快速开始#

本页使用一个不需要外部数据文件的简单解析 tokamak 平衡，带你走过 pyna 的三个 核心能力：场线追踪、Poincare 映射和磁岛拓扑。

Note

所有示例都使用 **Solov’ev 解析平衡**（Cerfon & Freidberg 2010），并缩放到接近 EAST 的参数（R₀ ≈ 1.86 m，B₀ = 5.3 T）。它是一个通用测试平台：精确的 Grad-Shafranov 解、闭式场分量，以及可调形状。

1. 构造解析平衡#

先导入平衡对象并查看它的基本参数：

import numpy as np
import matplotlib.pyplot as plt
from pyna.toroidal.equilibrium import solovev_iter_like

eq = solovev_iter_like(scale=0.3)          # EAST-like size
Rmaxis, Zmaxis = eq.magnetic_axis

print(f"R0 = {eq.R0:.2f} m   a = {eq.a:.2f} m   B0 = {eq.B0:.1f} T")
print(f"κ  = {eq.kappa:.2f}  δ = {eq.delta:.2f}  q0 = {eq.q0:.2f}")
print(f"Magnetic axis: R = {Rmaxis:.3f} m, Z = {Zmaxis:.3f} m")

返回的 eq 对象提供 eq.BR_BZ(R, Z)、eq.Bphi(R)、 eq.psi(R, Z)``（归一化磁通）和 ``eq.q_profile(psi)。

2. 追踪场线并累积 Poincare 截面点#

Poincare 截面记录场线每次穿过选定环向截面（这里为 φ = 0）时的 (R, Z) 坐标。 经过许多环向圈后，嵌套磁通面会表现为闭合曲线；磁岛则表现为一串离散截面点。

from pyna.flt import FieldLineTracer, get_backend
from pyna.topo.poincare import PoincareAccumulator, poincare_from_fieldlines
from pyna.topo.section import ToroidalSection

# Use the canonical topology section type; ``pyna.topo.poincare`` keeps
# backward-compatible aliases for accumulator-only workflows.
section = ToroidalSection(0.0)

# --- define the ODE right-hand side: dR/dφ, dZ/dφ ---
def field_rhs(phi, RZ):
    R, Z = RZ
    BR, BZ = eq.BR_BZ(R, Z)
    Bphi   = eq.Bphi(R)
    return [R * BR / Bphi, R * BZ / Bphi]

# --- seed 8 field lines radially outward from the axis ---
R_starts = np.linspace(Rmaxis + 0.05, Rmaxis + 0.45, 8)
Z_starts = np.zeros(8)

# --- integrate 300 toroidal turns per line ---
backend = get_backend('cpu')
flt = FieldLineTracer(field_rhs, backend=backend)
pacc = poincare_from_fieldlines(
    field_func=field_rhs,
    start_pts=np.column_stack([R_starts, Z_starts, np.zeros_like(R_starts)]),
    sections=[section],
    t_max=300 * 2 * np.pi,
    backend=flt,
)
poincare_pts = [pacc.crossing_array(0)[:, :2]]

# --- plot ---
fig, ax = plt.subplots(figsize=(6, 6))
for Rs, Zs in poincare_pts:
    ax.scatter(Rs, Zs, s=0.8, color='steelblue')
ax.set_xlabel('R (m)')
ax.set_ylabel('Z (m)')
ax.set_aspect('equal')
ax.set_title('Poincaré map — Solov\'ev equilibrium')
plt.tight_layout()
plt.show()
Poincaré map of a Solov'ev analytic equilibrium showing nested flux surfaces

图 1. Solov’ev 解析平衡的 Poincare 映射（接近 EAST 的参数，每条场线 250 次环向穿越）。每种颜色对应一条场线；嵌套闭合曲线是磁通面。红色叉号标出 磁轴；黑色曲线是最后闭合磁通面（LCFS, ψ = 1）。#

每个同心环对应一条绕磁通面缠绕的场线。q = m/n 有理面是共振扰动（例如 RMP 线圈）可以打开磁岛的位置。

3. 定位有理面并测量磁岛#

加入一个小的共振扰动后，q = 2/1 面上会打开磁岛。pyna 可以在一次调用中定位该面并 测量磁岛半宽：

from pyna.topo.toroidal_island import locate_rational_surface, island_halfwidth

# Build q(S) from PEST mesh
from pyna.toroidal.coords import build_PEST_mesh

nR, nZ = 100, 100
R_grid = np.linspace(0.3*eq.R0, 1.5*eq.R0, nR)
Z_grid = np.linspace(-eq.a*eq.kappa*1.3, eq.a*eq.kappa*1.3, nZ)
Rg, Zg  = np.meshgrid(R_grid, Z_grid, indexing='ij')

BR, BZ   = eq.BR_BZ(Rg, Zg)
Bphi     = eq.Bphi(Rg)
psi_norm = eq.psi(Rg, Zg)

S, TET, R_mesh, Z_mesh, q_iS = build_PEST_mesh(
    R_grid, Z_grid, BR, BZ, Bphi, psi_norm,
    Rmaxis, Zmaxis, ns=40, ntheta=181
)
S_values = S[1:]
q_values = q_iS[1:]
print(f"q range: {q_values[0]:.2f} → {q_values[-1]:.2f}")

# Locate q = 2/1 surface
res = locate_rational_surface(S_values, q_values, m=2, n=1)
print(f"q=2/1 surface at S = {res[0]:.4f}  (ψ_norm = {res[0]**2:.4f})")

返回的 S_res 值（S = √ψ_norm）会准确给出共振层位置。把它和受扰动的 Poincare 映射一起传给 island_halfwidth，即可得到以米为单位的磁岛宽度。

4. 一般有限维动力系统#

pyna 不局限于环形场线。同一套拓扑对象模型也可用于 Hamiltonian 系统、N-body 流、映射和 SDE 样本路径。

import numpy as np
from pyna.dynamics import (
    SeparableHamiltonianSystem,
    CallableMap,
    GeometricBrownianMotion,
)

oscillator = SeparableHamiltonianSystem(
    kinetic=lambda p, t: 0.5 * np.dot(p, p),
    potential=lambda q, t: 0.5 * np.dot(q, q),
    grad_kinetic=lambda p, t: p,
    grad_potential=lambda q, t: q,
    dof=1,
)
traj = oscillator.trajectory([1.0, 0.0], (0.0, 2*np.pi), dt=0.01)
print(traj.final)  # TimeSeriesSolution is a pyna.topo.core.Trajectory

linear_map = CallableMap(lambda x: np.array([2*x[0], 0.5*x[1]]), dim=2)
orbit = linear_map.orbit_geometry([1.0, 1.0], n_iter=5)
print(orbit.period_guess)

gbm = GeometricBrownianMotion(mu=[0.08], sigma=[0.2])
print(gbm.expected_log_growth())

当一条轨迹或映射轨道已经从采样数据提升为几何/拓扑对象时，可使用 pyna.topo.core 中的 Cycle、PeriodicOrbit、Tube 和 IslandChain 等对象。

5. 基于工作流的构造#

在较大的项目和教学 notebook 中，使用 TopologyWorkflow 可以让分析序列保持 显式，而不用在代码中散布临时构造器。

import numpy as np
from pyna.topo import TopologyWorkflow
from pyna.topo.section import HyperplaneSection

wf = TopologyWorkflow(closure_tol=1e-3)
flow = wf.system(
    "callable-flow",
    rhs=lambda x, t: np.array([x[1], -x[0]]),
    dim=2,
    coordinate_names=("q", "p"),
)

section = HyperplaneSection(np.array([1.0, 0.0]), 0.0, phase_dim=2)
pmap = wf.poincare_map(flow, section, dt=0.02)

closed_traj = wf.trajectory(flow, [1.0, 0.0], (0.0, 2*np.pi), dt=0.01)
cycle = wf.closed_cycle(closed_traj)

低层 adapter、builder、bridge 和 factory 仍然供库作者使用，但大多数 notebook 应从 workflow facade 开始。

6. 下一步#