(tokamak) plasmas

Tutorial Lecture on Fluctuation Measurements in

Laboratory (tokamak) Plasmas Anne White

Department of Nuclear Science and Engineering, MIT, Cambridge, MA, USA

A. White – Les Houches 2015 1 [email protected]

Fusion plasmas shown relative to some other plasmas of interest in n-T parameter space

A. White – Les Houches 2015 2

Outline


•  Introduction to tokamak plasmas

•  Summary of signal analysis techniques

•  Diagnostic techniques for the core of tokamak plasmas* •  Connection to gyrokinetic theory/simulations

•  Universal features of turbulence

*Note that S. Bale’s talk covered “in situ” diagnostics : Langmuir probes, magnetic probes, energy analyzers, etc.; physically inserted into plasmas

Magnetic confinement of plasma


In a magnetic field →B , a particle

with velocity → −v and charge q experiences the Lorentz force

→F = q→v ×

→B

NOTE: Lorentz force only acts on perpendicular motion In the direction parallel to the magnetic field the particle moves as it wants

Many magnetic bottles have been built to pursue fusion, some worked better than others…


Z-pinch (1950s-1960s) Reversed Field Pinch (1970s-present)

(present) Mirrors, pinches, combinations (1950s-Present)

Magnetic confinement of plasma in toroidal devices


Solution 1: Torus solves the end-loss problem Problem 2: In a simple toroidal field, particle drifts lead to charge separation

Solution 2: Add poloidal field, particles sample regions of inward and outward drift. Problem 3: Hoop stress from unequal magnetic and kinetic pressures.

Solution 3: Add vertical field, to counteract hoop stress. Tokamaks and stellarators are variations of this.

Plasma is confined on closed nested flux surfaces


•  Magnetic field lines are helical and lie on closed, nested surfaces – flux surfaces, Ψ = const.

•  Vertical ∇B drift averages to zero as particle follows helical field

•  To lowest order, particles are �stuck� on flux surfaces

Minor radius r

Best confinement achieved in toroidal geometry, helical field


�  Poloidal field from current in the plasma itself.

�  Intrinsically pulsed

�  Axisymmetric – good confinement

�  Current is source of instability

�  Poloidal field from external coils

�  Intrinsically steady-state

�  Non-axisymmetric – good confinement hard to achieve

�  More difficult to build

Tokamak Stellarator

Schematic view of tokamak plasma with magnetic field coils

A. White – Les Houches 2015

Image from http://www.generalfusion.com/magnetic fusion.html 9


The inside of C-Mod tokamak vessel, view from outboard midplane (looking inward along Rmaj)

<1 m

Tokamaks around the world: Explore physics of heating/transport with variety of tools


Turbulent transport in tokamaks limits performance


Problem: in current devices the required heating power to reach and maintain fusion temperatures exceeds the fusion power output Why? Confinement is not perfect, magnetic bottle can leak heat at a significant rate Turbulent cross-field transport is the primary cause of the “leak”

See T. Carter, lecture on driftwaves/turbulence last week


r/a 0 1

Density (1020 m-3) r/a ≥ 1.0 Scrape Off Layer (SOL) region. Dominated by density gradient driven interchange type turbulence

Last Closed Flux Surface (LCFS)

Tokamak plasmas often divided into three regions of interest

r/a > 0.9 Edge/Pedestal region Steep pressure gradients, Electromagnetic drift-wave-type turbulence

r/a < 0.9 Core region Often dominated by electrostatic/electromagnetic drift-wave type turbulence (ITG/TEM/ETG)

1

See Garbet, Carter, lectures on drift waves/modes last week

Fluctuations in core occur over very broad range of spatial-temporal scales


Gyrokinetics applicable in core r/a < 0.9 Turbulent fluctuations in parameters such as density, temperature, velocity, Eddies are on scale of electron and ion gyroradii, ρe ≈ 0.06mm, ρi ≈ 2.6mm at 10keV and 5.4 T Size of plasma a≈ 1m Frequencies in the 10-1000s of kHz Fluctuation amplitudes, ñ/n ≤ 0.1-1%

http://genecode.org/

Three common “types of turbulence” in core plasma (r/a < 0.9) classified by dominant linear instability characteristics


Figure from Garbet, lecture on drift waves/modes last week

•  In many cases, we can

consider just electrostatic turbulence

•  ion temperature gradient (ITG) mode

•  trapped electron mode (TEM)

•  electron temperature gradient (ETG) mode

Plasma parameters as function of radius in three H-mode plasmas from beta scan at C-Mod


a

Guttenfelder APS 2014

r/a

r/a

r/a

r/a

Plasma parameters near r/a = 0.8 in C-Mod L-mode plasma, large (1%) turbulent fluctuations were measured


Parameter C-Mod L-mode (r/a = 0.8)

ne (1020 m-3) 0.79

Te (keV) 1.06

Ti (keV) 1.14

Ti/Te 1.08

a/Lti = a/(T/|dT/dr|); a = 0.20 m 2.17

a/Lne 1.69

a/LTe 2.80

ρs (m) and ρ* = ρs/a 5e-4 and 0.0025

βe (%) 0.3

νei (kHz) 130 kHz

Toroidal flow velocity (km/sec) 10 km/sec (Mach ~ 0.05)

Zeff 2.75

Cs (km/s) and Cs/a (kHz) 206 and 1030

Outline




•  Diagnostic techniques for the core of tokamak plasmas •  Connection to gyrokinetic theory/simulations


Signal analysis references

•  Signal processing and data analysis techniques are very critical for turbulence studies

•  Do not destroy information, do not introduce information

•  Bendat, Julius S. and Piersol, Allan G., “Random Data: Analysis and Measurement Procedures”, Wiley-Interscience, New York, 1971.

•  Bendat, Julius S. and Piersol, Allan G., “Engineering Applications of Correlation and Spectral Analysis”, Wiley-lnterscience, New York, 1980.

•  Ritz and Powers, et al., Rev. Sci.lnstrum. 59 (8), August 1988 … From power spectrum to energy transfer between waves … Spectral analysis, correlation analysis, higher order spectral analysis


A few useful definitions for signal analysis: this talk will show a lot of power spectra


Can only calculate at discrete points, m = 0, ±1, ±2 ∆f = frequency spacing (1/∆t), ∆t = time interval between samples in the numerical integration, summed over finite time interval (Discrete Fourier Transform)

x(t) = time domain representation of the signal x Sx(f) = frequency domain representation of the signal x

Basic analysis in frequency and time domains as building blocks


Autopower spectrum

Autocorrelation function

Autocorrelation function (direct time domain analysis)

Cross-power spectrum

Coherence function

How to obtain frequency-wavenumber spectrum S(f,k)


Image from McKee APS 1999, see also PRL 2000. BES measurements in core (r/a ≈ 0.7) of DIII-D

•  Techniques just described give us S(f), from single point in space; accessible experimentally with wide variety of diagnostics (Taylor or “frozen flow” hypothesis: S(k) (see Alexandrova talk))


•  To obtain S(k) is more challenging: Measure fluctuating power at one point in space at different k, e.g. S(f, k1) and S(f, k2). Can integrate in f, construct S(k)


Image from Peebles TTF 2004. k-resolved power spectra from scattering diagnostic measuring core turbulence (line avg) at DIII-D



Image from GPI diagnostic (S(f,k) in edge (r/a ≈ 0.99) plasma at C-Mod tokamak), Sierchio APS 2014


•  To obtain S(k) is more challenging: Measure fluctuating power at one point in space at different k, e.g. S(f, k1) and S(f, k2). Can integrate in f, construct S(k)

•  S(f, k): Measure fluctuating power at several separated points in space and Fourier Transform in space and time. Beall, J. Appl. Phys. 53(6). June 1982

Also makes use of frozen flow hypothesis


Outline




•  Diagnostic techniques •  Connection to gyrokinetic theory/simulations


Diagnostics used to measure turbulence in three regions of tokamaks, table I


a

Name Region Parameters measured (fluctuations)

Basic Principle

Magnetic loop (in situ) Wall/SOL/far edge

B Magnetic induction

Langmuir probe (in situ) SOL/far edge ne, Te, φ, v, etc. Particle flux measurements

Gas Puff Imaging (GPI) SOL/edge ne Radiation from bound electrons

Beam Emission Spectroscopy (BES)

Edge/core ne Radiation from bound electrons

Diagnostics used to measure turbulence in three regions of tokamaks, table II (there are many more!)


a

Name Region Parameters measured (fluctuations)

Basic Principle

Reflectometer SOL/edge/core ne Changes in Index of refraction (cutoff)

High-k Scattering core ne Coherent scattering

Doppler Reflectometer/backscattering (DBS)

Edge/core ne Index of refraction/coherent scattering

Correlation electron cyclotron emission (CECE)

Edge/core Te Electromagnetic emission from free electrons

Polarimeter Core/edge (non-local)

B Changes in Index of refraction (Faraday rotation)

This talk will focus in detail on only a few diagnostics used to measure core plasma turbulence and compare with gyrokinetic codes


Reflectometer/Polarimeter (ITG/TEM) Low k k� ρσ< 0.5 High-k Coherent scattering (ETG) High k k� ρσ> 5 Doppler reflectometer (ITG/TEM) Intermediate k 1< k� ρσ < 10

Outline




•  Diagnostic techniques for the core of tokamak plasmas

Index of refraction measurements Scattering of electromagnetic waves

•  Connection to gyrokinetic theory/simulations


Measuring changes in plasma refractive index

•  n = ck/ω )•  c speed of light •  n = 1 for vacuum, n≠ 1 in plasma •  Changes in wave phase/propagation in plasma due to difference in n from

vacuum value gives information on density (and magnetic field) •  Cold plasma model assumptions

•  Small amplitude waves (we can linearize equations) •  Particles are initially at rest : No thermal motions, no pressure

•  Further assumption for Reflectometer diagnostic relevant waves •  High frequency (ω ∼ ωpe ) •  Ignore ion motion


Cold plasma waves •  Assume plane wave perturbations •  Linearize/Fourier transform the equation of motion and Maxwell’s equations •  Combine to write the wave equation as

•  Index of refraction ! n = ck/ω)•  Equation to solve for dispersion relation is

•  ε is plasma dielectric tensor and E1 is perturbed electric field, k-hat is unit vector in direction of k )

A. White – Les Houches 2015 31 Eqn. snapshots from “Lectures on basic plasma physics: Waves in cold plasmas” Aalto University, Dept. Applied Physics, Online notes

Dispersion relation for cold plasma waves with background magnetic field B0 •  Choose B0 in z-hat direction, and choose k in the x-z plane, i.e.

•  The dyad product is written as

•  And the equation to solve for the dispersion relation becomes


Dispersion relation for cold plasma waves with background magnetic field B


With S, P, D, having usual definitions Ωs = qsB/m ω 2ps = nqs

2/ε0ms

Non-trivial solution (determinant of matrix = 0) yields

Reflectometer: Waves launched from outboard midplane of tokamak (k perpendicular to B0), θ = π/2


•  Consider only high frequency waves (close to or above plasma frequency), and ignore the ion response

•  Two solutions are possible

•  Ordinary wave, with E1 || B0

•  Extraordinary wave, with E1 perp B0

Reflectometer diagnostics are “Radar” type diagnostics, exploiting wave cut-offs in plasma


•  If index of refraction n2 > 0, wave propagates, n2 < 0, wave is evanescent •  When n = 0, we encounter a cut-off as wavenumber goes to 0, wave will reflect at this point in plasma Example : Launch O-mode waves at 110GHz, 88GHz, and 75GHz from outside plasma with radially varying density profile, waves propagate until f = fpe

110GHz

88GHz

75GHz

Den

sity

Radius Figure adapted from Dominguez APS 2010

Antenna & transmission line & launch/receive electronics

Interpretation of returned signal

•  Reflectometer receiver electronics allow for recording of both the amplitude and phase of the returned wave with respect to reference

•  Generic detection schemes that do this is called “heterodyne” and provides two time series for any one signal, which are the real and imaginary parts

Sreal (t) = A(t) cos φ (t) Simag (t) = A(t) sin φ (t)

•  Combine to form Amplitude signal, A(t), or phase signal φ (t), or just analyse Sreal or Simag signal

A. White – Les Houches 2015 36 Rhodes, Plasma Phys. Control. Fusion 40 (1998) 493–510

Phase screen model for low amplitude fluctuations


Nazikian, Phys. Plasmas, Vol. 8, No. 5, May 2001 See also Valentina’s poster form AUG this week

•  Important region of wave-plasma interaction is compressed into thin screen.

•  Depth of the irregularities on the surface are directly related to the 1-D phase from geometric optics

•  Phase of geometric optics can be linearized for small density perturbations

•  Measured phase is then easily related to density fluctuation level, based on local density gradient

At low fluctuation levels measured phase fluctuations have physical spectra, easy to interpret


A cos φ

|S| =|A|

Rhodes, Plasma Phys. Control. Fusion 40 (1998) 493–510

Low fluctuation levels (H-mode), phase recovers turbulence info

phase screen model works well

Sreal (t) = A(t) cos φ (t)

But at high fluctuation levels measured phase fluctuations become unphysical, hard to interpret


A cos φ

A cos φ

|S| =|A|

|S| =|A|

Rhodes, Plasma Phys. Control. Fusion 40 (1998) 493–510

Low fluctuation levels (H-mode), phase recovers turbulence info

High fluctuation levels (L-mode), phase does not recover turbulence info

phase screen model breaks down phase screen model works well

Power spectrum from Langmuir probe signal and Sreal reflectometer signal agree very well


Conway, Plasma Phys. Control. Fusion 38 (1996) Conway, Rev. Sci. Instrum., Vol. 67, No. 11, November 1996 Rhodes, Plasma Phys. Control. Fusion 40 (1998) 493–510

Same data on log-log scale and semi-log scale

Probe

Probe

Multi-channel, fixed frequency reflectometers measure core/ edge turbulence simultaneously


Edge r/a ≈ 0.99: Spectrum has broad background feature in L-mode and I-mode, plus a narrowband feature in I-mode

Core r/a ≈ 0.55: Spectrum has broad background feature in L-mode and I-mode, with reduced fluctuation amplitude (-30%) in I-mode

Use the full signals, e.g. Sreal (t) = A(t) cos φ (t) , for the spectral analysis

Core and edge turbulence in L-mode is broadband; but near edge I-mode has structure


Polarimeter: Waves launched from outside tokamak (k parallel to B0), θ = 0


•  Three solutions are possible

•  1. E1 only in z-direction :

•  2. and 3. Ez = 0, and E1 is in x- and y- direction:

•  n2 = S + D, we obtain iE1x = E1y, which is right-handed circularly polarized wave (R-wave)

•  n2 = S − D, we obtain iE1x = −E1y, which is left-handed circularly polarized wave (L-wave)

Faraday rotation measurements are simple in principle, but can be obscured by Cotton-Mouton effect

•  Faraday rotation results from rotation of polarization of an electromagnetic wave due to magnetic field along the wave propagation direction (k||B)

•  Issues: component of B perpendicular

to the electromagnetic wave results in elliptization of a linearly polarized beam (Cotton-Mouton effect)


Faraday rotation can be measured by comparing the phase shift between the R-wave and L- wave propagating through plasma

•  Solution: Launch R-wave and L-wave; Phase shift between these two waveforms directly relates to the polarization change due to the magnetized plasma


Brower, Phys. Rev. Lett. 88,185005 (2002) Bergerson, RSI, 83, 10E316 (2012)

Schematic of polarimeter on C-Mod for B fluctuation measurements

A. White – Les Houches 2015 46 Figure from Bergerson, RSI, 83, 10E316 (2012)

Frequency launched is much higher that any cut-off frequency; f = 2.55 THz for C-Mod

•  Double pass configuration

•  Retro-reflectors mounted within the inner wall of the vacuum vessel

•  Six chords can be used to probe at different vertical locations

•  No spatial localization radially, line integrated measurement

Example spectra from C-Mod polarimeter showing broadband and coherent fluctuations in plasma

Problem with polarimeter is that it is a line integrated measurement – combines core/edge information

A. White – Les Houches 2015 47 Figure from Bergerson, RSI, 83, 10E316 (2012)

Summary: Index of refraction measurements foundation for several important turbulence diagnostics

•  Good: non-perturbative, (requires only port access for radiation to be launched/detected), excellent time resolution. Reflectometer has excellent spatial resolution, allows for measurement of radial correlation length of turbulence. Large body of theoretical work exists to interpret signals. Polarimeter gives access to B information.

•  Bad: Reflectometer is difficult to extract absolute fluctuation levels, except in some specific cases. Polarimeter has very poor spatial resolution.

•  Open question: fixed frequency reflectometer interpretation ?

•  New efforts: •  couple gyrokinetic code results with full-wave simulations to model

reflectometer data (e.g. Bañon-Navarro (UCLA), Lei Shi (PPPL) •  Cross-polarization scattering (O-mode/X-Mode differences probe B locally)

[Rhodes UCLA]


Outline




•  Diagnostic techniques for the core of tokamak plasmas

Index of refraction measurements Scattering of electromagnetic waves

•  Connection to gyrokinetic theory/simulations


Electromagnetic wave scattering by charged (elementary) particles

•  An incident electromagnetic wave Ei(r,t) = Ei exp i(ki ·r − ωi t) impinges on an electron at position r(t)

•  As a result of the electric and magnetic fields of the wave, the particle is accelerated.

•  The charged particle undergoing acceleration emits electromagnetic radiation in all directions: emitted radiation is the scattered wave.

See Hutchinson Principles of Plasma Diagnostics, 2002 for full derivation; equations are taken from Hutchinson’s online notes. See book and references therein for density turbulence details


We are interested in detecting the radiation from the scattering process

•  Electron produces a scattered field at a distant point x whose Fourier spectral component at scattered frequency ωs is

•  re is the classical electron radius, 2.818 ×10−15m

•  κ ≡ 1 − s-hat · β relates retarded and normal time via dt = κ′dt′ for a particle with speed v, β ≡ v/c

•  Π↔ is a tensor that transforms input wave polarization, e-hat ≡ Ei/Ei, to scattered wave polarization

•  ω is the "scattering" frequency, ωs−ωi; •  k is the "scattering" wave-vector, ks−ki


(S-1)

Coherent scattering: correlation between electrons must be taken into account


We need to add up contributions from electric fields (S-1) from all the electrons in the scattering region Assume the phases of all contributions are uncorrelated (incoherent scattering), perform incoherent summation, powers add P = Σ |Ej|2. Shielding effects of a plasma cause a test charge to be surrounded by shielding charges in a cloud of characteristic size approximately the Debye length, λD. The combination of the charge and its shield is referred to as a dressed particle

Incoherent vs coherent scattering depends on ratio of Debye length to wavelength of perturbations


The experimenter chooses the wavelength of of incident radiation, ki, and the scattering angle. The “scattering k”, k = ks –ki is selected If k λD << 1, the contribution from test particle and cloud will add up coherently since there is negligible phase different between them. Probe the collective behavior of the plasma (density turbulence) Have to sum over electric fields, not power.

Incoherent: kλD>>1, coherent kλD << 1

Coherent scattering widely used to diagnose density fluctuations in tokamaks


Measure the scattered power density from wave scattering off of density fluctuations Pi Incident power re classical electron radius Lz length of scattering volume Π polarization tensor e-hat direction of incident electric field V volume of scattering volume T observation time Scattering theory shows that the scattered power is proportional to the density fluctuations, ñe(k,ω), where S(k,ω) is the spectral density of plasma density fluctuations being probed

d 2PdΩdν

= Pire2Lz Π⋅ e

2 !ne(k,ω)2

VT


•  Gaussian Probe beam: 200 mW, 280 GHz, λ ~ 1.07 mm, beam radius = 3cm (1/e2 radius)

•  Propagation close to midplane => kr spectrum. •  Three wave-coupling between incident probe beam

(ki , ωi) and plasma (k , ω).

•  ωs ~ ωi and ks ~ ki, imposes Bragg condition

View from top of NSTX (D.R. Smith PhD thesis 2009) λD ~ 10-5 m k ~k┴<104 m-1 , kλD < 1

k=2kisin(θs/2)

ωs = ω + ωi k = +

NSTX high-k scattering diagnostic

ki ks

Multichannel scattering system has good wavenumber resolution, o.k. radial resolution

A. White – Les Houches 2015 56 J. Ruiz Ruiz, APS 2014

120 125 130 135 140 1450

1

2

3

4

5

6x 1013

R(cm)

n e(1013

cm−3

)

shot = 141767

0

0.2

0.4

0.6

0.8

T e(keV

)

shot = 141767

t=348mst=448mst=498mst=565ms

Scattering region

5 detection channels, gives Wavenumber range: kr ~ 5-30 cm-1 Wavenumber resolution: Δk = ± 0.7 cm-1 Radial coverage: R = 106-144 cm Radial resolution (determined by beam width, geometry): ΔR = ± 2 cm


t (s)

f(MHz

)

0.25 0.3 0.35 0.4 0.45 0.5 0.55−3−2−1

0123

t (s)f(M

Hz)

0.25 0.3 0.35 0.4 0.45 0.5 0.55−3−2−1

012

t (s)

f(MHz

)

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6−3−2−1

012

ch 1, k┴ρs~14-16

ch 2, k┴ρs~10-12

ch 3, k┴ρs~8-10

NSTX shot 141767 NSTX high-k scattering diagnostic

•  Choice of geometry/beam wavelength ki determines the probed k

•  By changing the scattering angle, each channel can measure a different value of k

•  Spectrograms at right show the variation in density fluctuations

at three different k in time

k=2kisin(θs/2)

J. Ruiz Ruiz, APS 2014


NSTX shot 141767 NSTX high-k scattering diagnostic

•  Measurement in stationary lab

frame of turbulence in rotating plasma frame: ωm = ωturb + ωD

•  NSTX: Doppler shift is mainly affected by toroidal rotation component ωD ~ kTvT

•  Peak near zero frequency is not plasma turbulence, but is due to detector picking up stray power from the launched beam

shot 141767, channel 1

f(MH

z)

t (s)0.3 0.35 0.4 0.45 0.5 0.55 0.6

−3−2−1

0123

−3000 −2000 −1000 0 1000 2000 3000−200

−180

−160

−140

−120

−100

f(kHz)

Pow

er (d

B)

t = 340 mst = 450 mst = 503 mst = 570 ms

120 125 130 135 140 1450

1

2

3

4

5

6x 1013

R(cm)

n e(1013

cm−3

)

shot = 141767

0

0.2

0.4

0.6

0.8

T e(keV

)

shot = 141767

t=348mst=448mst=498mst=565ms

J. Ruiz Ruiz, APS 2014

ch 1, k┴ρs~14-16

Multichannel scattering system allows for measurement of wavenumber spectrum of density fluctuations (as a function of time)


Figure from Yang Ren Nuclear Fusion (2013)

5 10 1510−10

10−9

10−8

10−7

k⊥ρs

(δ n/n

)2 (arb

.)

t=348 ms

t=398 ms

t=448 ms

t=482 ms

t=498 ms

Figure from Juan Ruiz Ruiz APS 2014

Summary: Electromagnetic wave scattering is powerful and widely used diagnostic technique for turbulence

•  Good: Theory well-developed, non-perturbative, (requires only port access for radiation to be launched/detected), excellent time resolution, good wavenumber resolution and good wavenumber range (ITG to ETG scales)

•  Bad: Spatial localization is determined by size of beam and scattering geometry/incident wavelength of radiation, and is often limited (i.e. not as local as you would like). Measurement is difficult to do.

•  High-k scattering system at NSTX: Used to study high-k turbulent density fluctuations associated with the ETG turbulence

•  These ETG type fluctuations are believed to be responsible for large levels of electron heat transport, limiting performance of spherical tokamaks like NSTX and MAST (EM turbulence also important)


Doppler reflectometry (Doppler Backscattering) combines spatial localization of reflectometer with clear interpretation/k-sensitivity of scattering measurements


Launch wave at fixed frequency, such that f = fpe in plasma (cutoff) If we detected blue beam, we have a reflectometer measurement… If we detect pink beam, we have a scattering measurement! Bragg condition satisfied, means only one k� is detected for each launched k0 (frequency) k� = kñ = 2k0 sin(θ)

See e.g. Conway, Plasma Phys. Control. Fusion 46 (2004) 951–970

Examples of turbulent wavenumber spectrum measured in Tore Supra with Doppler Reflectometer

A. White – Les Houches 2015 62 Vermare Le Houches 2011

Doppler Reflectometer measurements from stellerator TJK show variation with radius


F. Fernandez-Marina Nucl. Fusion, 54, (2014) 072001

r/a = 0.75

r/a = 0.65

Outline






Simulations predict turbulence over very broad range of spatial-temporal scales with very fine resolution


•  Diagnostics have limited spatial/wavenumber resolution

•  Must have model for a diagnostic response function to make meaningful comparisons with code results

•  Models are called “synthetic diagnostics” and are used to “filter”

the code results •  Apply synthetic diagnostic model as

post-processing of gyrokinetic code results

Figure adapted rom movie at http://genecode.org/

Simplest synthetic diagnostic will take spatial averaging and/or k sensitivity into account


Figure from Bravenec Rev. Sci. Instrum. 88 (l), January 1995

da << λ and db = λ are the chosen “sample volumes” As the wave propagates by the sampling locations, it will resolved in case a, while in case b, the wave will be completely undetected

Effect of applying synthetic diagnostic filter mimics k (or f ) response of instrument


Figures from White APS 2007, Numerical details in Holland PoP 2009

Tore Supra Reflectometer measurements compared with GYRO

A. White – Les Houches 2015 68 Bourdelle NF 2011

Fast sweeping (change frequency very fast) reflectometer avoids interpretation issues of density fluctuation amplitude and allows for quantitative measurements

Polarimeter spectra in C-Mod compared with GYRO results

A. White – Les Houches 2015 69 Guttenfelder APS 2014

Simulation More work needed to understand disagreements in amplitude and shape of spectra

Comparisons between Tore Supra Doppler Reflectometer and GYRO and GYSELA

A. White – Les Houches 2015 70 Casati PRL 2009

r/a = 0.75

spectral amplitude agrees – fluctuation levels are right Slope (fall off with k) agrees better with GYSELA than with GYRO

High-k scattering comparisons at NSTX with GTS and GYRO, shows some agreement, but more modeling is needed to understand disagreements

A. White – Les Houches 2015 71 Poli APS 2010

Outline






Notes on features of interest for linking tokamak turbulence to space/astrophysical turbulence


Core (r/a ≈ 0.5) density fluctuation measurements from DIII-D with BES

Shape of spectra varies strongly between core and edge

McKee, Plasma and Fusion Research, Volume 2, S1025 (2007)

Edge measurements near r/a ~ 1.0 do show what appear to be spectra with knee … power law(s)...


Density fluctuations measured at r/a = 0.99 Gas Puff Imaging Log-log plot NSTX (spherical tokamak)

White PoP 13, 072301 (2006)

Universality of edge turbulence spectra in tokamaks


Plasma edge fluctuation measurements carried out in different magnetic confinement devices (tokamaks & stellarators) support view that plasma turbulence/transport displays universality – Pedrosa et al., Phys. Rev. Lett. 82, 3621 (1999) • Also suggested that measurements of fluctuations exhibiting power spectra with exponential frequency dependence (Lorentzian spectral fit) over a broad frequency range – Maggs and Morales, Phys. Rev. Lett. 107, 185003 (2011)

Exponential fit works well in TJ-K, C-Mod and DIII-D edge plasmas, Lorentzian time pulses are found in time series data

•  Density fluctuation spectra from reflectometer S(t) = A(t) cos φ(t)

measured near r/a = 0.95 in L-mode •  Spectra are plotted linear-linear,

semi-log, and log-log

•  Exponential fit (red dashed) agrees with data over wide frequency range

•  In raw time series, Lorentzian

pulses are seen

•  FT(Lorentzian(t)) = exponential P(f)


Hornung, et al. Phys. Plasmas, 18, 082303 (2011) Winter APS 2012, TTF 2013 Maggs, et al. Phys. Plasmas 2015

C-Mod data r/a = 0.95

Intermittency, bursts in edge/SOL turbulence in tokamaks as seen in raw data

•  Extraction of coherent bursts from turbulent edge plasma in magnetic fusion devices using orthogonal wavelets, Farge et al. 13, 042304 (2006)

•  GPI data from NSTX


Measured signal from Edge of tokamak

Extracted “coherent” part

Extracted “incoherent” part

Movie of edge/SOL turbulence show “blobs”, intermittent bursts of density fluctuations


Gas Puff Imaging measures density fluctuations by recording fluctuations of D�α light from excitations of neutral gas puff introduced by puff in edge Gaussian PDF r/a < 1.0, Non-Gaussian PDF r/a > 1.0 NSTX data and movie, courtesy S. Zweben, see online movie collection t ≈ 0.245 sec L-H transition

Last Closed Flux Surface (LCFS)

Discussion and conclusions •  Great deal of progress measuring turbulence in core plasma and

comparing to gyrokinetic simulation in last decade

•  Overall good level of consistency with measured turbulence in core and gyrokinetic theory - Some discrepancies remain – active area of research (validation, synthetic diagnostics, diagnostic development)

•  “here there be dragons…” in some ways, the edge and SOL regions are easier to measure (in situ!)…but much more challenging to model

•  Development of kinetic/fluid turbulence models for edge of tokamaks

is frontier area of research

•  Connections to turbulence in other plasmas? Universality? Intermittency?


Thank you



O-mode and X-mode waves for reflectometry

Launched wave is polarized parallel (O-mode) to external magnetic field, then signal reflects from the plasma cutoff layer Polarized perpendicular (X-mode) to external magnetic field, signal reflects from the right-hand cutoff layer


Accessibility for launched waves from low field side of tokamak

Image: http://tempest.das.ucdavis.edu/pdg/reflect/

Comparison with Langmuir probes suggests that amplitude and Sreal (or Simag) contain useful information about the density fluctuations


Conway, Plasma Phys. Control. Fusion 38 (1996) Conway, Rev. Sci. Instrum., Vol. 67, No. 11, November 1996 Rhodes, Plasma Phys. Control. Fusion 40 (1998) 493–510

Sreal (t) = A(t) cos φ (t)

That was all inside last closed flux surface… in Scrape Off Layer (SOL), r/a ≥ 1, fluctuations are very intermittent

•  Gas Puff Imaging measures density fluctuations by recording fluctuations of D�α light from excitations/excitations of introduced neutral gas puff in edge

•  Gaussian PDF r/a < 1.0, Non-Gaussian PDF r/a > 1.0

A. White – Les Houches 2015 84 Data courtesy Jim Terry, GPI, MIT

Heating a tokamak plasma to fusion relevant (100,000,000K ~ 15 keV) temperatures


Ohmic heating : running current through conducting plasma Neutral Beam Injection (NBI) : inject high energy (100keV) neutral particles (D, T) into plasma, heats ions and electrons by collisions (also drives current and injects momentum) Electron Cyclotron Resonance Heating (ECRH): launch EM wave into plasma with single frequency (100-300GHz), wave transfers energy to electrons at resonance (ωc = qeB/me). Electrons are heated directly. D-T Ions heated by collisions Ion Cyclotron Resonance Heating (ICRH): Launch EM wave into plasma at single frequency (50-100 MHz) , wave transfers energy to minority ions at resonance (ωp = qpB/mp), e.g. protons in a D-T plasma. Minority ions are heated directly. Electrons and D-T Ions heated by collisions

C-Mod is a small tokamak, big performance


10x smaller than ITER in linear dimensions Runs at same field B = 5.4 T Runs at same density n = 1020-21 m-3

Runs at lower temperatures T < 10 keV

(tokamak) plasmas

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