Effect on plasma rotation of lower hybrid (LH) waves in Alcator C-Mod
J. P. Lee, M. Barnes, R. R. Parker, J. E. Rice, F. I. Parra, P. T. Bonoli, and M. L. Reinke
Citation: AIP Conference Proceedings 1580, 398 (2014); doi: 10.1063/1.4864572
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Effect on Plasma Rotation of Lower Hybrid (LH) Waves in
Alcator C-Mod
J. P. Lee, M. Barnes, R. R. Parker, J. E. Rice, F. I. Parra, P. T. Bonoli and M. L.
Reinke
Plasma Science and Fusion Center, MIT, Cambridge, USA
Abstract. The injection of LH waves for current drive into a tokamak changes the ion toroidal rotation. In Alcator C-Mod,
the direction of the steady state rotation change due to LH waves depends on the plasma current and the density. The change
in rotation can be estimated by balancing the external torque of lower hybrid waves with the turbulent radial transport of
the momentum. For high plasma current, the turbulent pinch and diffusion of the injected counter-current momentum are
sufficient to explain the rotation change. However, for low plasma current, the change in the the intrinsic momentum transport
(residual stress) for a non-rotating state is required to explain the co-current rotation change. Accordingly, we investigate the
intrinsic momentum transport for the non-rotating state when diamagnetic flow and ExB flow cancel each other. The change in
the intrinsic momentum transport due to lower hybrid waves is significant when the plasma current is low, which may explain
the rotation reversal for low plasma current. The effect of changed q (safety factor) profile by lower hybrid on the intrinsic
momentum transport is estimated by gyrokinetics.
Keywords: Tokamaks, Lower hybrid, Ion rotation
PACS: 52.55.Fa,52.55.Wq,07.60.Hv,52.35.Hr,52.25.Os,52.25.Os
INTRODUCTION
Lower hybrid waves are being investigated to drive non-inductive current and achieve a steady state operation in a
tokamak. In addition to the original purpose, current drive, it has been observed in many tokamaks that LH waves can
change the ion toroidal rotation [1, 2], which is beneficial to increase plasma confinement and stabilize the unexpected
disruption due to magnetohydrodynamic (MHD) instability. These measurements indicate that it may be possible to
control rotation with lower hybrid waves, but to do it, it is necessary to understand the mechanisms underlying the
rotation change.
In Alcator C-Mod, the toroidal rotation of the main ions is estimated by the Doppler shift of the line emission from
impurities using a high resolution imaging X-ray crystal spectrometer [1]. For a discharge with high plasma current
(I p 500kA), the ion toroidal rotation in the core is accelerated in the counter-current direction right after the lower
hybrid wave injection, and the acceleration is slowed down and almost saturated in O(100) msec. For a discharge with
low plasma current (Ip 500kA), the change in the steady state rotation after the wave injection is in the co-current
direction, which is opposite to the wave momentum input [2]. The initial acceleration after lower hybrid injection for
this low current case is still in the counter-current direction, but it changes direction to co-current at about 150 msec
after the wave injection. This reversal of the rotation change due to lower hybrid waves shows strong correlation with
the total plasma current (or the safety factor at the edge) and a relatively weaker correlation with the plasma density in
Alcator C-Mod [2].
Previously, the initial acceleration due to the external momentum source was theoretically investigated [3, 4]. In
this analysis, we focus on the change in the steady state rotation after LH wave injection. To attain steady-state ion
toroidal rotation, the toroidal angular momentum injected by lower hybrid waves should be transported out to the wall
by turbulence. Consequently, the momentum source term and the radial transport term balance in the equation,
1 ∂
(AΠ) + Tϕ = 0,
(1)
A ∂r
where A is the area of the flux surface, Π = mi d 3 v fi R(v · ϕ̂ )(v · r̂) s is the radial transport of ion toroidal angular
momentum, and Tϕ = me d 3 vR(v · ϕ̂ )Q( fe ) s is the external torque due to lower hybrid waves [5]. Here, fe and fi
are the electron and ion distribution function, me and mi are the electron and ion mass, ϕ and r is the toroidal and
−
Radiofrequency Power in Plasmas
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FIGURE 1. Radial profiles of rotation based on measurements (solid lines) and the reconstruction using only the momentum
pinch, diffusion and source in Eq. (3) (dashed lines) for (a) High current case (I p = 700kA) and (b) Low current case (I p = 350kA).
The black curves are the profiles before the lower hybrid wave injection (t=0.75 sec) and the red curves are the profiles after the
lower hybrid wave injection (t=1.25 sec for (a) and t=0.95 sec for (b)).
radial coordinate, respectively, R is the major radius, Q is the Kennel-Engelmann quasilinear operator, and ...s is the
flux surface average.
The momentum transport by microturbulence in a tokamak can be described by gyrokinetics [6, 7]. There are terms
of the gyrokinetic equation that depend on the toroidal flow and the radial gradient of the toroidal flow. They break
the symmetry of the turbulence and thus contribute to the momentum transport. The ion toroidal angular momentum
transport can be linearized in the subsonic regime (Mach number ∼ 0.1 − 0.2), giving an advective term and a diffusive
term that are proportional to the flow and the flow shear, respectively,
Π ≡ Πint − Pϕ ni mi R2 s Ωϕ − χϕ ni mi R2 s
∂ Ωϕ
,
∂r
(2)
where Ωϕ is the ion toroidal angular frequency and ni is the ion density. The advective term has the coefficient Pϕ
which is called momentum pinch coefficient, and the diffusive term is proportional to the momentum diffusivity χϕ .
Here, Πint is the intrinsic momentum transport (residual stress), which is the momentum flux generated even for zero
flow and flow shear (i.e. Ωϕ = 0 and ∂ Ωϕ /∂ r = 0)
The radial rotation profile can be obtained analytically from Eqs. (1-2), giving
a
Pϕ
dr
Ωϕ (r) = Ωϕ (a) exp
χϕ
r
r
a
Πint (r )
Pϕ
−
dr
exp
dr
χϕ ni mi R2 s
χϕ
r
r
r a
r
0 dr A(r )Tϕ (r )
Pϕ
,
(3)
dr
exp
dr
+
Aχϕ ni mi R2 s
χϕ
r
r
where a is the minor radius, the toroidal velocity at the last closed flux surface Ωϕ (a) is given as a boundary condition,
and zero momentum flux at the magnetic axis Π(r = 0) = 0 is used as the other boundary condition.
DIFFUSION AND PINCH OF THE INJECTED MOMENTUM
Figure 1 shows the radial rotation profile, Vϕ (r) Ωϕ (r)Rs , using Eq. (3) by assuming no contribution of intrinsic
momentum transport (i.e. Πint = 0). We evaluate the momentum diffusion and pinch coefficients using the gyrokinetic
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code (GS2) [8] with the experimental parameters at three different radii in Alcator C-Mod (i.e. (Pϕ /χϕ )Rs 1.0−3.0
and χϕ 0.2 − 0.6 [m2 /s] [5]). The toroidal velocity at the boundary is chosen to match the core rotation before the
lower hybrid wave injection (Vϕ (r = a) = −10km/s in Fig. 1 (a) and Vϕ (r = a) = −25km/s in Fig. 1 (b)), and the
external momentum source (Tϕ ) is evaluated by a full wave code (TORLH) and a Fokker-Planck code (CQL3D) [3].
The measured rotation change due to the lower hybrid waves near the magnetic axis agrees relatively well with the
change of the reconstructed profiles (see the red solid curve and the red dashed curve at r = 0 in Fig. 1 (a) and (b)).
For the low current case (b), the reconstructed profile with the momentum source is similar to the the radial profile at
t=0.95 sec in which the direction of the rotation change reverses from the counter-current direction to the co-current
direction.
The radial profiles obtained with only diffusion and pinch terms (black dashed curve) cannot explain the measured
intrinsic rotation (black solid curve) that has rapid spatial oscillations for r/a > 0.4. It requires intrinsic momentum
transport to be explained, although some rapid spatial oscillations are probably an artifact of the mapping between the
line integrated measurements and the radial profile of the toroidal rotation. In the low plasma current case in Fig. 1 (b),
the reconstructed profile (red dashed curve) cannot explain the saturated rotation profile (blue solid curve) that shows
a change of core rotation in the co-current direction.
CHANGE OF INTRINSIC MOMENTUM TRANSPORT DUE TO LH WAVES
We estimate the intrinsic momentum transport in the second term on the right hand side of Eq. (3). The normalized
intrinsic momentum transport required to explain the observed rotation profile is (Πint /Qi )(vti /R0 ) ∼ 0.1 where Qi is
the ion heat flux and vti is the ion thermal velocity [5]. The required size of the intrinsic momentum transport for the
low current case is larger than that for the high current case.
As an important source of the intrinsic momentum transport, we evaluate the diamagnetic effects. It is evaluated by
solving the higher order gyrokinetic equation implemented in GS2 [10] to obtain the intrinsic momentum transport
for a non-rotating plasma in which the E×B flow (Ωϕ ,E ) and the diamagnetic flow (Ωϕ ,d ) cancel each other (i.e.
Ωϕ = Ωϕ ,E + Ωϕ ,d = 0). We include the neoclassical distribution correction, which is calculated with NEO [9], to
the background distribution function in GS2. The numerical results for the intrinsic momentum transport due to the
diamagnetic effect are summarized in Table 5.2 in [5]. The diamagnetic effect results in the comparable size of the
required intrinsic momentum transport, (Πint /Qi )(vti /R0 ) ∼ 0.1. Because the diamagnetic flow for the low current case
is larger than the diamagnetic flow for the high current case, the intrinsic momentum transport due to the diamagnetic
effect for the low current case turns out to be larger than that for the high current case in the simulation results. It may
explain why the reversal of the acceleration direction due to the lower hybrid wave is observed only in the low plasma
current discharge.
Also, we numerically found the significant effect of the change in the magnetic shear on the intrinsic momentum
transport due to the diamagnetic effect for the low current case. For example, a 50% reduced magnetic shear results
in a significant reduction of the intrinsic momentum transport for the low current case, Δ(Πint /Qi )(vti /R0 ) ∼ −0.07.
This reduced momentum flux would correspond to a decrease of the absolute value of the counter-current rotation, as
observed in the low current case. The change in the radial profile of the safety factor is more significant for the low
current case than for the high current case if the lower hybrid wave driven current is the same for a given wave power
absorption. The effect of the plasma current on the momentum transport is also consistent with the observed time scale
for the reversal of the rotation change. Changes in the safety factor profile take a resistive current relaxation time. In
experiments, it takes around a resistive time scale (O(100) msec) for the rotation change to reverse.
ACKNOWLEDGMENTS
This work was supported in part by Samsung scholarship, by U.S. DoE FES Postdoctoral Fellowship, and by US DoE
Grant No. DE-SC008435. This research used computing resources of NERSC.
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