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)