5.1. Sheared Slab Model for Tokamak Magnetic Field Cartesian coordinates Toroidal coordinates:
x → r
y → rθ
z → Rφ
withB=Bφφˆ+Bθθ.ˆ
Safety factorq(r)'rBφ/RBθ characterizes pitch ofB.
Typically,q(r) is a monotonically increasing function ofr (positive shear plasma, magnetic shear ˆs=dlnq/dlnr). For a mode rational surface atrs,q(rs) =m/n.
δφ(r, θ, φ) =X
n,m
δφn,mei(nθ−mφ) wherenθ−mφ is the pitch of fluctuation, (m, n)∈Z. We can also express this asP
kexpi(kyy+kzz) so that kθ =m/r,kφ=−n/R.
Therefore,
kk = k·B
|B| = m r
Bθ B − n
R Bφ
B = Bθ
rB(m−nq(r))
kk = 0 at r=rs (q(rs) =m/n),m and nare fixed, but q(r) and kk vary with r.
q(r) =q(rs) + (r−rs)dq
dr(rs) +. . . , so that we replacem−nq by −n(dq/dr) (r−rs).
Magnitude of kk(r) =kk(x) increases with |x|wherex=r−rs. kk flips sign across x= 0 (r=rs).
cf. in reversed shear (RS) plasmas,q(r) can have a minimum value atr=rmin(not r= 0). Atqmin,dq/dr= 0 therefore q(r) =qmin+12 d2q/dr2
(r−rmin)2+. . . The magnetic field in sheared slab geometry is
B=B
ˆ z+ x
Ls
ˆ y
with the magnetic shear ˆs=r/q(dq/dr), Ls≡qR/ˆs.
This model is good for a single pair of (n, m), i.e., single helicity fluctuation.
kk ⇒ ky Ls
x k2x ⇒ − ∂2
∂x2
where the LHS are the local expressions from uniformB model and the RHS the sheared slab for uniformB.
kk = 0 at x= 0: “singular layer”.
We are interested in local stability around this location.
In this model,
δφ(x, t) =X
ky
δφky,ω(x)ei(kyy−ωt)
(After analyses in sheared slab, we should understand that results withky =nq/r and m=nq.)
Recall electrostatic linear drift wave for electrons: vT i .ω/kkvT e δne
n0
'exp
|e|δφ Te
−1' |e|δφ Te
Boltzmann response (adiabatic response forω/kk vthe).
For ions,
∂
∂tni =−∇ ·δ(nu) =−δuE · ∇n0− ∇ ·(n0δupol)− ∇ · n0uk
⇒ δni
n0 = ω∗e
ω −ρ2sk⊥2 +c2skk2 ω2
!|e|δφ Te
withω∗e≡kyρs/Ln (ω& from polarization, ω% from sound wave).
Quasi-neutrality (λλDe):
|e|δφ Te
= ω∗e
ω −ρ2sk2⊥+c2sk2k ω2
!|e|δφ Te
now eigenmode equation inx rather than local dispersion relation.
"
1 +ρ2sk2y−ρ2s ∂2
∂x2 − ω∗e
ω − c2s ω2
ky2 L2sx2
#
δφ(x) = 0 (Weber equation)
⇒Mathematically equivalent to time-independant Schr¨odinger equation for SHO
− ~ 2m
∂2
∂x2Ψ =
E−1 2mω02x2
Ψ Eigenmode equation for electrostatic drift wave:
−ρ2s ∂2
∂x2δφ=
"
−1−ρ2sky2+ ω∗e
ω + c2s ω2
c2s ω2
k2y L2sx2
# 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 δφin equations above is in realδφky,ω. We usedδφjust for simple notation.
(Nov. 7)
We can easily obtain eigenvalues, but effective potentialVeffhas 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,ωexp
−σx2 2
H` √
σx (H`: Hermitte polynomials). Here,
σ =± ikycs
Lsρsω
For`= 0,
δφky,ω(x) =δφky,ωexp
∓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: decaying at infinity is a consequence of causality condition, not a condition in itself.)
vgp,x≡ ∂ω
∂kx
(>0 forx→ ∞
<0 forx→ −∞
Then, what iskx? We recall an eikonal form exp iR
kxdx+ikyy−ωt kx =−i ∂
∂x =∓kycsx
Lsρsω or ω=∓ kycsx Lskxρs
Therefore,
vgp,x= ∂ω
∂kx
=± kycs Lsρskx2x
“outgoing wave” boundary condition (vgp >0 for x >0)
⇒ We choose (forky >0)
δφky,ω(x) =δφky,ωexp
−ikycs 2Lsω
x2 ρs
The corresponding eigenvalue is (recallE =~ω0(1/2 +N) for SHO in QM):
ω =ω∗e
1
1 +ky2ρ2s −i(2`+ 1) 1 +ky2ρ2s
Ln Ls
where the downshifting term is function ofky (k⊥ in the local term) and the 2nd term is the magnetic shear induced damping.
The eigenmode width ∆x is
∆x'
sLsω∗eρs kycs
∝ rLs
Ln
ρs (∼10ρs&ρs)
As ˆs%, ∆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 believed drift 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=Bφφˆ+Bθθˆ where |Btoroidal| 'Bφ= B0R0 R0+rcosθ Sheared slab magnetic field is
Bsheared slab=B0
ˆ z+ x
Lsyˆ
Therefore, sheared slab magnetic field is independent ofzandy: 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!
5.2. Mode Coupling in Toroidal Magnetic Field (Nov. 12)
In sheared slab geometry, ω =ω∗e
1
1+k2yρ2s −i(2`x+ 1) 1 +ky2ρ2s
Ln
Ls
where we haveω∗e=csρsky/Lnand theadiabatic electron responseand theshear- induced damping. We include a non-adiabatic electron response−iδe (function of kyρs,Ln,Te,νeff,r0/R, ˆs,q, ...)
δne n0
= (1−iδe)|e|δφ Te
⇒ω =ω∗e
1
1−iδe+ky2ρ2s −i . . .
for small δe1
=(ω) ω∗e
= γ ω∗e
= δe
1 +k2yρ2s2 −(2`x+ 1) 1 +k2yρ2s
Ln Ls Therefore,
δe> 1 +k2yρ2s
(2`x+ 1)Ln
Ls 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 to B =B0R0/(R0+rcosθ). We cannot get a single decoupled eigenmode equation forδφn,m(r). This will coupleδφn,m+1(r),δφn,m−1(r) sidebands!
Note thatkθρs=kyρs=nq(r)ρs/r0. γ is maximum for kθρs=O(1).
(∵ω∗e∝kθρs ⇒γ∝kyρs/(1 +ky2ρ2s))
Forr0 ∼102cm, ρs∼0.5cm,q(r0)∼2, we get n∼102 (small-scale)
→ Microinstability.
Recall our previous calculation in sheared slab geometrywas only for a single helicity, i.e., one pair of (m, n) e.g. n= 124, m= 206.
“Drift wave energy gets convected away from the mode rational surface where q rm/n
=m/nto large|x|region (x=r−rm/n) withvgp,x!”
We’ve learned from plasma electrodynamics (or electromagnetism) course that if E+=ˆyE0ei(kxx−ωt) : propagating to +x
E−=ˆyE0ei(−kxx−ωt): propagating to −x we get a standing wave
E++E− = ˆy2E0cos (kxx)e−iωt In this discussion ignoreδe ∵δe can also be a function of θ
(1−δe)|e|δφ Te =
ω∗e
ω −ky2ρ2s+ρ2s ∂2
∂x2 + c2s ω2
L2n L2sx2
|e|δφ Te where the RHS is the ion response from
∂
∂tδni+δuE· ∇n0+n0∇ ·δupol+n0∇ ·uk = 0 in uniform magnetic field.
In torus? ∇ ·δuE is no longer zero, thus we should considern0∇ ·δuE term.
(∇B effects are just equilibrium quantities.)
∇ ·δuE =∇ ·
cE×B B
=cb×δφ· ∇ 1
B
+c∇ ×b· ∇δφ B For uniform B,∇B1 = 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 (Tk =T⊥).
∇ 1
B
= 1 B0∇
1 + r
R0 cosθ
= 1
B0R0 (ercosθ−eθsinθ)
⇒n0∇ ·δuE =· · ·= 2ωde(θ)|e|δφ Te where
ωde= ρscs
R0
(kθcosθ+krsinθ) withkθ=nq(r0)/r0 and kr =−i∂/∂x.
(Nov. 14)
We add 2ωde/ω in the ion response. Then, the eigenmode equation become δne
n0 = |e|δφ Te =
ω∗e
ω + 2ωde
ω +ρ2s ∂2
∂x2 −kθ2ρ2s+ c2s ω2
L2n L2sx2
|e|δφ Te δφ(ζ, θ, r)⇒
A
A A X
n
δφn(θ, r)e−inζ =X
m
δφn,m(x)ei(mθ−nζ) whereζ is the toroidal angle andx=r−r0,q=m0/n.
“n” is a good quantum number (conserved), each toroidal harmonic decouples in linear theory ⇒einζ is a common factor.
Now, substituteδφ⇒P
mδφm,n(x)eimθ.
Each term in the eigenmode equation is proportional toeimθ, except (kθcosθ+krsinθ)X
m
δφn,m(x)eimθ
= 1 2
kθ
eiθ+e−iθ
+kr i
eiθ−e−iθ
X
m
δφn,m(x)eimθ Here,
eiθX
m
δφn,m(x)eimθ =X
m
δφn,m(x)ei(m+1)θ =X
m
δφn,m−1eimθ sincem is a dummy variable (from−∞ to∞). Likewise,
e−iθX
m
δφn,m(x)eimθ =X
m
δφn,m(x)ei(m−1)θ =X
m
δφn,m+1eimθ Now the equation has to be satisfied for every term multiplying “eimθ ”.
δne
n0
∝δφn,m0(x) = ω∗e
ω +ρ2s ∂2
∂x2 −kθ2ρ2s+ c2s ω2
L2n L2sx2
δφn,m0(x)
− csρs
ωR0 h
kθ(δφn,m0−1(x) +δφn,m0+1(x)) +kr
i (δφn,m0−1(x)−δφn,m0+1(x))i the coupled terms are called “sidebands”.
We have seen this kind of situation from classical mechanics.
infinite chain of coupled springs ⇒ eigenmodes?
Similar problem also appears in solid state physics with lattice structure.
Recall thatn'102 ⇒ every poloidal harmonics look almost the same. (Roughly speaking, 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
= ˆsq(r0)
r(m0+1)/n r0 −1
with Taylor expansion aroundr0. Therefore,
∆rn= r0 nqˆs = 1
kθsˆ
From quasi-translational invariance, we can getδφn,m0(x) by shiftingδφn,m0+1(x) to the left by ∆rn or by shifting δφn,m0−1(x) to the right by ∆rn, ... shifting δφn,m0+j to the left by j∆rn.
Let’s introduce a dimensionless variableX, 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
From (quasi-)translational invariance
δφn,m0+j(x) =δφn(X−j) which depend only onX−j !
Typicallynq(r0)∼102∼m0,|j| m0 (up toO(10)).
DefineX−j≡Z ∝x/∆rn. Rewriting the eigenmode equation, δne
n0 ∝δφn(Z) = ω∗e
ω + ρ2s (∆rn)2
∂2
∂Z2 −k2θρ2s+ c2s ω2
L2n
L2s (∆rn)2Z2
δφn(Z)
−ω∗e
ω Ln
R0 h
δφn(Z+ 1) +δφn(Z−1)
− 1 kθ∆rn
∂
∂Z(δφn(Z+ 1)−δφn(Z−1))i
1−ω∗e
ω −k2θρ2ssˆ2 ∂2
∂Z2 −k2θρ2s+ c2s ω2
L2n q2R20kθ2Z2
δφn(Z) +ω∗e
ω Ln R0
δφn(Z+ 1) +δφn(Z−1) + ˆs d
dZ (δφn(Z+ 1)−δφn(Z−1))
= 0 Finally, Fourier-transform to an extended poloidal angle “η”
(defined on−∞< η <∞)
δφˆn(η) = 1
√2π Z ∞
−∞
e−iZηδφn(Z)dZ δφn(η) = 1
√2π Z ∞
−∞
eiZηδφˆn(η)dη
⇒ −iZ = ∂
∂η, ∂
∂Z =iη : “conjugate (reciprocal) relations”
Then we obtain an one dimensional eigenmode equation in η ! (cosθ→cosη, sinθ→sinη)
The Fourier-transformed eigenmode equation is
1− ω∗e
ω +kθ2ρ2ssˆ2η2−kθ2ρ2s− c2s ω2
L2n q2R20k2θ
∂2
∂η2
δφˆn(η) +2ω∗e
ω Ln R0
h
cosηδφˆn(η) + ˆsηsinηδφˆn(η)i
= 0 The 2nd term only modifiesVeff of the Schr¨odinger equation.
“cosη” is the normal curvature, and “ηsinη” is the geodesic curvature ofB.
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.
5.3. Toroidal Drift Wave Eigenmode Equation (Nov. 19) d2
dη2 +ηs2Ω2Q(Ω, η)
δφˆn(η) = 0
with−∞< η <∞ the extended poloidal angle (ballooning coordinate).
Furthermore,
−Veff=Q(Ω, η) =bθ 1 + ˆs2η2
+ 1− 1 Ω+ 2n
Ω (cosη+ ˆsηsinη) withbθ ≡ky2ρ2s, Ω =ω/ω∗e,n=Ln/R0,ηs=qb1/2θ /2n
(ω∗e>0 for normal profile, ky >0).
This is a second-order differential equation which is similar to the time-independent Schr¨odinger equation.
The largeη asymptotic solutions are δφˆ(η)∼exp
h
±iΩηsb1/2θ ˆsη2/2 i
Which solution should we take? As we did in sheared slab geometry, we demand that, for unstable eigenmodes (=(Ω)> 0), δφˆshould decay as |η| → ∞. This is equivalent to taking the outgoing wave boundary condition, because
vgp ∝ ∂Ω
∂kη ⇒
ηsb1/2θ sηˆ −1
from eikonal representationδφˆ(η) = exp ±iR kηdη
(⇒kη = Ωηsb1/2θ sη).ˆ Therefore, we take the positive sign.
Note How to determine zero-point of η? There’s other solutions which are less unstable with θ−θ0, but most unstable solution is θ0 = 0 solution. ⇒ Let’s set the zero-point as the midpoint in low-field-side.
Veff is a function of η, but its shape depends on various dimensionless variables defined above.
1. Slab-like eigenmode
Veff(η)'Veff(0) +1 2
∂2
∂η2Veff
(0)η2
inverted SHO: anti-well⇒ Weber equation with eigenmodes (`= 0,1,2. . .).
Ω = 1−2n
1 +bθ −i(2`+ 1)n q(1 +bθ)
ˆ
s2+n(2ˆs−1) bθΩ
where the 2nd term is still the magnetic-shear-induced damping. We recover the results from sheared slab geometry if we ignore theblueterms (only quantitative modification). Slab-like eigenmodes are not likely to be unstable due to magnetic- shear-induced damping.
Toroidal coupling introduces modulations to the potential structure ⇒ new kind of 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 occurs through tunneling leakage and is very small!
Rough estimation of eigenfunction structure:
δφˆ(η)∝e−η2/2(∆η)2 electron folding length inη ⇒∆η≈π/√
2.
δφˆ(Z)∝ Z ∞
−∞
eiZηe−η2/2(∆η)2dη
∝ Z ∞
−∞
e−(η−iZ(∆η)2)2/2(∆η)2e−(∆η)2Z2/2dη
∝e−(∆η)2Z2/2 ≡eZ2/2(∆Z)2
Therefore ∆Z∼2/πand the un-normalized (physical) quantity is
∆x= (∆rn) ∆Z ∼ 2
π∆rn∼∆rn= r0
nqˆs
⇒ distance between neighboring rational surfaces for the samen.
Number of fingers ≈m0.
If translational invariance were exact, the chain of poloidal harmonics inside one finger (radially elongated eddy) will span the whole system size. However, trans- lational invariance is only approximate (quasi-translational-invariance).
Note thatω∗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 by considering these slow radial variations.