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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].
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)
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.
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.
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.
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 ·∇.
Perpendicular ion dynamics: the equilibrium particle drifts are diamagneticand magnetic (κ, ∇B0). Perturbed (fluctuation induced) particle drifts areδE ×B0, diamagnetic and polarization drifts
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.
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).
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.
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!
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.
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).
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...
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).
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.
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)
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.
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)
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
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.
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)
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).
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)
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
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.
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,
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
• Zonal structures scatter turbulence to shorter wavelength (stable) spectrum ⇒Mechanism for regulation of turbulence fluctuation level and turbulent transport.
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]
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
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].
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
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).