ICRF wave absorption in a hot plasma

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

6.2.4. ICRF wave absorption in a hot plasma

As shown by Stix [6.3], the absorbed power in the plasma Pabs can be determined by calculating the dissipated wave power Re(j E )

, which can be written in the form

I c.c

abs 16i

P  E K E



  

       (6.74) where the summation is over different modes, K

is the hot plasma dielectric tensor [6.3], I

is the unit diadic and c.c. represents the complex conjugate. Thus, the contribution will come from the anti-Hermitian part of K

and the Hermitian part will cancel.

6.2.4.1. Absorption on electrons

Now we will be interested in Landau-type interactions with electrons, namely  kvth e, . Examining the hot plasma dielectric tensor [6.3], we find that the following terms may contribute: Kzz, Kyy, Kyz and Kzy; and after some lengthy algebra, we obtain the power absorbed per unit volume per unit time (in cgs-Gaussian units)

2 2 2 2 2 2 2

1/2 2 2 2

4 v , e

abs pe y z y z e

ce th e

P k N E k E E E e

k k

 

   



    

 

 

  

 

     (6.75)

In Eq. (6.75) we defined v2th e, 2 /T me e, pe2 4n e me 2 / e, / v ,

e k th e

   and ce eB cm/ e. While Landau damping is the result of the force on a charge due to the parallel wave electric field (eEz), transit time magnetic pumping results from the force associated with the magnetic moment and the wave magnetic field ((B) with  being magnetic moment).

The spatial damping decrement is given by the ratio of the absorbed power Pabsand the Poynting flux S [6.3]

2kIm Pabs/S (6.76)

where we take S S  ~cN Ex| y| /82  with Nx ckx/, we assumed k2k2, and k kx. Evaluating Eq. (6.76), and considering the polarization of the fields, we obtain the effective spatial damping rate [6.47, 6.48]

1/2 2

Im Re 2

2 1 1

2 e e e

k k    e 



  

   

      (6.77)

where e n Te e/ (0B2) is the electron beta, k is in cm–1 and

 2 2  22

2 2 ci S

e zz

i pi

T N K

m c

 

 

  

   

 

   (6.78)

In the cold plasma limit e21, Eq. (6.78) reduces to

2 2 4

2 4

1 m ce T



 

 

   (6.79)

whereas in the hot plasma limit e21 and Eq. (6.78) becomes

 

2 4 4

2 4 4 2 4 4

1 1

1 0.86

e

e pi e e pi

m c N

T

 

        (6.80)

The term Eq. (6.79) has been noted previously by Moreau et al. [6.49]. For example, for e0.7 and the previously listed plasma parameters, 1/2 0.31. However, lowering the electron temperature from Te 6.0 keV to Te3.0 keV increases 2 to unity. At higher phase velocities (e21), 1/2 is less significant for Te 3.0 keV. Replacing kRe~ / v A and neglecting 1/2 to lowest order we get the simple result

1/2 4

Im 4 2

2 exp( )

2

pi e e e

pi

k c

     

    (6.81)

so that for e ~ 1, single pass absorption is proportional to , n3/2, Te and B–3. Again, k is in cm–1. Note that the maximum absorption occurs for

~ 0.7

e . For example, for present day machines, Te06.0 keV,

0 2.0 T

B  , ne0~ 5 10 m 19 3, e0.03, f = 76 MHz in a D plasma, e~ 0.7,

~ / 2 r a

 ~ 0.50 m, 2k r 0.62 and the single pass absorption is 1 exp( 2kIm x) 0.47

    

  . For example, this may be an achievable value in the DIII-D tokamak. After a few bounces, most of the power would be absorbed by electron Landau damping. The required parallel wavelength at the antenna would be   60 cm, which is very reasonable (toroidal wave propagation effects would reduce this to   44 cm near the centre of the plasma where e~ 0.7).

For a phased array (typically 4 elements) with 90° or 120° phasing, a directional spectrum would result which would drive significant on-axis toroidal current, of magnitude ne[10 m ] [MA] [m] / [MW] 0.1 [10 keV]20 –3 I R P  Te [6.47]. Thus, for R = 2 m, ne = 1 × 1020 m–3, Te = 10 keV, a power P = 10 MW would yield a current drive of I = 0.5 MA. In summary, the fast wave Landau absorption condition

/ vk th e, ~ 1

  is satisfied for [keV] 50 / 2

Te ≈ N (6.82)

which follows from Eq. (6.63) for θ =7 (quasi-linear Landau damping).

6.2.4.2. Absorption of ICRF waves on ion cyclotron harmonics

One of the competing mechanisms for the absorption of fast magnetosonic waves is absorption on ions near the fundamental, or the harmonics of the ion cyclotron frequency [6.3, 6.43, 6.44, 6.47, 6.48]. This may occur on bulk ions or fast ions due to simultaneous neutral beam injection or even on alpha particles in a reactor grade plasma. Ion Landau damping can occur either directly or through

the intermediary of mode conversion into an BW and subsequent ion Landau damping of the IBW. The latter process may dominate if N0. First, we shall assume that Nis finite (and in particular / vk th e, ~ 1 or Nc/ vth e, ) and that the fast wave power density is not high enough to distort the initial Maxwellian distribution of ions. When this condition is violated the situation becomes considerably more complex [6.43] and we need to consider quasi-linear theory [6.50].

Ion cyclotron harmonic absorption in the limit of near-perpendicular propagation may be obtained from the general result Eq. (6.74). After a considerable amount of algebra, for bi1, where bi is the finite Larmor radius, and l/ci, Eq. (6.74) reduces to [6.47, 6.48]

 2

2 2

1/2 ( 1) 2 2

, ,

( 1) exp

16 v ( 1)!2 v

ci

pi i

th i l th i

b l l

P E

k l k

  

  

  

  

     

(6.83) where we considered absorption only by the left hand polarized field component E.Here l is harmonic number, E ExiEy is the left hand polarized component of the wave electric field and bi k r2 2ci is the finite ion Larmor radius (with rci2v / 2ti 1/2ci and vth,i being the ion thermal velocity). This formula is valid for bi1, which is usually satisfied for not too high harmonics since

~ / vA

k  so that k r2 2ci ~l b2 i/ 2. We recall the result following Eq. (6.72), namely that at ω=ωci, E+ → 0 since the ions shield out the left hand polarized component of the electric field (i.e. the magnetosonic wave becomes purely right hand polarized for perpendicular propagation). As is well known [6.43, 6.44, 6.47, 6.48], if strong ion cyclotron absorption is desired, the polarization problem may be remedied by injecting a minority ion species into the plasma, typically a few per cent of H or 3He ions into a deuterium majority plasma. Thus ion cyclotron absorption becomes effective again at   cm, where the subscript m designates the minority species, since E(  cM) 0 (where M designates the usually heavier ion species). For  2CD CH (where CD, CH are deuterium and hydrogen cyclotron frequencies), E/Ey 21, while for a 3He minority resonance in a deuterium plasma 4 / 3CD and E/Ey 2 1/ 9. As a consequence, absorption is strongest in a deuterium plasma with H minority and 3He minority absorption is weaker by nearly an order of magnitude compared to D with H minority. However, in large tokamaks net absorption is sufficient as a result of multiple passes through the resonance layer. We can obtain the damping by integrating the power absorbed across a cyclotron harmonic resonance layer in a radially inhomogeneous magnetic field and divide the absorbed power by the Poynting flux S ( / 8 )c  N E| y|2 (see Eq. (6.76)). The dominant factor in the integral comes from the exponential factor, and integrating through the resonance [6.47, 6.48] in a single ion species plasma at harmonics of the ion cyclotron frequency we obtain

2 ( 1) ( 1)

2 2 4 ( 2)!

piR l i l l

c l

 

 

   

    (6.84)

where  is the transmission coefficient and R is the major radius of the tokamak at the cyclotron resonance layer, i is the ion beta, l is the harmonic number and pi is the angular ion plasma frequency. This result shows that at l = 1 the absorption is zero, at least when one uses cold plasma theory to calculate the polarization. It can be shown [6.47, 6.48] that if we were to improve on the polarization calculation, in a hot plasma at l = 1 we would obtain a small but finite absorption given by

2

2 pi 2i

i

RN T c m c

  (6.85)

In present day plasmas this is negligibly small, but in a reactor grade plasma (ITER or DEMO) this absorption could be of the order of 0.1. For the simple situation assumed above, the power transmission is given by Tr exp( 2 )  , and thus the absorption is given by

1 1 exp( 2 )

b r

A  T     (6.86)

where we neglected any reflection or mode conversion. This is a reasonable assumption as long as cyclotron Doppler broadening dominates over the distance between the cyclotron resonance layer and any kind of mode conversion layer, for example, for l2, iR/ 2kvth i,R/. If this condition is violated the problem becomes significantly more complicated, leading to mode conversion into Bernstein waves and reflection and partial power transmission by fast waves [6.44]. In the Doppler regime, for l2, the absorption from Eq. (6.84) is

2 2

2 4

piRl i

c

 

  (6.87)

which is proportional to the ion beta. We also see that absorption at the third harmonic would be reduced by a factor of i2. It is easy to see that for i0.01 at the second ion cyclotron harmonic in a JET or ITER size plasma a significant absorption can take place.

6.2.4.3. Minority ion cyclotron absorption

In the presence of a minority ion component with a Zi/mi different than the majority species (for example, H+ or 3He++ in a majority D+ or T+ plasma), absorption at the ion cyclotron resonance can be significant even in a relatively cold (low beta) plasma, as long as cyclotron Doppler broadening dominates.

Hence, this scenario is used most often in present day experiments, as well as

during startup in ITER plasmas. The absorption formula remains the same as given by Eq. (6.74), but the plasma parameters are calculated based on minority concentration. However, the polarization E /Ey 2 may still be calculated by the majority cold plasma parameters as previously. Since the wave frequency now is at the minority cyclotron frequency and not that of the majority species, the polarization term remains finite and significant absorption can result as indicated by the formula [6.44, 6.47]:

2

2 2

pM m m

M M y

n Z E c Rn Z E

 

   (6.88)

where subscripts m and M refer to the minority and majority species respectively and Z is the ion charge. The result for H+ minority in a D+ majority plasma is

2 2

pD H

D

Rn

c n

 

 (6.89)

where subscript D refers to the majority D species and subscript H to the minority H species. Thus, we see that absorption is proportional to the fraction of minority density, which is typically 5%, and in present day experiments with R = 1 m and a density of D nD  5 10 m19 –3, we obtain 2 1 or we have an e-folding power loss through the resonance layer. At higher densities or in larger machines single pass absorption in excess of 90% is obtained, even in a relatively cold (say 1 keV) initially ohmically heated plasma, as long as Doppler broadening dominates, namely if nH /nD kvth H, /.

Additional complications from hot plasma contributions to the polarization calculation may have to be considered otherwise and absorption would be reduced. We note that absorption in D+ plasma with 3He++ minority would be reduced by a factor of 1/9 as compared to that predicted by Eq. (6.89), due to the polarization factor given by Eq. (6.72). It should be also noted that for D minority in a H majority plasma the absorption would be reduced again as compared to Fig. 6.5 and significant reflection my occur due to a reversal of the relative radial position of the cyclotron resonance layer and the left hand cut-off layer.

Thus, scenarios in startup plasmas with H majority need careful examination, and modelling with modern full wave codes should be used for a quantitative prediction of absorption, reflection and transmission.