Electromagnetic mode conversion: Understanding waves that suddenly change their nature

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Electromagnetic mode conversion Understanding
waves that suddenly change their nature
D. B. Batchelor, L. A. Berry, M. D. Carter, E. F.
Jaeger ORNL Fusion Energy E. DAzevedo ORNL
Computer Science and Mathematics (OASCR
SSAP) C. K. Phillips, H. Okuda, N. Gorelenkov
PPPL P. T. Bonoli, J. C. Wright MIT D. N.
Smithe ATK Mission Research Corp. R. W. Harvey
CompX D. A. DIppolito, J. R. Myra Lodestar
Research Corporation M. Choi General
Atomics SciDAC PI Meeting June 26 - 30, 2005 San
Francisco

• In a magnetized plasma, such as in fusion
  devices or the Earths magnetosphere, several
  different kinds of waves can simultaneously
  exist, having very different physical properties.
  Under the right conditions one wave can quite
  suddenly convert to another type. Depending on
  the case, this can be either a great benefit or a
  problem for the use of waves to heat and control
  fusion plasmas. Understanding and accurately
  modeling such behavior is a major computational
  challenge

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Three minute introduction to magnetic confinement
fusion energy
Potential energy
En 14MeV deposited in heat exchangers
containing lithium for tritium breeding
Total potential
Electrical repulsion
Energy Yield EF 17.6 MeV
Ea 3.5 MEV deposited in plasma, provides self
heating
Nuclear attraction

• About 10 KeV of kinetic energy is required to
  overcome the Coulomb barrier to obtain nuclear
  reaction
• The nuclear interaction has short range whereas
  the Coulomb interaction is long range
• The fusion reaction rate of an energetic T in a D
  target is much less than the energy loss rate due
  to Coulomb scattering
• YOU CANT GET NET ENERGY GAIN BY USING AN
  ACCELLERATOR, SHOOTING INTO A COLD TARGET

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We can get net energy production from a
thermonuclear process

• We heat the particles so that the average energy
  is 10KeV ? 100,000,000 ? PLASMA
• Then we hold the fuel particles and energy long
  enough for many reactions to occur
• ne electron density
• tE energy confinement time
• Lawson breakeven criterion ne tE gt 1020 m-3s

Nuclear thermos bottle made of unobtainium alloy
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What can we really use for our nuclear thermos
bottle?

• Gravitational confinement it works for the sun
• Inertial confinement it works for H bombs, and
  maybe for laser fusion
• We use magnetic fields
• A uniform, straight magnetic field confines
  particles in the direction perpendicular to B,
  but allows free flow along the field (VD 700
  km/s at 10 KeV)
• To get confinement along the field we bend the
  field lines into a torus

ExampleB 5 TeslaE 10 KeV
rD 0.3 cm
re 0.05 mm
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Particle confinement in toroidal magnetic geometry
So we add a magnetic field component winding the
short way around ? poloidal field
A simple toroidal magnetic field doesnt provide
confinement

• ?B drift due to 1/R ? electrons ?, ions ?
• Vertical charge separation ? vertical E field
• E?B radial expansion ? rapid plasma loss

• Magnetic field lines lie on closed, nested
  surfaces flux surfaces, Y const.
• Vertical ?B drift averages to zero as particle
  follows field around poloidally

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Required poloidal magnetic field is produced
either by large internal plasma current (tokamak)
or external coils (stellarator)
Compact Stellaratornon-axisymmetric!!
Tokamak, axisymmetric
Magnetic flux surfaces, r const.
Magnetic axis
separatrix

• Tokamaks
• Axisymmetric ? very good plasma confinement
• Large internal current a problem ? Instability
  source, Inductive drive ? pulsed,
  non-inductive drive expensive
• Stellarators
• Non-axisymmetric ? not so good plasma confinement
• Small internal current ? Inherently steady state,
  less susceptible to current driven instability

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The next big step for the world fusion program is
to explore the physics of a burning plasma
ITER
ITER an international effort Japan, Europe, US,
Russia, China, Korea

• Fusion power 400MW
• Iplasma 15 MA, B0 5 Tesla T 10 keV, tE
  4 sec
• Large 30m tall, 20kTons
• Expensive 5B
• High level negotiations under way on site
  and cost-sharing
• First burning plasmas 2018

R0 6 m
Latest news http//www.iter.org
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Present fusion experiments are at the scientific
breakeven level of performance
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Plasma waves are essential processes in systems
ranging from the solar corona, to planetary
magnetospheres, to laboratory experiments, to
commercial devices
In fusion research, high power electromagnetic
waves (gt 107 W) are used to heat plasmas to
temperatures hotter than the sun and to control
non-linearly interacting plasma processes
heat drive electric currents drive plasma
flows create highly energetic particle
populations
DIII-D Tokamak
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We use plasma waves to heat fusion plasmas to
temperatures of 10keV (gt100 million? much hotter
than surface of the sun)
ECH launcher

• Plasma Control ? With waves we can
• Control plasma current profile
• Control plasma pressure profile
• Control plasma flow velocity
• Induce bulk plasma rotation
• Influence stability
• Ion cyclotron range of frequencies
• f 100 Mhz ( t 10-8 sec), l 10 cm
• Requires solution of wave equation
• Does not propagate in vacuum ? launcher near
  plasma
• Lower hybrid range of frequencies
• f .5 - 5 Ghz ( t 10-10 sec), l 1 cm
• Usually computed with geometrical optics
• Does not propagate in vacuum ? launcher near
  plasma
• Electron cyclotron range of frequencies
• f 100 Ghz ( t 10-11 sec), l 0.3 cm
• Can be computed with geometrical optics

ICRH or Lower hybrid launcher
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Modeling of waves in fusion devices requires a
number of interconnected components
3D Maxwell solver with simplified plasma boundary
conditions
Fokker Planck equation
Wave equation solver
Integrated transport code. Experimental data
Stand alone models
Our goal is to obtain quantitatively accurate,
predictive understanding of wave processes
important for heating, current drive, and
stability and transport applications
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We calculate the plasma response,
, from the Boltzmann equation
Nonlinear E and B driven by current and charge
described by f

• There are two very helpful approximations we can
  make for externally injected RF waves
• Separation of time scales - wave period 1/w ltlt
  time of equilibrium variation, tE
• The waves are stable (actually damped), so we can
  safely linearize the fast time equation

Gives fast time scale variation wave current
Contains Fokker Planck equation
Gives slow time scale variation of f0 power
deposition, equilibrium evolution
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Basic equations of wave propagation and absorption
plasma wave current an integral operator on E

• Time harmonic ? real w, coherent waves, spatial
  damping
• Jant antenna source current
• Boundary conditions bounded domain conducting
  or inhomogeneous source region
• Weakly non-linear, time average distribution
  function f0(v, t) evolves slowly
• Jp fluctuating plasma current due to wave
  non-local, integral operator on E
• Approximate operator locally by integrating along
  guiding center orbits
• Effectively uniform plasma conductivity (Stix) ?

slow, quasilinear time scale tE
Fast, RF time scale
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We are advancing two massively parallel wave
solver codes within our project for various
physics applications

• All Orders Spectral Algorithm (AORSA) 1D, 2D
  3D (Jaeger)
• Spectral in all 3 dimensions
• Cartesian/toroidal coordinates
• Includes all cyclotron harmonics
• No approximation of small particle gyro radius r
  compared to wavelength l
• Produces huge, dense, non-symmetric, indefinite,
  complex matrices
• TORIC 2D (Brambilla/Bonoli/Wright)
• Mixed spectral (toroidal, poloidal), finite
  element (radial)
• Flux coordinates
• Up 2nd cyclotron harmonic
• Expanded to 2nd order in r/l
• Sparse banded matrices

Blowup region
Slow ion cyclotron wave
Electrostatic ion Bernstein wave
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What are the computational and mathematical
challenges?

• High dimensionality p.d.e. in 2D or 3D for
  wave fields, up to 6D time for distribution
  function
• ? Large numbers of unknowns 105 ? gt106
• Complex medium
• Spatially non-uniform
• Anisotropic
• Non-local local plasma current is an integral
  operator over EM field at other locations at
  earlier times
• ? Use of spectral representations
• Wide range of length scales involved l L
  ? l ltlt L, length scales can interact in localized
  plasma regions ? mode conversion
• ? Need for adaptive (but spectral)
  representation
• Variety of physics mechanisms for absorption
• Non-linearity waves modify plasma on slow time
  scale, non-linear effects on waves
• Basic equations are non-symmetric and dissipative

QPS Compact Stellarator
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An example of the progress in understanding
plasma wave behavior is the process of mode
conversion

• In a magnetized plasma, such as in fusion devices
  or the Earths magnetosphere, several different
  kinds of waves can simultaneously exist, having
  very different physical properties
• Near the ion cyclotron frequency, there are two
  very different electromagnetic waves, similar to
  light waves, the fast magnetosonic wave and the
  slow ion cyclotron wave. In addition there is an
  electrostatic wave, similar to a sound wave,
  called the ion Bernstein wave.
• There are important differences in the way these
  3 waves interact with the plasma when they are
  absorbed.
• The fast magnetosonic wave tends to damp on
  energetic ions and drive a tail population of
  energetic ions
• The slow ion cyclotron wave tends to damp on
  lower energy ions and can drive bulk fluid flow
  of the plasma, influencing stability
• The electostatic ion Bernstein wave tends to damp
  on electrons and can drive electric current

Pre SciDAC state of the art required very severe
approximations to conductivity operator,
restricting to low frequency and long wavelength.
Computational limitations did not allow
resolution of the ion Bernstein wave (IBW)
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A beautiful story of science 2D effects on mode
conversion
Plasma waves have an unpleasant habit of changing
their character in the middle of a non-uniform
plasma
n S
Ion Bernstein Wave (IBW) conversion in 1D

• On the right (low magnetic field) the ion
  cyclotron wave (fast wave) has long wave length
  and the IBW has short, imaginary wavelength
  (evanescent)
• In the center (near the ion-ion hybrid resonance)
  the modes interact
• On the left (high magnetic field) the fast wave
  has long wave length, the IBW has short
  wavelength, which must be resolved, but is well
  separated from the fast wave.

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Surprise We find that fast, long wavelength
electromagnetic waves launched from the right can
be converted to slow electromagnetic
ion-cyclotron waves, as well as the previously
expected electrostatic ion Bernstein waves
Slow ion cyclotron wave
Electrostatic ion Bernstein wave
Blowup region

• Previous 1D analytic theory suggested that both
  conversions could occur, but gave no information
  about relative importance or actual field
  structure
• 2D theory gives complete, quantitative picture
• Evidence of conversion to slow ion cyclotron
  waves seen experimentally on Alcator Cmod at MIT

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These results are confirmed in experiments on
Alcator C-Mod tokamak with a new diagnostic
technique Phase Contrast Imaging
Contour Plot of Fourier Analyzed PCI Data
PCI measures line-integrated density fluctuations
along 32 vertical chords (separation 0.4 cm).
PCI Signal Structure
The laser is modulated at a frequency close to
the RF frequency, and the RF waves are detected
at the beat frequency.
E. Nelson-Melby et al, Phys. Rev. Letter, 90 (15)
155004 (2003)
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2-D density fluctuations calculated from TORIC
ICW exists on the mid-plane, Bpol/Btotal0.08
Density fluctuations are mainly from MC ICW and
MC IBW.
PCI Window
Y. Lin et al, 16th Topical Conference on RF Power
in Plasmas, 2005
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Good agreement in wave spatial structure and kR
spectrum
IBW/ICW
IBW and ICW appear as a broad peak in kR spectrum
Y. Lin et al, 16th Topical Conference on RF Power
in Plasmas, 2005
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First fully resolved 2D calculations of
conversion of fast waves to short wavelength
modes were obtained within our SciDAC project

• We have progressed from
• Simple, approximate, analytic theory (F.W.
  Perkins, 1977)
• Provided valuable paradigms for mode conversion
• Indicated several conversions were possible
• Did not give quantitative information for real 2D
  situations
• To numerical solutions in 1D (Smithe, 1997,
  Jaeger, 2000)
• Verified analytic calculations with much more
  inclusive physics
• Higher cyclotron harmonics, can treat short
  wavelengths
• To high-resolution solutions across the full
  plasma cross section
• Includes arbitrary cyclotron harmonics
• Very short wavelength structures limited by
  computer size and speed, not formulation

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All orders spectral technique has been extended
to 3D

• Preliminary calculation for Fast Wave minority
  heating on LHD stellarator 5 minority H in
  4He
• 16? 50?50 modes in f, x, y (10 independent
  solutions - one per field period)

Fast wave heating in LHD Stellarator

• Gigantic, dense linear system ? NERSC Seaborg,
  1600 processor IBM SP, 8 hr processor time at
  1.7 teraflops, memory 750Mb/processor 1,200
  Gb

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These studies are an excellent example of the
beneficial interaction of basic theory,
computational modeling and experiment

• The expectation was that fast waves would be
  converted to IBW propagating on the high magnetic
  field side of the conversion layer
• When the new codes first began to show short
  waves on the high magnetic field side the results
  were not understood and concerns were raised
  about the code validity.
• When the newly developed PCI diagnostic indicated
  waves on the low field side the results were not
  understood at first.
• Three decade old 1D analytic theory suggested
  looking toward the ICW conversion process.
• Detailed comparison of the computational results
  with experimental measurements lead to greatly
  increased confidence in our understanding of both
• These results are likely to have significant
  practical consequences because Bernstein waves
  are absorbed primarily by electrons and are
  effective at driving current, whereas the slow
  ion cyclotron wave can be absorbed by ions, which
  would be more effective at driving plasma flow
  and improving the ability of the magnetic field
  to hold the hot plasma.