Particle Energization and the Tearing-Driven Turbulent Cascade

John Sarff

A. Almagri, J. Anderson, A. DuBois, D. Craig*, D. Den Hartog, C. Forest, K. McCollam, M. Nornberg, P. Terry, D. Thuecks

and the MST Team(*Wheaton College)

Bringing Space Down to Earth • UCLA • Apr 10-12, 2017

The tearing-driven turbulent cascade and dissipation

• Connected challenges:– Multi-scale modeling framework that spans global instability to turbulent

dissipation– Diagnostics and methods that isolate and distinguish particle energization

processes

103 104 105 10610−6

10−5

10−4

10−3

10−2

10−1

100

101

Frequency (Hz)

Powe

r [J/(

m3 Hz)

]

Magnetic=B2/2μ0Kinetic=(1/2)mi ni V

2E× B

0

~

~

~

fci

The tearing-driven turbulent cascade and dissipation

103 104 105 10610−6

10−5

10−4

10−3

10−2

10−1

100

101

Frequency (Hz)

Powe

r [J/(

m3 Hz)

]

Magnetic=B2/2μ0Kinetic=(1/2)mi ni V

2E× B

0

~

~

~

fci

DrivingGradients

The tearing-driven turbulent cascade and dissipation

103 104 105 10610−6

10−5

10−4

10−3

10−2

10−1

100

101

Frequency (Hz)

Powe

r [J/(

m3 Hz)

]

Magnetic=B2/2μ0Kinetic=(1/2)mi ni V

2E× B

0

~

~

~

fci

DrivingGradients

Self-OrganizingFeedback

The tearing-driven turbulent cascade and dissipation

103 104 105 10610−6

10−5

10−4

10−3

10−2

10−1

100

101

Frequency (Hz)

Powe

r [J/(

m3 Hz)

]

Magnetic=B2/2μ0Kinetic=(1/2)mi ni V

2E× B

0

~

~

~

fci

DrivingGradients

Self-OrganizingFeedback

?

Dissipation:- mechanism(s)- scale(s)

-1.0 -0.5 0.0 0.5 1.0 1.50

200

400

600

800

1000

Tem

pera

ture

(eV

)

Time relative to reconnection (ms)

He2 +

D

H+

He2+

D

H+

MST reversed field pinch plasmas

• Magnetic induction is used to drive a large current in the plasma– Plasma current, Ip = 50-600 kA ; B < 0.5 T

– Inductive ohmic heating: 5-10 MW (input to electrons)– Ti ~ Te < 2 keV, despite weak i-e collisional coupling (n ~ 1019 m–3 )

– Minor radius, a = 50 cm ; ion gyroradius, ρi ≈ 1 cm ; c/ωpi ≈ 10 cmβ < 25% ; Lundquist number S = 5 ×104-6

q(r)

~ 0.2

minor radius, r0 a

conductin

g s

he

ll

m=1, n ≥ 6resonances

m=0,all n

1,61,7

1,8rBφRBθ

Minor Radius, r

Tearing instability drives a broadband turbulent cascade

Toroidicity allows distinct k|| = 0 resonant modes at many radii in the plasma:

0 = k ⋅B =mrBθ +

nRBφ

m = poloidal mode number

n = toroidal mode number

1 10 100 1000Frequency (kHz)

10–4

10–8

10–12

tearinginstability

ωci /2πρi ~ 1 cm

B2( f )(T2/Hz)

k ≥ 0.03 cm–1

Tearing instability drives a broadband turbulent cascade

Toroidicity allows distinct k|| = 0 resonant modes at many radii in the plasma:

0 = k ⋅B =mrBθ +

nRBφ

m = poloidal mode number

n = toroidal mode number

Linear Instability

Nonlinear Excitation

1 10 100 1000Frequency (kHz)

10–4

10–8

10–12

tearinginstability

ωci /2πρi ~ 1 cm

B2( f )(T2/Hz)

k ≥ 0.03 cm–1

The cascade is anisotropic and hints at a non-classical dissipation mechanism

• The k spectrum fits a dissipative cascade model (Y. Ren, PRL 2011; P. Terry, PoP 2009)

• Onset of exponential decay (dissipation) occurs at a smaller k than expected for classical dissipation

(T2-cm)

B2 (k)

(cm–1)

Wavenumber Power Spectrum

0 0.2 0.4 0.6 0.8 1.0 1.2

k

Tear

ing

Ran

ge

k||–5.4

10–4

10–6

10–8

10–10

10–12

10–14

New evidence for the onset of drift waves at intermediate scales

• Turbulence becomes kinetic energy dominant at • Signatures consistent with drift waves (Thuecks et al, PoP 2017)

103 104 105 10610−6

10−5

10−4

10−3

10−2

10−1

100

101

Frequency (Hz)

Pow

er [J

/(m3 H

z)]

Magnetic=B2/2μ0Kinetic=(1/2)mi ni V

2E× B

0

~

~

~

fci

104

105

106

−2 −1 0 1 2Fr

eque

ncy

(Hz)

k⊥ (cm−1)

2

4

6

8

10

k⊥ρs ≈ 0.2

Non-collisional Ion Energization

Powerful ion energization is associated with impulsive magnetic reconnection events

• Instantaneous heating rate up to 10 MeV/s (50 MW)

Time (ms)

(large-scale B)

Relative to Reconnection Event

Ti > Te

Heating is anisotropic and species dependent

• Ion distribution diagnostics on MST:– Rutherford scattering for majority ion temperature– Charge-exchange recombination spectroscopy (CHERS) for minority ions– Neutral particle energy analyzers

0.0

0.5

1.0

1.5

T (

keV

)

-2 -1 0 1 2

T

T||

C+6

Time (ms) Time (ms)

Heating is AnisotropicMinority Ions Hotter than Majority Ions

An energetic ion tail is generated and reinforced at each reconnection event

• Distribution is well-fit by a Maxwellian plus a power-law tail

BeforeEvent

AfterEvent

fD+(E) = A e –E/kT + B E–γ

Proposed Ion Heating Mechanisms

The models proposed for ion heating in the RFP are similar to those for the solar corona and wind

• Cyclotron-resonant heating:– Feeds off the turbulent cascade to gyro-scale– Preferential perpendicular heating, but with collisional relaxation– Preferential minority ion heating, since is larger where ωci is smaller– Mass scaling is predicted with dominant minority heating and collisional

relaxation

B2 (ωci )

Tangri et al., PoP 15 (2008)(similar to Cranmer et al)

Time (ms)

30 32 34 36 38 400

200

400( )

Tα

(eV

)

T||

[D]

T||

[C]

T⊥ [D]

T⊥ [C]

The models proposed for ion heating in the RFP are similar to those for the solar corona and wind

• Stochastic heating:– Feeds off large electric field fluctuations and a distinct chaotic diffusion

process– Monte Carlo modeling yields MST-like heating rates (Fiksel et al, PRL 2009)– Predicts mass scaling close to that observed

-0.6 -0.4 -0.2 -0.0 0.2 0.4 0.6Time From Sawtooth (ms)

200

400

600

800

1000

(V/m

)

103 104 105 106 107

Frequency (Hz)

102

10-2

10-6

J/m

3 -H

z

B2 / 2μ0

12miniV

2E×B0

ErRMS

Does non-Alfvenic cascade help make heating more powerful?

Viscous heating is not sufficient

• No experimental evidence for the required large sheared flow• Perpendicular flow is dominant for tearing modes for which the classical viscosity

is small• Difficult to achieve the large impulsive heating rates seen in MST plasmas• See, e.g., Svidzinski et al, PoP 15 (2009)

The need for multi-scale modeling

• Fluid treatment of tearing instability and the self-organizing feedback is mature but does not encompass small scales– Nonlinear MHD and two-fluid models (NIMROD, DEBS)– Typical spatial resolution 0.3– Braginskii dissipation, sometimes artificially enhanced for numerical purposes

• A large-scale electric field (dynamo flux) is created in the self-organizing feedback– Energization on the largest scale, e.g., energetic ions– Two-fluid nature allows ions and electrons to respond differently

• Boundary conditions are important– Unavoidable consequence of being a confined plasma– Coupling to other modes like drift waves, even if they are stable– Boundary interfaces are generic, e.g., corona or magnetosphere

kρi

(Some) next step RFP experimental opportunities

• Diagnostics and methods that discriminate heating mechanisms– Limited measurements suggest ion heating is preferentially perpendicular– Test bed for wave-particle correlation study– Diagnostic challenge is significant, so understanding probably hinges on

modeling predictions for turbulence characteristics

• Plasma control improvements, e.g., current magnitude and duration– Lower the Lundquist number to isolate MHD regime (versus two-fluid)

– Increase access for both intrusive and nonintrusive diagnostics

• Inject plasmoids– Mix with tearing-driven turbulence– Increase beta– Form shocks

Computational model for tearing-relaxation recently extended to include two-fluid effects

• Nonlinear multi-mode evolution solved using NIMROD• Motivated by measurements that suggest coupled electron and ion relaxation

E = −V×B+ 1neJ×B− 1

ne∇pe +ηJ+

mene2

∂J∂t

nmidVdt

= J×B−∇p−∇⋅Πgyro −∇⋅νnmiW

Ohm’s law:

Momentum:

Relaxation process couples electron and ion momentum balance

Two-fluid relaxation signatures are measured, e.g., Hall emf

Hall emf

MHD emf

0.6 0.7 0.8 0.9 1.0−40

−20

0

20

40

0.6 0.7 0.8 0.9 1.0Normalized Radius

−40

−20

0

20

40

V/m

E||−η J||<j×b>/noe−<v×b>, Inferred

MHD emf (inferred)

Hall emf

NIMROD Computation Deep-Insertion Magnetic Probe in MST

The turbulent cascade is not as strong when nonlinear coupling at the driving scale is disabled

Frequency (kHz)

˜ B 2

during event with m=0

before event

during event w/o m=0

tearing

q(r)

~ 0.2

minor radius, r0 a

conductin

g s

he

ll

m=1, n ≥ 6resonances

m=0,all n

1,61,7

1,8rBφRBθ

Minor Radius, r

(1,6)

(1,7)

(1,8)

(1,9)

(0,1)

(0,1)

(0,1)

etc.

Schematic of 3-wave cascade:

m ≥ 2 branchdrives the cascadeto smallest scales

Same m = 0couples tomany m = 1

Ti (eV)

Toroidalflux

Mag energy(kJ)

TearingAmplitude

(G)

n=6 n=1 (m=0)

Particle energization correlates with power flow into the cascade

Time (ms)

New power supplies enable four orders of magnitude in Lundquist number, S ≈ 3 104-8

MHDCodes

New ProgrammablePower Supplies

Existing Power Supplies

Freq

uenc

y (k

Hz)

Time (ms)10 12 14 16 18 20 22

50

100

150

20

15

10

5

Dens

ity fl

uctu

atio

n Po

wer

(1038

m-4

/kHz

) 10-4

10-5

10-6

10-7

10-8

10-9

a)

b)

c)

R 0‹| n

e| 0.8›/ ‹n

e,0›

Δ

–

n e| (1019

m-4

)

Δ|

0.90

0.80

0.70

0.60

8

6

4

2

0

r/aGyrokinetic modeling for the tearing cascade is a next challenge

• GENE modeling identifies standard drift mode branches in RFP equilibria• Initially motivated by improved-confinement regime, but gradients in standard

conditions are close to marginal stability (if not unstable)

ne

∇ne

Linear

Nonlinear

Nonlinearw. b-tilde

Summary

• Ion heating and acceleration associated with magnetic reconnection from tearing instability is a powerful process in the RFP laboratory plasma– Gyro-resonant and stochastic processes are likely candidates to support the

observed rapid heating and other features– Energetic tail formation for ions and electrons

• Global self-organization strongly coupled to turbulence and dissipation– Correlations in electric and magnetic field fluctuations are a hallmark of

dynamo feedback– Inhomogeneity on the system scale, e.g., strong edge gradients– Global magnetic flux change drives produces ion runaway energization– Impact of transport processes, which can be quite different for ions and

electrons