Simulating Ionospheric Simulating Ionospheric Plasma Physics Using Plasma Physics Using Millions and Millions of Millions and Millions of

ParticlesParticles

ByByMeers OppenheimMeers Oppenheim

Talk OutlineTalk Outline

What are simulations?What are simulations?What Ionospheric problems require What Ionospheric problems require simulations?simulations?Introduction to kinetic plasma Introduction to kinetic plasma physicsphysicsIntroduction to ParticleIntroduction to Particle--InIn--Cell (PIC) Cell (PIC) simulationssimulationsExample problemsExample problemsLimitations of these methodsLimitations of these methods

What are simulations?What are simulations?

The Encyclopedia Britannica says “the The Encyclopedia Britannica says “the use of a computer to represent the use of a computer to represent the dynamic responses of one system by the dynamic responses of one system by the behavior of another system modeled behavior of another system modeled after it. A after it. A simulationsimulation uses a uses a mathematical description, or model, of a mathematical description, or model, of a real system in the form of a computer real system in the form of a computer program.”program.”Contrasts with “Model” Contrasts with “Model” –– which we use which we use to mean any precise descriptionto mean any precise description

What use are simulations?What use are simulations?Views of nature:Views of nature:•• Physicists think that equations approximate the real Physicists think that equations approximate the real

world.world.•• Engineers think that the real world approximates Engineers think that the real world approximates

equations.equations.•• Mathematicians don’t care... Mathematicians don’t care...

Simulations explore the behavior of systems too Simulations explore the behavior of systems too complex for analytical theorycomplex for analytical theory•• Inhomogeneous systemsInhomogeneous systems•• Nonlinear systemsNonlinear systems•• TurbulenceTurbulence

Simulations give insight into complex systems!Simulations give insight into complex systems!

Where does one need simulations?Where does one need simulations?

The Auroral The Auroral Ionosphere: Ionosphere: Electrons Electrons accelerate accelerate from 3000from 3000--1500 km 1500 km altitude by altitude by unknown unknown mechanismsmechanisms

FAST Spacecraft FAST Spacecraft measures measures

turbulent auroral turbulent auroral plasmasplasmas

Radars Radars Measure Measure Electron Electron Density Density

Irregularities Irregularities in E & F in E & F Regions

99.6

119

90.0

94.8

104

109

114

124

0 4 8 12 16 20

Ran

ge

(km

)

Elapsed Time (min) since 23:04:42 LT of August 7, 1990

Altitude

Regions Altitude

Plasma Theory in 5 MinutesPlasma Theory in 5 Minutes1.1. Charged particles Charged particles

create fields: create fields: Maxwell’s Maxwell’s EquationsEquations

2.2. Lorentz Force Lorentz Force Accelerates Accelerates Particles:Particles: [ ]),(),( txBvtxE

mq

dtvd

ii

ii rvvv

×+=

JtEB

tBE

BnneE ei

vv

vvv

vv

vvvv

000

0

0 )(

εµµ

ε

+∂∂

=×∇∂∂

−=×∇

=•∇−=•∇

3.Collisions deflect particles (important in the lower ionosphere)

3.Too many particles – Need simplifications!

Particle SimulationsParticle Simulations

Ion

Particles move within a Particles move within a box:box:•• Position: Position: xxii•• Velocity: Velocity: vvii

Particles generate fields Particles generate fields which accelerate other which accelerate other particlesparticlesToo Slow! Speed Too Slow! Speed proportional to the proportional to the number of particles number of particles squared.

20 )(4 ji

jiij xx

qqF vvv

−=

πε

squared.

ParticleParticle--InIn--Cell (PIC) Steps:Cell (PIC) Steps:1.1. Gather to determine charge density, Gather to determine charge density, ρρ

2.2. Calculate Electric field:Calculate Electric field:3.3. Update velocities:Update velocities:

4.4. Update Positions: Update Positions: 5.5. Go to Step 1 Go to Step 1

0ερ=•∇ Evv

[ ]),(),( txBvtxEmq

dtvd

iiii

ii rvvv

×+=

ii vdtxd vv=

00 -1 -1

1 0 1 1

0 0 -12

-1 0 -1 0

Charge DensityParticle Trajectories

Electron Ion

Assumptions made by PICAssumptions made by PIC00 -1 -1

1 0 1 1

0 0 -12

-1 0 -1 0

Charge DensityParticle Trajectories

Electron Ion

Short range interactions eliminatedShort range interactions eliminated•• Simulators with a meshes cannot model behavior smaller than Simulators with a meshes cannot model behavior smaller than

the meshthe mesh•• Features must be bigger than the meshFeatures must be bigger than the mesh

Each PIC particle models the behavior of more than 10Each PIC particle models the behavior of more than 1066 real real particlesparticlesFull kinetic physics representedFull kinetic physics represented•• Particle trapping Particle trapping –– resonant accelerationresonant acceleration•• Landau damping Landau damping –– resonant wave dampingresonant wave damping

Fluid Simulators also use a meshFluid Simulators also use a mesh•• Only one velocity in one location (unlike kinetic simulators)Only one velocity in one location (unlike kinetic simulators)•• Misses some physics but is less costly (per cell)Misses some physics but is less costly (per cell)

One Problem with PICOne Problem with PIC

Particle noise from limited numbers of Particle noise from limited numbers of particlesparticles•• Random walk statistics: Random walk statistics: •• Example n=144 particles/cell Example n=144 particles/cell --> > σσnn=8.3%=8.3%•• Fixes:Fixes:

Nature reduces this through electrostatic shieldingNature reduces this through electrostatic shieldingUse nonUse non--point particlespoint particlesUse millions and millions of particlesUse millions and millions of particlesUse super computers!Use super computers!

cellparticlesn /∝σ

Macro-particles fill a volume

Boundary Conditions (BC)Boundary Conditions (BC)Simulations of all types require BCSimulations of all types require BCBC introduce limitations and, sometimes, errorBC introduce limitations and, sometimes, errorExample: Example: PeriodicPeriodic is the simplest BC is the simplest BC •• The right side connects to the leftThe right side connects to the left•• The top to the bottomThe top to the bottom•• Particles leaving the Left reenter on the Right and visa Particles leaving the Left reenter on the Right and visa

versaversa•• Particles leaving the top Particles leaving the top --> bottom …> bottom …

Ion

Solution: Parallel SupercomputingSolution: Parallel SupercomputingParticle Trajectories

Electron Ion

Processor 1 Processor 3

Processor 2 Processor 4

Domain Domain DecompositionDecomposition

Mesh Mesh Parallelization

Charge DensityParticle Trajectories

00 -1 -1

1 0 1 1

0 0 -12

-1 0 -1 0

Processor 1

Processor 2

+

-10 0 0

-1 1 0 0

-1 02

0 0 0 0

0

Parallelization

Boundary Conditions Cause Boundary Conditions Cause LimitationsLimitations

Periodic boundaries Periodic boundaries quantize the simulation:quantize the simulation:•• Only a full wave orOnly a full wave or•• Integer multiples allowedInteger multiples allowed•• Simulations must not focus Simulations must not focus

on waves spanning the on waves spanning the systemsystem

Other BC have other Other BC have other issuesissuesTrue in fluid simulators as True in fluid simulators as wellwell

0 1 2 3 4 5 6 7 8Grid Spacing

-1.0

-0.5

0.0

0.5

1.0

Ele

ctric

fiel

d (V

/m)

0 1 2 3 4 5 6 7 8Grid Spacing

-1.0

-0.5

0.0

0.5

1.0

Ele

ctric

fiel

d (V

/m)

Example in 1D

Example: 1D electron twoExample: 1D electron two--stream Instabilitystream Instability

e-phas

e sp

ace

den

sity

(m

3m

/s)-

1

~1 M

illio

npar

ticl

es

~1/6

4 M

illio

npar

ticl

es

velo

city

velo

city

Distance (128 Debye lengths) Distance (128 Debye lengths)

Expand grid spacing 10XExpand grid spacing 10XEliminate Beam

Distance (1280 Debye lengths)

velo

city

Distance (1280 Debye lengths)

velo

city

(8x longer simulation in time,shown 16x as fast)

Simulation LimitationsSimulation Limitations

Systematic:Systematic:•• Do the equations represent the physics?Do the equations represent the physics?•• Do you resolve the important scales?Do you resolve the important scales?

Numerical:Numerical:•• StabilityStability•• AccuracyAccuracy

Electron Electron Holes in Holes in auroral auroral

ionosphereionosphere

Electron Holes in 2DElectron Holes in 2DElectric Field

y (p

erp

to B

)

z (parallel to B)

Simulations enable us to:•Understand dynamics of plasmas and fluids•Study energy and momentum flow•Characterize Turbulence

Meteor Plasma wavesMeteor Plasma wavesLeonids picture from the shuttle

Large ApertureRadar Detectionof a Meteor

Time (s)

Altitude

(km

)

Meteor Meteor Plasma Plasma

SimulationSimulation

Den

sity

Ele

ctric

Fiel

d

ConclusionsConclusions

Simulations enable us to explore Simulations enable us to explore nonlinear systemsnonlinear systemsSimulations subject to systematic Simulations subject to systematic limitations and numerical errorslimitations and numerical errorsEnable us to better understand:Enable us to better understand:•• our models and our models and •• naturenature