Bayesian Inference of an FRC

§ Ion-scale modes have been shown to be stable in the C-2/C-2FRC core, in agreement with initial gyrokinetic simulation results, with a characteristic inverted toroidal wavenumber spectrum confirmed via Doppler Backscattering (DBS) measurements. Multi-scale turbulence is observed via DBS in the mirror-confined scrape-off layer plasma, in agreement with recent global gyrokinetic simulations which also indicate radial propagation into the outer layer of the FRC core. These observations have stimulated recent detailed analysis of the interaction of turbulence with large-scale E×B flow near the FRC separatrix. Radial inward turbulence propagation has been confirmed via DBS, however the radial turbulence correlation length exhibits a pronounced minimum just outside the separatrix at all wavenumbers, indicating effective radial transport barrier formation. An Advanced Doppler Backscattering diagnostic design for C-2W, and prospects for simultaneous measurements of magnetic fluctuations via cross-polarization scattering (CPS), using modified DBS microwave hardware, is discussed here.
Doppler Backscattering (DBS) Provides the Toroidal Density Fluctuation Wavenumber Spectrum ñ(kθ)
Axial Alignment is Crucial for Accurate Backscattering Spectra
General equation for scattered electric field Es:
r 2 2
E(ω,k)="kˆ × kˆ ×E $ 0 dωdtdkdx n!ωk e−x2/ax2e−y /aye−i(ωi+ωn−ωs)tei(ki+kn−ks)xe−iωsRd/c
FRC core plasma inside separatrix: Decreased fluctuation level at low kρs
Principle of Radial Correlation/ Delay Measurements
Launch two frequencies f1,f2 (radially separated turning points/cut-off layers):
excluded flux radius R : this is the location radial transport barrier formation.
DBS/CPS can probe toroidal wavenumbers kθ =1-15 cm-1 (via adjusting the toroidal launch angle). Axial FRC contraction cau- ses a finite axial mismatch angle that will be compensated via two-axis adjust- ment of the focusing mirror:
40 20 0
-20 -40
2
1
42473 2.5 ms
48 GHz
40 GHz
33 GHz
0 -10
2
(i) iε0ωpe ñ ωi + ! -
X,Y PD
3o Launch
Incident/scattered electric field E , E : (i) ii s
• C-2/C-2U FRC core: Ion modes stable due to FLR effects (large particle orbits), short connection length and grad-B drift reversal; only electron modes are weakly unstable (see also BP11.00059 by C. Lau)
• SOL: Moderate, multi-scale SOL turbulence observed/ predicted (driven by the radial density/electron temperature gradients, see also BP11.00060 by J. Bao)
• Turbulence is generated near Rs and propagates
inwards into the core as well as outwards into the SOL.
Reduced radial correlation near R due to E shear is sr
consistent with radial transport barrier formation .
lane Ray Trajectories
Ray Tajectories for varying axial mismatch angle φL
( s)&%c)(&%iεω)(s o∂t 0 i
-40 -20 0 20 40
x (cm)
ω =ω -k v
s i θ ExB
ks=ki-kθ
Separatrix E×B Shear Flows and Turbulence Propagation in the C-2U FRC; Diagnostic Upgrades for Density and Magnetic Field Fluctuation Measurements in C-2W
φ=0 φ=2o
J= E+ σσE×B/B ω n i εω2 , i .
-20 φ =4o
φ =8o
-30
-40 point
-50
-60 X-mode. 47.7 GHz
-70 42473
-2 0 2.7 7.3 12
z (cm)
ie ope
The second (highlit) term describes the
induced current in the opposite
polarization and is proportional to B r
In C-2W, the trajectories of launched and
backscattered X-mode and O-mode nearly
overlap (as ω >> ω ). Hence the CPS pe ce
(O-mode) component can be easily separated out via polarizers and detected via a dedicated receive horn
k I
ω,k
Turning (cutoff)
~
I
I
n n n
Probability Distribution of Radial Correlation Delay
Turbulence propagates inwards Turbulence propagates outwards
10 Shot Average Count (r < Rs)
120 80 40
Inwards Outwards
Cross-Polarization Scattering (CPS): Measuring Magnetic Fluctuations (Br)
Two-axis alignment of focusing mirror via dual stepper motor control
L. Schmitz1,2, D. Fulton1, C. Lau3, I Holod2, Z. Lin3, B. Deng1, H. Gota1, T. Tajima1,3, M. Binderbauer1, D. Gupta1, and the TAE Team 1TAE Technologies, Inc., 19631 Pauling, Foothill Ranch, CA 92610, 2University of California Los Angeles, 3University of California Irvine
sss#s s i%16π4aa n n nn ()∫(() )
Induced current J :
ω2$σ' ∂J(i)
$'
−∇×∇×E+ i 1− E=−iμ
xy
The measured radial correlation length shows a pronounced minimum at the
2 "22%
d I / d s ~ n! ( k , s ) e $ 2 '
Geometric Criterion for return of backscattered
radiation to (monostatic) receiver: φL: Mismatch angle of launched beam;
φ0: Mismatch angle at turning point
kn: Measured (resonant) density fluctuation
wavenumber;a Gaussianbeamwidth
Launch (DBS receive) X-mode DBS, CPS Launch (DBS receive)
Poloidal mismatch angle φ0:
n of maximum ExB shear [1,2] and indicates
x
structures likely form near Rs and propagate simultaneously inwards into the core and outwards into the SOL.
[1] M. Tuszewski et al. PRL 2012
[2] L. Schmitz et al., Nature Comm. 2016
0 k a n
x
s
inside Rs, indicating that turbulent
kS
ζ
X-Mode
DBS Launch/ Receive
k
O-Mode
Polarizer
DBS/CPS: Need to control the axial mismatch angle due to axial contraction of the FRC; this will be accomplished via a two-axis mount for the focusing mirror (under design for C-2W)
ki • B = cos(π /2-φ0 ) Typically requires
$− kn ax sin (φ0 ) '
2#& sf
Δr kI
kS
ζ
ωI,kI
sin (φ ) ≤ 2
φL ≤ 3-5o
The radial correlation delay is (stati- stically) positive outside R and negative
kθ
ωs=ωi-kθvExB ks=ki-kθ
θ
CPS receive
Low-k ion modes reduced by almost two orders of magnitude in the FRC
core
O-mode DBS launch/receive (40-60GHz)
Summary
X-mode DBS, CPS launch (DBS receive)
CPS O-mode receive
great potential; CPS is under development for C-2W.
25
20 15 10
inside Rs Inwards
Outwards 80 r < Rs 60
outside Rs Inwards
Outwards
42464 2.0-4.0 ms
r > Rs
5 00
Δt (μs)
A gyrokinetic simulation also shows inward
propagation inside Rs and outward propa- gation outside Rs (posters BP11.00059, BP 11.00060, by C. Lau et al., and J. Bao et al.)
Turbulence phase velocity from simulation:
directed inwards for r < Rs
-0.4
-0.2 0 0.2 0.4
-0.3 -0.2
-0.1
0 0.1 0.2 0.3 Δt (μs)
f1 2
ne (1019 m-3)
y (cm)
R (cm)
Counts
Counts
40
20 012
L
L
L
L
and outwards for r > Rs •
• Magnetic fluctuations measurements via CPS have