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Introduction to computational plasma physics 雷奕安 ，

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课程概况 iewtopic.php?t=77 上机 成绩评定为期末大作业

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Related disciplines Computation fluid dynamics (CFD) Applied mathematics, PDE, ODE Computational algorithms Programming language, C, Fortran Parallel programming, OpenMP, MPI Plasma physics, space, fusion, … Unix, Linux, …

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大规模数值模拟的特殊性 数值计算数学问题算法编程数值模拟物理问题数学模型算法编程 大规模数值模 拟 物理问题及 数学模型 相关学科研 究人员支持 超级计算机 软硬件系统

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Contents What is plasma Basic properties of plasma Plasma simulation challenges Simulation principles

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What is plasma Partially ionized gas, quasi-neutral Widely existed –Fire, lightning, ionosphere, polar aurora –Stars, solar wind, interplanetary (stellar, galactic) medium, accretion disc, nebula –Lamps, neon signs, ozone generator, fusion energy, electric arc, laser-material interaction Basic properties –Density, degree of ionization, temperature, conductivity, quasi-neutrality –magnetization

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Plasma vs gas PropertyGasPlasma ConductivityVery low, insulatorVery high, conductor SpeciesUsually oneAt least two, ion, electron DistributionUsually MaxwellianUsually non-Maxwellian InteractionBinary, short rangeCollective, long range

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Basic properties Temperature Quasi-neutrality Thermal speed Plasma frequency Plasma period

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Debye length System size and time Debye shielding λDλD U→0

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Debye lengths

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Plasma parameter Strong coupling Weak coupling

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Weakly coupled plasmas

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Collision frequency Mean-free-path Collisional plasma (Collisionless) Collisioning frequency

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Magnetized plasma Anisotropic Gyroradius Gyrofrequency Magnetization parameter Plasma beta

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Simulation challenges Problem size: ~ particles Debye sphere size: 10 2 ~ 10 6 particles Time steps: 10 4 ~ 10 6 Point particle, computational unstable, sigularities

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Solution No details, essence of the plasma One or two dimension to reduce the size No high frequency phenomenon, increase time step length Reduce N D, m i / m e Smoothing particle charge, clouds Fluidal approaches, single or double Kinetic approaches, f/f

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Simple Simulation Electrostatic 1 dimensional simulation, ES1 Self and applied electrostatic field Applied magnetic field Couple with both theory and experiment, and complementing them

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Basic model

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Field -> force -> motion -> field -> … Field: Maxwell's equations Force: Newton-Lorentz equations Discretized time and space Finite size particle Beware of nonphysical effects

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Computational cycle

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Equation of motion v i, p i, trajectory Integration method, fastest and least storage Runge-Kutta Leap-frog

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Planet Problem x0 = 1; vx0 = 0; y0 = 0; vy0 = 1 read (*,*) dt N = 30/dt do i = 0, N+3 x1 = x0 + vx0*dt y1 = y0 + vy0*dt r = sqrt(x0*x0 + y0*y0) fx = -x0/r**3 fy = -y0/r**3 vx1 = vx0 + fx*dt vy1 = vy0 + fy*dt ! if(mod(i,N/10).eq.2) write(*,*) x0, y0, -1/r+(vx0*vx0+vy0*vy0)/2 x0 = x1; y0 = y1; vx0 = vx1; vy0 = vy1 enddo end Forward differencing

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Planet Problem./a.out > data 0.1 $ gnuplot Gnuplot> plot “data” u 1:2

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Planet Problem./a.out > data 0.01 $ gnuplot Gnuplot> plot “data” u 1:2

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Planet Problem x0 = 1; vx0 = 0; y0 = 0; vy0 = 1 read (*,*) dt N = 30/dt x1 = x0 + vx0*dt y1 = y0 + vy0*dt xh0 = (x0+x1)/2; yh0 = (y0+y1)/2 do i = 0, N xh1 = xh0+vx0*dt; yh1 = yh0 + vy0*dt; r = sqrt(xh0*xh0 + yh0 *yh0 ) fx = -xh1/r**3 fy = -yh1/r**3 vx1 = vx0 + fx*dt vy1 = vy0 + fy*dt ! if(mod(i,N/100).eq.0) write(*,*) xh0, yh0, -1/r+(vx0*vx0+vy0*vy0)/2 xh0 = xh1; yh0 = yh1; vx0 = vx1; vy0 = vy1 enddo end Leap Frog

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Planet Problem./a.out > data 0.1 $ gnuplot Gnuplot> plot “data” u 1:2

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Planet Problem./a.out > data 0.01 $ gnuplot Gnuplot> plot “data” u 1:2

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Field equations Poisson’s equation

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Field equations Poisson’s equation is solvable In periodic boundary conditions, fast Fourier transform (FFT) is used, filtering the high frequency part (smoothing), is easy to calculate

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Particle and force weighting Particle positions are continuous, but fields and charge density are not, interpolating Zero-order weighting First-order weighting, cloud-in-cell

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Higher order weighting Quadratic or cubic splines, rounds of roughness, reduces noise, more computation

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Initial values Number of particles and cells Weighting method Initial distribution and perturbation The simplest case: perturbed cold plasma, with fixed ions. Warm plasma, set velocities

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Initial values

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Diagnostics Graphical snapshots of the history x, v, , , E, etc. Not all t i For particle quantities, phase space, velocity space, density in velocity For grid quantities, charge density, potential, electrical field, electrostatic energy distribution in k space

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Tests Compare with theory and experiment, with answer known Change nonphysical initial values (NP, NG, t, x, …) Simple test problems

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Server connection Ssh Host: , protocol: ssh2 Your username & password Vnc connection In ssh shell: “vncserver”, input vnc passwd, remember xwindow number Tightvnc: :xx (the xwindow number) Kill vncserver: “vncserver –kill :xx” (x-win no.)

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Xes1 Xes1 document Xgrafix already compiled in /usr/local Xes1 makefile make./xes1 -i inp/ee.inp LIBDIRS = -L/usr/local/lib -L/usr/lib -L/usr/X11R6/lib64

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Clients Ssh putty.exe Vncviewer exe Pscp:

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