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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
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)
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
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
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
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
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
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
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
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
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
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?
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
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:
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
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:
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
Tobias—July, 2015
Experimental observation of phase-locking
Tobias—July, 2015
Experimental observation of phase-locking
Conditions here
controlled by q95
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
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
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
Tobias—July, 2015
Below q95~4.5, modes’ pitch does not align
and modes do not remain phase-locked
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
Tobias—July, 2015
Rotation of the composite structure can be
evaluated in the plasma fluid frame
Tobias—July, 2015
Rotation of the composite structure can be
evaluated in the plasma fluid frame
Tobias—July, 2015
Rotation of the composite structure can be
evaluated in the plasma fluid frame
Dvpf
R=
-e
an0
Wq
Tobias—July, 2015
Rotation of the composite structure can be
evaluated in the plasma fluid frame
Dvpf
R=
-e
an0
Wq
Tobias—July, 2015
Rotation of the composite structure can be
evaluated in the plasma fluid frame
Dvpf
R=
-e
an0
Wq
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:
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)
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+ ?
Tobias—July, 2015
Tobias—July, 2015
R. Fitzpatrick, PoP (2015)
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