Fig. 1. Equilibrium with a circular cross section of A=100 and β_p=0.001. (a) Contours of ψ (magnetic surfaces). Solid lines are for ψ0 (plasma region), and broken lines are for ψ>0 (vacuum region). The outermost solid line is the plasma surface. (b) Profiles of the safety factor q (solid line) and the parallel current density j (broken line), where , q₀=1.32, q_a=2.89 and j|_a/j=0.21.

Fig. 2. The minimum eigenvalues λ₀ of n=2 modes for the equilibrium in Fig. 1 expressed as functions of b/a, where λ_0-M2D denotes the eigenvalue obtained by the MARG2D code, and λ_0-ERT is that obtained by ERATOJ (dotted line), respectively. The marginal stability locations of the wall determined by MARG2D–V.P., MARG2D–Bessel and ERATOJ are the same as each other b/a=1.23.

Fig. 3. Poloidal Fourier harmonics of the eigenfunctions belonging to λ₀ of the n=2 mode for b/a=1.24, at which the mode is unstable, but close to the marginal stability; (a) MARG2D–V.P. (λ_0-M2D=−7.51×10⁻⁴), (b) MARG2D–Bessel (λ_0-M2D=−7.52×10⁻⁴), and (c) ERATOJ (λ_0-ERT=−1.02×10⁻⁵). The harmonics Y_m obtained by these codes are nearly identical to each other, showing typical surface modes with the dominant harmonic (m,n)=(6,2) because nq_a=5.78.

Fig. 4. Equilibrium of A=3.3, κ=1.8, δ=0.45 and β_p=1.5, whose q₀=1.52, q_a=4.35 and j|_a/j_a=0.18. (a) Contours of ψ (magnetic surfaces). The outermost solid line shows the plasma surface. (b) Profiles of q (solid line) and j (broken line). (c) Profiles of the pressure p (solid line) and dp/dψ (broken line).

Fig. 5. β_p-dependence of λ₀ of the n=5 modes for the case of b/a=1.25. The broken line obtained by ERATOJ becomes convex as it approaches to the λ_0-ERT=0 line, and it is hard to determine β_p,cr. The solid line by MARG2D crosses the λ_0-M2D=0 line, and identifies the β_p limit as β_p,cr=1.5.

Fig. 6. Poloidal Fourier harmonics Y_m(s) of the eigenfunctions when β_p=1.5; the n=5 mode is close to the marginal stability. These are obtained by (a) MARG2D (λ_0-M2D=−5.55×10⁻⁴) and (b) ERATOJ (λ_0-ERT=−8.83×10⁻⁶). The mode structures shown in these figures are similar to each other. The dominant harmonic is m=22 peaking at the plasma surface, and the subdominant harmonic is m=12.

Fig. 7. Constant-height surfaces of Y₀(s,θ) of the n=5 mode when β_p=1.5 on the (R,Z) plane together with the contours of ψ. The values of Y₀(s,θ) is normalized to max(Y₀(s,θ)). (a) Range of the constant-height surface is set as −0.6Y₀0.6. The peeling mode structure whose m=22 is prominent. (b) Range of the constant-height surface is set as −0.3Y₀0.3. The solid lines show the contours of ψ=−0.8 (innermost contour), −0.5, and −0.2, and broken lines are for ψ0.2. A ballooning mode structure in the outboard bad curvature region is emphasized.

Fig. 8. Poloidal Fourier harmonics Y_m(s) of the eigenfunctions when β_p=2.2 obtained by (a) MARG2D (λ_0-M2D=−0.106) and (b) ERATOJ (λ_0-ERT=−4.13×10⁻²). The structure of Y_m(s) in Fig. 8(a) remains similar to that in Fig. 6(a). On the other hand, Y_m(s) structure in Fig. 8(b) is composed with an edge ballooning mode whose envelope peaks at s0.82, while the m=22 harmonic becomes subdominant.

Fig. 9. (a) β_p-dependence of λ₀ of the n=5 modes when b/a=1.25. The solid, broken and dotted lines are obtained by MARG2D, ERATOJ and ERATOJ-IN, respectively. (b) Poloidal Fourier harmonics Y_m(s) for β_p=2.2 obtained by ERATOJ-IN (λ_0-ERT-IN=−6.87×10⁻²). This mode structure is similar to that obtained by ERATOJ shown in Fig. 8(b).

Fig. 10. Equilibrium with a pedestal structure. The geometrical parameters are the same as those in Fig. 4, and β_p=1.4. (a) Profiles of q (solid line) and j (broken line), where q₀=1.7, q_a=4.27, and j|_a/j_a=0.135, respectively. (b) Profiles of p (solid line) and dp/dψ (broken line).

Fig. 11. Poloidal Fourier harmonics of the eigenfunction of a n=40 mode (λ_0-M2D=−7.77×10⁻³) as functions of (a) and (b) nq, where q_a=4.27 (nq_a=170.8), β_p=1.4, and b/a=1.2. We observe that the dominant poloidal mode number is m=171 and that an edge ballooning mode structure constructs the envelope whose maxima is nq150.

Fig. 12. Constant-height surfaces of Y₀(r,θ) of the n=40 mode for q_a=4.27, β_p=1.4 and b/a=1.2. (a) The range of the constant-height surface is set as −0.6Y₀0.6. An edge ballooning structure appears in the outboard bad curvature region. (b) The range is set as −0.1Y₀0.1. Contours of ψ are also shown as in Fig. 7(b). A peeling component is emphasized at the plasma surface, especially at the top and bottom of the surface.

Fig. 13. Dependence of dλ_0-M2D on the reciprocal of the square of total radial mesh number (NPSI_M2D+NV_M2D) for the equilibrium shown in Fig. 10; dλ_0-M2D is the difference of λ_0-M2D from that computed with the mesh number NPSI_M2D+NV_M2D=880. The NPSI_M2D+NV_M2D value changes from 880 to 3960. Good quadratic convergence is observed for all values of n (=1,5,10,20,40).

Fig. 14. β_p dependence of λ_0-M2D for the different b/a cases. (a) The n=1 external mode is sensitive to the wall position. When (b) n=40, the stability is virtually determined in the no-wall condition when b/a1.07.

Fig. 15. Dependence of the CPU time and the parallelization efficiency C_eff on the number of processors n_PE in the cases of (a) n=1 and (b) n=30. The parameters in the MARG2D code are shown above each figure. The CPU time reduces about 25 times (n=1) and 36 times (n=30) by increasing n_PE from 2 to 128, and C_eff in n_PE=128 keeps about 40% (n=1) and 56% (n=30), respectively.

Fig. 16. Poloidal cross section for defining geometrical parameters of a tokamak equilibrium.

Parameters in Eq. (62) for each β_p equilibrium for investigating n=5 stability

Parameters in the MARG2D code for checking a convergence property