Motion in a toroidal magnetic field

Chapter 2: Theories

2.4. Single-particle orbit motions

2.4.4. Motion in a toroidal magnetic field

Inevitably, there is terminal loss in the magnetic mirror confinements. The terminal loss can be avoided by using closed magnetic field configuration such as the toroidal magnetic field configuration as shown in Fig. 2.7. In the toroidal magnetic field configuration, the magnetic-force lines are formed by circling around the toroidal direction for several cycles. The magnetic surfaces are nested structure. The most central magnetic surface degenerates into a magnetic-force line closed around one cycle which is called the magnetic axis. The distribution of the magnetic field is getting smaller and smaller outward, and getting larger and larger inward along the radius R of the device.

Therefore, curvature drift will lead to charge separation to create electric filed which is perpendicular to 𝑩. Then, 𝐄 × 𝑩 drift lead to particle loss. The simple ring magnetic field is not enough to confine the thermonuclear plasma.[48]

In tokamak and stellarator, the rotation transformation is adopted to make the magnetic field line not close after one circle and rotate form up side to down side to offset charge separation as shown in Fig. 2.8. In this magnetic field configuration, ι = 2π_𝑚^𝑛 is the rotation transformation angle, where m and n mean that it returns to its original position after going around the circle m times in the polar direction of a magnetic field line, which happens to go around the circle n times in toroidal direction. In order to realize the rotation transformation, a polar magnetic field 𝑩_𝜃 needs to be added to the toroidal magnetic field 𝑩_𝜑 as shown in Fig. 2.8. Thus the magnetic field in the plasma is the

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superposition of these two fields, and the magnetic-field lines become helical. In stellarator, 𝑩_𝜃 is supplied by coil. In tokamak, 𝑩_𝜃 is generated by plasma current. The stability factor (safety factor) of magnetohydrodynamics (MHD) is also defined as:

𝑞 =^𝑩_𝑩^𝜑

𝜃 𝑟

𝑅 =^2𝜋_ι (2.4.9)

where R is the radius of magnetic axis, r is the small radius of the helical magnetic-field line. [48]

Fig. 2.7 Drift in a toroidal magnetic field.

Fig. 2.8 The rotation transform of magnetic field line.

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In this kind of magnetic-field configurations, the motion of charged particles can be divided into two categories. One is a passing particle whose angle between the particle velocity and the magnetic field B is small, that is the parallel component 𝒗_⫽ is larger.

The result is that the particle can pass through the strong field without being bounced back. The other type is the trapped particle. This kind of particles with a small 𝒗_⫽ cannot pass through a region of strong magnetic field, but can only bounce back and forth between two adjacent regions of strong magnetic field. Its orbit is like the shape of a banana. Therefore, it is also called a banana particle. [48]

Fig. 2.9 Passing particle cyclotron motion.

If there is no drift, the passage particle moves only along the magnetic surface. Fig.

2.9 shows that a passing particle moves along with a magnetic surface centered at O and drift. The motion of a particle guiding-center is decomposed into the motion along the magnetic-field line and the drift caused by the non-uniform magnetic field 𝑩 = 𝑩_𝜑𝒆_𝜑+ 𝑩_𝜃𝒆_𝜃, where 𝑩_𝜑= 𝑩₀^𝑅_𝑅⁰.

Magnetic surface Particle drift surface

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𝒗_𝐷 = 𝑚^2𝒗_2𝑒𝑩^{⫽2+𝒗⟘2}

𝜑𝑅 𝒆_𝑦 (2.4.10)

The motion equation of the projection on the xy plane of the guiding center is:

{

𝑑𝑥

𝑑𝑡 = 𝒗_⫽^𝑩_𝑩^𝜃𝑦_𝑟

𝑑𝑦

𝑑𝑡 = −𝒗_⫽^𝑩_𝑩^𝜃𝑥_𝑟+ 𝒗_𝐷 (2.4.11)

By using 𝑑𝑟 =¹_𝑟(𝑥𝑑𝑥 + 𝑦𝑑𝑦) , EQ(2.4.11) change to ^𝑑𝑟_𝑑𝑥=_𝒗^𝒗𝐷𝑩

⫽𝑩_𝜃= 𝑎₀ (for passing particles 𝑎₀ ≪ 1) whose solution is 𝑟 = 𝑟₀+ 𝑎₀𝑥. Therefore, (𝑥 − 𝑎₀𝑟₀)²+ 𝑦² ≈ 𝑟₀², which means that the entire drift surface of the co-going (clockwise) particle orbit moves a distance ∆𝑟 to the right relative to the magnetic field.

∆𝑟 = 𝑎₀𝑟₀ ≈ 𝑞𝑟_𝑐 (2.4.12)

Similarly, the center of the entire drift surface of a counter-going (counterclockwise) particle orbit is shifted a distance ∆𝑟 to the left relative to the magnetic surface. [48]

If drift is not taken into account, the trapped particle only moves back and forth in two local magnetic-mirror fields of the rotating transformation-magnetic-field configuration, where the projection of its guiding-center on the xy plane is only an arc between two reflection points M1 and M2 on the magnetic surface as shown in Fig. 2.10.

After considering the drift, the orbit projection of the guiding center is ^𝑑𝑟_𝑑𝑥= ±_𝒗^𝒗𝐷𝑩

⫽𝑩_𝜃. Due to 𝒗₀² = 𝒗_⟘²+ 𝒗_⫽²= 𝒗_⟘𝑀² and 𝜇 =^𝑚𝒗_2𝑩^⟘2 =^𝑚𝒗_2𝑩⁰²

𝑀 , 𝒗_⫽= 𝒗₀√1 −_𝑩^𝑩

𝑀 ≅ 𝒗₀√^𝑥−𝑥_𝑅 ^𝑀

0 , where 𝒗_⟘𝑀 is perpendicular velocity at the maximum magnetic field 𝑩_𝑀. Therefore, the maximum deviation between the particle drift surface and the magnetic surface is:

∆𝑟_𝑇 = ^2𝒗_𝒗^𝐷𝑩0

0𝑩_𝜃 √𝑅₀𝑟₀(1 − cos 𝜃_𝑀) ≈^{2𝑞𝑟𝑐√𝜀(1−cos 𝜃}_𝜀 ^𝑀) <^{2𝑞𝑟𝑐}

√𝜀 (2.4.13)

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where 𝜃_𝑀 is the angle between 𝑟₀ and the x-axis when the particle returns to the magnetic surface. ∆𝑟_𝑇 is half Banana orbit width.

𝜀 = _𝑅^𝑟0

0 ≈_𝑅^𝑎

0 (2.4.14)

𝜀⁻¹ is a ratio of transverse and longitudinal. [48]

Fig. 2.10 Banana orbit.

When the guiding center of the particle rotates clockwise along the magnetic field line, r > r0, it drift outward; When the guiding center of the particle rotates counterclockwise along the magnetic field line, r<r0, it drift inward.

The critical angle 𝜃_𝑐 of the toroidal magnetic field can be obtained by the loss cone similar to the magnetic mirror. The minimum and maximum values of the magnetic field are 𝑩_𝑚 ≅ 𝑩₀(1 − 𝜀) and 𝑩_𝑀 ≅ 𝑩₀(1 + 𝜀), respectively.

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sin 𝜃_𝑐 = √^𝑩_𝑩^𝑚

𝑀 = √^1−𝜀_1+𝜀 (2.4.15)

According to the proportion of the escape cone solid angle to the total solid angle 4π is P = 1 − cos 𝜃_𝑐. The ratio of trapped particles to total particles is:

𝑓_𝑇 = 1 − 𝑃 = √_1+𝜀^2𝜀 ≈ √2𝜀 (2.4.16)

Therefore, when 𝒗_⫽ 𝒗_⟘

⁄ > √2𝜀 , and 𝜃 < 𝜃_𝑐 , the particle is passing particle. When 𝒗_⫽

𝒗_⟘

⁄ < √2𝜀, and 𝜃 > 𝜃_𝑐, the particle is trapped particle. [48]

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