2. NON-UNIFORMITY MITIGATION METHOD

In this study, we analyze the implosion non-uniformity by the arbitrary Lagrangian–Eulerian method hydrodynamics simulation (Hirt et al., Reference Hirt, Amsden and Cook1974). The target structure is shown in Figure 1. The 32 Pb⁺ ion beams are illuminated in the arrangement shown in Figure 2 to the target (Skupsky & Lee, Reference Skupsky and Lee1983; Ogoyski et al., Reference Ogoyski, Someya and Kawata2004, Reference Ogoyski, Kawata and Popov2010). The HIB particle density has the Gaussian distribution, and the transverse beam emittance is 3.2 mm-mrad.

Fig. 1.

Target structure.

Fig. 2.

32HIBs system.

When the instability driver wobbles uniformly in time, the imposed perturbation δg(t) for a gravity g ₀ at t = τ may be written as

(1) $$g = g_0 + {\rm \delta} g\left( t \right) = g_0 + {\rm \delta} ge^{i\Omega {\rm \tau}} e^{{\rm \gamma} ({\rm t} - {\rm \tau} ) + i\vec k \cdot \vec x}.$$

Here, δg is the amplitude, Ω is the wobbling or oscillation frequency defined actively by the dribing wobbler, and Ωτ is the phase shift of superimposed perturbations. At each time t, the wobbler or the modulated driver provides a new perturbation with the phase and the amplitude actively defined by the driving wobbler itself. The superposition of the perturbations provides the actual perturbation at t as follows:

(2) $$\int_0^t {d{\rm \tau} {\rm \delta} ge^{i\Omega {\rm \tau}} e^{{\rm \gamma} (t - {\rm \tau} ) + i\vec k \cdot \vec x}} \propto \displaystyle{{{\rm \gamma} + i\Omega} \over {{\rm \gamma} ^2 + \Omega ^2}}{\rm \delta} ge^{{\rm \gamma} t}e^{i\vec k \cdot \vec x}.$$

When Ω ≫ γ, the perturbation amplitude is reduced by the factor of γ/Ω, compared with the pure instability growth (Ω = 0) based on the energy deposition non-uniformity. The result in Eq. (2) presents that the perturbation phase should oscillate with Ω ≳ γ for the effective amplitude reduction.

In the simulations, we realize the oscillating δg and the mitigation mechanism by the following pulsating foot and main HIB pulses. The foot pulse power P _foot and the main pulse power P _main (see Fig. 3) are represented by the following equations:

(3) $$P_{{\rm foot\; \; \;}} = 5.80\left[ {{\rm TW}} \right]\left( {1 + A\sin \left( {\displaystyle{{2{\rm \pi} t} \over T} + \displaystyle{{2{\rm \pi} {\rm \xi}} \over {360}}} \right)} \right),$$

(4) $$P_{{\rm main}} = 320\left[ {{\rm TW}} \right]\left( {1 + A\sin \left( {\displaystyle{{2{\rm \pi} t} \over T} + \displaystyle{{2{\rm \pi} {\rm \xi}} \over {360}}} \right)} \right).$$

Here A is the amplitude of the input pulse, T is the pulsation period and ξ is the phase of each pulsating HIB. In this case, we employ A = 0.100 and T = 1.00 ns for our simulations. The phase ξ for each HIB is listed in Table 1.

Fig. 3.

Beam power pulsation.

Table 1.

Beam power phase ξ for each HIB.

3. EVALUATION METHOD FOR NON-UNIFORMITY

In this study, we use the root mean square (RMS) shown by the following equation for the non-uniformity evaluation:

(5) $${\rm \sigma} _{\rm i}^{{\rm rms}} = \displaystyle{1 \over F}\sqrt {\displaystyle{{\mathop \sum \nolimits_j {(F_{ij} - F_{ij})}^2} \over {{\rm \theta} _{{\rm mesh}}}}}. $$

F _ij is a physical quantity, 〈F _ij 〉 is the average of physical quantity of circumferential direction on a certain radius, θ_mesh is the total mesh number in the θ-direction, and (i, j) is the mesh numbers for the radial direction and the azimuthal direction, respectively.

We also perform the mode analysis for the non-uniformity f(θ) based on the Legendre polynomial P _n . Here the amplitude of the mode n is obtained by the following equation:

(6) $$A_n = \displaystyle{{2n + 1} \over 2}\int_0^{\rm \pi} {\,f\left( {\cos {\rm \theta}} \right)P_n\left( {\cos {\rm \theta}} \right)\sin {\rm \theta} d{\rm \theta}}. $$

The Legendre polynomial P _n (P ₀ ~ P ₅) is shown in Figure 4 for reference.

Fig. 4.

Legendre polynomial P _n(P ₀ ~ P ₅).

4. NON-UNIFORMITY MITIGATION IN HIF TARGET IMPLOSION

First, we examine the effect of the pulsating and dephasing HIBs illumination on the target implosion acceleration. Figures 5 shows the implosion acceleration histories at θ = 0°, 74.6°, and 152° for the deuterium and tritium (DT) layer. In Figure 5a, the pulsating HIBs are in phase, and so the implosion acceleration is also in phase. However, in Figure 5b the pulsating HIBs’ phases are controlled as shown in Table 1. Figure 5b demonstrates that the pulsating and controlled dephasing HIBs illumination creates the DT fuel implosion acceleration oscillation of δg.

Fig. 5.

Time histories of the DT fuel acceleration. (a) In-phase HIBs illumination. (b) Dephasing HIBs illumination.

Table 2 shows the summary of the implosion simulation results for the in-phase HIBs illumination and for the dephasing HIBs.

Table 2.

Implosion result summary for the in-phase HIBs and for the dephasing HIBs.

Figure 6 shows the non-uniformity histories of the density ρ, the ion temperature T _i, the pressure P and the radial direction speed V _r of the DT layer. The solid line shows the non-uniformities by the in-phase HIBs’ illumination, and the dotted line shows them by the dephasing pulsating HIBs’ illumination. Figure 6 presents that the dephasing and pulsating HIBs reduce the non-uniformities successfully.

Fig. 6.

Time histories of the DT fuel non-uniformities for the pulsating and dephasing HIBs and for the HIBs without the pulsation.

Figures 7a, 7b show the non-uniformity mode analyses results for the averaged ion temperature T _i of the DT layer at t = 38 ns). Figure 7 presents that the dephasing and pulsating HIBs reduce the largest mode of the “Mode 2” significantly.

Fig. 7.

Modes of the ion temperature T _i in the DT layer at t = 38 ns. (a) W/o HIBs pulsation and (b) with HIBs pulsation.

In Table 1, and Figures 5, 6, and 7b, the oscillation amplitude A of the HIBs input power in Eqs (3) and (4) was A = 0.100. Figure 8 shows the relation of the fuel target gain versus the oscillation amplitude A of the HIBs input power. When the HIBs input power oscillation of A is 0.1, the fuel target gain becomes the maximum. The target gain becomes 0, when A exceeds 0.140. The results in Figure 8 demonstrate that the dephasing and pulsating HIBs illumination realizes the better uniformity in the DT fuel implosion, and consequently leads a higher gain.

Fig. 8.

Target fuel gain versus the HIBs pulsating amplitude A.

We also study the robustness against the displacement dz (see Fig. 9) of the target misalignment in a fusion reactor. Figure 10 shows the relation between the fuel target gain and the displacement dz for each input pulse modulation amplitude A. Figure 10 presents that the dephasing and pulsating HIBs illumination is robust against the target misalignment dz. When dz increases, the implosion non-uniformity increases (Kawata et al., Reference Kawata, Karino and Ogoyski2016). Therefore, the relatively larger dz does not induce the fuel ignition, and for the larger dz no fusion output energy is obtained as shown in Figure 10. The gain curve tendency in Figure 10 should be studied further in the near future.

Fig. 9.

Target displacement dz.

Fig. 10.

Gain versus the target misalignment dz.