ocw.snu.ac.krtopics in fusion and plasma studies 459.666a 004 part ii. plasma turbulence and...
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Topics in Fusion and Plasma Studies
459.666A 004
Part II. Plasma Turbulence and Turbulent Transport
T.S. HahmDepartment of Nuclear Engineering
Seoul National University
Fall 2012
References:
• J. Wesson, Tokamaks
• R. J. Goldston and P. H. Rutherford, Introduction to Plasma Physics
Topics:
• Microinstabilities in tokamaks
• Anomalous transport
Fluid Approach Kinetic approach
− Linear − Linear
−Nonlinear −Nonlinear
0. Magnetic confinement (Sept. 3)
⇒ Ti(r), Te(r), n(r) are radially inhomogeneous.
• ∇Ti, ∇Te, ∇n act as an expansion free energy to drive⇒ any physical system tends to have homogeneous physical quantities
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⇒ various waves unstable⇒ instabilities, with various wavelengths and frequencies
• Each one has its own characteristics and conditions for excitation.
• It’s unlikely to suppress (make stable) all instabilities simultaneously.
This is the reason why tokamak transport rate is higher than prediction basedon Coulomb collisions (in toroidal geometry, called “neoclassical theory”), even inthe absence of large scale MHD instabilities.
1. Generic features of tokamak microturbulence
• “δn” density fluctuations are commonly observed from all tokamaks.
• There’s a general trend that:
– When δn, χe, D and τE .
– When δn, χe, D and τE .
(χe: electron thermal diffusivity, D: particle diffusivity, τE : energy confinement)δn is easier to measure than other parameters.
|e| δφ/Te measured frequently at edge using Langmuir probe (cf. at core HeavyIon Beam Probe, but expensive diagnostic).
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δTe/Te sometimes measured at core and edge (Electron Cyclotron Emission, spec-troscopy).
If one constructs a contour of ne(r) (level curve), we can see the evolution of“turbulent eddys”. This is a self-organized nonlinear structure which originatesfrom specific instabilities with wave-like characters.
Wavelengths of instabilities (≈ eddy size):
λr '2π/kr
λθ '2π/kθ
If “∆t” large ⇒ coherent structure (not called turbulent structure).From measurements (microwave scattering, beam emission spectroscopy):
*Until ’90, using microwave scattering one could see only high-k part.(Beam emission spectroscopy: R. Fonck ’93.)
Frequency spectrum of fluctuations: δnk,ω ∼ e−iωt+ik·xIf a fluctuation satisfies a linear dispersion relation ω = ω(k):
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But experiments show that: S(ω) has a broad peak (frequency broadening).“∆ω” significant ⇒ we should face nonlinearity.
Note Weak turbulent theory is nice since it assumes small higher order effects sothat one just add higher terms after 0th and 1st order terms. But now we shouldconsider strong turbulence theory.
2. Examples of Basic Microinstabilities (Sept. 5)
Consider a uniform magnetic field B = B0z, nonuniform density profile n0 =n0(x), periodic system in y. Let’s consider uniform temperatures for simplicity.In this simple geometry, any perturbed quantities can be Fourier-decomposed in
y and z directions. E.g.,
δφ(x, y, z) =∑k,ω
δφk,ω(x) exp i(kyy + kzz − ωt)
δn(x, y, z) =∑k,ω
δnk,ω(x) exp i(kyy + kzz − ωt)
We’ll pursuing a local theory at first (at one point in x).
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2.1. Electron Drift Wave
Electron drift wave was discussed seriously in theoretical community, since it canbe driven only by density gradient and can be easily unstable.Let’s search for an “electrostatic” (i.e. ∇×E = 0⇒ E = ∇φ) wave with a phasevelocity satisfying vT i ω/kz vTe. Here vTe =
√Te/me, vT i =
√Ti/Mi, kz =
k‖ ≡ B · k/ |B|.
A.Electron response
Since electrons move fast (can cover the system size during one wave period!k‖vTe ω), we can consider them in a thermal equilibrium in the presence ofelectrostatic fluctuation δφ.
Maxwell-Boltzmann Statistics⇒ fe(E) ∝ exp (−E/Te) = exp (−(mev
2 − |e| δφ)/Te)⇒ ne =
∫d3vfe(E) = ne0 exp (|e| δφ/Te) : Boltzmann relation
“δne = ne − ne0 = ne0
[1 + |e|δφ
Te+O
(( |e|δφTe
)2)− 1]
= ne0|e|δφTe
”
δne/ne0 = |e| δφ/Te : Electrons obey Boltzmann response.
This Boltzmann response is also called the adiabatic response. “Adiabatic” hererefers to a slow time variation of a wave.From a fluid description;
mened
dtue = −ne |e|
(E +
1
cue ×B
)−∇pe
Linearize (assuming ue0 = 0):
mene∂
∂tδue = −ne |e|
(δE +
1
cδue ×B
)−∇δpe
Take b· and ignore electron inertia; me → 0 (recall vTe =√Te/me very fast)
⇒ |e|n0∇‖δφ− Te∇‖δne = 0
δne/ne0 = |e| δφ/Te (1)
(∇‖ = b · ∇ = ∂/∂z, we assumed uniform Te ⇒ δpe = Teδne)
B. Ion Response
Ions which satisfy ω/k‖ vT i, we further assume “cold ions”,
i.e. k⊥ρi 1 (ignore FLR effect), but Ti Te ⇒ k⊥ρs ∼ 1 .
Here ρi = vT i/Ωci, Ωci ≡ |e|B0/Mic, ρs = cs/Ωci =√Te/Tiρi, cs =
√Te/Mi.
k⊥ = ky in this simple system. In tokamak, Ohmically heated, or ECR heatedplasma satisfy Ti Te, but here just for simplicity.In this situation, most of ions move slowly enough. From the wave’s point of view,
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they more or less move together like a fluid.
mened
dtui = −ni |e|
(E +
1
cui ×B
)−∇pi
Linearize and drop ∇pi (cold ions!) (with ui0 = 0)
⇒Mini∂
∂tδui = ni |e|
(δE +
1
cδui ×B
)(2)
along B⇒
Min0∂
∂tδui‖ = n0 |e| δE‖ = −n0 |e| ∇‖δφ (3)
across B⇒we can solve Equation (2) via iteration knowing (or assuming) ω/Ωci 1.(We are dealing with “low frequency” microinstabilities.)
(ω/Ωci ∼ ω/ |e|BMic∝ |Mi/e| 1)
1st order : RHS=0.
δE +1
cδu
(1)i ×B = 0
⇒ δu(1)⊥ = δuE =
cb×∇δφB
(x-direction here)
2nd order :
Min0∂
∂tδuE = n0 |e|
1
cδu
(2)⊥ ×B (4)
δu(2)⊥ = δupolarization drift =
Mic2
|e|B2
∂
∂tE⊥ = −Mic
2
|e|B2
∂
∂t∇⊥δφ (5)
(1) ⇒
δnen0
=|e| δφTe
(3), (4), (5) with ∂∂tni +∇ ·
(niui
)= 0 ⇒ linearize to get
∂
∂tδni + δuE · ∇n0 + n0∇ · δupol + n0∇‖δui‖ = 0
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Here you can check ∇ · δuE and other contributions are even smaller or vanish.Fourier decompose, i.e. ∼ exp (kyy + kzz − ωt) ⇒
∂
∂t→ −iω, ∇‖ → ik‖, ∇⊥ → ik⊥,but ∇n = −x n
Ln
Here Lnk⊥ 1, i.e. (system size) (⊥ wave size), and therefore δupol ·∇n0 termis dropped in the linearized continuity equation.
⇒ δnin0
=(ω∗eω
+k2‖c
2s
ω2− k2⊥ρ
2s
) |e| δφTe
Here the 1st term on RHS is from δuE · ∇n0, and the 2nd term and the 3rd termare from ∇‖δui‖ and ∇ · δupol.For long enough wavelength λ λDe, the Poisson equation can be approximatedby the quasi-neutrality equation : δne = δni⇒ we obtain the linear dispersion relation for the electron drift wave.
1 + k2⊥ρ
2s −
ω∗eω−k2‖c
2s
ω2= 0
Here ω∗e = kyρsLncs = kyv∗e is electron diamagnetic frequency, where v∗e is electron
diamagnetic drift velocity.(cf. the notation vde could be confused with ∇B or curvature drift.)
2.2. Electron Drift Wave in Uniform Magnetic Field (Sept. 10)
Linear dispersion relation:
1 + k2⊥ρ
2s −
ω∗eω−k2‖c
2s
ω2= 0
Here, k2⊥ = k2
x + k2y, B = B0z and the diamagnetic drift frequency is ω∗e ≡
(kyρs/Ln) cs = kyv∗e. We assumed
δφ (x, y, z) =∑k,ω
δφk,ω (x) exp i (kyy + kzz − ωt)
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and used the WKB approximation,
δφk,ω (x) = δφ (x) exp i
∫ x
kx (x) dx,
which is valid for kxLn 1. Here, the eikonal actor ei∫ x kx(x)dx captures the fast
variation in x and δφ (x) and kx (x) are slowly varying in x. To the lowest orderin the 1/kxLn expansion, there is only a local value in kx in the linear dispersionrelation.Let’s calculate the particle flux in the x direction carried by a drift wave.
Γptl = 〈δneδvx〉
Here, 〈. . . 〉 is an ensemble average, or a long time average. Practically, it’s replacedby an average over ignorable coordinate(s) (i.e., direction of symmetry).In this simple slab geometry, both y and z are ignorable coordinates. (In tokamakgeometry, only the toroidal angle “φ” is an ignorable coordinate.) The E×B driftis
δvx =cδEyB2
Bz = − c
B
∂
∂yδφ
and the density fluctuation is
δn =|e| δφTe
n0
Therefore,
Γptl = 〈δnδvx〉 = − c |e|BTe
⟨δφ
∂
∂yδφ
⟩= − c
2B
|e|Ten0
⟨∂
∂y(δφ)2
⟩= 0
with
〈. . . 〉 =1
Ly
∮ Ly
0dy (. . . )⇐ 1
2π
∮dθ (. . . ) rdθ = dy.
Therefore, electron drift waves (not instabilities) cannot carry any particle flux.The underlying reason is that δvx and δne are π/2 out of phase (because δne ∝ δφ).Also note that there’s no instability in this simple limit. Electron drift wavejust wobbles without growing in almplitude or driving a net particle flux. For aninstability and net particle flux, there should be a phase-shift between δne and δφ.
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Note We can controll only the initial condition of macroscopic quantities (like Ip,B, etc). Thus we hope an ensemble average can describe plasma phenomena. Butin nonlinear phenomena, like in chaotic system, slightly different initial conditioncauses very different result. ⇒ ansemble average method cannot be used.
2.3. Electron Drift Instability in Uniform Magnetic Field
Linear dispersion relation:
1−iδk,ω + k2⊥ρ
2s −
ω∗eω−k2‖c
2s
ω2= 0
δnen0
= (1−iδk,ω)|e| δφTe
δk,ω = δk,ω (k, ω) , |δk,ω| 1
where the minus sign comes from the inverse-dissipation of the electron drift wave(i.e., wave gain something) and δk,ω is the corresponding phase-shift.⇒ = (ω) ∝ δk,ωω∗e for (k⊥ρs)
2 1 and k2‖c
2s/ω
2 1.Then what mechanism then will lead to the inverse dissipation?
Electron drift instabilities are classified according to the specific inverse dissipationmechanism which destabilizes the electron drift wave.
• Collisionless (electron) drift instability (“universal instability” because onlyrequires n0 (x), B0z and electron Landau damping) γ > 0
Due to Inverse Landau damping of (passing) electrons ω/k‖ ∼ v‖:these can be discussed in the context of an uniform magnetic field.
• Collisional (resistive) drift instability γ < 0 (’78 ∼’79)
Due to Magnetic shear induced damping of drift waves (via ion Landaudamping): this requires introducing of sheared magnetic field (for simplemodel of Bp, B = B0[z + (x/Ls)y]).
• Trapped Electron(-driven electron instability) Mode (TEM) γ > 0
This requires treating magnetically trapped electrons (banana orbits) intoroidal geometry (resonance with precession of banana orbits ω ∼ kφvprec).
Furthermore, ∇B and curvature drift of ions will couple neighboring poloidal har-monics of drift waves and render magnetic shear induced damping of drift wavesineffective.
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2.4. Inverse Landau Damping in 1-D Plasma
(uniform n0, B, near-Maxwellian electron distribution fe)The net result is that the wave will lose energy (damping) by accelerating moreparticles than deccelerating. ⇒ “surfing” the wave.When can we get inverse Landau damping? With a Bump On (the distribution)Tail (∂fe/∂v > 0). ⇒ B.O.T. instability.
More particles losing energy than gaining energy from the wave. This is inverseLandau damping.For collisionless (universal) electron drift instability,
fe (x, v) ∝ n0 (x) e−v2/2v2Te
⇒ how on Earth can we get the inverse Landau damping?It comes from the nonuniform density n0 (x).
2.5. Drift-kinetic Equation
The drift kinetic equation is a simplification of the Vlasov equation for
ω
Ωce 1, k⊥ρe 1
(ρeλ⊥ 1
),
1
k⊥Ln 1
(λ⊥Ln 1
),ρeLn 1
in a strong magnetic field. (λ⊥ ∼ 1cm, Ln ∼ 102cm ∼ a, ρe ≪ 1mm)
• Electrons gyrate very very fast: Ωce ∼ 10GHz
• while drift waves oscillate slowly: ω ∼ ∆ω ∼ 100kHz
fe (x, y, z, vx, vy, vz, t)→ fe,gc(xgc, ygc, zgc, ξ, µ, v‖, t
)Gyrophase (ξ) angle dependance can be eliminated (not simply ignored).For ions, krρi ∼ 1 will modify the situation ⇒ gyrokinetic equation.
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2.5.1. Electron Drift-kinetic Equation (Sept. 12)
fe (x, y, z, vx, vy, vz, t)→ fe,gc(xgc, v⊥, v‖, t
)for uniform B. (xgc: position of guiding center, no gyro-angle dependance)Let’s consider a Maxwellian fgc,0
fgc,0 = n0 (x)
(m
2πTe
)3/2
exp
(−mv
2
2Te
)For an application for drift-waves, the existence of diamagnetic drift flow was es-sential : v∗ey = (ρs/Ln) csy.
Homework 1 Discuss how we can treat drift wave with a Maxwellian equilibriumdistribution fgc,0 which is symmetric in vy (Hint: Maxwellian for “guiding center”).Consider both electron and ion contributions.
2.5.2. Derivation of the Drift-kinetic Equation
The total number of electron guiding centers in a 6-D phase-space volume V is
Ne =
∫fed
3xd3v =
∫fedV
Here, fe is the guiding-center distribution function.In effect, for guiding centers in an uniform magnetic field,
dV = d3xgc2πv⊥dv⊥dv‖
The 2π factor is there because the gyro-dependance has been eliminated, and thesystem has a cylindrical symmetry in v-space, so (vx, vy, vz)⇒
(v‖, v⊥
).
Conservation of the number of guiding centers :
0 =d
dtNe =
∫ (∂fe∂t
)dV +
∫SfeU · dS
=
∫ (∂fe∂t
+∇ · (feU)
)dV
where the 2nd term is the flux out of the volume V through the phase-space surfaceS which bounds volume V . Since V is arbitrary, the integrand is zero:
∂fe∂t
+∇ · (feU) =∂fe∂t
+∇x · (xfe) +∇v · (vfe) = 0
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with
∇v · (vfe) =1
v⊥
∂
∂v⊥(v⊥ ˙v⊥fe) +
∂
∂v‖
(˙v‖fe
)˙xgc =
d
dtxgc = v‖z + vE×B
Here,
vE×B =cδE×B
B2=cb×∇δφ
B
(Note that ∇ · vE×B = 0 for uniform B: incompressible)There’s no ∇B and curvature drift, since we are assuming uniform B.Then, how about vpol?⇒ actually, in the modern derivation of drift-kinetic equation, vpol can be elimi-nated when eliminating gyrophase angle dependency. (In ancient years, pioneersof drift-kinetic equation prefered to put vpol term in their derivation.) Physically,we are considering just electron and vpol ∝ m, so vpol is neglected.
˙v⊥ = 0 (since B is uniform, the magnetic moment µ =˙(v2⊥2B
)= v⊥
B ˙v⊥ = 0).
This is the first adiabatic invariant which is a constant for
ω Ωce, ρe LB ≡∣∣∣∂|B|∂x
∣∣∣−1=∞ for uniform magnetic field.
For v, we only need to consider
v‖ =dv‖
dt=|e|me
δE‖ =|e|me
∂
∂zδφ
Therefore, the electron drift-kinetic equation with electrostatic fluctuation in anuniform magnetic field is given by
∂fe∂t
+
(v‖z +
cb×∇δφB
)· ∇fe +
|e|me
∂
∂zδφ∂fe∂v‖
= 0
Let’s linearize it around a maxellian f0: f = f0 + δfe
• 0th order, (∂
∂t+ v‖
∂
∂z
)f0 = 0 satisfied.
• 1st order,
∂
∂tδfe + v‖
∂
∂zδfe +
cb×∇δφB
· ∇(f0 +δfe
)+|e|me
∂
∂zδφ
∂
∂v‖
(f0 +δfe
)= 0
The E×B nonlinearity and the velocity-space nonlinearity are neglected.
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Therefore, the linear kinetic electron response is given by “δfe”.Since δφ, δne, δfe · · · ∝ e−iωt+ik·x, the linear (unperturbed) propagator is
−i(ω − k‖z
)δfe = −cz
B× y ∂
∂yδφ · x∂f0
∂x− i |e|
mek‖δφ
∂f0
∂v‖
= +c
Bikyδφ
∂n0
∂x
f0
n0− i |e|
mek‖δφ
(−v‖
v2Te
)f0
(Note that f0 only depends on x in real space.)The 1st term is the relaxation of expansion free energy in configuration space, andthe 2nd is a “heating term”. For non Maxwellian (e.g. B.O.T.), this correspondsto the relaxation of velocity space free energy.Therefore, with k‖ ≡ kz,
δfe =kyv∗e − k‖v‖ω − k‖v‖
|e| δφme
f0 =
(1− ω − ω∗e
ω − k‖v‖
)|e| δφTe
f0
δne =
∫d3vδfe =
|e| δφTe−∫d3v
(ω − ω∗eω − k‖v‖
)f0|e| δφTe
Defining 1-D distribution function F0
(v‖)dv‖ = f0
(x, v‖
)2πv⊥dv⊥dv‖,
the 2nd term contains ∫d3v
f0
ω − k‖v‖=
∫ ∞−∞
dv‖F0
(v‖)
ω − k‖v‖
with
F0
(v‖)
= n0
(m
2πTe
)3/2
exp
(−mv2‖
2Te
)How should we treat an apparent singularity at v‖ = ω/k‖?
Following Landau’s prescription, we’ll evaluate the integral as if ω had a real andimaginary part ω ⇒ ω + iε ≡ ω + i0+ (|ε| 1, ε > 0). Therefore, in a complexv‖-plane, we’ll chose a contour which passes below the pole at v‖ = ω/k‖.(This is true for k‖ > 0 which is a usual convention.)
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∫ ∞−∞
dzf (z)
z − i0+= Pr
∫ ∞−∞
dzf (z)
z+ iπf (0)
1
z − i0+= Pr
(1
z
)+ iπδ (z)
(Pr: principal value)
Physical interpretation: since δφ ∝ e−iωt, for ω = ω + iε, δφ ∝ e−iωteεt.For t→ −∞, this is extremely small.Justifying linearization: on the other hand, t → ∞ is irrelevant since we facenonlinearities at finite time (present even if δne/ne ∼ 10−2).
2.6. Electron Response to Drift-wave Fluctuation,Including Contribution from Wave-particle Interaction (Sept. 17)
Contribution from the pole at vz = ω/k‖ according to Landau’s prescription:
• Resonant part:
Res
∫ ∞−∞
Fe0 (vz) dvzω − k‖vz
= − iπ∣∣k‖∣∣Fe0 (vz)
= −i(π
2
)1/2 ne0∣∣k‖∣∣ vTe exp
(− ω2
2k2‖v
2Te
)
' −i(π
2
)1/2 ne0∣∣k‖∣∣ vTe• Principal value of the integral (from the rest of the real axis):
Pr
∫ ∞−∞
Fe0 (vz) dvzω − k‖vz
= Pr
∫ ∞0
Fe0 (vz) dvzω − k‖vz
+ Pr
∫ 0
−∞
Fe0 (vz) dvzω − k‖vz
= Pr
∫ ∞−∞
2ω
ω2 − k2‖v
2z
dvz ∼ O
(ne0ω
k2‖v
2Te
)
which is negligible.
Therefore,
δnene0
=|e| δφTe
[1 + i
(π2
)1/2 ω − ω∗e∣∣k‖∣∣ vTe]
where the first term corresponds to the adiabatic (Boltzmann) response and thesecond term to the non-adiabatic response (ω is linked to the gradient in velocityspace and ω∗e to the gradient in configuration space).
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Now, we can discuss the collisionless drift instability (26.34).
1 + k2⊥ρ
2s −
ω∗eω−k2‖c
2s
ω2= −i
(π2
)1/2 ω − ω∗e∣∣k‖∣∣ vTeIn most cases, k2
⊥ρ2s 1 and k‖cs ω
⇒ we can solve Eq. (26.34) perturbatively, with ω ' ω∗e.The small corrections to this are
ω ' ω∗e1− k2
⊥ρ2s
' ω∗e(1− k2
⊥ρ2s
)where k2
⊥ρ2s proceeds from the ion polarization drift with Mi me,
< (ω) ' ω∗e(1− k2
⊥ρ2s
)+k2‖c
2s
ω∗e
γ
ω∗e== (ω)
ω∗e=(π
2
)1/2 ω∗e∣∣k‖∣∣ vTe(k2⊥ρ
2s −
k2‖c
2s
ω2∗e
)For an instability, downward shift of ω below ω∗e is required. In this simple limit,
k2⊥ρ
2s
k2‖c
2s
ω2∗e
Note For wave-particle interaction to be important,
• The plasma needs to be collisionless enough( ∫∞−∞
Fe0(vz)dvzω−k‖vz−iνe
)• < (ω) = (ω)
Otherwise the instability is “reactive” (i.e. can be obtained from fluid description).
2.7. Physical meaning of i(π2
)1/2 “ω−ω′′∗e|k‖|vTe
term
Recall the expression for δfe from the drift-kinetic equation (26.20):
−i(ω − k‖vz
)δfe = −cδEy
B0
∂
∂xFe0 +
e
meδEz
∂
∂vzFe0
where the 1st term is proportional to ∂n0∂x and associated with ω∗e, and the 2 term
is proportional to k‖vz =(k‖vz − ω
)+ω after substracting the adiabatic response.
Note that
δvx =cδEyB0
= −ickyδφB0
dδvzdt
= −eδEzme
= iek‖δφ
me
Since δEy/δEz = ky/kz > 0, δvx and ddtδvy are 180 out of phase!
15
• For the first term:
– δvx > 0⇒ ddtδvz < 0, electrons lose energy
– δvx < 0⇒ ddtδvz > 0, electrons gain energy
– Since ∂∂xn0 (x) < 0, there are more electrons which lose energy to drift
wave ⇒ destabilizing! (∝ ω∗e)
• For the second term, since ∂∂vz
Fe0 < 0, the wave-particle response ends upheating electrons, i.e. drift waves gives energy to particles ⇒ stabilizing!(∝ ω)
In summary,
• k2⊥ρ
2s ⇒ ω , destabilizing
• k2‖c
2s ⇒ ω , stabilizing
And we have
ω < ω∗e ⇒ k2⊥ρ
2s >
k2‖c
2s
ω2∗e
Since
ω∗e =ρ2s
L2n
k2yc
2s
the RHS becomes (k‖
ky
)2L2n
ρ2s
Since k2⊥ρ
2s . 1 and Ln/ρs 1, we need
(k‖/ky
)2 1 to have a drift instability,i.e. λ‖ λ⊥.
2.8. Effect of Electron Temperature Gradient on Drift Instability(Sept. 19)
Now, we consider an equilibrium Maxwellian distribution
Fe0 = ne0 (x)
(m
2πTe (x)
)3/2
exp
[− me
2Te (x)
(v2⊥ + v2
‖
)](with Te = Te0)
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Then the expansion free energy is related to
∂
∂xFe0 = Fe0
∂
∂xlnFe0 =
Fe0ne0
dne0dx− Fe0
Te
dTedx
(3
2− v2
2v2Te
)=Fe0ne0
dne0dx
[1− ηe
(3
2− v2
2v2Te
)]where v2
Te = Te/me and
ηe ≡LnLTe
=1
Te
dTedx
(1
ne0
dne0dx
)−1
, LTe ≡ −1
Te
dTedx
Note ηe →∞ for a flat density and finite LTe .We can repeat the same calculation we did with an additional term related to ηe.
δfe =|e| δφTe
Fe0 −1
ω − kzv‖
ω − ω∗e
[1− ηe
(3
2− v2
2v2Te
)]|e| δφTe
Fe0
δnene0
=1
ne0
∫d3vδfe = . . .
=|e| δφTe− |e| δφ
Te
∫ ∞−∞
Fe0(v‖)dv‖
ω − kzv‖
ω − ω∗e
[1− ηe
(3
2−1− v2
2v2Te
)]
where the−1 comes from the integration∫∞
0 dv⊥v⊥ (. . . )[3/2−
(v2⊥ + v2
‖
)/2v2
Te
].
Here, Fe0 is a one-dimensional Maxwellian distribution. Evaluating a resonantcontribution from a pole at v‖ = ω/k‖ as before,
δnene0
=|e| δφTe
[1 + i
(π2
)1/2 ω − ω∗e (1− ηe/2)∣∣k‖∣∣]
Note dTedx does not affect ion dynamics!
Therefore, the linear dispersion relation is
1 + k2⊥ρ
2s −
ω∗eω−k2‖c
2s
ω2= i(π
2
)1/2 ω − ω∗e (1− ηe/2)∣∣k‖∣∣ vTeEvaluating = (ω) for ω/ω∗e ' 1− k2
⊥ρ2s + k2
‖c2s/ω
2∗e (ηe does not affect < (ω)),
we get γ (ω):
γ
ω∗e≡(π
2
)1/2 ω∗e∣∣k‖∣∣ vTe[k2⊥ρ
2s −
k2‖c
2s
ω2∗e− ηe
2
]as in Eq. (26.46) with typo for the 1/2 factor for ηe.In this case, ∇Te is a stabilizing influence on drift instability!
17
Here, we are only considering a specific (particular) example, in which ω/k‖ vTe(i.e. v‖resonant vTe) and ω ' ω∗e.
δne ∼ Res
∫ ∞−∞
dv‖1
ω − k‖v‖ne0
T1/2e (x)
(. . . )
ω − ω∗e
[1− ηe
(1
2−v2‖
2vTe
)]
where Fe0(v‖)∝ ne0/T 1/2
e (x) and v2‖/2v
2Te ∼ v2
‖resonant/2v2Te 1.
In this limit, ne0 (x) /T1/2e (x) is the effective density which characterizes the ex-
pansion free energy!
Of course, in other examples with different frequency ordering, the electron tem-perature gradient can destabilize a wave and excite an instability. e.g., for Elec-tron Temperature Gradient (ETG) instability (in a fluid limit: ω/k‖ & vTe) or IonTemperature Gradient (ITG) instability driven by ∇Ti (ω/k‖ & vT i).
2.9. Effect of Electron Current on Drift Instability
Now, we consider
Fe0 = ne0 (x)
(m
2πTe
)3/2
exp
[−me
2Te
(v2⊥ +
(v‖ − ue0
)2)]i.e. 1-D shifted Maxwellian. Here we ignore∇Te effect for simplicity, and Te = Te0.(The electron current is j‖ = − |e|ne0ue0 = − |e| 2π
∫∞0 dv⊥v⊥
∫∞−∞ dv‖Fe0.)
Then, in the electron drift equation,
∂
∂v‖Fe0 = −
v‖ − ue0v2Te
Fe0
We again repeat a similar calculation for δne.
δfe =|e| δφTe
Fe0 −ω − ω∗e − k‖ue0
ω − k‖v‖|e| δφTe
Fe0
Do 2π∫∞
0 dv⊥v⊥∫∞−∞ dv‖,
δnene0
=|e| δφTe− |e| δφ
Te
(ω − ω∗e − k‖ue0
) ∫ ∞−∞
Fe0(v‖)
ω − k‖v‖dv‖
Taking the resonant contribution from the integral, we obtain
δnene0
=|e| δφTe
[1 + i
(π2
)1/2 ω − ω∗e − k‖ue0∣∣k‖∣∣ vTe]
18
for ue0 vTe, since exp[−(v‖resonant − ue0
)2/2v2
Te
]' 1.
The linear dispersion equation becomes
1 + k2⊥ρ
2s −
ω∗eω−k2‖c
2s
ω2= −i
(π2
)1/2 ω − ω∗e − k‖ue0∣∣k‖∣∣ vTeNote the effect ω − k‖ue0 can be regarded as a Doppler shifted frequency in theframe moving with ue0.
γ
ω∗e'(π
2
)1/2 ω∗e∣∣k‖∣∣ vTe(k2⊥ρ
2s −
k2‖c
2s
ω2∗e
+k‖ue0
ω∗e
)
With our sign convention (k‖ > 0, ue0 > 0), ue0 is destabilizing.
Homework 2 Problem 26.3 and 26.4 of Goldston and Rutherford.
3. Ion Temperature Gradient Instability (Sept. 24)
High central ion temperature is required for magnetic nuclear fusion→ ∇Ti existsand ion heat transport is a topic of primary interest.
In uniform B, ∇Ti alone can excite ITG instability by driving the ion acousticwave unstable. ∇n influences conditions for excitation and growth rate. This canhappen even with Boltzmann (adiabatic) electron response (i.e. ω k‖vTe).
Let’s take Ti0(x) = T0(x), Ti = T0 + δTi, B = B0]z. Now, we generalize iondynamics (still assuming ω k‖vT i and k2
⊥ρ2i 1→ fluid description is justified).
Linearize
⇒ ∂
∂tδn+ δuE · ∇n0 + n0∇‖δu‖ + n0
∇⊥ · δupol = 0 (6)
We’ll relax cold ion assumption (Te Ti) which was used before for electron driftwave. ⇒ Ti ∼ Te, but this leads to k2
⊥ρ2s 1 (ρs = cs/Ωci, cs =
√Te/Mi).
Mi∂
∂tδu‖ = − |e| ∇‖δφ−
1
n0∇‖δpi (7)
and
∂
∂tδpi + δuE · ∇p0 + Γn0∇‖δu‖ = 0 (8)
Here, “Γ” is an “adiabatic” exponent which appears in the equation of state:
p
ρΓ= const
19
Here the term adiabatic implies “thermal insulation” in thermodynamics.E.g. for a sound wave in a gas,(
period of vibration
)
( relaxation time for volume element of gasto exchange energy with the rest of fluid
through heat flow
)
As before, we assumed perturbed quantities ∝ exp (−iωt+ ik · x)→ ∂
∂t ⇒ −iω, ∇‖ ⇒ ik‖ when they are operated on perturbed quantities.
Let
∂
∂xTi(x)
∣∣∣x=x0
= − 1
LT iTi(x)
∣∣∣x=x0
ω∗e =kyρsLn
cs > 0
pi = n0Ti ω∗pi = −kyρsLpi
vT i < 0
⇒ 1
Lpi=
1
LT i+
1
Ln, τ ≡ Te
Tiω∗T i = −kyρs
LT ivT i < 0
⇒ with a proper normalization, Equations (6), (7) and (8) become
−iω δnn0
+ iω∗e|e| δφTe
+ ik‖cs
(δu‖cs
)= 0
−iωδu‖
cs+ ik‖cs
|e| δφTe
+ ik‖cs
(TiTe
)δpip0
= 0
−iω δpip0
+ iω∗piτ
|e| δφTe
+ ik‖csδu‖
cs= 0
⇒ with δn/n0 = |e| δφ/Te, −i(ω − ω∗e) ik‖cs 0
ik‖cs −iω ik‖cs1τ
−iωpi
τ ik‖csΓ −iω
δn/n0
δu‖/csδpi/po
= 0
Determinant [3×3] = 0 ⇒ Cubic algebraic equation for “ω”
1− ω∗eω−k2‖c
2s
ω2
(1 +
Γ
τ− ω∗pi
ω
)= 0 (9)
Note that as τ →∞, LT i →∞; we recover
1− ω∗eω−k2‖c
2s
ω2= 0
(Since se assumed k2⊥ρ
2s 1, there’s no k⊥ term.) Now, we consider more inter-
esting case with strong gradient, 1/LT i , 1/Ln . Then, ω∗e, |ω∗pi| and the2nd and 5th terms of Equation (9) become dominant.
ω∗eω'k2‖c
2s
ω2
ω∗piω
= −ω∗eω
k2‖c
2s
ω2
1 + ηiτ
20
Here ηi ≡ Ln/LT i.
ω2 = −(1 + ηi
τ
)k2‖c
2s ⇒ γ =
(1 + ηiτ
)1/2k‖cs (10)
(This dissipation relation is valid for ηi 1, k‖ very small, 1/Ln relatively high.)⇒ Purely growing (reactive) instability! (without a help from wave-particle reso-nance) i.e. <(ω) ' 0,=(ω) > 0.↔ Electron drift instability which is “resonant”. Let’s check the validity regime.
|ω| k‖vT i ⇒(1 + ηi
τ
)1/2 1
But
ω∗e|ω| 1⇒ k‖cs
(1 + ηiτ
)1/2 kyρs
Lncs ( ⇒ “k‖ should be very small′′)
Equation (10) is the most pessimistic estimation of ITG linear growth rate.
So far we are addressing only the “local” stability or instability. To address morerealistic spatial variation of perturbation, we need to consider non local theory(δφ ∼ δφ(x) exp i(kyy + kzz)).
3.1. Sound Wave
“Who derived the sound wave velocity at first?”⇒ I. Newton first derived the equation of sound wave even before measuring it.
• In plasma, ω2 = k2‖c
2s
• In gas, v2ph = ω2/k2
‖ = Γp0/ρ0 (Newton’s answer was for Γ = 1)
Newton used Boyle’s law in the derivation (Γ = 1, pV = const. i.e., isothermal)rather than a proper equation of state p/ρΓ = const.
21
3.2. Ion Temperature Gradient Instability in Uniform MagneticField (Sept. 26)
We obtainedω2
k2‖
= −(
1 + ηiτ
)c2s
when (1 + ηi) /τ 1, k‖ very small, ω/k‖ vT i. The dispersion relation clearlyshows the “sound-wave-like” character of ITG instability.Let’s review a sound wave in a gas:
ρ0∂
∂tδu‖ = − ∂
∂zδp (11)
∂
∂tδp = −Γp0
∂
∂zδu‖ (12)
The determinant of this [2×2] system is
ω2
k2‖
=Γp0
ρ0
Note that the linearized continuity equation ∂∂tδp+ρ0
∂∂z δu‖ = 0 and p/ρΓ = const
yields Equation (12).Let’s check how this is related to the compressibility.
κs = −V −1
(∂V
∂p
)s
is the adiabatic compressibility (in thermal insulation or at constant entropy “s”).
pV Γ = p0VΓ
0 ⇒ p = p0VΓ
0 V−Γ(
∂p
∂V
)s
= −Γp0VΓ
0 V−Γ−1 ⇒ V
(∂p
∂V
)s
= −Γp0
Therefore,
κs ≡ −V −1
(∂V
∂p
)s
=
[−V
(∂p
∂V
)s
]−1
= (Γp0)−1
22
so that (ω
k‖
)2
= (ρ0κs)−1
From this, we can interpret that the effective “compressibility” of the ITG insta-bility is a “negative”.Instability mechanism: P when V in a gas, but the opposite happens here!
Previous example: the 2nd and 5th terms of Equation (9) are dominant.Now, if Ln →∞ (flat density): ω∗e/ω 0.Therefore, |ω∗e/ω| 1 so that the 1st term should balance the 5th term. Also,we need |ω∗T i/ω| 1 and k2
‖c2s/ω
2 1 to make this dominant balance justified.
1 =k2‖c
2s
ω2
ω∗T iω
= −k2‖c
2s |ω∗T i|ω3
This cubic equation has 3 roots.
Now the ITG is no longer purely growing. < (ω) < 0 in the ω∗i direction.
ω = ei2πN/3 |ω∗T i|1/3(k‖cs
)2/3with N = 0, 1, 2.Actually, for a tokamak core plasma, this limit is usually more relevant thananother limiting case, ω ∼ i [(1 + ηi) /τ ]1/2 k‖cs.
ITG has a hybrid character of a drift wave and a sound wave. Its frequency is alsoof opposite sign to the electron drift wave.
23
Now, back to (k2⊥ρ
2i 1):
1− ω∗eω−k2‖c
2s
ω2
(1 +
Γ
τ− ω∗pi
ω
)= 0
We have ω∗e ∝ 1/Ln, ω∗pi ∝ 1/Lpi, 1/Lpi = 1/Ln + 1/LT i and (neglecting1/Ln) ω∗pi ⇒ ω∗T i. Also, for generality, Γ/τ = O (1). Therefore, if 1/LT i → 0,τ ≡ Te/Ti →∞, ω∗T i → 0 and we recover the electron dispersion relation.
Recall the evolution equations determining dynamics:
∂
∂tpi + δuE · ∇p0 + Γp0∇ · δu‖ = 0
Mi∂
∂tδu‖ = −e∇‖δφ−
1
n0∇‖δpi
∂
∂tδn+ δuE · ∇n0 + n0∇ · δu‖ = 0
with sound and drift wave contributions and δn/n0 = |e| δφ/Te.In these determining equations, the terms of sound wave and drift wave co-exist.⇒ In this sense, we can say that drift wave and sound wavea are “coupled”.
What happens if 1/LT i → 0?Even for very weak ITG (1/LT i → 0), it remains unstable! which is unphysical(does not match with experimental results). We need to re-examine the approxi-mations we’ve used. In particular, ω/k‖ vT i will break down as |ω| → 0, so weneed to consider ion kinetic effects, including ion Landau damping.
Inclusion os ion Landau damping using ion drift kinetic equation (similar to whatwe did for electrons for the electron drift wave) leads to a conclusion that ITGmode is unstable for ηi ≡ Ln/LT i > 2 (for (k⊥ρi)
2 1).The condition 1
LTi(1− ηi/2) is similar to the free energy relaxation in the electron
drift wave.
Question when does the condition ηi > 2 become unphysical? (exam)
24
4. Basic Plasma Physics for Microinstabilities (Oct. 18)
4.1. Review of Single Particle Motion in a Strong Magnetic Field
4.1.1. Adiabatic Invariant
When magnetic field varies in space smoothly (i.e. ρi LB ≡ |∇ lnB|−1), we canidentify approximate constants of motion.“Adiabatic” in here means slow variation in time and space. This is well illustratedfrom the point of view of Quantum Mechanics (QM).Let’s consider a Simple Harmonic Oscillator (SHO): Schrodinger Equation is
Hψ(x) =(− ~2
2m
∂2
∂x2+
1
2mω2
0x2)ψ(x) = Eψ(x)
Here the eigenvalues are
E = ~ω0
(N +
1
2
)where N is quantum number (N = 0, 1, 2, . . . ).
Suppose that potential well is changed very (very) slowly in time (τ 1/ω0).In this adiabatic process, what remains constant is “N” (eigenstate is preserved).Energy (eigenvalue) changes in time. “N” is an example of adiabatic invariant.
N = E/ω0 : classical limit
4.1.2. Adiabatic Invariant in Classical Limit
In Classical Mechanics (CM), the Hamiltonian of SHO is given by
H(p, q) =p2
2m+
1
2mω2
0q2 = E
25
Adiabatic invariant in CM is coupled to the conservation of the volume in thephase space for appropriate action-angle variables.
I =
∮dq p
This is also called “Action Invariant”.
For SHO the action invariant is
I = πLqLp = π
√2mE
2E
mω20
= 2πE
ω0
Thus except for a numerical factor 2π, we recover N = E/ω0 from SHO in QM.(The useful formula N = E/ω0 represents the “Duality of Wave and Particles”.)This illustration of geometric meaning of action invariant can be extended toquasi-periodic motion (recall τ 1/ω0).
4.1.3. Gyromotion in Slowly Varying Magnetic Field
Consider the gyrating motion of charged particles in slowly varying magnetic fieldin time (1/ω) and space (LB).
For this gyration, the corresponding action invariant is the magnetic moment (the1st adiabatic invariant).
• Energy corresponding to gyration: E⊥ = mv2⊥/2
• Frequency corresponding to gyration: Ωc = eB/mc
26
⇒ µ ∝ 1
2mv2⊥/(eBmc
)∝( v2⊥
2B
): not exact constant
What is the error or precision of the statement?What is the expansion parameter or smallness parameter in this motion?If εω ≡ ω/Ωc 1 and εB ≡ ρi/LB 1,the adiabatic invariant is good up to arbitrary order!
Error = O(
exp (−const
εB), exp (−const
εω))
4.1.4. Charged Particle Motion in Inhomogeneous Magnetic Field
E =1
2mv2⊥ +
1
2mv2‖ = µB +
1
2mv2‖
If µ is a constant,
E = µB(x) +1
2mv2‖
Mathematically, µB(x) acts as “potential” energy for parallel motion: Veff(x).
Whether a particle passes or be trapped depend on E and µ.
Example Cosmic ray in spcae: abundance of high energy cosmic ray
Fermi explained this using particle trapping in [Phys. Rev. 75, 1169 (1949)].1. Particles get trapped in magnetic field2. Magnetic field changes3. Particles are accelerated⇒ “Magnetic mirror”
In tokamaks;
|B| = B =R0
RB ∼= B0
(1− r
R0cos θ
)(R = R0 + r cos θ)
27
(Let’s say, E = 0)We know that in inhomogeneous magnetic field, there’re ∇B and curvature drift.Question Are these only the 1st order drift in ρi/LB?
Egc =1
2mv2‖ + µB
Is ddtEgc = 0 satisfied if we use d
dtRgc = bv‖ + v∇B + vcurv?
d
dtEgc = mv‖
dv‖
dt+ µ
dB
dt
with chain rule
dB
dt=dRgc
dt· ∇B
Homework 3 Check if ddtEgc = 0 and discuss the result.
From elementary considerations for each case separately:
v∇B+curv =(mv2
⊥2
+mv2‖
)b×∇BqB2
(in currentless case)
But is there a unified (common) derivation for both?
mdv
dt= Feff + q
v ×B
c⇒ vFeff =
Feff ×B
qB2
For ∇B 6= 0, Veff = µB ⇒ Feff = −∇Veff = −µ∇B: Mirror force
⇒ v∇B =c
qB2
(− µ∇B ×B
)For curvature, Feff = mv2
‖Rc/R2c : Centrifugal force
where Rc/R2c = −b · ∇b ≡ κ and for currentless B, b · ∇b is related to ∇B.
4.2. Magnetically Trapped Particles in Tokamaks (Oct. 22)
B =B0
1 + (r/R0) cos θφ+
ε
q
B0
1 + (r/R0) cos θθ
ε = r/R0: inverse aspect ratioq (r) ' rBφ/RBθ: safety factorB ≡ |B| ∼= Bφ = B0R0/R = B0/
(1 + (r/R0) cos θ
)Axisymmetric equilibrium ⇒“H0” unperturbed Hamiltonian of single particle is independant of φ.Therefore,
d
dtPφ = − ∂
∂φH0 = 0
28
⇒ Canonical angular momentum Pφ is conserved.
Pφ = R(mvφ +
q
cAφ
)≈ R
(mv‖ +
q
cAφ
)= Rmv‖ −
q
cψ
where ψ: poloidal flux function (dψ = RBθdr)Unperturbed particle orbits in a tokamak stay on the surface of constant Pφ,constant µ and constant E.Let’s consider a guiding-center motion along B. Let ` be a distance along B,
d`2 = R2dφ2 + r2dθ2 = (qR0)2 dθ2 + r2dθ2 ≈ (qR0)2 dθ2
where q = dφ/dθ = rBφ/RBθ, therefore v‖ = d`/dt = qR0dθ/dt.For r/R0 1 (and assuming no equilibrium electric field Φ0 = 0, E = 0),
E =1
2mv2‖ +
1
2mv2⊥ =
1
2mq2R0
(dθ
dt
)2
+ µB0
(1− r
R0cos θ
)The 2nd term is an effective potential Veff for motion along B projected to θ.µ is different for different particles, it is a parameter.
For trapped particles, there is a θT such that
E = µB (θ = θT ) =1
2mv2‖ (θ = 0) + µB (θ = 0)
(at θ = θT , v‖ = 0; at θ = 0, v‖ is maximum and B is minimum)For passing particles, v‖ 6= 0 even for v‖ (θ = ±π), where B is maximum and v‖ isminimum as a function of θ.Therefore, trapped-passing boundary is determined from
1
2mv2‖ (θ = ±π) + µB0
(1 +
r
R0
)=
1
2mv2‖ (θ = 0) + µB0
(1− r
R0
)by requiring v‖ (θ = ±π) ≥ 0, i.e.
v2‖ (θ = 0) ≥ 4
r
R0
µB0
m
29
fraction of trapped particles ≡ # of trapped particles / # of total particles ∝√ε
for isotropic distribution in v.From Pφ conservation, we can estimate the radial widths of orbits(deviation from flux surface)
(constant for each ptl) = Pφ = −qcψ +mRv‖ = −q
cψ0
for trapped particles. Taking values at θ = 0 and recalling dψ = RBθdr,
∆b =2∆ψ
RBθ=
2
RBθ
mc
qRv‖ (θ = 0) = 2
mc
qBθv‖ (θ = 0) ' 2
mc
qB
BφBθ
√εv
(for a typical trapped particle)
'√ερi
q
ε' q√
ερi
Meanwhile, for strongly passing particles, variation of v‖ along B is not large,
⇒ ∆ ' 2∆ψ
RBθ' 2
RBθ
mc
qv‖ (θ = 0) ∆R ' 2mc
qBθ
r
Rv‖ ' 2qρi
since v‖ v⊥ ⇒ v‖ ' v.
Note Why we calculate ∆b or ∆?⇒ by Coulomb collision, trapped particles diffuse as ∆b-scale, not as ∆-scale.
4.3. Trapped particles’ bounce motion and precession
We’ve learned about the 1st adiabatic invariant µ = mv2⊥/2B from particle gyro-
motion,
µ ∝∮dl · v⊥ ∝
∮dq⊥ · p⊥
From trapped particles’ bounce motion (which is also quasi-periodic), we canobtain the 2nd adiabatic invariant
J =
∮mv‖d` ∝
∮dq‖ · p‖
since A‖’s contribution = 0.Using this we can simplify the derivation of trapped particles’ precession frequency,bounce frequency, etc.From Hamiltonian mechanics: with L = p · q−H
dp
dt=− ∂
∂qH (p,q)
dq
dt=∂
∂pH (p,q)
30
This can be extended to action-angle variables,
L = µdΘ
dt−HR
where µ is the 1st adiabatic invariant and Θ the gyrophase angle, with a reducedHamiltonian HR which is independant of Θ.
dΘ
dt=∂
∂µHR = Ωc
dµ
dt=− ∂
∂ΘHR = 0!
where Ωc is the gyrofrequency and we have conservation of µ. Extending thisto bounce-motion (for application to problems which deal with longer time scalecompared to bounce time),
L = JdΨ
dt+ µ
dΘ
dt−HB
with a further reduced Hamiltonian HB which is independant of “bounce-angle”Ψ.
dΨ
dt=∂
∂JHB ≡ ωb
dJ
dt=− ∂
∂ΨHB = 0
where ωb is the bounce frequency and we have conservation of J .
By introducing
• β = ψ: poloidal flux (radial direction)
• α = φ−q (r) θ: bi-normal direction (perpendicular to B and radial direction)
We can represent the Lagrangian and the Hamiltonian as “Clebsch form”
L =q
cαdβ
dt+ J
dΨ
dt−HB
Hamilton’s equations include
dα
dt=− c
q
∂
∂βHB
dβ
dt=c
q
∂
∂αHB
where HB = HB(α, β, J).
31
Since there’s no secularity in the particle’s motion in θ, the motion in “α” describesthe precession in the toroidal direction.
dα
dt= ωprecession = − c
q
∂
∂ψHB = − c
q
∂J
∂ψ
∂HB
∂J= − c
qωb∂J
∂ψ
J = 4
∫ θT
0mv‖qR0
(1− r
R0cos θ
)dθ ' 8
√2εqR0v
[E (κ)−
(1− κ2
)K (κ)
]with elliptic functions of the 2nd (E) and 1st (K) kind and pitch-angle variable
κ ≡v2‖ (θ = 0)
2εv2⊥ (θ = 0)
We also have with ωb = ∂HB/∂J
τb =2π
ωb=∂J
∂E=
4qR0
v√ε
√2K (κ)
We can also obtain
ωprecession = − cqωb∂J
∂ψ=v2
Ωc
q
rRG (κ, q, s)
where
G (κ, q, s) =E (κ)
K (κ)− 1
2+
2q′r
q
(E (κ)
K (κ)− 1 + κ2
)
Trapped particles’ motion can resonate with a wave and excite instabilities.
• Electrons meet drift wave ⇒ Trapped electron instability (TEM)
• Energetic ions meet internal kink (shear Alfvenic) ⇒ Fishbone instability
This happens when ωprecession > 0 unfavorable direction.Note that precession reversal (ωprecession < 0) happens for barely trapped particles.More particles reverse precession direction for q′ ∝ s < 0 and for Shafranov shift.
32
4.4. Introduction to Nonlinear Gyrokinetic Theory (Oct. 24)
The most general equation used for kinetic description of collisionless plasma isVlasov equation.[ ∂
∂t+ v · ∂
∂x+
q
m
(E +
v
c×B
)· ∂∂v
]f(x,v, t) = 0
Direct numerical soluation of actual size fusion plasmas in realistic geometry us-ing the primitive nonlinear Vlasov equation is still far beyond the computationalcapability foreseable future.
4.4.1. Gyrokinetic Vlasov Equation
For turbulence and transport problems in fusion plasmas, we are interested in
• “time scale” Ω−1c :
turbulent eddy turn-over time (∼ lifetime of eddy) or γ−1lin
• ρi . “spatial scale′′ R ∼∣∣ ∂∂r lnB
∣∣−1 ≡ LB:size of eddy in ⊥ direction or width of eigenfunction of fluctuation.
Note Eddy: contour of constant density
Plasma density changes in time or space in a turbulent fashion. From linear theorypoint of view, these eddys develop from instabilities characterized by eigenmodes(eiganfunctions).
We can obtain most of relevant information we need from a reduced descriptionin which details of gyration (such as gyrophase angle) are ignored.
33
Derivation of gyrokinetic equation from Vlasov equation consists of “dealing” ofgyromotion and achieving a kinetic equation in a reduced dimension (5-D in whichthe 1st adiabatic invariant “µ” entires as a parameter).“Gyrophase angle” θ becomes an ignorable coordinate.The gyrokinetic equation is,( ∂
∂t+
R
dt· ∂∂R
+dv‖
dt
∂
∂v‖
)〈f〉 (R, µ, v‖, t) = 0
dµ
dt= 0 and
∂
∂θ〈f〉 = 0
cf. Recall that if φ is ignorable
dPφdt
= 0 = −∂H0
∂φ
Assumptions (nonlinear gyrokinetic ordering):
ω Ωci, k‖ k⊥ ∼ ρ−1i ,
( ω
Ωci∼ ρiL∼)δff0∼ δn
n0∼ |e| δφ
T 1
Let’s consider uniform magnetic field B = Bb for simplicity, and fluctuatingelectric field δE = −∇δφ, but δB = 0 and E0 = 0.Our goal is to derive a kinetic equation in guiding center coodinates (R, µ, v‖, Aθ).(x,v)⇒ (R, µ, v‖, Aθ): “Guiding Center (GC) coordinate tranform”
R = x− ρ, ρ =b× v
Ω, Ω =
qB
mc, v‖ = v · b, µ =
v2⊥
2Bv = v‖b + v⊥e⊥, e⊥ = −e2 cos θ − e1 sin θ, eρ = e1 cos θ − e2 sin θ
34
First step is to express ∂∂x and ∂
∂v in guiding center coordinates.
∂
∂x=∂R
∂x· ∂∂R
+∂µ
∂x
∂
∂µ+∂v‖
∂x
∂
∂v‖+∂θ
∂x
∂
∂θ
∂
∂v=∂R
∂v· ∂∂R
+∂µ
∂v
∂
∂µ+∂v‖
∂v
∂
∂v‖+∂θ
∂x
∂
∂θ
This look trivial and boring, but one should be careful in remembering “whatvariables” are held constant when taking derivatives!Many expressions are “0”: such as
∂µ
∂x
∣∣∣v=const
=∂v‖
∂x
∣∣∣v
=∂θ
∂x
∣∣∣v
= 0
But
∂
∂x=
∂
∂R,∂
∂vv‖ =
∂
∂v(b · v) = b,
∂
∂vµ =
v⊥B,∂
∂vR =
∂
∂v
(x− b× v
Ω
)=
I× b
Ω
⇒ ∂
∂v= b
∂
∂v‖+
v⊥B
∂
∂µ− b× e⊥
v⊥
∂
∂θ+
I× b
Ω· ∂∂R
where
I =
1 0 00 1 00 0 1
(unit dyadic)
From [ ∂∂t
+ v · ∂∂x
+q
m
(E +
v
c×B
)· ∂∂v
]f(x,v, t) = 0
The 1st term ⇒
v‖b ·∂
∂R+
v⊥ ·∂
∂R
The 2nd term ⇒q
m
(δE‖
∂
∂v‖+δE · v⊥B
∂
∂µ− δE · (b× v⊥)
B
∂
∂θ+cδE×B
B2· ∂∂R
)The 3rd term ⇒
−Ω(v ×B) · (B× v)
B2v2⊥
∂
∂θ+ Ω
(v × b)× b
Ω· ∂∂R
= Ω∂
∂θ−
v⊥ ·∂
∂R
We also want to express δφ(x) and δE(x) in terms of GC variables.
δφ(x) = δφ(R + ρ(θ))
∂
∂θδφ =
∂x
∂θ
∣∣∣R· ∂δφ∂x
=∂ρ
∂θ· ∂δφ∂x
=v⊥Ω· ∂δφ∂x
= −δE · v⊥Ω
35
Therefore,
q
m
δE · v⊥B
∂
∂µ= −1
c
( qm
)2∂δφ
∂θ
∂
∂µ
Collecting contributions from the 1st, the 2nd and the 3rd terms,[ ∂∂t
+ v‖b ·∂
∂R+ c
δE×B
B2· ∂∂R− q
m∇‖δφ
∂
∂v‖+ Ω
∂
∂θ
∼ −iω k‖v‖ δvE · k⊥ k‖v‖|e| δφTe
Ω
− qΩ
mB
∂δφ
∂θ
∂
∂µ−Ω
δvE · v⊥v2⊥
∂
∂θ
]f = 0
mess!
(The estimation of each term is for cases when it operates on perturbation.)First three terms are obviously important, not only in terms of magnitudes, butalso from physics they are representing!The 4th term is a bit similar than the first three terms, but this represents velocityspace nonlinearity, nonlinear exchange of energy between particles and waves.The 5th term is the largest term! since Ω ω.The 6th term is correction of µ due to δφ ⇒ keep it!The 7th term is correction of Ω due to δφ ⇒not only small, we’ve not interested on this term for turbulent problems.
Now, we’ll use the gyrokinetic ordering and perform perturbation theory, sinceΩ ω, the 5th term is the largest term.Expanding f = f (0) + f (1) + · · · with an expansion parameter ε = ω/Ωci.The lowest order equation is given by
Ω∂
∂θf (0) = 0
and the next order equation is
Ω∂
∂θf (1) +
( ∂∂t
+ v‖b ·∂
∂R+ c
δE×B
B2· ∂∂R
+q
mE‖
∂
∂v‖− qΩ
mB
∂δφ
∂θ
∂
∂µ
)f (0) = 0
For 0th order equation; f (0) is independent of θ!
∴ f = 〈f〉+ f(1)AC in which f
(1)AC 〈f〉 .
The 1st and the last term on 1st order equation can be combinated into
Ω∂
∂θ
[fAC −
q
mBδφ
∂
∂µ〈f〉]
36
(Oct. 29)Now, we can represent the tranformed Vlasov equation in the following form
Ω∂
∂θ
(fAC −
q
mBδφ
∂
∂µ〈f〉)
+
∂
∂t+(v‖b + vE
)· ∂∂R− q
m∇‖δφ
∂
∂v‖
〈f〉 = 0
(13)
with the average over the gyrophase angle
〈. . . 〉 = (2π)−1∮dθ (. . . )
By taking 〈. . . 〉 (i.e., the gyrophase average) on the equaion above, we obtain∂
∂t+(v‖b +
c
Bb×∇〈δφ〉
)· ∂∂R− q
mb · ∂
∂R〈δφ〉 ∂
∂v‖
〈f〉 ≡
(d
dt
)(0)
〈f〉 = 0
(14)
This is electrostatic nonlinear GyroKinetic (GK) Vlasov equation for uniform B.This derivation is a bit modernized version of the original papers up to early 80s.Note 〈δφ〉 6= δφ for gyrokinetic equations (〈δφ〉 = δφ in drift-kinetic ordering).
There was a widespread misconception about gyrokinetics:
“Gyrokinetic Theory throws away the gyrophase information”
Reasons: conventional (old-fashioned) derivation is rather opaque. It was hard toidentify the role or necessity of θ-dependant information.
Now, we show that fAC (θ) can be obtained from gyrokinetics and it can playan important role. (Gyrokinetic Vlasov equation obtained above cannot describediamagnetic flow, since it’s not origined from motion of guiding centers.)
Recall that:
• Vlasov equation, E,B⇒ evolution of f (x,v, t)
• Maxwell equation, f (x,v, t)⇒ ρ,J⇒ evolution of E,B
We have derived gyrokinetic Vlasov equation: E,B ⇒ evolution of f(R, v‖, µ, t).What about the Maxwell equation? This is still valid, but we need to expressρi = ni (x) |e|, etc. in terms of f
(R, v‖, µ, t
)(not f (x,v, t)).
For the electrostatic problem, Poisson equation is
∇2δφ = −4πe [ni (x)− ne (x)]
37
Electron Larmor radius is usually ignored (xe = Re+ZZρe ), but we need to considerfinite Larmor radius effect for ions.For electrons, we usually use adiabatic (Boltzmann) response for electrons forpure Ion Temperature Gradient mode, or drift-kinetic equation or bounce-kineticequation for trapped electrons. In any case the waves studied have a wavelengthvery large compared to the electron Larmor radius.
4.4.2. Bessel Function Representation of FLR Effects
We assume∣∣k‖∣∣ k⊥.
δφ =∑k
δφkeik⊥·(R+ρ) =
∑k
δφkeik⊥·Reik⊥ρi sin θ
(Fourier-Bessel expansion: eik⊥ρi sin θ =∑∞
n=−∞ J0 (k⊥ρi) einθ)
⇒ δφ =∑k
δφkeik⊥·R
∞∑n=−∞
J0 (k⊥ρi) einθ
Therefore,
〈δφ〉 =∑k
J0 (k⊥ρi) δφkeik⊥·R
This is FLR reduction of the bare potential δφ.The Bessel function (|J0| ≤ 1) represent FLR effect from (x,v) ⇒
(R, v‖, µ, θ
),
push-forward transformation.(k⊥ → 0⇒ J0 → 1, go back to drift-kinetic equation.)
4.4.3. Determination of the Formula for fAC(θ)
Now, let’s go back to Equations (13) and (14).
Ωc∂
∂θ
(fAC −
q
mBδφ
∂
∂µ〈f〉)
= −
d
dt−(d
dt
)(0)〈f〉
Integrate this equation,∫ θ
0 dθ′ . . . to give
0 ' 1
Ωc
b×∇B
∫ θ
0dθ′ (δφ− 〈δφ〉) · ∇ 〈f〉+ fAC (θ)− fAC (0)
− q
mB(δφ (θ)− δφ (0))
∂
∂µ〈f〉
We can show that (for k⊥ k‖) (i)/(ii) ∼ k⊥ρi ρiLp 1
⇒ fAC (θ)− fAC (0) =q
mB(δφ (θ)− δφ (0))
∂
∂µ〈f〉
38
Since 〈fAC〉 = 0, we can determine constants of integration “fAC(0)”, “δφ(0)” and
fAC (θ) ' q
mB(δφ− 〈δφ〉) ∂
∂µ〈f〉
4.4.4. Polarization Effect (Oct. 31)
In Poisson equation
∇2δφ = −4πe (ni (x)− ne (x))
the LHS is small if |k| kDebye i.e. λ λDebye Debye shielding term.⇒ we have Quasi-neutral plasma.
ni (x) 6=∫dµdv‖dθB 〈fi〉
(R, µ, v‖, θ, t
)=
∫d3vfi (x,v, t)
=
∫d3v
∫d3x′fi (x,v, t) δ
(x′ − x
)=
∫dRdµdv‖dθBfi
(R, µ, v‖, θ, t
)δ (R + ρ− x)
ni (x) =
∫dRdµdv‖dθB
〈f〉+
q
mB(δφ (θ)− 〈δφ〉) ∂
∂µ〈f〉δ (R + ρ− x)
= ni,gc
(=
∫dRdµdv‖dθB 〈f〉 δ (R + ρ− x)
)+ ni,pol
(=
∫dRdµdv‖dθB
q
mB(δφ (θ)− 〈δφ〉) ∂
∂µ〈f〉 δ (R + ρ− x)
)For illustration, let’s evaluate δni (x) the linear response to δφ (x).Let 〈f〉 = F0 + δf .
δni (x, t) = δni,gc (x, t) + δni,pol (x, t)
where
δni,gc (x, t) =
∫dRdµdv‖dθBδ (R + ρ− x) δf (15)
δni,pol (x, t) =
∫dRdµdv‖dθBδ (R + ρ− x)
q
mB[δφ− 〈δφ〉] ∂
∂µF0 (16)
39
Here, δf satisfies the linearized gyrokinetic Vlasov equation.(∂
∂t+ v‖b · ∇
)δf +
c
Bb×∇〈δφ〉+
q
mb · ∂
∂R〈δφ〉 ∂
∂v‖F0
For Maxwellian F0, ∂∂µF0 = −mB
TIF0, so the 1st term in Equation (16) becomes
q
mB
∫dµdv‖dθBδφ (x)
(−mBTi
)F0 = − q
Tin0 (x) δφ (x)
The 2nd term of Equation (16) is∫d3Rdµdv‖dθ
∑k
δφkeik⊥·RJ0 (k⊥ρi) δ (R + ρ− x)
(− q
Ti
)F0
=
∫dµdv‖dθ
∑k
δφkeik⊥·(x−ρ)J0 (k⊥ρi)
(− q
Ti
)F0
J0 (k⊥ρi) here came from the (x,v)→(R, µ, v‖, θ
)push-forward transformation,
now 12π
∮dθe−ik⊥·ρ = J0 (k⊥ρi) comes from
(R, µ, v‖, θ
)→ (x,v) pull-back trans-
formation.
= 2π
∫dµdv‖
∑k
δφkJ20 (k⊥ρi)
(− q
Ti
)n0
(2πTi/m)3/2e−mv2‖/Tie−mµB/Ti
(integral formula:∫∞
0 dt te−pt2J2
0 (at) = 12pe−a2/2pI0
(a2/2p
), where I0 is the mod-
ified Bessel function.)
= −qn0
Ti
∑k
δφkeik⊥·xΓ0 (bi)
where Γ0 (bi) = I0 (bi) e−bi and bi ≡ k2
⊥ρ2i .
δni,pol (x, t) = −qn0
Ti
∑k
(1− Γ0 (bi)) eik⊥·xδφk
Γ0 (bi) ' 1− bi +O(b2i)
for bi ≡ k2⊥ρ
2i 1 (long wavelength limit), therefore
δni,pol (x, t) = −qn0
Ti
∑k
ρ2i k
2⊥e
ik⊥·xδφk
=qn0
Tiρ2i∇2⊥δφ =
qn0
Teρ2s∇2⊥δφ
with ρs ≡ cs/Ωc, cs ≡ Te/Mi.The RHS is independent of Ti, so non-zero even for cold-ion limit Te Ti.Here, ρi = vth,i/Ωc (cf. in J0 (k⊥ρ), ρ = v⊥/Ωc for each particle).Poisson equation here is
∇2δφ = −4πe [δni,gc (x) + δni,pol (x)− δne (x)]
40
Since the term ∇2δφ in Poisson equation is Debye-shielding term, it is very small.The quasi-neutrality condition can be written as
en0 (x)
Teρ2s∇2⊥δφ = δne (x, t)− δni,gc (x, t)
which is the GK Poisson equation.LHS=Polarization Shielding term (λ ρi ∼ ρs λDe typically!).Mathematically, polarization shielding term is similar to Debye shielding term.
Even before the invention of gyrokinetics, many computational methods for solv-ing the Vlasov-Poisson system have developed (e.g., particle-in-cell method by J.Dawson, Bunemann, Birdsall, ...). These techniques can be used for gyrokineticsystem with minor modifications.
Why is it named “polarization density”?From
mid
dtδv = q
(δE +
δv ×B
c
)we can derive
δv⊥ = cδE⊥ ×B
B− mic
2
qB
∂
∂t∇⊥δφ
where the 2nd term is the polarization drift (linear version). One can show that
∂
∂tδni,pol +∇ · (n0δvpol) = 0
Recall that δvpol did not appear in guiding center drift (i.e. ddtR = v‖b+δvE×B+
. . . ), but δnpol appears in the GK-Poisson equation.
In summary, the electrostatic gyrokinetic system consists of[∂
∂t+ v‖b · ∇ −
c
B〈δφ〉 × b
∂
∂R− q
mb · ∇ 〈δφ〉 ∂
∂v‖
]〈f〉 = 0
the GK Vlasov equation, and
en0 (x)
Teρ2s∇2⊥δφ = δne (x, t)− δni,gc (x, t)
the GK Poisson equation (more general form of ρ2s∇2⊥δφ with 1− Γ0).
Since we’ve ignored any dissipation in the system, if it’s closed, the total energyof the system should be conserved.Any reliable plasma dynamical model should have a conserved quantity in theabsence of dissipation (δB→ 0),
E =1
8π
∫d3x |∇δφ|2 +
∫d3xd3vfe
1
2mv2
+
∫d3Rdv‖dµdθB
[1
2mv2‖ + µB
]〈f〉+
1
2Ti
∑k
q2n0 (1− Γ0) |δφ|2
41
Here the 4th term is sloshing energy. The 1st term is related to Debye shielding,so it is relatively small. The 4th term is related to polarization shielding, and itactually plays the role of electrostatic energy density!
5. Drift Wave Eigenmode Equation (Nov. 5)
5.1. Sheared Slab Model for Tokamak Magnetic Field
Cartesian coordinates Toroidal coordinates:
x → r
y → rθ
z → Rφ
with B = Bφφ+Bθθ.Safety factor q (r) ' rBφ/RBθ characterizes pitch of B.Typically, q (r) is a monotonically increasing function of r (positive shear plasma,magnetic shear s = d ln q/d ln r). For a mode rational surface at rs, q (rs) = m/n.
δφ (r, θ, φ) =∑n,m
δφn,mei(nθ−mφ)
where nθ −mφ is the pitch of fluctuation, (m,n) ∈ Z.We can also express this as
∑k exp i (kyy + kzz) so that kθ = m/r, kφ = −n/R.
Therefore,
k‖ =k ·B|B|
=m
r
BθB− n
R
BφB
=BθrB
(m− nq (r))
k‖ = 0 at r = rs (q (rs) = m/n), m and n are fixed, but q (r) and k‖ vary with r.
q (r) = q (rs) + (r − rs)dq
dr(rs) + . . . ,
so that we replace m− nq by −n (dq/dr) (r − rs).Magnitude of k‖ (r) = k‖ (x) increases with |x| where x = r − rs.k‖ flips sign across x = 0 (r = rs).
42
cf. in reversed shear (RS) plasmas, q (r) can have a minimum value at r = rmin (notr = 0). At qmin, dq/dr = 0 therefore q (r) = qmin + 1
2
(d2q/dr2
)(r − rmin)2 + . . .
The magnetic field in sheared slab geometry is
B = B
(z +
x
Lsy
)with the magnetic shear s = r/q (dq/dr), Ls ≡ qR/s.
This model is good for a single pair of (n,m), i.e., single helicity fluctuation.
k‖ ⇒ kyLsx
k2x ⇒ − ∂2
∂x2
where the LHS are the local expressions from uniform B model and the RHS thesheared slab for uniform B.k‖ = 0 at x = 0: “singular layer”.We are interested in local stability around this location.In this model,
δφ (x, t) =∑ky
δφky ,ω (x) ei(kyy−ωt)
(After analyses in sheared slab, we should understand that results with ky = nq/rand m = nq.)Recall electrostatic linear drift wave for electrons: vT i . ω/k‖ vTe
δnen0' exp
(|e| δφTe
)− 1 ' |e| δφ
Te
Boltzmann response (adiabatic response for ω/k‖ vthe).For ions,
∂
∂tni = −∇ · δ (nu) = −δuE · ∇n0 −∇ · (n0δupol)−∇ ·
(n0u‖
)⇒ δni
n0=
(ω∗eω− ρ2
sk2⊥ +
c2sk
2‖
ω2
)|e| δφTe
43
with ω∗e ≡ kyρs/Ln (ω from polarization, ω from sound wave).Quasi-neutrality (λ λDe):
|e| δφTe
=
(ω∗eω− ρ2
sk2⊥ +
c2sk
2‖
ω2
)|e| δφTe
now eigenmode equation in x rather than local dispersion relation.[1 + ρ2
sk2y − ρ2
s
∂2
∂x2− ω∗e
ω− c2
s
ω2
k2y
L2s
x2
]δφ (x) = 0 (Weber equation)
⇒ Mathematically equivalent to time-independant Schrodinger equation for SHO
− ~2m
∂2
∂x2Ψ =
(E − 1
2mω2
0x2
)Ψ
Eigenmode equation for electrostatic drift wave:
−ρ2s
∂2
∂x2δφ =
[−1− ρ2
sk2y +
ω∗eω
+c2s
ω2
c2s
ω2
k2y
L2s
x2
]δφ
where the first three terms of the RHS correpond to the energy and the last term−Veff the opposite of the effective potential.Note δφ in equations above is in real δφky ,ω. We used δφ just for simple notation.
(Nov. 7)We can easily obtain eigenvalues, but effective potential Veff has an anti-well (hill)structure for |< (ω)| > |= (ω)|, a typical case for drift wave.
We’ve learned from QM how to solve this Weber equation.
δφky ,ω (x) = δφky ,ω exp
[−σx
2
2
]H`
(√σx)
(H`: Hermitte polynomials). Here,
σ = ± ikycsLsρsω
44
For ` = 0,
δφky ,ω (x) = δφky ,ω exp
[∓ ikycs
2Lsω
x2
ρs
]Which solution should we take?This is already decided by a “causality” condition.For = (ω) > 0 (unstable solution), lim|x|→∞ |δφ (x)| = 0.
While the fluctuation grows locally (in space, i.e. at “one” space) as time goes by,at a given time, it should decay in space as |x| → ∞.This is equivalent to the “outgoing wave” boundary condition. (Note: decayingat infinity is a consequence of causality condition, not a condition in itself.)
vgp,x ≡∂ω
∂kx
> 0 for x→∞< 0 for x→ −∞
Then, what is kx? We recall an eikonal form exp[i∫kxdx+ ikyy − ωt
]kx = −i ∂
∂x= ∓ kycsx
Lsρsωor ω = ∓ kycsx
Lskxρs
Therefore,
vgp,x =∂ω
∂kx= ± kycs
Lsρsk2x
x
“outgoing wave” boundary condition (vgp > 0 for x > 0)⇒ We choose (for ky > 0)
δφky ,ω (x) = δφky ,ω exp
[− ikycs
2Lsω
x2
ρs
]The corresponding eigenvalue is (recall E = ~ω0 (1/2 +N) for SHO in QM):
ω = ω∗e
1
1 + k2yρ
2s
− i (2`+ 1)
1 + k2yρ
2s
LnLs
where the downshifting term is function of ky (k⊥ in the local term) and the 2ndterm is the magnetic shear induced damping.The eigenmode width ∆x is
∆x '
√Lsω∗eρskycs
∝√LsLn
ρs (∼ 10ρs & ρs)
45
As s, ∆x.Technique Just take unstable solution (i.e. =(ω) > 0) when getting eigenfunc-
tion, regardless whether eigenvalue is stabilizing or destabilizing.
“Universal” instability of drift wave required only ∇n and inverse-Landau damp-ing of electrons in collisionless plasmas. Then, until late 70s, most people believeddrift wave should be stable due to magnetic shear induced damping (e.g., Pearl-stein and Berk, [Phys. Rev. Lett. 23, 220 (1969)])
We can introduce destabilizing effect of electrons,
• Resonance of passing electrons with drift wave ⇒ universal instability
• Resonance of trapped electrons with drift wave ⇒ collisionless TEM
• Collisions of electrons ⇒ dissipative drift instability - dissipative TEM
Note All this discussion were for a single set of (n,m).
What will happen in toroidal geometry?
Btoroidal = Bφφ+Bθθ where |Btoroidal| ' Bφ =B0R0
R0 + r cos θ
Sheared slab magnetic field is
Bsheared slab = B0
(z +
x
Lsy
)Therefore, sheared slab magnetic field is independent of z and y: no mode coupling.However in toroidal magnetic field, there are crucial difference for θ dependance.⇒ Sheared slab model should be modified. m’s are coupled!
46
5.2. Mode Coupling in Toroidal Magnetic Field (Nov. 12)
In sheared slab geometry,
ω = ω∗e
1
1 + k2yρ
2s
− i (2`x + 1)
1 + k2yρ
2s
LnLs
where we have ω∗e = csρsky/Ln and the adiabatic electron response and the shear-induced damping. We include a non-adiabatic electron response −iδe (function ofkyρs, Ln, Te, νeff, r0/R, s, q, ...)
δnen0
= (1− iδe)|e| δφTe
⇒ ω = ω∗e
1
1− iδe + k2yρ
2s
− i . . .
for small δe 1
= (ω)
ω∗e=
γ
ω∗e=
δe(1 + k2
yρ2s
)2 − (2`x + 1)
1 + k2yρ
2s
LnLs
Therefore,
δe >(1 + k2
yρ2s
)(2`x + 1)
LnLs
for instability
Lower mode numbers (Gaussian-like eigenfunctions) are more likely to be unstable.
How is this picture modified in toroidal geometry?where m (poloidal mode number) is no longer a good quantum number due toB = B0R0/ (R0 + r cos θ). We cannot get a single decoupled eigenmode equationfor δφn,m (r). This will couple δφn,m+1 (r), δφn,m−1 (r) sidebands!
Note that kθρs = kyρs = nq (r) ρs/r0. γ is maximum for kθρs = O (1).(∵ ω∗e ∝ kθρs ⇒ γ ∝ kyρs/(1 + k2
yρ2s))
For r0 ∼ 102cm, ρs ∼ 0.5cm, q (r0) ∼ 2, we get n ∼ 102 (small-scale)→ Microinstability.
47
Recall our previous calculation in sheared slab geometrywas only for a singlehelicity, i.e., one pair of (m,n) e.g. n = 124,m = 206.“Drift wave energy gets convected away from the mode rational surface whereq(rm/n
)= m/n to large |x| region (x = r − rm/n) with vgp,x!”
We’ve learned from plasma electrodynamics (or electromagnetism) course that if
E+ =yE0ei(kxx−ωt) : propagating to + x
E− =yE0ei(−kxx−ωt) : propagating to − x
we get a standing wave
E+ + E− = y2E0 cos (kxx) e−iωt
In this discussion ignore δe ∵ δe can also be a function of θ
(1− δe)|e| δφTe
=
(ω∗eω− k2
yρ2s + ρ2
s
∂2
∂x2+c2s
ω2
L2n
L2s
x2
)|e| δφTe
where the RHS is the ion response from
∂
∂tδni + δuE · ∇n0 + n0∇ · δupol + n0∇ · u‖ = 0
in uniform magnetic field.In torus? ∇ · δuE is no longer zero, thus we should consider n0∇ · δuE term.(∇B effects are just equilibrium quantities.)
∇ · δuE = ∇ ·(cE×B
B
)= cb× δφ · ∇
(1
B
)+c∇× b · ∇δφ
B
For uniform B, ∇ 1B = 0 and ∇× b = 0.
Now, we decompose into parallel and perpendicular components
∇× b = b (b · ∇ × b)− b× (b×∇× b) = b (b · ∇ × b) + b× (b · ∇) b
We are considering ∇δφ ≈ i (krer + kθeθ) δφ ∴ eζ-direction doesn’t contribute.For low-β equilibrium v∇B ' vcurv for thermal particles (T‖ = T⊥).
∇(
1
B
)=
1
B0∇(
1 +r
R0cos θ
)=
1
B0R0(er cos θ − eθ sin θ)
⇒ n0∇ · δuE = · · · = 2ωde (θ)|e| δφTe
whereωde =
ρscsR0
(kθ cos θ + kr sin θ)
with kθ = nq (r0) /r0 and kr = −i∂/∂x.
48
(Nov. 14)We add 2ωde/ω in the ion response. Then, the eigenmode equation become
δnen0
=|e| δφTe
=
(ω∗eω
+ 2ωdeω
+ ρ2s
∂2
∂x2− k2
θρ2s +
c2s
ω2
L2n
L2s
x2
)|e| δφTe
δφ (ζ, θ, r)⇒AAA
∑n
δφn (θ, r) e−inζ =∑m
δφn,m (x) ei(mθ−nζ)
where ζ is the toroidal angle and x = r − r0, q = m0/n.“n” is a good quantum number (conserved), each toroidal harmonic decouples inlinear theory ⇒ einζ is a common factor.Now, substitute δφ⇒
∑m δφm,n(x)eimθ.
Each term in the eigenmode equation is proportional to eimθ, except
(kθ cos θ + kr sin θ)∑m
δφn,m (x) eimθ
=1
2
kθ
(eiθ + e−iθ
)+kri
(eiθ − e−iθ
)∑m
δφn,m (x) eimθ
Here,
eiθ∑m
δφn,m (x) eimθ =∑m
δφn,m (x) ei(m+1)θ =∑m
δφn,m−1eimθ
since m is a dummy variable (from −∞ to ∞). Likewise,
e−iθ∑m
δφn,m (x) eimθ =∑m
δφn,m (x) ei(m−1)θ =∑m
δφn,m+1eimθ
Now the equation has to be satisfied for every term multiplying “ eimθ ”.
δnen0∝ δφn,m0 (x) =
(ω∗eω
+ ρ2s
∂2
∂x2− k2
θρ2s +
c2s
ω2
L2n
L2s
x2
)δφn,m0 (x)
− csρsωR0
[kθ (δφn,m0−1 (x) + δφn,m0+1 (x))
+kri
(δφn,m0−1 (x)− δφn,m0+1 (x))]
the coupled terms are called “sidebands”.
We have seen this kind of situation from classical mechanics.
49
infinite chain of coupled springs ⇒ eigenmodes?Similar problem also appears in solid state physics with lattice structure.
Recall that n ' 102 ⇒ every poloidal harmonics look almost the same. (Roughlyspeaking, eigenstructures are approximately same for neighboring poloidal har-monics |∆m| ∼ 101)
⇒ Lattice symmetry! “quasi-translational invariance” (almost, not perfect)Each poloidal harmonics are packed very closely to each other radially.What is the typical distance between neighboring harmonics?i.e.,
r(m0+1)/n − rm0/n ≡ ∆rn, where rm0/n ≡ r0
1
n= q(r(m0+1)/n)− q(r0) '
(r(m0+1)/n − r0
)(∂q∂r
)r0
= sq (r0)
(r(m0+1)/n
r0− 1
)with Taylor expansion around r0. Therefore,
∆rn =r0
nqs=
1
kθs
From quasi-translational invariance, we can get δφn,m0 (x) by shifting δφn,m0+1 (x)to the left by ∆rn or by shifting δφn,m0−1 (x) to the right by ∆rn, ... shiftingδφn,m0+j to the left by j∆rn.Let’s introduce a dimensionless variable X,
X ≡ x
∆rn=r − r0
∆rn
⇒r − r(m0+1)/n
∆rn=r − r0 + r0 − r(m0+1)/n
∆rn= X − 1
r − r(m0−1)/n
∆rn= · · · = X + 1
50
From (quasi-)translational invariance
δφn,m0+j (x) = δφn (X − j)
which depend only on X − j !Typically nq (r0) ∼ 102 ∼ m0, |j| m0 (up to O (10)).Define X − j ≡ Z ∝ x/∆rn. Rewriting the eigenmode equation,
δnen0∝ δφn (Z) =
(ω∗eω
+ρ2s
(∆rn)2
∂2
∂Z2− k2
θρ2s +
c2s
ω2
L2n
L2s
(∆rn)2 Z2
)δφn (Z)
− ω∗eω
LnR0
[δφn (Z + 1) + δφn (Z − 1)
− 1
kθ∆rn
∂
∂Z(δφn (Z + 1)− δφn (Z − 1))
](
1− ω∗eω− k2
θρ2s s
2 ∂2
∂Z2− k2
θρ2s +
c2s
ω2
L2n
q2R20k
2θ
Z2
)δφn (Z)
+ω∗eω
LnR0
[δφn (Z + 1) + δφn (Z − 1) + s
d
dZ(δφn (Z + 1)− δφn (Z − 1))
]= 0
Finally, Fourier-transform to an extended poloidal angle “η”(defined on −∞ < η <∞)
δφn (η) =1√2π
∫ ∞−∞
e−iZηδφn (Z) dZ
δφn (η) =1√2π
∫ ∞−∞
eiZηδφn (η) dη
⇒ −iZ =∂
∂η,
∂
∂Z= iη : “conjugate (reciprocal) relations”
Then we obtain an one dimensional eigenmode equation in η !(cos θ → cos η, sin θ → sin η)The Fourier-transformed eigenmode equation is(
1− ω∗eω
+ k2θρ
2s s
2η2 − k2θρ
2s −
c2s
ω2
L2n
q2R20k
2θ
∂2
∂η2
)δφn (η)
+2ω∗eω
LnR0
[cos ηδφn (η) + sη sin ηδφn (η)
]= 0
The 2nd term only modifies Veff of the Schrodinger equation.“cos η” is the normal curvature, and “η sin η” is the geodesic curvature of B.
Note You can get along-the-field property from this transformed eigenmode equa-tion. If you calculate inverse Fourier transform, you can get radial dependence.
51
5.3. Toroidal Drift Wave Eigenmode Equation (Nov. 19)[d2
dη2+ η2
sΩ2Q (Ω, η)
]δφn (η) = 0
with −∞ < η <∞ the extended poloidal angle (ballooning coordinate).Furthermore,
−Veff = Q (Ω, η) = bθ(1 + s2η2
)+ 1− 1
Ω+
2εnΩ
(cos η + sη sin η)
with bθ ≡ k2yρ
2s, Ω = ω/ω∗e, εn = Ln/R0, ηs = qb
1/2θ /ε2n
(ω∗e > 0 for normal profile, ky > 0).This is a second-order differential equation which is similar to the time-independentSchrodinger equation.
The large η asymptotic solutions are
δφ (η) ∼ exp[±iΩηsb1/2θ sη2/2
]
Which solution should we take? As we did in sheared slab geometry, we demandthat, for unstable eigenmodes (= (Ω) > 0), δφ should decay as |η| → ∞. This isequivalent to taking the outgoing wave boundary condition, because
vgp ∝∂Ω
∂kη⇒(ηsb
1/2θ sη
)−1
from eikonal representation δφ (η) = exp(±i∫kηdη
)(⇒ kη = Ωηsb
1/2θ sη).
Therefore, we take the positive sign.
Note How to determine zero-point of η? There’s other solutions which are lessunstable with θ − θ0, but most unstable solution is θ0 = 0 solution. ⇒ Let’s setthe zero-point as the midpoint in low-field-side.
Veff is a function of η, but its shape depends on various dimensionless variablesdefined above.
1. Slab-like eigenmode
52
Veff (η) ' Veff (0) +1
2
(∂2
∂η2Veff
)(0) η2
inverted SHO: anti-well ⇒ Weber equation with eigenmodes (` = 0, 1, 2 . . . ).
Ω =1− 2εn1 + bθ
− i(2`+ 1) εnq (1 + bθ)
(s2 +
εn (2s− 1)
bθΩ
)where the 2nd term is still the magnetic-shear-induced damping. We recover theresults from sheared slab geometry if we ignore the blue terms (only quantitativemodification). Slab-like eigenmodes are not likely to be unstable due to magnetic-shear-induced damping.
Toroidal coupling introduces modulations to the potential structure ⇒ new kindof eigenfunctions (@ in slab) can exist quasi-bounded by local potential wells.
2. Weak toroidicity-induced mode.
3. Strong toroidicity-induced mode.
For toroidicity-induced eigenmodes, magnetic shear-induced damping only occursthrough tunneling leakage and is very small!
53
Rough estimation of eigenfunction structure:
δφ (η) ∝ e−η2/2(∆η)2
electron folding length in η ⇒ ∆η ≈ π/√
2.
δφ (Z) ∝∫ ∞−∞
eiZηe−η2/2(∆η)2dη
∝∫ ∞−∞
e−(η−iZ(∆η)2)2/2(∆η)2e−(∆η)2Z2/2dη
∝ e−(∆η)2Z2/2 ≡ eZ2/2(∆Z)2
Therefore ∆Z ∼ 2/π and the un-normalized (physical) quantity is
∆x = (∆rn) ∆Z ∼ 2
π∆rn ∼ ∆rn =
r0
nqs
⇒ distance between neighboring rational surfaces for the same n.
Number of fingers ≈ m0.If translational invariance were exact, the chain of poloidal harmonics inside onefinger (radially elongated eddy) will span the whole system size. However, trans-lational invariance is only approximate (quasi-translational-invariance).Note that ω∗e = ω∗e (r), n0 = n0 (r), Te = Te (r), Ln = Ln (r)⇒ one can estimate how many poloidal side bands are contained in each eddy byconsidering these slow radial variations.
54
6. Microinstability and Turbulent Transport
6.1. Revisiting Ion Temperature Gradient Instability (Nov. 26)
In uniform B and flat density profile (∇n = 0 for simplicity), ∇Ti can make theacoustic wave unstable (effective compressibility was negative).
⇒( linear growth rate
real frequency
)∝ |ω∗T i|1/3
(k‖cs
)2/3where ω∗i = −kyρivT i/Ln ≤ 0, ω∗Ti = −kyρivT i/LT i < 0.But ITG is unstable for extremely weak ∇Ti ⇒ How this can be happened?In the derivation, ω∗T i ω k‖vT i was assumed. However, as |∇Ti| , ω so that ω k‖vTi breaks down.⇒ For weaker ∇Ti, we need to perform kinetic calculation, keeping wave-particleresonant interaction ⇒ Landau damping.We can linearize the electrostatic ion gyrokinetic equation. It’s convenient to sep-arate the ion adiabatic response, before transforming guiding center coordinates.
〈δφ〉 = J0 (k⊥ρi) δφ, f = F0 + δf, δf = −|e| δφTi
F0 + δh
⇒(∂
∂t+ v‖∇‖
)δh+
(−|e|Ti
∂
∂t〈δφ〉+
c
Bb×∇〈δφ〉 · ∇
)F0 = 0
(from Kadomtsev and Pogutse, Review of Plasma Physics Vol. 5 (1970))
δh =ω∗i
1 + ηi
(v2
2v2Ti− 3
2
)+ ω
ω − k‖v‖|e| δφTi
J0F0
with ηi ≡ d lnTi/d lnn0 = Ln/LT i: measure of relative peakness.
Origin or the blue term and the red term1. The blue term−vT iρiky ∂
∂x lnF0 ⇒ TiTeω∗e
1 + ηi
(v2
2v2Ti− 3
2
): gradient in configuration space.
“velocity-dependent free energy” (potentially destabilizing.)2. The red termFrom ∂
∂t 〈δφ〉; i.e. “heating term” due to relaxation in velocity space (stabilizing!).k‖∂F0/∂v‖ω−k‖v‖
∝ k‖v‖ω−k‖v‖
= ωω−k‖v‖
− 1 where 1 is adiabatic response.
Instability Threshold (Magnetically) is determined by requiring “zero” relaxationof the velocity-dependent free energy (in the limit <(ω)→ 0) i.e.,
=∫d3v−vT iρiky ∂
∂x lnF0
ω − k‖v‖= 0
55
limω→0=∫ ∞−∞
dv‖
∫ ∞0
dv⊥v⊥1
ω − k‖v‖∂
∂r
(n0
T3/2i
J20 e−v2/2v2Ti
)
= limω→0=∫ ∞−∞
dv‖e−v2‖/2v
2Ti
ω − k‖v‖∂
∂r
(n0
T3/2i
∫ ∞0
dv⊥v⊥e−v2⊥/2v
2TiJ2
0 (k⊥ρi)
)
= − π∣∣k‖∣∣ limω→0
∂
∂r
[e−v
2Res/2v
2Ti
n0
T1/20
Γ0 (bi)
](ρi =
v⊥Ωci
, notvT iΩci
!)
where vRes = ω/k‖ → 0, bi ≡ k2yρ
2T i.
We used Plenelji formula
=(
1
ω − k‖v‖
)= − π∣∣k‖∣∣δ
(ω
k‖− v‖
)Therefore, for vRes → 0,
∂
∂r
[n0(r)
T1/20 (r)
Γ0
(k2⊥ρ
2i
)]= 0
⇒ The most simple magnitude profile.
For k⊥ρi → 0 (long wavelength limit),
∂
∂r
[n0(r)
T1/20 (r)
]= 0
Note For ∂∂r
[n0(r)
T1/20 (r)
]= 0, Ti(r) ∝ n0(r)2 radially.
⇒ Ti(edge) determines Ti(core) if transport is stiff.
If ITG instability is violent enough, it can throw out ion heat rapidly to outsideas soon as Ti gets heated (e.g. by NBI) above the threshold condition.For a known density profile n0(r) and the wavelength of fluctuation k⊥, we candetermine the ion temperature profile.
56
⇒ ITG turbulence is also called “stiff turbulence”.No matter what the input power is, profile go back to marginal profile!Of course, it’s most pessimistic scenario.ITG instability is hard to handle, so it’s one of the reason why people are focusedon the improvement at the edge.
0.7 < ρ = r/a < 0.85: core-edge connection region (No man’s land!)
6.2. Onset Condition of ITG Instability (Nov. 28)
The onset condition here refers the linear threshold.
ηi =d lnTid lnn0
=LnLTi≥ 2
1 + 2bi [1− I1 (bi) /I0 (bi)]with bi = k2
⊥ρ2i
in a uniform B field (Kadomtsev and Pogutse).
“eta-i mode”:For flat density profile Ln →∞, there’s an instability for very weak ∇Ti.There should be a threshold condition (for Ln →∞) in terms of 1/LTi ≥ (1/LTi)crit
or R0/LTi ≥ (R0/LTi)crit. In the long wavelength limit, bi ≡ (k⊥ρi)2 → 0 and the
57
gyrokinetic equation simplifies to the drift-kinetic equation∂t + v‖ b · ∇+ vd · ∇+ δuE · ∇ −
1
m
(q∇‖δφ+ µ∇‖B
)∂v‖
F = 0
where vd · ∇ come from gradient and curvature drifts and µ∇‖B is the mirrorforce. After linearization,(
∂t + v‖ b · ∇+ vd · ∇)δf ⇒ −i
(ω − k‖v‖ − ωdi
)δf
which is the linear propagator and ωdi = k · vd (cf. for Vlasov, we would have anadditional Ωci). Therefore, the wave can resonate with particles’ motion along B(v‖) and/or particles’ drift motion across B (vd). It’s very difficult to keep bothresonances analytically, however each resonance can be handled.
• Sheared slab: ωdi is ignored in ω − k‖v‖ − ωdi
• Simplified toroidal calculation: k‖v‖ is ignored in ω − k‖v‖ − ωdi
Keeping only ω−k‖v‖ resonance, with k‖ = kyx/Ls, Ls = qR0/s and x = r−rm/n,
q
s
R0
LTi≥(LsLTi
)crit
= 1.9
(TiTe
+ 1
)for Ln →∞. Favorable role of Ti/Te (hot ion mode).
Keeping only ω − ωdi resonance, with ωdi = − (cTi/eB) 1/R0,
R0
LTi≥ 4
3
(TiTe
+ 1
)Likewise, favorable role of Ti/Te.
More complicated formulas exist for finite Ln.
Steve Scott [Phys. Rev. Lett. 29, 531 (1990)]: ηi & ηcriti of above formulas
χi ∼ χφ (ion thermal diffusivity ∼ momentum diffusivity)(NBI plasmas in TFTR, PPPL)
In retrospect, it’s hard to understand why most people were reluctant to accept apossibility that χφ is anomalous! χneo
φ χneoi ! (Mattor and Diamond ’88)
Note Why diamagnetic drift frequency is contained in GK equation, which isfor guiding center? ⇒ Information is contained in equilibrium distribution whichdescribe many particles i.e. F0 → n0. In propagator which describes motion ofone particle, there’s no diamagnetic drift!
58
6.3. Basic Properties of ITG Instability in Toroidal Geometry
Unstable ITG should have characteristics related to Rayleigh-Taylor instability(contrast to negative compressibility ITG in slab geometry).
⇒ Motivates a “local” theory at bad curvature side (large R).Each guiding-center drift:
v∇B+curv = v∇B+curv
(v2‖ + µB
v2T i
)
High energy ions drift faster!
0. Seed perturbation in pi
1. vdi ∝(v2‖ + µB
)⇒ hot particles drift faster and density build up occurs below hot spots.
2. E builds up in toroidal direction due to ion surplus (deficiency) at high (low) n
3. radial E×B drift results → “0.”,reinforce the initial seed perturbation → instability
59
6.4. Fluid Description of ITG Instability in Toroidal Geometry(Dec. 3)
Pursue a local theory in the bad curvature region.
In the previous lecture, we’ve learned that energy dependence of∇B and curvaturedrift led to an instability using figures. This time, we derive a simplified lineardispersion relation.Linearized drift-kinetic equation (krρi 1):(
∂
∂t+ v‖∇‖ + vdi · ∇
)δf +
(c
B∇δφ× b · ∇ − q
m∇‖δφ
∂
∂v‖
)F0 ' 0
where vdi is the gradient and curvature drift. If we take a velocity moment,∫d3v = 2π
∫dµdv‖B of this, we obtain a continuity equation for δni.
∂
∂tni + δuE · ∇n0 + n0∇‖δu‖ + i
n0
TiωdiδTi + · · · = 0
where the underlined term is a consequence of the energy dependence of vdi, i.e.,
vdi ' vdi,th(v2‖ + µB
)/v2
th,i → δni couples to δTi!
We can also derive an evolution equation for δpi by taking∫d3v
(12mv
2)
momentof the drift-kinetic equation.The simplest model for the evolution of δTi is (note that for linear physics δpi =Tiδni + niδTi),
∂∂tδTi + δuE · ∇Ti ' 0, i.e., an E×B convection. Then we have
∂
∂tδTi + iω∗Ti |e| δφ ' 0
with
ωdi ≡ −cTieBR0
ky
(at bad curvature side), and
ω∗Ti ≡ −cTieB
kyLTi
the ion temperature diamagnetic frequency, where we haveω∗e = (cTe/eB) (ky/Ln).Taking a limit Ln → ∞ and k‖ → 0 (simplified toroidal calculation), with
60
δni/n0 = δne/n0 = |e| δφ/Te (Boltzmann response for electrons, adiabatic),
− iω δnin0
+ iωdiδTiTi0' 0
− iω δTiTi0
+
(TeTi
)iω∗Ti
|e| δφTe
' 0
2x2 equations for δTi and δφ ⇒ det [ ] = 0.
ω2 = −TeTi|ωdiω∗Ti | or γ2 =
TeTi|ωdiω∗Ti |
Instability, since γ > 0 !This simplified toroidal instability shows a hybrid nature, i.e. we need both
• ion temperature gradient ∼ |ω∗Ti |
• bad curvature ∼ |ωdi|
A similar example has been considered for MHD description of Rayleigh-Taylorinstability.Can we justify a local consideration at bad curvature side?→ ignoring good curvature side (high-B side) requires λ‖ ≤ 2πqR ⇒ k‖ ≥ 1/qR.This is incompatible with an approximation used for the derivation, k‖ → 0.Why did we spend so much time dealing with magnetic geometry and BallooningMode Formalism?In uniform B field and sheared slab model, we kept k‖. We’ve learned that for aninstability to exist, k‖ cannot be too large. Otherwise, the ion Landau damping,shear-induced damping can stabilize the mode.⇒ k‖ = 2π/λ‖ should be kept small enough⇒ Flute-like mode structure: λ‖ > 2πqR !
The mode structure needs to be “ballooning at the outside” and “flute-like alongB” simultaneously.
Why do we care about the mode structure of instabilities?Mode width can be regarded as an approximation of the turbulent eddy size.Turbulent eddy size is related to the anomalous transport rate due to turbulence.Using a random walk argument,
Dturb ∼(∆x)2
∆t∼ (eddy size)2
eddy life time (circulation time)
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If we’re forced to quantities which are available from linear theory,
∆x ∼λx ∼ k−1x
∆t ∼γ−1lin
Dturb ∼(∆x)2
∆t∼ λ2
xγlin ∼γlin
k2⊥
at a conceptual level, many transport calculations are more quantitative elabora-tions of this consideration. This is why λx, γlin of an instability are quantities ofpractical interest for magnetic fusion.
6.5. Simple Estimation of Turbulent Transport (Dec. 5)
Note very rough estimation, could be off by a factor!If we’re forced to quantities which are available from linear theory,
Dturb ∼(∆x)2
∆t∼ γlin
k2x
This is also related toγeff = γlin − k2
xDturb
where the correcting factor is the damping due to nonlinear coupling to othermodes (sometimes people put k⊥ instead of kx).
(Kadomtsev)Turbulent diffusion increases with λx ∼ 1/kx and γlin.If we take
γlin ∼ ω∗ ∼kyρivT iL
where
“macroscopic length” L ⇒ LT i for ITG
⇒ Ln for electron drift instability
⇒ LTe for TEM
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and so on,
Dturb ∼kyk2xρi
(ρiL
)(cTieB
)Therefore from k-spectra of turbulence, we can guess the scaling of transport.For instance, if kx ∼ ky ∝ ρ−1
i i.e. λx ∼ λy ∝ ρi,
Dturb ∼(ρiL
)(cTieB
): gyro(reduced)-Bohm scaling
If λx ∼ λy ∝ a ∼ L,
Dturb ∝(cTieB
): Bohm scaling
Define ρ∗ = ρi/a, and it’s value is 103 for ITER, 1/300 for JET, 1/100 − 1/200for KSTAR/DIII-D/AUG...Why has Bohm scaling been observed in the early days (∼ 60s, 70s) of magneticconfinement research?
(− ~
2m
∂2
∂x2+ V (x)
)ψ = Eψ
for SHO in QM, ψ ∝ ψ`x (x) ∝ H`x (√σx) e−σx
2/2 (quantization condition).In wave guides, quantization is based on system size. Same thing there.
There was very small a, and relatively weak B⇒ ρi/a was NOT very small (& 1/10)⇒ it’s likely that even drift wave eigenmodes have low quantum number `x⇒ λx, λy = fraction of system size ∝ a∝ρi
From experiments, Bohm-like scaling persisted even up to present days, in partic-ular (for ion heat transport) in NBI-heated L-mode plasmas.In those days, Te Ti. From experiments, Bohm-like scaling persisted even up topresent days, in particular for ion heat transport in NBI-heated L-mode plasmas.This even though ρ∗ < 10−2 in DIII-D for instance!
Most ITG or drift wave theory for tokamaks lead to gyro-Bohm.From the k-spectra of turbulence, we can guess the scaling of transport.
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In the early 90’s, radially elongated eddys (or global toroidal eigenmodes) wereused to explain Bohm-like transport. In toroidal geometry, θ is no longer sym-metric: the poloidal harmonics m couple with each other!The radial width of a global toroidal eigenmode is determined by profile variationswhich break translational invariance. For example, if ω∗Ti (r) varies with L∗:
ω∗Ti (r) = ω∗Ti (r0)
(1− (r − r0)2
L2∗
)⇒ ∆r ∝
√L∗ρi while ky ∝ ρ−1
i
Then
k2x ∼ (ρiL∗)
−1 ⇒ Dturb ∼(cTieB
)even for tokamak plasmas.
cf. from magnetic shear in sheared slab model,
∆rDW ∼ ρs√Ls/Ln ⇒ still rDW ∼ ρi
6.6. Zonal flows (Dec. 10)
Dominance of radially elongated eddys, global toroidal eigenmodes, streamers⇒ high turbulence level and Bohm-like transport.
However, since 90’s, it has been observed from simulations that turbulent eddysget broken up by self-generated (turbulence-generated) zonal flows.
δΦZF = δΦZF (r, t) ; independant of θ and φ (ζ)
Zonal flow is represented by fluctuation.
uZF =cb×∇δΦZF
B
⇒ binormal (mostly poloidal) direction→ E×B zonal flows or radial mode (kr 6= 0, kθ = kφ = 0)
Note that uZF,r = 0 !→ Zonal flows are linearly “stable”.They cannot tap free energy in (mean) ∂Ti
∂r and ∂n∂r .
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→ They can only be driven nonlinearly.
Reference Review of zonal flows [Diamond et al., PPCF 47, R35 (2005)]
1. Global simulations
density fluctuation contours from ITG turbulence⇒ lower level for turbulence and transport ∼ gyroBohm!
2. Flux-tube simulations (quasi-global) until early 90’s
• ω∗Ti (r) , ω∗e (r) = constant in radius
• q (r) , s = constant in radius
• Radially periodic boundary conditions for turbulence
⇒ Efficient! ∆rturb L ∼ a→ They have found the importance of ZFs ∼’93
Until early 90’s: global gyrokinetic simulation, ρ−1∗ . 100, 125. The ZFs from
these simulations were either in system-size or suppressed!
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χGyrofluidi > χGK
i ? χFlux tube, GKi > χGlobal, GK
i ?
(most flux-tube simulations were gyro-fluid)
1996, J. Glanz Science article:ITER (based on old design) will fail like the Titanic! Based on (mostly) ITGtransport model heavily relying on nonlinear gyrofluid simulations.
1998, Rosenbluth and Hinton, Phys. Rev. Lett. 80, 724 (residual zonal flow):Gyrofluid equations underestimates zonal flows (overestimates zonal flow damp-ing) ⇒ importance of ZFs.
Before this work, most simulation codes were bench-marked for γlin of unsta-ble ITG. After this, this “RH” ZF damping test is widely used as well. Fromsimulations (numerical experiments), one can suppress ZF articicially, or keep itnaturally. ⇒ contrast two sets of simulations, to isolate the effects of zonal flows.
Effects of zonal flows on (ITG) turbulence
• Zonal flows reduce “turbulence eddy size”
• Zonal flows reduce turbulence amplitude
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• Therefore, zonal flows reduce turbulent transport!
⇒ Paradigm shift: need to incorporate ZFs which regulate ITG turbulence.
Outstanding (frequently asked) questions about zonal flows:
1. How do turbulence eddys get broken up? (beyond movies showing it)
2. How do ZFs get generated?
• For 1, it’s useful to consider the effects of mean E×B shear flows on turbu-lence. (related to H-mode and ITB physics.)
• For the understanding of 2, consideration of “conservation laws” can provideuseful insight.Of course, one can do a non-linear mode-coupling analysis for this, but weran out of time for this class.
6.7. Duality of Eddy Shearing (due to Zonal Flows) and Sponta-neous Zonal Flow Generation (Dec. 12)
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For this two-component system (turbulence + zonal flows), the total energy shouldbe conserved. (Diamond et al. ’95)
Etot = EZF + Eturb =1
2
∑q
|vZF,q|2 +1
2
∑k
Ek
where q = (qr, 0, 0) is the wave number for ZFs and
Ek =Te2
∑k
(1 + k2
⊥ρ2s
)|δφk|2
(k2⊥ = k2
r + k2y) for a drift wave.
ωDW =kyv∗e
1 + k2⊥ρ
2s
≡ ω∗e1 + k2
⊥ρ2s
ωZF ωDW
⇒ adiabatic invariant, i.e. drift-wave “action” density Nk is conserved.Quasi-particle picture (duality of wave and particle) ⇒ Ek = Nkωk.Note that streaming of eddys ⇒ ∆r , kr therefore ωDW ⇒ Ek (∵ Nk is conserved) ⇒ ZF energy (i.e., ZF generation/growth)
6.8. Outstanding Issues of Turbulent Transport in TokamaksQiQeΓφΓp
= −
χi . . . . . . . . .. . . χe . . . . . .. . . . . . χφ . . .. . . . . . . . . D
(∇Ti)r(∇Te)r(∇Uφ)r(∇n)r
Generalization of Fick’s law Γp = −D ∂n
∂x (→ diffusion equation).Exist off-diagonal pinch terms.
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τE with Ip more strongly than Bp!
Ion Thermal Transport (typically due to ITG contribution)
One candidate: ZF shearing gets ineffective if geodesic acoustic side band (of ZF)with ωGAM ∼ cs/R0 gets stronger at the expense of the main (ωZF ∼ 0) ZF.GAMs can be Landau damped with γdamping ∼ −e−q
2/2 where q ' rBφ/RBθ.∴ large q → strong GAM → strong turbulence.
Electron Thermal Transport
No dominant candidate for every case. Depending on parameters.
A. Trapped electron mode (TEM)
A strong evidence from AUG ECH experiments (F. Ryter, PRL ’05)DTEM (dissipative TEM) ⇒ Neo Alcator scaling τE ∝ neaR2
(’82 for ohmic plasmas), but GK codes find ITG more unstable than TEM!?
B. Electron temperature gradient mode (ETG)
Associated ZFs are relatively weak→ radially elongated eddys→ streamer could dominateOtherwise ∆r ∼ several ρe ⇒ transport small
Microtearing Mode
Instead of E×B transport due to electrostatic fluctuation
δvr =cb×∇δφ
B
transport mechanism due to magnetic flutter
δvr =δBrB0
v‖
Momentum Transport
Spontaneous/intrisic rotation of plasmas in the absence of external torque inputfrom NBI, ICRH ... (even for Ohmic plasmas!)NBI not efficient in driving rotation in ITER. → rotation of ITER plasma?(for resistive wall mode, turbulence, etc.)
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H-mode Transition∣∣v′E×B∣∣ ⇒ turbulence δn ⇒ transportwhich component, uθ, uφ or ∇p?uθ deviates from neoclassical prediction (DIII-D ’94) even in L-mode plasmas(but deviation from uθ is very small in NSTX).
Existence of Turbulence in the Absence of Local Drive due to RadialGradient
“Turbulence spreading” from linearly unstable zone to linearly stable zone.
Edge-core coupling region (0.7 < ρ = r/a < 0.85): “No Man’s Land”
Isotopic Dependance of Transport
χGB ∼ρia
cTieB∝√Mi
but from experiments,
χDeuterium < χHydrogen (TFTR: Tritium even better)
See fig.
“Simulations & Modeling” should focus more on identification of physical mecha-nism rather than case by case agreements in numbers.
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