Self Organization in Fusion Plasma Physics ChuanSheng Liu University of Maryland

Congratulations to ASIPP 40th Anniversary From ground zero to forefront of world fusion Major support for ITER The longest discharge in Tokamak plasmas over 50 seconds Fully superconducting Tokamak

We have come a long way ITER zeta stellerator Still long way to go

Progress and Problems in Tokamak Research Present-day large tokamak experiments have currents up to 7 MA and the confinement time of nearly 2 seconds provides nτe of 1020 m-3s. Fusion output power has reached ~ 1.7 MW. Physics problems remaining: a) electron heat transport is still two orders of magnitude higher than classical. b) physics of L to H transition c) disruption control d) steady state operation Engineering and material problems are important

Stages of Confinement Studies 50-60: Search for stable magnetic confinement configurations. Minimum B for stabilization of interchange. Trapped particle instability (Kadamtzev) Maximum J configuration (Rosenbluth) 60-70: Universal instabilities: drift wave and temperature gradient instabilities 70-80: How to live with instabilities 80-90: Spontaneous onset of H mode: competition between coherent motion and turbulence After 90: EAST and DIIID to ITER

Trapped Ion Mode B Coppi, R Galvao, R Pellat, M Rosenbluth and P Rutherford, Sov. J. Plasma Phys. 2, 533 (1976) Ion temperature gradient mode in a slab plasma with sheared magnetic field CS Liu, Phys. Fluids 12, 1489 (1969) Ion temperature gradient mode in toroidal geometry In octopole (minimum B) and in maximum J configuration, ∇T is still unstable

Importance of Trapped Electron Mode

Electron Temperature gradient driven modes and anomalous transport in tokamaks Guzdar, Dong, Lee, Drake, Liu, Rosenbluth, Gladd and Chang. Theory of Fusion Plasmas (1987) Trapped Electron Modes Microtearing Modes ηe Modes This mode is Ohkawa formula for Alcator scaling for Ohmic plasma Type equation here. 𝝌 𝒆 =𝟎.𝟏 𝒄 𝟐 𝝎 𝒑𝒆 𝟐 𝒗 𝒆 𝟐 𝑹𝒒 𝜼 𝒆 𝟏+ 𝜼 𝒆

Electron Heat Transport by Plasma Waves Liu and Rosenbluth, Phys. Fluids (1976) Plasma waves, unlike electrons, are not confined by the magnetic field. They can propagate across the field with long mean free path. They can transport energy across magnetic field comparable to electron heat conduction.

Inertial Fusion Laser fusion is a competitor with magnetic fusion. Laser power has increased one thousand times every decade since 1960. First laser has 100 W power, now we have tens of petawatts. ELI, Korea, China, USA all have high power lasers for fusion and other studies.

National Ignition Facility (NIF) Indirect Drive National Ignition Facility (NIF) 1.85 MJ, 192 Lasers, 500 trillion watts, 3.5 billion dollar

Raman Scattering: a reason for NIF failure to ignite 30% of laser being backscattered by stimulated Raman instability How it was all started in 1972 Excitation of Plasma Waves by Two Laser Beams. M. N. Rosenbluth and C. S. Liu. Phys. Rev. Lett. 29, 701 (1972) Parametric Backscattering Instabilities of Electromagnetic Waves in Underdense, Inhomogeneous Plasmas COO3237 (1972) C. S. Liu and M. N. Rosenbluth Raman and Brillouin scattering of electromagnetic waves in inhomogeneous plasmas. Physics of Fluids 17, 1211 (1974) C. S. Liu, Marshall N. Rosenbluth, and Roscoe B. White Parametric instabilities of electromagnetic waves in plasmas Physics of Fluids 17, 778 (1974) J. F. Drake, P. K. Kaw, Y. C. Lee, and G. Schmidt, C. S. Liu and Marshall N. Rosenbluth

SRS in Plasma 𝑘 0 = 𝑘 𝑝 + 𝑘 𝑠 𝜔 0 = 𝜔 𝑝 + 𝜔 𝑠 Instability Laser Scattered wave Plasma wave Parametric instability of three wave interaction Instability 𝜔 0 = 𝜔 𝑝 + 𝜔 𝑠 𝑘 0 = 𝑘 𝑝 + 𝑘 𝑠 1. Electron oscillates in the pump laser: 𝑑 𝑣 0 𝑑𝑡 = 𝑒𝐸 𝑚 𝑒 −𝑖 𝜔 0 𝑡+𝑖 𝑘 0 𝑥 Plasma wave has density oscillate 𝛿𝑛 = 𝛿𝑛 0 𝑒 −𝑖 𝜔 𝑝 𝑡+𝑖 𝑘 𝑝 𝑥 Together these produce scattered EM wave with a current resonantly. 𝛿 𝑗 𝑠 =𝑒𝛿𝑛∙ 𝑣 𝑜 2. Pump laser and scattered wave produce pondermotive force via Lorentz force e( 𝑣 0 𝐵 1 + 𝑣 1 𝐵 0 ) to drive plasma wave resonantly.

SRS Growth Rate 𝛾= 𝑘 𝑝 𝑣 0 2 𝜔 𝑝 𝜔 0 1/2 𝑘 𝑝 =2 𝑘 0 𝑘 0 𝑘 𝑝 𝛾= 𝑘 𝑝 𝑣 0 2 𝜔 𝑝 𝜔 0 1/2 𝑘 p 𝜆 D = 𝑣 e / 𝑣 ph Maximum growth for backscatter 𝑘 𝑝 =2 𝑘 0 𝑘 0 𝑘 𝑝 𝛾 𝑚𝑎𝑥 = 𝑣 0 𝑐 𝜔 𝑝 𝜔 0 1/2 𝑘 𝑠 =− 𝑘 0

Raman and Brillouin effect on propagation of EM wave 1 4 𝑛 𝑐 𝑛 𝑐 Raman backscatter Brillouin backscatter 𝑘=( 𝜔 2 − 𝜔 𝑝 2 𝑐 2 ) 1/2 Linear theory 𝑘: Real for wave propagation 𝑘=0 for wave reflection 𝑘:imaginary evanescent Convective Raman makes the wave backscattered at density below nc/4 At nc/4, Raman is absolute. Drake & Lee, Phy. Rev. Lett. (1973)

Review Article David S. Montgomery, Two decades of progress in understanding and control of laser plasma instabilities in indirect drive inertial fusion, Phys. Plasmas 23, 055601 (2016). Trident laser plasma system: Highly reproducible plasma formed in the laser hot spot for laser plasma interaction studies.

Inflation of Raman Reflectivity D. S. Montgomery, POP 23,055601(2016).

Nonlinear Transition from Convective to Absolute Instability Convective instability Absolute instability

Simple Derivation of Absolute Threshold 𝜕 2 𝑎 𝜕𝑥 2 + 𝛾 0 2 𝜐 1 𝜐 2 − 1 4 𝑝+ 𝜈 1 𝜐 1 + 𝑝+ 𝜈 2 𝜐 2 2 𝑎=0 𝑎=𝐴 sin 𝑘𝑥 𝑎 0 =𝑎 𝐿 =0, 𝑘𝐿=𝑛𝜋 𝛾 0 2 𝜐 1 𝜐 2 − 1 4 𝑝+ 𝜈 1 𝜐 1 + 𝑝+ 𝜈 2 𝜐 2 2 = 𝑛𝜋 𝐿 2 𝑝= 2 𝛾 0 2 𝜐 1 𝜐 2 − 𝜋 𝐿 2 − 𝜈 1 𝜐 1 + 𝜈 2 𝜐 2 1 𝜐 1 + 1 𝜐 2 solving for 𝑝 𝑛=1 Condition for absolute instability positive real 𝑝 solution to exist 𝛾 0 2 𝜐 1 𝜐 2 > 𝜋 𝐿 2 + 1 4 𝜈 1 𝜐 1 + 𝜈 2 𝜐 2 2 M. N. Rosenbluth R.White and CS Liu,PRL (1973).

Distribution function and Landau damping of plasma wave Trappe electron effect Convective instability Absolute instability

Convective instability Absolute instability

Distribution function Plasma wave growth 𝐼=3× 10 15 𝑊 𝑐𝑚 2 Absolute instability Distribution function Plasma wave growth

Growth of Reflectivity

Saturation of growth Plasma wave breaking 𝐼=3× 10 15 𝑊 𝑐𝑚 2

Experimental data：D. S. Montgomery et al. POP 9,2311 (2002) absolute 𝑇 𝑒 =0.5𝑘𝑒𝑉 𝑛 𝑒 =0.025 𝑛 𝑐 𝑘 𝜆 𝐷 =0.35 𝐼=1× 10 15 −−2× 10 16 𝑊 𝑐𝑚 2 𝜆 0 =0.527𝜇𝑚 transition convective

Transition from Landau damping to bounce resonance damping Vlasov equation in action angel variable : C. S. Liu, J. Plasma Physics (1972)

Nonlinear effects: soliton 𝜕 2 𝐴 𝜕𝑡 2 =− 𝜔 𝑝0 2 𝐴+3 𝜐 2 𝜕 2 𝐴 𝜕𝑥 2 𝐴=𝑎 𝑥,𝑡 𝑒 −𝑖 𝜔 𝑝 𝑡 −2𝑖 𝜔 𝑝 𝜕𝑎 𝜕𝑡 − 𝜔 𝑝 2 𝑎+ 𝜔 𝑝0 2 𝑎−3 𝜐 2 𝜕 2 𝑎 𝜕𝑥 2 =0 𝑚= 𝑚 0 1− 𝜐 0 𝑐 2 𝜔 𝑝 2 = 𝜔 𝑝0 2 1− 1 2 𝜐 0 𝑐 2 Nonlinear Schrontinger equation: −2𝑖 𝜔 𝑝 𝜕𝑎 𝜕𝑡 − 3𝜐 2 𝜕 2 𝑎 𝜕𝑥 2 + 1 2 𝑒 2 𝑚 2 𝑎 2 𝑎=0

Soliton Acceleration in Inhomogeneous Plasma Wave package soliton acceleration by plasma inhomogeneity

In the book “Nonlinear and Relativistic Plasmas” 𝜔 2 = 𝜂 1 2 − 𝜂 2 2 𝑃 2

Transition from Soliton to Chaos In the range 0 ≤ P ≤ 0.575, the solution is steady state much like the Airy function for P = 0. This solution is well known. 0.575 < P < 1.088. Periodic emission of solitons to the underdense region is observed. 1.088 < P < 1.155. The time evolution of the wave amplitude at x = 0 undergoes period doubling. 1.1575 < P < 1.255. A small frequency modulation appears. P > 1.255. The emission of soliton shows chaotic behavior.

Magnetic Confinement Studies I started fusion research in 1968. Inspired by the Russian Tokamak results of 1 keV temperature measured by the British team. I was asked by B. Fried to give talks on neoclassical transport and instabilities. I wrote a paper on temperature gradient instability in toroidal plasma showing even in maximum J configuration proposed by Rosenbluth to stabilize Kadamtsez trapped particle mode, there are still instabilities. C.S. Liu, Phys. Fluids 69 We must live with instabilities

My Involvement in GA I met Ohkawa in 1969 at APS meeting. He told me that there are fluctuations even in Octopole, a minimum B device for suppressing interchange instability. So I mentioned the ion temperature mode as a possible explanation. He then invited me to spend a year in General Atomic. At that time, there are only six people in the fusion group of GA. So I became the seventh samurai. That year (1969-1970) I was very productive. Ohkawa was a great mentor and a most innovative physicist. In 1970 I went to Princeton to work with Rosenbluth. But in 1980, Ohkawa asked me to come back to GA as director of theoretical and computational group. I served there for 5 years.