Lecture 4 - Access · Lecture 4 Relationship of low frequency shear Alfv´en spectrum to MHD and microturbulence Fulvio Zonca ... space notations; δφstands for δΦ etc (ˆ Lecture

Max-Planck-Institut fur Plasmaphysik Lecture Series-Winter 2013 Lecture 4 – 1

Lecture 4

Relationship of low frequency shear Alfven spectrum to MHD and

microturbulence

Fulvio Zonca

http://www.afs.enea.it/zonca

Associazione Euratom-ENEA sulla Fusione, C.R. Frascati, C.P. 65 - 00044 - Frascati, Italy.

Institute for Fusion Theory and Simulation, Zhejiang University, Hangzhou 310027, P.R.C.

February 21.st, 2013

Max-Planck-Institut fur Plasmaphysik Lecture Series-Winter 2013,Kinetic theory of meso- and micro-scale Alfvenic fluctuations in fusion plasmas

19–22 February 2013, IPP, Garching

F. Zonca


The roles of low frequencies

Lecture 2 assessed the important role of SAW for energetic particle transportin burning plasmas magnetically confined in toroidal geometry

Lecture 3 constructed the general theoretical framework for investigatingSAW fluctuations based on the general fishbone like dispersion relation

This Lecture: look at specific applications, favoring low frequencies, sincethey are the natural ones on which both energetic as well as thermal particlescan resonantly excite collective modes characterized by the respective scalelengths: fast ions ⇒ mesoscales, thermal particles ⇒ microscales.

Similar temporal scales of disparate phenomena facilitates their interplayand dictates long time-scale nonlinear dynamic response (importance forthe fusion burn)

F. Zonca

http://www.afs.enea.it/zonca/references/seminars/IPPGarching_winter13/lecture2.pdf



! Reduced kinetic SAW model equations.Assume !ds ! !ds = (msc/es)(µ+ v2!/B)!!, !! = (k" b) · "Vorticity # Quasineutrality are:

B ·$!

k2"

k2"B

2B ·$#$

"

+!(! % !#pi)

v2A

k2"

k2"

#%%

#

$

s=i

4&esk2"c

2!!ds#Ks

%

+4&

k2"B

2k" b · $

&

P" +P!

'

!!#$ =

(

4&eEk2"c

2J0(k"'E)!!dE#KE

)

,

#

$

s$=E

e2sms

(F0s

(E

%

(#%% #$) +$

s=i

es &#Ks' = 0 ,

[!tr(# % i (! % !d)]s #Ks = i* e

m

+

sQF0s

,

J0(k"'s) (#%% #$) +

!

!d

!

"

s

J0(k"'s)#$

-

.

F. Zonca


BAE dispersion relation and high-frequency fish-

bones

Assume massless electrons and decompose drift-kinetic ion response as

δfi =( e

m

)

i

[

∂F0

∂E δφ− QF0i

ωδψ

]

+ δKi

Definitions: F0i the equilibrium distribution function, E = v2/2 the energyper unit mass, QF0i = (ω∂E + ω∗)iF0i, ω∗iF0i = (mic/eB)(k× b)·∇F0i.

Scalar fields are δφ and δψ, related with the vector potential fluctuationδA‖ by δA‖ ≡ −i (c/ω)b · ∇δψ (δE‖ = −b · ∇(δφ − δψ)). Fields areobtained from quasi-neutrality and vorticity equations. Simplify ballooningspace notations; δφ stands for δΦ etc (Lecture 1).

F. Zonca



The particle distribution function δKi is derived from the drift-kinetic equa-tion

[ωtr∂θ − i (ω − ωd)]i δKi = i( e

m

)

iQF0i

[

(δφ− δψ) +(ωd

ω

)

iδψ]

Definitions: ωtr = v‖/qR is the transit frequency, k2⊥ = k2ϑ[1+(sθ−α sin θ)2]and ωdi is the magnetic drift frequency ωdi(θ) = g(θ)kϑmic(v

2⊥/2+v

2‖)/eBR,

g(θ) = cos θ + [sθ − α sin θ] sin θ.

Quasi-neutrality and vorticity equations are

Bb · ∇[

1

B

k2⊥k2ϑ

b · ∇δψ]

+ω2

v2A

(

1− ω∗pi

ω

) k2⊥k2ϑδφ+

α

q2R2g(θ)δψ =

⟨

4πe

k2ϑc2ωωdiδKi

⟩

(

1 +1

τ

)

(δφ− δψ) =Tine

〈δKi〉

F. Zonca


BAE are important since they can be excited by both fast ions (longwavelengths) as well by thermal ions (short wavelengths; AITG). This waspredicted theoretically (PPCF96 2011) and then observed experimentally(Nazikian etal PRL 2006).

For BAE one typically has ω ≈ ωti ≈ ω∗pi ≈ k‖vA. Important effects areexpected from kinetic interaction in the radial local region (kinetic layer)k‖qR0 ≈ β1/2.

Since (nq − m) and ballooning angle θ are conjugate variables w.r.t. theFourier integral transform (Lecture 1), relevant kinetic physics is expectedat large |θ|.

Typical paradigmatic case for application of BF to drift Alfven turbulenceas well as MHD (no specification of mode number so far; just k‖qR0 ≈ β1/2).

Derivation of layer equation via asymptotic expansion in β1/2. Lowest or-der solution yields δK

(0)i = −(e/m)i(QF0i/ω)(δφ

(0) − δψ(0)), which yieldsδψ(0) = δφ(0) when substituted back into the quasi-neutrality condition.

F. Zonca



At the next order, δφ(1) = eiθδφ(1+) + e−iθδφ(1−) with a correspondingδK

(1)i = eiθδK

(1+)i + e−iθδK

(1−)i . Let ω

(±)tr = [1± (nq −m)] (v‖/qR0). Then

i[

±ω(±)tr − ω

]

δK(1±)i = i

( e

m

)

iQF

(∓)0i δφ(1±) ∓ i

( e

m

)

iQF0i

v2⊥/2 + v2‖R0ωciω

kr2iδφ(0)

Note that the dominant effect comes from the geodesic curvature (largekr ∝ θ), causing radial magnetic drifts. Sideband generation is evident in

the wave-particle interaction as well as in the ω∗ effect in QF(∓)0i , which is

computed at m∓ 1 [Zonca NF 09, Lauber PPCF 09].

Substituting back into the quasi-neutrality and letting ω(±)ti =

(2Ti/mi)1/2 [1± (nq −m)] /qR0

δφ(1±) = ∓i cTieB0

Nm(ω/ω(±)ti )

Dm∓1(ω/ω(±)ti )

krωR0

δφ(0)

F. Zonca


Definitions: Z(x) = π−1/2∫∞

−∞e−y2/(y − x)dy and subscripts m and m∓ 1

indicate poloidal mode numbers to compute

N(x) =(

1− ω∗ni

ω

)

[

x+(

1/2 + x2)

Z(x)]

− ω∗T i

ω

[

x(

1/2 + x2)

+(

1/4 + x4)

Z(x)]

,

D(x) =

(

1

x

)(

1 +1

τ

)

+(

1− ω∗ni

ω

)

Z(x)− ω∗T i

ω

[

x+(

x2 − 1/2)

Z(x)]

,

Corresponding solutions of the drift-kinetic equation are

δK(1±)i = ∓i cTi

eB0

e/mi

ω ∓ ω(±)tr

[

mi

2Ti

(

v2⊥2

+ v2‖

)

QF0i −Nm(ω/ω

(±)ti )


QF(∓)0i

]

krωR0

δφ(0)

F. Zonca


The large |θ| vorticity equation becomes finally (canonical form yielding thegeneral fishbone like dispersion relation)

(

∂2

∂θ2+ Λ2

)

δΨ(0) = 0

Definitions: δΨ(0) = [1 + (sθ − α sin θ)2]1/2δψ(0)

Λ2 =ω2

ω2A

(

1− ω∗pi

ω

)

+ q2ω2ti

2ω2A

[(

1− ω∗ni

ω

)(

(ω/ω(+)ti )F (ω/ω

(+)ti ) + (ω/ω

(−)ti )F (ω/ω

(−)ti ))

−ω∗T i

ω

(

(ω/ω(+)ti )G(ω/ω

(+)ti ) + (ω/ω

(−)ti )G(ω/ω

(−)ti ))

−(

(ω/ω(+)ti )Nm(ω/ω

(+)ti )

Nm−1(ω/ω(+)ti )

Dm−1(ω/ω(+)ti )

+ (ω/ω(−)ti )Nm(ω/ω

(−)ti )

Nm+1(ω/ω(−)ti )

Dm+1(ω/ω(−)ti )

)]

F (x) = x(

x2 + 3/2)

+(

x4 + x2 + 1/2)

Z(x) ,

G(x) = x(

x4 + x2 + 2)

+(

x6 + x4/2 + x2 + 3/4)

Z(x) ,

F. Zonca


Details of sideband generation (ω∗) are unnecessary for drift Alfven waves;they are needed for moderate wavelength MHD-type modes

BAE dispersion relation is obtained by asymptotic matching of the large |θ|solution with internal region

∂2θδΨ(0) − (s− α cos θ)2

[

1 + (sθ − α sin θ)2]2 δΨ

(0) +α cos θ

[

1 + (sθ − α sin θ)2]δΨ(0) = 0 .

Definition: δWf as MHD-fluid potential energy

δWf =1

2

∫ ∞

−∞

dθ

[

∣

∣

∣∂θδΨ

(0)ID

∣

∣

∣

2

+

(

(s− α cos θ)2

[

1 + (sθ − α sin θ)2]2 − α cos θ

[

1 + (sθ − α sin θ)2]

)

∣

∣

∣δΨ

(0)ID

∣

∣

∣

2]

.

General fishbone-like dispersion relation (δΨ(0) ≃ eiΛ|θ| for |θ| ≫ 1))

iΛ = δWf + δWk (fast ions included)

F. Zonca


Reminder from Lecture 3: Λ2 = k2‖q2R2

0 represents the SAW continuum;

i.e., frequency gap is at Λ2 < 0

To demonstrate that Λ2 expression contains the physics of low-frequency(BAE) gap formation in the SAW continuum, take the fluid limit in whichMHD is valid, |ω| ≫ ωti

Large argument expansion of the plasma Z function yields:

Λ2 =1

ω2A

[

ω2 −(

7

4+TeTi

)

q2ω2ti

]

+ i√πq2e−ω2/ω2

tiω2

ω2A

(

ωti

ω− ω∗T i

ωti

)(

ω2

ω2ti

+TeTi

)2

.

From solution of Lecture 2 Exercise, the low frequency gap in MHD shouldoccur at ω2 < γ(Ti + Te)/(miR

20). Comparing results with kinetic theory:

γe = 2 electrons behave as adiabatic massless fluid in 2D

γi = 7/2???

F. Zonca




Result is not surprising since MHD has many simplifications; includingisotropic pressure response

Using a generalized expression of the inertia term Λ(ω) one readily demon-strates the existence of a high-frequency kink-fishbone branch at the fre-quencies of the Geodesic Acoustic Mode (GAM) [Zonca etal PPCF 07].(ω ≫ ω∗pi, ωti and neglecting damping)

iω

ωA

[

1−(

7

4+TeTi

)

q2ω2ti

ω2

]1/2

= δWf + δWk

When compression effects are important [Zonca et al NF 09], it is generallyimportant to take into account the deviation of the mode structure fromthe usual rigid plasma displacement [Kolesnichenko et al NF 10].

Anticipating BAE/GAM degeneracy ...

F. Zonca


Experimental observations: JET Observation of finite frequency fishbone oscillations at the GAM frequency

(F. Nabais, et al. 2005, PoP 12 102509) and “low-frequency feature” ofAlfven Cascades (B.N. Breizman, et al. 2005, PoP 12 112506).

Λ2 = k2‖0v2A

F. Zonca


Micro- and meso- scale excitation of low-frequency

AE/EPM

With the general expression of Λ (k‖ = 0)

Λ2 =ω2

ω2A

(

1− ω∗pi

ω

)

+ q2ωωti

ω2A

[(

1− ω∗ni

ω

)

F (ω/ωti)−ω∗T i

ωG(ω/ωti)

−Nm(ω/ωti)

2

(

Nm+1(ω/ωti)

Dm+1(ω/ωti)+Nm−1(ω/ωti)

Dm−1(ω/ωti)

)]

low frequency AE/EPM can be excited by both thermal ions (micro-scale)and energetic ions (meso-scales)

iΛ = δWf + δWk (fast ions included)

F. Zonca


Excitation of low-frequency AE by thermal ions is most eas-ily seen using the simplified expression of Λ derived earlier:

AITG excitation mechanism

Λ2 =1

ω2A

[

ω2 −(

7

4+TeTi

)

q2ω2ti

]

+ i√πq2e−ω2/ω2

tiω2

ω2A

(

ωti

ω− ω∗T i

ωti

)(

ω2

ω2ti

+TeTi

)2

.

When ωω∗T i > ω2ti accumulation point becomes unstable! The unstable

continuum is not a concern (E: compute how much the mode can growbefore mode converting to KAW. Hint: compute how long takes for a wave-packet born at small θ-ballooning to reach large θ where KAW physics isimportant).

When ωω∗T i > ω2ti and equilibrium effects localize the AE, the Alfvenic ITG

mode is excited (AITG)

F. Zonca

Max-Planck-Institut fur Plasmaphysik Lecture Series-Winter 2013 Lecture 4 – 16R.Nazik

ian,et

al.06,

PRL96,105006

R. Nazikian, et al. 06, PRL 96, 105006

F. Zonca


BAE – GAM degeneracy Kinetic expression of the GAM dispersion relation is degenerate with that

of the low frequency shear Alfven accumulation point (BAE) in the longwavelength limit (no diamagnetic effects).

This degeneracy is not accidental [Zonca&Chen PPCF 2006, IAEA 2006,NF 2007]and is due to the identical dynamics of GAM (n = m = 0) ands.A. wave near the mode rational surface (nq ≃ m) under the action ofgeodesic curvature. The difference between the two branches is in the modepolarization: GAM, e.s. with small magnetic component; BAE, e.m. withAlfvenic structure.

In reference to experimental observations of modes at the GAM frequency,besides measuring the mode frequency, it is necessary to measure polariza-tion and toroidal mode number to clearly identify the mode.

F. Zonca


BAE excitation: n ≃ m 6= 0 ⇒ excitation by both energetic ions (at thelongest wavelengths) as well as via the AITG mechanism (at the shortestwavelengths) [Zonca etal. POP 1999]. Confirmed by observations on DIII-D[Nazikian etal. PRL 2006].

GAM excitation: n = m = 0 ⇒ no linear excitation mechanism by spa-tial non-uniformity. Only instability mechanism is via velocity space: e.g.,intense high-speed drifting beam such that ∂Fb/∂v‖ > 0.

EGAM excitation by a radially narrow [Fu PRL 2008] (weak coupling toGAM continuum) or broad [Qiu et al POP 2012] (weak coupling to GAMcontinuum) energetic particle beam. Mode conversion to kinetic GAM[Zonca and Chen EPL 2008].

F. Zonca


Experimental observations: fast ion driven GAM

in JET Observation of frequency chirping oscillations at the GAM frequency excited

in the presence of fast ion tails due to HFS ICRH (H.L. Berk, et al. 2006,NF 46 S888)⇒ large orbits ...

F. Zonca


GAM continuous spectrum

In realistic plasmas: Te(r), Ti(r), q(r)

• ⇒ ω2GAM ≃ 2Ti(r)/(miR

20) (7/4 + Te(r)/Ti(r)) = ω2

GAM (r)

• ωGAM varies radially

• ω2GAM(r) forms a continuous spectrum

∂rδJr(r, t) = 0 ⇒ BAE-GAM degeneracy

∂

∂r

N0(r)

[

ω2 − (7Ti/2 + 2Te) (r)

miR20

]

δEr

= 0

⇒ Singular solution at ω2 = ω2GAM (r)

⇒ Generally ∂r(

N0(r)Λ2(ω)δEr

)

= 0 [Zonca&Chen PPCF 1996]

⇒ Similar to Alfven resonance [Chen&Hasegawa POF 1974]

F. Zonca


Kinetic GAM

ion and electron

δEr singular at r0 where ω2 = ω2GAM

⇒ |kr| → ∞ finite ion Larmor ra-dius effects!

⇒ Linear mode conversion to Ki-netic GAM (KGAM) ⇒ prop-agating radially outward

⇒ Similar to, e.g., KineticAlfven Wave (KAW)[Hasegawa&Chen POF 1976]

Dispersion relation of KGAM

ω2 = ω2GAM (r) + Cbi C > 0 , bi = k2rρ

2i

F. Zonca


C > 0, complicated expression, lengthy: can be obtained using the degen-eracy of BAE and GAM spectra [Zonca&Chen 2006, 2007, 2008]

⇒ bi > 0 when ω2 > ω2GAM : propagation

⇒ bi < 0 when ω2 < ω2GAM : cut-off

Radial wave equation and mode con-version of GAM

• In nonuniform plasma kr =−i∂/∂r

⇒ Radial wave equation

∂

∂r

N0(r)

[

ρ2i (r)C(r)∂2

∂r2+ ω2 − ω2

GAM (r)

]

δEr

= 0

⇒ Same as that for mode con-version of shear Alfven wave[Hasegawa&Chen POF 1976]

Evidence of outward propagating GAMin JFT-2M [Ido etal. PPCF 2006]

F. Zonca


Fishbone modes: a celebrated example of EPM

A celebrated example of EPM is the fishbone instability [Chen etal PRL 84;Coppi etal PRL 86], where

i|s|[(

R20/v

2A

)

ω (ω − ω∗pi) (1 + ∆)]1/2

= δWf + δWk ,

ω∗pi is the core ion diamagnetic frequency and ∆ ∝ q2 is the enhancementof plasma inertia due to geodesic curvature [Glasser etal PF 75; Graves etalPPCF 00].

∆ ∼ 1.6q2(R0/r)1/2 inertia enhancement is not the classic GGJ factor ob-

tained in MHD [Glasser etal PF 75] and is mainly determined by trappedparticle dynamics [Graves etal PPCF 00]. Wave-particle resonances withtrapped particles can be crucial for determining the kink/fishbone stabilityin burning plasmas [Hu etal POP 06]

F. Zonca


The ∆ ∼ 1.6q2(R0/r)1/2 is identical to Zonal Flow polarizability (Rosen-

bluth and Hinton 1998): not accidental! Similar argument to BAE/GAMdegeneracy.

Analytical expression of Λ2 recently obtained, smoothly connecting MHDto BAE/GAM frequencies (Chavdarovski 2009).

Trapped particle compressions generally lower the BAE accumulation pointfrequency [Chavdarovski and Zonca PPCF 09] and are needed for a correctdescription of experimental observations [Lauber et al PPCF 09].

RP: Repeat the calculation of Λ2 including the degeneracy removal betweenpoloidal sidebands due to both finite m and finite k‖qR0. Do the same for com-puting the terms involving Finite Larmor Radius and Finite drift Orbit Width

RP: Use the above results to derive a very general dispersion relation for SAW atlow frequency, including discretization of the SAW continuum by FLR/FOW inthe frequency range of the so called finite Beta induced Alfven Acoustic Eigen-modes ω<∼ ωti

F. Zonca


Acoustic wave couplings

The notion of Beta induced Alfven Acoustic Eigenmode (BAAE) was orig-inally formulated on the basis of a fluid approach, which cannot be appliedto collisionless plasmas of fusion interest. This approach was later on ex-tended to kinetic analyses of circulating thermal particles [Gorelenkov etalPOP 2009].

Gorelenkov N N, Van Zeeland M A, Berk H L et al. 2009 Phys. Plasmas 16 056107

____m m+1n n

__n

m−1

A

q

Ω2

a

continuum

cylindricaluncoupled

m__n

q

∼β4/3

BAAE gap

RSAE

BAE gap ~ β

GAM/BAE

BAAEs

TAE gap ~ εtorus

FIG. 1. ! Color" Schematic of the low-frequency Alfvénic and acoustic con-tinuum in a cylinder! a" and in a torus! b" .

F. Zonca


Simple derivation of kinetic BAAE. The derivation closely follows that forBAE/KBM with circulating particle response only.

For Alfvenic polarization, the dominant component is flute-like and satisfiesδψ(0) = δφ(0) (ideal Ohm’s law). For appreciable acoustic polarization, wemust consider an O(1) sideband δφs = eiθδφ(+)+e−iθδφ(−); i.e., the sidebanddoes not enter as first order modulation as for BAE/KBM. Note that theδψs sideband component is negligible at low-β, due to the vorticity equationconstraints.

At the lowest order of the β1/2 asymptotic expansion, solution of the quasi-neutrality condition implies δψ(0) = δφ(0) (as for the BAE/KBM problem)and

Dm∓1(ω/ω(±)ti )δφ(±) = 0

So, no connection is provided at the lowest order between δφ(±) and δφ(0).

This shows that, at the lowest order, the “true” acoustic mode (noSAW coupling) is merely the usual (sideband) e.s. drift-wave, given by

Dm∓1(ω/ω(±)ti ) = 0.

F. Zonca


At the next O(β1/2) order, the solution of the quasi-neutrality conditiongives

δφ(±) = ∓i cTieB0

Nm(ω/ω(±)ti )


krωR0

δφ(0)

i.e. the same link between sidebands and flute-like component that waswritten for the BAE/KBM case.

The only difference is that, in the present case, we may generally haveδφ(±) ≈ δφ(0) for Dm∓1(ω/ω

(±)ti ) ≈ β1/2, or even |δφ(±)| ≫ |δφ(0)| for a

purely electrostatic polarization [Zonca and Chen PPCF 96].

The final consistency condition is obtained from the solution of the vorticityequation up to O(β), which yields the same expression for Λ2, obtainedabove.

F. Zonca


Summarizing:

• The BAAE dispersion relation reduces trivially to the GFLDR in thelimit where trapped particle dynamics are neglected; however, theyplay crucial roles for |ω|<∼ ωti [Chavdarovski PPCF 09, Lauber et alPPCF 09].

• Both SAW and acoustic polarizations are considered on the same foot-ing in the GFLDR framework. The notion of the low frequency KineticThermal Ion (KTI) gap [Chen and Zonca NF 07] is the most generaland appropriate for interpretation of experimental data.

• The acoustic polarization has stronger Landau damping due to thetypically lower fluctuation frequency and the stronger a.c. electricfield component

F. Zonca


Numerical solutions for the kinetic spectrum

Note the stronger damping of the acoustic/mixed polarization branch (log-scale), consistent with theoretical predictions. This is the main reason whythe Alfvenic branch is the one of practical interest for interpreting experi-mental observations for meso-scale fluctuations, driven by energetic parti-cles. Micro-scales typical of micro-turbulence are not treated here.

Fixed parameters are v2

Ti/v2

A= 0.01, ω

∗ni/ωTi = 0.1, ω∗Ti/ωTi = 0.2, q = 2 and τ = 2.

F. Zonca


From theory to experiment: one single view ...

A variety of experimental observations have renewed the interest in thedetailed structures of the Alfven continuum at low frequencies:

• finite frequency fishbone oscillations at the GAM frequency and “low-frequency feature” of Alfven Cascades (JET)

• observation of a broad band discrete Alfven spectrum (DIII-D withn ∼ 2 ÷ 40) excited by both energetic ions (low-n) and thermal ions(high-n)

• excitations of BAE modes by finite amplitude magnetic islands (FTU,TEXTOR)

• evidence of GAM structures (DIII-D, CHS, JFT-2M, HL-2A, AUG,T10, TEXT)

F. Zonca


New interest was attracted by observations of BAAE [Gorelenkov etal 07] atfrequencies below the “BAE accumulation point” and more recent evidenceof “Sierpes modes” in ASDEX Upgrade [Garcia-Munoz PRL 08] interpretedas BAE excited by energetic ions generated by ICRH (ICRH) [Ph. LauberNF 08], and of ICRH driven BAE in Tore Supra [R. Sabot et al. NF 09].

For interpretation of these experimental data, it is necessary to use kinetictheories for the proper treatment of wave-particle resonant interactions withthermal particles.

All these observations well fit within the present theoretical understand-ing, which poses new challenging questions to be addressed by next stepexperiments and theories of burning plasmas.

Due to – e.g. – the degeneracy of GAM and BAE accumulation points,these questions encompass issues that impact macroscopic MHD as well asplasma micro-turbulence in a subtle way.

F. Zonca


Structures of the low-frequency SAW spectrum Low-frequency Shear Alfven Wave (SAW) gap: ω ∼ ω∗i ∼ ωti; Λ

2(ω) = k2‖v2A

⇒ (ideal MHD) accumulation point (at ω = 0) shifted by thermal ionkinetic effects (F. Zonca, et al. 1996, PPCF 38 2011)

⇒ new low-freq. gap! Kinetic Thermal Ion (KTI) gap (L. Chen 2007,NF 47 S727)

Diamagnetic drift: KBM (H. Biglari, et al. 1991, PRL 67 3681) Thermal ion compress.: BAE (W.W. Heidbrink, et al. 1993,PRL 71 855)

∇Ti and wave-part. resonances: AITG (F. Zonca, et al. 1999,POP 6 1917)

⇒ unstable SAW accumulation point

⇒ “localization” ⇒ unstable discrete AITG mode

For physics analogy: BAE – GAM degeneracy (F. Zonca, et al. 2006, PPCF48 B15); (L. Chen, et al. 2007, NF 47 S727).

F. Zonca


Collective modes and DW turbulence

E.m. plasma turbulence: theory predicts excitation of Alfenic fluctuationsin a wide range of mode numbers near the low frequency accumulation pointof s.A. continuum, ω ≃ (7/4 + Te/Ti)

1/2(2Ti/mi)1/2/R (F. Zonca, L. Chen,

et al. 96, PPCF 38, 2011; ... 99, PoP 6, 1917):

• by energetic ions at long wavelength: finite Beta AE (BAE)/EPM

• by thermal ions at short wavelength: Alfven ITG

Magnetic flutter: may be relevant for electron transport (B.D. Scott2005,NJP 7, 92; V. Naulin , et al. 2005, PoP 12, 052515)

Recent observations on DIII-D confirm these predictions (R. Nazikian, etal. 06, PRL 96, 105006)

F. Zonca

Max-Planck-Institut fur Plasmaphysik Lecture Series-Winter 2013 Lecture 4 – 16R.Nazik

ian,et

al.06,

PRL96,105006

R. Nazikian, et al. 06, PRL 96, 105006

F. Zonca


Collective modes and DW turbulence

E.m. plasma turbulence: theory predicts excitation of Alfenic fluctuationsin a wide range of mode numbers near the low frequency accumulation pointof s.A. continuum, ω ≃ (7/4 + Te/Ti)

1/2(2Ti/mi)1/2/R (F. Zonca, L. Chen,

et al. 96, PPCF 38, 2011; ... 99, PoP 6, 1917):

• by energetic ions at long wavelength: finite Beta AE (BAE)/EPM

• by thermal ions at short wavelength: Alfven ITG

Magnetic flutter: may be relevant for electron transport (B.D. Scott2005,NJP 7, 92; V. Naulin , et al. 2005, PoP 12, 052515)

Recent observations on DIII-D confirm these predictions (R. Nazikian, etal. 06, PRL 96, 105006)

Theory describes well the nonlinear excitation of BAE modes in FTUby magnetic islands (S.V. Annibaldi et al. 07, PPCF 49, 475), whenFLR/FOW effects are included (F. Zonca, et al. 98, PPCF 40, 2009).

F. Zonca


The same modes are excited by a large amplitude magnetic island on FTU(P. Buratti, et al. 2005, NF 45 1446; S. Annibaldi, et al. 2007, PPCF 49475). ⇒ Recent theoretical descriptions by [A. Biancalani et al. 10, 11].

n=-1, m=-2 tearing mode Locking & unlocking

n = -1 HF mode

n=+1

P. Smeulders, et al. 2002, ECA 26B, D5.016

F. Zonca


Zonal Flows and Zonal Structures

Very disparate space-time scales of AE/EPM, MHD modes and plasmaturbulence: complex self-organized behaviors of burning plasmas will belikely dominated by their nonlinear interplay via zonal flows and fields

Crucial role of toroidal geometry for Alfvenic fluctuations: fundamentalimportance of magnetic curvature couplings in both linear and nonlineardynamics (B.D. Scott 2005,NJP 7, 92; V. Naulin , et al. 2005, PoP 12,052515)

Long time scale behaviors of zonal structures are important for the overallburning plasma performance: generators of nonlinear equilibria

F. Zonca


Long time scale behaviors Depending on proximity to marginal stability, AE and EPM nonlinear evo-

lutions can be predominantly affected by

• spontaneous generation of zonal flows and fields (L. Chen, et al. 2001,NF 41, 747; P.N. Guzdar, et al. 2001, PRL 87, 015001)

• radial modulations in the fast ion profiles (F. Zonca, et al. 2000,Theory of Fusion Plasmas, 17) EPM NL dynamics (Lecture 6)

AITG and strongly driven MHD modes behave similarly

F. Zonca



Zonal Flows and Zonal Structures

Very disparate space-time scales of AE/EPM, MHD modes and plasmaturbulence: complex self-organized behaviors of burning plasmas will belikely dominated by their nonlinear interplay via zonal flows and fields

Crucial role of toroidal geometry for Alfvenic fluctuations: fundamentalimportance of magnetic curvature couplings in both linear and nonlineardynamics (B.D. Scott 2005,NJP 7, 92; V. Naulin , et al. 2005, PoP 12,052515)

Long time scale behaviors of zonal structures are important for the overallburning plasma performance: generators of nonlinear equilibria

The corresponding stability determines the dynamics underlying the dissipa-tion of zonal structures in collision-less plasmas and the nonlinear up-shiftof thresholds for turbulent transport (L. Chen, et al. 2006)

Impact on burning plasma performance

F. Zonca


Crucial points and Summary

There is unified theoretical framework that allows explaining a variety ofexperimental observations of shear Alfven waves with one single “fishbone-like” dispersion relation. The various phenomenologies are different onlyapparently.

The existence of continuous spectra made of radial singular structures, bothfor GAM and SAW, plays a crucial role in experimental observations thatdepend on the source spatial profile and temporal coherence

There is a relationship of MHD and SAW in the kinetic thermal ionfrequency gap with microturbulence, Zonal Flows and Geodesic AcousticModes, which has importance in determining long time scale dynamic be-haviors in burning plasmas.

(Un)expected behaviors: ITG turbulence enhanced by EGAM [D. Zarzoso,submitted to PRL].

F. Zonca


References and reading material

F. Zonca and L. Chen, “Structures of the low frequency Alfven continuous spec-trum and their consequences on MHD and micro-turbulence”, CP1069, Theoryof Fusion plasmas: Joint Varenna-Lausanne International Workshop edited by O.Sauter, X. Garbet and E. Sindoni (AIP, 2008) p. 355-60.

L. Chen and A. Hasegawa, Phys. Fluids 17, 1399, (1974).

A. Hasegawa and L. Chen, Phys. Fluids 19, 1924, (1976).

L. Chen, R.B. White and M.N. Rosenbluth, Phys. Rev. Lett. 52, 1122, (1984)

F. Zonca et al, Plasma Phys. Control. Fusion 38, 2011, (1996)

F. Zonca et al, Phys. Plasmas 6, 1917, (1999)

F. Zonca, S. Briguglio, L. Chen, G. Fogaccia and G. Vlad, “Theoretical Aspectsof Collective Mode Excitations by Energetic Ions in Tokamaks”, Theory of FusionPlasmas, pp. 17-30, J.W. Connor, O. Sauter and E. Sindoni (Eds.), SIF, Bologna,

F. Zonca


(2000).

L. Chen and F. Zonca, Nucl. Fusion 47, S727, (2007)

G. Fu, Phys. Rev. Lett. 101, 185002 (2008).

F. Zonca and L. Chen, Europhys. Lett. 83, 35001 (2008).

Z. Qiu, F. Zonca and L. Chen, Phys. Plasmas 19, 082507 (2012).

I. Chavdarovski and F. Zonca, Plasma Phys. Controlled Fusion 51, 115001(2009).

Ph. Lauber et al, Plasma Phys. Controlled Fusion 51, 124009 (2009).

Ya. I. Kolesnichenko, V. V. Lutsenko and R. B. White, Nucl. Fusion 50, 084017(2010).

N. N. Gorelenkov et al., Phys. Plasmas 16, 056107 (2009).

N. N. Gorelenkov, H. L. Berk, E. Fredrickson and S. E. Sharapov, Phys. Lett. A370, 70 (2007)

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M. Garcia-Munoz et al., Phys. Rev. Lett. 100, 055005 (2008).

Ph. Lauber and S. Gunter, Nucl. Fusion 48 084002 (2008).

R. Sabot et al, Nucl. Fusion 49 085033 (2009).

F. Zonca

Documents

Lecture 4 - Access · Lecture 4 Relationship of low frequency shear Alfv´en spectrum to MHD and microturbulence Fulvio Zonca ... space notations; δφstands for δΦ etc (ˆ Lecture