1 Banana kinetic equation and plasma transport in tokamaks K. C. Shaing 1 , M. S. Chu 1 , S. A. Sabbagh 2 , and J. Seol 3 1 National Cheng Kung University. Tainan City 701, Taiwan, Republic of China 2 Columbia University, New York, NY 10027, USA 3 National Fusion Research Institute, Daejeon, 305-333 Korea In the core of fusion grade tokamak plasmas, where ν * = νRq ε 32 v t ( ) is much less than unity, banana kinetics becomes important for modes with wavelengths comparable to or shorter than the width of bananas, and with frequency ω lower than the gyro-frequency Ω = eB Mc ( ) and the bounce frequency of bananas [1-3]. Here, ν is the typical collision frequency, R is the major radius, q is the safety factor, ε is the inverse aspect ratio, v t = 2TM is the thermal speed, T is the temperature, e is the charge, c is the speed of light, B is the magnetic field strength, and M is the mass. Because the width of the gyro-orbits ρ is assumed to be much less than the wavelengths of the modes, gyro-kinetics is neglected and the drift kinetic equation is used for the banana kinetics. To treat banana kinetics, we choose p ζ ,θ ,ζ 0 , E, μ, t ( ) as independent variables [1-3]. Here, p ζ is the toroidal component of the canonical momentum, θ is the poloidal angle, ζ 0 = q θ - ζ is the field line label, ζ is the toroidal angle, E is the particle energy per unit mass, and μ is the magnetic moment per unit mass. We employ Hamada coordinates [4], in which the equilibrium magnetic field is expressed as B 0 = ʹ ψ ∇V ×∇θ - ʹ χ ∇V ×∇ζ , where ʹ ψ = B 0 • ∇ζ , χ and ψ are respectively the poloidal and toroidal flux divided by 2 π , , ʹ χ = B 0 • ∇θ , and prime denotes d dV . The covariant representation for B 0 is B 0 = G ∇θ + F ∇ζ + ∇ϕ , where F = F χ ( ) , and G χ ( ) are the poloidal current outside, and the toroidal current inside a magnetic surface multiplied by c 2 , respectively. The function ϕ satisfies the equation B 0 • ∇ϕ = B 0 2 - B 0 2 [5], where angular brackets denote flux surface average. We choose the electrostatic and vector potentials to represent perturbed electromagnetic fields. For the electrostatic potential φ , φ = φ 0 + φ 1 = φ 0 V ( ) + φ lmn lmn ∑ e imθ − nζ ( ) + ilk χ χ − iωt , where φ 0 V ( ) is the equilibrium potential, φ 1 is the perturbed potential, φ imn is the Fourier amplitude, k χ is the radial wave vector in terms of χ , and l , m, n ( ) are respectively the radial, poloidal, and toroidal mode numbers. For ω < ω b = 45 th EPS Conference on Plasma Physics P1.1002
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Banana kinetic equation and plasma transport in tokamaks
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1
Banana kinetic equation and plasma transport in tokamaks
K. C. Shaing1, M. S. Chu1, S. A. Sabbagh2, and J. Seol3
1National Cheng Kung University. Tainan City 701, Taiwan, Republic of China 2Columbia University, New York, NY 10027, USA
3National Fusion Research Institute, Daejeon, 305-333 Korea
In the core of fusion grade tokamak plasmas, where
€
ν* =
€
νRq ε 3 2vt( ) is much less
than unity, banana kinetics becomes important for modes with wavelengths comparable to
or shorter than the width of bananas, and with frequency
€
ω lower than the gyro-frequency
€
Ω =
€
eB Mc( ) and the bounce frequency of bananas [1-3]. Here,
€
ν is the typical collision
frequency, R is the major radius, q is the safety factor,
€
ε is the inverse aspect ratio,
€
vt =
€
2T M is the thermal speed, T is the temperature, e is the charge, c is the speed of light, B
is the magnetic field strength, and M is the mass. Because the width of the gyro-orbits
€
ρ is
assumed to be much less than the wavelengths of the modes, gyro-kinetics is neglected and
the drift kinetic equation is used for the banana kinetics.
To treat banana kinetics, we choose
€
pζ ,θ,ζ0,E,µ,t( ) as independent variables [1-3].
Here,
€
pζ is the toroidal component of the canonical momentum,
€
θ is the poloidal angle,
€
ζ0
= q
€
θ -
€
ζ is the field line label,
€
ζ is the toroidal angle, E is the particle energy per unit mass,
and
€
µ is the magnetic moment per unit mass. We employ Hamada coordinates [4], in which
the equilibrium magnetic field is expressed as B0 =
€
ʹ ψ
€
∇V × ∇θ -
€
ʹ χ
€
∇V × ∇ζ , where
€
ʹ ψ =
€
B0 •∇ζ ,
€
χ and
€
ψ are respectively the poloidal and toroidal flux divided by 2
€
π , ,
€
ʹ χ =
€
B0 •∇θ , and prime denotes
€
d dV . The covariant representation for B0 is B0 = G
€
∇θ + F
€
∇ζ +
€
∇ϕ , where F =
€
F χ( ), and
€
G χ( ) are the poloidal current outside, and the toroidal
current inside a magnetic surface multiplied by
€
c 2 , respectively. The function
€
ϕ satisfies
the equation
€
B0 •∇ϕ=
€
B02 -
€
B02 [5], where angular brackets denote flux surface average.
We choose the electrostatic and vector potentials to represent perturbed
electromagnetic fields. For the electrostatic potential
€
φ ,
€
φ =
€
φ0 +
€
φ1 =
€
φ0 V( ) +
€
φlmnlmn∑ ei mθ −nζ( )+ ilkχ χ − iωt , where
€
φ0 V( ) is the equilibrium potential,
€
φ1 is the perturbed
potential,
€
φimn is the Fourier amplitude,
€
kχ is the radial wave vector in terms of
€
χ, and
€
l,m,n( ) are respectively the radial, poloidal, and toroidal mode numbers. For
€
ω <
€
ω b=
45th EPS Conference on Plasma Physics P1.1002
2
€
vt ε Rq( ) , the bounce frequency of bananas, it is convenient to use
€
ζ0, and
€
φ becomes
€
φ
=
€
φ0 V( ) +
€
φln θ( )ln
∑ einζ 0 + ilkχ χ − iωt . Because
€
φ1 is real,
€
φlmn =
€
φ−l−m−n* , and
€
φln θ( ) =
€
φ−l−n* θ( ),
where the superscript * indicates the complex conjugate. Similarly, the perturbed vector
potential A is expressed either as A =
€
Almnlmn∑ ei mθ −nζ( )+ ilkχ χ − iωt , or as A =
€
Aln θ( )ln
∑ einζ 0 + ilkχ χ − iωt . The A and its Fourier amplitudes are decomposed as A =
€
A|| n0 +
€
A⊥,
where n0= B0/|B0|, and the subscripts || and
€
⊥ indicate the components parallel and
perpendicular to B0, respectively. Also,
€
Almn =
€
A−l−m−n* , and
€
Aln θ( ) =
€
A−l−n* θ( ) .
We derive the banana kinetic equation using the following orderings. We adopt
€
k⊥ρ
< 1, where
€
k⊥ is the magnitude of the wave vector perpendicular to B0. For maximum
ordering, we choose
€
kχρpRBp ∼ 1, with
€
ρp , the poloidal gyro-radius, and
€
Bp , the
equilibrium poloidal magnetic field strength. We still assume that
€
ρp < L, with L, the
equilibrium radial gradient scale length.
Following the procedure in [1-3], orbit averaging the drift kinetic equation when
€
ω b
is the dominant frequency yields the banana kinetic equation