Lower hybrid 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

6.2.2. Lower hybrid wave propagation and accessibility

The second frequency regime of interest is that of the so-called lower hybrid range of frequencies (see Fig. 6.3). In the usual case of 2pici2,

2 2

pe ce

  , the lower hybrid frequency LH is near the ion plasma frequency ( / )1/2

pi m me i pe

   and is a mixture of electron and ion contribution. Taking the limit cos 0 in Eq. (6.28) we get S = 0 or

2 2

LH (1 2 / 2)

pi

pe ce

 

 

  (6.47)

where ci2 ce2(m me/ i) was neglected in comparison with 2pi. The dispersion relationship for lower hybrid waves in the cold plasma is obtained from Eq. (6.28), by multiplying the equation with k2 k2 k2, and in the present limit one obtains

2 1/2

LH 1 2 i

e

k m k m

    

 

 (6.48)

where we also assumed k2k2, valid in the plasma core but not at the plasma edge. This dispersion relation predicts that kk  at   LH if k is fixed at the surface by a slow-wave structure. This result has been verified in linear machine experiments in the 1970s by Bellan and Porkolab [6.7]. It was also

shown that the wave energy propagates along trajectories of the group velocity which define so-called “resonance cones” [6.6], emanating from the launching structure at the plasma edge (phase array waveguides, or “grill”, see Fig. 6.3).

FIG. 6.3. Penetration of lower hybrid waves into an inhomogeneous plasma. Regions A, B, D correspond to the cold resonance cone regimes, region C to the mode conversion region, and E to mode converted (hot ion plasma or Bernstein ) waves; 0 is the RF source frequency [6.1].

Reprinted from Ref. [6.1]. Copyright (2011), with permission from Ref. Elsevier.

The resonance cone angle is defined by the trajectory of the group velocity, and it is perpendicular to the phase velocity as given by Eq. (6.48). Thus, the cone angle is

2 2 2 2

LH LH

tan  (m mi / e)(  ) / (6.49)

In Fig. 6.3 it is assumed that the plasma is uniform in the z direction and the density is increasing with radius from the edge (a) to the plasma centre (r0) and k(i.e. N) isfixed by the antenna array phasing. Note that the theory breaks down at r1(z1) where k2 approaches infinity and finite temperature effects must be taken into account. Owing to their low N,rays D are not accessible (see Fig. 6.3) as explained below. Note that in the toroidal direction the path of the wave penetration is long, and in particular Eq. (6.49) shows that Lz ~ (a m me/ )i 1/2 (where a is the plasma radius and Lz is the path length along the magnetic field). Since the electric field strengths are strong (Ex10 kV.cm1), non-linear effects in the outer plasma layers could impact the lower hybrid or “slow wave” propagation. For example, parametric instabilities, soliton formation and scattering by drift-wave fluctuations may occur [6.29–6.32]. Although these processes have been observed in experiments, at present there is no clear understanding of these phenomena and therefore usually linear theory is assumed to hold as long as the frequency is well above the lower hybrid frequency.

To describe wave propagation, accessibility and mode conversion at the lower hybrid layer in an inhomogeneous plasma column, we must keep electro- magnetic terms from Eq. (6.31). Then, in the lower hybrid regime for B24AC we have

2 P( 2 S) D2

S S

Ns   N   (6.50)

2 2 2

2 2 2

D ( S)

( S) D / P

f

N N

 N

 

  





(6.51) where the subscripts s and f refer to the slow and fast branches and P, S and D were defined earlier. In particular, in the lower hybrid limit of plasma parameters, we have

2 2

2 2

S 1 pe pi

ce

 

 

   (6.52)

2

P 1 pe2

   (6.53)

2

D pe

ce



 (6.54)

Resonance for the slow wave (n2s  ) exists at the location where S = 0 or

2 1

2 2 1

pe ce ci

ce

  

 

 

   (6.55)

that is 2LH2 . In the limit n21 2 2pe /ce2 2pi /20(1) the slow wave is nearly electrostatic and is referred to as the lower hybrid(Eq. (6.48)).

If B2 4AC, the solution of the dispersion relation predicts two distinct branches, as shown in Eqs (6.50) and (6.51), and in Fig. 6.4: a slow wave and a fast wave. In general, these roots are distinct as long as B2 4AC. However, for a sufficiently low value of N, the condition B2 4AC may be satisfied at two distinct values of the density, ne and ne, which are both lower than the density at the lower hybrid resonance layer. At these critical densities, the slow and fast modes coalesce, as illustrated in Fig. 6.4. The critical densities are given by the relationship

2 2 1/2

/ 1 ( 1)

pi N g N g

        (6.56)

where g2  2/ ce ci. Equation (6.56) specifies the minimum value of N required for a wave packet to propagate to a density and magnetic field as specified by pi( )n and g(B). The minus sign corresponds to the density ne (see Fig. 6.4) and is relevant for electron Landau heating and thus current drive applications. The plus sign corresponds to ne, the mode trapped in the plasma core, and is of relevance only for conversion to the finite temperature ion plasma wave and applications to ion heating. In particular, if we wish to have accessibility all the way to the lower hybrid resonance layer and heat ions, Eq. (6.56) reduces to N2 1/ (1g2) and if the lower hybrid resonance layer is in the plasma, accessibility to it is given by

LH

2 1 2pe / ce2

N      (6.57)

This relation was first obtained in Ref. [6.33]. Since typically 2pece2, we see that Ncrit 1.4 and a slow wave structure is necessary to launch a wave from the surface of the plasma column [6.7, 6.34].

FIG. 6.4. Accessibility diagram of lower hybrid waves for two different values of N. Left: The Nis too low for accessibility to the lower hybrid layer; however, accessibility to the slow–fast wave mode conversion layer (ne) is available for electron Landau absorption and thus current drive applications. The ns is the critical density, where the slow wave is cut off, namely   pe. Right: The Nis high enough to provide accessibility to the lower hybrid resonance layer and thus mode conversion to ion plasma waves and ion Landau absorption [6.35].

It is worth noting that near the edge of the plasma, Eq. (6.50) reduces to

2 (1 2pe/ 2)(1 2)

N    N and since 1N2, propagation is only possible if

2 2

1pe/ , or the plasma has to be “overdense” (see Fig. 6.4, density ns). This

can present difficulty in launching waves with low scrape-off layer densities, and will be discussed in Section 6.3.4.

When the lower hybrid resonance does not exist in the plasma (  LH), electron heating and/or current drive may still take place. In this case, the condition for wave penetration to the maximum density is still given by Eq. (6.56), if g2 1 and 2pi(0) / 2 g2/ (1g2). Otherwise, accessibility is determined by the negative root of Eq. (6.56), and the critical value of n needed for the slow wave to penetrate to a given density without mode conversion to the fast wave is

2 2pe/ ce2 (1 2pe/ ce2 2pi / 2 1/2)

n        (6.58)

For applications to RF current drive (which is of the greatest interest in connection with lower hybrid waves), absorption by electron Landau damping is of importance, which occurs due to interaction parallel to the confining magnetic field [6.36]. We note that Landau damping is the most important collisionless damping process occurring in high temperature plasmas and is due to collective effects associated with phase mixing of waves and interaction of waves with a distribution of particle velocities. Physically, Landau damping occurs since particles moving with velocities slightly less than the phase velocity of the wave are accelerated by the wave, and particles moving slightly faster than the wave are decelerated by the wave. Thus, if there are more particles moving slightly slower than the wave than faster, net energy is transferred from the wave to the particles, and the wave is damped [6.37]. Such is the case for the Maxwellian velocity distribution. In the presence of even rare collisions, the absorbed energy is thermalized (i.e. the plasma is heated). We also note that electron Landau damping of lower hybrid waves occurs when /k~ 3vth e, [6.38, 6.39].

Specifically, one relies on the fact that Te peaks at the centre of the plasma column, and if the k spectrum is such that in the outer layers of the plasma

3vth e, /k (6.59)

and near the plasma core

,

vth e 3 k

 



(6.60) then absorption by electron Landau damping occurs near the centre. The damping rate can be obtained by adding the imaginary part due to Landau damping to the electrostatic dispersion relationship Eq. (6.48), and is given by the finite Te correction as

2 2 2 2 Im

Sk Pk 0

k  k i  (6.61)

where S 1 2pe/ce2 2pi/2, P 1 2pe/2, k2 is fixed by the antenna, k2 is complex and its imaginary part (damping term) is obtained from [6.7, 6.39]

1/2 2

Im 2 2 2 2

, ,

vth eexp v

D k th e

k k

  

 

 

  

 

 

(6.62) where D2 v2th e, / 22pe is the Debye length (squared) and vth e, 2 /T me e1/2. One should note that for an accurate treatment of Landau damping one should include the so-called quasi-linear effects; namely, the deformation of the particle distribution function due to the expected strong electric fields must be considered in a self-consistent manner, balancing RF diffusion versus collisional drag, leading to net momentum input parallel to the magnetic field in the case of an asymmetrically launched wave spectrum [6.39]. Significant electron heating should take place if [6.38–6.40]

e

N T

 

 (6.63)

where Te is the electron temperature in keV and  5 for linear Landau damping and  7 for quasi-linear damping. Subsequently, resonant electrons diffuse in velocity space to higher energies (as determined by the spectrum of waves) to create a suprathermal electron tail. Thus, we see that for a given value of N (usually determined by accessibility) quasi-linear Landau damping allows wave penetration to higher temperatures than linear Landau damping due to the diffusion of resonant electrons by the RF waves. If the antenna launches a unidirectional spectrum with respect to the magnetic field, the suprathermal electron tail is preferentially populated in the direction of wave propagation, and an electron current may be generated in the toroidal direction; this is the principle of RF current drive [6.40].

Regarding the ion plasma wave, in high temperature plasmas it is not accessible and we shall not discuss it here. The interested reader is referred to the extensive literature describing the mode conversion process to such waves [6.2, 6.3, 6.7]. From a practical point of view, there is a competition between accessibility and Landau absorption and we find that for reactor grade plasmas for 2pe ce2 , the accessible N1.8 and the electron Landau absorption limit is about 12 keV, which limits LH wave penetration slightly beyond the top of the pedestal in an H mode plasma. This is fortuitous since alpha particles would absorb LH waves in the core of a burning plasma.