Introduction to computational plasma physics

雷奕安62755208 ， yalei@pku.edu.cn

课程概况• http://www.phy.pku.edu.cn/~fusion/forum/

viewtopic.php?t=77

• 上机• 成绩评定为期末大作业

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, …

大规模数值模拟的特殊性

Contents

• What is plasma

• Basic properties of plasma

• Plasma simulation challenges

• Simulation principles

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

Plasma vs gas

Property Gas Plasma

Conductivity Very low, insulator Very high, conductor

Species Usually one At least two, ion, electron

Distribution Usually Maxwellian Usually non-Maxwellian

Interaction Binary, short range Collective, long range

Basic properties

• Temperature

• Quasi-neutrality

• Thermal speed

• Plasma frequency

• Plasma period

Debye length

• System size and time

• Debye shielding

λD

U→0

Debye lengths

Plasma parameter

• Strong coupling

• Weak coupling

Weakly coupled plasmas

Collision frequency

• Mean-free-path

• Collisional plasma

• (Collisionless)

• Collisioning frequency

Magnetized plasma

• Anisotropic

• Gyroradius

• Gyrofrequency

• Magnetization parameter

• Plasma beta

Simulation challenges

• Problem size: 1014 ~ 1024 particles

• Debye sphere size: 102 ~ 106 particles

• Time steps: 104 ~ 106

• Point particle, computational unstable, sigularities

Solution

• No details, essence of the plasma

• One or two dimension to reduce the size

• No high frequency phenomenon, increase time step length

• Reduce ND, mi / me

• Smoothing particle charge, clouds

• Fluidal approaches, single or double

• Kinetic approaches, f/f

Simple Simulation

• Electrostatic 1 dimensional simulation, ES1

• Self and applied electrostatic field

• Applied magnetic field

• Couple with both theory and experiment, and complementing them

Basic model

Basic model

Basic model

• Field -> force -> motion -> field -> …

• Field: Maxwell's equations

• Force: Newton-Lorentz equations

• Discretized time and space

• Finite size particle

• Beware of nonphysical effects

Computational cycle

Equation of motion

• vi, pi, trajectory

• Integration method, fastest and least storage• Runge-Kutta• Leap-frog

Planet Problem

tdt

d ii

1

x0 = 1; vx0 = 0; y0 = 0; vy0 = 1read (*,*) dtN = 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 = vy1enddoend

Forward differencing

t

xx

dt

dx ii

1

Planet Problem

./a.out > data

0.1

$ gnuplot

Gnuplot> plot “data” u 1:2

Planet Problem

./a.out > data

0.01

$ gnuplot

Gnuplot> plot “data” u 1:2

Planet Problemx0 = 1; vx0 = 0; y0 = 0; vy0 = 1read (*,*) dtN = 30/dt

x1 = x0 + vx0*dty1 = y0 + vy0*dtxh0 = (x0+x1)/2; yh0 = (y0+y1)/2do 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 = vy1enddoend

Leap Frog

tdt

d ii

1

t

xx

dt

dx ii

2123

Planet Problem

./a.out > data

0.1

$ gnuplot

Gnuplot> plot “data” u 1:2

Planet Problem

./a.out > data

0.01

$ gnuplot

Gnuplot> plot “data” u 1:2

Field equations

• Poisson’s equation

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

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

Higher order weighting

• Quadratic or cubic splines, rounds of roughness, reduces noise, more computation

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

Initial values

Diagnostics

• Graphical snapshots of the history

• x, v, , , E, etc.

• Not all ti

• For particle quantities, phase space, velocity space, density in velocity

• For grid quantities, charge density, potential, electrical field, electrostatic energy distribution in k space

Tests

• Compare with theory and experiment, with answer known

• Change nonphysical initial values (NP, NG, t, x, …)• Simple test problems

Server connection

• SshHost: 162.105.23.110, protocol: ssh2

• Your username & password• Vnc connection

In ssh shell: “vncserver”, input vnc passwd, remember xwindow number

• Tightvnc: 162.105.23.110:xx (the xwindow number)

• Kill vncserver: “vncserver –kill :xx” (x-win no.)

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

Clients

• Sshputty.exe

• Vncviewerhttp://www.phy.pku.edu.cn/~lei/vncviewer.exe

• Pscp:

• http://www.phy.pku.edu.cn/~lei/pscp.exe