Methods of determining drift orbits

5.6 Methods of determining drift orbits

If neglecting the magnetic drift, a guiding-center orbit is along a magnetic ﬁeld lines, i.e., there is no derivation from the magnetic surface where a guiding center is initially located. Taking the magnetic drift into account, a guiding-center orbit will deviate from the initial magnetic surface, giving an orbit of nozero width in the poloidal plane.

Whether a guiding-center will drift radially outward or inward from a local magnetic surface near the midplane can be determined in the following way. First note that the zero-order approximation of the guiding-center orbit (zero-width orbit) is either parallel or anti-parallel to the local magnetic ﬁeld, depending on the sign of v_∥. Further note the direction of the magnetic drift ( -Z-
|Z| B×∇B and curvature drift) is approximatedly vertical, which can be either up or down, depending on the charge sign and direction of the toroidal magnetic ﬁeld. Finally, by imposing the magnetic drift on the zero-width orbit, we can determine whether the guiding center will drift inward or outward from the local magnetic surface. Figures 17-19 plots the drift orbits for all the possible combinitions of tokamak magnetic conﬁgurations and particle initial conditions (assume particles of positive charge, i.e., Z∕|Z| > 0).

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Figure 17: Projection of trapped orbits on the poloidal plane for 4 magnetic conﬁgurations. A Deuterium ion of 20keV is launched from the low-ﬁeld-side midplane (R_initial = 2.15m,Z_initial = 0m) with pitch angle 𝜃 = 75^∘(v_∥ > 0) and 𝜃 = 105^∘(v_∥ < 0). Note that v_∥ > 0 implies that the zero-width orbit in the poloidal plane is along the direction of the poloidal magnetic ﬁeld, which is in turn determined by the direction of the toroidal plasma current. The magnetic equilibrium is from EAST discharge #62585@2.8s (gﬁle provided by ZhengZheng). The direction of toroidal plasma current, magnetic ﬁeld, and the corresponding magnetic drift are indicated on the ﬁgures.

Figure 17 can be used to identify the direction of the bootstrap current due to the radial density gradient of trapped particles. Examining all the cases in Fig. 17, one ﬁnds that the boostrap current is always along the direction of plasma current, and the bootstrapt current direction is independent of the charge sign.

Next, consider passing particles launched from the low-ﬁeld-side midplane. Figure 18 plots all the 4 possible cases.

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Figure 18: Projection of passing orbits on the poloidal plane for various magnetic conﬁgurations. A Deuterium ion of 20keV is launched from the low-ﬁeld-side midplane (R_ini = 2.15m,Z_ini = 0m) with pitch angle 𝜃 = 50^∘(v_∥ > 0) and 𝜃 = 130^∘(v_∥ < 0). The magnetic equilibrium is from EAST discharge #62585@2.8s (gﬁle provided by ZhengZheng). The direction of plasma current, magnetic ﬁeld, and the corresponding magnetic drift are indicated on the ﬁgures.

Next, consier a particle launched from the high ﬁeld side midplane, which must be a passing particle. Figure 19 plots all the 4 possible cases.

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Figure 19: Projection of passing orbits on the poloidal plane for various magnetic conﬁgurations. A Deuterium ion of 20keV is launched from the high-ﬁeld-side midplane (R_ini = 1.6452m,Z_ini = 0m) with pitch angle 𝜃 = 50^∘(v_∥ > 0) and 𝜃 = 130^∘(v_∥ < 0). The magnetic equilibrium is from EAST discharge #62585@2.8s (gﬁle provided by ZhengZheng). The direction of plasma current, magnetic ﬁeld, and the corresponding magnetic drift are indicated on the ﬁgures.

Examining the above results, one ﬁnds that reversing the direction of the toroidal magnetic ﬁeld does not change the projection of orbits on the poloidal plane, i.e., the location and shape of the poloidal orbits remain the same. However the direction of the poloidal motion is changed from clockwise (anti-clockwise) to anti-clockwise (clockwise). (This is because v_∥ of a particle changes sign when the toroidal ﬁeld is reversed and thus the direction of the poloidal motion changes).

Examining the above results, we can also ﬁnd that, for particles launched from low-ﬁeld-side midplane, co-current partilces have their orbits inside the magnetic surface at which the particle is iniitally located, and counter-current particles have thier orbits outside of the magnetic surface. For particles launched from the high-ﬁeld-side midplane, the conclusion is reversed, i.e., co-current partilces have their orbits outside the magnetic surface where they are initally located, and counter-current particles have thier orbits inside of the magnetic surface.

These conclusions have important implications for the neutral beam injection, where orbits outside a reference magnetic surface (birth location) are more likely to be lost to the wall of the machine. If the neutral beam injection (NBI) is along the same direction of the plasma current, it is called the co-current injection. Otherwise it is called the counter-current injection. Using the above conclusions, we know that, for co-current injection, ions ionized at the low-ﬁeld-side have better conﬁnement compared with those ionized at the high-ﬁeld-side. For the counter-current injection, ions ionized at the high-ﬁeld-side have better conﬁnement compared with those ionized at the low-ﬁeld-side. Whether the overall conﬁnment of ions due to co-current injection is better or worse than that of the counter-current injection depends on the ratio of number of ions deposited at the low-ﬁeld-side to that deposited at the high-ﬁeld side. For the shine-through loss to be small, most neutral must ionize at the low-ﬁeld-side (most neutrals ionizing at the the high-ﬁeld side usually means a very high shine-through loss fraction ( >50%)). Therefore, with the assumption that most neutral beam particles are ionized on the low-ﬁeld-side, co-curent injection is better than counter-current injection in terms of the ﬁrst-orbit loss.

Figure 20 and 21 compares the poloidal orbits of energetic Deuterium particles ionized at the low-ﬁeld-side midplane due to co-current and counter-current injection. The results indicate again that the counter-injected particles ionized at the low-ﬁeld-side midplane are easy to be lost from the plasma because their orbits are outside the ﬂux surface where they are ionized, and thus are more likely to touch the ﬁrst wall.

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Figure 20: Poloidal orbits of Deuterium particles of 50keV ionized at the low-ﬁeld-side midplane (R = 2.25m,Z = 0m) with a birth pitch angle 𝜃 = 125^∘ (red), 𝜃 = 105^∘ (blue), 𝜃 = 75^∘ (green), and 𝜃 = 65^∘ (violet). Pitch angle 𝜃 is the included angle between the magnetic ﬁeld and the velocity of particles. Since the magnetic ﬁeld and the plasma current are in the same direction for this case, 𝜃 > 90^∘ means counter-current injection and 𝜃 < 90^∘ means co-current injection. The counter-injected particles are easy to be lost from the plasma because their orbits are outside the ﬂux surface where they are ionized, and thus are more likely to touch the ﬁrst wall. The magnetic equilibrium is for EAST discharge #62585@2.8s, which is a upper single-null conﬁguration (gﬁle provided by ZhengZheng).

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Figure 21: Poloidal orbits of Deuterium particles of 50keV ionized at the low-ﬁeld-side midplane (R = 2.25m,Z = 0m) with a birth pitch angle 𝜃 = 125^∘ (blue), 𝜃 = 105^∘ (red), 𝜃 = 75^∘ (violet), and 𝜃 = 60^∘ (green). Since the magnetic ﬁeld and the plasma current are in the opposite direction for this case, 𝜃 > 90^∘ means co-current injection and 𝜃 < 90^∘ means counter-current injection. The magnetic equilibrium is from EAST discharge #62585@2.8s but with the direction of the toroidal magnetic ﬁeld reversed.

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Figure 22: The resonant layer of 50MHz electromagnetic wave with the third harmonic of ₁¹H ion cyclotron frequency on EAST tokamak. The toroidal magnetic ﬁeld of EAST is approximately given by B_ϕ = 4.160 × 10⁻⁴I_s∕R, where I_s is the current in a single turn of the TF coils, which in in the range from 8000A to 10000A for usual EAST discharges. The ion cyclotron angular frequency is given by ω_ci = B_φe∕m_i. The small Dopper frequency shift k_∥v_∥ is not included in the estimation of the resonant layer.

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Figure 23: I_s = 11kA, B_ϕ = 4.160 × 10⁻⁴I_s∕R fundament harmonic of ₁¹H ion cyclotron frequency, 37MHZ = Ω∕2π, where Ω = B_ϕe∕m.

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