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

€

∂f0∂t

+

€

vd •∇ζ0 ob

€

∂f0∂ζ0

+

€

eM∂φ∂t

−eMc

v||n •∂A∂t

⎛

⎝ ⎜

⎞

⎠ ⎟

ob

€

∂f0∂E

+

€

˙ p ζ ob

∂f0

∂pζ =

€

C f0( )ob

, (1)

where

€

f0 =

€

f0

€

pζ ,ζ0,E,µ,t( ),

€

vd = -

€

v||n × ∇ v|| Ω( ) +

€

v||2 BΩ( )[ ] ∇ × B( )⊥ +

€

v|| Ω( )n×

€

∂n ∂t( )

is the drift velocity, n =

€

B0 + B1( ) B ,

€

B1=

€

∇ × A,

€

pζ =

€

χ -

€

Fv|| Ω -

€

∇V × ∇θ • A ,

€

˙ p ζ =

€

v|| Ω( ) ∂ v||B2 Ω( ) ∂ζ0 +∂ B0 • A( ) ∂ζ0[ ] , the orbit averaging operator is

€

• ob =

€

dθB •( ) v|| ʹ χ ( )θ t1

θ t 2∫σ

∑⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 dθB v|| ʹ χ ( )θ t1

θ t 2∫⎛ ⎝ ⎜ ⎞

⎠ ⎟ ,

€

σ∑ is a sum over

€

σ ,

€

σ is the sign of the

parallel particle speed

€

v|| , and

€

θ t1 and

€

θ t2 are two turning points at which

€

v|| = 0. The

€

θ

integrals in

€

• ob are performed holding

€

pζ ,ζ0,E,µ,t( ) constant. Fast bounce motion of

bananas smoothens out the

€

θ variation in

€

f0 . Because

€

ερp L < 1, the difference between

€

pζ and V on the equilibrium quantities is neglected. Ordering

€

ω∼

€

ω d =

€

vd •∇ζ0 ob∼

€

ν >

€

ω dpζ, a reduced banana kinetic equation is obtained, i.e.,

€

∂f01∂t

+

€

vd •∇ζ0 ob

€

∂f01∂ζ0

-

€

C f01( )ob

= -

€

eM∂φ∂t

−eMc

v||n •∂A∂t

⎛

⎝ ⎜

⎞

⎠ ⎟

ob

€

∂f00∂E

-

€

˙ p ζ ob

∂f00

ʹ χ ∂V, (2)

where

€

f00 =

€

N π 3 2vt3( )−1e− ME T −eφ 0 T( ) is a Maxwellian distribution, and N is density. Equation

(2) is used to calculate the fluxes in all the low collisionality regimes as outlined in the

45th EPS Conference on Plasma Physics P1.1002

3

theory for neoclassical toroidal plasma viscosity [6]. The explicit expression for the right

side of Eq.(2) is

-

€

eM∂φ∂t

−eMc

v||n •∂A∂t

⎛

⎝ ⎜

⎞

⎠ ⎟

ob

€

∂f00∂E

-

€

˙ p ζ ob

∂f00

ʹ χ ∂V

=

€

cfMʹ χ

€

in −ωn

e ʹ χ cT

+ʹ p

p+

e ʹ φ 0T

+ x 2 − 52

⎛

⎝ ⎜

⎞

⎠ ⎟ ʹ ́ T

T⎡

⎣ ⎢

⎤

⎦ ⎥ l,n

∑

€

φln −v||cA||,ln

ob

€

ei( lkχ pζ +nζ 0 −ωt ), (3)

where p is plasma pressure, x =

€

v vt , and the consequences of the banana kinetics can be

found in the orbit averaged potentials

€

φln −v||cA||,ln

ob

=

€

dθ Bv|| ʹ χ θ t1

θ t 2∫⎛

⎝ ⎜

⎞

⎠ ⎟

−1

×

€

dθ Bv|| ʹ χ θ t1

θ t 2∫ φln θ( )coslkχFvΩ

v||v

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ − i

v||vvvtvtcA||,ln θ( )sin

lkχFvΩ

v||v

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ . (4)

It is obvious that bananas sample the perturbed fields along their trajectories. We emphasize

that the contribution of the vector potential to the banana kinetic equation remains finite

unless the width of bananas vanishes.

We solve Eq.(2) in the superbanana plateau regime caused by the drift resonance,

€

ω - n

€

ω d = 0. (5)

The ensemble averaged particle flux

€

Γ•∇V en and heat flux

€

q •∇V en T are, for j = 1- 3,

€

Γ•∇V en

q •∇V en T

⎛

⎝ ⎜

⎞

⎠ ⎟ = -

€

2ecπM ʹ χ

€

Nvt−2

ʹ ε

€

BM Bm −1

€

η1η2

⎛

⎝ ⎜

⎞

⎠ ⎟ −

ωn

e ʹ ́ χ cT

+ʹ p

p+

e ʹ Φ T

⎛

⎝ ⎜

⎞

⎠ ⎟ +

η2η3

⎛

⎝ ⎜

⎞

⎠ ⎟ ʹ T

T

⎡

⎣ ⎢

⎤

⎦ ⎥ , (6)

and

€

η j =

€

dxxmin

∞

∫ x 2 − 5 2( ) j−1e−x 2l,n∑

€

1+ kr2 BM Bm −1( )[ ]

−3 2

€

n φln − v||A||,ln cob kr

2∂Gk ∂kr

2 =1×

€

dθθ t1

θ t 2∫ B Bm( ) kr2 − B Bm −1( ) BM Bm −1( )[ ]−1 2

. The resonance function

€

Gk k( ) =

€

λ ×

€

dθθ t1

θ t 2∫B Bm ʹ ε ( )

k 2 − B Bm −1( ) BM Bm −1( )

€

2 BM

Bm

−1⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢ k 2 − B Bm −1

BM Bm −1⎛

⎝ ⎜

⎞

⎠ ⎟

ʹ ́ χ

ʹ χ − B2( )ʹ 2B2( )

⎛

⎝ ⎜

⎞

⎠ ⎟ +

€

BBm

B2( )ʹ 2B2( )⎤

⎦ ⎥

€

dθθ L

θU∫ B Bm

k 2 − B Bm −1( ) BM Bm −1( )

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

−1

, where

€

ʹ ́ χ

ʹ χ =

€

B2 ʹ

B2 +

€

4πPʹB2

- F

€

ʹ q

€

χʹ

B2,

€

λ =

€

µBm Ek , Ek =

€

v2 2 is the kinetic energy of the particle per unit mass, P is total

plasma pressure, and

€

BM and

€

Bm are respectively the maximum and minimum values of B

45th EPS Conference on Plasma Physics P1.1002

4

along the banana bouncing trajectories [7]. The trapping state is characterized by the pitch

angle parameter

€

k 2 =

€

1− λ( ) λ BM Bm −1( )[ ] : for trapped particles, 1

€

≥

€

k 2

€

≥ 0 and for

circulating particles,

€

k 2

€

≥ 1. The resonance pitch angle

€

kr2 is the solution of the drift

resonance condition, i.e., Eq.(5), for a given energy. The lower limit of the energy integral

€

xmin depends on the sign of Gk(k); only those particles with the normalized energy larger

than

€

xmin can resonate. In the region G(k) > 0,

€

xmin =

€

2e ʹ χ Mcvt2 ʹ ε ( )[ ] ω n − c ʹ Φ 0 ʹ χ ( ) Max Gk k( )[ ]{ }

−1 1 2

, and in the region Gk(k) < 0,

€

xmin =

€

2e ʹ χ Mcvt2 ʹ ε ( )[ ] ω n − c ʹ Φ 0 ʹ χ ( ) Min Gk k( )[ ]{ }

−1 1 2

. The Max and Min are defined respectively

as the maximum and minimum values of the argument.

The transport fluxes in Eq.(6) can be used to model transport behaviors of thermal

particles and energetic alpha particles in the presence of electrostatic turbulence, Alfvenic

waves, and chaotic magnetic fields when magnetic field correlation length

€

LM > Rq in

tokamaks. For electrostatic turbulence, we set A = 0, and banana kinetics is stabilizing or

improves confinement. For Alfvenic waves, it is the interference between the electrostatic

and vector potentials that determines if banana kinetics improves or degrades plasma

confinement. In chaotic magnetic fields, we set

€

φ1 = 0 and

€

ω = 0, and when

€

kχρpRBp ε ∼

1, drift resonance enhances transport losses over transit or bounce frequency resonance

induced losses, e.g., Rechester-Rosenbluth coefficient [8], by a factor of

€

L ρp > 1. The

banana kinetics can be used to selectively pump out undesirable species. Acknowledgement

This work was supported by Taiwan Ministry of Science and Technology under Grant No. 100-2112-M-006-

004-MY3.

References

[1] K. C. Shaing, and S. A. Sabbagh Phys. Plasmas 23, 072511 (2016).

[2] K C. Shaing, Phys. Plasmas 24, 122504 (2017).

[3] K. C. Shaing, M. S. Chu, S. A. Sabbagh, and J. Seol, Phys. Plasmas 25, 032501 (2018)

[4] S. Hamada, Nucl. Fusion 2, 23 (1962).

[5] M. Yu. Isaev, M. I. Mikhailov, and V. D. Shafranov, Plasma Physics Reports 20, 319 (1994).

[6] K. C. Shaing, K. Ida, and S. A. Sabbagh, Nucl. Fusion 55, 125001 (2015).

[7] K. C. Shaing, J Plasma Phys. 81, 905810203 (2015).

[8] A. B. Rechester, and M. N. Rosenbluth, Phys. Rev. Lett. 40, 38 (1978).

45th EPS Conference on Plasma Physics P1.1002