Lecture 11 - Access€¦ · Nonlinear coherent interaction and “avalanche” transport: nonlinear...

MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture 11 – 1

Lecture 11

Instabilities and transport in burning plasmas

Fulvio Zonca

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

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

March 28.th, 2013

“Tor Vergata” University of Rome,MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico

20–28 March 2013

Fulvio Zonca


Lecture PlanPart I

1. Drift and drift Alfven waves (fluid and kinetic instabilities)

2. Charged fusion products as source of instability in thermonuclear plasmas

3. Kinetic energy principle applied to shear Alfven and MHD waves: unifieddescription of low frequency waves

Part II

4. Qualitative estimate of fluctuation induced transport: “random walk” and“mixing length” and their intrinsic limits

5. Role of zonal structures: new paradigm for turbulent transport

6. Resonant particle transport: the “fishbone” mode case

7. Nonlinear coherent interaction and “avalanche” transport: nonlinear Schro-dinger equation models

Fulvio Zonca


Teaching Material

Sources: Large portions of these lecture notes are based on Lectures I gavefor the IFTS Intensive Courses on Advanced Plasma Physics-Spring 2009 and2010 at the Institute for Fusion Theory and Simulation, Zhejiang University,Hangzhou, China. These Lecture Notes are all available online by clicking onthe given hyperlinks. Similar material, also available online, can be obtainedfrom the Lecture Series-Winter 2013 on Kinetic theory of meso- and micro-scale

Alfvenic fluctuations in fusion plasmas, at the Max-Planck-Institut fur Plasma-physik, Garching, February 19-22, (2013).

Lecture Notes: Available at electronic form on my personal webpage (followhyperlinks) and on the 2013 ENEAMASTER webpage, hosted by the Tor VergataUniversity of Rome. At the end of the lecture, a list of specific reading materialis given explicitly. Should you have difficulty in finding literatures and papers,please contact me at [email protected].

Fulvio Zonca

http://www.afs.enea.it/zonca/references/seminars/IFTS_spring09

http://www.afs.enea.it/zonca/references/seminars/IFTS_spring10

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

http://www.afs.enea.it/zonca/references/seminars/zonca_eneamaster13.pdf

mailto:[email protected]


Review of ideal MHD (MASTER Lectures L2,L4,L5,L6)

• Continuity equation:d

dtρ+ ρ∇ · v = 0

d

dt=

∂

∂t+ v · ∇

• Momentum equation: ρd

dtv =

J×B

c−∇p

• Equation of state:d

dtp+ γp∇ · v = 0

• Ohm’s law: E′ = E+v ×B

c= 0

• Faraday’s law: ∇× E = −1

c

∂B

∂t

• Ampere’s law: ∇×B =4π

cJ

Fulvio Zonca


Waves in an infinite homogeneous magnetized plasma

Uniform B = B0z (no eq. current), v0 = 0, but finite ρ0 and p0.

Three purely oscillation modes are found: (v2A = B20/(4πρ0), v

2S = γp0/ρ0)

• Shear Alfven Wave (SAW): ω2 = k2‖v2A (ρ1 = p1 = ∇ · v1 = 0)

• Fast and Slow Magnetoacoustic Waves (FMW/SMW):ω2 = (k2/2)(v2A + v2S)

[1± (1− α2)1/2

]

(α2 = 4(k2‖/k2)v2Av

2S/(v

2A + v2S)

2)

E: Derive the three branches above. Show that, for β = v2S/v2A ≪ 1, the

FMW becomes the Compressional Alfven Wave (CAW) with ω2 = k2v2A and4πp1/(B0B1z) = O(β); meanwhile the SMW becomes the Sound Wave (SW)with ω2 = k2‖v

2S and v1y/v1z = O(β).

E: Ideal MHD implies E‖ = 0; yet it gives the SW! Solve the apparent paradox.

Fulvio Zonca


1. Drift and drift Alfven waves

Roles of plasma nonuniformity and equilibrium magnetic geometry:

• Uniform plasmas: purely oscillating MHD waves

• Nonuniform plasmas: density, temperature, etc. gradients are freeenergy sources which can drive unstable fluctuations

• In toroidal systems, the situation is further complicated by equilib-rium magnetic geometry: magnetic drifts and new particle orbits im-pact instabilities (MASTER L6) and transport (MASTER L7)

Drift and drift Alfven waves: micro-instabilities that dominate fluctuationinduced transport

• Micro-instabilities: dominated by wavelengths of the order of (elec-tron/ion) Larmor radii; much smaller than macroscopic system size

• Modification at short scale of SW (nearly e.s.) and SAW (δE‖ ≃ 0)

Fulvio Zonca


The importance of toroidal geometry

Toroidal geometry must be ac-counted for, since it cruciallymodifies

• Particle motions and trans-port processes

• Linear and nonlinear modestructures

• Plasma response: effectiveinertia, dispersiveness, etc.

Fulvio Zonca


Tokamak: 2D axisymmetric equilibrium (MAST.L2)

Jacobian J = (∇ψ ×∇θ · ∇φ)−1

B·∇ψ = 0 ⇒ B = B1∇ψ×∇θ+B2∇φ×∇ψ∇ ·B = 0 ⇒ ∂θB2 = 0 ⇒ B2 = B2(ψ)

Choose ψ = Φp/2π ⇒ B2 = 1;Φp =

∫JB · ∇θdψdφ = 2π

∫B2(ψ)dψ

B = F (ψ, θ)∇φ+∇φ×∇ψ

F = F (ψ) only if there are notoroidal forces on the system!

Fulvio Zonca


(4π/c)J = ∇F×∇φ+R2∇φ∇·(R−2∇ψ)(4π/c)J×B = −∇ψ∇ · (R−2∇ψ)

−R−2F∇F + J−1∇φ∂θFF = F (ψ) only if there are no toroidalforces on the system!

E: What is the effect of rotation? Andof anisotropic pressure tensor (still diag-onal)?

Straight Field line flux coordinates2D with Generic ζ = φ− ν(θ, ψ)

0

2π

0 2π

θ

ζ

...

Choose ν(θ, ψ) such thatq = B · ∇ζ/B · ∇θ = q(ψ)

Clebsch representation for BB = ∇α(ψ, θ, ζ)×∇ψ

= ∇ (ζ − qθ)×∇ψE: Show that magnetic field lines areintersections of surfaces α(ψ, θ, ζ) =α0 and ψ = ψ0.

Further freedom of choosing θ is usedto select J = (∇ψ ×∇θ · ∇ζ)−1

Examples are:J = JH(ψ): Hamada coordinatesJ = JB(ψ)/B

2: Boozer coordinates

Fulvio Zonca


Particle drifts in nonuniform systems

Strongly magnetized plasmas (MASTER L7): for (equilibrium) scale lengthsmuch larger than ion Larmor radius and for (fluctuation) time scalesmuch longer than the cyclotron period, particle motions are essentially freestreaming along B0 and gyromotion

v = v‖b+ v⊥ (e1 cosα+ e2 sinα)

v2⊥/2B0 = µ = conste1 × e2 = b = B0/B0

α = Ω = eB0/(mc)

Particle drifts: at next order (in the drift parameter ρL/R, MASTER L7)particle drifts are

• E ×B drift: vE = (c/B20)E ×B0

• Magnetic drift (curvature and ∇B):vd = Ω−1b× (µ∇B0 + v2‖κ) κ ≡ b ·∇b (curvature)

E: Show that particle drifts enter at the first order in the drift parameter.

Fulvio Zonca


∇B - drift

Fulvio Zonca


Fluid drifts in nonuniform systems

From fluid equations (MASTER L4),

mndu

dt= F + n

e

cu×B

d

dt≡ ∂

∂t+ u ·∇

it is generally possible to show that, under a force density F (not includingLorentz), the fluid plasma responds with a drift

vF =c

neB0

F × b

In the case of inertia, F = −mnu, the fluid drift is called polarization drift

vp =1

Ωb× du

dt

E: Show that particle drifts enter at the first order in the drift parameter.

Fulvio Zonca


∇T - diamagnatic drift and ∇N - diamagnatic drift

In the following: intuitive derivation of ion temperature gradient (ITG)drift-wave dispersion relation, based on particle and fluid drifts and on theelectrostatic approximation (MASTER L10).

δE ≃ −∇δφ

E: Demonstrate that electrostatic approximation does not imply δB = 0.

Fulvio Zonca


Ion Temperature Gradient (ITG) driven modes

ITG are one of the dominant drift wave instabilities that are responsibleof turbulent transport, together with Trapped Electron Modes (TEM) andElectron Temperature Gradient (ETG) driven modes

They are essentially electrostatic in nature and are characterized by micro-scales (see later) that range from electron to ion Larmor radii.

Fulvio Zonca


ITG disp. relation in the local e.s. approximation

Electrostatic approximation: δE ≃ −∇δφ: need only one scalar field equa-tion in order to derive the wave dispersion relation.

Micro-instabilities in the long wavelength approximation: assume thatλ⊥<∼ ρi and λ⊥ ≫ λD (Debye length; MASTER L4)

Adiabatic electron response:

ne = ne0 exp

(e

Teδφ

)

⇒ ne

ne0

≃ e

Teδφ

Poisson’s equation (unit charge ions):

∇2δφ = 4πne0e2

Te

(

1− Teeδφ

δni

ne0

)

δφ ⇒ LHS

RHS∼ k2⊥λ

2D

Fulvio Zonca


Quasi-neutrality condition (unit charge ions) yields the relevant field equa-tion and mode dispersion relation:

δni

ne0

=eδφ

Te

Need only to solve for ion dynamics (adiabatic electrons). From lin-earized (in the fluctuation strength) ion continuity equation, assumingδφ ∝ exp(−iωt) (stability problem):

−iωδni +∇ · δ (niv⊥i) + B0∇‖δ

(ni

B0

v‖i

)

= 0

E: Derive the (quasi-neutrality) equation above step by step, noting that ⊥ and‖ are defined with respect to the equilibrium magnetic field B0 and ∇‖ = b ·∇.

Fulvio Zonca


Perpendicular ion dynamics: the equilibrium particle drifts are diamagneticand magnetic (κ, ∇B0). Perturbed (fluctuation induced) particle drifts areδE ×B0, diamagnetic and polarization drifts

• δE ×B0-drift: δ(niv⊥iE) = (ni0c/B0)b×∇δφ ≡ ni0δvE

• diamagnetic drift: δ(niv⊥i∗) = (c/eB0)b×∇δpi ≡ ni0δv∗pi

• polarization drift: δ(niv⊥ip) = (ni0/Ωi)b× ∂t(δvE + δv∗pi)

E: Justify the perturbed particle drifts derived above, filling in the details stepby step. Can you give a simple justification of why polarization drift is higherorder with respect to δE ×B0 and diamagnetic drifts?

The perturbed particle drifts are incompressible to the lowest order: for thisreason polarization drift plays an important role even if it enters at higherorder.

Fulvio Zonca


Using the formal presence of b×∇ in the δE×B0 and diamagnetic drifts:

∇ · [δ(niv⊥iE) + δ(niv⊥i∗)] = ni0c

(

∇δφ+1

ni0e∇δpi

)

·(

∇× b

B0

)

︸︷︷︸

magnetic curvature+c

B0

b×∇δφ ·∇ni0

︸︷︷︸

convective effect

Ion pressure perturbation is derived from the equation of state, assuming∇ ·u ≃ 0, i.e., nearly incompressible response. In fact, this is shown aboveand is intrinsically connected with the minimization of potential energy forMHD modes (MASTER L5 and L6; see also later).

∂

∂tδpi + δvE ·∇p0i = 0 ⇒ δpi

pi0= −ω∗pi

ω

e

Tiδφ E : Derive this!

ω∗pi = ω∗pi + ω∗pi =cTieB0

b×(∇ni0

ni0

+∇TiTi

)

· (−i∇) (operator)

Fulvio Zonca


E: Show that in a low-pressure toroidal plasma, given the magnetic curvatureκ = b ·∇b (

∇× b

B0

)

≃ 1

B0

b×(

κ+∇B0

B0

)

≃ 2

B0

b× κ

Introducing the definition of magnetic drift frequency

ωdi =cTie

(

∇× b

B0

)

· (−i∇) ≃ 2cTieB0

b× κ · (−i∇) (operator)

the divergence of perpendicular ion particle flow becomes (ρ2i ≡ Ω−2i (Ti/mi)

∇ · δ (niv⊥i) = ini0e

Ti

[(

1− ω∗pi

ω

)

ωdi − ω∗ni + ω(

1− ω∗pi

ω

)

ρ2i∇2]

δφ

E: Using the results given above for the polarization drift and other particledrifts, fill in the details leading the above expression and, in particular, the (last)polarization contribution. Hint: remember that λ⊥<∼ ρi ≪ R0.

Fulvio Zonca


Parallel ion dynamics: from the parallel force balance equation (MASTERL4), noting v‖i0 = 0 and letting v2ti ≡ Ti/mi

−ini0miωδv‖i = −∇‖δpi − eni0∇‖δφ

B0∇‖δ

(ni

B0

v‖i

)

= −ini0e

Ti

v2tiω

(

1− ω∗pi

ω

)

B0∇‖

(1

B0

∇‖δφ

)

Note: In general λ⊥<∼ ρi ≪ R0 but λ‖ ∼ R0 ≫ ρi: two scale lengthstructures of plasma turbulence. In summary:

Tie

δni

ni0

=

[(

1− ω∗pi

ω

) ωdi

ω︸︷︷︸

− ω∗ni

ω︸︷︷︸

+(

1− ω∗pi

ω

)

ρ2i∇2

︸︷︷︸

]

δφ−v2ti

ω2

(

1− ω∗pi

ω

)

B0∇‖

(1

B0

∇‖δφ

)

︸︷︷︸

curvature convection FLR ‖ compression (SW)

Fulvio Zonca


This equation is substituted back into the quasineutrality condition with adi-abatic electrons, δni/ne0 = eδφ/Te (p.16), and – with equilibrium quasineu-trality ne0 = ni0 – yields the e.s. ITG wave equation

[(

1− ω∗pi

ω

) ωdi

ω−(ω∗ni

ω+TeTi

)

+(

1− ω∗pi

ω

)

ρ2i∇2

]

δφ−v2ti

ω2

(

1− ω∗pi

ω

)

B0∇‖

(1

B0

∇‖δφ

)

= 0

The role of plasma nonuniformity and (toroidal) equilibrium geometry iscrucial. Difficult problem to solve even for linear stability analyses. Non-linear theory and simulation becomes very challenging.

In the local limit, ∇ ⇒ ik ⇒ local ITG dispersion relation (k2⊥ ≫ k2‖)

[(

1− ω∗pi

ω

) ωdi

ω−(ω∗ni

ω+TiTe

)

−(

1− ω∗pi

ω

)

k2⊥ρ2i

]

+k2‖v

2ti

ω2

(

1− ω∗pi

ω

)

= 0

Recovers the sound wave (SW) in the uniform plasma limit. E: Show this!

Fulvio Zonca


Assume cold ions, Ti = 0, no curvature and finite density gradient, withω∗ne = −(Te/Ti)ω∗ni, c

2s = Te/mi and ρ

2s = c2s/Ω

2i ⇒ electron DW

ω∗ne

ω− 1− k2⊥ρ

2s +

k2‖c2s

ω2= 0 ⇒ ω =

ω∗ne ±√

ω2∗ne + 4(1 + k2⊥ρ

2s)k

2‖c

2s

2(1 + k2⊥ρ2s)

Fulvio Zonca


Assume ω∗ni = k⊥ρi = ωdi = 0 and strong pressure gradient |ω∗pi/ω| ≫ 1 ⇒ITG mode

1 +k2‖c

2s

ω2

ω∗pi

ω= 0 ⇒ 3 roots

ω =(−k2‖c2sω∗pi

)1/3 ⇒ neutrally stable!

ω =(−k2‖c2sω∗pi

)1/3

(

−1± i√3

2

)

⇒ one unstable root!

More generally, in toroidal geometry, the situation is more complex: beyondthe scope of the present lecture.

Modes with higher ω∗pi grow faster ⇒ increased transport for short wave-length modes (λ⊥<∼ ρi) ⇒ micro-instabilities.

Growth rates of the order of the mode frequency ⇒ broad band turbulencespectrum yielding turbulent transport.

Fulvio Zonca


Fluid and kinetic instabilities

Drift waves (DW) are essentially e.s. and result from the effect of diamag-netic drifts and magnetic field geometry in nonuniform toroidal plasmas offusion interest.

DW are characterized by short wavelength (λ⊥<∼ ρi) ⇒ Need of kineticdescriptions.

Kinetic descriptions are also needed for proper treatment of Landau damp-ing (MASTER L10). DW, as modification of the SW branch, are affectedby Landau damping ⇒ Additional motivation for DW to prefer short wave-lengths.

In addition to ITG, other DW are important for determining turbulenttransport in tokamaks: TEM and ETG (p.14; beyond the scope of this lec-ture). For these DW turbulences, all considerations above apply in general.

Fulvio Zonca


From MHD, we know of another branch, the shear Alfven wave (SAW;p.5), which is nearly incompressible (unlike the fast magneto-acoustic wave;MASTER L5 and L6).

SAW are less affected by Landau damping than SW, since δE‖ ≃ 0 andfrequency is generally (but not always) higher ⇒ SAW can produce bothdrift Alfven turbulence, as well as longer wavelength fluctuations,which caninteract with and be destabilized by energetic particles that are presentin burning plasmas: as charged fusion products and/or from additionalheating/current-drive (MASTER L1, L10).

Fulvio Zonca


2. Energetic particles and Shear Alfven Waves

Possible detrimental effect of shear Alfven instabilities on energetic ions infusion plasmas was recognized theoretically before experimental evidencewas clear

• SAW have group velocity ≃ vA ‖ B and of the same order of EPcharacteristic speed, vE ≈ vA

• Mikhailovskii 75 and Rosenbluth and Rutherford 75 conjecture theSAW excitation by resonant wave-particle interaction with MeV ions

• Topic of Lecture 2, Lecture 3 and Lecture 4 of theLecture Series-Winter 2013

Main interest in the 60’s focused on the (electron) beam plasma system:O’Neil, Malmberg, Mazitov, Shapiro, Ichimaru...

Fulvio Zonca

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





The beam-plasma system vs. EP-SAW interac-

tions in tokamaks

Similarities can be drawn but strong differences and peculiarities emergedepending on the strength of the drive:

• Advantages of using a simple 1-D system for complex dynamics studies

• Roles of mode structures, non-uniformity and geometry in determin-ing nonlinear behaviors (Lecture 5 and Lecture 6 of L.S.-Winter 2013)

Fulvio Zonca





Conceptual breakthrough

From p.25: SAW are less affected by Landau damping than SW, since δE‖ ≃0 and frequency is generally (but not always) higher: ω2 = k2‖v

2A ≫ k2‖v

2ti for

βi = 2v2ti/v2A ≪ 1 ⇒ SAW can produce both drift Alfven turbulence, as well

as longer wavelength fluctuations,which can interact with and be destabi-lized by energetic particles that are present in burning plasmas: as chargedfusion products and/or from additional heating/current-drive (MASTERL1, L10).

Drift Alfven waves at short wavelength (λ⊥<∼ ρi), as DW (p. 23), may havegrowth rates of the order of the mode frequency ⇒ broad band turbulencespectrum yielding turbulent transport.

However, longer wavelengths (λ⊥>∼ ρE; energetic particle Larmor radius)are also easily excited by energetic particles (see above).

Fulvio Zonca


Energetic particle driven SAW have longer wavelength and typically smallergrowth rate, |γL/ω| ≪ 1, than drift Alfven turbulence ⇒ Important role ofresonant particle transport (MASTER L8).

Roles of the continuous SAW spectrum and self-consistent interplay of en-ergetic particle transport and mode nonlinear dynamics. ⇒ Part II.

Fulvio Zonca


SAW equation in low-β nonuniform plasmas

Express perturbed magnetic field and velocity in terms of a vector and scalarpotential (|(ω/c)δA⊥| ≪ |∇δφ|):

δB = ∇× δA

δv⊥ =c

B2B ×∇⊥δφ

Derive SAW equation from: i) quasineutrality condition ∇ · δJ = 0 and ii)parallel Ampere’s law

~∇ · δ ~J⊥ + ~B · ~∇(δJ‖/B

)= 0

For SAW δA ≃ δA‖b (E: show it!) k2‖ ≪ k2⊥ (see later)

δJ‖ ≃ −(c/4π)∇2⊥ δA‖

Fulvio Zonca


The perpendicular current is obtained from the perpendicular momentumequation

δJ⊥ =c

B2B ×

d

dtδv⊥ +

c

B2B ×∇δp +

+J‖BδB⊥ − (δB‖B)

B2J⊥

Neglect the coupling with FMW/CAW 4πδp+BδB‖ = 0 (E: show it!) andconsider parallel Ohm’s law, −b ·∇δφ−(1/c)∂δA‖/∂t = 0. Then substituteinto ∇ · δJ = 0 (E: fill in missing algebra) and obtain (κ = b ·∇b)

∇ ·(ω2

v2A∇⊥δφ

)

+(

B0 · ∇

)[ 1

B20

∇2⊥

(

B0 · ∇

)]

δφ−

−4πγP0

B20

[(

B0×2κ

B0

)

·∇⊥

]2

δφ+4π[(

B0×2κ

B20

)

·∇⊥

][(

B0×∇P0

B20

)

·∇⊥

]

δφ = 0

Fulvio Zonca


Take the limit P0/B20 → 0

∇ ·(ω2

v2A∇⊥δφ

)

+(

B0 · ∇

)[ 1

B20

∇2⊥

(

B0 · ∇

)]

δφ = 0

In a 1D equilibrium (cylinder or slab), take the Fourier decompositionδφ = exp (ikzz − imθ) δφm(r). For ξm(r) = mc/(rωB0)δφm(r), this equa-tion takes the form of the Hain-Lust equation (K Hain and R Lust 1958, Z.Naturforsch. 13a 956)

d

dr

(

α(r − r0)d

dr

)

ξm(r) + βξm(r) + γ(r − r0)ξm(r) = 0

For r → r0, ω2 − ω2

A(r) ≃ α(r − r0) → 0, ω2A(r) = k2‖(r)v

2A(r). Where the

2nd order ODE sees the highest order derivative vanish, we expect boundarylayer with singular structures in the fluctuating field.Sign of SAW continuous spectrum.E: Show that boundary layer appears when r → r0.

Fulvio Zonca


Canonical (Fourier) form of the Hain-Lust equation; ω2A(r) = k2‖(r)v

2A(r)

1

r3d

dr

(

r3(ω2A(r)− ω2)

d

dr

)

ξm(r)−m2 − 1

r2(ω2

A(r)− ω2)ξm(r) = 0

Solution is singular near r0 where ǫA(r0) = ω2A(r0) − ω2 = 0. So ξm(r)

(radial displacement) has two scale: fast (singular) x = (r − r0) and slow(equilibrium) r, with |∂x| ≫ |∂r| formally

Near r0, ǫA = ǫ′A(r)x and lowest order solution is (ξm = ξ(0)m + ξ

(1)m + . . .)

ξ(0)m = C0(r)/ǫ′A(r) lnx

At lowest order ∇ · ξ(0)m = 0; thus,

ξ(0)mθ = −i r

m∂xξ

(0)m = −i rC0(r)

mǫ′A(r)x

Fulvio Zonca


Initial value form of Hain-Lust equation; localized solutions r2/m2|∂2x| ≫ 1

∂

∂r

(

ω2A(r) +

∂2

∂t2

) ∂

∂rξ(r, t) = 0

Solution is found as ∂rξ(r, t) = C0(r) exp(±iωA(r)t). Time asymptotically,∂x ≃ ±iω′

A(r)t. Thus,

ξ(r, t) = ∓i C0(r)

ω′A(r)t

exp(±iωA(r)t)

Poloidal displacement oscillates indefinitely. From ∇ · ξ = 0

ξθ(r, t) =i

kθ∂rξ(r, t) = i

C0(r)

kθexp(±iωA(r)t)

Fulvio Zonca


Satellite measurements of SAW

Engebretson et al. 1987

Fulvio Zonca


Torus: Shear Alfven continuous spectrum with

gaps

1.50.5-0.50

1

gap

ω

nq-m

ω o

+

-

ω

ω

4

2

2

2

2

SAW continuum damping∝ ∆r(d/dr)ωA (Chen and Hasegawa1974), with ∆r the typical modewidth

SAW are most easily exited near(d/dr)ωA = 0 (SAW continuum ac-cumulation points)

Equilibrium non-uniformity createthe local potential well for boundstates to exist

Frequency gap is due to to formationof standing waves by two degeneratecounter-propagating SAW: k‖,m+1 =−k‖,m, i.e. nq − (m + 1) = −nq +m ⇒ q = (2m + 1)/(2n). (Kierasand Tataronis 1982)

Fulvio Zonca


The Alfven Wave Zoology (Heidbrink 2002)

EPM

Various non-uniformity effects al-low “different varieties” of the sameSAW “species” to exist (Zoology;Heidbrink 2002).

Unique and general theoreticalframework for explanation of thisvariety and interpretation of obser-vations: the general fishbone-likedispersion relation (Zonca and Chen2006) ⇒ Lecture 3 in theLecture Series-Winter 2013

Fulvio Zonca




Typical space time scales of low frequency plasma

waves

Consider a magnetized plasma with a sheared magnetic field: 2D equilibrium

Magnetic shear ⇒ k‖ = k‖(ψp); ψp ≡ magnetic flux.

In order to minimize kinetic damping mechanisms, compression and fieldline bending effects λ‖ ≈ L, with L the system size

Perpendicular wavelength λ⊥ ≈ Lp/n can be significantly shorter than thecharacteristic scale length of the equilibrium profile Lp for sufficiently highmode number n.

Using the ordering k‖/k⊥ ≪ 1 and k⊥Lp ≫ 1, the 2D problem of plasmawave propagation can be cast into the form of two nested 1D wave equations:parallel mode structure ⊕ radial wave envelope.

Fulvio Zonca


Fourier harmonics δφm(r, t) have two scale structures:

• ≈ (nq′)−1 due to −1<∼ k‖qR = (nq −m)<∼ 1: ‖ mode-structure

• ≈ LA ≪ Lp due to equilibrium variation: radial envelope

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Eikonal Ansatz for the radial envelope make it possible to solve the 2Dproblem of plasma wave propagation in the form of two nested 1D waveequations: provided (kr ≡ nq′θk)

∣∣∣∣

nq′θ′k(nq′θk)

2

∣∣∣∣≪ 1

2DPDE L(∂t, ∂r, ∂θ; r, θ)δφ = 0︷︸︸︷

symmetric ⇓1DODE L(∂t, ∂η, θk; r, η)A(r, t)δΦ(η, r, t) = 0

︷︸︸︷

symmetric︸︷︷︸

⇓∫ ∞

−∞

δΦ(η, r, t)L(∂t, ∂η, θk; r, η)A(r, t)δΦ(η, r, t)dη = 0

⇓D = δWf + δWk − iΛ ⇐ 1DΨDE D(∂t, θk; r)A(r, t) = 0

Fulvio Zonca


3. General (unified) kinetic energy principle

In general, demonstrate that the mode dispersion relation can be alwayswritten in the form of a fishbone-like dispersion relation (Chen et al 1984)

−iΛ + δWf + δWk = 0 ,

where δWf and δWk play the role of fluid (core plasma) and kinetic (fast ion)contribution to the potential energy, while Λ represents a generalized inertia term.

The generalized fishbone-like dispersion relation can be derived by asymp-totic matching the regular (ideal MHD) mode structure with the general(known) form of the SA wave field in the singular (inertial) region, as thespatial location of the shear Alfven resonance, ω2 = k2‖v

2A, is approached.

Examples are : Λ2 = ω(ω − ω∗pi)/ω2A for |k‖qR0| ≪ 1 and Λ2 = (ω2

l −ω2)/(ω2

u − ω2) for |k‖qR0| ≈ 1/2, with ωl and ωu the lower and upperaccumulation points of the shear Alfven continuous spectrum toroidal gap(Chen 94).

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δWf is generally real, whereas δWk is characterized by complex values, thereal part accounting for non-resonant and the imaginary part for resonantwave particle interactions with energetic ions.

The fishbone-like dispersion relation demonstrates the existence of two typesof modes (note: Λ2 = k2‖q

2R20 is SAW continuum; see later):

• a discrete gap mode, or Alfven Eigenmode (AE), for IReΛ2 < 0;

• an Energetic Particle continuum Mode (EPM) for IReΛ2 > 0.

For EPM, the iΛ term represents continuum damping. Near marginal sta-bility (Chen et al 84, Chen 94)

IReδWk(ωr) + δWf = 0 determines ωr

γ/ωr = (−ωr∂ωrIReδWk)

−1(IImδWk − Λ) determines γ/ωr

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For AE, the non-resonant fast ion response provides a real frequency shift,i.e. it removes the degeneracy with the continuum accumulation point,while the resonant wave-particle interaction gives the mode drive. Causalitycondition imposes

• δWf + IReδWk > 0 when AE frequency is above the continuum accu-mulation point: inertia in excess w.r.t. field line bending

Λ2 = λ20(ωℓ − ω) ; ω > ωℓ ⇒ Λ → −i√−Λ2

• δWf + IReδWk < 0 when AE frequency is below the continuum accu-mulation point: inertia in lower than field line bending

Λ2 = λ20(ω − ωu) ; ω < ωu ⇒ Λ → i√−Λ2

Fulvio Zonca


For AE, iΛ represents the shift of mode frequency from the accumulationpoint

For both AE and EPM, the SAW accumulation point is the natu-ral gateway through which modes are born at marginal stability (note,however, that unstable continuum may exist; see Lecture 4 in theLecture Series-Winter 2013).

For EPM, ω is set by the relevant energetic ion characteristic frequency andmode excitation requires the drive exceeding a threshold due to continuumdamping. However, the non-resonant fast ion response is crucially impor-tant as well, since it provides the compression effect that is necessary forbalancing the positive MHD potential energy of the wave.

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4. Qualitative estimate of fluctuation induced

transport

Assuming a turbulent bath of fluctuations δφ(k) (here k stands for both k

and ωk), the growth of δφ(k) due to the available free energy source (in theabsence of fluctuation induced transport), may be written as

d

dtδφ(k) = γ(k)δφ(k)

In the presence of fluctuations, turbulent transport competes with the fluc-tuation growth. Assuming that transport is diffusive, with effective diffusioncoefficient D, one may estimate the reduction to the fluctuation growth asDk2⊥. Thus,

d

dtδφ(k) =

(γ(k)−Dk2⊥

)δφ(k)

Fulvio Zonca


In stationary conditions, the previous equation allows to estimate the fluc-tuation induced transport coefficient as

D ∼ maxk

(γ(k)

k2⊥

)

mixing length estimate

From MASTER L7, this result could be obtained from a random walktransport process characterized by a typical step-size ∼ k−1

⊥ and a time-step ∼ γ(k)−1.

This picture is by far too simplistic:

• Mixing length estimate predicts transport only in regions that areunstable to fluctuations: this is in contrast with experimental obser-vations and numerical simulation results

• Transport in stable regions may be due to turbulence spreading: longtime scale behaviors in burning plasmas and complex systems

• Special role of zonal structures: linearly stable structures with poloidaland toroidal symmetric nature ⇒ nonlinear equilibria

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5. The effect of zonal structures on turbulent

transport

Zonal structures are toroidally and poloidally symmetric structures: notablyzonal flows, zonal currents, corrugation of radial profiles.

⇒

Zonal Flow Zonal Current[More generally: phase-space zonal structures]

• Zonal structures scatter turbulence to shorter wavelength (stable) spectrum ⇒Mechanism for regulation of turbulence fluctuation level and turbulent transport.

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Zonal Flows (ZF) are common in plasmas

Zonal Flows on Jupiter

Drift Waves

Drift waves+

Zonal flows

Paradigm ChangeP.H. Diamond, et al. 2005

PPCF 47, R35

ZFs peculiarities

• No direct radial transport

• No linear instability

• Turbulence driven

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ZFs regulate turbulence: effect on transport

Z. Lin, et al. 1998, Science 281, 1835

Transport: local process predicted to scale as ∝ I: confirmed by numericalsimulations (Z. Lin, etal. 1999, PRL83, 3645; 2004, PoP11, 1099)

Drift wave intensity is determined by globalequilibrium properties: turbulence spreading (X.Garbet, et al. 1994, NF 34, 963; P.H. Diamond,et al. 1995, PoP 2, 3685)

Any size scaling of turbulent transport can bereduced to dependence of I on L.

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6. Resonant particle transport: the fishbone case

Experimental observation of fish-bones in PDX [McGuire et al. 83]with macroscopic losses of ⊥ in-jected fast ions ...

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Followed by numerical simulation of the mode-particle pumping (secular)loss mechanism [White et al 83] ...

... and the theoretical explanation of the resonant internal kink excitationby energetic particles and the (model) dynamic description of the fishbonecycle [Chen, White, Rosenbluth 84]

Fulvio Zonca


7. Coherent self-consistent nonlinear interactions

Within the approach of p.40, it is possible to systematically generate stan-dard NL equations in the form (expand wave-packet propagation aboutenvelope ray trajectories) of nonlinear Schrodinger equations:

D(ω + i∂t, r, ∂r)A(r, t) = NL TERMS

drive/damping︸︷︷︸

potential well︸︷︷︸

ω−1∂t −γ

ω− ξ

nq′θk∂r + i(λ+ ξ) + i

λ

(nq′θk)2∂2r

A(r, t) = NLTERMS

︷︸︸︷

group vel.︷︸︸︷

(de)focusing

θk solution of DR(r, ω, θk) = 0 and

λ =

(θ2k2

)∂2DR/∂θ

2k

ω∂DR/∂ω; ξ =

θk(∂DR/∂θk)− θ2k(∂2DR/∂θ

2k)

ω∂DR/∂ω; γ =

−DI

∂DR/∂ω

Fulvio Zonca

MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture 11 – 53Zonca et al. IAEA, (2002)

Avalanches and NL EPM dynamics|φm,n(r)|

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

x 10-3

r/a

8, 4 9, 4

10, 411, 412, 413, 414, 415, 416, 4

- 4

- 2

0

2

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

δαH

r/a

= 60.00t/τA0

X1t=60

NL distor tion of free ener gy SR C

|φm,n(r)|

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

x 10-2

r/a

8, 4 9, 4

10, 411, 412, 413, 414, 415, 416, 4

- 4

- 2

0

2

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

δαH

r/a

= 75.00t/τA0

X10t=75

|φm,n(r)|

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

.001

.002

.003

.004

.005

.006

.007

.008

.009

r/a

8, 4 9, 4

10, 411, 412, 413, 414, 415, 416, 4

- 4

- 2

0

2

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

δαH

r/a

= 90.00t/τA0

X30t=90

Importance of toroidal geometry on wave-packet propagation and shape

Fulvio Zonca

MASTER ENEA Modulo 1 Fisica del Plasma. Confinamento Magnetico Lecture 11 – 54Vlad et al. IAEA-TCM, (2003)

Propagation of the unstable front

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0 50 100 150 200 250 300

rmax

[d(rnH

)/dr]

t/τAlinear

phase convectivephase

diffusivephase

0.025

0.030

0.035

0.040

0.045

0.050

0.055

0.060

0 50 100 150 200 250 300

[d(rnH

)/dr]max

t/τAlinear

phase convectivephase

diffusivephase

Gradient steepening and relaxation: spreading ... similar to turbulence

Fulvio Zonca


EPM solitons and convective amplification

Detailed analyses are given in Lecture 6 in the Lecture Series-Winter 2013.

Look for solutions that can be expressed as the convectively amplified prop-agating (self-similar) wave-packet

A(ξ, t) = U(ξ)e∫t γ(t′)dt′ ≡ W (ξ)eiϕ(ξ)+

∫t

0γ(t′)dt′ ,

ξ − ξ0 ≡kn0|skϑ|

(x− x0) ≡kn0|skϑ|

(

x− |skϑ|∫ t

0

vg(t′)dt′

)

,

with kn0 denoting the nonlinear wave-vector and vg the nonlinear groupvelocity.

Adopt the general procedure for isolating the behavior of the wave-packetsoliton; i.e., balance the nonlinear term with the linear dispersiveness.

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This optimal balance fixes kn0vg, while the nonlinear group velocity is setby maximum wave-particle power transfer.

k2n0v2g =

3π(r/R0)1/2αH

2√2κ(s)

ω0

ωdF

k4ϑv2Eρ

2LE exp

(

2

∫ t

0

γ(t′)dt′)

,

The shape U(ξ) of the self-similar wave-packet satisfies the NL equation

∂2ξU = λ0U − 2iU |U |2 .

The value λ0 ≃ −0.47+ i1.32 correspondsto the ground state of the correspondingcomplex nonlinear oscillator.

Fulvio Zonca


On the nonlinear Schrodinger equation

The EPM soliton formation and convective wave-packet amplification yield

∂2ξU = λ0U − 2iU |U |2 .

This is similar but not the same as the equation a nonlinear oscillator inthe so-called “Sagdeev potential” V = (−U2 + U4)/2, which generates theequation of motion

∂2ξU = U − 2U3 ,

and gives U = sech(ξ).

This is the solution that appears in the problem of ITG turbulence spreadingvia soliton formation [Z. Guo et al PRL 09].

Fulvio Zonca


More generally, this form appears in soliton-like solutions of the nonlinearSchrodinger equations; e.g.

• the Gross-Pitaevsky equation describing the ground state of a quan-tum system of identical bosons [Gross 61; Pitaevsky 61]

• the envelope of modulated water wave groups [Zakharov 68]

• the propagation of the short optical pulse of a FEL in the superradiantregime [Bonifacio 90]

The complex nature of EPM equation is novel and connected with the pecu-liar role of wave-particle resonances, which dominate the nonlinear dynamicsof EPMs via resonant wave-particle power exchange, whose maximizationyields two effects:

• the mode radial localization, similar to the analogous mechanism dis-cussed for the linear EPM mode structure

• the strengthening of mode drive Imλ0 > 0, connected with the steep-ening of pressure gradient, convectively propagating with the EPMwave-packet

Fulvio Zonca


References and reading material

J.P Freidberg, “Ideal Magneto Hydrodynamics”, Plenum Press, 1987

L. Chen, “Waves and Instabilities in Plasmas” World Scientific Publication Co.,1987

R. B. White, “Theory of Tokamak Plasmas”, North Holland, 1989

T.M. O’Neil, Phys. Fluids 8, 2255 (1965).

T.M. O’Neil and J.H. Malmberg, Phys. Fluids 11, 1754 (1968).

T.M. O’Neil, J.H. Winfrey and J.H. Malmberg, Phys. Fluids 14, 1204 (1971).

M.J. Engebretson et al., J. Geophys. Res. 92, 10053 (1987).

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

C. E. Kieras and J. A. Tataronis, J. Plasma Phys. 28, 395 (1982).

W. W. Heidbrink, Phys. Plasmas 9, 2113 (2002).

Fulvio Zonca


F. Zonca and L. Chen, Plasma Phys. Control. Fusion 48, 537 (2006).

L. Chen, Phys. Plasmas 1, 1519 (1994).

R.B. White, R.J. Goldston, K. McGuire K. et al., Phys. Fluids 26, 2958, (1983)

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

S. T. Tsai and L. Chen, Phys. Fluids B 5, 3284 (1993)

F. Zonca and L. Chen, Phys. Plasmas 3, 323, (1996)

F. Zonca, S. Briguglio, L. Chen, G. Fogaccia and G. Vlad, Nucl. Fusion 45, 477,(2005)

F. Zonca, S. Briguglio, L. Chen, G. Fogaccia and G. Vlad, “Theoretical Aspectsof Collective Mode Excitations by Energetic Ions in Tokamaks”; in Theory ofFusion Plasmas, pp. 17-30, J.W. Connor, O. Sauter and E. Sindoni (Eds.), SIF,Bologna, (2000).

Z. Guo, L. Chen and F. Zonca, Phys. Rev. Lett. 103, 055002 (2009).

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F. Zonca and L. Chen, Phys. Plasmas 7, 4600 (2000).

E. P. Gross, Nuovo Cimento 20, 454 (1961).

L. P. Pitaevsky, Sov. Phys. JETP 13, 451 (1961).

V. E. Zakharov, J. Appl. Mech. Tech. Phys. 9, 190 (1968).

R. Bonifacio, L. De Salvo, P. Pierini and N. Piovella, Nucl. Instrum. MethodsPhys. Res., Sect. A 296, 358 (1990).

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