

By putting finite temperature into the equations, the mode conversion leads to a new hot ion plasma branch, that of ion Bernstein waves (see Fig. 6.6 left and Ref. [6.43]). Furthermore, if magnetic shear is taken into account off the midplane, a kinetic form of the slow ion cyclotron wave is excited by mode conversion (see Fig. 6.6 right and Ref. [6.44]). We see that in general, the fast wave propagates to the left hand cut-off layer, tunnels through the evanescent layer and, depending on plasma parameters, partially reflects, partially converts to the IBW just beyond the resonance layer, or to the ICW before the left hand cut-off layer, and partially continues its propagation as the fast wave (FW) branch. However, if the minority concentration is low (of the order of a few per cent), then the ion-ion hybrid resonance layer is near the minority ion cyclotron layer, and due to the finite value of k and Doppler shift, dominant ion cyclotron absorption can take place [6.3, 6.4]. So the situation can be quite complicated and, in general, the Budden tunneling problem must be solved with dissipation taken into account (due to finite N and cyclotron absorption), a process best handled by modern full wave codes (see Section 6.2.8).

FIG. 6.6. Dispersion relationships with mode conversion in a multi-ion species plasma as a function of major radius. Left: Without shear. Indicates excitation of ion Bernstein waves (IBWs) [6.45]. Right: With shear. Note that shear introduces a new mode converted wave, the ion cyclotron wave (ICW), in addition to IBW [6.46].