T HEORETICAL STUDY ON ION ROTATION BY
LOWER HYBRID WAVE IN A TOKAMAK
Jungpyo(J.P.) Lee, F.I. Parra, M. Barnes, D.Ernst, P.J. Catto, J.C. Wright,
P.T. Bonoli, R.R. Parker, Y.Podpaly
MIT Plasma Science and Fusion Center
T URBULENT MOMENTUM TRANSPORT
A BSTRACT
The effect of Lower hybrid wave to ion toroidal rotation in a tokamak is investigated
theoretically by evaluating a wave momentum source term and a turbulent driven radial
transport of toroidal angular momentum.
W HY rotation IMPORTANT ?
C-M OD MEASUREMENT
• It increases confinement time by velocity shear reducing turbulence radial wavelength.
• It inhibits toroidal assymetric MHD
unstable modes.
• X-ray spectroscopy and CXRS are
used for ion rotation measurement
• Momentum flow:
Asymmetric
toroidal antenna spectrum → LH
wave momentum (count-current
direction) → electrons(by electron Landau damping) → ions(by
collisions or radial pinch)
• Saturation rotation direction:
◦ For high current(Ip > 400kA):
Counter direction
◦ For low current (Ip < 400kA):
Co-direction(Unexpected ! Why? )
W HY LH wave IS NEEDED ?
LH wave is used for non-inductive current
drive for a steady state tokamak. Power
and momentum is transferred to electrons
by electron Landau damping. The radial profile of safety factor(q) can be controlled by LH wave damping location.
For a perturbed non-adiabatic distribution function, h, by an electrostatic potential ϕ,
a gyro- kinetic equation(GKE) can be modified in the low flow order correction. The
correction is in a poloidal rhostar B/Bp δ higher order than traditional GKE in the first
order δ = ρi /a. i.e. h = h1 + h2 , ϕ = ϕ1 + ϕ2 where h1 /FM ∼ eϕ1 /Te ∼ O(δ) and
h2 /FM ∼ eϕ2 /Te ∼ O(B/Bp δ 2 ).
∂
d
+ (vk b̂ + vD + vC + vδE ) ·
+C h
dt
∂R
NC
mv
c
e
d
I
∂H
k
1
∗
0
=
FM
− iωT − i kθ
Ωφ
J0 ϕ − vδE ·
+ ... (4)
T
dt
eB B
∂R
{z
}
|
O(B/Bp δ 2 )correction
Without the Coriolis term vC , the velocity shear term with Ω0φ and the O(B/Bp δ 2 )
correction terms, the radial flux of the toroidal angular momentum, Π is antisymmetic in a
inversion (i.e. Π(−θ, −vk , −kx , ky ) = −Π(θ, vk , kx , ky )), since −h(−θ, −vk , −kx , ky )
and ϕ(−θ, −vk , −kx , ky ) are solutions of GKE if h(θ, vk , kx , ky ) and ϕ(θ, vk , kx , ky )
do. Here,
the
momentum
radial
flux
is
defined
as
R
R
Π=
d3 v (mi Rvk )vE×B hi
s
∼ Re iky φ(kx , ky )
d3 v(mi Rvk )h1 (−kx , −ky )J0
S YMMETRY BREAKING IN LINEAR GS2
Before a rotation is generated, the Coroiols term and the velocity shear term are zero. Intrinsic rotation occurs due to the symmetry breaking terms in the O(B/Bp δ 2 ) correction.
1.Without correction: Π(mom flux)/Q(heat
flux) are antisymmetic in kx →no momentum transport. The slope of Π/Q in proportional to the size of drift term(toroidicity
effect)
Courtesy of Y. Podpaly
2.With low flow correction: the line of
Π/Q are shifted according to the size
R 3 ∂H1N C
of the correction term∼ dv vk ∂ψ ∼
R
∂
dv 3 vk2 (I/Ω) ∂ψ
∂ ln(Pi )
∂ψ
+
∂ ln(Ti )
A ∂ψ
F OKKER -P LANCK EQUATIONS
The proper way to evaluate the ion rotation is to use the momentum equation combining
the electron (1) and ion (2) Fokker-Plank equations.
e
∂fe
+ v · ∇fe +
∇Φ + Ωe v × b̂ · ∇v fe
∂t
me
∂fi
Ze
+ v · ∇fi + −
∇Φ + Ωi v × b̂ · ∇v fi
∂t
mi
=
Cee {fe } + Cei {fe , fi } + ∇v · Dql · ∇v fe (1)
=
Cii {fi } + Cie {fi , fe }
(2)
Here, f is a the distribution function, Φ is an electrostatic potential, C is a collision
operator, and Dql is the quasilinear diffusion tensor by LH wave. Flux surface averaging
of the toroidal component from (1) and (2) yields (3). Assuming ambipolarity, the ion
toroidal rotation change in L.H.S is determined by two terms, the off-diagonal stress
tensor Π(viscosity) and the toroidal momentum source term (the last term in R.H.S).
GS2 RESULT FOR A C-M OD SHOT (L- MODE )
A positive momentum flux results in an inward radial flux of counter-current direction rotation, To explain measured countercurrent rotation with a core-peak and a
Z
mach
number
(∼
0.3),
we
need
to
consider
nφ
∂
1 ∂
0
3 1
2
mi RnViφ s = −
(V
Π)
+
hJ
·
∇ψi
+
R
d
v
m
v
∇v · Dql · ∇v fe
(3) Corioli’s effect for an inward pinch and a
e
s
0
∂t
V ∂ψ
c
2
| {z }
s
|
{z
}
|
{z
}
|
{z
}
diffusion
effect
by
velocity
shear
term
as
0
transient
turbulent viscosity
momentum source
well.
NC
dH
1
Π ∼ aΩφ + bΩ0φ + c dψ
Notice: Wave momentum is transferred
not
only
parallel
direction
but
also
perpendic
R 3
ular direction, so the source is not
d vme Rvϕ ∇v · Dql · ∇v fe s , and its difference
R EFERENCES
induces the radial electron pinch. Other sources for the non-ambipolar electron pinch are
derived from a trapped electron Ware-like pinch and the different locations in terms of
[1] Lee J P, wright J C, , Bonoli P T, et. al. 2011 19th Topical Conf. on RF Power in Plasmas 1406 459
major radius between wave power damping and collisions. However, the contribution of
[2] Parra F and Catto P J 2010 Plasma Physics of Controlled Fusion 52 045004
all radial pinch to the momentum transfer can be negligible compared to the collisions.
[3] Parra F, Barnes M, Peeters A Physics of Plasmas 18 062501
FM