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5.4 Bounce frequency of deeply trapped particles

Let us analytically estimate the bounce frequency of deeply trapped particles. The time evolution of the paralell velocity of a guiding center is given by Eq. (2), i.e.,

dv∥ μ B⋆ -dt = −m- B⋆-⋅∇B, ∥
(85)

which can be approximately written as

dv∥ dt = −μ- mb ⋅∇B,

which can be further written as

2 2 d-l2 = −-v⊥-dB- dt 2B dl
(86)

where dl is the arc lengh along the magnetic field. In a large aspect ratio tokamak with circular flux surfaces, the magnetic field can be written approximatedly as

B B = -------0-----, 1+ (r∕R0)cos𝜃
(87)

The equation of magnetic field is written

B dl rd𝜃 -𝜃-= -p-= ---, B dl dl
(88)

which can be written

B dl =---rd𝜃 B 𝜃
(89)

Using Eqs. (89) and (87), the paralel derivative of the magnetic field is written as

dB B𝜃 dB B 𝜃 B0 r -dl = rB-d𝜃-= rB-[1-+-(r∕R--)cos𝜃]2R--sin𝜃, 0 0
(90)

Plug this into equation (86), then we obtain

d2l v2 B B r v2 B 1 --2 = −-⊥---𝜃--------0-----2 --sin𝜃 = −-⊥--𝜃----sin𝜃 dt 2B rB [1+ (r∕R0)cos𝜃] R0 2 B0 R0
(91)

Consider deeply trapped particles (particles are trapped in a very small region near the low-field-side midplane), i.e., 𝜃 ≈ 0, then we have sin𝜃 ≈ 𝜃. Using this, the above equation is written as

2 2 d2l≈ − v⊥B-𝜃-1-𝜃 dt 2 B0 R0
(92)

Assume the orbit is along the magnetic field line (i.e. zero-width orbit approximation), then the equation of magnetic field (89) is also satisfied by the orbit. In the linear approximation, we have 𝜃 ≈ B𝜃∕(Br)l. Using this in Eq. (92), we obtain

2 2 2 d-l2 = − rv⊥-B2𝜃2l dt 2R0 B r
(93)

Using the definition of safety factor, q = rB0∕R0B𝜃, the above quation is written

2 2 d-l2 = −-v2⊥2--r-l dt q R02R0
(94)

Define

( )1∕2 ωb = -v⊥- -r-- , qR0 2R0
(95)

(for deeply trapped particles, the variation of v⊥ during one poloidal period is small, and thus can be considered constant, and thus ωb can also be considered constant), then Eq. (94) is written

2 d-l= − ω2 l, dt2 b
(96)

which indicates that the motion of a deeply trapped particle is a harmonic oscillation with an angular frequency ωb. Equations (95) and (96) agree with Eqs. (3.12.3) and (3.12.4) in Wesson’s book “Tokmaks”[2]. I have test the accuracy of formula (95) by comparing it with the numerical results, which indicates the formula can usually give a reasonable estimation of the bounce frequency (for example, 28kHz is obtained numerically while the analytical formula gives 24kHz for a not very deeply trapped orbit).

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