Learning objectives / 学习目标
学完本课后，你应能：
- 对 cold symmetric two-stream model 推导 dispersion relation、unstable band 与 maximum growth rate，并说明 warm-beam/Penrose boundary；
- 识别 density/temperature gradients、diamagnetic drift 与 unfavorable curvature 驱动的 drift/interchange free energy；
- 用 azimuthal mode number \(m\) 区分 sausage \(m=0\) 与 kink \(m=1\)，并在 simple cylinder 中应用 Kruskal–Shafranov criterion；
- 正确使用 \(\omega=\omega_r+i\gamma\)、e-folding time、threshold 与 ideal-MHD energy principle，避免把任何 positive estimate 当 universal criterion；
- 由 fluctuation correlation \(\Gamma_r=\langle\widetilde n\widetilde v_r\rangle\) 计算 turbulent flux 与 effective diffusivity，并解释 phase、saturation、cascade 与 zonal-flow regulation；
- 将 linear growth、nonlinear saturation、transport feedback 与 profile relaxation串成闭环。
Prerequisites / 先修知识
- Plasma Waves：dispersion branches、polarization、phase/group velocity；
- Kinetic Theory：resonant slope、Landau growth/damping、phase-space distribution；
- MHD Models：magnetic tension、flux freezing、\(v_A\) 与 current sheets；
- eigenvalues、complex exponentials、small perturbations、correlation 与 Fourier spectrum。
Scope and conventions / 范围与约定
- perturbation convention \(X_1\propto\exp[i(\mathbf k\cdot\mathbf r-\omega t)]\)，\(\omega=\omega_r+i\gamma\)；\(\gamma>0\) 为 temporal growth。
- 每个 criterion 都绑定明确 equilibrium、geometry、boundary 与 closure；本课不把 slab, cylinder 与 torus thresholds 混为同一 universal law。
- two-stream 推导用 equal-density cold electron beams \(\pm u_0\) 与 immobile ions；finite temperature 可稳定部分 modes。
- drift/interchange 用 local slab/curvature analog 讲 free-energy mechanism；完整 toroidal microinstability 需要 gyrokinetics。
- kink/sausage 用 straight periodic cylindrical column；tokamak safety factor 与 wall/pressure/shear effects 只作接口。
- turbulence 中 Reynolds decomposition \(n=\overline n+\widetilde n\)，time/ensemble average 满足 \(\langle\widetilde n\rangle=0\)。
Notation and modeling contract / 符号与建模契约
| Symbol | Meaning | SI / convention |
|---|---|---|
| \(u_0\) | beam speed magnitude | \(\mathrm{m\,s^{-1}}\) |
| \(\gamma\) | temporal growth rate | \(\mathrm{s^{-1}}\) |
| \(\omega_{*s}\) | diamagnetic/drift frequency | \(\mathrm{s^{-1}}\) |
| \(L_n^{-1}=\lvert\nabla n\rvert/n\) | density-gradient inverse scale | \(\mathrm{m^{-1}}\) |
| \(m\) | cylindrical azimuthal mode number | integer |
| \(q(a)\) | edge safety factor in simple periodic cylinder | dimensionless |
| \(\delta W\) | second variation of potential energy | J |
| \(\Gamma_r\) | radial particle flux | \(\mathrm{m^{-2}\,s^{-1}}\) |
| \(D_{\rm eff}\) | inferred effective diffusivity | \(\mathrm{m^2\,s^{-1}}\) |
1. Two-stream instability / 双流不稳定性
Knowledge ID: P13-U03-L02-K01
1.1 Velocity-space free energy
两个 counter-streaming populations 的 distribution 有 multiple maxima 与中间 minimum。若 wave phase velocity 落在 positive-slope region，resonant particles 可把 directed beam energy 交给 wave；这与 Maxwellian negative-slope Landau damping 相反。
图 1 的 two peaks 是 free-energy reservoir；正斜率是常见必要提示，但实际 stability 仍由完整 dielectric/Penrose criterion 决定。
1.2 Cold symmetric-beam dispersion
总 electron density 对应 \(\omega_{pe}\)；每束 density 为 \(n_0/2\)。cold electrostatic dielectric：
\[1-\frac{\omega_{pe}^2/2}{(\omega-ku_0)^2} -\frac{\omega_{pe}^2/2}{(\omega+ku_0)^2}=0.\]
令 \(a=ku_0/\omega_{pe}\)、\(x=\omega^2/\omega_{pe}^2\)，得到
\[x_{\pm}=\frac{1+2a^2\pm\sqrt{1+8a^2}}{2}.\]
lower branch 在 \(0<a<1\) 有 \(x_-<0\)，即 \(\omega=i\gamma\)：
\[\frac{\gamma}{\omega_{pe}}= \sqrt{\frac{\sqrt{1+8a^2}-1-2a^2}{2}}.\]
maximum 在 \(a^2=3/8\)：\(\gamma_{\max}=\omega_{pe}/(2\sqrt2)\)。finite thermal spread 会缩小 unstable band；cold delta beams 不是 universal threshold。
图 2 保留 exact endpoints 与 maximum ordering；横向 coordinate 仅作 schematic mapping，不声称厘米长度等于 \(a\)。
1.3 Nonlinear trapping
当 electric potential 足够大，resonant particles 在 wave frame 中被 trapping，phase-space 出现 vortices；distribution 在 resonance 附近 flatten，linear source 减弱。saturation 不是 free energy 消失，而是它被重排到 particle heating、plateau、sidebands 或 subsequent turbulence。
图 3 是 saturated trapping topology；ellipses 不是 physical-space circular orbits。
2. Drift and interchange instabilities / 漂移与交换不稳定性
Knowledge ID: P13-U03-L02-K02
2.1 Drift-wave mechanism
cross-field density gradient 与 magnetized species response 产生 diamagnetic frequency。local slab 中一种 magnitude estimate：
\[\lvert\omega_{*e}\rvert =\frac{k_y k_B T_e}{e B L_n}.\]
理想 Boltzmann electrons 与 nondissipative response 可给 neutral drift wave；resistivity、collisional/nonadiabatic electron response、magnetic shear 或 kinetic phase shift 使 density 与 electrostatic potential 不再 exactly in phase，波便能从 equilibrium gradient 抽取 free energy。必须区分 diamagnetic flow frequency 与单一 guiding center 的 material advection。
图 4 展示 finite cross-phase；若相位使 \(\langle\widetilde n\widetilde v_r\rangle\ne0\)，便有 net radial transport。
2.2 Interchange/flute mechanism
unfavorable magnetic curvature 对 pressure gradient 的作用类比 Rayleigh–Taylor：高-pressure plasma 被 curved magnetic field 支撑时，交换相邻 flux tubes 若降低 total potential energy，则 perturbation grows。local estimate 常写
\[\gamma^2\sim\frac{g_{\rm eff}}{L_p}-k_\parallel^2v_A^2,\]
第一项是 curvature/pressure drive，第二项是 field-line bending stabilization。exact coefficient、sign 与 \(g_{\rm eff}\) 定义依 geometry；不能将此量级式当 tokamak universal formula。flute perturbations 倾向 \(k_\parallel\simeq0\)，以减少 bending energy。
图 5 是 unfavorable-curvature energy analogy，not to scale；颜色不代表 actual mass density without a stated model。
3. Kink and sausage modes / 扭曲与箍缩模
Knowledge ID: P13-U03-L02-K03
在 cylindrical plasma column 中 perturbation \(\propto\exp[i(m\theta+k_z z-\omega t)]\)：
- sausage \(m=0\)： axisymmetric radius modulation，narrow section 的 magnetic pressure/current concentration 可 amplify necking；
- kink \(m=1\)： column axis 整体 helical displacement，current 与 self-field 可推动 bend 增长；
- \(m\ge2\)：更细的 fluting/shape harmonics。
图 6 用 two panels 区分 radial necking 与 axis displacement；它们共享 current-driven free energy，但 eigenfunctions 不同。
对 length \(L\) 的 periodic straight column，edge safety factor
\[q(a)=\frac{2\pi aB_z}{LB_\theta(a)}.\]
simple Kruskal–Shafranov ideal external-kink condition 常表为 \(q(a)>1\) 较稳定，等价于 field line axial turn 不过度 twisted。conducting wall、line tying、pressure/current profiles、flow shear 和 toroidal geometry 会修改 threshold；不可把 \(q=1\) 当所有 kink 的唯一判据。
4. Stability criteria and growth rates / 稳定判据与增长率
Knowledge ID: P13-U03-L02-K04
4.1 Linear eigenvalue audit
从 equilibrium \(X_0\) 加 \(X_1\)，linearized equations 形成 eigenproblem：
\[\mathcal L(\omega,\mathbf k;X_0)X_1=0.\]
在本课 convention 下：
\[\gamma=\operatorname{Im}\omega \begin{cases} <0,&\text{damped},\\ =0,&\text{marginal/oscillatory},\\ 0,&\text{unstable}. \end{cases}\]
e-folding time \(\tau_g=1/\gamma\)，amplitude \(A=A_0e^{\gamma t}\)，energy-like quadratic quantity grows \(\propto e^{2\gamma t}\)。必须说明报告的是 amplitude rate 还是 energy rate。
图 7 在 semilog axes 上区分 growth/damping；straight lines 只属于 linear constant-\(\gamma\) interval。
4.2 Energy principle
ideal MHD 可写 displacement \(\boldsymbol\xi\) 的 quadratic kinetic energy与 potential-energy variation。若对所有 allowed perturbations \(\delta W>0\)，equilibrium stable；若找到一个满足 boundary constraints 且 \(\delta W<0\) 的 trial displacement，则证明存在 instability。概念上
\[\gamma^2\sim-\frac{\delta W}{K[\boldsymbol\xi]}, \qquad K[\boldsymbol\xi]>0.\]
energy principle 是充分/必要关系需依据 operator self-adjointness 与 model；resistive/kinetic/non-normal systems 需要其他 criteria（Nyquist、Penrose、transient growth）。
图 8 是 one-coordinate analogy；真实 \(\delta W\) 是 function space quadratic form，不是一维 potential。
5. Turbulent transport / 湍流输运
Knowledge ID: P13-U03-L02-K05
5.1 Correlated fluctuations, not random decoration
radial particle flux：
\[\Gamma_r=\langle n v_r\rangle =\overline n\,\overline v_r +\langle\widetilde n\widetilde v_r\rangle.\]
若 mean radial flow 小，turbulent term dominates。对于 sinusoidal rms fluctuations：
\[\Gamma_r=\widetilde n_{\rm rms}\widetilde v_{r,\rm rms}\cos\Delta\phi.\]
所以 amplitude 大不保证 net transport；cross-phase \(\Delta\phi=90^\circ\) 时 average flux 为零。若用 Fick-like diagnostic definition：
\[D_{\rm eff}=-\frac{\Gamma_r}{d\overline n/dr},\]
它可能依 radius、time、gradient，甚至在 nonlocal/pinch transport 中不是正 constant material property。
图 9 的 blue curves 是 mean-flux surfaces，red arrows 是一组 outward-correlated radial-velocity samples。向外样本有 \(\widetilde n>0,\widetilde v_r>0\)，向内样本有 \(\widetilde n<0,\widetilde v_r<0\)；两类都给 \(\widetilde n\widetilde v_r>0\)，所以 eddy velocity 本身并非永远同向。图只作 conditional-correlation 示意，不声称未筛选的每个 sample 都贡献正 flux。
5.2 Saturation and mixing-length estimate
linear instability drives selected \(k\) modes；nonlinear advection couples modes、distorts the driving profile、traps particles或生成 zonal flows。一个 dimensional saturation estimate：eddy turnover/decorrelation \(k_\perp\widetilde v_E\sim\gamma\)，correlation length \(\ell_\perp\sim1/k_\perp\)，于是
\[D_{\rm mix}\sim\widetilde v_E\ell_\perp \sim\frac{\gamma}{k_\perp^2}.\]
这是 order-of-magnitude mixing-length closure，不是 first-principles universal coefficient。zonal \(\mathbf E\times\mathbf B\) shear 可 decorrelate eddies，减少 radial flux；profile flattening 则削弱原 gradient drive。
图 10 是 spectral energy budget；注明 cascade direction is model-dependent，避免把一条箭头冒充所有 plasma turbulence。
Worked examples / 完整例题
Worked Example 1 — Cold symmetric two-stream maximum growth
已知： total electron plasma frequency \(\omega_{pe}=1.00\times10^9\ \mathrm{s^{-1}}\)，beam speeds \(\pm u_0\) with \(u_0=2.00\times10^6\ \mathrm{m\,s^{-1}}\)。
研究对象： two equal cold electron beams；immobile neutralizing ions。
方向： \(k>0\)；symmetric distribution has zero mean flow。
模型： cold 1-D electrostatic two-stream dispersion。
maximum at \(a=ku_0/\omega_{pe}=\sqrt{3/8}=0.612\)：
\[k_{\max}=\sqrt{\frac38}\frac{\omega_{pe}}{u_0} =3.06\times10^2\ \mathrm{m^{-1}}.\]
\[\gamma_{\max}=\frac{\omega_{pe}}{2\sqrt2} =3.54\times10^8\ \mathrm{s^{-1}}, \qquad \tau_g=\gamma^{-1}=2.83\ \mathrm{ns}.\]
检查： \(k_{\max}u_0=0.612\omega_{pe}<\omega_{pe}\)，位于 unstable band。finite beam temperature、collisions 与 boundaries 通常降低/修改 growth；此处不外推到 warm beam。
Worked Example 2 — Electron diamagnetic frequency scale
已知： \(T_e=20.0\ \mathrm{eV}\)，\(B=0.500\ \mathrm T\)，\(L_n=0.100\ \mathrm m\)，\(k_y=50.0\ \mathrm{m^{-1}}\)。
坐标： \(\mathbf B=B\hat z\)，density gradient radial；这里只求 magnitude。
模型： local slab diamagnetic-frequency estimate。
因 \(k_BT_e/e=20.0\ \mathrm V\)：
\[\lvert\omega_{*e}\rvert =\frac{k_y(k_BT_e/e)}{BL_n} =\frac{(50.0)(20.0)}{(0.500)(0.100)} =2.00\times10^4\ \mathrm{s^{-1}}.\]
相应 \(y\)-phase-speed scale \(\lvert\omega_*/k_y\rvert=400\ \mathrm{m\,s^{-1}}\)。
检查： sign 要由 \(\mathbf B,\nabla n,q_e\) 的完整 cross product 决定；magnitude alone 不给 growth rate，nonadiabatic phase response 才决定 drive/damping。
Worked Example 3 — Simple Kruskal–Shafranov audit
已知： periodic cylinder radius \(a=0.100\ \mathrm m\)、length \(L=5.00\ \mathrm m\)，\(B_z=2.00\ \mathrm T\)，edge \(B_\theta=0.200\ \mathrm T\)。
系统边界： straight periodic column，no conducting-wall stabilization。
模型： simple external ideal-kink criterion \(q(a)>1\)。
\[q(a)=\frac{2\pi aB_z}{LB_\theta} =\frac{2\pi(0.100)(2.00)}{(5.00)(0.200)} =1.26.\]
在该 simplified criterion 下位于 stable side。若 current increase 使 \(B_\theta=0.300\ \mathrm T\)：
\[q(a)=0.838<1,\]
进入 kink-unstable side。
检查： \(q\) dimensionless；增加 azimuthal self-field 增加 twist 并降低 \(q\)。真实 threshold 还依 profiles、wall、line tying、pressure 与 toroidal geometry。
Worked Example 4 — 从 fluctuation correlation 求 turbulent flux
已知： \(\overline n=1.00\times10^{19}\ \mathrm{m^{-3}}\)，density rms \(\widetilde n_{\rm rms}=0.0500\overline n\)，\(\widetilde v_{r,\rm rms}=200\ \mathrm{m\,s^{-1}}\)，\(\cos\Delta\phi=0.600\)，\(L_n=0.100\ \mathrm m\)。另有 linear estimate \(\gamma=1.50\times10^4\ \mathrm{s^{-1}}\)、\(k_\perp=150\ \mathrm{m^{-1}}\)。
系统： local radial patch；outward flux positive。
模型： stationary correlation and Fick-like diagnostic \(\lvert d\bar n/dr\rvert=\bar n/L_n\)。
\[\Gamma_r=(0.0500)(1.00\times10^{19})(200)(0.600) =6.00\times10^{19}\ \mathrm{m^{-2}\,s^{-1}}.\]
\[D_{\rm eff}=\frac{\Gamma_r}{\overline n/L_n} =0.600\ \mathrm{m^2\,s^{-1}}.\]
mixing-length comparison：
\[D_{\rm mix}\sim\frac{\gamma}{k_\perp^2} =0.667\ \mathrm{m^2\,s^{-1}}.\]
检查： 两 estimates 同量级，但不证明 mixing-length model 唯一正确；若 cross-phase 变为 \(90^\circ\)，flux 为零，即使 rms amplitudes 不变。
Lesson synthesis / 全课整合
instability-to-transport loop：指定 equilibrium/free-energy source → 选 kinetic/fluid/MHD model 与 boundary → linearize并求 complex \(\omega\) → 用 criterion/energy source 审核 \(\gamma\) sign → 验证 eigenfunction/polarization → 追踪最先达到 nonlinear amplitude 的 modes → 识别 trapping、profile flattening、mode coupling、shear flows 等 saturation → 用 cross-correlations 计算 flux → 更新 equilibrium gradients，重复 stability analysis。
Formula-and-decision table / 公式与决策表
| Need | Relation | Use only when | Main caution |
|---|---|---|---|
| cold two-stream | \(1-\omega_{pe}^2/[2(\omega-ku_0)^2]-\omega_{pe}^2/[2(\omega+ku_0)^2]=0\) | equal cold beams | finite spread changes threshold |
| maximum cold growth | \(\gamma_{\max}=\omega_{pe}/(2\sqrt2)\) | same symmetric model | not universal beam growth |
| drift scale | \(\lvert\omega_{*e}\rvert=k_y k_B T_e/(e B L_n)\) | local slab magnitude | frequency is not growth rate |
| interchange estimate | \(\gamma^2\sim g_{eff}/L_p-k_\parallel^2v_A^2\) | stated curvature model | coefficients/sign depend on geometry |
| cylinder twist | \(q(a)=2\pi aB_z/(LB_\theta)\) | periodic straight cylinder | wall/profile/toroid modify criterion |
| growth time | \(\tau_g=1/\gamma\) | constant positive linear \(\gamma\) | energy rate is \(2\gamma\) |
| turbulent flux | \(\Gamma_r=\langle\widetilde n\widetilde v_r\rangle\) | defined average | cross-phase controls sign/magnitude |
| mixing length | \(D\sim\gamma/k_\perp^2\) | order-of-magnitude saturation | not a universal transport law |
Model boundaries / 模型边界
- Cold two-stream delta functions exaggerate sharp resonances；warm distributions require kinetic dielectric/Penrose or numerical solution。
- Drift and interchange labels cover families；collisional drift wave、ITG、TEM、ballooning and resistive interchange have distinct equations。
- Simple \(q>1\) applies to one ideal cylindrical kink problem；internal/external/resistive kink thresholds differ。
- Ideal-MHD \(\delta W\) cannot decide collisionless velocity-space instability or all resistive/non-normal growth。
- Linear growth says nothing by itself about saturation amplitude or final transport；nonlinear energy conservation and profile evolution matter。
- \(D_{\rm eff}\) can hide pinch, convection, avalanches and nonlocality；report the defining flux-gradient relation。
- Turbulence spectra may transfer energy forward, inverse, or across fields/velocity space；one schematic cascade is not universal。
Misconception clinic / 易错概念诊所
| Misconception | Correction |
|---|---|
| any two beams are unstable at all \(k\) | even cold symmetric beams have a finite unstable band; temperature modifies it |
| diamagnetic frequency equals radial particle flow | it is a wave/free-energy frequency; net radial flux needs cross-phase |
| curvature always destabilizes | favorable curvature and field-line bending can stabilize |
| kink and sausage are the same necking | kink displaces axis (\(m=1\)); sausage changes radius axisymmetrically (\(m=0\)) |
| \(\gamma>0\) means amplitude instantly large | linear growth needs several e-foldings and starts from a seed |
| stable eigenvalues guarantee no transient response | non-normal forcing/transient growth can exist outside simple modal criterion |
| large fluctuations guarantee large flux | \(\Gamma\) depends on correlation and phase, not amplitudes alone |
| turbulence is an adjustable diffusion coefficient | it is nonlinear dynamics; an effective \(D\) is a diagnostic closure with limits |
Practice and retrieval / 练习与检索
- [K01] symmetric cold beams 的 \(a=ku_0/\omega_{pe}=1.20\) 是否在 unstable band？
Answer 1
否；该 model 的 unstable band 是 \(0<a<1\)，\(a=1.20\) 时 lower \(\omega^2\) 不再为负。
- [K01] two-stream nonlinear saturation 后为何常见 phase-space vortices？
Answer 2
resonant particles 被 finite-amplitude wave potential trapping，在 wave frame 做 bounce motion，形成 closed phase-space islands/vortices，并 flatten resonant slope。
- [K02] \(L_n\) 减半且其他量固定，\(\lvert\omega_{*e}\rvert\) 如何变化？
Answer 3
与 \(1/L_n\) 成正比，所以加倍；但 growth rate 不一定同倍变化。
- [K02] interchange mode 为什么偏好 small \(k_\parallel\)？
Answer 4
小 \(k_\parallel\) 减少 field-line bending/tension energy，因此更容易利用 unfavorable-curvature pressure drive。
- [K03] axisymmetric radius necking 对应哪个 \(m\)？
Answer 5
\(m=0\) sausage mode；\(m=1\) 是 axis kink displacement。
- [K03] simple cylinder 中增大 \(B_z\) 对 edge \(q\) 与 kink tendency 怎样影响？
Answer 6
\(q\propto B_z\) 增大；在其他量 fixed 的 Kruskal–Shafranov picture 中 twist 相对减小，kink 更稳定。
- [K04] amplitude growth rate \(\gamma=4.0\times10^3\ \mathrm{s^{-1}}\)，增长 5 e-foldings 需多久？
Answer 7
\(t=5/\gamma=1.25\ \mathrm{ms}\)；amplitude 乘 \(e^5\approx148\)。
- [K04] 找到一个 allowed displacement 使 ideal-MHD \(\delta W<0\) 可推出什么？
Answer 8
在该 equilibrium、boundary 与 ideal-MHD model 内可证明存在 instability；不能自动推出其 nonlinear outcome 或 kinetic growth rate。
- [K05] \(\widetilde n\) 与 \(\widetilde v_r\) 相差 \(90^\circ\) 时 average turbulent flux？
Answer 9
对 sinusoidal stationary fluctuations，\(\Gamma_r=n_{rms}v_{r,rms}\cos90^\circ=0\)。
- [综合] profile gradient 被 turbulence flatten 后，原 instability growth 通常如何反馈？
Answer 10
free-energy gradient 减弱，linear drive/growth 往往下降；外部 fueling/heating 可重建 gradient，形成 drive–transport–profile feedback cycle。
Exit ticket / 迁移检验
对一条声称“发现新 plasma instability”的 growth curve，列出最少五项可复核信息。
Exit-ticket benchmark
equilibrium/distribution 与 free-energy source；geometry/boundary；model/closure；Fourier sign convention 与 amplitude/energy rate；unstable \(k\) range/eigenfunction；numerical convergence/noise；nonlinear saturation and conserved-energy budget。
References and source notes / 参考资料与使用说明
以下权威直接来源均在 2026-08-31 实际打开核对；所有 diagrams 与 function sketches 均原创，未复制来源 figures。
- MIT OpenCourseWare 22.611J, Chapter 5 — Plasma Waves and Instabilities — two-stream dispersion、kinetic waves and instability context。
- MIT OpenCourseWare 22.611J, Chapter 4 — Fluid Description and MHD Stability — energy principle、kink、sausage and interchange/flute mechanisms。
- MIT OpenCourseWare 22.615, Lecture 17 — Interchange, Flute, and Sausage Stability — MHD mode criteria and geometry。
- MIT OpenCourseWare 6.642, Lecture 9 — Plasma Stability (z–theta pinch) — cylindrical equilibrium, perturbations and pinch stability。
- Richard Fitzpatrick, UT Austin, Velocity-Space Instabilities — resonant slope and Penrose criterion。
- Richard Fitzpatrick, UT Austin, Counter-Propagating Beam Instability — warm counter-streaming threshold and growth interpretation。
- Princeton Plasma Physics Laboratory, Modeling 2D Drift-Wave Turbulence — linear drift-wave growth, nonlinear vortices, zonal flow and particle flux diagnostics。
- Princeton Plasma Physics Laboratory, PPPL-5178 — Statistical Closure and Drift-Wave Turbulence — mixing-length context、closures and zonal-flow feedback。
- Princeton Plasma Physics Laboratory, AST 552 General Plasma Physics II — energy principle、fusion-plasma instabilities、reconnection and turbulence course map。
Source-to-content map
| Source | Used for | Knowledge ID / Section | Model boundary or convention |
|---|---|---|---|
| MIT 22.611J Ch. 5 | cold two-stream derivation | K01, K04 | electrostatic equal cold beams |
| MIT 22.611J Ch. 4 | MHD free-energy mechanisms | K02–K04 | ideal cylindrical/slab teaching models |
| MIT 22.615 L17 | interchange/sausage stability | K02–K04 | MHD equilibrium and boundary dependent |
| MIT 6.642 L9 | pinch perturbation framework | K03–K04 | z–theta cylindrical geometry |
| Fitzpatrick Velocity-Space | slope/Penrose logic | K01, K04 | one-dimensional kinetic distributions |
| Fitzpatrick Counter-Beams | warm-beam boundary | K01 | Cauchy-beam example, not cold formula |
| PPPL SULI turbulence | saturation and flux | K02, K05 | reduced 2-D drift-wave model |
| PPPL-5178 | mixing length and closure caveats | K05 | statistical turbulence theory |
| PPPL AST552 | curriculum cross-check | K02–K05 | graduate MHD/kinetic scope |
复习与拓展
- Lesson 13.4.1 将把 kink/interchange/turbulent transport 置入 magnetic-confinement performance 与 Lawson energy balance。
- Lesson 13.4.2 将连接 two-stream、shock、reconnection 与 turbulence 到 solar-wind/magnetospheric/astrophysical plasmas。
- 可扩展：Nyquist/Penrose construction、ballooning/tearing/plasmoid、gyrokinetic ITG/TEM、quasilinear diffusion 与 zonal-flow Dimits shift。