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5.7 Toroidal procession

Figure 24 plots the guiding centers orbits for particles launched at the low-field-side of the midplane with different values of the pitch angle 𝜃.

pict pict

Figure 24: Left: projection of guiding center orbits of trapped particles on (R,Z) plane. Right: Toroidal motion of guiding center of the trapped particles. All particles have the same kinetic energy 𝜀 = 10keV and are launched from the low-field side midplane of the reference magnetic surface (R = 2.1,Z = 0). The equilibrium is a Solovev equilibrium with R0 = 1.9m,B0 = 2.0Tesla, κ0 = 1.5, Ze0 = 1.5, g = 3.8mT.

Figure 24 shows (1) the toroidal procesion of deeply trapped particles is faster than that of the shallowly trapped particles and (2) the direction of the toroidal precession of the particle with 𝜃 = 65∘ is different from the others.

Procession angular frequency of a trapped particles (from Porcelli’s slide):

-v2--- --Ek--- ωD = 2ΩcRr = BZeRr ,
(100)

where Ek is the kinetic energy of the particle. Equation () indicates that the procession frequency is proportional to the energy of the particle.

--v2-- ----v2---- ---v2-- ωD = 2ΩcRr = 2B-ΩnRrL2n = Ωn8π2BRr-
(101)

Compared with the results given by the numerical code, the above results seems to be roughly correct when the orbit is not near the magnetic axis.

-2 v2⊥- 2B-μ -2Bn-μ --- v⊥ = v2n = mv2n = B mv2n = 2Bμ
(102)

 

 

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