# Presentation: 179852

## 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. D’Azevedo– 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. D’Ippolito, 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 Earth’s 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

**Three minute introduction to magnetic confinement fusion energy**

**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 CAN’T GET NET ENERGY GAIN BY USING AN ACCELLERATOR, SHOOTING INTO A COLD TARGET**

***E*****n***** *****= 14MeV**
**deposited in heat exchangers containing lithium for tritium breeding**

***E********** *****= 3.5 MEV**
** ****deposited in plasma, provides self heating**

**Nuclear attraction**

**Electrical repulsion**

**Energy Yield**** *****E*****F**** = 17.6 MeV**

**Total potential**

**Potential energy**

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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**
*** ******n******e***** = electron density**
** ********E**** = energy confinement time**
**Lawson breakeven criterion *****n******e *********E****  > 10****20**** m****-3****s**

**Nuclear thermos bottle made of ‘unobtainium’ alloy**

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## 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

**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 (V****D**** ~ 700 km/s at 10 KeV)**
**To get confinement along the field we bend the field lines into a torus**

**What can we really use for our nuclear thermos bottle?**

******D**** ~ 0.3 cm**

******e**** ~ 0.05 mm**

**Example:****B = 5 Tesla****E = 10 KeV**

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## Particle confinement in toroidal magnetic geometry

******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, ******** = const.**
**Vertical ********B drift averages to zero as particle follows field around poloidally**

**A simple toroidal magnetic field doesn’t provide confinement**

**So we add a magnetic field component winding the short way around ******** poloidal field**

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## Required poloidal magnetic field is produced either by large internal plasma current (tokamak) or external coils (stellarator)

**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**

**Compact Stellarator****non-axisymmetric!!**

**Tokamak, axisymmetric**

- Magnetic flux surfaces,  const.

- Magnetic axis

- separatrix

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## The next big step for the world fusion program is to explore the physics of a “burning” plasma – ITER

*Latest news    ****http://www.iter.org***

**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**
** ****I****plasma**** = 15 MA, B****0**** = 5 Tesla****  T ~ 10 keV, ********E**** ~ 4 sec**
** ****Large – 30m tall, 20kTons**
**• ****Expensive ~ $5B+**
**• ****High level negotiations under****   way on site and cost-sharing **
**• ****First burning plasmas ~2018**

***R******0***** = 6 m**

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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 (> 10****7**** 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 (>100 million much hotter than surface of the sun)

- ICRH or Lower hybrid launcher

- 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 ( ******** ~ 10****-8**** sec), ******** ~ 10 cm**
**Requires solution of wave equation**
**Does not propagate in vacuum ******** launcher near plasma**
**Lower hybrid range of frequencies**
**f ~ .5 - 5 Ghz ( ******** ~ 10****-10**** sec), ******** ~ 1 cm**
**Usually computed with geometrical optics**
**Does not propagate in vacuum ******** launcher near plasma**
**Electron cyclotron range of frequencies**
**f ~ 100 Ghz ( ******** ~ 10****-11**** sec), ******** ~ 0.3 cm**
**Can be computed with geometrical optics**

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## Modeling of waves in fusion devices requires a number of interconnected components

**3D Maxwell solver with simplified plasma boundary conditions**

**Wave equation solver**

**Fokker Planck equation**

**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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**Notes:**
Coupling to transport -> turbulent and collisional

## We calculate the plasma response,                   , from the Boltzmann equation 

**There are two very helpful approximations we can make for externally injected RF waves**
**Separation of time scales - wave period 1/******** <<  time of equilibrium variation, **********
**The waves are stable (actually damped), so we can safely linearize the fast time equation:**

**Gives fast time scale variation – wave current**

**Gives slow time scale variation of *****f*****0**** – power deposition, equilibrium evolution**

**Contains Fokker Planck equation**

**Nonlinear– *****E***** and *****B***** driven by current and charge described by *****f***

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## Basic equations of wave propagation and absorption

**Time harmonic ******** real ********coherent waves,  spatial damping**
**J****ant**** = antenna source current**
**Boundary conditions: bounded domain – conducting or inhomogeneous source region **
**Weakly non-linear, time average distribution function *****f*****0****(v, t) evolves slowly:**
**J****p**** = 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) ******** **

**plasma wave current: an integral operator on E**

**slow, quasilinear time scale ~ ********E**

**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 ******** compared to wavelength ******
**Produces huge, dense, non-symmetric, indefinite, complex matrices **
**TORIC –  2D (Brambilla/Bonoli/Wright)**
**Mixed spectral (toroidal, poloidal), finite element (radial)**
**Flux coordinates**
**Up 2****nd**** cyclotron harmonic**
**Expanded to 2****nd**** order in ******
**Sparse banded matrices**

**Blowup region**

**Slow ion ****cyclotron wave**

**Electrostatic ion ****Bernstein wave**

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## What are the computational and mathematical challenges?

**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 10****5**** ******** >10****6**
**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, 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 Earth’s 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

**Ion Bernstein Wave (IBW) conversion in 1D**

- Plasma waves have an unpleasant habit of changing their character in the middle of a non-uniform plasma

**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.**

**n****||**** = S**

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## 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

**2D theory gives complete, quantitative picture**
**Evidence of conversion to slow ion cyclotron waves seen experimentally on Alcator Cmod at MIT**

**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**

**Blowup region**

**Electrostatic ion Bernstein wave**

**Slow ion cyclotron wave**

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## These results are confirmed in experiments on Alcator C-Mod tokamak with a new diagnostic technique – Phase Contrast Imaging

**PCI measures line-integrated density fluctuations along 32 vertical chords (separation ~ 0.4 cm). **

**The laser is modulated at a frequency close to the RF frequency, and the RF waves are detected at the beat frequency.**

**Contour Plot of Fourier Analyzed PCI Data**

**PCI Signal Structure**

**E. Nelson-Melby *****et al*****, Phys. Rev. Letter, 90 (15) 155004 (2003)**

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## 2-D density fluctuations calculated from TORIC

**2-D density fluctuations calculated from TORIC**

**PCI Window**

**Density fluctuations are mainly from MC ICW and MC IBW. **

**ICW exists on the mid-plane, B****pol****/B****total****~0.08**

**Y. Lin *****et al*****, 16****th**** 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 k****R**** spectrum**

**Y. Lin *****et al*****, 16****th**** 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 ****4****He**
**16******** 50********50 modes in ********, x, y (10 independent solutions - one per field period)**

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

- Fast wave heating in LHD Stellarator

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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.

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