rotation dynamics of coupled ntms theory and simulations... · tobias—july, 2015 •on diii-d,...
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![Page 1: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/1.jpg)
Tobias—July, 2015
Benjamin John Tobias Princeton Plasma Physics Laboratory
with M. Chen1, C.W. Domier1, R.
Fitzpatrick2,
B.A. Grierson3, N.C. Luhmann, Jr.1,
M. Okabayashi3, K.E.J. Oloffson4,
C. Paz-Soldan4, and the DIII-D team
1University of California at Davis 2University of Texas at Austin 3Princeton Plasma Physics Laboratory 4General Atomics
TSD Workshop
Princeton, NJ
July 13th, 2015
Rotation dynamics of coupled NTMs
![Page 2: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/2.jpg)
Tobias—July, 2015
• On DIII-D, multiple core tearing instabilities are
often observed at elevated βN
• Coupling amongst multiple modes has a
nonlinear impact on rotation and stability
• At least 2 regimes of MHD-induced momentum
transport can be observed in non-disruptive, Hybrid scenario discharges:
– Phase-locking (flattens core rotation)
– Hollowing (flow shear reversal)
![Page 3: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/3.jpg)
Tobias—July, 2015
Phase-locking precedes mode locking
• Growing modes
trigger NBI feedback
• Rotation collapses
from the core outward
• The discharge
ultimately decelerates as a
rigid body
![Page 4: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/4.jpg)
Tobias—July, 2015
Differential rotation closely tied to disruptivity
stable > 100 ms
disrupting < 50 ms
stable > 50 ms
ITER-similar 147008 – 147106
(q95 ~ 3.2, saturated 3/2 mode)
differential rotation (kHz) (CER at q=2 surface vs. 3/2 mode)
CE
R a
t q
=2
su
rfa
ce
(kH
z)
• Low absolute rotation can be sustained,
but loss of differential
rotation indicates
trouble
– Rotation shear is
classically stabilizing
– Differential rotation
decouples surfaces
![Page 5: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/5.jpg)
Tobias—July, 2015
Phase-locking and rigid deceleration
• Mode spectra evolve toward phase-locking
(toroidal co-rotation)
• Rotation collapses first in
the core (large momentum
diffusivity)
• 2/1 mode is not triggered
until the core becomes a rigid body
![Page 6: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/6.jpg)
Tobias—July, 2015
Phase-locking and rigid deceleration
• Mode spectra evolve toward phase-locking
(toroidal co-rotation)
• Rotation collapses first in
the core (large momentum
diffusivity)
• 2/1 mode is not triggered
until the core becomes a rigid body
![Page 7: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/7.jpg)
Tobias—July, 2015
Phase-locking explored in non-disruptive
discharges—ITER hybrid scenario on DIII-D
• Modes initially rotate
freely
• Sudden bifurcation
to phase-locking
– Sometimes briefly
‘unlocked’ by ELMs
• Modes saturate and
discharges survive to
normal shutdown
![Page 8: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/8.jpg)
Tobias—July, 2015
3-wave mode coupling: the RFP ‘slinky mode’
• Single-helicity
modes couple
through the
Shafranov shift
• Single-fluid model of
3-wave coupling
describes salient
features:
– Bifurcation of torque
balance
– Phase-locking
R. Fitzpatrick, PoP (2015)
DY2,1 = EY2,1 +C Y2,1,Y2,1,Y4,2
DY3,2 = EY3,2 +C Y1,0 ,Y3,2 ,Y4,2
DY4,2 = EY4,2 +C Y2,1,Y2,1,Y4,2 +C Y1,0 ,Y3,2 ,Y4,2
![Page 9: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/9.jpg)
Tobias—July, 2015
3-wave mode coupling: the RFP ‘slinky mode’
R. Fitzpatrick, PoP (2015)
DTm,n
EM =2p 2Bf
2a4n
m0R0
Im DYm,nYm,n
*éë ùû
DTm,n
V = 4p 2R0
3 mr¶
¶rDWf( )é
ëêù
ûúrm ,n-
rm ,n+
• Single-helicity
modes couple
through the
Shafranov shift
• Single-fluid model of
3-wave coupling
describes salient
features:
– Bifurcation of torque
balance
– Phase-locking
slip frequency
torq
ue
TEM
TV
![Page 10: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/10.jpg)
Tobias—July, 2015
3-wave mode coupling: the RFP ‘slinky mode’
slip frequency
torq
ue
TEM
TV
high-slip regime low-slip
wbifurcation = w0 2
• Single-helicity
modes couple
through the
Shafranov shift
• Single-fluid model of
3-wave coupling
describes salient
features:
– Bifurcation of torque
balance
– Phase-locking
![Page 11: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/11.jpg)
Tobias—July, 2015
3-wave mode coupling: the RFP ‘slinky mode’
slip frequency
torq
ue
TEM
TV
high-slip regime low-slip
• Single-helicity
modes couple
through the
Shafranov shift
• Single-fluid model of
3-wave coupling
describes salient
features:
– Bifurcation of torque
balance
– Phase-locking
![Page 12: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/12.jpg)
Tobias—July, 2015
Internal geometry of the ‘tokamak slinky:’
• Mode-coupling theory developed in a cylinder,
mode structure in a tokamak very different
– How does this impact the physics?
– How does it modify the experimental observables?
![Page 13: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/13.jpg)
Tobias—July, 2015
Outer layer approximated by parameterization
of local poloidal perturbation wavenumber
Contours of constant
eigenmode phase
Fitted ‘local mode
number’
Experiment Model kq 0 r( ) º m
r
r0, R ³ R0
![Page 14: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/14.jpg)
Tobias—July, 2015
Expands upon single-helicity mode spectrum by
encompassing additional features
n1 = an0
kq1 = a kq 0 + e
e r( ) = kq1 r( ) -a kq 0 r( ) = rm1
r1-
am0
r0
é
ëê
ù
ûú, r ³ r0
am0 m1 = r0 r1
r0 r1 = 4 3
Two
modes:
Become
resonant
when:
Otherwise:
![Page 15: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/15.jpg)
Tobias—July, 2015
Coupling of Shafranov shift no longer explicit
DY2,1 = EY2,1 +C Y2,1,Y2,1,Y4,2 + ¢C Y2,1,Y2,1,Y3,2
DY3,2 = EY3,2 + ¢C Y2,1,Y2,1,Y3,2 + ¢D Y3,2 ,Y4,2
DY4,2 = EY4,2 +C Y2,1,Y2,1,Y4,2 + ¢D Y3,2 ,Y4,2
DY2,1 = EY2,1 +C Y2,1,Y2,1,Y4,2
DY3,2 = EY3,2 +C Y1,0 ,Y3,2 ,Y4,2
DY4,2 = EY4,2 +C Y2,1,Y2,1,Y4,2 +C Y1,0 ,Y3,2 ,Y4,2
Single-helicity modes in cylinder Generalized toroidal modes
![Page 16: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/16.jpg)
Tobias—July, 2015
This parameterization also yields observable
phase velocities (cylindrical torus)
v = VΩ =vpf
vpq
é
ë
êê
ù
û
úú
=
R -Rkq
n
-rn
kq
r
é
ë
êêêêê
ù
û
úúúúú
Wf
Wq
é
ë
êê
ù
û
úú
Dvpf
Dvpq
é
ë
êê
ù
û
úú
=
R -Ra kq 0 + e
an0
-ran0
a kq 0 + er
é
ë
êêêêê
ù
û
úúúúú
Wf1
Wq1
é
ë
êê
ù
û
úú-
R -Rkq 0
n0
-rn0
kq 0
r
é
ë
êêêêê
ù
û
úúúúú
Wf0
Wq 0
é
ë
êê
ù
û
úú
Dvpfs
Dvpqs
é
ë
êê
ù
û
úú
=
R1 -R0
-r1n1
m1
r0n0
m0
é
ë
êêê
ù
û
úúú
Wf1
Wf0
é
ë
êê
ù
û
úú-
R1
m1
n1
-R0
m0
n0
-r1 r0
é
ë
êêê
ù
û
úúú
Wq1
Wq 0
é
ë
êê
ù
û
úú
Two modes on any one, given surface:
Comparing the rotation of two rational surfaces:
Any given mode:
![Page 17: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/17.jpg)
Tobias—July, 2015
This parameterization also yields observable
phase velocities (cylindrical torus)
v = VΩ =vpf
vpq
é
ë
êê
ù
û
úú
=
R -Rkq
n
-rn
kq
r
é
ë
êêêêê
ù
û
úúúúú
Wf
Wq
é
ë
êê
ù
û
úú
Dvpf
Dvpq
é
ë
êê
ù
û
úú
=
R -Ra kq 0 + e
an0
-ran0
a kq 0 + er
é
ë
êêêêê
ù
û
úúúúú
Wf1
Wq1
é
ë
êê
ù
û
úú-
R -Rkq 0
n0
-rn0
kq 0
r
é
ë
êêêêê
ù
û
úúúúú
Wf0
Wq 0
é
ë
êê
ù
û
úú
Dvpfs
Dvpqs
é
ë
êê
ù
û
úú
=
R1 -R0
-r1n1
m1
r0n0
m0
é
ë
êêê
ù
û
úúú
Wf1
Wf0
é
ë
êê
ù
û
úú-
R1
m1
n1
-R0
m0
n0
-r1 r0
é
ë
êêê
ù
û
úúú
Wq1
Wq 0
é
ë
êê
ù
û
úú
Two modes on any one, given surface:
Comparing the rotation of two rational surfaces:
Any given mode: Singular matrices
![Page 18: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/18.jpg)
Tobias—July, 2015
Experimental observation of phase-locking
![Page 19: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/19.jpg)
Tobias—July, 2015
Experimental observation of phase-locking
Conditions here
controlled by q95
![Page 20: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/20.jpg)
Tobias—July, 2015
Momentum transport increases at lower q95 To
roid
al ro
tatio
n
Mo
m.
Diffu
siv
ity
To
rqu
e d
en
sity
![Page 21: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/21.jpg)
Tobias—July, 2015
Momentum transport increases at lower q95 To
roid
al ro
tatio
n
Mo
m.
Diffu
siv
ity
To
rqu
e d
en
sity
Phase-locked
![Page 22: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/22.jpg)
Tobias—July, 2015
Momentum convection arises at q95 ~ 4.5 To
roid
al ro
tatio
n
Mo
m.
Diffu
siv
ity
To
rqu
e d
en
sity
Phase-locked Unlocked
![Page 23: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/23.jpg)
Tobias—July, 2015
Below q95~4.5, modes’ pitch does not align
and modes do not remain phase-locked
![Page 24: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/24.jpg)
Tobias—July, 2015
Differential pitch of the modes coincides with
differential toroidal and poloidal phase velocity
• Toroidal phase velocity
from Mirnov array:
• Poloidal phase velocity from microwave imaging:
Dvpf =-R
n0
e
aWq 0
Dvpq =r
k0
e
aWq 0 - Wq1 -
an0
ak0 + eWf1
æ
èçö
ø÷é
ëê
ù
ûú
Consistent
with Fitzpatrick
![Page 25: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/25.jpg)
Tobias—July, 2015
Rotation of the composite structure can be
evaluated in the plasma fluid frame
![Page 26: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/26.jpg)
Tobias—July, 2015
Rotation of the composite structure can be
evaluated in the plasma fluid frame
![Page 27: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/27.jpg)
Tobias—July, 2015
Rotation of the composite structure can be
evaluated in the plasma fluid frame
Dvpf
R=
-e
an0
Wq
![Page 28: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/28.jpg)
Tobias—July, 2015
Rotation of the composite structure can be
evaluated in the plasma fluid frame
Dvpf
R=
-e
an0
Wq
![Page 29: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/29.jpg)
Tobias—July, 2015
Rotation of the composite structure can be
evaluated in the plasma fluid frame
Dvpf
R=
-e
an0
Wq
![Page 30: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/30.jpg)
Tobias—July, 2015
Rotation of the composite structure can be
evaluated in the plasma fluid frame
Dvpf
R=
-e
an0
WqPoloidal rotation can break
phase-locking
0 =akq 0 +e( ) -an0
-kq 0 n0
é
ë
êêê
ù
û
úúú
DWq
DWf
é
ë
êê
ù
û
úú
…but these modes can still be
‘co-propagating’ with the fluid on
a given flux surface
w1 =aw0 +eWq
n1DW = k0G0 - k1G1
G j =kq1 jBq j
r j
-n1Bf j
Rj
In tokamak geometry using flux-
conserved rotation variables, the
condition of co-propagation becomes:
![Page 31: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/31.jpg)
Tobias—July, 2015
Poloidal rotation lags behind rotation of the
combined mode structures
Toroidal Rotation Poloidal Rotation (vertical CER)
Rotation of the composite
structure is ~ 5 km/s at either
rational surface (about 10x
measured Carbon flow)
![Page 32: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/32.jpg)
Tobias—July, 2015
Negative differential rotation implies a regime of
further non-ambipolarity
• Points of constructive perturbation phase in the
composite mode structure propagate ahead of
the measured Carbon (and estimated main ion)
fluid in the ion diamagnetic direction
– Past the point of EM force balance predicted by theory and overcoming fluid viscosity
• Computed Er (quasi-neutral force balance) has
been reduced; are non-ambipolar terms
required?
– Next avenue of investigation:
Er = vfBq - vqBf +ÑPi
Zieni+ ?
![Page 33: Rotation dynamics of coupled NTMs Theory and Simulations... · Tobias—July, 2015 •On DIII-D, multiple core tearing instabilities are often observed at elevated β N •Coupling](https://reader036.vdocument.in/reader036/viewer/2022071212/6026f8c53d65a801950f36d5/html5/thumbnails/33.jpg)
Tobias—July, 2015
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Tobias—July, 2015
R. Fitzpatrick, PoP (2015)