Numerical Studies of Test Particle Dynamics in Turbulent Force Fields

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Numerical Studies of Test Particle Dynamics in
Turbulent Force Fields
INTERNATIONAL SCHOOL OF SPACE SCIENCE, lAquila,
27.3 1.4 2006

• Kaspar Arzner
• Paul Scherrer Institut / ETH

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Why Numerics?

• The evolution of turbulent fields and of (test-)
  particles orbits therein is highly nonlinear.
• Thus we have only very limited analytical tools.
• Often, the interest is in particle population
  averages.
• The problem may then be simplified by replacing
  the complicated (exact) orbit equations by
  randomized ones.
• This is an approximation, and should be checked
  by numerical experiments!

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Topics

• Turbulent environments
• From deterministic motion in random media to
  random motion in deterministic media
• Limitations of the diffusion approximation
• Exact orbits
• Benchmarking and Diagnostics
• Summary

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Construction of Turbulent Fields

• Direct numerical
• simulations

B
Gaussian (random-phase) proxies
B
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Direct Numerical Simulation(not a desktop task)

• Solve the MHD equations
• forward in time. Two main approaches
• Pseudo-spectral operates in Fourier space, goes
  back to real space to evaluate the nonlinear
  terms. Periodic boundaries.
• Real-space allows for arbitrary boundaries,
  adaptive mesh refinement, and a natural domain
  decomposition.

(?t u.?) u - ?p j x b ? ?u
?tb ? x (u x b) ? ?b
EXPENSIVE !
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Random-phase (Gauss field) proxies

• Let the fluctuations be collected in z (u, b,
  ?, ...)
• Statistically homogeneous random-phase field
• z(x,t) ?k? ?(k,?) cosx.k ? ?t
  ?(k,?)
• Sij(k) must respect the underlying physics,
• Sbb(k,?).k 0 (?.b 0),
• and should account for the linearized
  dynamics (?t z ?z)
• Sij(k,?) 0 unless det i?
  - ?(k) 0
• Sij(k,?).? 0 unless (i? -
  ?(k))? 0.
• Example 1 linearly polarized Alfvén waves z
  (u,b), ? B0.k, and ? ? (k x B0, k x
  B0).
• Example 2 linear force-free perturbations k2
  ?2.
• Sij(k) should agree with observations.

uniform in 0,2?
Gaussian with zero mean and covariance Sij(k)
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Observational constraints
Matthaeus et al. (2005) magnetic single-time
two-point functions of the solar wind, using
multi-spacecraft observations (WIND, ACE,
Cluster).
?B(x).B(xr)?
Hnat et al. (2003) distributions of X(t?)-X(t)
with X B, v, B2, v2, ?v2 (WIND data)
scales with ? -? .
Numerical experiments non-gaussian PDFs
(Sorriso-Valvo et al. 1999, 2000) and structure
functions (Politano et al. 1998).
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Further Reading, Turbulence

• Batchelor (1982) Theory of homogeneous
  turbulence.
• McComb (1990) The physics of fluid turbulence.
• Falkovich et al. (Rev. Mod. Phys. 73, 913 2001)
  review transport of n-point functions in fluid
  (and i.e. Kraichnan-) turbulence.
• Verma (Physics Reports 401, 229 2004) reviews
  recent developments in MHD turbulence.

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Test particles

• Do not interact with each other
• Do not feed back to the plasma

Natural applications

• Rare species (He, O, ...)
• Collisionless high-energy tails,
• Runaway electrons.

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Coulomb Collisions

• Eqn. of motion d2r/dt kr/r3
• Scale r by ?, and t by ?
• ? ??-2 d2r/dt2 - ?-2 kr/r3
• ? unchanged if ?3?2 (Kepler).
• Since v (?/?),
• v2 ? const. for similar orbits.
• Now, ? ? and thus ?2 v-4
• Thus the collision rate
• ? n ?2 v nv-3 decreases with v!

?2
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Test particles

• Do not interact with each other
• Do not feed back to the plasma

Natural applications

• Rare species (He, O, ...)
• Collisionless high-energy tails

2 CLASSES OF FORCES
2. Coarse-grained forces from local densities
currents
e-
p
1. Collisions
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Basic Techniques for Test Particles
Microscopic
ODEs and SDEs Comparably simple, stable, easily
parallelized, but expensive

• Exact orbit integration
• Stochastic (approximate) differential equations

Macroscopic
PDEs More efficient, but potentially unstable,
parallelization requires careful design.

• Exact transport equations
• Diffusion (Fokker-Planck) approximations
• Others (e.g. fractional diffusion)

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A little systematics
Wiener process
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Wiener Process (Brownian motion)
Numerical Sample paths
Wn1 Wn ?t1/2? where ? is standard normal
?W2? t
Average
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In the limit ?t ? 0, W(t) is continuous but
nowhere differentiable
So what does the stochastic differential
equation dX a(X)dt b(X)dW really mean?
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Stochastic Interpretation
At which X should b(X) be evaluated?
dX a(X) dt b(X) dW
Stratonovich at (X-X)/2
Itô at X-
Example dX X dt X dW (leapfrog same W as
before)
Milshtein (1974), to order dt (dW2) do begin
dW sqrt(dt)randomn(seed) x x a(x)dt
b(x)dW b(x)db_dx(x)dW2/2 ...... Strat. x
x a(x)dt b(x)dW b(x)db_dx(x)(dW2-dt)/2
.. Ito t t dt end do
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Itô and Stratonovich have the same diffusion but
different drifts. The difference is not just
cosmetic
?tf ??x a f ½ ?x ?x b b f ?tf ??x a f ½
?x b ?x b f
Classical physics (continuous sample paths, red
noise approximated by white noise) usually leads
to Stratonovich.
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Some Astrophysical Applications
Literature on SDEs
General

• Gardiner Handbook of stochastic methods
• Risken The Fokker-Planck equation
• Higham (SIAM Rev 43, 525-546, 2001) A
  practical introduction
• Brissaud Frisch (J. Math. Phys, 15, 525,
  1974) Linear SDEs

• Krülls Achterberg (1994) Cosmic Ray
  acceleration.
• Chalov, Fahr Izmodenov (1995) ion pickup in
  the termination shock.
• Arzner Magun (1999) Coronal radio photon
  scattering.
• Marcowith Kirk (1999) diffusive shock
  acceleration.
• Schücker et al. (2001) cosmological mass
  clumping, described as a diffusion in k-space
• ...

Other Applications

• Black-Scholes (1973) stock options.
• Quantum physics, Biology, ...

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From field correlations to particle diffusion
coefficients
t
t
? requires suitable coordinates
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Analytic Coding Example (Maple)
Toy model dx/dt v, dv/dt ???, where
?(r) is a gaussian random field with
gaussian two-point function C(r) ??(0)?(r)?
Velocity diffusion Coefficient D ? dt
???(0)??(R)?
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D ? Fokker-Planck Equations

• Space science applications
• - Cosmic rays (Parker, Jokipii, Schlickeiser,
  ...)
• - Particle acceleration in solar flares and
  in the solar wind (Miller, Petrosian, Park,
  Karimabadi, Le Roux, ...) diffusion in momentum
  space to higher and higher energies. Different
  wave modes (constraints on Sij(k)) used for
  different particles.

• However, Fokker-Planck equations do not capture
  all features of the deterministic motion!
• Crude example dX/dt F(X) ? f(X). A change of
  sign does not affect D ?f f?, and has thus no
  effect if ?f? 0.

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More subtle ...
projector along v
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... Now ,
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2D case discrete flip about v

d2x/dt2 ?R(v)??(x)
d2x/dt2 ???(x)
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3D case continuous rotation by ?
Trapping at ?max
Displacement
Velocity
Remember, all ? have the same formal velocity
diffusion coefficients!
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Thus,

• There is need to resolve the full non-linear
  dynamics!
• All the more as the diffusion tensors involve
  only the two-point functions of the force fields,
  and higher-order effects (which are distinctive
  for turbulence) are disregarded.

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Test particle orbit integration (a desktop task!)

• Traditional Forward-Euler, Leapfrog,
  Runge-Kutta adaptive time steps (e.g. Cash-Karp)
• Boris scheme for gyrating particles
• Symplectic (enforce exact conservation of motion
  integrals in finite time steps)
• more ...

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Benchmarking

• Crucial!
• Judge by eye (IDL/matlab helpful)
• Check convergence as ?t ? 0
• Long-term behaviour which time step is optimal
  for the envisaged duration?
• Systematic approach Enforce symmetry and check
  the corresponding conservation law
• Compare to existing code

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Symmetries and conservation laws
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Explicitly, ...
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Example (ii)
energy
Should be conserved ?!
a0.(qxp)
a0
B(q) 4q2 a0 ? q
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Forward Euler systematically increases the enregy
v ?t v ? B2 v2 ?t2 v ? B2
Many better schemes are known Boris scheme for
gyration x ? x ?t v v ? v ½ ? v
.... (v2/2 ?(x) const) v ? RB(x) v
................ Rotate by B ?t v ? v
½ ? v Runge-Kutta higher-order accurate. Idea
improve forward-Euler by evaluating dx/dt and
dv/dt at fractions of ?t, and linearly combine
the results such as to eliminate the error of
(?x,?v) to highest possible order in
?t. Symplectic operate on (q,p) such as to
exactly conserve H over finite ?t.
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Example (ii) again, Boris scheme
energy
OK!
a0
a0.(qxp)
For comparison fwd Euler
Boris convergence as ?t ? 0 (end position)
?t t x y
z 5.0e-03 10.000000 -0.041466860 -1.1607201
1.1547736 1.0e-03 10.001000 -0.027493344
-1.1936583 1.1049999 5.0e-04 10.000000
-0.025199815 -1.1974548 1.1009599 1.0e-04
10.000100 -0.024044465 -1.2008377
1.0965715 5.0e-05 10.000050 -0.023854222
-1.2012338 1.0961140 1.0e-05 10.000010
-0.023702114 -1.2015505 1.0957489 5.0e-06
10.000005 -0.023683106 -1.2015901 1.0957033
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RK4 operating on (x,v)
But Invariants exhibit systematic drifts!
????t 0.02
a0
Positions OK
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Gyrokinetics

• If rL ? ln B 1 and ?t ln B ?g , then
  the gyrophase can be eliminated (fast variable),
  and
• the motion described
• by (Xg, v, v-).
• OK for electrons in the solar corona.
• Adiabatic invariants ? p-2/B v2/2 ? E.dl
• Many formal approaches drift equations (Alfvén,
  1950) gyro-averaging (Morotov Solovev, 1966)
  canonical transformations (Gardner 1959) Lie
  transform methods (Littlejohn, 1979 ff)
  Variational formulations (Brizard 2001), yielding
  different gyrokinetic equations.

B
ExB/B2
Xg
Gyro centre
E
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Literature
General

• Press et al Numerical Recipes (mandatory!)
• Sarlet, SIAM Rev 4, 567 (1981) Noether
  theorems in classical mechanics
• Symplectic integrators (intro) Donelly
  Rogers, Am. J. Phys. 73, 938

Space Science Applications of test particle
approach

• Ambrosiano et al. (1984,88) test particle
  acceleration in turbulent reconnecting magnetic
  fields.
• Mace et al. (2000) Verification of
  weak-trubulence perpendicular diffusion
  coefficients by test particle simulations
• Matthaeus (2003) nonlinear perpendicular
  diffusion, test particle simulations versus
  theory.

Pitch angle
simulation
theory

• Dmitruk et al (2003) test ptcl acceleration
  in compressible MHD turbulence.
• Vlahos et al. (2004) acceleration in random
  localized electric fields.
• Arzner et al. (2006) effect of coherent
  structures on ptcl acceleration.

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Diagnostics

• 1. match theory
• Verify diffusion approximations
• Sub/Superdiffusion ?
• Moments and n-point functions of particles
• Statistical conservation laws

• 2. match observations
• Photon emission
• Instrumental response
• Error statistics likelihood function
• What are the observables? Are there
• sufficient statistics ?
• Regularisation and a priori information

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Observational Diagnostics Photons

• Electrons
• - (Gyro)synchrotron (microwaves)
• - Plasma radiation (radio)
• - Transition radiation (radio)
• - Bremsstrahlung (X-Rays)
• Protons
• - nuclear de-exitation lines (gt 1MeV)
• - child products neutron capture, ...

Radiative Transfer Faraday effect, Scattering,
Cutoffs, Absorption, ...
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My Gift An alternative and less known
formulation of MHD (Frenkel, 1982)

• Advantage immediate access to (new!)
  conservation laws ? f(?,?) d3r const.

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Summary
Test Particle
Turbulence
Intermittency, dynamic alignment
Gauss-Field Proxies
Exact particle dynamics, conservation laws
Exact orbit integration
Test Particle Diffusion Coefficients
Choose Stochastic Interpretation
Fokker-Planck
SDEs
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And, test particle orbits can be done on labtops!
Our understanding of particle orbits in irregular
force fields is still far from complete.